[
  {
    "id": "2512.00318v1",
    "paper_link": "http://arxiv.org/abs/2512.00318v1",
    "theorems_cnt": 7,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main1}\nLet $\\mathcal {A}$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement for any $n$.",
      "start_pos": 28627,
      "end_pos": 29134,
      "label": "thm:main1"
    },
    "ref_dict": {
      "def:admissible1": "\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}",
      "thm:main1": "\\begin{theorem}\n\\label{thm:main1}\nLet $\\ca$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\ca$ is a $K(\\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ca$ is a $K(\\pi,1)$ arrangement for any $n$.\n\\end{theorem}",
      "subsec:deligne complex": "\\label{subsec:deligne complex}\n\n\\subsection{Falk complexes}\n\nIn \\cite{falk1995k}, for each affine arrangement in $\\mathbb C^2$ which is the complexification of a real arrangement, Falk described a loc",
      "conj:dn": "\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}",
      "cor:AB": "\\begin{cor}\n\\label{cor:AB}\nSuppose $\\ca$ is a complete finite shape affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}",
      "thm:main": "\\begin{thm}\n\\label{thm:main}\nSuppose $\\ca$ is a complete admissible affine arrangement in $\\mathbb R^n$ with finite shape. Suppose Conjecture~\\ref{conj:dn} holds. Then $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{thm}",
      "prop:key": "\\begin{prop}\n\\label{prop:key}\nThe poset $((\\Delta'_{\\ca(3)})^0,<)$ is bowtie free and upward flag.\n\\end{prop}",
      "thm:examples": "\\begin{thm}\n\\label{thm:examples}\nSuppose Conjecture~\\ref{conj:dn} holds in dimension $n$. Then $(\\falk_{\\ch_{k,n}},d_\\infty)$ and $(\\falk_{\\ck_{k,n}},d_\\infty)$ are injective metric spaces, and \n$\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $k\\ge 1$. \n\nThus by Theorem~\\ref{thm:dn dim 3 and 4}, for $n=2,3,4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. \n\\end{thm}",
      "rmk:contrast": "\\begin{remark}\n\\label{rmk:contrast}\n    There is an interesting contrast between this proposition and Theorem~\\ref{thm:bowtie free An}, as the poset in Theorem~\\ref{thm:bowtie free An} is not upward flag.\n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 3961,
    "pre_theorem_intro_text": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}. \n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n    \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n    \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n    \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n    \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.",
    "context": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}.\n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\n\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}\n\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}",
    "full_context": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}.\n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\n\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}\n\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nGiven an affine arrangement $\\ca$, an \\emph{$\\ca$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\ca$. The \\emph{local arrangement} at an $\\ca$-vertex $x$ is the collection of all hyperplanes in $\\ca$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\ca$, one can detect the asphericity of $M(\\ca)$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\ca)$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\nIn the situation of the above theorem, we have a free action of $\\mathbb Z^n$ on $M(\\ca)$. Then the fundamental group of $M(\\ca)/\\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\\ca$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group).\n\n\\begin{thm}(=Theorem~\\ref{thm:examples})\nFor $n\\le 4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $n,k$.\n\\end{thm}\n\n\\begin{cor} \\textup{(= \\Cref{cor:AB})}\n\\label{cor:AB intro}\nSuppose $\\ca$ is a complete, finite shape, affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}\n\nThe Coxeter diagram $\\Lambda$ of type $D_n$ (for $n\\ge 3$) is shown in Figure~\\ref{fig:dn}, where all edges are labeled by $3$. The associated reflection arrangement is $\\{x_i=\\pm x_j\\}_{1\\le i < j\\le n}$ in $\\mathbb R^n$.\nWe subdivide each edge of $\\Delta_\\Lambda$ connecting a vertex of type $\\hat s_n$ and a vertex of type $\\hat s_{n-1}$, and declare the middle point of such edge is of type $m$. Cut each top dimensional simplex in $\\Delta_\\Lambda$ into two simplices along the codimension 1 simplex spanned by vertices of type $m$ and $\\{\\hat s_i\\}_{i=1}^{n-2}$. This gives a new simplicial complex, denoted by $\\Delta'_\\Lambda$. Define a map $t$ from $(\\Delta'_\\Lambda)^0$ to $\\{1,2,\\ldots,n\\}$ by sending vertices of type $\\hat s_i$ to $i$ for $1\\le i\\le n-2$, vertices of type $m$ to $n-1$, and vertices of type $\\hat s_n$ and $\\hat s_{n-1}$ to $n$. We define a relation $<$ on $(\\Delta'_\\Lambda)^0$ as follows. For two vertices $x,y$ of $\\Delta'_\\Lambda$, $x<y$ if $x$ and $y$ are adjacent in $\\Delta'_\\Lambda$ and $t(x)<t(y)$. The simplicial complex $\\Delta'_{\\Lambda}$, together with this relation, is called the \\emph{$(s_n,s_{n-1})$-subdivision of $\\Delta_\\Lambda$}. Similarly, we can define $(s_n,s_{n-1})$-subdivision $\\bC'_\\Lambda$ of $\\bC_\\Lambda$. Note that $\\bC'_\\Lambda$ is isomorphic to the Coxeter complex of type $B_n$ via a type-preserving isomorphism. While $\\Delta'_\\Lambda$ is not isomophic to the Artin complex of type $B_n$, the following conjecture, due to Haettel, predicts that these two complexes share the following property -- his original motivation for this conjecture is that it leads an alternative proof that Artin group of type $\\widetilde D_n$ satisfies the $K(\\pi,1)$-conjecture.\n\n\\subsection{Falk complexes for admissible arrangements}\n\\label{subsec:admissible}\nFor $i=1,2$, let $\\ca_i$ be an affine hyperplane arrangement in $\\mathbb R^{m_i}$. Then the \\emph{product} of $\\ca_1$ and $\\ca_2$ is an arrangement $\\ca$ in $\\mathbb R^{m_1+m_2}$ whose hyperplanes are of form $H\\times \\mathbb R^{m_2}$ for $H\\in \\ca_1$ or $\\mathbb R^{m_1}\\times H$ for $H\\in \\ca_2$. \n\\begin{definition}\n\\label{def:admissible}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nThe next result can be proved in the same way as Theorem~\\ref{thm:main}, using Proposition~\\ref{prop:key} and Theorem~\\ref{thm:Bnflag}.\n\\begin{cor}\n\\label{cor:AB}\nSuppose $\\ca$ is a complete finite shape affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}",
    "post_theorem_intro_text_len": 5858,
    "post_theorem_intro_text": "In the situation of the above theorem, we have a free action of $\\mathbb Z^n$ on $M(\\mathcal {A})$. Then the fundamental group of $M(\\mathcal {A})/\\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\\mathcal {A}$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group). \n\n\\begin{cor}\nUnder the assumption of Theorem~\\ref{thm:main1}, the manifold $M(\\mathcal {A})/\\mathbb Z^n$ is homotopy equivalent to a finite aspherical cell complex, which is the quotient of the Salvetti complex of $\\mathcal {A}$ (\\cite{s87}) by a free action of $\\mathbb Z^n$. In particular, the fundamental group of $M(\\mathcal {A})/\\mathbb Z^n$ is of type $F$.\n\\end{cor}\n\n\\Cref{thm:main1} is a special case of a more general statement which does not require $\\mathcal {A}$ to be $\\mathbb Z^n$-invariant, see \\Cref{thm:main}.\n\nNow we give more concrete examples of arrangements where Theorem~\\ref{thm:main1} (or more specifically, \\Cref{thm:main}) applies. For each dimension $n$, we will construct an infinite family of finite affine arrangements $\\ch_{k,n}$, and an infinite family of $\\mathbb Z^n$-invariant affine arrangements $\\ck_{k,n}$. These arrangements are not Coxeter arrangements, and their complexified complement do not admit iterated fibration structure in an obvious way. The finite arrangements are not simplicial. So the $K(\\pi,1)$ results for these arrangements are new. \n\n\\begin{definition}\nFor $k\\ge 1$, let $\\ch_{k,n}$ be the affine hyperplane arrangement in $\\mathbb R^n$ given by $x_i \\in \\{-2k-1, -2k+1, \\dots,-3, -1, 1, 3, \\dots, 2k - 1, 2k + 1\\}$ for $1 \\le i \\le n$ and $x_i \\pm x_j = 0$ for $1\\le i\\neq j\\le n$. This family $\\ch_{k,n}$ generalizes Falk's \\cite[Example 3.13]{falk1995k} which is neither supersolvable or simpicial.\n\nFor $k\\ge 1$, let $\\ck_{k,n}$ be the affine hyperplane arrangement in $\\mathbb R^n$ given by $x_i\\in \\mathbb Z$ for $1\\le i\\le n$, and $x_i+x_j \\in 2k\\mathbb Z+1$, $x_i-x_j \\in 2k\\mathbb Z$ for $1\\le i\\neq j\\le n$.\n\\end{definition}\n\n\\begin{thm}(=Theorem~\\ref{thm:examples})\nFor $n\\le 4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $n,k$.\n\\end{thm}\n\nFor $n=2$, Falk \\cite{falk1995k} constructed a (locally infinite) 2-dimensional complex $\\falk_\\mathcal {A}$ that is homotopy equivalent to the universal cover of $M(\\mathcal {A})$. This construction of Falk can be generalized to higher dimensions without too much difficulty \\cite{huang2024cycles}. Later, we show $\\falk_\\mathcal {A}$ is homotopy equivalent to the universal cover of $M(\\mathcal {A})$ whenever all the local arrangements are $K(\\pi,1)$. Thus all the above results rely on showing the complex $\\falk_\\mathcal {A}$ is contractible. \n\nRecall that a geodesic metric space $X$ is \\emph{injective} if any pairwise intersecting closed metric balls in $X$ have non-empty common intersection. For example, $\\mathbb R^n$ equipped with the $\\ell^\\infty$ metric is injective. Injective metric spaces are contractible, and they are connected to the above theorems in the following way.\n\n\\begin{thm}\n\\textup{(= \\Cref{thm:main} and \\Cref{thm:examples})}\nUnder the assumptions of any of the previous theorems, the Falk complex $\\falk_\\mathcal {A}$ admits a metric which makes it an injective metric space.\n\\end{thm}\n\nThe reason for us to consider skewed type $A_n$ arrangements as local arrangements, rather than the standard $A_n$-arrangements, is for the compatibility of arranging injective metrics on $\\falk_\\mathcal {A}$. A substantial part of the article (Section~\\ref{sec:check links}) is devoted to checking such compatibility (Proposition~\\ref{prop:key} and Remark~\\ref{rmk:contrast}). \nIt is a topic of independent interest to produce natural examples of injective metric spaces arising from group theory, and the above theorem gives many such examples. As the above theorem becomes conditional for $n\\ge 5$, we also have the following variation that is unconditional for all dimensions, where we allow the local arrangements to be a mixture of $A_n$-type and $B_n$-type.\nThe following corollary is more interesting in terms of providing new examples of injective metric spaces, rather than $K(\\pi,1)$ results, as one can prove the arrangements in the following corollary are $K(\\pi,1)$ via an iterated fibration argument.\n\n\\begin{cor} \\textup{(= \\Cref{cor:AB})}\n\\label{cor:AB intro}\nSuppose $\\mathcal {A}$ is a complete, finite shape, affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\mathcal {A},d_\\infty)$ is an injective metric space and $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}\n\n\\subsection*{Structure of the article} In Section~\\ref{sec:prelim} we collect some background material. In Section~\\ref{subsec:deligne complex} we define Falk complexes of affine hyperplane arrangements and prove some of them admit injective metrics, modulo a key proposition (Proposition~\\ref{prop:key}) about skewed $A_n$ arrangements. Section~\\ref{sec:check links} is devoted to this key proposition. Section~\\ref{sec:example} contains some new, concrete examples of affine arrangements where our results apply.",
    "sketch": "The introduction explains that the results (including Theorem~\\ref{thm:main1}) “rely on showing the complex $\\falk_\\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\\falk_\\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\\falk_\\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\\mathcal{A})$ whenever all the local arrangements are $K(\\pi,1)$,” so contractibility of $\\falk_\\mathcal{A}$ implies $M(\\mathcal{A})$ is aspherical.\n\nThey then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, under the assumptions of the main theorems, “the Falk complex $\\falk_\\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\\falk_\\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility (via “Proposition~\\ref{prop:key}”).",
    "expanded_sketch": "The introduction explains that the results (including the main theorem) “rely on showing the complex $\\falk_\\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\\falk_\\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\\falk_\\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\\mathcal{A})$ whenever all the local arrangements are $K(\\pi,1)$,” so contractibility of $\\falk_\\mathcal{A}$ implies $M(\\mathcal{A})$ is aspherical.\n\nThey then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, in establishing the main theorem, “the Falk complex $\\falk_\\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\\falk_\\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility via the following proposition.\n\n\\begin{prop}\n\\label{prop:key}\nThe poset $((\\Delta'_{\\ca(3)})^0,<)$ is bowtie free and upward flag.\n\\end{prop}",
    "expanded_theorem": "\\label{thm:main1}\nLet $\\mathcal {A}$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement. More generally, modulo the following group theoretical conjecture on the spherical Artin group of type $D_n$, $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement for any $n$. \\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal A\\) be an affine hyperplane arrangement in \\(\\mathbb R^n\\). For an \\(\\mathcal A\\)-vertex \\(x\\) (a point that is an intersection of hyperplanes of \\(\\mathcal A\\)), the local arrangement at \\(x\\) is the collection of hyperplanes of \\(\\mathcal A\\) that contain \\(x\\). Call \\(\\mathcal A\\) admissible if, at every \\(\\mathcal A\\)-vertex \\(x\\), the local arrangement at \\(x\\) is a translate of one of the following: (i) type \\(B_n\\): \\(x_i\\pm x_j=0\\) for \\(1\\le i\\ne j\\le n\\) and \\(x_i=0\\) for \\(1\\le i\\le n\\); (ii) type \\(D_n\\): \\(x_i\\pm x_j=0\\) for \\(1\\le i\\ne j\\le n\\); (iii) skewed type \\(A_n\\): \\(x_i=0\\) for \\(1\\le i\\le n\\) and \\(x_i=x_j\\) for \\(1\\le i\\ne j\\le n\\), or any image of this arrangement under the \\((\\mathbb Z/2\\mathbb Z)^n\\)-action on \\(\\mathbb R^n\\) by reflections in the coordinate hyperplanes; or (iv) a product of arrangements of the previous types, where the product of arrangements in \\(\\mathbb R^{m_1}\\) and \\(\\mathbb R^{m_2}\\) has hyperplanes of the form \\(H\\times \\mathbb R^{m_2}\\) or \\(\\mathbb R^{m_1}\\times H\\). Assume also that \\(\\mathcal A\\) is invariant under the action of a discrete translation subgroup of \\(\\mathbb R^n\\) isomorphic to \\(\\mathbb Z^n\\) (not necessarily the standard lattice). Let \\(M(\\mathcal A)=\\mathbb C^n\\setminus \\bigcup_{H\\in\\mathcal A}(H\\otimes\\mathbb C)\\), and call \\(\\mathcal A\\) a \\(K(\\pi,1)\\) arrangement when \\(M(\\mathcal A)\\) is aspherical. Which statement holds for every such \\(\\mathcal A\\)?",
      "correct_choice": {
        "label": "A",
        "text": "If \\(n\\le 4\\), then \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement. More generally, assuming Haettel's conjecture that for every type \\(D_n\\) with \\(n\\ge 3\\), the associated poset \\(((\\Delta'_\\Lambda)^0,<)\\) is bowtie free and upward flag, \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement for every \\(n\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "If \\(n<4\\), then \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement. More generally, assuming Haettel's conjecture that for every type \\(D_n\\) with \\(n\\ge 3\\), the associated poset \\(((\\Delta'_\\Lambda)^0,<)\\) is bowtie free and upward flag, \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement for every \\(n<4\\)."
        },
        {
          "label": "C",
          "text": "Assuming Haettel's conjecture that for every type \\(D_n\\) with \\(n\\ge 3\\), the associated poset \\(((\\Delta'_\\Lambda)^0,<)\\) is bowtie free and upward flag, \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement for every \\(n\\)."
        },
        {
          "label": "D",
          "text": "If \\(n\\le 4\\), then \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement. More generally, for every \\(n\\), if each type \\(D_n\\) local arrangement has associated poset \\(((\\Delta'_\\Lambda)^0,<)\\) bowtie free and upward flag, then \\(\\mathcal A\\) is a \\(K(\\pi,1)\\) arrangement."
        },
        {
          "label": "E",
          "text": "If \\(n\\le 4\\), then \\(M(\\mathcal A)\\) is contractible. More generally, assuming Haettel's conjecture that for every type \\(D_n\\) with \\(n\\ge 3\\), the associated poset \\(((\\Delta'_\\Lambda)^0,<)\\) is bowtie free and upward flag, \\(M(\\mathcal A)\\) is contractible for every \\(n\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "dimension bound n\\le 4 replaced by n<4",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the unconditional low-dimensional conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "global Haettel conjecture weakened to a condition only on type D_n local arrangements of the given arrangement",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "conclusion upgraded from asphericity/K(\\pi,1) to contractibility of M(\\mathcal A)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only definitions and hypotheses; it does not state the theorem’s conclusion or otherwise signal option A directly. There is no explicit or trivial answer leakage from the wording of the question itself."
      },
      "TAS": {
        "score": 2,
        "justification": "Although the item is theorem-focused, it is not a mere restatement with an obvious equivalent option. The choices vary in logical strength, dimension bounds, quantifier scope, and even the topological conclusion, so the student must distinguish among genuinely competing conclusions."
      },
      "GPS": {
        "score": 2,
        "justification": "The item creates real reasoning pressure: one must track unconditional vs conjectural parts, compare a stronger true statement against a weaker true one, and reject subtle quantifier shifts and an overly strong contractibility claim. This goes beyond spotting a paraphrase."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful. B tests boundary sensitivity (n≤4 vs n<4), C is a weaker true statement, D probes quantifier dependence, and E tests confusion between asphericity and contractibility. These align well with common failure modes."
      },
      "total_score": 8,
      "overall_assessment": "A strong MCQ: no answer leakage, non-tautological structure, substantial logical reasoning required, and high-quality distractors."
    }
  },
  {
    "id": "2512.00348v1",
    "paper_link": "http://arxiv.org/abs/2512.00348v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\mathcal{A}\\subseteq\\NN^{n}$ is not exposed if and only if $r=\\R_+ \\mathbf{x}^\\alpb$ for some $\\alpb\\in \\mathcal{A}$, and there exists a circuit $(S,\\betab)$ on $\\mathcal{A}$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.",
      "start_pos": 12353,
      "end_pos": 12661,
      "label": "thm:main"
    },
    "ref_dict": {
      "prop:unexposed": "\\begin{proposition}\\label{prop:unexposed} Let $\\cA\\subseteq \\nonneg{n}$ be a finite set, and let $\\gammab \\in \\cA\\cap \\even{n}$ be such that there exists a circuit $(S,\\betab)$ on $\\cA$ with $|S|>1$,  $\\gammab\\in S$, and $\\betab$ even, then the extreme ray $r= \\R_+ \\xb^\\gammab$ of $\\SONC{\\cA}$ is not exposed.\n\\end{proposition}",
      "prop:monomial": "\\begin{proposition}\\label{prop:monomial}\n    If $r = \\R_+ \\xb^{\\gammab}$ is an extreme ray of $\\SONC{\\cA}$ on a finite set $\\cA\\subseteq \\nonneg{n}$, and there is no circuit $(S,\\betab)$ on $\\cA$ such that $|S|>1$,  $\\gammab\\in S$, and $\\betab$ even, then $r$ is exposed.\n\\end{proposition}",
      "subsec:SONC": "\\begin{equation}\\label{eq:defcircuitnumber}\n\\Theta_f  \\coloneqq \\prod_{\\alp \\in S} \\left( \\frac{c_\\alpb}{\\lambda_\\alpb}\\right)^{\\lambda_\\alpb}.\n\\end{equation}\n\nThis invariant can be used to easily check if a circuit polynomial is nonnegative, see \\cite[Theorem 1.1]{iliman2016amoebas}. Let $f$ be a circuit polynomial as in~\\eqref{eq:defcircuit}, then $f$ is nonnegative if and only if \n $f$ is a sum of monomial squares, or\n $   |d|\\leq \\Theta_f$. \n\n\\subsection{The SONC cone and its extreme rays}\\label{subsec:SONC} \nIf a polynomial can be written as a sum of nonnegative circuit polynomials, we call it a \\struc{\\emph{SONC polynomial}}.\nThe \\struc{\\emph{SONC cone $ \\SONC{\\cA}$}} on some finite ground set $\\cA\\subseteq \\nonneg{n}$ is the set of all conic (finite nonnegative) combinations of nonnegative circuit polynomials in $\\Pspace\\cA$. It is not necessary to consider all possible circuits on $\\cA$ to represent each polynomial in $\\SONC{\\cA}$ as a conic combination of nonnegative circuit polynomials; instead, the representation can be restricted to reduced circuits. We adjust the original definition of reduced circuits given in \\cite{katthan2021unified}  to our less general setting as follows.\n\n\\begin{definition} Let $\\cA\\subseteq \\nonneg{n}$ be finite. A circuit $(S,\\betab)$ on $\\cA$ is \\struc{\\emph{reduced}} (with respect to $\\cA$) if \n\\[\n\\conv (S) \\cap \\cA \\cap \\even{n} = S\\cup \\{\\betab\\} \\cap \\even{n}.\n\\]  \n\\end{definition}\n\nIn other words, a circuit is reduced if there are no even points in its convex hull apart from the points in $S\\cup\\{\\betab\\}$.\n\n\\smallskip\nRecall that a ray $r=\\R_+\\cdot v$ (for some $v\\neq 0$) is an \\struc{\\emph{extreme ray}} of a convex cone $K$ if for any $y,z\\in K$ such that $x = y+z$, we have $y,z\\in \\R_+\\cdot v$. In other words, an extreme ray is a one-dimensional face of a pointed cone $K$.\n\nThe concept of reduced circuits leads to a complete characterization of the extreme rays of the SONC cone.\n\n\\begin{proposition}[\\cite{katthan2021unified}, Corollary~4.6]\\label{prop:extreme} \nLet $r$ be an extreme ray of $\\SONC{\\cA}$. Then $r = \\RR_+ f$, where $f\\in \\Pspace\\cA$ belongs to one of the following three types of nonnegative circuit polynomials:\n\\begin{itemize}\n    \\item[1.] $f(\\xb) = \\xb^\\betab, \\quad \\betab\\in \\cA\\cap \\even{n}$;\n    \\item[2.] $f$ is supported on some (even or odd) reduced circuit $(S,\\betab)$, with $|S|>1$ and  \n    \\[\n    f(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \\xb^\\alpb - \\Theta_f \\xb^\\betab,\\quad c_\\alpb >0\\quad  \\text{ for all } \\; \\alpb \\in S;\n    \\]\n    \\item[3.] $f$ is supported on an odd reduced circuit $(S,\\betab)$, with $|S|>1$ and  \n    \\[\n    f(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \\xb^\\alpb +  \\Theta_f \\xb^\\betab,\\quad c_\\alpb >0\\quad  \\text{ for all } \\; \\alpb \\in S.\n    \\]\n\\end{itemize}\n\\end{proposition}\n\n\\section{Unexposed rays}\\label{sec:unexposed}\n\nIn this section we prove that the extreme rays of type 1 given in Proposition~\\ref{prop:extreme} that satisfy the conditions of Theorem~\\ref{thm:main} are unexposed. Together with the positive results of the next Section (showing that the remaining extreme rays are exposed) this proves Theorem~\\ref{thm:main}.\n\nRecall that a face $F$ of a convex cone $K\\in X$ (where $X$ is a linear vector space) is \\struc{\\emph{exposed}} if there exists a linear function $\\struc{l}\\colon X\\to \\R$  such that\n\\begin{equation}\\label{eq:defexposed}    \nl(x)=0\\; \\, \\text{ for all }\\; x\\in F,\\quad \\text{ and } \\quad l(y)>0\\;\\, \\text{ for all }\\, y\\in K\\setminus F. \n\\end{equation}",
      "thm:main": "\\begin{theorem}\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\cA\\subseteq\\nonneg{n}$ is not exposed if and only if $r=\\R_+ \\xb^\\alpb$ for some $\\alpb\\in \\cA$, and there exists a circuit $(S,\\betab)$ on $\\cA$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.\n\\end{theorem}",
      "prop:nonmonomial": "\\begin{proposition}\\label{prop:nonmonomial} Every non-monomial extreme ray of the SONC cone on a finite ground set $\\cA$ is exposed.\n\\end{proposition}",
      "subsec:nncircuits": "\\label{subsec:nncircuits}\nRecall that a \\struc{\\emph{circuit}} is a set that is minimal affine dependent, see e.g. \\cite{GKZ:Discriminants}. If a circuit $S\\cup \\{\\betab\\}\\subseteq \\NN^n$ is such that"
    },
    "pre_theorem_intro_text_len": 3769,
    "pre_theorem_intro_text": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.   \n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding  nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.",
    "context": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.\n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.",
    "full_context": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.\n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.\n\nA polynomial $f(\\xb)=\\sum_{\\alpb \\in S} c_{\\alpb} \\xb^{\\alpb} + d \\xb^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\xb^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $\\even{n}$ with $\\betab\\in \\nonneg{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\cA\\subseteq \\nonneg{n}$ as the set of all SONC polynomials supported on $\\cA$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\cA$ (i.e., $S\\subset \\cA$ and $\\betab\\in \\cA$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.\n\nThis result is proved by showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing, and constructing explicit exposing normals for all of the remaining cases. The construction of these normals relies on a graded partition of the ground set $\\cA$ that allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\cA$.\n\n\\subsection{Nonnegative circuit polynomials}\\label{subsec:nncircuits}\nRecall that a \\struc{\\emph{circuit}} is a set that is minimal affine dependent, see e.g. \\cite{GKZ:Discriminants}. If a circuit $S\\cup \\{\\betab\\}\\subseteq \\NN^n$ is such that $S$ is affinely independent and $\\betab$ belongs to the relative interior of $\\conv(S)$, we call the circuit \\emph{simplicial}. Since we are only dealing with simplicial circuits, we just refer to this case as a circuit hereafter. We will also use the notation \\struc{$(S,\\betab)$} to denote such circuits to streamline our exposition. \nFurther, we say a circuit $(S,\\betab)$ is \\struc{\\emph{odd}} or \\struc{\\emph{even}} according to the parity of $\\betab$. \nNote that for a circuit, $\\conv(S)$ is a simplex, and hence there is a unique way to represent $\\betab$ as a convex combination of $S$. That is, there exists unique \\struc{\\emph{barycentric coordinates $(\\lambda_{\\alpb})_{\\alpb \\in S}$}} (with respect to $S$) such that $\\betab = \\sum_{\\alpb \\in S} \\lambda_\\alpb \\alpb$, with $\\lambda_\\alpb>0$ for all $\\alpb \\in S$ and $\\sum_{\\alpb \\in S} \\lambda_\\alpb = 1$.\n\nThe following result contributes to the proof of the negative part of Theorem~\\ref{thm:main}. \n\\begin{proposition}\\label{prop:unexposed} Let $\\cA\\subseteq \\nonneg{n}$ be a finite set, and let $\\gammab \\in \\cA\\cap \\even{n}$ be such that there exists a circuit $(S,\\betab)$ on $\\cA$ with $|S|>1$, $\\gammab\\in S$, and $\\betab$ even, then the extreme ray $r= \\R_+ \\xb^\\gammab$ of $\\SONC{\\cA}$ is not exposed.\n\\end{proposition}\n\\begin{proof}\nSuppose to the contrary that under the assumptions of the proposition, the extreme ray $r=\\R_+ \\xb^\\gammab$ is exposed. Let $(\\lambda_\\alpb)_{\\alpb \\in S}$ be the barycentric coordinates of $\\betab$ with respect to $S$, and let\n\\[\nf_t (\\xb) \\coloneqq \\lambda_\\gammab \\xb^{\\gammab}+ \\sum_{\\alpb \\in S\\setminus \\{\\gammab\\}} \\lambda_{\\alpb} t^{\\frac{1}{1-\\lambda_{\\gammab}}}\\xb^\\alpb - t \\xb^\\betab. \n\\]\nNotice that for any $t>0$, the polynomial $f_t$ is a circuit polynomial on $(S,\\betab)$ with $\\Theta_{f_t} =t$, and therefore $f_t\\in \\SONC{\\cA}$.\n\n\\begin{lemma}\\label{lem:graded} For a finite set $\\cA\\subseteq \\nonneg{n}$ and a reduced circuit $(S,\\betab)$ on $\\cA$, there exists a \\struc{\\emph{graded partition}} $\\struc{L_0}, \\struc{L_1},\\dots, \\struc{L_K}$ of $(\\cA\\cap \\even{n})\\cup \\{\\betab\\}$ such that $L_0 = S\\cup \\{\\betab\\}$,\n\\begin{itemize}\n \\item[(i)]$L_0\\cup L_1\\cup \\cdots \\cup L_K = (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$;\\quad $L_i\\cap L_j = \\emptyset$ $\\text{ for all } i\\neq j$; and\n \\item[(ii)] for any circuit $(S',\\gammab)\\neq (S,\\betab)$, $|S'|>1$ on $\\cA$ with $\\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$ there exists $\\alpb \\in S'$ such that $\\alpb\\in L_j$, $\\gammab\\in L_i$ with $j>i$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proposition}\\label{prop:monomial}\n If $r = \\R_+ \\xb^{\\gammab}$ is an extreme ray of $\\SONC{\\cA}$ on a finite set $\\cA\\subseteq \\nonneg{n}$, and there is no circuit $(S,\\betab)$ on $\\cA$ such that $|S|>1$, $\\gammab\\in S$, and $\\betab$ even, then $r$ is exposed.\n\\end{proposition}\n\\begin{proof} Let $\\xb^\\gammab$ and $\\cA$ satisfy the assumptions of the proposition. We prove that the extreme ray $r = \\R_+ \\xb^\\gammab$ is exposed by constructing an explicit exposing linear mapping $l\\colon \\Pspace{\\cA}\\to \\R$.\n\n\\begin{proof} Let $(S,\\betab)$ be a reduced circuit with $|S|>1$, and let $f$ be a circuit polynomial on this circuit, generating one of the non-monomial extreme rays (that is, one of the extreme rays given in Proposition~\\ref{prop:extreme} items 2 and 3). We have \n\\[\nf(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \n\\xb^\\alpb \\pm \\Theta_f \\xb^\\betab,\n\\]\nwith $c_\\alpb>0$ for all $\\alpb \\in S$. To show that $r = \\R_+ f$ is an exposed ray, we construct the exposing linear mapping $l$ explicitly using a similar idea to the proof of Proposition~\\ref{prop:monomial}. Choose some $\\sigma, \\delta>0$ such that \n\\[\n\\sigma < \\min\\left \\{\\min_{\\alpb\\in S} \\frac{\\lambda_\\alpb}{c_\\alpb}, \\Theta^{-1}_f\\right\\},\\quad \n\\delta> \\max \\left\\{\\max_{\\alpb\\in S} \\frac{\\lambda_\\alpb}{c_\\alpb}, \\Theta^{-1}_f\n\\right\\}.\\]\nBy Lemma~\\ref{lem:graded}, there exists a graded partition $L_0,L_1,\\dots, L_K$ of $(\\cA\\cap \\even{n})\\cup \\{\\betab\\} $ such that $L_0 = S\\cup \\{\\betab\\}$ and for any circuit $(S',\\gammab)$, where $\\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$, there is an $\\alpb'\\in S'$ such that $\\gammab\\in L_i$, $\\alpb'\\in L_j$ with $j>i$. \nLet \n\\begin{equation}\\label{eq:constrNormals34}\nl(\\xb^\\alpb) \\coloneqq \\begin{cases}\n \\sigma^{-1} \\frac{\\lambda_\\alpb}{c_\\alpb}, & \\text{ if } \\alpb \\in S;\\\\\n \\mp \\sigma^{-1}\\Theta^{-1}_f, & \\text{ if } \\alpb = \\betab;\\\\\n (\\sigma^{-1}\\delta)^{\\Lambda^i}, & \\text{ if } \\alpb \\in L_i, \\; i \\in \\{1,\\dots, K\\};\\\\ \n 0, & \\text{otherwise (when } \\alpb \\in \\cA\\setminus (\\even{n}\\cup \\{\\betab\\}) \\text{)}.\n\\end{cases}\n\\end{equation}\nIn the case $\\alpb=\\betab$ we specifically want the value $l(\\xb^\\betab)$ to have the opposite sign to the coefficient of $f$ at $\\xb^\\betab$. In particular, this means that the only case when $l(\\xb^\\gammab)$ is negative is when $\\gammab = \\betab$ and $\\betab$ is odd (see Proposition~\\ref{prop:extreme}, items 2 and 3). Observe that due to our choice of $\\sigma$ and $\\delta$, we have \n\\[\n|l(\\xb^\\gammab)|> 1 \\quad \\text{ for all } \\, \\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}.\n\\]\nTo prove that $r = \\R_+ f$ is an exposed ray of $\\SONC{\\cA}$, by Lemma~\\ref{lem:extremeChar} it is sufficient to show that for any extreme ray $q = \\R_+ g$ of $\\SONC{\\cA}$, we have $l(g)>0$, unless $g\\in \\R_{+}f$, in which case we must have $l(g) =0$. We verify this by systematically considering all possible extreme rays of $\\SONC{\\cA}$ listed in Proposition~\\ref{prop:extreme}.",
    "post_theorem_intro_text_len": 1546,
    "post_theorem_intro_text": "This result is proved by showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing, and constructing explicit exposing normals for all of the remaining cases. The construction of these normals relies on a graded partition of the ground set $\\mathcal{A}$ that allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\mathcal{A}$.\n\n\\smallskip\nWhile it is conceivable that the existence of such exposing normals could be deduced from general convex analysis arguments (as discussed in Section~\\ref{sec:open}), our proof is fully constructive in the sense that we provide explicit normals certifying exposure of each ray. \nThis constructive approach not only clarifies the geometry of the SONC cone but also lends itself to algorithmic implementation, potentially contributing to practical optimization frameworks that incorporate SONC certificates.\n\n\\medskip\nOur paper is organized as follows. Section~\\ref{sec:preliminaries} sets up the notation and reviews key properties of the SONC cone, including the characterization of its extreme rays.  In Section~\\ref{sec:unexposed} we establish the negative part of the result in Proposition~\\ref{prop:unexposed}. In Section~\\ref{sec:exposed} we show that the remaining extreme rays are exposed in Propositions~\\ref{prop:monomial} and~\\ref{prop:nonmonomial}, and finalize the proof of Theorem~\\ref{thm:main}. We close with a brief discussion of open problems in Section~\\ref{sec:open}.",
    "sketch": "The result is proved by (i) \"showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing,\" and (ii) \"constructing explicit exposing normals for all of the remaining cases.\" The construction of exposing normals \"relies on a graded partition of the ground set $\\mathcal{A}$\" which \"allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\mathcal{A}$.\" The proof is \"fully constructive\" in that it provides \"explicit normals certifying exposure of each ray.\" Concretely, the paper \"establish[es] the negative part of the result\" in Proposition~\\ref{prop:unexposed}, then proves the remaining extreme rays are exposed in Propositions~\\ref{prop:monomial} and~\\ref{prop:nonmonomial}, and \"finalize[s] the proof of Theorem~\\ref{thm:main}.\"",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\mathcal{A}\\subseteq\\NN^{n}$ is not exposed if and only if $r=\\R_+ \\mathbf{x}^\\alpb$ for some $\\alpb\\in \\mathcal{A}$, and there exists a circuit $(S,\\betab)$ on $\\mathcal{A}$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal A\\subseteq \\mathbb N^n\\) be finite, and let \\(\\operatorname{SONC}(\\mathcal A)\\) denote the cone of sums of nonnegative circuit polynomials supported on \\(\\mathcal A\\). Here a circuit on \\(\\mathcal A\\) means a pair \\((S,\\beta)\\) with \\(S\\subseteq \\mathcal A\\cap (2\\mathbb N)^n\\) affinely independent and \\(\\beta\\in \\mathcal A\\cap \\operatorname{relint}(\\operatorname{conv}(S))\\). Which extreme rays of \\(\\operatorname{SONC}(\\mathcal A)\\) are not exposed?",
      "correct_choice": {
        "label": "A",
        "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which there exists a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\) and \\(\\beta\\neq \\alpha\\) even; equivalently, an extreme ray of \\(\\operatorname{SONC}(\\mathcal A)\\) is not exposed if and only if it is of this form."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which there exists a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\) and \\(\\beta\\neq \\alpha\\); equivalently, the parity of \\(\\beta\\) is irrelevant."
        },
        {
          "label": "C",
          "text": "Every extreme ray of \\(\\operatorname{SONC}(\\mathcal A)\\) that is not exposed is a monomial ray \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for some \\(\\alpha\\in\\mathcal A\\)."
        },
        {
          "label": "D",
          "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which for every circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\), one has \\(\\beta\\neq \\alpha\\) even; equivalently, an extreme ray is not exposed precisely when \\(\\alpha\\) belongs to no circuit with odd inner exponent."
        },
        {
          "label": "E",
          "text": "An extreme ray \\(r\\) of \\(\\operatorname{SONC}(\\mathcal A)\\) is not exposed if and only if either \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for some \\(\\alpha\\in\\mathcal A\\) with a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) satisfying \\(\\alpha\\in S\\) and \\(\\beta\\neq\\alpha\\) even, or else \\(r\\) is itself generated by a non-monomial circuit polynomial whose inner exponent \\(\\beta\\) is even."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "evenness_of_\\beta",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "existence_of_circuit_witness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "existential_vs_universal_circuit_condition",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "monomial_only_classification",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct characterization. It asks for an equivalent condition and provides only background definitions, so the correct answer is not leaked from the wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: it asks for the statement equivalent to non-exposedness of an extreme ray, which closely matches a characterization theorem rather than requiring application in a new setting."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle logical features such as existential vs. universal quantification, omission of the evenness condition, and confusion between monomial and circuit-generated rays. However, the item mainly tests recall of the exact theorem statement rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one is a weaker true statement, one drops a necessary condition, one alters the quantifier, and one switches the ray type. These reflect realistic failure modes and are clearly distinct."
      },
      "total_score": 5,
      "overall_assessment": "A technically strong recall-style MCQ with no answer leakage and high-quality distractors, but it is largely a direct theorem-characterization question rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.00348v1",
    "paper_link": "http://arxiv.org/abs/2512.00348v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\mathcal{A}\\subseteq\\NN^{n}$ is not exposed if and only if $r=\\R_+ \\mathbf{x}^\\alpb$ for some $\\alpb\\in \\mathcal{A}$, and there exists a circuit $(S,\\betab)$ on $\\mathcal{A}$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.",
      "start_pos": 12353,
      "end_pos": 12661,
      "label": "thm:main"
    },
    "ref_dict": {
      "prop:unexposed": "\\begin{proposition}\\label{prop:unexposed} Let $\\cA\\subseteq \\nonneg{n}$ be a finite set, and let $\\gammab \\in \\cA\\cap \\even{n}$ be such that there exists a circuit $(S,\\betab)$ on $\\cA$ with $|S|>1$,  $\\gammab\\in S$, and $\\betab$ even, then the extreme ray $r= \\R_+ \\xb^\\gammab$ of $\\SONC{\\cA}$ is not exposed.\n\\end{proposition}",
      "prop:monomial": "\\begin{proposition}\\label{prop:monomial}\n    If $r = \\R_+ \\xb^{\\gammab}$ is an extreme ray of $\\SONC{\\cA}$ on a finite set $\\cA\\subseteq \\nonneg{n}$, and there is no circuit $(S,\\betab)$ on $\\cA$ such that $|S|>1$,  $\\gammab\\in S$, and $\\betab$ even, then $r$ is exposed.\n\\end{proposition}",
      "subsec:SONC": "\\begin{equation}\\label{eq:defcircuitnumber}\n\\Theta_f  \\coloneqq \\prod_{\\alp \\in S} \\left( \\frac{c_\\alpb}{\\lambda_\\alpb}\\right)^{\\lambda_\\alpb}.\n\\end{equation}\n\nThis invariant can be used to easily check if a circuit polynomial is nonnegative, see \\cite[Theorem 1.1]{iliman2016amoebas}. Let $f$ be a circuit polynomial as in~\\eqref{eq:defcircuit}, then $f$ is nonnegative if and only if \n $f$ is a sum of monomial squares, or\n $   |d|\\leq \\Theta_f$. \n\n\\subsection{The SONC cone and its extreme rays}\\label{subsec:SONC} \nIf a polynomial can be written as a sum of nonnegative circuit polynomials, we call it a \\struc{\\emph{SONC polynomial}}.\nThe \\struc{\\emph{SONC cone $ \\SONC{\\cA}$}} on some finite ground set $\\cA\\subseteq \\nonneg{n}$ is the set of all conic (finite nonnegative) combinations of nonnegative circuit polynomials in $\\Pspace\\cA$. It is not necessary to consider all possible circuits on $\\cA$ to represent each polynomial in $\\SONC{\\cA}$ as a conic combination of nonnegative circuit polynomials; instead, the representation can be restricted to reduced circuits. We adjust the original definition of reduced circuits given in \\cite{katthan2021unified}  to our less general setting as follows.\n\n\\begin{definition} Let $\\cA\\subseteq \\nonneg{n}$ be finite. A circuit $(S,\\betab)$ on $\\cA$ is \\struc{\\emph{reduced}} (with respect to $\\cA$) if \n\\[\n\\conv (S) \\cap \\cA \\cap \\even{n} = S\\cup \\{\\betab\\} \\cap \\even{n}.\n\\]  \n\\end{definition}\n\nIn other words, a circuit is reduced if there are no even points in its convex hull apart from the points in $S\\cup\\{\\betab\\}$.\n\n\\smallskip\nRecall that a ray $r=\\R_+\\cdot v$ (for some $v\\neq 0$) is an \\struc{\\emph{extreme ray}} of a convex cone $K$ if for any $y,z\\in K$ such that $x = y+z$, we have $y,z\\in \\R_+\\cdot v$. In other words, an extreme ray is a one-dimensional face of a pointed cone $K$.\n\nThe concept of reduced circuits leads to a complete characterization of the extreme rays of the SONC cone.\n\n\\begin{proposition}[\\cite{katthan2021unified}, Corollary~4.6]\\label{prop:extreme} \nLet $r$ be an extreme ray of $\\SONC{\\cA}$. Then $r = \\RR_+ f$, where $f\\in \\Pspace\\cA$ belongs to one of the following three types of nonnegative circuit polynomials:\n\\begin{itemize}\n    \\item[1.] $f(\\xb) = \\xb^\\betab, \\quad \\betab\\in \\cA\\cap \\even{n}$;\n    \\item[2.] $f$ is supported on some (even or odd) reduced circuit $(S,\\betab)$, with $|S|>1$ and  \n    \\[\n    f(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \\xb^\\alpb - \\Theta_f \\xb^\\betab,\\quad c_\\alpb >0\\quad  \\text{ for all } \\; \\alpb \\in S;\n    \\]\n    \\item[3.] $f$ is supported on an odd reduced circuit $(S,\\betab)$, with $|S|>1$ and  \n    \\[\n    f(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \\xb^\\alpb +  \\Theta_f \\xb^\\betab,\\quad c_\\alpb >0\\quad  \\text{ for all } \\; \\alpb \\in S.\n    \\]\n\\end{itemize}\n\\end{proposition}\n\n\\section{Unexposed rays}\\label{sec:unexposed}\n\nIn this section we prove that the extreme rays of type 1 given in Proposition~\\ref{prop:extreme} that satisfy the conditions of Theorem~\\ref{thm:main} are unexposed. Together with the positive results of the next Section (showing that the remaining extreme rays are exposed) this proves Theorem~\\ref{thm:main}.\n\nRecall that a face $F$ of a convex cone $K\\in X$ (where $X$ is a linear vector space) is \\struc{\\emph{exposed}} if there exists a linear function $\\struc{l}\\colon X\\to \\R$  such that\n\\begin{equation}\\label{eq:defexposed}    \nl(x)=0\\; \\, \\text{ for all }\\; x\\in F,\\quad \\text{ and } \\quad l(y)>0\\;\\, \\text{ for all }\\, y\\in K\\setminus F. \n\\end{equation}",
      "thm:main": "\\begin{theorem}\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\cA\\subseteq\\nonneg{n}$ is not exposed if and only if $r=\\R_+ \\xb^\\alpb$ for some $\\alpb\\in \\cA$, and there exists a circuit $(S,\\betab)$ on $\\cA$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.\n\\end{theorem}",
      "prop:nonmonomial": "\\begin{proposition}\\label{prop:nonmonomial} Every non-monomial extreme ray of the SONC cone on a finite ground set $\\cA$ is exposed.\n\\end{proposition}",
      "subsec:nncircuits": "\\label{subsec:nncircuits}\nRecall that a \\struc{\\emph{circuit}} is a set that is minimal affine dependent, see e.g. \\cite{GKZ:Discriminants}. If a circuit $S\\cup \\{\\betab\\}\\subseteq \\NN^n$ is such that"
    },
    "pre_theorem_intro_text_len": 3769,
    "pre_theorem_intro_text": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.   \n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding  nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.",
    "context": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.\n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.",
    "full_context": "A central objective in real algebraic geometry, \nand a driving force in polynomial optimization,\nis to understand the structure of the cone of nonnegative polynomials and, crucially, its \\textit{tractable} subcones, that is, those for which membership can be efficiently verified.\nThe most prominent such subcone is the cone of sums of squares (SOS), whose relationship to nonnegativity has been extensively studied since Hilbert’s seminal 1888 work~\\cite{Hilbert1888}.\nOver the past decades, the facial geometry of the cone of nonnegative polynomials and the SOS cone has received sustained attention, including analyses of faces, extreme and exposed rays, the boundary, and rank structure, see e.g. \\cite{Blekherman:NNSOS,Blekherman:etal:Boundaries,Blekherman:Iliman:Kubitzke,Blekherman:Parrilo:Thomas}.\nThese works have substantially advanced the understanding of the structure and interplay of these cones, yet a complete understanding is still lacking.\n\n\\smallskip\nA more recent example of a tractable subcone, which is independent of the SOS cone, is the cone of sums of nonnegative circuit (SONC) polynomials. \nFirst formally introduced in~\\cite{iliman2016amoebas} and extending the earlier notion of agiforms from~\\cite{reznick1989forms}, the SONC cone\nprovides a sparsity-preserving and combinatorially governed alternative to SOS-based certificates. An equivalent construction in the signomial setting, known as the cone of sums of arithmetic-geometric exponentials (SAGE), is introduced in~\\cite{ChandrasekaranShah_REPforSignomialOptimization}, and related but independently developed frameworks can be found in~\\cite{Fidalgo:Kovacec,PanteaEtAL_GlobalInjectivityAndMultipleEquilibria}.\n\nDespite several recent works that have substantially advanced our knowledge of the geometry of the SONC cone, see e.g.~\\cite{katthan2021unified,dressler2021real,AlgBound}, its facial structure remains far from fully understood. In particular, it is still unknown which of its faces are exposed. In~\\cite{dressler2021real}, the first author initiated the study of exposed faces of the SONC cone, hereby focusing on exposed faces arising from polynomials vanishing at a finite set of points in $\\RR^n$ and providing explicit classifications in low dimensions together with general dimension bounds.\n\n\\smallskip\nIn this work, we (partially) close this gap by fully characterizing the exposed extreme rays of the SONC cone.\n\nA polynomial $f(\\mathbf{x})=\\sum_{\\alpb \\in S} c_{\\alpb} \\mathbf{x}^{\\alpb} + d \\mathbf{x}^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\mathbf{x}^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $2 \\NN^{n}$ with $\\betab\\in \\NN^{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\mathcal{A}\\subseteq \\NN^{n}$ as the set of all SONC polynomials supported on $\\mathcal{A}$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\mathcal{A}$ (i.e., $S\\subset \\mathcal{A}$ and $\\betab\\in \\mathcal{A}$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.\n\nA polynomial $f(\\xb)=\\sum_{\\alpb \\in S} c_{\\alpb} \\xb^{\\alpb} + d \\xb^{\\betab}$, (where we use the standard shorthand for writing multivariate monomials, $\\xb^\\alpb \\coloneqq x_1^{\\alp_1} x_2^{\\alp_2} \\cdots x_n^{\\alp_n}$), where $S$ is an affinely independent subset of $\\even{n}$ with $\\betab\\in \\nonneg{n}$ in the relative interior of the convex hull of $S$ and the coefficients $c_{\\alpb}>0$, is called a circuit polynomial; see also Section~\\ref{subsec:nncircuits}. Deciding nonnegativity of a circuit polynomial is equivalent to solving a system of linear equations, a fact that follows directly from the arithmetic-geometric inequality. The distinguishing feature of this approach is its sparsity-preserving nature, making it particularly well-suited for handling high-degree polynomials with sparse support. This motivates us to consider the SONC cone on a given finite ground set $\\cA\\subseteq \\nonneg{n}$ as the set of all SONC polynomials supported on $\\cA$, that is, all nonnegative combinations of circuit polynomials with ``circuits\" on $\\cA$ (i.e., $S\\subset \\cA$ and $\\betab\\in \\cA$). For the explicit definition, see Sections~\\ref{subsec:nncircuits} and \\ref{subsec:SONC}.\n\n\\medskip\nOur main result is the following characterization of exposed rays of the SONC cone.\n\nThis result is proved by showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing, and constructing explicit exposing normals for all of the remaining cases. The construction of these normals relies on a graded partition of the ground set $\\cA$ that allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\cA$.\n\n\\subsection{Nonnegative circuit polynomials}\\label{subsec:nncircuits}\nRecall that a \\struc{\\emph{circuit}} is a set that is minimal affine dependent, see e.g. \\cite{GKZ:Discriminants}. If a circuit $S\\cup \\{\\betab\\}\\subseteq \\NN^n$ is such that $S$ is affinely independent and $\\betab$ belongs to the relative interior of $\\conv(S)$, we call the circuit \\emph{simplicial}. Since we are only dealing with simplicial circuits, we just refer to this case as a circuit hereafter. We will also use the notation \\struc{$(S,\\betab)$} to denote such circuits to streamline our exposition. \nFurther, we say a circuit $(S,\\betab)$ is \\struc{\\emph{odd}} or \\struc{\\emph{even}} according to the parity of $\\betab$. \nNote that for a circuit, $\\conv(S)$ is a simplex, and hence there is a unique way to represent $\\betab$ as a convex combination of $S$. That is, there exists unique \\struc{\\emph{barycentric coordinates $(\\lambda_{\\alpb})_{\\alpb \\in S}$}} (with respect to $S$) such that $\\betab = \\sum_{\\alpb \\in S} \\lambda_\\alpb \\alpb$, with $\\lambda_\\alpb>0$ for all $\\alpb \\in S$ and $\\sum_{\\alpb \\in S} \\lambda_\\alpb = 1$.\n\nThe following result contributes to the proof of the negative part of Theorem~\\ref{thm:main}. \n\\begin{proposition}\\label{prop:unexposed} Let $\\cA\\subseteq \\nonneg{n}$ be a finite set, and let $\\gammab \\in \\cA\\cap \\even{n}$ be such that there exists a circuit $(S,\\betab)$ on $\\cA$ with $|S|>1$, $\\gammab\\in S$, and $\\betab$ even, then the extreme ray $r= \\R_+ \\xb^\\gammab$ of $\\SONC{\\cA}$ is not exposed.\n\\end{proposition}\n\\begin{proof}\nSuppose to the contrary that under the assumptions of the proposition, the extreme ray $r=\\R_+ \\xb^\\gammab$ is exposed. Let $(\\lambda_\\alpb)_{\\alpb \\in S}$ be the barycentric coordinates of $\\betab$ with respect to $S$, and let\n\\[\nf_t (\\xb) \\coloneqq \\lambda_\\gammab \\xb^{\\gammab}+ \\sum_{\\alpb \\in S\\setminus \\{\\gammab\\}} \\lambda_{\\alpb} t^{\\frac{1}{1-\\lambda_{\\gammab}}}\\xb^\\alpb - t \\xb^\\betab. \n\\]\nNotice that for any $t>0$, the polynomial $f_t$ is a circuit polynomial on $(S,\\betab)$ with $\\Theta_{f_t} =t$, and therefore $f_t\\in \\SONC{\\cA}$.\n\n\\begin{lemma}\\label{lem:graded} For a finite set $\\cA\\subseteq \\nonneg{n}$ and a reduced circuit $(S,\\betab)$ on $\\cA$, there exists a \\struc{\\emph{graded partition}} $\\struc{L_0}, \\struc{L_1},\\dots, \\struc{L_K}$ of $(\\cA\\cap \\even{n})\\cup \\{\\betab\\}$ such that $L_0 = S\\cup \\{\\betab\\}$,\n\\begin{itemize}\n \\item[(i)]$L_0\\cup L_1\\cup \\cdots \\cup L_K = (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$;\\quad $L_i\\cap L_j = \\emptyset$ $\\text{ for all } i\\neq j$; and\n \\item[(ii)] for any circuit $(S',\\gammab)\\neq (S,\\betab)$, $|S'|>1$ on $\\cA$ with $\\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$ there exists $\\alpb \\in S'$ such that $\\alpb\\in L_j$, $\\gammab\\in L_i$ with $j>i$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proposition}\\label{prop:monomial}\n If $r = \\R_+ \\xb^{\\gammab}$ is an extreme ray of $\\SONC{\\cA}$ on a finite set $\\cA\\subseteq \\nonneg{n}$, and there is no circuit $(S,\\betab)$ on $\\cA$ such that $|S|>1$, $\\gammab\\in S$, and $\\betab$ even, then $r$ is exposed.\n\\end{proposition}\n\\begin{proof} Let $\\xb^\\gammab$ and $\\cA$ satisfy the assumptions of the proposition. We prove that the extreme ray $r = \\R_+ \\xb^\\gammab$ is exposed by constructing an explicit exposing linear mapping $l\\colon \\Pspace{\\cA}\\to \\R$.\n\n\\begin{proof} Let $(S,\\betab)$ be a reduced circuit with $|S|>1$, and let $f$ be a circuit polynomial on this circuit, generating one of the non-monomial extreme rays (that is, one of the extreme rays given in Proposition~\\ref{prop:extreme} items 2 and 3). We have \n\\[\nf(\\xb) = \\sum_{\\alpb \\in S} c_\\alpb \n\\xb^\\alpb \\pm \\Theta_f \\xb^\\betab,\n\\]\nwith $c_\\alpb>0$ for all $\\alpb \\in S$. To show that $r = \\R_+ f$ is an exposed ray, we construct the exposing linear mapping $l$ explicitly using a similar idea to the proof of Proposition~\\ref{prop:monomial}. Choose some $\\sigma, \\delta>0$ such that \n\\[\n\\sigma < \\min\\left \\{\\min_{\\alpb\\in S} \\frac{\\lambda_\\alpb}{c_\\alpb}, \\Theta^{-1}_f\\right\\},\\quad \n\\delta> \\max \\left\\{\\max_{\\alpb\\in S} \\frac{\\lambda_\\alpb}{c_\\alpb}, \\Theta^{-1}_f\n\\right\\}.\\]\nBy Lemma~\\ref{lem:graded}, there exists a graded partition $L_0,L_1,\\dots, L_K$ of $(\\cA\\cap \\even{n})\\cup \\{\\betab\\} $ such that $L_0 = S\\cup \\{\\betab\\}$ and for any circuit $(S',\\gammab)$, where $\\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}$, there is an $\\alpb'\\in S'$ such that $\\gammab\\in L_i$, $\\alpb'\\in L_j$ with $j>i$. \nLet \n\\begin{equation}\\label{eq:constrNormals34}\nl(\\xb^\\alpb) \\coloneqq \\begin{cases}\n \\sigma^{-1} \\frac{\\lambda_\\alpb}{c_\\alpb}, & \\text{ if } \\alpb \\in S;\\\\\n \\mp \\sigma^{-1}\\Theta^{-1}_f, & \\text{ if } \\alpb = \\betab;\\\\\n (\\sigma^{-1}\\delta)^{\\Lambda^i}, & \\text{ if } \\alpb \\in L_i, \\; i \\in \\{1,\\dots, K\\};\\\\ \n 0, & \\text{otherwise (when } \\alpb \\in \\cA\\setminus (\\even{n}\\cup \\{\\betab\\}) \\text{)}.\n\\end{cases}\n\\end{equation}\nIn the case $\\alpb=\\betab$ we specifically want the value $l(\\xb^\\betab)$ to have the opposite sign to the coefficient of $f$ at $\\xb^\\betab$. In particular, this means that the only case when $l(\\xb^\\gammab)$ is negative is when $\\gammab = \\betab$ and $\\betab$ is odd (see Proposition~\\ref{prop:extreme}, items 2 and 3). Observe that due to our choice of $\\sigma$ and $\\delta$, we have \n\\[\n|l(\\xb^\\gammab)|> 1 \\quad \\text{ for all } \\, \\gammab\\in (\\cA\\cap \\even{n})\\cup \\{\\betab\\}.\n\\]\nTo prove that $r = \\R_+ f$ is an exposed ray of $\\SONC{\\cA}$, by Lemma~\\ref{lem:extremeChar} it is sufficient to show that for any extreme ray $q = \\R_+ g$ of $\\SONC{\\cA}$, we have $l(g)>0$, unless $g\\in \\R_{+}f$, in which case we must have $l(g) =0$. We verify this by systematically considering all possible extreme rays of $\\SONC{\\cA}$ listed in Proposition~\\ref{prop:extreme}.",
    "post_theorem_intro_text_len": 1546,
    "post_theorem_intro_text": "This result is proved by showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing, and constructing explicit exposing normals for all of the remaining cases. The construction of these normals relies on a graded partition of the ground set $\\mathcal{A}$ that allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\mathcal{A}$.\n\n\\smallskip\nWhile it is conceivable that the existence of such exposing normals could be deduced from general convex analysis arguments (as discussed in Section~\\ref{sec:open}), our proof is fully constructive in the sense that we provide explicit normals certifying exposure of each ray. \nThis constructive approach not only clarifies the geometry of the SONC cone but also lends itself to algorithmic implementation, potentially contributing to practical optimization frameworks that incorporate SONC certificates.\n\n\\medskip\nOur paper is organized as follows. Section~\\ref{sec:preliminaries} sets up the notation and reviews key properties of the SONC cone, including the characterization of its extreme rays.  In Section~\\ref{sec:unexposed} we establish the negative part of the result in Proposition~\\ref{prop:unexposed}. In Section~\\ref{sec:exposed} we show that the remaining extreme rays are exposed in Propositions~\\ref{prop:monomial} and~\\ref{prop:nonmonomial}, and finalize the proof of Theorem~\\ref{thm:main}. We close with a brief discussion of open problems in Section~\\ref{sec:open}.",
    "sketch": "The result is proved by (i) \"showing separately that the extreme rays that correspond to the conditions in the theorem are not exposing,\" and (ii) \"constructing explicit exposing normals for all of the remaining cases.\" The construction of exposing normals \"relies on a graded partition of the ground set $\\mathcal{A}$\" which \"allows precise control over the relative magnitudes of the normal entries and reflects the combinatorial layering inherent in $\\mathcal{A}$.\" The proof is \"fully constructive\" in that it provides \"explicit normals certifying exposure of each ray.\" Concretely, the paper \"establish[es] the negative part of the result\" in Proposition~\\ref{prop:unexposed}, then proves the remaining extreme rays are exposed in Propositions~\\ref{prop:monomial} and~\\ref{prop:nonmonomial}, and \"finalize[s] the proof of Theorem~\\ref{thm:main}.\"",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main} An extreme ray $r$ of the SONC cone on a finite ground set $\\mathcal{A}\\subseteq\\NN^{n}$ is not exposed if and only if $r=\\R_+ \\mathbf{x}^\\alpb$ for some $\\alpb\\in \\mathcal{A}$, and there exists a circuit $(S,\\betab)$ on $\\mathcal{A}$ such that $\\alpb \\in S$ and $\\betab\\neq \\alpb$ is even.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal A\\subseteq \\mathbb N^n\\) be finite, and let \\(\\operatorname{SONC}(\\mathcal A)\\) denote the cone of sums of nonnegative circuit polynomials supported on \\(\\mathcal A\\). Here a circuit on \\(\\mathcal A\\) means a pair \\((S,\\beta)\\) with \\(S\\subseteq \\mathcal A\\cap (2\\mathbb N)^n\\) affinely independent and \\(\\beta\\in \\mathcal A\\cap \\operatorname{relint}(\\operatorname{conv}(S))\\). Which extreme rays of \\(\\operatorname{SONC}(\\mathcal A)\\) are not exposed?",
      "correct_choice": {
        "label": "A",
        "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which there exists a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\) and \\(\\beta\\neq \\alpha\\) even; equivalently, an extreme ray of \\(\\operatorname{SONC}(\\mathcal A)\\) is not exposed if and only if it is of this form."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which there exists a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\) and \\(\\beta\\neq \\alpha\\); equivalently, the parity of \\(\\beta\\) is irrelevant."
        },
        {
          "label": "C",
          "text": "Every extreme ray of \\(\\operatorname{SONC}(\\mathcal A)\\) that is not exposed is a monomial ray \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for some \\(\\alpha\\in\\mathcal A\\)."
        },
        {
          "label": "D",
          "text": "Exactly the monomial rays \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for which for every circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) with \\(\\alpha\\in S\\), one has \\(\\beta\\neq \\alpha\\) even; equivalently, an extreme ray is not exposed precisely when \\(\\alpha\\) belongs to no circuit with odd inner exponent."
        },
        {
          "label": "E",
          "text": "An extreme ray \\(r\\) of \\(\\operatorname{SONC}(\\mathcal A)\\) is not exposed if and only if either \\(r=\\mathbb R_+\\mathbf x^{\\alpha}\\) for some \\(\\alpha\\in\\mathcal A\\) with a circuit \\((S,\\beta)\\) on \\(\\mathcal A\\) satisfying \\(\\alpha\\in S\\) and \\(\\beta\\neq\\alpha\\) even, or else \\(r\\) is itself generated by a non-monomial circuit polynomial whose inner exponent \\(\\beta\\) is even."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "evenness_of_\\beta",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "existence_of_circuit_witness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "existential_vs_universal_circuit_condition",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "monomial_only_classification",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and asks for a classification, but it does not state or strongly hint at the specific characterization in choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct statement-of-theorem question: it asks exactly which extreme rays are not exposed, and the correct option gives the theorem-level iff classification almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the distractors vary by parity, quantifiers, and overgeneralization. However, the item mainly tests recognition/recall of the precise classification rather than derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: dropping the evenness condition, weakening an iff to a one-way statement, swapping existential/universal quantifiers, and overextending to non-monomial rays."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically well-constructed recall-style MCQ with strong distractors and no answer leakage, but it is largely a direct theorem-restatement rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.00445v1",
    "paper_link": "http://arxiv.org/abs/2512.00445v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "teo",
      "content": "\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\mathbb{R}^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.",
      "start_pos": 9781,
      "end_pos": 10728,
      "label": "maintheorem0"
    },
    "ref_dict": {
      "H_0": "\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}",
      "maintheorem0": "\\begin{teo}\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\R \\times \\R \\to \\R$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\R^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.\n\\end{teo}",
      "sistema0": "\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}",
      "sistema_anterior": "\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1  u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\quad  \\qquad \\qquad \\qquad  + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2  v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\  v(x,0) = v_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation}",
      "H2": "\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5900,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nFor any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\nThe present system \\eqref{sistema0} aims to analyze the null controllability within a broader class of equations that extends the model below considered in the previous article of the authors \\cite{Limaco-Lobosco-Yapu_24}: \n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1  u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\quad  \\qquad \\qquad \\qquad  + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2  v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\  v(x,0) = v_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and  $q_{i1}+q_{i2}=0$. \n\nProblems of the type \\eqref{sistema_anterior} were studied, for instance, when the control $\\nu$ is not considered and \n$$J_i(x) = \\lambda_i C_i e^{-\\frac{(x-d_i)^2}{2k_i^2}},$$\nwith $\\lambda_i$, $C_i$, $d_i$ and $k_i$ are constants. In that case, we have the model proposed by \\cite{acoplado} for pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. This model generalized the classical work \\cite{BS} where the Black-Scholes equation was established for derivative pricing of European options. Due to the complexity of financial markets, that generalization became necessary in order to have the ability of efficiently interpret the economic cycles and the changes in the financial time series data due to the regime shifts, as was analysed in \\cite{acoplado}. For other financial options pricing models, see the articles \\cite{BS2, BS7, BS14, FM1}. \n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time,  Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition  \\eqref{H_0} below.  \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.  \n\nIn the present article, we investigate whether the controllability properties established for the original system remain valid under the introduction of additional terms $F$ and $G$, thereby providing a natural generalization of the earlier results. In fact, the conditions imposed in the funcions $F$ and $G$ for the system \\eqref{sistema0} permit to include the terms of system \\eqref{sistema_anterior} that do not appear explicitly in system \\eqref{sistema0} and to introduce nonlinearities. \n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.   \n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:",
    "context": "For any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1 u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\qquad \\qquad \\qquad + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2 v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\ v(x,0) = v_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and $q_{i1}+q_{i2}=0$.\n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time, Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition \\eqref{H_0} below. \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.\n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}",
    "full_context": "For any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1 u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\qquad \\qquad \\qquad + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2 v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\ v(x,0) = v_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and $q_{i1}+q_{i2}=0$.\n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time, Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition \\eqref{H_0} below. \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.\n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\n\nFor any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nThis paper is organized as follows. In Section~\\ref{sec:linear_case0} we formulate the linear problem associated with system~\\eqref{sistema0}. This system differs from that in~\\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\\ref{sec:add_coments} we present related problems and additional remarks.\n\n\\begin{teo}\\label{theorem_linear} \nIf the hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied, $\\tilde c \\in W^{2,\\infty}(Q)$ with $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{eq:linearized_fixed_point} verifies $y(x,T)=z(x,T)=0$ and\n for a constant $C=C(M,\\omega,T)$ we have an estimate for the control of the form\n\\begin{equation}\\label{ControlEstimate}\n\\|\\nu\\|_{L^2(\\omega\\times(0,T))} \\leq C (\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)}). \n\\end{equation}\n\\end{teo}\n\n\\begin{prop} \\label{carleman_um_controle0}\nConsider the adjoint system \\eqref{sistema_adj0} with $\\tilde c \\in W^{2,\\infty}(Q)$, $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$ and $\\phi_T \\in L^2(\\Omega)$, $\\psi_T \\in L^2(\\Omega)$. Moreover assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0} and suppose that the kernel $\\lambda_1(t) J_1(x)$ satisfies \\eqref{H2}. Then, there exist constants $C=C(\\omega,a_1,a_2,b_1,c_1,T,\\tilde c, \\tilde a)$, $\\kappa_1=C(\\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\\omega)(aT+ (aT)^2+\\overline{K}^\\frac{2}{3}T^2)$, where $\\overline{K}$ was defined in \\eqref{H_0} and $a = \\max(a_1,a_2)$, such that the solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} satisfy\n\\begin{equation}\\label{Carleman2_coeff33}\nI(\\phi) + I(\\psi) \\leq Cs^7 \\kappa^8\\left(\\iint_{\\omega \\times (0,T)} e^{-2s\\alpha}\\xi^7 \\vert \\phi \\vert^2dxdt \\right),\n\\end{equation}\nfor any constants $\\kappa>\\kappa_1$ and $s>s_1$.\n\n\\begin{prop}[\\cite{GlobalCarleman}]\\label{prop2.2}\nThere exist positive constants $C=C(\\Omega,\\omega,a)$, $s_1 = C(\\Omega,\\omega)(aT+ (aT)^2)$ and $\\kappa_1=C(\\Omega,\\omega)\n$ such that, for any $s>s_1$, $\\kappa > \\kappa_1$, $F \\in L^2(Q)$, $z_T\\in L^2(\\Omega)$ and $a>0$ being a positive constant, the solution of the equation \n\\begin{equation} \\label{sistema4}\n\\begin{cases}\n\\displaystyle z_t+a\\Delta z = F, \\ \\ &\\mbox{ in } \\ \\ 0 < x < 1,\\text{ } 0<t<T, \\\\\nz(0,t) = z(1,t) = 0 ,\\ \\ &\\mbox{ in } 0<t<T,\\\\\n\\displaystyle z(x,T) = z_T(x), &\\mbox{ in } 0 < x < 1,\n\\end{cases}\n\\end{equation}\nsatisfy\n\\begin{equation}\\label{Carleman11}\nI(z)\\leq C \\left[ s^3 \\kappa^4 \\iint_{\\omega\\times(0,T)}e^{-2s\\alpha}\\xi^3 \\vert z \\vert^2 dxdt + \\int_0^T \\int_0^1 e^{-2s\\alpha} \\vert F \\vert^2 dx dt \\right].\n\\end{equation}\n\\end{prop}\n\\begin{proof}\n\n\\begin{prop}\\label{Carleman1}\nLet $\\phi_T, \\psi_T \\in L^2(\\Omega)$ be given and assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0}. Then, there exist constants $C=C(\\Omega,\\omega,a_1,a_2)$, $\\kappa_1=C(\\Omega,\\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\\Omega,\\omega)(aT+ (aT)^2+\\overline{K}^\\frac{2}{3}T^2)$, where $\\overline{K}$ was defined in \\eqref{H_0} and $a = \\max(a_1,a_2)$, such that a solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} with final conditions $\\phi_T$ and $\\psi_T$, satisfy\n\\begin{equation}\n\\label{eq:carleman_dois_controles}\nI(\\phi) + I(\\psi) \\leq Cs^3 \\kappa^4\\left(\\iint_{\\omega \\times (0,T)} \\! \\! \\! \\! \\! e^{-2s\\alpha}\\xi^3 \\vert \\phi \\vert^2dxdt+\\iint_{\\omega \\times (0,T)} \\! \\! \\! \\! e^{-2s\\alpha}\\xi^3 \\vert \\psi \\vert^2dxdt\\right)\n\\end{equation}\nfor any constants $\\kappa > \\kappa_1$ and $s>s_1$.\n\\end{prop}\n\nAs a consequence of the Carleman estimate, we prove now the observability inequality.\n\\begin{prop}\\label{prop_observabilidade}\nConsider the adjoint system \\eqref{sistema_adj0} with $\\tilde c \\in W^{2,\\infty}(Q)$, $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$ and $\\phi_T \\in L^2(\\Omega)$, $\\psi_T \\in L^2(\\Omega)$. Moreover, assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0} and suppose that the kernel $\\lambda_1(t) J_1(x)$ satisfies \\eqref{H2}. Then, there exist constants $C=C(M,\\Omega,\\omega,a_1,a_2,b_1,b_2,c_1,c_2,T,\\tilde c, \\overline{K})$, where $\\overline{K}$ was defined in \\eqref{H_0}, such that the solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} satisfy \\begin{equation}\\label{obsin}\n \\parallel\\phi(0)\\parallel^2_{L^2(\\Omega)} + \\parallel\\psi(0)\\parallel^2_{L^2(\\Omega)} \\leq C \\iint_{\\omega \\times (0,T)} \\vert \\phi \\vert^2 dxdt.\n \\end{equation}\n\\end{prop}\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{teo}\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\R \\times \\R \\to \\R$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\R^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.\n\\end{teo}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}",
    "post_theorem_intro_text_len": 903,
    "post_theorem_intro_text": "This paper is organized as follows. In Section~\\ref{sec:linear_case0} we formulate the linear problem associated with system~\\eqref{sistema0}. This system differs from that in~\\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\\ref{sec:add_coments} we present related problems and additional remarks.\n\nThroughout the paper, we use \\(C\\) and \\(C_i\\) (\\(i=1,2,\\ldots\\)) to denote generic positive constants whose value may change from line to line. When it is necessary to emphasize parameter dependence, we indicate it explicitly; for example, we write \\(C=C(\\omega,T)\\) to mean that \\(C\\) depends only on \\(\\omega\\) and \\(T\\).",
    "sketch": "In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem.",
    "expanded_sketch": "Next we prove the main result, To prove the main theorem, for our system\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\nvia Kakutani’s fixed-point theorem.",
    "expanded_theorem": "\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nand\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nare satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\mathbb{R}^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\\\  z(x,0) = z_0(x), \\\\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\n$F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system above verifies $y(x,T)=0$ and $z(x,T)=0$.",
    "theorem_type": [
      "Existential–Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let $T>0$, $\\Omega=(0,1)$, $Q=\\Omega\\times(0,T)$, and let $\\omega\\subset(0,1)$ be a nonempty open set. Consider the semilinearly coupled parabolic system with one control\n\\[\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x+c_1 y+\\lambda_1(t)\\displaystyle\\int_0^1 J_1(\\zeta-x)y(\\zeta,t)\\,d\\zeta+F(y,z)+\\nu 1_\\omega, & \\text{in }Q,\\\\[1mm]\nz_t=a_2 z_{xx}+b_2 z_x+c_2 z+\\lambda_2(t)\\displaystyle\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)\\,d\\zeta+G(y,z), & \\text{in }Q,\\\\\ny(0,t)=y(1,t)=z(0,t)=z(1,t)=0, & 0<t<T,\\\\\ny(x,0)=y_0(x),\\quad z(x,0)=z_0(x), & x\\in\\Omega,\n\\end{cases}\n\\]\nwhere $a_i,b_i,c_i$ are given real constants with $a_i>0$, the control is $\\nu\\in L^2(\\omega\\times(0,T))$, and $J_i,\\lambda_i$ satisfy the kernel assumptions\n\\[\n\\overline K=\\sup_{t\\in[0,T]}\\exp\\!\\left(\\frac{2\\sigma^-}{t(T-t)}\\right)\\lambda_i(t)^2\\int_{-1}^1 |J_i(z)|^2\\,dz<\\infty \\qquad (i=1,2),\n\\]\nand\n\\[\n\\sup_{(x,t)\\in[0,1]\\times[0,T]}\\exp\\!\\bigl(2s\\alpha^-(t)\\bigr)\\lambda_1(t)^2\\int_0^1 |J_1(x-z)|^2\\,dz<\\overline\\delta.\n\\]\nAssume moreover that $F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}$ are $C^1$ functions such that $F(0,0)=0$, $G(0,0)=0$,\n\\[\n\\max\\left\\{\\left|\\frac{\\partial F}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial F}{\\partial s}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial s}(r,s)\\right|\\right\\}\\le M\n\\quad \\text{for all }(r,s)\\in\\mathbb{R}^2,\n\\]\nand $\\dfrac{\\partial G}{\\partial r}(r,s)\\neq 0$. Which conclusion about the system is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "There exists $\\delta=\\delta(T)>0$ such that, for every initial datum $y_0,z_0\\in L^2(\\Omega)$ with\n\\[\n\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta,\n\\]\nand for every pair of nonlinearities $F,G$ satisfying all the stated assumptions, one can find a control $\\nu\\in L^2(\\omega\\times(0,T))$ for which the corresponding solution of the above system satisfies\n\\[\ny(x,T)=0\\quad\\text{and}\\quad z(x,T)=0 \\qquad \\text{in }\\Omega.\n\\]\nEquivalently, the system is locally null controllable to $(0,0)$ at time $T$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists $\\delta=\\delta(T)>0$ such that, for every initial datum $y_0,z_0\\in L^2(\\Omega)$ with\n\\[\n\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta,\n\\]\nand for every pair of nonlinearities $F,G$ satisfying all the stated assumptions, one can find a control $\\nu\\in L^2(\\omega\\times(0,T))$ for which the corresponding solution of the above system satisfies\n\\[\ny(x,T)=0\\quad\\text{and}\\quad z(x,T)=0 \\qquad \\text{in }\\Omega,\n\\]\nwithout needing the additional smallness condition\n\\[\n\\sup_{(x,t)\\in[0,1]\\times[0,T]}\\exp\\!\\bigl(2s\\alpha^-(t)\\bigr)\\lambda_1(t)^2\\int_0^1 |J_1(x-z)|^2\\,dz<\\overline\\delta.\n\\]"
        },
        {
          "label": "C",
          "text": "For every pair of nonlinearities $F,G$ satisfying all the stated assumptions and every initial datum $y_0,z_0\\in L^2(\\Omega)$ with\n\\[\n\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta,\n\\]\nfor some $\\delta=\\delta(T)>0$, there exists a control $\\nu\\in L^2(\\omega\\times(0,T))$ such that the corresponding solution satisfies\n\\[\ny(x,T)=0 \\qquad \\text{in }\\Omega.\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a constant $\\delta>0$, independent of $T$, such that for every $T>0$, every initial datum $y_0,z_0\\in L^2(\\Omega)$ with\n\\[\n\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta,\n\\]\nand every pair of nonlinearities $F,G$ satisfying all the stated assumptions, one can find a control $\\nu\\in L^2(\\omega\\times(0,T))$ such that the corresponding solution satisfies\n\\[\ny(x,T)=0\\quad\\text{and}\\quad z(x,T)=0 \\qquad \\text{in }\\Omega.\n\\]"
        },
        {
          "label": "E",
          "text": "There exists $\\delta=\\delta(T)>0$ such that, for every initial datum $y_0,z_0\\in L^2(\\Omega)$ with\n\\[\n\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta,\n\\]\nand for every pair of nonlinearities $F,G$ satisfying all the stated assumptions, one can find a control $\\nu\\in L^2(\\omega\\times(0,T))$ for which the corresponding solution of the above system satisfies\n\\[\ny(x,T)=0\\quad\\text{and}\\quad z(x,T)=0 \\qquad \\text{in }\\Omega,\n\\]\nand moreover this remains true even if the coupling condition $\\dfrac{\\partial G}{\\partial r}(r,s)\\neq 0$ is replaced by $\\dfrac{\\partial F}{\\partial s}(r,s)\\neq 0$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "smallness_condition_H2_on_first_kernel",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "simultaneous_null_control_of_both_components",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence_of_delta_on_T",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "specific_nonlinear_coupling_nonvanishing_condition_on_partial_r_G",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It presents hypotheses and asks for the valid conclusion, without directly stating local null controllability."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: given a full list of assumptions, the correct choice is essentially the theorem's conclusion. The distractors introduce altered quantifiers and hypotheses, so it is not a pure verbatim restatement, but it remains only mildly non-tautological."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare subtle variants: simultaneous control of both components, dependence of δ on T, necessity of the kernel smallness condition, and the specific nonvanishing coupling derivative. However, the task mainly rewards recognition of the exact theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: dropping a key smallness assumption, weakening the conclusion to one component, changing quantifier dependence on T, and altering the crucial coupling condition. They are distinct and relevant."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-matching MCQ. It avoids direct answer leakage and has strong distractors, but it primarily tests recall/recognition of a precise controllability theorem rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.00568v1",
    "paper_link": "http://arxiv.org/abs/2512.00568v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "lettertheorem",
      "content": "\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.",
      "start_pos": 28636,
      "end_pos": 29022,
      "label": "thm:main-theorem-intro"
    },
    "ref_dict": {
      "thm:main-theorem-intro": "\\begin{lettertheorem}\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.\n\\end{lettertheorem}",
      "def:signature-of-formal-point-mod-p": "\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 6158,
    "pre_theorem_intro_text": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n    \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely. \n\nAs opposed to the case of curves, the Albanese map need not be injective and its kernel, denoted by $F^2(X)$, is known to vary wildly. For instance it is finite when $K$ is finite \\cite{KatoSaito1983}, but can be uncountable, when $K=\\mathbb{C}$ \\cite{Mumford1969}. When $K$ is a number field the problem of determining $F^2(X)$ remains wide open. In this direction, there are far-reaching conjectures by Beilinson and Bloch \\cite{Beilinson1985,Bloch1984} that suggest that $F^2(X)$ is finite.\n\nA historically fruitful testing ground for such conjectures is the case when $K$ is a $p$-adic field, i.e. a finite extension of $\\mathbb{Q}_p$. In this case, we have the following conjecture of Colliot-Th\\'el\\`ene, later refined by Raskind and Spiess.\n\n\\begin{letterconjecture}\\label{conj:CT}\n  (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nAn important advancement in this direction is the work of Saito and Sato \\cite{SaitoSato2010} where they show, under some mild technical conditions, that the larger group $F^1(X)$ is the direct sum of a finite group and a group that is divisible by every integer coprime to $p$. Thus the key unresolved aspect of Colliot-Th\\'el\\`ene's conjecture is proving that the quotients $F^2(X)/p^n$ are finite for all $n\\geq1$, along with determining the asymptotic behavior of their sizes as $n\\rightarrow\\infty$.\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n    \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nFrom this description, one can see why determining the size of $F^2(X)/p^n$ is difficult. Even showing that $F^2(X)/p$ is finite is still very hard. Indeed, each summand $E_1(L)/p\\otimes E_2(L)/p$ of $K(K;E_1,E_2)/p$ can be quite large, requiring many relations to offset the shear number of generators. Furthermore, the structure of each summand depends sensitively on both the ramification of $K$ and on the reduction type of the elliptic curves.\n\nWhen $E_1$ and $E_2$ have good reduction and the absolute ramification index $e$ of $K$ is small, i.e. $e<p-1$, it is expected that\n\\[\n    K(k;E_1,E_2)/p=0.\n\\]\nThis expectation comes from the vanishing of a certain cycle map on $K(K;E_1,E_2)/p$ \\cite[Corollary 8.1]{Gazaki2018}. So in order to prove a statement of this type, one must be able to produce enough relations in $K(K;E_1,E_2)$ to cancel the generators of the Somekawa $K$-group modulo $p$. In this paper, we focus on the ``first level'' of $K(K;E_1,E_2)/p$, that is the subgroup of $K/K$-symbols.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.",
    "context": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.\n\n\\begin{letterconjecture}\\label{conj:CT}\n (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}",
    "full_context": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.\n\n\\begin{letterconjecture}\\label{conj:CT}\n (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\wh{P}_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\nThe WR relations produced in Theorem \\ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\\times E_2$, and a choice of rational function $f\\in K(C)^\\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\\f_1:C\\ra E_1$ and $\\f_2:C\\ra E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\\f_1,\\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.\n\n\\begin{definition}\n Let $K$ be a $p$-adic field and let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Let $\\{P_1,P_2\\}_{K/K}$ be a symbol in $K(K;E_1,E_2)$, where $P_\\ell\\in E_\\ell(L)$ for $\\ell=1,2$. The \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$ is defined as the pair of integers\n \\[\n \\sS(\\{P_1,P_2\\}_{K/K})=\\left(\\nu(\\wh{P}_1),\\nu(\\wh{P}_2)\\right)\n \\]\n where $\\wh{P}_\\ell\\in\\E_\\ell(\\m)$ is the associated formal point of $P_\\ell$ (see Definition \\ref{def:associated-formal-point}).\n\\end{definition}\n\n\\begin{proposition}\\label{prop:necessary-conditions-for-PhiL-0}\n Let $K$ be a $p$-adic field with absolute ramification index $e<p-1$. Let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Suppose we have a set of $K/K$-symbols\n \\[\n \\left\\{\\{P_{ij},Q_{i'j'}\\}_{K/K}\\in K(K;E_1,E_2)\\mid 1\\leq i,i'\\leq e,1\\leq j,j'\\leq f\\right\\}\n \\]\n such that\n \\begin{enumerate}[label=(\\roman*)]\n \\item\\label{item:1-Phi_L} $\\{P_{ij},Q_{i'j'}\\}_{K/K}\\equiv 0\\pmod{p}$ for all $i,i',j,'$.\n \\item\\label{item:2-Phi_L} $\\sS(\\{P_{ij},Q_{i'j'}\\}_{K/K})=(i,i')$,\n \\item\\label{item:3-Phi_L} Both $\\{\\wh{P}_{i1}+\\E_1(\\m^{i+1}),\\ldots,\\wh{P}_{if}+\\E_1(\\m^{i+1})\\}$ and $\\{\\wh{Q}_{i'1}+\\E_2(\\m^{i+1}),\\ldots,\\wh{Q}_{i'f}+\\E_1(\\m^{i+1})\\}$ are $\\F_p$-basis of $k\\cong\\E_\\ell(\\m^i)/\\E_\\ell(\\m^{i+1})$ ($\\ell=1,2$) for all $i$ and $i'$.\n \\end{enumerate}\n Then $\\Symb_{K/K}=0$.\n\\end{proposition}\n\n\\begin{lemma}\\label{lemma:p-3mod4-and-legendre-form}\n Suppose $K$ is a totally ramified extension of $\\Q_p$ with $p>3$. Let $E$ be an elliptic curve will full 2-torsion and supersingular reduction. Then\n \\begin{enumerate}[label=(\\roman*)]\n \\item $p\\equiv3\\pmod{4}$, in particular $-1$ is not a square in $K^\\times$.\n \\item There exists $\\l\\in\\O$ such that $\\l\\not\\equiv 0,1\\spmod{\\m}$ and such that $E$ is isomorphic over $K$ to a Legendre form elliptic curve\n \\[\n E_\\l:y^2=x(x-1)(x-\\l).\n \\]\n \\item The $j$-invariant $j(E)$ satisfies $j(E)\\not\\equiv 0\\spmod{\\m}$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}$\\;$\n\nThe above lemma tells us that if we have elliptic curves $E_1$ and $E_2$ over a totally ramified field $K$ with full 2-torsion and supersingular reduction, then we may assume that $E_1=E_\\l$ and $E_2=E_\\mu$ for some $\\l,\\mu\\in\\O$. The Legendre form of an elliptic curve is not unique up to isomorphism, so choosing a different form will produce a different genus 2 curve.\n\n\\begin{lemma}\\label{lemma:existence-of-good-span}\n Let $K/\\Q_p$ be a totally ramified extension with $p>3$. Let $E_1$ and $E_2$ be elliptic curves over $K$ with full 2-torsion and supersingular reduction. Then there exists a smooth genus 2 curve $C$ that spans $E_1$ and $E_2$. The curve $C$ has defining equation of the form\n \\[\n C:Y^2=u(X^2-r)(X^2-s)(X^2-t),\n \\]\n where $u,r,s,t\\in\\O^\\times$, and the spanning maps $\\s_1:C\\ra E_1$ and $\\s_2:C\\ra E_2$ have the following form\n \\[\n \\s_1(X,Y)=(u(X^2-v),uY),\\quad \\s_2(X,Y)=(ust(X^2-r)X^{-2},-urst X^{-3}Y),\n \\]\n where $v$ is a unit. Furthermore, $C$ has a $K$-rational Weierstrass point $W=(w,0)$ where $w\\in \\O^\\times$.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm:main-theorem}\n Let $K$ be a totally ramified extension of $\\Q_p$, with $p>3$. Let $E_1$ and $E_2$ be elliptic curves with full 2-torsion and supersingular reduction. Then for every positive integer $N$, there exist points $P_N\\in \\E_1(K)$ and $Q_N\\in \\E_2(K)$ such that the $K/K$-symbol $\\{P_N,Q_N\\}_{K/K}\\in K(K;E_1,E_2)$ has signature $(N,N)$ and such that\n \\[\n \\{P_N,Q_N\\}_{K/K}\\equiv 0\\pmod{p}.\n \\]\n\\end{theorem}\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}\n\n\\begin{lettertheorem}\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.\n\\end{lettertheorem}",
    "post_theorem_intro_text_len": 1659,
    "post_theorem_intro_text": "The WR relations produced in Theorem \\ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\\times E_2$, and a choice of rational function $f\\in K(C)^\\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\\f_1:C\\rightarrow E_1$ and $\\f_2:C\\rightarrow E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\\f_1,\\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.\n\nThis paper is organized into three sections. In section \\ref{sec:prelim}, we review all the basic theory we will need about formal groups of elliptic curves, supersingular reduction, and the Somekawa $K$-group for products of elliptic curves. Along the way we define the signature of a $K/K$-symbol and determine the type and quantity of WR relations one needs to show the vanishing of all $K/K$-symbols modulo $p$. In section \\ref{sec:scholten_curves} we review the construction of the hyperelliptic curves that map to $E_1\\times E_2$, compute the WR relations they produce, and use them to prove Theorem \\ref{thm:main-theorem-intro}. Lastly, in section \\ref{sec:computations}, we focus on the case of quadratic ramified extensions $K$ and present computational evidence that the genus 2 spans of section \\ref{sec:scholten_curves} produce the necessary WR relations to show that all $K/K$-symbols vanish modulo $p$.",
    "sketch": "The WR relations in Theorem \\ref{thm:main-theorem-intro} are obtained from genus 2 covers of $E_1$ and $E_2$: there is a WR relation in $K(K;E_1,E_2)$ for every curve $C/K$ whose Jacobian maps into $E_1\\times E_2$, together with a choice of $f\\in K(C)^\\times$. The paper uses J. Scholten's construction of a genus 2 (hyperelliptic) curve $C$ and explicit nonconstant maps $\\f_1:C\\to E_1$ and $\\f_2:C\\to E_2$. Because $C$ is hyperelliptic, it has \"particularly simple principal divisors\", so $f$ can be chosen flexibly. With a suitable choice of $C,\\f_1,\\f_2$ and $f$, the resulting WR relation is then $p$-adically approximated \"to determine its signature\" (yielding signature $(n,n)$).",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.",
    "theorem_type": [
      "Existence",
      "Universal"
    ],
    "mcq": {
      "question": "Let p>3 be a prime, let K be a p-adic field totally ramified over \\(\\mathbb{Q}_p\\), and let \\(E_1\\) and \\(E_2\\) be elliptic curves over K with full K-rational 2-torsion and supersingular reduction. Let \\(K(K;E_1,E_2)\\) denote the Somekawa K-group, whose generators include K/K-symbols \\(\\{P_1,P_2\\}_{K/K}\\) with \\(P_i\\in E_i(K)\\). A WR relation means a vanishing relation in this group coming from Weil reciprocity. For such a K/K-symbol, its signature is the pair \\((n_1,n_2)\\) obtained from the valuations of the corresponding formal-group representatives modulo p; thus saying the symbol has signature \\((n,n)\\) means both valuation entries equal n. Which statement holds for every such choice of p, K, \\(E_1\\), \\(E_2\\), and every positive integer n?",
      "correct_choice": {
        "label": "A",
        "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,n)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,m)\\) for every pair of positive integers \\(n,m\\)."
        },
        {
          "label": "C",
          "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n_1,n_2)\\) for some positive integers \\(n_1,n_2\\)."
        },
        {
          "label": "D",
          "text": "There exists a single pair of points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\), independent of \\(n\\), such that for every positive integer \\(n\\) the relation \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,n)\\)."
        },
        {
          "label": "E",
          "text": "For every positive integer \\(n\\), there exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that the \\(K/K\\)-symbol \\(\\{P_1,P_2\\}_{K/K}\\in K(K;E_1,E_2)\\) has signature \\((n,n)\\) and satisfies \\(\\{P_1,P_2\\}_{K/K}\\equiv 0\\pmod p\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "diagonal-signature-only",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "exact prescribed signature (n,n)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "existential dependence on n",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "WR-vanishing replaced by congruence mod p",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the target conclusion or quote the exact existence statement. It provides setup and definitions, but no direct hint that the correct answer is the strongest universal diagonal statement."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to asking for the precise theorem conclusion under the stated hypotheses, so it is partly a theorem-restatement question. However, it is not completely tautological because the options vary in quantifiers, dependence on n, and the exact vanishing property."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the strongest valid claim from weaker or subtly altered alternatives, especially regarding 'for every n' versus 'there exists n', diagonal signature versus sum condition, and zero versus p-torsion. Still, success mainly depends on recognizing the theorem statement rather than generating a substantial argument."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful. They reflect common failure modes: weakening universal quantifiers, confusing existential strength, mishandling dependence on parameters, and replacing exact vanishing by p-torsion."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it only moderately tests generative reasoning because it is fairly close to selecting the exact theorem conclusion."
    }
  },
  {
    "id": "2512.00690v3",
    "paper_link": "http://arxiv.org/abs/2512.00690v3",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "[{= Theorem \\ref{infty integrable thm}}]\\label{main thm1}\n    There exists $M>0$ (depending on $R$) such that\n$$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$",
      "start_pos": 15413,
      "end_pos": 15677,
      "label": "main thm1"
    },
    "ref_dict": {
      "localization ider cor": "\\begin{corollary}\\label{localization ider cor}\n Let $R$ be a $k$-algebra essentially of finite type. Then $\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\\operatorname{Ider}_k(S^{-1}R)=S^{-1}\\operatorname{Ider}_k(R)$$\nas $S^{-1}R$-modules for any multiplicatively closed subset $S\\subset R$.\n\\end{corollary}",
      "eq finie thm": "\\begin{theorem}\\label{eq finie thm}\n        Let $e$ be an integer with\n    $$  e\\ge C_1+ C_2 + C_3\\cdot q+C_4\\cdot q+q$$ where the $C_i$ are as in \\ref{hom lem}-\\ref{j lem}. Let $\\operatorname{Der}_k^{q}(R)$ be the module of $q$-integrable derivations of $R$. Then any element of\n    $\\mathfrak{n}^{e(q+1)+1} \\cdot \\operatorname{Der}_k^{q}(R)$\n    is $m$-integrable for any positive integer $m$. In particular, the set of leaps produced by $\\operatorname{Der}_k^{q}(R)$ is finite.\n\\end{theorem}",
      "infty integrable thm schemes": "\\begin{theorem}\\label{infty integrable thm schemes}\n    Let $X$ be a $k$-scheme essentially of finite type. Then there exists $M>0$ such that\n$$ \\operatorname{Der}_k^M(\\mathcal{O}_X)= \\operatorname{Der}_k^{M+1}(\\mathcal{O}_X)=\\dots=\\operatorname{Ider}_k(\\mathcal{O}_X).$$\n\\end{theorem}",
      "main thm1": "\\begin{theorem}[{= Theorem \\ref{infty integrable thm}}]\\label{main thm1}\n    There exists $M>0$ (depending on $R$) such that\n$$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$\n\\end{theorem}",
      "Her thm": "\\begin{theorem}\\label{Her thm}\n        Let $M$ be a sub-$R$-module of $\\operatorname{Der}_k(R,R)$ and $\\mathfrak{l}\\subset \\mathfrak{n}$ and ideal such that $R/\\mathfrak{l}$ is of finite length. Assume that any $d\\in \\mathfrak{l}\\cdot M$ is $m$-integrable for any positive integer $m$. Let $\\operatorname{Leaps}(M)\\subset \\mathbb{N}$ be the set of leaps produced by $M$. Then $$\\# \\operatorname{Leaps}(M)\\le \\operatorname{length}_R(M/ \\mathfrak{l}M).$$ In particular, $\\operatorname{Leaps}(M)$ is finite.\n    \\end{theorem}",
      "duality": "\\begin{theorem}[{= \\cite[Theorem 2.25]{miy2025}}]\\label{duality}\nLet $X$ be a scheme essentially of finite type over $k$. Then there exist two filtrations of sheaves on $X$ such that\n$$\\operatorname{Der}_k(\\mathcal{O}_X)\\supset\\operatorname{Der}_k^p(\\mathcal{O}_X)\\supset\\operatorname{Der}_k^{p^2}(\\mathcal{O}_X)\\supset\\dots$$\nand that\n$$0\\subset \\operatorname{Ob}^p_X\\subset\\operatorname{Ob}^{p^2}_X\\subset\\dots\\subset T^1_{X/k}.$$ For every $i=1,2,\\dots,$ we have\n$$\\operatorname{Der}_k^{p^{i-1}}(\\mathcal{O}_X)/\\operatorname{Der}_k^{p^i}(\\mathcal{O}_X)\\simeq \\operatorname{Ob}_X^{p^i}/\\operatorname{Ob}_X^{p^{i-1}}. $$\n\\end{theorem}",
      "mn int theorem": "\\begin{theorem}\\label{mn int theorem}\n    Let $R$ be a $k$-algebra essentially of finite type, let $n\\ge 1$ be an integer, and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}",
      "main cor": "\\begin{corollary}[{=Corollary \\ref{localization ider cor}}]\\label{main cor}\nThe module $\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\\operatorname{Ider}_k(S^{-1}R)=S^{-1}\\operatorname{Ider}_k(R).$$\n\\end{corollary}",
      "localization": "\\begin{theorem}[= {\\cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}}]\n\\label{localization}\n    Let $R$ be a $k$-algebra essentially of finite type over $k$. Let $m\\ge 1$ an integer. Let $S\\subset R$ a multiplicatively closed subset. Then we have a canonical isomorphism $$\\alpha: S^{-1}\\operatorname{Der}_k^m(R)\\overset{\\sim}{\\to} \\operatorname{Der}_k^m(S^{-1}R).$$\n\\end{theorem}",
      "infty integrable thm": "\\begin{theorem}\\label{infty integrable thm}\n     Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that\n     $$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$\n\\end{theorem}",
      "main thm global": "\\begin{theorem}\\label{main thm global}\n    Let $X$ be a $k$-scheme essentially of finite type. Then $X$ satisfies $\\FL{\\tau}$ for a sufficiently large integer $\\tau$. In particular, the set of leaps of $X$ is finite.\n\\end{theorem}",
      "FL def": "\\begin{definition}\\label{FL def}\n    Let $R$ [resp.\\ $X$] be a $k$-algebra essentially of finite type [resp.\\ $k$-scheme essentially of finite type]. Let $i\\ge 1$ be an integer. Then we say that $R$ [resp.\\ $X$] satisfies $\\FL{i}$ if we have\n    $$\\operatorname{Ob}_R^{p^i}=\\operatorname{Ob}_R^{p^{i+1}}=\\operatorname{Ob}_R^{p^{i+2}}=\\dots $$\n    [resp. $\\operatorname{Ob}_X^{p^i}=\\operatorname{Ob}_X^{p^{i+1}}=\\dots $].\n\\end{definition}",
      "main thm2": "\\begin{theorem}[{= Theorem \\ref{mn int theorem}}]\\label{main thm2}\n    Let $n\\ge 1$ be an integer and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}",
      "mn int lem": "\\begin{lemma}\\label{mn int lem}\n    Let $R$ be a $k$-algebra essentially of finite type, let $m, \\tau\\ge 1$ be integers, and let $q:=p^\\tau$. Suppose that $R$ satisfies $\\FL{\\tau}$. Let $D\\in \\HS_k^m(R)$. Let $l_1,l_2,\\dots$ be the sequence of integers defined as in \\ref{l_n def}. Then there exists $D^+\\in \\HS_k^{m+1}(R)$ such that $D$ and $D^+$ coincide up to $l_{m+1}-1$, i.e., $D_1=D^+_1,\\dots, D_{l_{m+1}-1}= D_{l_{m+1}-1}^+$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 2546,
    "pre_theorem_intro_text": "Hasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \\cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \\cite[\\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\\operatorname{HS} _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\\![t]\\!]/(t^{m+1})$-algebra automorphisms\n$$R[\\![t]\\!]/(t^{m+1}) \\to R[\\![t]\\!]/(t^{m+1})$$\nthat become the identity when restricted to $ R[\\![t]\\!]/(t^{m+1}) \\otimes_{A_m} k=R$. For $m=\\infty$, we regard $t^\\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\\operatorname{HS} _k^1(R)$ and the module $\\operatorname{Der}_k(R,R)$. Under this identification, the image of\n$$\\operatorname{HS} _k^m(R)\\to \\operatorname{HS} _k^1(R)\\simeq \\operatorname{Der}_k(R,R)$$\nis called the module of $m$-integrable derivations and denoted by $\\operatorname{Der}_k^m(R)$, which has been studied in \\cite{matsumura1982integrable}, \\cite{NARVAEZMACARRO20122712}, \\cite{NARVAEZMACARRO2024109758}. The module $\\operatorname{Der}_k^\\infty(R)$ is also denoted by $\\operatorname{Ider}_k(R)$. Note that if $k$ contains $\\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k(R)$ for any $m\\le \\infty$ (see \\cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain\n$$ \\operatorname{Der}_k(R)\\supset \\operatorname{Der}_k^2(R)\\supset \\dots \\supset \\operatorname{Der}_k^m(R) \\supset\\dots.$$\nThe question of when the inclusion $\\operatorname{Der}_k^{m-1}(R)\\supset \\operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).\nThe aim of this paper is to show, using results from \\cite[\\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:",
    "context": "Hasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \\cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \\cite[\\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\\operatorname{HS} _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\\![t]\\!]/(t^{m+1})$-algebra automorphisms\n$$R[\\![t]\\!]/(t^{m+1}) \\to R[\\![t]\\!]/(t^{m+1})$$\nthat become the identity when restricted to $ R[\\![t]\\!]/(t^{m+1}) \\otimes_{A_m} k=R$. For $m=\\infty$, we regard $t^\\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\\operatorname{HS} _k^1(R)$ and the module $\\operatorname{Der}_k(R,R)$. Under this identification, the image of\n$$\\operatorname{HS} _k^m(R)\\to \\operatorname{HS} _k^1(R)\\simeq \\operatorname{Der}_k(R,R)$$\nis called the module of $m$-integrable derivations and denoted by $\\operatorname{Der}_k^m(R)$, which has been studied in \\cite{matsumura1982integrable}, \\cite{NARVAEZMACARRO20122712}, \\cite{NARVAEZMACARRO2024109758}. The module $\\operatorname{Der}_k^\\infty(R)$ is also denoted by $\\operatorname{Ider}_k(R)$. Note that if $k$ contains $\\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k(R)$ for any $m\\le \\infty$ (see \\cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain\n$$ \\operatorname{Der}_k(R)\\supset \\operatorname{Der}_k^2(R)\\supset \\dots \\supset \\operatorname{Der}_k^m(R) \\supset\\dots.$$\nThe question of when the inclusion $\\operatorname{Der}_k^{m-1}(R)\\supset \\operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).\nThe aim of this paper is to show, using results from \\cite[\\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:\n\n\\begin{theorem}\\label{infty integrable thm}\n     Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that\n     $$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$\n\\end{theorem}",
    "full_context": "Hasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \\cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \\cite[\\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\\operatorname{HS} _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\\![t]\\!]/(t^{m+1})$-algebra automorphisms\n$$R[\\![t]\\!]/(t^{m+1}) \\to R[\\![t]\\!]/(t^{m+1})$$\nthat become the identity when restricted to $ R[\\![t]\\!]/(t^{m+1}) \\otimes_{A_m} k=R$. For $m=\\infty$, we regard $t^\\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\\operatorname{HS} _k^1(R)$ and the module $\\operatorname{Der}_k(R,R)$. Under this identification, the image of\n$$\\operatorname{HS} _k^m(R)\\to \\operatorname{HS} _k^1(R)\\simeq \\operatorname{Der}_k(R,R)$$\nis called the module of $m$-integrable derivations and denoted by $\\operatorname{Der}_k^m(R)$, which has been studied in \\cite{matsumura1982integrable}, \\cite{NARVAEZMACARRO20122712}, \\cite{NARVAEZMACARRO2024109758}. The module $\\operatorname{Der}_k^\\infty(R)$ is also denoted by $\\operatorname{Ider}_k(R)$. Note that if $k$ contains $\\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k(R)$ for any $m\\le \\infty$ (see \\cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain\n$$ \\operatorname{Der}_k(R)\\supset \\operatorname{Der}_k^2(R)\\supset \\dots \\supset \\operatorname{Der}_k^m(R) \\supset\\dots.$$\nThe question of when the inclusion $\\operatorname{Der}_k^{m-1}(R)\\supset \\operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).\nThe aim of this paper is to show, using results from \\cite[\\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:\n\n\\begin{theorem}\\label{infty integrable thm}\n     Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that\n     $$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$\n\\end{theorem}\n\n\\section{Introduction}\nHasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \\cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \\cite[\\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\\HS _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\\![t]\\!]/(t^{m+1})$-algebra automorphisms\n$$R[\\![t]\\!]/(t^{m+1}) \\to R[\\![t]\\!]/(t^{m+1})$$\nthat become the identity when restricted to $ R[\\![t]\\!]/(t^{m+1}) \\otimes_{A_m} k=R$. For $m=\\infty$, we regard $t^\\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\\HS _k^1(R)$ and the module $\\operatorname{Der}_k(R,R)$. Under this identification, the image of\n$$\\HS _k^m(R)\\to \\HS _k^1(R)\\simeq \\operatorname{Der}_k(R,R)$$\nis called the module of $m$-integrable derivations and denoted by $\\operatorname{Der}_k^m(R)$, which has been studied in \\cite{matsumura1982integrable}, \\cite{NARVAEZMACARRO20122712}, \\cite{NARVAEZMACARRO2024109758}. The module $\\operatorname{Der}_k^\\infty(R)$ is also denoted by $\\operatorname{Ider}_k(R)$. Note that if $k$ contains $\\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k(R)$ for any $m\\le \\infty$ (see \\cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain\n$$ \\operatorname{Der}_k(R)\\supset \\operatorname{Der}_k^2(R)\\supset \\dots \\supset \\operatorname{Der}_k^m(R) \\supset\\dots.$$\nThe question of when the inclusion $\\operatorname{Der}_k^{m-1}(R)\\supset \\operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).\nThe aim of this paper is to show, using results from \\cite[\\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:\n\nThis theorem provides, for the first time, a general finiteness result for leaps in arbitrary dimensions, thereby giving an affirmative answer to Macarro’s question [8, Question 3.6.5] and contributing to a better structural understanding of HS-derivations. Since leaps are stable under separable base changes by \\cite[Theorem 3.27]{hernandez2020behavior}, this also implies the finiteness of leaps over any perfect field. We also give an affirmative answer to another question \\cite[Question 3.6.1]{NARVAEZMACARRO20122712} by the following:\n\n\\begin{theorem}[{= Theorem \\ref{mn int theorem}}]\\label{main thm2}\n Let $n\\ge 1$ be an integer and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}\nIn fact, Theorem \\ref{main thm1} is a special case of Theorem \\ref{main thm2} where we put $n=1$. As an application of Theorem \\ref{main thm1}, we obtain:\n\n\\begin{corollary}[{=Corollary \\ref{localization ider cor}}]\\label{main cor}\nThe module $\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\\operatorname{Ider}_k(S^{-1}R)=S^{-1}\\operatorname{Ider}_k(R).$$\n\\end{corollary}\nThis is the first coherence result for $\\operatorname{Ider}_k(R)$ to the best of our knowledge. For $m<\\infty$, a similar statement has been proven in \\cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}. The case $m=\\infty$ is more subtle because the map in \\cite[Corollary 1.3.6]{NARVAEZMACARRO20122712} is not surjective for $m=\\infty$ while $m$ is assumed to be an integer there (cf. \\cite[Example 1.4]{MR1764573}). Note that Corollary \\ref{main cor} implies that $\\operatorname{Ider}_k(R)$ constitutes a coherent sheaf $\\operatorname{Ider}_k(\\mathcal{O}_X)$ on an algebraic scheme $X$ over $k$. In particular, it is expected to behave better than the usual tangent sheaf. In fact, some pathological phenomena of derivations peculiar to positive characteristic do not occur if we replace it by integrable derivations. For example, the local rank of $\\operatorname{Ider}_k(R)$ is at most $\\operatorname{dim}R$, while it is not true for $\\operatorname{Der}_k(R)$ (see \\cite{MR554764}). This pathology does not occur in characteristic zero (see \\cite{MR366909} when $R$ is a domain and \\cite{MR498523} in general). We can also state Theorem \\ref{main thm1} in terms of schemes:\n\\begin{theorem}[{= Theorem \\ref{infty integrable thm schemes}}]\\label{main thm3}\n Let $X$ be a $k$-scheme essentially of finite type (=locally essentially of finite type and quasi-compact). Then there exists $M>0$ such that\n$$ \\operatorname{Der}_k^M(\\mathcal{O}_X)= \\operatorname{Der}_k^{M+1}(\\mathcal{O}_X)=\\dots=\\operatorname{Ider}_k(\\mathcal{O}_X),$$\nwhere $\\operatorname{Der}_k^m(\\mathcal{O}_X)$ denotes the sheaf of $m$-integrable derivations (see Theorem \\ref{localization}).\n\\end{theorem}\n\n\\begin{proposition}\\label{a to b extension prop}\n Let $e$ be an integer with\n $$e\\ge C_1+C_3\\cdot q+C_4\\cdot q+q$$\n where $C_1,C_3,C_4$ are as in \\ref{hom lem}-\\ref{j lem}. Let $m$ be a positive integer and $D\\in \\HS_k^\\infty(T)$ be such that:\n \\begin{enumerate}\n \\item $D$ is $I$-logarithmic up to $m$, thus induces $\\widetilde{D}\\in \\HS_k^m(R),$\n \\item $D$ is logarithmically $a[e,m,q]$-bounded.\n \\end{enumerate}\n Then there exists $E\\in \\HS_k^\\infty(T)$ such that:\n \\begin{enumerate}\n \\item $E$ is $I$-logarithmic up to $m$, thus induces $\\widetilde{E}\\in \\HS_k^m(R),$\n \\item $\\operatorname{ob}_{m+1}(\\widetilde{E})=0$,\n \\item $E$ is logarithmically $b[e,m+1,q]$-bounded,\n \\item $D$ and $E$ coincide up to $l_{m+1}-1$, i.e., $D_1=E_1,\\dots,D_{l_{m+1}-1}=E_{l_{m+1}-1}$.\n \\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{b to a extension prop}\n Let $e$ be an integer with\n $$e\\ge C_1+C_2+1$$\n where $C_1,C_2$ are as in \\ref{hom lem}, \\ref{G lem}. Let $m$ be a positive integer and $E\\in \\HS_k^\\infty(T)$ such that:\n \\begin{enumerate}\n \\item $E$ is $I$-logarithmic up to $m$, thus induces $\\widetilde{E}\\in \\HS_k^m(R),$\n \\item $\\operatorname{ob}_{m+1}(\\widetilde{E})=0$,\n \\item $E$ is logarithmically $b[e,m+1,q]$-bounded.\n \\end{enumerate}\n Then there exists $D^+\\in \\HS_k^\\infty (T)$ such that:\n \\begin{enumerate}\n \\item $D^+$ is $I$-logarithmic up to $m+1$,\n \\item $D^+$ is logarithmically $a[e,m+1,q]$-bounded,\n \\item $D^+$ and $E$ coincide up to $m$, i.e., $D^+_1=E_1,\\dots,D^+_{m}=E_{m}$.\n \\end{enumerate}\n\\end{proposition}\n\nSetting $n=1$ in the above theorem, we obtain the following.\n\\begin{theorem}\\label{infty integrable thm}\n Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that\n $$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$\n\\end{theorem}\n\n\\begin{theorem}\\label{infty integrable thm schemes}\n Let $X$ be a $k$-scheme essentially of finite type. Then there exists $M>0$ such that\n$$ \\operatorname{Der}_k^M(\\mathcal{O}_X)= \\operatorname{Der}_k^{M+1}(\\mathcal{O}_X)=\\dots=\\operatorname{Ider}_k(\\mathcal{O}_X).$$\n\\end{theorem}",
    "post_theorem_intro_text_len": 5485,
    "post_theorem_intro_text": "This theorem provides, for the first time, a general finiteness result for leaps in arbitrary dimensions, thereby giving an affirmative answer to Macarro’s question [8, Question 3.6.5] and contributing to a better structural understanding of HS-derivations. Since leaps are stable under separable base changes by \\cite[Theorem 3.27]{hernandez2020behavior}, this also implies the finiteness of leaps over any perfect field. We also give an affirmative answer to another question \\cite[Question 3.6.1]{NARVAEZMACARRO20122712} by the following:\n\n\\begin{theorem}[{= Theorem \\ref{mn int theorem}}]\\label{main thm2}\n    Let $n\\ge 1$ be an integer and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}\nIn fact, Theorem \\ref{main thm1} is a special case of Theorem \\ref{main thm2} where we put $n=1$. As an application of Theorem \\ref{main thm1}, we obtain:\n\n\\begin{corollary}[{=Corollary \\ref{localization ider cor}}]\\label{main cor}\nThe module $\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\\operatorname{Ider}_k(S^{-1}R)=S^{-1}\\operatorname{Ider}_k(R).$$\n\\end{corollary}\nThis is the first coherence result for $\\operatorname{Ider}_k(R)$ to the best of our knowledge. For $m<\\infty$, a similar statement has been proven in \\cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}. The case $m=\\infty$ is more subtle because the map in \\cite[Corollary 1.3.6]{NARVAEZMACARRO20122712} is not surjective for $m=\\infty$ while $m$ is assumed to be an integer there (cf. \\cite[Example 1.4]{MR1764573}). Note that Corollary \\ref{main cor} implies that $\\operatorname{Ider}_k(R)$ constitutes a coherent sheaf $\\operatorname{Ider}_k(\\mathcal{O}_X)$ on an algebraic scheme $X$ over $k$. In particular, it is expected to behave better than the usual tangent sheaf. In fact, some pathological phenomena of derivations peculiar to positive characteristic do not occur if we replace it by integrable derivations. For example, the local rank of $\\operatorname{Ider}_k(R)$ is at most $\\operatorname{dim}R$, while it is not true for $\\operatorname{Der}_k(R)$ (see \\cite{MR554764}). This pathology does not occur in characteristic zero (see \\cite{MR366909} when $R$ is a domain and \\cite{MR498523} in general). We can also state Theorem \\ref{main thm1} in terms of schemes:\n\\begin{theorem}[{= Theorem \\ref{infty integrable thm schemes}}]\\label{main thm3}\n    Let $X$ be a $k$-scheme essentially of finite type (=locally essentially of finite type and quasi-compact). Then there exists $M>0$ such that\n$$ \\operatorname{Der}_k^M(\\mathcal{O}_X)= \\operatorname{Der}_k^{M+1}(\\mathcal{O}_X)=\\dots=\\operatorname{Ider}_k(\\mathcal{O}_X),$$\nwhere $\\operatorname{Der}_k^m(\\mathcal{O}_X)$ denotes the sheaf of $m$-integrable derivations (see Theorem \\ref{localization}).\n\\end{theorem}\n\nWe briefly outline the contents of the paper.\n\nIn Section 2, we recall the definition of HS-derivations and results that will  be used later. A key relation is that the obstructions to extending HS-derivations are canonically dual to $m$-integrable derivations (Theorem \\ref{duality}). We also define the property $\\operatorname{FL}(i)$ for algebras and schemes, which states that leaps do not occur at every $m> p^i$ (Definition \\ref{FL def}).\n\nIn Section 3, we introduce the notion of $a$-boundedness for HS-derivations. Intuitively, this concept measures how each term of an HS-derivation shrinks with respect to the $\\mathfrak{m}$-adic topology.\n\nIn Section 4, we prove the finiteness of leaps in the local case. In this section we fix a local $k$-algebra $(R,\\mathfrak{n})$ such that $\\operatorname{Spec}R\\setminus \\{\\mathfrak{n}\\}$ satisfies $\\operatorname{FL}(\\tau)$ for $\\tau>0$, and we will prove that $\\operatorname{Spec}R$ itself satisfies the finiteness of leaps. The idea is that, starting from a bounded HS-derivation, we construct another one that is less bounded and coincides with the original one in small degrees. We use the Artin-Rees lemma to obtain positive constants which control the bounds. The essential step is Theorem \\ref{eq finie thm}, which states that elements of $$\\mathfrak{n}^{e(q+1)+1} \\cdot \\operatorname{Der}_k^{q}(R)$$ do not leap, where $q=p^\\tau$ and $e\\gg 0$. Then the fact that $R/\\mathfrak{n}^{e(q+1)+1} $ is of finite length implies the finiteness of leaps of $R$ (Theorems \\ref{Her thm}, \\ref{sec 4 main thm}).\n\nIn Section 5, we prove the finiteness of leaps in the global case and discuss its consequences. We use noetherian induction to reduce to the case of Section 4 (Theorem \\ref{main thm global}). The finiteness of leaps of $m$-integrable derivations implies that of obstruction modules. This enables us to extend HS-derivations step by step (Lemma \\ref{mn int lem}), and taking a limit, we obtain a HS-derivation of infinite length: This method yields a characterization of $\\infty$-integrability (Theorem \\ref{mn int theorem}).\n\nThroughout this paper, we fix an algebraically closed field $k$ of positive characteristic $p$. We assume that any $k$-algebra is unital and commutative. We say that a $k$-scheme $X$ is essentially of finite type if there exists a finite open affine covering $\\{U_i=\\operatorname{Spec}R_i\\}_{i=1,\\dots,n}$ for $X$ where each $R_i$ is a $k$-algebra essentially of finite type. We assume $\\mathbb{N}:=\\{0,1,\\dots\\}$.",
    "sketch": "In Section 5, the authors say they \"prove the finiteness of leaps in the global case\" and \"use noetherian induction to reduce to the case of Section 4 (Theorem \\ref{main thm global}).\" Then, since \"the finiteness of leaps of $m$-integrable derivations implies that of obstruction modules,\" they can \"extend HS-derivations step by step (Lemma \\ref{mn int lem}), and taking a limit, we obtain a HS-derivation of infinite length.\" This \"method yields a characterization of $\\infty$-integrability (Theorem \\ref{mn int theorem}),\" and they note that \"Theorem \\ref{main thm1} is a special case of Theorem \\ref{main thm2} where we put $n=1$.\"",
    "expanded_sketch": "Next, the authors prove the finiteness of leaps in the global case and use noetherian induction to reduce to the local case, as stated in the following result.\n\n\\begin{theorem}\\label{main thm global}\n    Let $X$ be a $k$-scheme essentially of finite type. Then $X$ satisfies $\\FL{\\tau}$ for a sufficiently large integer $\\tau$. In particular, the set of leaps of $X$ is finite.\n\\end{theorem}\n\nThen, since the finiteness of leaps of $m$-integrable derivations implies that of obstruction modules, they extend HS-derivations step by step using the following lemma.\n\n\\begin{lemma}\\label{mn int lem}\n    Let $R$ be a $k$-algebra essentially of finite type, let $m, \\tau\\ge 1$ be integers, and let $q:=p^\\tau$. Suppose that $R$ satisfies $\\FL{\\tau}$. Let $D\\in \\HS_k^m(R)$. Let $l_1,l_2,\\dots$ be the sequence of integers defined as in \\ref{l_n def}. Then there exists $D^+\\in \\HS_k^{m+1}(R)$ such that $D$ and $D^+$ coincide up to $l_{m+1}-1$, i.e., $D_1=D^+_1,\\dots, D_{l_{m+1}-1}= D_{l_{m+1}-1}^+$.\n\\end{lemma}\n\nTaking a limit, they obtain a HS-derivation of infinite length. This method yields the following characterization of $\\infty$-integrability.\n\n\\begin{theorem}\\label{mn int theorem}\n    Let $R$ be a $k$-algebra essentially of finite type, let $n\\ge 1$ be an integer, and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}\n\nFinally, to prove the main theorem, they note that it is a special case of\n\\begin{theorem}[{= Theorem \\ref{mn int theorem}}]\\label{main thm2}\n    Let $n\\ge 1$ be an integer and let $D\\in \\HS_k^n(R)$. Then $D$ is $\\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\\HS_k^m(R)\\to \\HS_k^n(R)$.\n\\end{theorem}\nwhere one puts $n=1$.",
    "expanded_theorem": "[{= To prove the main theorem, } ]\\label{main thm1}\n    There exists $M>0$ (depending on $R$) such that\n$$ \\operatorname{Der}_k^M(R)= \\operatorname{Der}_k^{M+1}(R)=\\dots= \\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$,",
    "theorem_type": [
      "Existential–Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $k$ be an algebraically closed field of positive characteristic, and let $R$ be a $k$-algebra essentially of finite type. For each $m\\in \\mathbb{Z}_{>0}\\cup\\{\\infty\\}$, let $\\operatorname{Der}_k^m(R)$ denote the module of $m$-integrable $k$-derivations, i.e. the image of the natural map from length-$m$ Hasse--Schmidt derivations $\\operatorname{HS}_k^m(R)$ to $\\operatorname{Der}_k(R,R)\\simeq \\operatorname{HS}_k^1(R)$, and write $\\operatorname{Ider}_k(R)=\\operatorname{Der}_k^\\infty(R)$. Which statement holds about the sequence\n$$\\operatorname{Der}_k(R)\\supset \\operatorname{Der}_k^2(R)\\supset \\cdots \\supset \\operatorname{Der}_k^m(R)\\supset \\cdots ?$$",
      "correct_choice": {
        "label": "A",
        "text": "There exists an integer $M>0$, depending on $R$, such that for every $m\\ge M$ one has\n$$\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R),$$\nthat is,\n$$\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k^{M+1}(R)=\\cdots=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an integer $M>0$, depending only on $\\operatorname{char}(k)$ and not on $R$, such that for every $m\\ge M$ one has\n$$\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R),$$\nthat is,\n$$\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k^{M+1}(R)=\\cdots=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$"
        },
        {
          "label": "C",
          "text": "There exists an integer $M>0$, depending on $R$, such that\n$$\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R).$$"
        },
        {
          "label": "D",
          "text": "For every integer $m>0$ one has\n$$\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k^\\infty(R)=\\operatorname{Ider}_k(R),$$\nthat is,\n$$\\operatorname{Der}_k(R)=\\operatorname{Der}_k^2(R)=\\cdots=\\operatorname{Der}_k^\\infty(R).$$"
        },
        {
          "label": "E",
          "text": "There exists an integer $M>0$, depending on $R$, such that for every $m\\ge M$ one has\n$$\\operatorname{Der}_k^m(R)=\\operatorname{Der}_k^M(R),$$\nand moreover every derivation is eventually integrable, i.e.\n$$\\operatorname{Der}_k^M(R)=\\operatorname{Der}_k(R).$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence_of_M_on_R",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "eventual_constancy_for_all_m_ge_M",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "only_eventual_not_immediate_stabilization",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "limit_equals_Ider_not_full_Der",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal option A. It presents definitions and asks for the correct statement about the chain, without directly stating the stabilization conclusion."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct option is essentially an explicit stabilization theorem about the chain. Still, the alternatives vary in quantifiers and strength, so it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish dependence on R versus dependence only on the characteristic, and eventual stabilization versus stronger claims. However, the pressure is weakened because the task mainly tests recognition of the exact theorem statement rather than deeper derivation."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically meaningful and target natural mistakes (uniformity in p, premature stabilization, elementwise vs modulewise criteria). But option C is a weaker statement that is also true if A is true, so the distractor set fails uniqueness and is only of mixed quality."
      },
      "total_score": 5,
      "overall_assessment": "Reasonably well-formed and free of answer leakage, but only moderately generative and significantly weakened by a non-unique answer set, since option C is also true."
    }
  },
  {
    "id": "2512.00787v1",
    "paper_link": "http://arxiv.org/abs/2512.00787v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.",
      "start_pos": 11662,
      "end_pos": 12798,
      "label": "theorem: intrinsic torsion"
    },
    "ref_dict": {
      "prop:two-def-same": "\\begin{proposition}\\label{prop:two-def-same}\nLet $v$ be a discrete valuation on $k$, and $O$ its valuation ring.\nLet $\\sX \\to \\Spec O$ be a smooth proper morphism with geometrically connected fibers,\nand denote by $X$ the generic fiber.\nThen we have\n\\[ \\Im(\\langle \\cdot, \\cdot \\rangle : \n\\CH^1(X)_\\Tor \\times A_0(X) \\to k^\\times \\otimes \\Q/\\Z)\n\\subset\n\\Im(O^\\times \\otimes \\Q/\\Z \\to k^\\times \\otimes \\Q/\\Z).\n\\]\n\\end{proposition}",
      "prop:TateCv": "\\begin{proposition}\\label{prop:TateCv}\n\\begin{enumerate}\n\\item \nLet $a, b \\in k^\\times$ and $m>0$.\nIf $[a],  [b] \\in E_q(k)[m]$, we have\n\\[ \\langle [a], [b] \\rangle \n= a^{-s_m([b])} \\otimes \\frac{1}{m}\n= b^{-s_m([a])} \\otimes \\frac{1}{m}\n= q^{-s_m([a])s_m([b])} \\otimes \\frac{1}{m^2}.\n\\]\n\\item\nWe have $E_q(k)_\\Tor^\\is = \\{ [\\zeta] \\mid \\zeta \\in \\mu(k) \\}$.\n\\end{enumerate}\n\\end{proposition}",
      "cor:kodairatype": "\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}",
      "eq:pairing-intro": "\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}",
      "theorem: intrinsic torsion": "\\begin{theorem} \\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\Q$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\Z/2\\Z & 1,2\\\\\n    \\Z/3\\Z & 1,3\\\\\n    \\Z/4\\Z & 1,2,4 \\\\\n    \\Z/5\\Z & 1,5 \\\\\n    \\Z/6\\Z & 1,2,3 \\\\\n    \\Z/7\\Z & 1 \\\\\n    \\Z/8\\Z & 1,2 \\\\\n    \\Z/9\\Z & 1 \\\\\n    \\Z/10\\Z & 1\\\\\n    \\Z/12\\Z & 1 \\\\\n    \\Z/2\\Z\\times\\Z/2\\Z & 1,2\\\\\n    \\Z/4\\Z\\times\\Z/2\\Z & 1,2 \\ \\\\\n    \\Z/6\\Z\\times\\Z/2\\Z & 1 \\\\\n    \\Z/8\\Z\\times\\Z/2\\Z & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\Z/4\\Z \\times \\Z/2\\Z$,\nthere are two $\\Aut(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\Q$ \nfor which there is an isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$\nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n\\end{theorem}",
      "rem:FreyRuck": "\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}",
      "proposition: pairing 1-mod": "\\begin{theorem}\\label{proposition: pairing 1-mod}\n  Assume that $N\\in\\{4,5,6,7,8,9,10,12\\}$ and\n  let $M$ be a positive divisor of $N$.\n  Denote by $X_1(MN,N)$ the modular curve over $\\Q$ \n  associated to the congruence subgroup\n    $$\n    \\Gamma_1(MN,N):= \\Gamma_0(MN) \\cap \\Gamma_1(N)\n    =\\left\\{\\M abcd\\in\\SL(2,\\Z) : \\ \n      a,d\\equiv1\\bmod N, \\ MN|c \\right\\},\n    $$\n  on which the cusp $0$ is $\\Q$-rational.\n  Then $X_1(N)_M^+$ is isomorphic to $X_1(MN,N)$ over $\\Q$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2339,
    "pre_theorem_intro_text": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$)  is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.",
    "context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}",
    "full_context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\Pic(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\gen{P,Q}:=\\gen{[P-O], [Q-O]}$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\Pic(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\Q)_\\Tor,E(\\Q)_\\Tor^\\is)$ for an elliptic curve $E$ over $\\Q$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\Q)_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\Z/2N\\Z) \\times (\\Z/2\\Z)$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\Q)_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\nWe briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\Q$ on\nwhich the cusp $0$ is $\\Q$-rational. For a subfield $k$ of $\\C$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\C/(\\Z\\tau+\\Z),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\Q)$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\Z):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\ul{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve \n$X_1^0(N, 2)_{\\ul{M}}^\\pm$ over $\\Q$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\Q)$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\Q$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\ul{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\n\\begin{lemma} \\label{lemma: ENt}\n \\begin{enumerate}\n \\item Assume that the characteristic of $k$ is not $2$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational point\n $P$ of order two, then $(E,P)$ is isomorphic to $(E_{2,t,a},(0,0))$\n for some $a,t\\in k^\\times$, where\n \\begin{equation} \\label{eq: E2ta}\n E_{2,t,a}:\n \\begin{cases}\n \\displaystyle y^2=x\\left(x^2+ax+\\frac{a^2t}{4(t+1)}\\right),\n &\\text{if }t\\neq-1, \\\\\n y^2=x(x^2+a), &\\text{if }t=-1.\n \\end{cases}\n \\end{equation}\n Moreover, two elliptic curves $(E_{2,t,a},(0,0))$ and\n $(E_{2,t',a'},(0,0))$ are isomorphic over $k$ if and only\n if $t=t'\\neq-1$ and $a/a'$ is a square in $k^\\times$\n or $t=t'=-1$ and $a/a'\\in(k^\\times)^4$ . \n \\item Assume that the characteristic of $k$ is not $3$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational\n point $P$ of order $3$, then $(E,P)$ is isomorphic to $(E_{3,t},(0,0))$\n for some $t\\neq-1$ in $k^\\times$, where\n \\begin{equation} \\label{eq: E3t}\n E_{3,t}:y^2+xy+\\frac t{27(t+1)}y=x^3\n \\end{equation}\n or isomorphic to $(E_{3,-1,a},(0,0))$ for some $a\\in k^\\times$,\n where\n \\begin{equation} \\label{eq: E3a}\n E_{3,-1,a}:y^2+ay=x^3.\n \\end{equation}\n Two elliptic curves $(E_{3,t},(0,0))$ and\n $(E_{3,t'},(0,0))$ with $t,t'\\neq-1$ are isomorphic if and only\n if $t=t'$. Also, $(E_{3,-1,a},(0,0))\\simeq(E_{3,-1,a'},(0,0))$\n if and only if $a/a'\\in(k^\\times)^3$. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Consider first the case $N=2$. \nWe start with the equation \\eqref{eq: E2}\nand introduce a new parameter\n $t=4b/(a^2-4b)$ (so $b=a^2t/4(t+1)$). When $a\\neq0$, we have\n $t\\neq-1$ and the equation \\eqref{eq: E2} in the new parameters is\n $y^2=x(x^2+ax+a^2t/4(t+1))$. When $a=0$, we change the notation $b$\n in \\eqref{eq: E2} to $-a$ and get $y^2=x(x^2-a)$. Now it is easy to\n see that an isomorphism\n $\\phi:(E_{2,t,a},(0,0))\\to(E_{2,t',a'},(0,0))$ must be of the form\n $\\phi(x,y)=(u^2x,u^3y)$ (c.f. \\cite[Proposition 3.1]{sil}). From\n this, we immediately get the condition for $E_{2,t,a}$ and\n $E_{2,t',a'}$ to be isomorphic.\n\nThis leaves us with the cases\n$(N, M) \\in \\{ (6,6), (8, 4), (9,3), (10, 2), (12, 2) \\}$ with $\\epsilon=-$.\n(For $(N, M)=(9,3)$, the value of $\\epsilon$ does not matter\nbecause of Remark \\ref{rem:plus-minus} (1).)\nIn all cases, there exists \nan isomorphism $\\alpha : E \\to X_1(N)_M^\\epsilon$,\nwhere\n$E$ is an elliptic curve over $\\Q$\nwith finitely many $\\Q$-rational points,\nas exhibited in the following table,\nwhere we set\n$h_1(x)=(1-x)(1+x)^4(1+x-x^2)^2$ and\n$h_2(x)=(1+x)(1+x+x^2)(1+x^2)^2(1-x)^3$.\n $$ \\extrarowheight3pt\n \\begin{array}{c|c|c|c} \\hline\\hline\n (N, M) & E & E(\\Q) \\setminus \\{ O \\} & \n(s, t)=\\alpha(x,y) \\\\ \\hline\n(6,6) & y^2=x^3-1 & (1,0) &\n\\left(-\\frac{xy}{1-x^3},\\frac1{1-x^3}\\right)\n\\\\\n(8,4) & y^2=x^3+x & (0,0) &\n\\left(\\frac y{x^2}(1-\\frac{1}{x^2}),-\\frac1{x^2}\\right)\n\\\\\n(9,3) & y^2+y=x^3 & (0,0), (0,-1) &\n\\left(xy(y^2+y+1),y+1\\right)\n\\\\\n(10,2) & y^2=x^3-x^2-x & (0,0) &\n\\left(yh_1(x),x\\right)\n\\\\\n(12,2) & y^2=x^3+x^2+x & (0,0) &\n\\left(yh_2(x),x\\right)\n\\\\\n \\hline\\hline\n \\end{array}\n $$\nSince all points of $\\alpha(E(\\Q))$ are cusps \nby Lemma \\ref{lemma: uniformizers},\nwe get $Y_1(\\Q)^\\epsilon_M=\\emptyset$ in these cases as well.\nWe are done.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}\n\\begin{proof}\nThe assumption implies that\nthe base change of $E$ to $k_v$ is \nisomorphic to the Tate curve $E_q=\\G_m/q^\\Z$\nfor some $q \\in k_v$ such that $v(q)=n$\n(\\cite[Chapter V, Theorem 5.3]{sil})\nLet us write the image of $P$ in $E_q(k_v)$ as $[a]$ with $a \\in k_v$.\nSince $\\mu_m(k_v)= \\{ 1 \\}$, \nwe find from \\eqref{eq:ex-TateCv} that\n$E_q(k_v)[m]$ is a cyclic group of order $m$ generated by $[a]$.\nIn particular, $s:= s_m([a])$ is invertible in $\\Z/m\\Z$.\nWe have $\\langle [a], [a] \\rangle = 0$ since $P \\in E(k)_\\Tor^\\is$.\nOn the other hand,\nProposition \\ref{prop:TateCv} (1) shows\n$\\langle [a], [a] \\rangle = q^{-s^2} \\otimes (1/m^2)$.\nBy looking at its image under the map\n$v \\otimes \\id_{\\Q/\\Z} : k_v^\\times \\otimes \\Q/\\Z \\to \\Q/\\Z$,\nwe conclude that $-s^2 n/m^2 = 0$ holds in $\\Q/\\Z$,\nthat is, $m^2$ divides $n$.\n\\end{proof}",
    "post_theorem_intro_text_len": 4805,
    "post_theorem_intro_text": "We briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\mathbb{Q}$ on\nwhich the cusp $0$ is $\\mathbb{Q}$-rational. For a subfield $k$ of $\\mathbb{C}$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\mathbb H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\mathbb{C}/(\\mathbb{Z}\\tau+\\mathbb{Z}),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$. \n\nFor a pair $(N, M)$ of $N \\in \\{4, \\dots, 10, 12\\}$ and a positive divisor $M$ of $N$,\nwe will construct a smooth projective curve $X_1 (N)^\\pm_M$ over $\\mathbb{Q}$ equipped with\na finite morphism $X_1(N)_M^\\pm \\to X_1(N)$\nthat enjoys the following property:\nwhen $t \\in X_1(N)(\\mathbb{Q})$ corresponds to a pair $(E, P)$ as above (with $k=\\mathbb{Q}$),\none has \n\\[\n\\langle P, (N/M)P \\rangle = 0 \\Leftrightarrow\nt \\in \\operatorname{Im}(X_1(N)^\\pm_M(\\mathbb{Q}) \\to X_1(N)(\\mathbb{Q})).\n\\]\nIt is worth  mentioning that\n$X_1(N)^+_M$ is isomorphic to the modular curve\n$X_1(MN, N)$ associated to $\\Gamma_1(MN, N):=\\Gamma_0(MN) \\cap \\Gamma_1(N)$\n(Theorem \\ref{proposition: pairing 1-mod})\nand $X_1(N)^-_M$ is its twist.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\mathbb{Z}):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\underline{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve  \n$X_1^0(N, 2)_{\\underline{M}}^\\pm$ over $\\mathbb{Q}$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\mathbb{Q})$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\underline{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\nIn the last section \\S \\ref{sect:high-dim},\nwe generalize the pairing \\eqref{eq:pairing-intro}\nand the intrinsic subgroup \nto higher dimensional varieties.\nThis new construction enables us to prove that,\nif $X$ has good reduction with respect to a discrete valuation of $k$,\nthere is a strong restriction on the values of the pairing \n(see Proposition \\ref{prop:two-def-same}).\nFinally, as a sample for the case of bad reduction,\nwe compute the intrinsic subgroup of \nTate elliptic curves (see Proposition \\ref{prop:TateCv}).\nAs an application,\nwe show that the intrinsic subgroup of an elliptic curve over a number field\nimposes some constraint on its reduction type\n(see Corollary \\ref{cor:kodairatype}).\n\n\\subsection*{Acknowledgement}\nIt is Kenneth Ribet who formulated \\eqref{eq:pairing-intro} as a pairing \nand asked if it is symmetric.\nThe authors express deep gratitude to him for asking this question,\nwhich was a starting point of our work.\n\nThe first author (T. Y.) is supported by JSPS KAKENHI Grant (25K06961). \nThe second author (Y. Y.) is supported by \nGrant 113-2115-M-002-003-MY3 of the National Science and Technology\n  Council of the Republic of China (Taiwan). \nThe third author (H. Y.) is supported by National Research Foundation of Korea(NRF) grant \nfunded by the Korea government(MSIT) (No. RS-2023-00239918  and No. 2020R1A5A1016126).\nThe fourth author (M. Y.) is supported by the National Research Foundation of Korea (NRF) grant \nfunded by the\nKorea government (MSIT) (RS-2025-23525445).\n\n\\subsection*{Notation and convention}\nFor an abelian group $A$, \nwe write $A[n]$ and $A/n$ for the kernel and cokernel\nof $A \\to A, \\ a \\mapsto na$ for each $n \\in \\mathbb{Z}$,\nand put $A_\\Tor := \\cup_{n > 0} A[n]$.\nFor a field $k$,\nwe set $\\mu_n(k):=(k^\\times)[n]$ and $\\mu(k):=(k^\\times)_\\Tor$.\n\nThe identity element of an elliptic curve is denoted by $O$.",
    "sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that “Therefore Theorem~\\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$,” and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”",
    "expanded_sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that, in establishing the main theorem, everything is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$, and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”",
    "expanded_theorem": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.,",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let $E$ be an elliptic curve over $\\mathbb{Q}$ with identity element $O$. Via the identification $P \\mapsto [P-O]$, identify $E(\\mathbb{Q})_{\\mathrm{Tor}}$ with $\\operatorname{Pic}(E)_{\\mathrm{Tor}}$. Let\n\\[\n\\langle\\cdot,\\cdot\\rangle: E(\\mathbb{Q})_{\\mathrm{Tor}}\\times E(\\mathbb{Q})_{\\mathrm{Tor}}\\to \\mathbb{Q}^\\times\\otimes \\mathbb{Q}/\\mathbb{Z}\n\\]\nbe the induced biadditive symmetric pairing, and define the intrinsic subgroup by\n\\[\nE(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}:=\\{P\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\mid \\langle P,Q\\rangle=0\\text{ for all }Q\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\}.\n\\]\nWhich existence statement holds for the pair consisting of the torsion subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}$ and its intrinsic subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}$?",
      "correct_choice": {
        "label": "A",
        "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exists at least one elliptic curve $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        },
        {
          "label": "C",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are among the following: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$."
        },
        {
          "label": "D",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        },
        {
          "label": "E",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and both classes occur simultaneously for every elliptic curve $E/\\mathbb{Q}$ with $E(\\mathbb{Q})_{\\mathrm{Tor}}\\cong A$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "infinitely_many_realizations",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped_exactness_and_infinite_realization_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "computational_check",
          "tampered_component": "order_4_subgroup_for_Z4xZ2",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "outer_automorphism",
          "tampered_component": "existential_occurrence_of_two_Aut_conjugacy_classes",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the classification or signal the correct option. It only defines the intrinsic subgroup and asks for the complete abstract classification."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: it asks for the exact classification statement itself rather than an application or consequence. However, it is not a pure tautology because the choices contain subtly different candidate classifications."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact classification from near-miss variants (overly strong, weaker true, or incomplete converse statements). Still, success depends largely on recall of the theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are plausible, structurally similar to the correct theorem, and reflect realistic failure modes such as overgeneralizing admissible subgroups, adding forbidden cases, or weakening the realization statement."
      },
      "total_score": 6,
      "overall_assessment": "A solid high-level theorem-classification MCQ with little answer leakage and strong distractors, but it mainly tests precise recall/discrimination rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.00787v1",
    "paper_link": "http://arxiv.org/abs/2512.00787v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.",
      "start_pos": 11662,
      "end_pos": 12798,
      "label": "theorem: intrinsic torsion"
    },
    "ref_dict": {
      "prop:two-def-same": "\\begin{proposition}\\label{prop:two-def-same}\nLet $v$ be a discrete valuation on $k$, and $O$ its valuation ring.\nLet $\\sX \\to \\Spec O$ be a smooth proper morphism with geometrically connected fibers,\nand denote by $X$ the generic fiber.\nThen we have\n\\[ \\Im(\\langle \\cdot, \\cdot \\rangle : \n\\CH^1(X)_\\Tor \\times A_0(X) \\to k^\\times \\otimes \\Q/\\Z)\n\\subset\n\\Im(O^\\times \\otimes \\Q/\\Z \\to k^\\times \\otimes \\Q/\\Z).\n\\]\n\\end{proposition}",
      "prop:TateCv": "\\begin{proposition}\\label{prop:TateCv}\n\\begin{enumerate}\n\\item \nLet $a, b \\in k^\\times$ and $m>0$.\nIf $[a],  [b] \\in E_q(k)[m]$, we have\n\\[ \\langle [a], [b] \\rangle \n= a^{-s_m([b])} \\otimes \\frac{1}{m}\n= b^{-s_m([a])} \\otimes \\frac{1}{m}\n= q^{-s_m([a])s_m([b])} \\otimes \\frac{1}{m^2}.\n\\]\n\\item\nWe have $E_q(k)_\\Tor^\\is = \\{ [\\zeta] \\mid \\zeta \\in \\mu(k) \\}$.\n\\end{enumerate}\n\\end{proposition}",
      "cor:kodairatype": "\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}",
      "eq:pairing-intro": "\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}",
      "theorem: intrinsic torsion": "\\begin{theorem} \\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\Q$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\Z/2\\Z & 1,2\\\\\n    \\Z/3\\Z & 1,3\\\\\n    \\Z/4\\Z & 1,2,4 \\\\\n    \\Z/5\\Z & 1,5 \\\\\n    \\Z/6\\Z & 1,2,3 \\\\\n    \\Z/7\\Z & 1 \\\\\n    \\Z/8\\Z & 1,2 \\\\\n    \\Z/9\\Z & 1 \\\\\n    \\Z/10\\Z & 1\\\\\n    \\Z/12\\Z & 1 \\\\\n    \\Z/2\\Z\\times\\Z/2\\Z & 1,2\\\\\n    \\Z/4\\Z\\times\\Z/2\\Z & 1,2 \\ \\\\\n    \\Z/6\\Z\\times\\Z/2\\Z & 1 \\\\\n    \\Z/8\\Z\\times\\Z/2\\Z & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\Z/4\\Z \\times \\Z/2\\Z$,\nthere are two $\\Aut(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\Q$ \nfor which there is an isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$\nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n\\end{theorem}",
      "rem:FreyRuck": "\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}",
      "proposition: pairing 1-mod": "\\begin{theorem}\\label{proposition: pairing 1-mod}\n  Assume that $N\\in\\{4,5,6,7,8,9,10,12\\}$ and\n  let $M$ be a positive divisor of $N$.\n  Denote by $X_1(MN,N)$ the modular curve over $\\Q$ \n  associated to the congruence subgroup\n    $$\n    \\Gamma_1(MN,N):= \\Gamma_0(MN) \\cap \\Gamma_1(N)\n    =\\left\\{\\M abcd\\in\\SL(2,\\Z) : \\ \n      a,d\\equiv1\\bmod N, \\ MN|c \\right\\},\n    $$\n  on which the cusp $0$ is $\\Q$-rational.\n  Then $X_1(N)_M^+$ is isomorphic to $X_1(MN,N)$ over $\\Q$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2339,
    "pre_theorem_intro_text": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$)  is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.",
    "context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}",
    "full_context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\Pic(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\gen{P,Q}:=\\gen{[P-O], [Q-O]}$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\Pic(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\Q)_\\Tor,E(\\Q)_\\Tor^\\is)$ for an elliptic curve $E$ over $\\Q$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\Q)_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\Z/2N\\Z) \\times (\\Z/2\\Z)$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\Q)_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\nWe briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\Q$ on\nwhich the cusp $0$ is $\\Q$-rational. For a subfield $k$ of $\\C$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\C/(\\Z\\tau+\\Z),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\Q)$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\Z):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\ul{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve \n$X_1^0(N, 2)_{\\ul{M}}^\\pm$ over $\\Q$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\Q)$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\Q$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\ul{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\n\\begin{lemma} \\label{lemma: ENt}\n \\begin{enumerate}\n \\item Assume that the characteristic of $k$ is not $2$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational point\n $P$ of order two, then $(E,P)$ is isomorphic to $(E_{2,t,a},(0,0))$\n for some $a,t\\in k^\\times$, where\n \\begin{equation} \\label{eq: E2ta}\n E_{2,t,a}:\n \\begin{cases}\n \\displaystyle y^2=x\\left(x^2+ax+\\frac{a^2t}{4(t+1)}\\right),\n &\\text{if }t\\neq-1, \\\\\n y^2=x(x^2+a), &\\text{if }t=-1.\n \\end{cases}\n \\end{equation}\n Moreover, two elliptic curves $(E_{2,t,a},(0,0))$ and\n $(E_{2,t',a'},(0,0))$ are isomorphic over $k$ if and only\n if $t=t'\\neq-1$ and $a/a'$ is a square in $k^\\times$\n or $t=t'=-1$ and $a/a'\\in(k^\\times)^4$ . \n \\item Assume that the characteristic of $k$ is not $3$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational\n point $P$ of order $3$, then $(E,P)$ is isomorphic to $(E_{3,t},(0,0))$\n for some $t\\neq-1$ in $k^\\times$, where\n \\begin{equation} \\label{eq: E3t}\n E_{3,t}:y^2+xy+\\frac t{27(t+1)}y=x^3\n \\end{equation}\n or isomorphic to $(E_{3,-1,a},(0,0))$ for some $a\\in k^\\times$,\n where\n \\begin{equation} \\label{eq: E3a}\n E_{3,-1,a}:y^2+ay=x^3.\n \\end{equation}\n Two elliptic curves $(E_{3,t},(0,0))$ and\n $(E_{3,t'},(0,0))$ with $t,t'\\neq-1$ are isomorphic if and only\n if $t=t'$. Also, $(E_{3,-1,a},(0,0))\\simeq(E_{3,-1,a'},(0,0))$\n if and only if $a/a'\\in(k^\\times)^3$. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Consider first the case $N=2$. \nWe start with the equation \\eqref{eq: E2}\nand introduce a new parameter\n $t=4b/(a^2-4b)$ (so $b=a^2t/4(t+1)$). When $a\\neq0$, we have\n $t\\neq-1$ and the equation \\eqref{eq: E2} in the new parameters is\n $y^2=x(x^2+ax+a^2t/4(t+1))$. When $a=0$, we change the notation $b$\n in \\eqref{eq: E2} to $-a$ and get $y^2=x(x^2-a)$. Now it is easy to\n see that an isomorphism\n $\\phi:(E_{2,t,a},(0,0))\\to(E_{2,t',a'},(0,0))$ must be of the form\n $\\phi(x,y)=(u^2x,u^3y)$ (c.f. \\cite[Proposition 3.1]{sil}). From\n this, we immediately get the condition for $E_{2,t,a}$ and\n $E_{2,t',a'}$ to be isomorphic.\n\nThis leaves us with the cases\n$(N, M) \\in \\{ (6,6), (8, 4), (9,3), (10, 2), (12, 2) \\}$ with $\\epsilon=-$.\n(For $(N, M)=(9,3)$, the value of $\\epsilon$ does not matter\nbecause of Remark \\ref{rem:plus-minus} (1).)\nIn all cases, there exists \nan isomorphism $\\alpha : E \\to X_1(N)_M^\\epsilon$,\nwhere\n$E$ is an elliptic curve over $\\Q$\nwith finitely many $\\Q$-rational points,\nas exhibited in the following table,\nwhere we set\n$h_1(x)=(1-x)(1+x)^4(1+x-x^2)^2$ and\n$h_2(x)=(1+x)(1+x+x^2)(1+x^2)^2(1-x)^3$.\n $$ \\extrarowheight3pt\n \\begin{array}{c|c|c|c} \\hline\\hline\n (N, M) & E & E(\\Q) \\setminus \\{ O \\} & \n(s, t)=\\alpha(x,y) \\\\ \\hline\n(6,6) & y^2=x^3-1 & (1,0) &\n\\left(-\\frac{xy}{1-x^3},\\frac1{1-x^3}\\right)\n\\\\\n(8,4) & y^2=x^3+x & (0,0) &\n\\left(\\frac y{x^2}(1-\\frac{1}{x^2}),-\\frac1{x^2}\\right)\n\\\\\n(9,3) & y^2+y=x^3 & (0,0), (0,-1) &\n\\left(xy(y^2+y+1),y+1\\right)\n\\\\\n(10,2) & y^2=x^3-x^2-x & (0,0) &\n\\left(yh_1(x),x\\right)\n\\\\\n(12,2) & y^2=x^3+x^2+x & (0,0) &\n\\left(yh_2(x),x\\right)\n\\\\\n \\hline\\hline\n \\end{array}\n $$\nSince all points of $\\alpha(E(\\Q))$ are cusps \nby Lemma \\ref{lemma: uniformizers},\nwe get $Y_1(\\Q)^\\epsilon_M=\\emptyset$ in these cases as well.\nWe are done.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}\n\\begin{proof}\nThe assumption implies that\nthe base change of $E$ to $k_v$ is \nisomorphic to the Tate curve $E_q=\\G_m/q^\\Z$\nfor some $q \\in k_v$ such that $v(q)=n$\n(\\cite[Chapter V, Theorem 5.3]{sil})\nLet us write the image of $P$ in $E_q(k_v)$ as $[a]$ with $a \\in k_v$.\nSince $\\mu_m(k_v)= \\{ 1 \\}$, \nwe find from \\eqref{eq:ex-TateCv} that\n$E_q(k_v)[m]$ is a cyclic group of order $m$ generated by $[a]$.\nIn particular, $s:= s_m([a])$ is invertible in $\\Z/m\\Z$.\nWe have $\\langle [a], [a] \\rangle = 0$ since $P \\in E(k)_\\Tor^\\is$.\nOn the other hand,\nProposition \\ref{prop:TateCv} (1) shows\n$\\langle [a], [a] \\rangle = q^{-s^2} \\otimes (1/m^2)$.\nBy looking at its image under the map\n$v \\otimes \\id_{\\Q/\\Z} : k_v^\\times \\otimes \\Q/\\Z \\to \\Q/\\Z$,\nwe conclude that $-s^2 n/m^2 = 0$ holds in $\\Q/\\Z$,\nthat is, $m^2$ divides $n$.\n\\end{proof}",
    "post_theorem_intro_text_len": 4805,
    "post_theorem_intro_text": "We briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\mathbb{Q}$ on\nwhich the cusp $0$ is $\\mathbb{Q}$-rational. For a subfield $k$ of $\\mathbb{C}$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\mathbb H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\mathbb{C}/(\\mathbb{Z}\\tau+\\mathbb{Z}),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$. \n\nFor a pair $(N, M)$ of $N \\in \\{4, \\dots, 10, 12\\}$ and a positive divisor $M$ of $N$,\nwe will construct a smooth projective curve $X_1 (N)^\\pm_M$ over $\\mathbb{Q}$ equipped with\na finite morphism $X_1(N)_M^\\pm \\to X_1(N)$\nthat enjoys the following property:\nwhen $t \\in X_1(N)(\\mathbb{Q})$ corresponds to a pair $(E, P)$ as above (with $k=\\mathbb{Q}$),\none has \n\\[\n\\langle P, (N/M)P \\rangle = 0 \\Leftrightarrow\nt \\in \\operatorname{Im}(X_1(N)^\\pm_M(\\mathbb{Q}) \\to X_1(N)(\\mathbb{Q})).\n\\]\nIt is worth  mentioning that\n$X_1(N)^+_M$ is isomorphic to the modular curve\n$X_1(MN, N)$ associated to $\\Gamma_1(MN, N):=\\Gamma_0(MN) \\cap \\Gamma_1(N)$\n(Theorem \\ref{proposition: pairing 1-mod})\nand $X_1(N)^-_M$ is its twist.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\mathbb{Z}):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\underline{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve  \n$X_1^0(N, 2)_{\\underline{M}}^\\pm$ over $\\mathbb{Q}$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\mathbb{Q})$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\underline{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\nIn the last section \\S \\ref{sect:high-dim},\nwe generalize the pairing \\eqref{eq:pairing-intro}\nand the intrinsic subgroup \nto higher dimensional varieties.\nThis new construction enables us to prove that,\nif $X$ has good reduction with respect to a discrete valuation of $k$,\nthere is a strong restriction on the values of the pairing \n(see Proposition \\ref{prop:two-def-same}).\nFinally, as a sample for the case of bad reduction,\nwe compute the intrinsic subgroup of \nTate elliptic curves (see Proposition \\ref{prop:TateCv}).\nAs an application,\nwe show that the intrinsic subgroup of an elliptic curve over a number field\nimposes some constraint on its reduction type\n(see Corollary \\ref{cor:kodairatype}).\n\n\\subsection*{Acknowledgement}\nIt is Kenneth Ribet who formulated \\eqref{eq:pairing-intro} as a pairing \nand asked if it is symmetric.\nThe authors express deep gratitude to him for asking this question,\nwhich was a starting point of our work.\n\nThe first author (T. Y.) is supported by JSPS KAKENHI Grant (25K06961). \nThe second author (Y. Y.) is supported by \nGrant 113-2115-M-002-003-MY3 of the National Science and Technology\n  Council of the Republic of China (Taiwan). \nThe third author (H. Y.) is supported by National Research Foundation of Korea(NRF) grant \nfunded by the Korea government(MSIT) (No. RS-2023-00239918  and No. 2020R1A5A1016126).\nThe fourth author (M. Y.) is supported by the National Research Foundation of Korea (NRF) grant \nfunded by the\nKorea government (MSIT) (RS-2025-23525445).\n\n\\subsection*{Notation and convention}\nFor an abelian group $A$, \nwe write $A[n]$ and $A/n$ for the kernel and cokernel\nof $A \\to A, \\ a \\mapsto na$ for each $n \\in \\mathbb{Z}$,\nand put $A_\\Tor := \\cup_{n > 0} A[n]$.\nFor a field $k$,\nwe set $\\mu_n(k):=(k^\\times)[n]$ and $\\mu(k):=(k^\\times)_\\Tor$.\n\nThe identity element of an elliptic curve is denoted by $O$.",
    "sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that “Therefore Theorem~\\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$,” and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”",
    "expanded_sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that, in establishing the main theorem, everything is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$, and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”",
    "expanded_theorem": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.,",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let $E$ be an elliptic curve over $\\mathbb{Q}$ with identity element $O$. Via the identification $P \\mapsto [P-O]$, identify $E(\\mathbb{Q})_{\\mathrm{Tor}}$ with $\\operatorname{Pic}(E)_{\\mathrm{Tor}}$. Let\n\\[\n\\langle\\cdot,\\cdot\\rangle: E(\\mathbb{Q})_{\\mathrm{Tor}}\\times E(\\mathbb{Q})_{\\mathrm{Tor}}\\to \\mathbb{Q}^\\times\\otimes \\mathbb{Q}/\\mathbb{Z}\n\\]\nbe the induced biadditive symmetric pairing, and define the intrinsic subgroup by\n\\[\nE(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}:=\\{P\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\mid \\langle P,Q\\rangle=0\\text{ for all }Q\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\}.\n\\]\nWhich existence statement holds for the pair consisting of the torsion subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}$ and its intrinsic subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}$?",
      "correct_choice": {
        "label": "A",
        "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exists at least one elliptic curve $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        },
        {
          "label": "C",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are among the following: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$."
        },
        {
          "label": "D",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        },
        {
          "label": "E",
          "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and both classes occur simultaneously for every elliptic curve $E/\\mathbb{Q}$ with $E(\\mathbb{Q})_{\\mathrm{Tor}}\\cong A$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "infinitely_many_realizations",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped_exactness_and_infinite_realization_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "computational_check",
          "tampered_component": "order_4_subgroup_for_Z4xZ2",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "outer_automorphism",
          "tampered_component": "existential_occurrence_of_two_Aut_conjugacy_classes",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the pairing and intrinsic subgroup but does not reveal the classification or the correct existence statement. No explicit or obvious hint singles out the correct option."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially a theorem-recognition question: the correct choice is a near-verbatim classification/existence statement, and the task is to spot the exact theorem among lightly altered variants."
      },
      "GPS": {
        "score": 1,
        "justification": "The options are subtle and require careful comparison, but the pressure is mainly on recalling or recognizing a detailed theorem statement rather than generating a conclusion from the definitions in the stem."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they weaken exactness, alter realizability, change allowed subgroup orders, or overstate conjugacy-class behavior. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-recognition MCQ with no answer leakage and strong distractors, but it scores lower on tautology avoidance and genuine generative reasoning because it mainly tests recall of an exact classification statement."
    }
  },
  {
    "id": "2512.00802v1",
    "paper_link": "http://arxiv.org/abs/2512.00802v1",
    "theorems_cnt": 8,
    "theorem": {
      "env_name": "thm",
      "content": "[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.",
      "start_pos": 5733,
      "end_pos": 5899,
      "label": "thm:Ar"
    },
    "ref_dict": {
      "thm:Ar": "\\begin{thm}[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set. \n\\end{thm}",
      "thm:F": "\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}",
      "thm:main": "\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}",
      "def:Arak": "\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}"
    },
    "pre_theorem_intro_text_len": 2314,
    "pre_theorem_intro_text": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible. \n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace  f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace  f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}",
    "context": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}",
    "full_context": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}\n\n\\begin{abstract}\nArakelian's classical approximation theorem \\cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\\subset \\mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem. \n\\end{abstract}\n\nApproximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}\n\n\\subsection{Main result}\n\nWe can now state the main result:\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}\nAt first glance, the existence of logarithmic branches of functions $f\\in A(F)$ might seem to have little to do with the conditions in Definition \\ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \\cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\\supset F$ whose connected components are simply connected.\n\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}\nAt least one of the directions in Theorem \\ref{thm:main} is clearer now. If a function $f\\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.\n\nTheorem \\ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\\subset \\mathbb{C}$ \\cite{Ar2} and non-planar Riemann surfaces of finite genus \\cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \\ref{def:Arak}) are not always sets of uniform approximation, see \\cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \\cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.\n\nThe one direction of the characterization in Theorem \\ref{thm:main} is contained in \\cite{GP}. We include the proof here for the sake of completeness.\n\\begin{prp}[Gauthier-Pouryayevali \\cite{GP}]\\label{prop:GP}\nSuppose that the closed set $F\\subset \\mathbb{C}$ is an Arakelian set. For every function $f\\in A(F)$ with no zeros in $F$, there exists a function $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{prp}\n\\begin{proof}\nBy Tietze's extension theorem, there exists a continuous extension of $f$ in $\\mathbb{C}$, which we\ndenote by $\\widetilde{f}:\\mathbb{C}\\to\\C$. Our assumption implies that the open set $U=\\mathbb{C}\\setminus \\widetilde{f}^{-1}(\\{0\\})$ contains $F$.\nThus, by Theorem \\ref{thm:F} there exists a simply connected open set $V$ with $F\\subset V\\subset U$.\nIf we consider the covering map $\\exp: \\mathbb{C}\\to\\mathbb{C}\\setminus\\{0\\}$, then the latter implies that\n$\\widetilde{f}_{\\big|V}:V\\to\\mathbb{C}\\setminus\\{0\\}$ can be lifted\nto a continuous function $\\widetilde{g}:V\\to\\mathbb{C}$, such that $\\widetilde{f}_{\\big|V}=e^{\\widetilde{g}}$.\nThe function $g=\\widetilde{g}_{\\big|F}$ is obviously continuous and $e^g=f$ in $F$.\nSince $f_{\\big|F^0}$ is holomorphic, by Lemma \\ref{lem:Car}, $g$ is also holomorphic in $F^0$.\nThus, $g\\in A(F)$ and the proof of the proposition is complete.\n\\end{proof}\nThe reverse direction is contained in the following proposition that, combined with Proposition \\ref{prop:GP}, completes the proof of Theorem \\ref{thm:main}. \n\\begin{prp}\\label{prop:FNP}\nLet $F\\subset\\mathbb{C}$ be a closed set, such that for every $f\\in A(F)$ with no zeros in $F$, there exists a $g\\in A(F)$ satisfying $e^g=f$ in $F$. Then $F$ is an Arakelian set.\n\\end{prp}\n\\begin{proof}\n{\\it Step 1. $F$ has no holes.} Arguing by contradiction, suppose that $F$ has a hole $B$. Let $\\zeta\\in B$ and let $f(z)=z-\\zeta$. Since $f$ has no zeros in $F$, there exists $g\\in A(F)$, such that $e^g=f$ in $F$.\n\n\\begin{thm}[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set. \n\\end{thm}\n\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}",
    "post_theorem_intro_text_len": 3824,
    "post_theorem_intro_text": "\\subsection{Main result}\n\nWe can now state the main result:\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}\nAt first glance, the existence of logarithmic branches of functions $f\\in A(F)$ might seem to have little to do with the conditions in Definition \\ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \\cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\\supset F$ whose connected components are simply connected.\n\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}\nAt least one of the directions in Theorem \\ref{thm:main} is clearer now. If a function $f\\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.\n\nWe should give credit here to the influential work of Gauthier-Pouryayevali \\cite{GP}, which inspired us to pursue the above characterizations. In fact, the directions in both Theorems \\ref{thm:main}, \\ref{thm:F} where we assume that $F$ is an Arakelian set, are contained in \\cite{GP}.\n\nOur contribution in the present paper is to prove the reverse direction stated in Theorem \\ref{thm:main}. We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption. Interestingly, the function $f$ that we use is entire and it is produced by invoking the Weierstrass factorization theorem, unlike the meromorphic functions that are usually employed in such proofs \\cite[Chapter IV, \\S2.$C_2$]{Gaier}. \n\n\\subsection{Outlook}\n\nTheorem \\ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\\subset \\mathbb{C}$ \\cite{Ar2} and non-planar Riemann surfaces of finite genus \\cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \\ref{def:Arak}) are not always sets of uniform approximation, see \\cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \\cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface. \n\nIt would be interesting to seek potential extensions of Theorem \\ref{thm:main} to Arakelian sets of planar domains $G\\subset \\mathbb{C}$, as in \\cite{Ar2}. The present logarithmic characterization and our method of proof could be naturally extended to Arakelian sets of simply connected domains $G\\subset\\mathbb{C}$, see \\cite[Corollary 2.15]{F} for one of the directions. However, we do not know how it could be generalized to planar domains $G$ with holes, even when $G$ is an annulus. Finally, coming up with a generalization of our characterization that would take into account the analytic structure of a given domain, could hopefully be of use for characterizing uniform approximation sets in Riemann surfaces.\n\n\\subsection{Acknowledgments}\n\nWe would like to thank Paul M. Gauthier for his interest in this work and insightful comments.\nG.F. and S.P. would like to acknowledge the support of the ERC starting grant 101078061 SINGinGR, under the European Union's Horizon Europe program for research and innovation, and the H.F.R.I. grant 7126, under the 3rd call for H.F.R.I. research projects to support post-doctoral researchers.",
    "sketch": "For the direction assuming $F$ is Arakelian, the text explains that using Theorem~\\ref{thm:F} (Arakelian $\\Leftrightarrow$ a neighborhood basis of simply connected open sets), “at least one of the directions in Theorem \\ref{thm:main} is clearer now”: if $f\\in A(F)$ “could be extended to a neighborhood of $F$,” then “the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter,” yielding $g$ with $e^g=f$ on $F$.\n\nFor the reverse direction in Theorem~\\ref{thm:main}, the authors state: “We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption.” They add that this $f$ “is entire and it is produced by invoking the Weierstrass factorization theorem,” in contrast to “the meromorphic functions that are usually employed in such proofs.”",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.",
    "theorem_type": [
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Let $F\\subset \\mathbb{C}$ be a closed set, and let\n\\[\nA(F)=\\{f:F\\to\\mathbb{C}: f \\text{ is holomorphic on } F^\\circ \\text{ and continuous on } F\\}.\n\\]\nSay that $F$ is a set of uniform approximation if for every $\\varepsilon>0$ and every $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\varepsilon$ for all $z\\in F$. Which of the following statements is equivalent to $F$ being a set of uniform approximation?",
      "correct_choice": {
        "label": "A",
        "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\\subset \\mathbb{C}$, the union of all holes of $F\\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and there exists a compact set $K\\subset \\mathbb{C}$ such that the union of all holes of $F\\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."
        },
        {
          "label": "C",
          "text": "$F$ has no holes; equivalently, $\\mathbb{C}\\setminus F$ has no bounded connected components."
        },
        {
          "label": "D",
          "text": "$F$ is an Arakelian set; that is, $\\mathbb{C}\\setminus F$ is connected and, for every compact set $K\\subset \\mathbb{C}$, the complement $\\mathbb{C}\\setminus (F\\cup K)$ is connected."
        },
        {
          "label": "E",
          "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\\subset \\mathbb{C}$, the union of all connected components of $\\mathbb{C}\\setminus (F\\cup K)$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "universal quantifier over compact sets",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the condition on holes of $F\\cup K$ for all compact $K$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "replaced boundedness of holes of $F\\cup K$ by connectedness of full complement after adjoining $K$",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "boundedness applies only to holes, not to all components of the complement",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the approximation property but does not explicitly name or describe the Arakelian characterization. The correct answer is not leaked directly; the test-taker must know or infer the geometric criterion."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-recall item: one side of the equivalence is given in the stem and the other side is to be selected. It is not a verbatim restatement, but it is still a fairly direct reformulation of Arakelian's theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the distractors differ by subtle quantifier changes and related topological conditions. However, the item mainly rewards recognition of the exact theorem statement rather than deeper generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible: one weakens a universal quantifier to an existential one, one gives a weaker necessary condition, one confuses bounded-hole conditions with connectedness, and one overstates boundedness to all complement components."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-characterization MCQ with good distractors and no obvious answer leakage, but it leans more toward precise recall/discrimination than genuine generative reasoning."
    }
  },
  {
    "id": "2512.01137v1",
    "paper_link": "http://arxiv.org/abs/2512.01137v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.",
      "start_pos": 17951,
      "end_pos": 18076,
      "label": "th:lambda-to-0"
    },
    "ref_dict": {
      "r:comb-nondegenerate": "\\begin{remark}\\label{r:comb-nondegenerate}\nNote that\nthis triangulation, as well as a triangulation from theorem~\\ref{th:lambda-to-0}, is combinatorial\n(the star of every vertex has simplicial embedding into $\\R^n$, so the stars form a $PL$-atlas of~$\\SS^n$),\nsince the join of combinatorial triangulations of spheres is a combinatorial triangulation of a sphere.\n\nMoreover, these triangulations are isomorphic to the boundary of a convex polytope simplicial in $\\R^{n+1}$.\nIndeed, if $P\\subset\\R^{k+1}$ and $Q\\subset\\R^{m+1}$ are convex polytopes containing the origins in their interiors,\nthen the join $S=\\partial P\\star\\partial Q$ is canonically embedded into $\\R^{k+m+2}=\\R^{k+1}\\times\\R^{m+1}$.\nIt remains to notice that $S$ is the boundary of the cone over $S$ with the vertex at the origin\n(which in fact coincides with $\\mathrm{conv}(S)\\subset\\R^{k+m+2}$).\n\\end{remark}",
      "th:f-vectors": "\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\SS^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}",
      "th:lambda-to-0": "\\begin{theorem}\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2052,
    "pre_theorem_intro_text": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:",
    "context": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:",
    "full_context": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:\n\n\\begin{abstract}\nFor positive integers $n,d$, let $\\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere\nwhich admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex.\nWe show that $\\lim_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$ for any $n\\ge3$,\ndisproving O.\\,Musin's conjecture.\nUsing similar idea, for any $C$ we construct a triangulation of $\\SS^n$, $n\\ge3$,\nfor which $\\frac{f_j}{f_i}>C$,\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac{n-1}2\\rfloor$.\nAll triangulations we obtain are isomorphic to boundaries of convex polytopes in $\\R^{n+1}$.\n\\end{abstract}\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:\n\nThe key idea of the proof of theorem~\\ref{th:lambda-to-0}\nis to construct a triangulation of $\\SS^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.\nThis idea can also be applied to show the following fact.\n\n\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\SS^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}\n\n\\begin{proof}[Proof of theorem~\\ref{th:lambda-to-0}]\nLet us first prove the case $n>3$.\nBy corollary~\\ref{c:n-est}, for any $k_1,k_2>0$ we have\n$$\\frac{\\lambda(n,k_1k_2)}{k_1k_2}\\le\\frac{3k_1+3k_2+n-2}{k_1k_2}.$$\nIn particular, for any $k>0$, the value\n$$\\frac{\\lambda(n,k^2)}{k^2}\\le\\frac{6k+n-2}{k^2}$$\nis arbitrary small.\nNamely, take any $\\eps>0$.\nIf $k^2>4\\cdot\\frac{n-2}\\eps$ and $k>\\frac{24}\\eps$,\nthen\n$$\\frac{\\lambda(n,k^2)}{k^2}\\le\\frac{6k+n-2}{k^2}=\\frac{n-2}{k^2}+\\frac6k<\\textstyle\\frac24\\eps.$$\nFurther, note that for $l=0,1,\\ldots,k-1$ we have\n$$\n\\frac{\\lambda(n,k(k+l))}{k(k+l)}\n\\le\n\\frac{\\lambda(n,k^2)}{k(k+l)}+\\frac{3l}{k(k+l)}\n\\le\n\\frac{\\lambda(n,k^2)}{k^2}+\\frac{3}{k}\n<\\textstyle\\frac34\\eps.\n$$\nFinally, for any $s=0,1,\\ldots,k-1$\nwe can apply proposition~\\ref{pr:d+1} $s$ times, to obtain\n$$\n\\frac{\\lambda(n,k(k+l)+s)}{k(k+l)+s}\n\\le\n\\frac{\\lambda(n,k(k+l))}{k(k+l)}+\\frac{3s}{k(k+1)}\n<\\eps.\n$$\nWe have shown that $\\frac{\\lambda(n,d)}d<\\eps$ for every $d>\\max\\{4,\\frac{n-2}\\eps,\\frac{24}\\eps\\}$.\n\n\\begin{remark}\nIn order to prove theorem~\\ref{th:lambda-to-0},\none can use the join of $h$ copies of $\\SS^1$, for $h=\\lfloor\\frac{n+1}2\\rfloor$\n(as in the proof of theorem~\\ref{th:f-vectors}), with the additional join with $\\SS^0$ for even $n$.\nIn this way one can show that $\\lim_{d\\to\\infty}\\frac{\\lambda(n,k)^{h-1}}d=0$.\n\\end{remark}\n\nWe have estimated the asymptotic of $\\lambda(n,d)$ as $d\\to\\infty$.\nHowever, finding its exact values appears to be a difficult problem.\nThe estimates for $\\lambda(n,d)$ obtained in the proof of theorem~\\ref{th:lambda-to-0}\nare apparently far from the minimum.\n\n\\begin{question}\nIs it true that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)^{\\lfloor\\frac{n+1}2\\rfloor}}{d}\\ne0$?\n\\end{question}\n\n\\begin{theorem}\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 954,
    "post_theorem_intro_text": "The key idea of the proof of theorem~\\ref{th:lambda-to-0}\nis to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.\nThis idea can also be applied to show the following fact.\n\nFor a finite simplicial complex $K$,\nits {\\it $f$-vector} is the sequence $f_0,f_1,\\ldots$,\nwhere $f_k$ is the number of $k$-simplices in $K$.\n\n\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\mathbb{S}^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}\n\nNote that the estimations from theorems~\\ref{th:lambda-to-0} and \\ref{th:f-vectors}\nhold even if we\nconsider\nonly {\\it combinatorial} triangulations.\nMoreover, we can consider only triangulations which are {\\it isomorphic to the boundary of a convex simplicial polytope in $\\mathbb{R}^{n+1}$}.\nSee remark~\\ref{r:comb-nondegenerate}.",
    "sketch": "The key idea of the proof of theorem~\\ref{th:lambda-to-0} is to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.",
    "expanded_sketch": "The key idea in establishing the main theorem is to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.,",
    "expanded_theorem": "\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.",
    "theorem_type": [
      "Asymptotic or Limit",
      "Universal"
    ],
    "mcq": {
      "question": "For positive integers n and d, let λ(n,d) denote the minimal number of vertices of a triangulation K of the n-sphere S^n such that K admits a degree-d simplicial map to the standard n-sphere S^n_{n+2}, equivalently to the boundary of an (n+1)-simplex. Which statement holds for every integer n \\ge 3?",
      "correct_choice": {
        "label": "A",
        "text": "For every n \\ge 3, \\(\\lim_{d\\to\\infty} \\frac{\\lambda(n,d)}{d} = 0\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(n \\ge 3\\), there exists a constant \\(c_n>0\\) such that for all sufficiently large \\(d\\), \\(\\frac{\\lambda(n,d)}{d} \\ge c_n\\)."
        },
        {
          "label": "C",
          "text": "For every \\(n \\ge 3\\), there exists a sequence \\(d_m\\to\\infty\\) such that \\(\\lim_{m\\to\\infty} \\frac{\\lambda(n,d_m)}{d_m}=0\\)."
        },
        {
          "label": "D",
          "text": "There exists a constant \\(C>0\\) such that for every \\(n \\ge 3\\) and all sufficiently large \\(d\\), \\(\\frac{\\lambda(n,d)}{d} \\le C\\)."
        },
        {
          "label": "E",
          "text": "For every \\(n \\ge 3\\), there exists an integer \\(d_0=d_0(n)\\) such that for every \\(d\\ge d_0\\) one has \\(\\lambda(n,d)=o\\!\\left(d\\right)\\) uniformly in \\(d\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "sublinear-vanishing asymptotic replaced by positive linear lower bound",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "full limit over all d weakened to convergence along a subsequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "vanishing ratio weakened to merely bounded ratio with uniform constant",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "misstated asymptotic notion by imposing eventual equality with an o(d) condition",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the asymptotic conclusion; it only defines the quantity and asks which limiting statement is true."
      },
      "TAS": {
        "score": 0,
        "justification": "The item appears to ask almost exactly for the theorem's main asymptotic conclusion, with choice A giving the direct statement rather than requiring application in a new setting."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish a full limit from weaker or altered asymptotic claims (e.g. liminf, limsup, positive constant, uniform-in-n bound), but the task is still largely theorem recognition rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and distinct: they test common confusions between limit/liminf/limsup, zero versus positive asymptotic rate, and pointwise-in-n versus uniform-in-n statements."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with good distractors and no answer leakage, but it is fairly tautological and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.01137v1",
    "paper_link": "http://arxiv.org/abs/2512.01137v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.",
      "start_pos": 17951,
      "end_pos": 18076,
      "label": "th:lambda-to-0"
    },
    "ref_dict": {
      "r:comb-nondegenerate": "\\begin{remark}\\label{r:comb-nondegenerate}\nNote that\nthis triangulation, as well as a triangulation from theorem~\\ref{th:lambda-to-0}, is combinatorial\n(the star of every vertex has simplicial embedding into $\\R^n$, so the stars form a $PL$-atlas of~$\\SS^n$),\nsince the join of combinatorial triangulations of spheres is a combinatorial triangulation of a sphere.\n\nMoreover, these triangulations are isomorphic to the boundary of a convex polytope simplicial in $\\R^{n+1}$.\nIndeed, if $P\\subset\\R^{k+1}$ and $Q\\subset\\R^{m+1}$ are convex polytopes containing the origins in their interiors,\nthen the join $S=\\partial P\\star\\partial Q$ is canonically embedded into $\\R^{k+m+2}=\\R^{k+1}\\times\\R^{m+1}$.\nIt remains to notice that $S$ is the boundary of the cone over $S$ with the vertex at the origin\n(which in fact coincides with $\\mathrm{conv}(S)\\subset\\R^{k+m+2}$).\n\\end{remark}",
      "th:f-vectors": "\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\SS^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}",
      "th:lambda-to-0": "\\begin{theorem}\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2052,
    "pre_theorem_intro_text": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:",
    "context": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:",
    "full_context": "\\label{s:introduction}\n\nSuppose $f:M\\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\\ge1$.\nThe {\\it degree} $\\deg f$ is an integer $d$ such that the $n$-th homology homomorphism\n$f_*:H_n(M;\\mathbb{Z})=\\mathbb{Z}\\to\\mathbb{Z}=H_n(N;\\mathbb{Z})$ is the multiplication by $d$.\nIf the map $f$ is smooth, then $\\deg f$ equals the number of preimages of any regular value of $f$,\nwhere the points at which $df$ preserves the orientation are counted as ``$1$'',\nand the points at which $df$ reverses the orientation are counted as ``$-1$''.\nIf $f$ is a simplicial map of triangulated manifolds,\nthen the degree can be counted in a similar way via preimages of an $n$-simplex.\n\nNamely, we call an $n$-simplex $\\sigma\\subset M$ {\\it positive/negative},\nif $f(\\sigma)\\subset N$ is non-degenerate $n$-simplex and $f|_\\sigma$ preserves/reverses orientation.\nThen $\\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\\tau\\subset N$.\nThis difference does not depend on $\\tau$.\n\nLet $\\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices\nand suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\\mathbb{S}^n$.\nSuppose there is a simplicial map $K\\to\\mathbb{S}^n_{n+2}$ of degree $d$.\nOne may ask: {\\it given $n$ and $d$, how few vertices can $K$ have?}\nDenote the minimal number of vertices of such $K$ by $\\lambda(n,d)$.\n\nEvidently, if $d=0$ or $d=\\pm1$, we have $\\lambda(n,d)=n+2$,\nsince a triangulation of the $n$-sphere cannot have fewer vertices.\nIt is also easy to see that $\\lambda(1,d)=3d$.\nFor $n=2$ and $d\\ge3$ it is known by \\cite{sarkaria} that $\\lambda(2,d)=2d+2$\n(the lower bound for $\\lambda(2,d)$ obviously follows from the Euler's formula: \n$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:\n\n\\begin{abstract}\nFor positive integers $n,d$, let $\\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere\nwhich admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex.\nWe show that $\\lim_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$ for any $n\\ge3$,\ndisproving O.\\,Musin's conjecture.\nUsing similar idea, for any $C$ we construct a triangulation of $\\SS^n$, $n\\ge3$,\nfor which $\\frac{f_j}{f_i}>C$,\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac{n-1}2\\rfloor$.\nAll triangulations we obtain are isomorphic to boundaries of convex polytopes in $\\R^{n+1}$.\n\\end{abstract}\n\nIn \\cite{musin-deg} K.\\,Apolonskaya and O.\\,Musin\nconjectured that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)}d=\\frac{n+2}n$ for all $n>0$.\nWe disprove this conjecture:\n\nThe key idea of the proof of theorem~\\ref{th:lambda-to-0}\nis to construct a triangulation of $\\SS^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.\nThis idea can also be applied to show the following fact.\n\n\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\SS^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}\n\n\\begin{proof}[Proof of theorem~\\ref{th:lambda-to-0}]\nLet us first prove the case $n>3$.\nBy corollary~\\ref{c:n-est}, for any $k_1,k_2>0$ we have\n$$\\frac{\\lambda(n,k_1k_2)}{k_1k_2}\\le\\frac{3k_1+3k_2+n-2}{k_1k_2}.$$\nIn particular, for any $k>0$, the value\n$$\\frac{\\lambda(n,k^2)}{k^2}\\le\\frac{6k+n-2}{k^2}$$\nis arbitrary small.\nNamely, take any $\\eps>0$.\nIf $k^2>4\\cdot\\frac{n-2}\\eps$ and $k>\\frac{24}\\eps$,\nthen\n$$\\frac{\\lambda(n,k^2)}{k^2}\\le\\frac{6k+n-2}{k^2}=\\frac{n-2}{k^2}+\\frac6k<\\textstyle\\frac24\\eps.$$\nFurther, note that for $l=0,1,\\ldots,k-1$ we have\n$$\n\\frac{\\lambda(n,k(k+l))}{k(k+l)}\n\\le\n\\frac{\\lambda(n,k^2)}{k(k+l)}+\\frac{3l}{k(k+l)}\n\\le\n\\frac{\\lambda(n,k^2)}{k^2}+\\frac{3}{k}\n<\\textstyle\\frac34\\eps.\n$$\nFinally, for any $s=0,1,\\ldots,k-1$\nwe can apply proposition~\\ref{pr:d+1} $s$ times, to obtain\n$$\n\\frac{\\lambda(n,k(k+l)+s)}{k(k+l)+s}\n\\le\n\\frac{\\lambda(n,k(k+l))}{k(k+l)}+\\frac{3s}{k(k+1)}\n<\\eps.\n$$\nWe have shown that $\\frac{\\lambda(n,d)}d<\\eps$ for every $d>\\max\\{4,\\frac{n-2}\\eps,\\frac{24}\\eps\\}$.\n\n\\begin{remark}\nIn order to prove theorem~\\ref{th:lambda-to-0},\none can use the join of $h$ copies of $\\SS^1$, for $h=\\lfloor\\frac{n+1}2\\rfloor$\n(as in the proof of theorem~\\ref{th:f-vectors}), with the additional join with $\\SS^0$ for even $n$.\nIn this way one can show that $\\lim_{d\\to\\infty}\\frac{\\lambda(n,k)^{h-1}}d=0$.\n\\end{remark}\n\nWe have estimated the asymptotic of $\\lambda(n,d)$ as $d\\to\\infty$.\nHowever, finding its exact values appears to be a difficult problem.\nThe estimates for $\\lambda(n,d)$ obtained in the proof of theorem~\\ref{th:lambda-to-0}\nare apparently far from the minimum.\n\n\\begin{question}\nIs it true that $\\lim\\sup_{d\\to\\infty}\\frac{\\lambda(n,d)^{\\lfloor\\frac{n+1}2\\rfloor}}{d}\\ne0$?\n\\end{question}\n\n\\begin{theorem}\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 954,
    "post_theorem_intro_text": "The key idea of the proof of theorem~\\ref{th:lambda-to-0}\nis to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.\nThis idea can also be applied to show the following fact.\n\nFor a finite simplicial complex $K$,\nits {\\it $f$-vector} is the sequence $f_0,f_1,\\ldots$,\nwhere $f_k$ is the number of $k$-simplices in $K$.\n\n\\begin{theorem}\\label{th:f-vectors}\nTake any $n\\ge3$, and any $C>0$.\nThen there is a triangulation of $\\mathbb{S}^n$ for which $\\frac{f_j}{f_i}>C$\nfor any $0\\le i<j\\le n$ such that $i<\\lfloor\\frac {n-1}2\\rfloor$.\n\\end{theorem}\n\nNote that the estimations from theorems~\\ref{th:lambda-to-0} and \\ref{th:f-vectors}\nhold even if we\nconsider\nonly {\\it combinatorial} triangulations.\nMoreover, we can consider only triangulations which are {\\it isomorphic to the boundary of a convex simplicial polytope in $\\mathbb{R}^{n+1}$}.\nSee remark~\\ref{r:comb-nondegenerate}.",
    "sketch": "The key idea of the proof of theorem~\\ref{th:lambda-to-0} is to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.",
    "expanded_sketch": "The key idea in establishing the main theorem is to construct a triangulation of $\\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.,",
    "expanded_theorem": "\\label{th:lambda-to-0}\nFor any $n\\ge3$ we have $\\lim\\limits_{d\\to\\infty}\\frac{\\lambda(n,d)}d=0$.",
    "theorem_type": [
      "Asymptotic or Limit",
      "Universal"
    ],
    "mcq": {
      "question": "For positive integers n and d, let λ(n,d) denote the minimal number of vertices of a triangulation K of the n-sphere S^n such that K admits a degree-d simplicial map to the standard n-sphere S^n_{n+2}, equivalently to the boundary of an (n+1)-simplex. Which statement holds for every integer n \\ge 3?",
      "correct_choice": {
        "label": "A",
        "text": "For every n \\ge 3, \\(\\lim_{d\\to\\infty} \\frac{\\lambda(n,d)}{d} = 0\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(n \\ge 3\\), there exists a constant \\(c_n>0\\) such that for all sufficiently large \\(d\\), \\(\\frac{\\lambda(n,d)}{d} \\ge c_n\\)."
        },
        {
          "label": "C",
          "text": "For every \\(n \\ge 3\\), there exists a sequence \\(d_m\\to\\infty\\) such that \\(\\lim_{m\\to\\infty} \\frac{\\lambda(n,d_m)}{d_m}=0\\)."
        },
        {
          "label": "D",
          "text": "There exists a constant \\(C>0\\) such that for every \\(n \\ge 3\\) and all sufficiently large \\(d\\), \\(\\frac{\\lambda(n,d)}{d} \\le C\\)."
        },
        {
          "label": "E",
          "text": "For every \\(n \\ge 3\\), there exists an integer \\(d_0=d_0(n)\\) such that for every \\(d\\ge d_0\\) one has \\(\\lambda(n,d)=o\\!\\left(d\\right)\\) uniformly in \\(d\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "sublinear-vanishing asymptotic replaced by positive linear lower bound",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "full limit over all d weakened to convergence along a subsequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "vanishing ratio weakened to merely bounded ratio with uniform constant",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "misstated asymptotic notion by imposing eventual equality with an o(d) condition",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only defines λ(n,d) and asks which asymptotic statement is true; it does not explicitly or implicitly reveal option A."
      },
      "TAS": {
        "score": 2,
        "justification": "The item is not a bare restatement in the stem itself; the student must distinguish among several asymptotic formulations with different quantifiers and strengths."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ subtly in limit strength, subsequences, and uniformity. However, this is weakened because option C is a logically weaker consequence of A, so the item does not cleanly force a unique strongest conclusion."
      },
      "DQS": {
        "score": 0,
        "justification": "Distractor quality is poor overall: C is also true if A is true, so the question is not genuinely single-correct; E is mathematically malformed/awkward ('λ(n,d)=o(d)' as an eventual equality statement, 'uniformly in d'), which makes it less plausible as a serious distractor."
      },
      "total_score": 5,
      "overall_assessment": "The question avoids answer leakage and has some asymptotic reasoning content, but it is fundamentally flawed as an MCQ because at least one distractor is also true and another is poorly formulated."
    }
  },
  {
    "id": "2512.03141v1",
    "paper_link": "http://arxiv.org/abs/2512.03141v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "maintheorem",
      "content": "[Dynamical Non-Commutative Bifurcation]\\label{thm:main_bifurcation}\nLet $P_0(x)$ be a central polynomial over $\\mathbb{A} \\in \\{\\mathbb{H}, \\mathbb{O}\\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\\rm M} > 0$. Let $P_\\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\\epsilon = Z(P_\\epsilon)$. Let $d_{\\rm A} = \\dim(\\mathbb{A})$.\n\\begin{enumerate}\n    \\item \\textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\\epsilon=0$.\n    \\item \\textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\\rank(J_{P_0}(x)) = d_{\\rm A} - d_{\\rm M}$.\n    \\item \\textbf{Dynamical Retraction:} Let $\\mathcal{V}_\\epsilon(x) = \\|P_\\epsilon(x)\\|^2$. The gradient flow $\\dot{x} = -\\nabla\\mathcal{V}_\\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\\rm collapse} \\propto \\epsilon^{-2}$.\n    \\item \\textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\\epsilon,T)$ exhibits a discontinuity at $(\\epsilon, T) = (0, 0)$.\n\\end{enumerate}",
      "start_pos": 403055,
      "end_pos": 404309,
      "label": "thm:main_bifurcation"
    },
    "ref_dict": {
      "prop:lojasiewicz_exponent": "\\begin{proposition}[Lojasiewicz Exponent]\\label{prop:lojasiewicz_exponent}\nFor a generic analytic perturbation (Morse function $f_\\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\\theta = 1/2$. This yields the quadratic timescale $T_{\\rm collapse} \\propto \\epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\\theta$ may be smaller ($0 < \\theta < 1/2$), leading to slower collapse timescales $T_{\\rm collapse} \\propto \\epsilon^{-(1+\\theta)/\\theta}$.\n\\end{proposition}",
      "def:order_parameter": "\\begin{definition}[Alignment Order Parameter]\\label{def:order_parameter}\nLet $\\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\\nu$ is the normalized mean square alignment of the coordinate along $\\nu$ under the Gibbs measure. For $\\HH$ ($d_{\\rm A}=4$) aligned along $i$ (coordinate $b$):\n\\begin{equation}\nm(\\epsilon, T) = \\frac{\\langle b^2 \\rangle_{\\mu_T(\\epsilon)}}{\\langle b^2+c^2+d^2 \\rangle_{\\mu_T(\\epsilon)}}.\n\\end{equation}\n\\end{definition}",
      "cor:inflation_law": "\\begin{corollary}[Dimensional Inflation Law]\\label{cor:inflation_law}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$) of dimension $d_{\\rm A} \\ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\\rm M} = d_{\\rm A} - 2$.\n\\end{corollary}",
      "def:gibbs_measure": "\\begin{definition}[Gibbs Measure and Partition Function]\\label{def:gibbs_measure}\nThe Gibbs probability measure $\\mu_T$ on $\\A$ is defined by\n\\begin{equation}\nd\\mu_T(x) = \\frac{1}{Z(T)} \\exp\\left(-\\frac{\\mathcal{V}(x)}{T}\\right) dx,\n\\end{equation}\nwhere $Z(T) = \\int_{\\A} \\exp(-\\mathcal{V}(x)/T) dx$ is the partition function.\n\\end{definition}",
      "thm:phase_transition": "\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n    \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n    \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}",
      "thm:generalized_symmetry_reduction": "\\begin{theorem}[Generalized Symmetry Reduction Theorem]\\label{thm:generalized_symmetry_reduction}\nThe root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.\n\\end{theorem}",
      "def:aut_stabilizer": "\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
      "thm:deformation_retract": "\\begin{theorem}[Deformation Retract]\\label{thm:deformation_retract}\nThe gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\\epsilon$. (Part 3 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}",
      "thm:collapse_timescale": "\\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\\label{thm:collapse_timescale}\nThe characteristic relaxation time $T_{\\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\\epsilon$ scales as $T_{\\rm collapse} \\propto 1/\\epsilon^2$.\n\\end{theorem}",
      "thm:jacobian_singularity": "\\begin{theorem}[Jacobian Singularity of the Central Fiber]\\label{thm:jacobian_singularity}\nLet $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\\R^{d_{\\rm A}} \\to \\R^{d_{\\rm A}}$) is singular for all $x \\in V_0$. (Part 2 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}",
      "prop:collapse": "\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}",
      "thm:algebraic_alignment": "\\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\\label{thm:algebraic_alignment}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$). Let $P(x) \\in \\A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\\A(P)$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3769,
    "pre_theorem_intro_text": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n    \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n    \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n    \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\n\\subsection{Main Results}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}",
    "context": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
    "full_context": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}\n\n\\abstract{We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($\\HH$ and $\\OO$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $\\HH$, $G_2$ for $\\OO$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($\\Delta=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $\\mathcal{V}(x) = \\|P(x)\\|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_{\\rm collapse} \\propto \\epsilon^{-2}$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.}\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\A$ be a real division algebra. Let $P_0(x) \\in \\R[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\A[x]$ parameterized by $\\epsilon \\in \\R$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{proof}\nWe synthesize the results established in the subsequent sections.\n\n\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\textwidth]{topological_collapse.pdf}\n \\caption{Visualization of Topological Collapse via Gradient Flow (in $\\HH$). The sequence illustrates the transition from $S^2$ to $S^0$ under a non-central perturbation aligned with the $i$-axis. (1) Initial state $V_0$. (2-4) Evolution under the gradient flow $-\\nabla\\mathcal{V}_\\epsilon$. (5) Final state $V_\\epsilon$: The manifold has collapsed onto the isolated roots (Alignment Principle). An animation of the collapse is shown in \\url{http://youtu.be/yaJQOuftZjE}}\n \\label{fig:topological_collapse}\n\\end{figure}\n\n\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}\n\nThe analysis reveals a unified picture across the real normed division algebras:\n\\begin{enumerate}\n \\item The geometry of central roots is determined by the automorphism group of the algebra (Generalized Inflation Theorem).\n \\item Central dynamics are governed by the auxiliary polynomial and its discriminant, independent of associativity (due to power-associativity).\n \\item Topological collapse is driven by symmetry reduction (Generalized Symmetry Reduction Theorem) and the Alignment Principle (Localization Theorem), which holds due to alternativity.\n \\item The dynamics of collapse follow a gradient flow characterized by Critical Slowing Down ($T \\propto \\epsilon^{-2}$).\n \\item The collapse can be rigorously characterized as a thermodynamic phase transition (Entropy Scaling Law and Order Parameter analysis).\n\\end{enumerate}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
    "post_theorem_intro_text_len": 1792,
    "post_theorem_intro_text": "\\begin{proof}\nWe synthesize the results established in the subsequent sections.\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, by the Dimensional Inflation Law (Corollary~\\ref{cor:inflation_law}), $d_{\\rm M} = d_{\\rm A} - 2$. For $\\epsilon \\neq 0$, $G_{P_\\epsilon}$ is trivial. By the Generalized Symmetry Reduction Theorem (Theorem~\\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\\rm M}=0$). The discontinuity follows (Proposition~\\ref{prop:collapse}).\n\n\\textbf{Part 2 (Algebraic Singularity):} The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$. The rank deficiency follows from the positive dimension of the manifold (Theorem~\\ref{thm:jacobian_singularity}).\n\n\\textbf{Part 3 (Dynamical Retraction):} The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive. The Lojasiewicz gradient inequality \\cite{lojasiewicz1963propriete} guarantees convergence of the gradient flow (Theorem~\\ref{thm:deformation_retract}). The timescale scaling $T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\\ref{prop:lojasiewicz_exponent}).\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} We define the Gibbs measure (Definition~\\ref{def:gibbs_measure}) and the alignment order parameter $m(\\epsilon, T)$ (Definition~\\ref{def:order_parameter}). In the limit $T \\to 0$, the measure concentrates on $V_\\epsilon$. For $\\epsilon=0$, symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$. For $\\epsilon \\neq 0$, the Localization Theorem (Theorem~\\ref{thm:algebraic_alignment}) implies alignment, $m(\\epsilon, 0) = 1$. The discontinuity characterizes the phase transition (Theorem~\\ref{thm:phase_transition}).\n\\end{proof}",
    "sketch": "To prove Theorem~\\ref{thm:main_bifurcation} the proof \\\"synthesizes the results established in the subsequent sections\\\" and proceeds by parts:\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, \\\"by the Dimensional Inflation Law (Corollary~\\ref{cor:inflation_law}), $d_{\\rm M} = d_{\\rm A} - 2$.\\\" For $\\epsilon\\neq 0$, \\\"$G_{P_\\epsilon}$ is trivial\\\" and \\\"by the Generalized Symmetry Reduction Theorem (Theorem~\\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\\rm M}=0$).\\\" Hence \\\"the discontinuity follows (Proposition~\\ref{prop:collapse}).\\\"\n\n\\textbf{Part 2 (Algebraic Singularity):} \\\"The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$.\\\" Therefore \\\"the rank deficiency follows from the positive dimension of the manifold (Theorem~\\ref{thm:jacobian_singularity}).\\\"\n\n\\textbf{Part 3 (Dynamical Retraction):} \\\"The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive.\\\" Then \\\"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow (Theorem~\\ref{thm:deformation_retract}).\\\" The scaling \\\"$T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\\ref{prop:lojasiewicz_exponent}).\\\"\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} \\\"We define the Gibbs measure (Definition~\\ref{def:gibbs_measure}) and the alignment order parameter $m(\\epsilon, T)$ (Definition~\\ref{def:order_parameter}).\\\" As \\\"$T\\to 0$, the measure concentrates on $V_\\epsilon$.\\\" For $\\epsilon=0$, \\\"symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$.\\\" For $\\epsilon\\neq 0$, \\\"the Localization Theorem (Theorem~\\ref{thm:algebraic_alignment}) implies alignment, $m(\\epsilon, 0) = 1$.\\\" \\\"The discontinuity characterizes the phase transition (Theorem~\\ref{thm:phase_transition}).\\\"",
    "expanded_sketch": "To prove the main theorem, the proof \"synthesizes the results established in the subsequent sections\" and proceeds by parts:\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, by the following result.\n\n\\begin{corollary}[Dimensional Inflation Law]\\label{cor:inflation_law}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$) of dimension $d_{\\rm A} \\ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\\rm M} = d_{\\rm A} - 2$.\n\\end{corollary}\n\nFor $\\epsilon\\neq 0$, \"$G_{P_\\epsilon}$ is trivial\" and by the following theorem, the roots are isolated ($d_{\\rm M}=0$).\n\n\\begin{theorem}[Generalized Symmetry Reduction Theorem]\\label{thm:generalized_symmetry_reduction}\nThe root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.\n\\end{theorem}\n\nHence the discontinuity follows from the following proposition.\n\n\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}\n\n\\textbf{Part 2 (Algebraic Singularity):} \"The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$.\" Therefore \"the rank deficiency follows from the positive dimension of the manifold\" by the following theorem.\n\n\\begin{theorem}[Jacobian Singularity of the Central Fiber]\\label{thm:jacobian_singularity}\nLet $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\\R^{d_{\\rm A}} \\to \\R^{d_{\\rm A}}$) is singular for all $x \\in V_0$. (Part 2 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}\n\n\\textbf{Part 3 (Dynamical Retraction):} \"The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive.\" Then \"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow\" via the following theorem.\n\n\\begin{theorem}[Deformation Retract]\\label{thm:deformation_retract}\nThe gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\\epsilon$. (Part 3 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}\n\nThe scaling \"$T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential\" using the following theorem and proposition.\n\n\\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\\label{thm:collapse_timescale}\nThe characteristic relaxation time $T_{\\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\\epsilon$ scales as $T_{\\rm collapse} \\propto 1/\\epsilon^2$.\n\\end{theorem}\n\n\\begin{proposition}[Lojasiewicz Exponent]\\label{prop:lojasiewicz_exponent}\nFor a generic analytic perturbation (Morse function $f_\\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\\theta = 1/2$. This yields the quadratic timescale $T_{\\rm collapse} \\propto \\epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\\theta$ may be smaller ($0 < \\theta < 1/2$), leading to slower collapse timescales $T_{\\rm collapse} \\propto \\epsilon^{-(1+\\theta)/\\theta}$.\n\\end{proposition}\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} \"We define the Gibbs measure\" as follows.\n\n\\begin{definition}[Gibbs Measure and Partition Function]\\label{def:gibbs_measure}\nThe Gibbs probability measure $\\mu_T$ on $\\A$ is defined by\n\\begin{equation}\nd\\mu_T(x) = \\frac{1}{Z(T)} \\exp\\left(-\\frac{\\mathcal{V}(x)}{T}\\right) dx,\n\\end{equation}\nwhere $Z(T) = \\int_{\\A} \\exp(-\\mathcal{V}(x)/T) dx$ is the partition function.\n\\end{definition}\n\nand \"the alignment order parameter $m(\\epsilon, T)$\" as follows.\n\n\\begin{definition}[Alignment Order Parameter]\\label{def:order_parameter}\nLet $\\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\\nu$ is the normalized mean square alignment of the coordinate along $\\nu$ under the Gibbs measure. For $\\HH$ ($d_{\\rm A}=4$) aligned along $i$ (coordinate $b$):\n\\begin{equation}\nm(\\epsilon, T) = \\frac{\\langle b^2 \\rangle_{\\mu_T(\\epsilon)}}{\\langle b^2+c^2+d^2 \\rangle_{\\mu_T(\\epsilon)}}.\n\\end{equation}\n\\end{definition}\n\nAs \"$T\\to 0$, the measure concentrates on $V_\\epsilon$.\" For $\\epsilon=0$, \"symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$.\" For $\\epsilon\\neq 0$, the following theorem implies alignment, $m(\\epsilon, 0) = 1$.\n\n\\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\\label{thm:algebraic_alignment}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$). Let $P(x) \\in \\A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\\A(P)$.\n\\end{theorem}\n\nThe discontinuity characterizes the phase transition by the following theorem.\n\n\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n    \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n    \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}\n",
    "expanded_theorem": "[Dynamical Non-Commutative Bifurcation]\\label{thm:main_bifurcation}\nLet $P_0(x)$ be a central polynomial over $\\mathbb{A} \\in \\{\\mathbb{H}, \\mathbb{O}\\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\\rm M} > 0$. Let $P_\\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\\epsilon = Z(P_\\epsilon)$. Let $d_{\\rm A} = \\dim(\\mathbb{A})$.\n\\begin{enumerate}\n    \\item \\textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\\epsilon=0$.\n    \\item \\textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\\rank(J_{P_0}(x)) = d_{\\rm A} - d_{\\rm M}$.\n    \\item \\textbf{Dynamical Retraction:} Let $\\mathcal{V}_\\epsilon(x) = \\|P_\\epsilon(x)\\|^2$. The gradient flow $\\dot{x} = -\\nabla\\mathcal{V}_\\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\\rm collapse} \\propto \\epsilon^{-2}$.\n    \\item \\textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\\epsilon,T)$ exhibits a discontinuity at $(\\epsilon, T) = (0, 0)$.\n\\end{enumerate}",
    "theorem_type": [
      "Universal",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb{A}\\in\\{\\mathbb{H},\\mathbb{O}\\}\\) and let \\(d_{\\rm A}=\\dim(\\mathbb{A})\\). Suppose \\(P_0(x)\\) is a central polynomial over \\(\\mathbb{A}\\), meaning its coefficients lie in \\(\\mathbb{R}\\), and let its root variety be \\(V_0=Z(P_0)\\). Assume \\(V_0\\) contains a non-real manifold of dimension \\(d_{\\rm M}>0\\). Let \\(P_\\epsilon(x)\\in \\mathbb{A}[x]\\) be a generically non-central analytic deformation of \\(P_0\\), meaning that the coefficients of \\(P_\\epsilon\\) depend analytically on \\(\\epsilon\\in\\mathbb{R}\\), one has \\(P_\\epsilon=P_0\\) at \\(\\epsilon=0\\), and for \\(\\epsilon\\neq 0\\) in some punctured neighborhood of \\(0\\), the automorphism stabilizer \\(G_{P_\\epsilon}=\\{g\\in \\operatorname{Aut}(\\mathbb{A})\\mid g(a_k)=a_k\\text{ for all coefficients }a_k\\}\\) is trivial. Write \\(V_\\epsilon=Z(P_\\epsilon)\\), and let \\(\\mathcal{V}_\\epsilon(x)=\\|P_\\epsilon(x)\\|^2\\). Which statement holds for every such choice of \\(\\mathbb{A}\\), \\(P_0\\), and deformation \\(P_\\epsilon\\)?",
      "correct_choice": {
        "label": "A",
        "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
      },
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          "label": "B",
          "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\) for every analytic deformation; and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
        },
        {
          "label": "C",
          "text": "The Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\), the central fiber \\(V_0\\) has Jacobian rank deficiency, and the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract."
        },
        {
          "label": "D",
          "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at every point of the line \\(T=0\\)."
        },
        {
          "label": "E",
          "text": "All of the following hold: (1) for \\(\\epsilon\\neq 0\\) sufficiently small, the root variety \\(V_\\epsilon\\) still contains a non-real manifold of dimension \\(d_{\\rm A}-2\\), so the Hausdorff dimension varies continuously through \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
        }
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          "sketch_hook_type": "regularity",
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          "template_used": "uniformity_effectivity"
        },
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          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_explicit_epsilon^{-2}_scaling_and_thermodynamic_discontinuity_clause",
          "template_used": "weaker_true"
        },
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          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "location_of_phase_transition_discontinuity",
          "template_used": "boundary_range"
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          "sketch_hook_type": "outer_automorphism",
          "tampered_component": "trivial_stabilizer_implies_isolated_roots",
          "template_used": "wildcard"
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      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the resulting asymptotic statement, but the decisive claims in A (dimension jump, exact rank formula, epsilon^{-2} scaling, discontinuous order parameter) are not stated in the prompt itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-statement recognition item: the correct choice reproduces a full bundled conclusion under the given hypotheses. The task is mainly to identify the exact theorem-like conclusion rather than reason through a genuinely new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "The item does require some discrimination among subtle variants (continuity vs discontinuity, exact vs weaker Jacobian claim, deformation-dependent vs uniform constant), so the answer is not obvious. However, the pressure is mostly on recalling or matching a theorem rather than generating the conclusion from first principles."
      },
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        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: reversing continuity behavior, giving a weaker true statement, overclaiming uniformity in constants, or inserting unjustified quantitative/statistical assertions."
      },
      "total_score": 5,
      "overall_assessment": "Technically sophisticated item with strong distractors and no clear answer leakage, but it functions largely as theorem-recognition rather than a non-tautological test of generative mathematical reasoning."
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  },
  {
    "id": "2512.03141v1",
    "paper_link": "http://arxiv.org/abs/2512.03141v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "maintheorem",
      "content": "[Dynamical Non-Commutative Bifurcation]\\label{thm:main_bifurcation}\nLet $P_0(x)$ be a central polynomial over $\\mathbb{A} \\in \\{\\mathbb{H}, \\mathbb{O}\\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\\rm M} > 0$. Let $P_\\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\\epsilon = Z(P_\\epsilon)$. Let $d_{\\rm A} = \\dim(\\mathbb{A})$.\n\\begin{enumerate}\n    \\item \\textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\\epsilon=0$.\n    \\item \\textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\\rank(J_{P_0}(x)) = d_{\\rm A} - d_{\\rm M}$.\n    \\item \\textbf{Dynamical Retraction:} Let $\\mathcal{V}_\\epsilon(x) = \\|P_\\epsilon(x)\\|^2$. The gradient flow $\\dot{x} = -\\nabla\\mathcal{V}_\\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\\rm collapse} \\propto \\epsilon^{-2}$.\n    \\item \\textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\\epsilon,T)$ exhibits a discontinuity at $(\\epsilon, T) = (0, 0)$.\n\\end{enumerate}",
      "start_pos": 403055,
      "end_pos": 404309,
      "label": "thm:main_bifurcation"
    },
    "ref_dict": {
      "prop:lojasiewicz_exponent": "\\begin{proposition}[Lojasiewicz Exponent]\\label{prop:lojasiewicz_exponent}\nFor a generic analytic perturbation (Morse function $f_\\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\\theta = 1/2$. This yields the quadratic timescale $T_{\\rm collapse} \\propto \\epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\\theta$ may be smaller ($0 < \\theta < 1/2$), leading to slower collapse timescales $T_{\\rm collapse} \\propto \\epsilon^{-(1+\\theta)/\\theta}$.\n\\end{proposition}",
      "def:order_parameter": "\\begin{definition}[Alignment Order Parameter]\\label{def:order_parameter}\nLet $\\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\\nu$ is the normalized mean square alignment of the coordinate along $\\nu$ under the Gibbs measure. For $\\HH$ ($d_{\\rm A}=4$) aligned along $i$ (coordinate $b$):\n\\begin{equation}\nm(\\epsilon, T) = \\frac{\\langle b^2 \\rangle_{\\mu_T(\\epsilon)}}{\\langle b^2+c^2+d^2 \\rangle_{\\mu_T(\\epsilon)}}.\n\\end{equation}\n\\end{definition}",
      "cor:inflation_law": "\\begin{corollary}[Dimensional Inflation Law]\\label{cor:inflation_law}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$) of dimension $d_{\\rm A} \\ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\\rm M} = d_{\\rm A} - 2$.\n\\end{corollary}",
      "def:gibbs_measure": "\\begin{definition}[Gibbs Measure and Partition Function]\\label{def:gibbs_measure}\nThe Gibbs probability measure $\\mu_T$ on $\\A$ is defined by\n\\begin{equation}\nd\\mu_T(x) = \\frac{1}{Z(T)} \\exp\\left(-\\frac{\\mathcal{V}(x)}{T}\\right) dx,\n\\end{equation}\nwhere $Z(T) = \\int_{\\A} \\exp(-\\mathcal{V}(x)/T) dx$ is the partition function.\n\\end{definition}",
      "thm:phase_transition": "\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n    \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n    \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}",
      "thm:generalized_symmetry_reduction": "\\begin{theorem}[Generalized Symmetry Reduction Theorem]\\label{thm:generalized_symmetry_reduction}\nThe root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.\n\\end{theorem}",
      "def:aut_stabilizer": "\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
      "thm:deformation_retract": "\\begin{theorem}[Deformation Retract]\\label{thm:deformation_retract}\nThe gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\\epsilon$. (Part 3 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}",
      "thm:collapse_timescale": "\\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\\label{thm:collapse_timescale}\nThe characteristic relaxation time $T_{\\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\\epsilon$ scales as $T_{\\rm collapse} \\propto 1/\\epsilon^2$.\n\\end{theorem}",
      "thm:jacobian_singularity": "\\begin{theorem}[Jacobian Singularity of the Central Fiber]\\label{thm:jacobian_singularity}\nLet $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\\R^{d_{\\rm A}} \\to \\R^{d_{\\rm A}}$) is singular for all $x \\in V_0$. (Part 2 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}",
      "prop:collapse": "\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}",
      "thm:algebraic_alignment": "\\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\\label{thm:algebraic_alignment}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$). Let $P(x) \\in \\A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\\A(P)$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3769,
    "pre_theorem_intro_text": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n    \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n    \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n    \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\n\\subsection{Main Results}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}",
    "context": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
    "full_context": "The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\\mathbb{H}$) by Niven \\cite{niven1941equations} and Eilenberg and Niven \\cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \\cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \\cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\\mathbb{R}$), 0-dimensional sets in $\\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\\mathbb{H}$, $S^6$ in $\\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.\n\nThis paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \\cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \\cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \\cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.\n\nThe motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \\cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \\cite{baez2002octonions}.\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\nThe central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\mathbb{A}$ be a real division algebra. Let $P_0(x) \\in \\mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\mathbb{A}[x]$ parameterized by $\\epsilon \\in \\mathbb{R}$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}\n\n\\abstract{We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($\\HH$ and $\\OO$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $\\HH$, $G_2$ for $\\OO$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($\\Delta=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $\\mathcal{V}(x) = \\|P(x)\\|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_{\\rm collapse} \\propto \\epsilon^{-2}$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.}\n\nOur main original contributions are:\n\\begin{enumerate}\n \\item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).\n \\item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.\n \\item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.\n\\end{enumerate}\n\n\\begin{definition}[Analytic Deformation]\\label{def:analytic_deformation}\nLet $\\A$ be a real division algebra. Let $P_0(x) \\in \\R[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\\epsilon(x) \\in \\A[x]$ parameterized by $\\epsilon \\in \\R$, such that the coefficients depend analytically on $\\epsilon$, $P_\\epsilon(x) = P_0(x)$ when $\\epsilon=0$, and $G_{P_\\epsilon}$ is trivial for $\\epsilon \\neq 0$ in a punctured neighborhood of $\\epsilon=0$.\n\\end{definition}\n\n\\begin{proof}\nWe synthesize the results established in the subsequent sections.\n\n\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\textwidth]{topological_collapse.pdf}\n \\caption{Visualization of Topological Collapse via Gradient Flow (in $\\HH$). The sequence illustrates the transition from $S^2$ to $S^0$ under a non-central perturbation aligned with the $i$-axis. (1) Initial state $V_0$. (2-4) Evolution under the gradient flow $-\\nabla\\mathcal{V}_\\epsilon$. (5) Final state $V_\\epsilon$: The manifold has collapsed onto the isolated roots (Alignment Principle). An animation of the collapse is shown in \\url{http://youtu.be/yaJQOuftZjE}}\n \\label{fig:topological_collapse}\n\\end{figure}\n\n\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}\n\nThe analysis reveals a unified picture across the real normed division algebras:\n\\begin{enumerate}\n \\item The geometry of central roots is determined by the automorphism group of the algebra (Generalized Inflation Theorem).\n \\item Central dynamics are governed by the auxiliary polynomial and its discriminant, independent of associativity (due to power-associativity).\n \\item Topological collapse is driven by symmetry reduction (Generalized Symmetry Reduction Theorem) and the Alignment Principle (Localization Theorem), which holds due to alternativity.\n \\item The dynamics of collapse follow a gradient flow characterized by Critical Slowing Down ($T \\propto \\epsilon^{-2}$).\n \\item The collapse can be rigorously characterized as a thermodynamic phase transition (Entropy Scaling Law and Order Parameter analysis).\n\\end{enumerate}\n\n\\begin{definition}[Automorphism Stabilizer of the Polynomial]\\label{def:aut_stabilizer}\nLet $P(x) \\in \\A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\\Aut(\\A)$ that fixes all coefficients:\n\\begin{equation}\nG_P = \\{g \\in \\Aut(\\A) \\mid g(a_k) = a_k \\text{ for all } k\\}.\n\\end{equation}\n\\end{definition}",
    "post_theorem_intro_text_len": 1792,
    "post_theorem_intro_text": "\\begin{proof}\nWe synthesize the results established in the subsequent sections.\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, by the Dimensional Inflation Law (Corollary~\\ref{cor:inflation_law}), $d_{\\rm M} = d_{\\rm A} - 2$. For $\\epsilon \\neq 0$, $G_{P_\\epsilon}$ is trivial. By the Generalized Symmetry Reduction Theorem (Theorem~\\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\\rm M}=0$). The discontinuity follows (Proposition~\\ref{prop:collapse}).\n\n\\textbf{Part 2 (Algebraic Singularity):} The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$. The rank deficiency follows from the positive dimension of the manifold (Theorem~\\ref{thm:jacobian_singularity}).\n\n\\textbf{Part 3 (Dynamical Retraction):} The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive. The Lojasiewicz gradient inequality \\cite{lojasiewicz1963propriete} guarantees convergence of the gradient flow (Theorem~\\ref{thm:deformation_retract}). The timescale scaling $T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\\ref{prop:lojasiewicz_exponent}).\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} We define the Gibbs measure (Definition~\\ref{def:gibbs_measure}) and the alignment order parameter $m(\\epsilon, T)$ (Definition~\\ref{def:order_parameter}). In the limit $T \\to 0$, the measure concentrates on $V_\\epsilon$. For $\\epsilon=0$, symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$. For $\\epsilon \\neq 0$, the Localization Theorem (Theorem~\\ref{thm:algebraic_alignment}) implies alignment, $m(\\epsilon, 0) = 1$. The discontinuity characterizes the phase transition (Theorem~\\ref{thm:phase_transition}).\n\\end{proof}",
    "sketch": "To prove Theorem~\\ref{thm:main_bifurcation} the proof \\\"synthesizes the results established in the subsequent sections\\\" and proceeds by parts:\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, \\\"by the Dimensional Inflation Law (Corollary~\\ref{cor:inflation_law}), $d_{\\rm M} = d_{\\rm A} - 2$.\\\" For $\\epsilon\\neq 0$, \\\"$G_{P_\\epsilon}$ is trivial\\\" and \\\"by the Generalized Symmetry Reduction Theorem (Theorem~\\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\\rm M}=0$).\\\" Hence \\\"the discontinuity follows (Proposition~\\ref{prop:collapse}).\\\"\n\n\\textbf{Part 2 (Algebraic Singularity):} \\\"The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$.\\\" Therefore \\\"the rank deficiency follows from the positive dimension of the manifold (Theorem~\\ref{thm:jacobian_singularity}).\\\"\n\n\\textbf{Part 3 (Dynamical Retraction):} \\\"The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive.\\\" Then \\\"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow (Theorem~\\ref{thm:deformation_retract}).\\\" The scaling \\\"$T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\\ref{prop:lojasiewicz_exponent}).\\\"\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} \\\"We define the Gibbs measure (Definition~\\ref{def:gibbs_measure}) and the alignment order parameter $m(\\epsilon, T)$ (Definition~\\ref{def:order_parameter}).\\\" As \\\"$T\\to 0$, the measure concentrates on $V_\\epsilon$.\\\" For $\\epsilon=0$, \\\"symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$.\\\" For $\\epsilon\\neq 0$, \\\"the Localization Theorem (Theorem~\\ref{thm:algebraic_alignment}) implies alignment, $m(\\epsilon, 0) = 1$.\\\" \\\"The discontinuity characterizes the phase transition (Theorem~\\ref{thm:phase_transition}).\\\"",
    "expanded_sketch": "To prove the main theorem, the proof \"synthesizes the results established in the subsequent sections\" and proceeds by parts:\n\n\\textbf{Part 1 (Topological Collapse):} For $\\epsilon=0$, by the following result.\n\n\\begin{corollary}[Dimensional Inflation Law]\\label{cor:inflation_law}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$) of dimension $d_{\\rm A} \\ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\\rm M} = d_{\\rm A} - 2$.\n\\end{corollary}\n\nFor $\\epsilon\\neq 0$, \"$G_{P_\\epsilon}$ is trivial\" and by the following theorem, the roots are isolated ($d_{\\rm M}=0$).\n\n\\begin{theorem}[Generalized Symmetry Reduction Theorem]\\label{thm:generalized_symmetry_reduction}\nThe root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.\n\\end{theorem}\n\nHence the discontinuity follows from the following proposition.\n\n\\begin{proposition}[Topological Collapse]\\label{prop:collapse}\nFor a generically non-central perturbation $P_\\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\\epsilon)$ changes discontinuously at $\\epsilon=0$. (Part 1 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{proposition}\n\n\\textbf{Part 2 (Algebraic Singularity):} \"The tangent space $T_x V_0$ is related to $\\Ker(J_{P_0}(x))$.\" Therefore \"the rank deficiency follows from the positive dimension of the manifold\" by the following theorem.\n\n\\begin{theorem}[Jacobian Singularity of the Central Fiber]\\label{thm:jacobian_singularity}\nLet $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\\R^{d_{\\rm A}} \\to \\R^{d_{\\rm A}}$) is singular for all $x \\in V_0$. (Part 2 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}\n\n\\textbf{Part 3 (Dynamical Retraction):} \"The potential $\\mathcal{V}_\\epsilon(x)$ is real-analytic and coercive.\" Then \"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow\" via the following theorem.\n\n\\begin{theorem}[Deformation Retract]\\label{thm:deformation_retract}\nThe gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\\epsilon$. (Part 3 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\end{theorem}\n\nThe scaling \"$T_{\\rm collapse} \\propto \\epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential\" using the following theorem and proposition.\n\n\\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\\label{thm:collapse_timescale}\nThe characteristic relaxation time $T_{\\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\\epsilon$ scales as $T_{\\rm collapse} \\propto 1/\\epsilon^2$.\n\\end{theorem}\n\n\\begin{proposition}[Lojasiewicz Exponent]\\label{prop:lojasiewicz_exponent}\nFor a generic analytic perturbation (Morse function $f_\\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\\theta = 1/2$. This yields the quadratic timescale $T_{\\rm collapse} \\propto \\epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\\theta$ may be smaller ($0 < \\theta < 1/2$), leading to slower collapse timescales $T_{\\rm collapse} \\propto \\epsilon^{-(1+\\theta)/\\theta}$.\n\\end{proposition}\n\n\\textbf{Part 4 (Thermodynamic Phase Transition):} \"We define the Gibbs measure\" as follows.\n\n\\begin{definition}[Gibbs Measure and Partition Function]\\label{def:gibbs_measure}\nThe Gibbs probability measure $\\mu_T$ on $\\A$ is defined by\n\\begin{equation}\nd\\mu_T(x) = \\frac{1}{Z(T)} \\exp\\left(-\\frac{\\mathcal{V}(x)}{T}\\right) dx,\n\\end{equation}\nwhere $Z(T) = \\int_{\\A} \\exp(-\\mathcal{V}(x)/T) dx$ is the partition function.\n\\end{definition}\n\nand \"the alignment order parameter $m(\\epsilon, T)$\" as follows.\n\n\\begin{definition}[Alignment Order Parameter]\\label{def:order_parameter}\nLet $\\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\\nu$ is the normalized mean square alignment of the coordinate along $\\nu$ under the Gibbs measure. For $\\HH$ ($d_{\\rm A}=4$) aligned along $i$ (coordinate $b$):\n\\begin{equation}\nm(\\epsilon, T) = \\frac{\\langle b^2 \\rangle_{\\mu_T(\\epsilon)}}{\\langle b^2+c^2+d^2 \\rangle_{\\mu_T(\\epsilon)}}.\n\\end{equation}\n\\end{definition}\n\nAs \"$T\\to 0$, the measure concentrates on $V_\\epsilon$.\" For $\\epsilon=0$, \"symmetry implies $m(0, 0) = 1/(d_{\\rm A}-1)$.\" For $\\epsilon\\neq 0$, the following theorem implies alignment, $m(\\epsilon, 0) = 1$.\n\n\\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\\label{thm:algebraic_alignment}\nLet $\\A$ be a real normed division algebra ($\\C, \\HH, \\OO$). Let $P(x) \\in \\A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\\A(P)$.\n\\end{theorem}\n\nThe discontinuity characterizes the phase transition by the following theorem.\n\n\\begin{theorem}[Symmetry Breaking and Phase Transition]\\label{thm:phase_transition}\nConsider a perturbation $P_\\epsilon(x)$ breaking the symmetry along the $\\nu$-axis. We analyze the behavior in the zero temperature limit ($T \\to 0$). (Part 4 of Main Theorem~\\ref{thm:main_bifurcation}).\n\\begin{enumerate}\n    \\item \\textbf{Symmetric Phase (Disordered):} For $\\epsilon=0$, the measure concentrates uniformly on $S^{d_{\\rm A}-2}$. $m(0, 0) = 1/(d_{\\rm A}-1)$.\n    \\item \\textbf{Ordered Phase (Aligned):} For $\\epsilon \\neq 0$, the measure concentrates on the isolated roots aligned with the $\\nu$-axis. $m(\\epsilon, 0) = 1$.\n\\end{enumerate}\n\\end{theorem}\n",
    "expanded_theorem": "[Dynamical Non-Commutative Bifurcation]\\label{thm:main_bifurcation}\nLet $P_0(x)$ be a central polynomial over $\\mathbb{A} \\in \\{\\mathbb{H}, \\mathbb{O}\\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\\rm M} > 0$. Let $P_\\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\\epsilon = Z(P_\\epsilon)$. Let $d_{\\rm A} = \\dim(\\mathbb{A})$.\n\\begin{enumerate}\n    \\item \\textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\\epsilon=0$.\n    \\item \\textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\\rank(J_{P_0}(x)) = d_{\\rm A} - d_{\\rm M}$.\n    \\item \\textbf{Dynamical Retraction:} Let $\\mathcal{V}_\\epsilon(x) = \\|P_\\epsilon(x)\\|^2$. The gradient flow $\\dot{x} = -\\nabla\\mathcal{V}_\\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\\rm collapse} \\propto \\epsilon^{-2}$.\n    \\item \\textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\\epsilon,T)$ exhibits a discontinuity at $(\\epsilon, T) = (0, 0)$.\n\\end{enumerate}",
    "theorem_type": [
      "Universal",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb{A}\\in\\{\\mathbb{H},\\mathbb{O}\\}\\) and let \\(d_{\\rm A}=\\dim(\\mathbb{A})\\). Suppose \\(P_0(x)\\) is a central polynomial over \\(\\mathbb{A}\\), meaning its coefficients lie in \\(\\mathbb{R}\\), and let its root variety be \\(V_0=Z(P_0)\\). Assume \\(V_0\\) contains a non-real manifold of dimension \\(d_{\\rm M}>0\\). Let \\(P_\\epsilon(x)\\in \\mathbb{A}[x]\\) be a generically non-central analytic deformation of \\(P_0\\), meaning that the coefficients of \\(P_\\epsilon\\) depend analytically on \\(\\epsilon\\in\\mathbb{R}\\), one has \\(P_\\epsilon=P_0\\) at \\(\\epsilon=0\\), and for \\(\\epsilon\\neq 0\\) in some punctured neighborhood of \\(0\\), the automorphism stabilizer \\(G_{P_\\epsilon}=\\{g\\in \\operatorname{Aut}(\\mathbb{A})\\mid g(a_k)=a_k\\text{ for all coefficients }a_k\\}\\) is trivial. Write \\(V_\\epsilon=Z(P_\\epsilon)\\), and let \\(\\mathcal{V}_\\epsilon(x)=\\|P_\\epsilon(x)\\|^2\\). Which statement holds for every such choice of \\(\\mathbb{A}\\), \\(P_0\\), and deformation \\(P_\\epsilon\\)?",
      "correct_choice": {
        "label": "A",
        "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\) for every analytic deformation; and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
        },
        {
          "label": "C",
          "text": "The Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\), the central fiber \\(V_0\\) has Jacobian rank deficiency, and the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract."
        },
        {
          "label": "D",
          "text": "All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at every point of the line \\(T=0\\)."
        },
        {
          "label": "E",
          "text": "All of the following hold: (1) for \\(\\epsilon\\neq 0\\) sufficiently small, the root variety \\(V_\\epsilon\\) still contains a non-real manifold of dimension \\(d_{\\rm A}-2\\), so the Hausdorff dimension varies continuously through \\(\\epsilon=0\\); (2) the central fiber \\(V_0\\) is characterized by a Jacobian rank deficiency, namely \\(\\operatorname{rank}(J_{P_0}(x))=d_{\\rm A}-d_{\\rm M}\\); (3) the gradient flow \\(\\dot x=-\\nabla \\mathcal{V}_\\epsilon(x)\\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \\(T_{\\rm collapse}\\propto \\epsilon^{-2}\\); and (4) in a statistical ensemble at temperature \\(T\\), the collapse appears as a phase transition, with an alignment order parameter \\(m(\\epsilon,T)\\) that is discontinuous at \\((\\epsilon,T)=(0,0)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "genericity_required_for_quadratic_timescale",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_explicit_epsilon^{-2}_scaling_and_thermodynamic_discontinuity_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "location_of_phase_transition_discontinuity",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "outer_automorphism",
          "tampered_component": "trivial_stabilizer_implies_isolated_roots",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option and does not contain an obvious lexical cue pointing to A. It sets up hypotheses and asks for the valid universal conclusion without giving away the specific strengthened clauses."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to a theorem-recall/restatement question: the correct option bundles several theorem-style conclusions almost verbatim, while the distractors are mostly slight perturbations of quantifiers or boundary conditions. It is not purely tautological because there are competing variants, but it is only a mild reformulation."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare subtle alternatives such as generic versus uniform claims, exact locus of discontinuity, and continuity versus collapse. However, the task is largely recognition of the strongest theorem statement rather than genuine derivation, and the presence of a weaker true option reduces clear generative pressure."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and mathematically targeted: overgeneralization in B, wrong phase-transition locus in D, and failure of collapse in E. But C appears to be a genuinely true weaker statement, which makes the single-best-answer format ambiguous and lowers distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated but only moderately strong MCQ: it avoids direct answer leakage and uses plausible theorem-adjacent distractors, but it is close to theorem restatement and is weakened by an apparently true weaker alternative, making the item not fully well-posed as a single-answer question."
    }
  },
  {
    "id": "2512.03188v1",
    "paper_link": "http://arxiv.org/abs/2512.03188v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "prop",
      "content": "\\label{prop:bound_a}\n    If $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.",
      "start_pos": 5961,
      "end_pos": 6112,
      "label": "prop:bound_a"
    },
    "ref_dict": {
      "cor:SerreSet": "\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}"
    },
    "pre_theorem_intro_text_len": 3235,
    "pre_theorem_intro_text": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$?  This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s.  It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$.  In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker.  In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero.  Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$.  Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$.  Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}).  In the definition above, and throughout the sequel, we assume that $a < b$.  It is straightforward to see that $a \\neq b$ for nontrivial solutions.  Indeed, if $a=b$, then $c!=a!^2$.  However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$.  The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem.  Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero.  Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.",
    "context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}",
    "full_context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\section{Solutions Modulo a Prime} \\label{sec:primes}\n\n\\begin{prop} There are no class $2$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $5 \\pmod{6}$.\n\\end{prop}\n\\begin{proof} \nNote that $c-b=2$ and $a!b!=c!$ implies $a! = c(c-1)$. Suppose $a!=c(c-1)$ where $a+1=p$ is a prime congruent to $5 \\pmod{6}$. Then\n$$\nc^2 - c - a! = 0\n$$ \nso we have $c = \\frac{1\\pm \\sqrt{1-4(-a!)}}{2}$. Thus, $1+4a! = 1 + 4(p-1)!$ is a perfect square. By Wilson's Theorem, $1+4(p-1)! \\equiv 1+4(-1) = -3 \\pmod{p}$, so $-3$ is a quadratic residue modulo $p$. However, this contradicts Lemma \\ref{lem:-3}.\n\\end{proof}\n\n\\begin{prop} \\label{thm:pIs4} There are no class $4$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $2$ or $3 \\pmod{5}$.\n\\end{prop}\n\\begin{proof}\nSuppose $a! = c(c-1)(c-2)(c-3)$, where $a=p-1$ for some prime $p$. Note that $b>0$ implies $c=b+4>4$, so $c!/b! \\geq 24 > 3!$, so we may assume $a>3$ and so $p>4$; furthermore, if $b>0$ then $4! = c(c-1)(c-2)(c-3)$ only has the solution $c=4$, but then $b = c-4 = 0$, a contradiction, so $p \\neq 5$. Let $r \\equiv c/2 \\pmod{p}$, and apply Wilson's Theorem to conclude that\n$$\n2r(2r-1)(2r-2)(2r-3) \\equiv -1 \\pmod{p}.\n$$\nNow, \n$$\n2r(2r-1)(2r-2)(2r-3) = (4r^2-6r+1)^2-1,\n$$\nso \n$$\n(4r^2-6r+1)^2 \\equiv 0 \\pmod{p},\n$$\ni.e., $p | 4r^2 - 6r + 1$. However, $(2r-3/2)^2 - 5/4 = 4r^2 - 6r + 1$, so $5/4$ is a quadratic residue mod $p$, which happens iff $5$ is a quadratic residue mod $p$. It is known this occurs only for primes $p \\equiv 1,4 \\pmod{5}$.\n\\end{proof}\n\n\\begin{theorem} \\label{thm:PositiveDensityIfIrreducible}Suppose $F_k(x) = x(x-1)\\cdots (x-k+1)+1$ is irreducible over $\\mathbb{Z}$. Let $P_0$ be the set of primes so that, if $a=p-1$ for some $p \\in P_0$, then $a!b!=c!$ has no class $k$ solutions in positive integers. Then the natural density of $P_0$ in the primes is at least $1/k$.\n\\end{theorem}\n\\begin{proof}\n Suppose $a!b!=c!$ is class $k$ and $a = p-1$ for some prime $p$. Then $c = b+k$ and $a! = c!/b! = F_k(c)-1$. Then, reducing this equation modulo $p$ yields $c^{\\underline{k}} \\equiv a! \\pmod{p}$, and by Wilson's Theorem, this is equivalent to $F_k(x) \\equiv 0 \\pmod{p}$. However, by Theorem 1 and 2 of \\cite{Serre03}, the set $P_0$ of primes $p$ for which $F_k(x)$ does not have solution modulo $p$ has positive density, and the density is at least $1/k$.\n\\end{proof}\n\n\\begin{prop}\\label{thm:WestlundFluge}\n For $k \\geq 2$, $F_k(x)$ is reducible over $\\mathbb{Z}$ iff $k=4$. \n\\end{prop}\n\\begin{proof}\n This is an immediate consequence of a result by Westlund and Fl\\\"{u}ge, answering a question of Schur, as discussed in \\cite{DorwartOre33}.\n\\end{proof}\n\n\\begin{cor} \\label{cor:SerreSet}\n Fix $k > 1$. For at least a $1/k$ natural density subset of primes $p$, the equation\n $$\n (p-1)! b! = (b+k)!\n $$\n has no solutions for integers $b$.\n\\end{cor}\n\\begin{proof}\n By Theorem \\ref{thm:WestlundFluge}, $F_k$ is irreducible if $k \\neq 4$, so Theorem \\ref{thm:PositiveDensityIfIrreducible} implies that $a!b!=c!$ has no solutions in the integers when $c = b+k$ and $a=p-1$, for an at least $1/k$ density subset of the primes. Proposition \\ref{thm:pIs4} yields the remaining case, by Dirichlet's Theorem on primes in arithmetic progressions.\n\\end{proof}\n\n\\begin{prop}\n For $p$ prime, the equation\n $$\n (p-1)!b! = (b+k)!\n $$\n has at most $nk/p + 1$ solutions in the integers with $b \\leq n$.\n\\end{prop}\n\\begin{proof}\n Suppose $(p-1)! b! = (b+k)!$. Then, $(p-1)! = F_k(b)-1$. Taking this equation modulo $p$ and applying Wilson's Theorem gives\n $$\n -1 \\equiv F_k(b) - 1 \\pmod{p},\n $$\n i.e., $F_k(b) \\equiv 0 \\pmod{p}$. However, this equation has at most $k$ solutions $A_k \\subseteq \\mathbb{Z}/p\\mathbb{Z}$. If $b \\pmod{p} \\not \\in A_k$, then $(p-1)!b! \\neq (b+k)!$. Since the set of $b \\in [1,n]$ so that $b \\pmod{p} \\in A_k$ has cardinality $\\lceil nk/p \\rceil \\leq nk/p + 1$, the conclusion follows.\n\\end{proof}",
    "post_theorem_intro_text_len": 874,
    "post_theorem_intro_text": "\\begin{proof}\n    Write $s_q(n)$ for the sum of the base-$q$ digits of $n$. Clearly, $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$.  Applying Legendre's Formula, we obtain\n    $$\n    \\frac{a-s_p(a)}{p-1} + \\frac{b-s_p(b)}{p-1} = \\frac{c-s_p(c)}{p-1},\n    $$\n    from which it follows that $a-(c-b) = s_p(a)+s_p(b)-s_p(c)$ for every prime $p$.\n    Since $a < b < c$, in particular\n    $$\n    | s_2(a)+s_2(b)-s_2(c) | < 2 \\lceil \\log_2 c \\rceil .\n    $$\n    Thus, $|a-k| < 2 \\lfloor \\log_2 c \\rfloor$.  Now, we argue that $a > k$.  First,\n    \\begin{align*}\n    a! & = \\frac{c!}{b!} = c(c-1)\\cdots(c-k+1) \\\\\n    & > (c-k+1)^k = (b+1)^k \\geq (a+2)^k.\n    \\end{align*}\n    Then taking the log of both sides above and applying Stirling's approximation (that $\\log a! < a \\log a$) gives\n    $$\n    a \\log a > \\log a! > k \\log (a+2)\n    $$\n    so that $k < a \\log a / \\log(a+2) < a$.\n\\end{proof}",
    "sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.",
    "expanded_sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.",
    "expanded_theorem": "\\label{prop:bound_a}\nIf $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\\,b!=c!$. Such a solution is said to be of class $k$ if $c-b=k$. Under these assumptions, which quantitative estimate for $a$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k\\le a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        },
        {
          "label": "C",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        },
        {
          "label": "D",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+2\\left\\lceil \\log_2 a\\right\\rceil.\\]"
        },
        {
          "label": "E",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "strict_lower_bound_from_stirling",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_lower_bound_k<a",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "dependence_on_c_in_digit_sum_bound",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "factor_2_from_base_2_digit_sum_estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct choice. It introduces necessary notation and hypotheses, but does not single out the exact bound in A beyond what is needed to read the options."
      },
      "TAS": {
        "score": 0,
        "justification": "The item appears to ask for essentially the exact theorem conclusion, with the correct option being a near-verbatim statement rather than a conclusion derived from a fresh scenario."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways (strict vs. weak inequality, dependence on c vs. a, special role of base 2, weaker true statement). However, the task is still mainly recognition of the precise theorem statement rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are well designed: B tests boundary sharpness, C is a weaker true statement, D swaps the parameter inside the logarithm, and E overgeneralizes the base-2 feature to all primes. These are plausible mathematical failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is weak on tautology avoidance because it largely restates the target result instead of requiring deeper generative reasoning."
    }
  },
  {
    "id": "2512.03188v1",
    "paper_link": "http://arxiv.org/abs/2512.03188v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "prop",
      "content": "\\label{prop:bound_a}\n    If $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.",
      "start_pos": 5961,
      "end_pos": 6112,
      "label": "prop:bound_a"
    },
    "ref_dict": {
      "cor:SerreSet": "\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}"
    },
    "pre_theorem_intro_text_len": 3235,
    "pre_theorem_intro_text": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$?  This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s.  It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$.  In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker.  In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero.  Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$.  Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$.  Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}).  In the definition above, and throughout the sequel, we assume that $a < b$.  It is straightforward to see that $a \\neq b$ for nontrivial solutions.  Indeed, if $a=b$, then $c!=a!^2$.  However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$.  The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem.  Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero.  Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.",
    "context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}",
    "full_context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\section{Solutions Modulo a Prime} \\label{sec:primes}\n\n\\begin{prop} There are no class $2$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $5 \\pmod{6}$.\n\\end{prop}\n\\begin{proof} \nNote that $c-b=2$ and $a!b!=c!$ implies $a! = c(c-1)$. Suppose $a!=c(c-1)$ where $a+1=p$ is a prime congruent to $5 \\pmod{6}$. Then\n$$\nc^2 - c - a! = 0\n$$ \nso we have $c = \\frac{1\\pm \\sqrt{1-4(-a!)}}{2}$. Thus, $1+4a! = 1 + 4(p-1)!$ is a perfect square. By Wilson's Theorem, $1+4(p-1)! \\equiv 1+4(-1) = -3 \\pmod{p}$, so $-3$ is a quadratic residue modulo $p$. However, this contradicts Lemma \\ref{lem:-3}.\n\\end{proof}\n\n\\begin{prop} \\label{thm:pIs4} There are no class $4$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $2$ or $3 \\pmod{5}$.\n\\end{prop}\n\\begin{proof}\nSuppose $a! = c(c-1)(c-2)(c-3)$, where $a=p-1$ for some prime $p$. Note that $b>0$ implies $c=b+4>4$, so $c!/b! \\geq 24 > 3!$, so we may assume $a>3$ and so $p>4$; furthermore, if $b>0$ then $4! = c(c-1)(c-2)(c-3)$ only has the solution $c=4$, but then $b = c-4 = 0$, a contradiction, so $p \\neq 5$. Let $r \\equiv c/2 \\pmod{p}$, and apply Wilson's Theorem to conclude that\n$$\n2r(2r-1)(2r-2)(2r-3) \\equiv -1 \\pmod{p}.\n$$\nNow, \n$$\n2r(2r-1)(2r-2)(2r-3) = (4r^2-6r+1)^2-1,\n$$\nso \n$$\n(4r^2-6r+1)^2 \\equiv 0 \\pmod{p},\n$$\ni.e., $p | 4r^2 - 6r + 1$. However, $(2r-3/2)^2 - 5/4 = 4r^2 - 6r + 1$, so $5/4$ is a quadratic residue mod $p$, which happens iff $5$ is a quadratic residue mod $p$. It is known this occurs only for primes $p \\equiv 1,4 \\pmod{5}$.\n\\end{proof}\n\n\\begin{theorem} \\label{thm:PositiveDensityIfIrreducible}Suppose $F_k(x) = x(x-1)\\cdots (x-k+1)+1$ is irreducible over $\\mathbb{Z}$. Let $P_0$ be the set of primes so that, if $a=p-1$ for some $p \\in P_0$, then $a!b!=c!$ has no class $k$ solutions in positive integers. Then the natural density of $P_0$ in the primes is at least $1/k$.\n\\end{theorem}\n\\begin{proof}\n Suppose $a!b!=c!$ is class $k$ and $a = p-1$ for some prime $p$. Then $c = b+k$ and $a! = c!/b! = F_k(c)-1$. Then, reducing this equation modulo $p$ yields $c^{\\underline{k}} \\equiv a! \\pmod{p}$, and by Wilson's Theorem, this is equivalent to $F_k(x) \\equiv 0 \\pmod{p}$. However, by Theorem 1 and 2 of \\cite{Serre03}, the set $P_0$ of primes $p$ for which $F_k(x)$ does not have solution modulo $p$ has positive density, and the density is at least $1/k$.\n\\end{proof}\n\n\\begin{prop}\\label{thm:WestlundFluge}\n For $k \\geq 2$, $F_k(x)$ is reducible over $\\mathbb{Z}$ iff $k=4$. \n\\end{prop}\n\\begin{proof}\n This is an immediate consequence of a result by Westlund and Fl\\\"{u}ge, answering a question of Schur, as discussed in \\cite{DorwartOre33}.\n\\end{proof}\n\n\\begin{cor} \\label{cor:SerreSet}\n Fix $k > 1$. For at least a $1/k$ natural density subset of primes $p$, the equation\n $$\n (p-1)! b! = (b+k)!\n $$\n has no solutions for integers $b$.\n\\end{cor}\n\\begin{proof}\n By Theorem \\ref{thm:WestlundFluge}, $F_k$ is irreducible if $k \\neq 4$, so Theorem \\ref{thm:PositiveDensityIfIrreducible} implies that $a!b!=c!$ has no solutions in the integers when $c = b+k$ and $a=p-1$, for an at least $1/k$ density subset of the primes. Proposition \\ref{thm:pIs4} yields the remaining case, by Dirichlet's Theorem on primes in arithmetic progressions.\n\\end{proof}\n\n\\begin{prop}\n For $p$ prime, the equation\n $$\n (p-1)!b! = (b+k)!\n $$\n has at most $nk/p + 1$ solutions in the integers with $b \\leq n$.\n\\end{prop}\n\\begin{proof}\n Suppose $(p-1)! b! = (b+k)!$. Then, $(p-1)! = F_k(b)-1$. Taking this equation modulo $p$ and applying Wilson's Theorem gives\n $$\n -1 \\equiv F_k(b) - 1 \\pmod{p},\n $$\n i.e., $F_k(b) \\equiv 0 \\pmod{p}$. However, this equation has at most $k$ solutions $A_k \\subseteq \\mathbb{Z}/p\\mathbb{Z}$. If $b \\pmod{p} \\not \\in A_k$, then $(p-1)!b! \\neq (b+k)!$. Since the set of $b \\in [1,n]$ so that $b \\pmod{p} \\in A_k$ has cardinality $\\lceil nk/p \\rceil \\leq nk/p + 1$, the conclusion follows.\n\\end{proof}",
    "post_theorem_intro_text_len": 874,
    "post_theorem_intro_text": "\\begin{proof}\n    Write $s_q(n)$ for the sum of the base-$q$ digits of $n$. Clearly, $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$.  Applying Legendre's Formula, we obtain\n    $$\n    \\frac{a-s_p(a)}{p-1} + \\frac{b-s_p(b)}{p-1} = \\frac{c-s_p(c)}{p-1},\n    $$\n    from which it follows that $a-(c-b) = s_p(a)+s_p(b)-s_p(c)$ for every prime $p$.\n    Since $a < b < c$, in particular\n    $$\n    | s_2(a)+s_2(b)-s_2(c) | < 2 \\lceil \\log_2 c \\rceil .\n    $$\n    Thus, $|a-k| < 2 \\lfloor \\log_2 c \\rfloor$.  Now, we argue that $a > k$.  First,\n    \\begin{align*}\n    a! & = \\frac{c!}{b!} = c(c-1)\\cdots(c-k+1) \\\\\n    & > (c-k+1)^k = (b+1)^k \\geq (a+2)^k.\n    \\end{align*}\n    Then taking the log of both sides above and applying Stirling's approximation (that $\\log a! < a \\log a$) gives\n    $$\n    a \\log a > \\log a! > k \\log (a+2)\n    $$\n    so that $k < a \\log a / \\log(a+2) < a$.\n\\end{proof}",
    "sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.",
    "expanded_sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.",
    "expanded_theorem": "\\label{prop:bound_a}\nIf $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\\,b!=c!$. Such a solution is said to be of class $k$ if $c-b=k$. Under these assumptions, which quantitative estimate for $a$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k\\le a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        },
        {
          "label": "C",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[a<k+2\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        },
        {
          "label": "D",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+2\\left\\lceil \\log_2 a\\right\\rceil.\\]"
        },
        {
          "label": "E",
          "text": "If $a!\\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \\[k<a<k+\\left\\lceil \\log_2 c\\right\\rceil.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "strict_lower_bound_from_stirling",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_lower_bound_k<a",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "dependence_on_c_in_digit_sum_bound",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "factor_2_from_base_2_digit_sum_estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the target estimate or give away the correct option. It only defines the setup and asks which bound on a holds."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially asking the test-taker to identify the correct formal estimate among nearby variants of a theorem-like statement. It is not a verbatim restatement in the stem, but it mainly tests recognition/recall of the exact bound rather than deriving a new conclusion from substantive premises."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways: strict versus weak inequality, dependence on c versus a, and the factor 2. However, without intermediate data or a proof context, success is driven more by memory of the result than by strong generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically well targeted. They reflect common failure modes: weakening a strict bound, dropping the lower bound, replacing c by a in the logarithm, and losing the factor 2. They are distinct and close enough to the correct statement to discriminate understanding."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with no answer leakage and strong distractors, but only moderate generative depth because it mainly tests recall of the exact estimate rather than substantial reasoning."
    }
  },
  {
    "id": "2512.03294v1",
    "paper_link": "http://arxiv.org/abs/2512.03294v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: main thm 2 and 3}\nFor $k=2,3$, \na shifted \n$k$-uniform \nhypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an \ninitial lex-segment.",
      "start_pos": 7957,
      "end_pos": 8174,
      "label": "thm: main thm 2 and 3"
    },
    "ref_dict": {
      "thm: main thm 2 and 3": "\\begin{theorem} \\label{thm: main thm 2 and 3}\nFor $k=2,3$, \na shifted \n$k$-uniform \nhypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an \ninitial lex-segment.\n\\end{theorem}",
      "main theorem": "\\begin{theorem} \\label{main theorem}\nLet $k\\ge 4$ and\nlet $H\\subseteq\\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \\eqref{S for case 1}-\\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.\n\\end{theorem}",
      "S for case 2": "\\begin{equation} \\label{S for case 2}\ns_H=\\{a,\\ldots,b,c-1,n-m-l,n-m-l+1,\\ldots,n-m-1,n-m,\\ldots,n\\}.\n\\end{equation}",
      "thm: symmetric case for graphs": "\\begin{proposition} \\label{thm: symmetric case for graphs}\n  A simple graph $H\\subseteq\\binom{[n]}{2}$ is s-matroidal if and only if the edges of $H$ form an initial lex-segment.  \n\\end{proposition}",
      "S for case 1": "\\begin{equation} \\label{S for case 1}\ns_H=\\{a,\\ldots,b,c-1,n-(l+1),n-l,\\ldots,n-1\\}.\n\\end{equation}",
      "conj:any_k": "\\begin{conjecture}\\label{conj:any_k}\nFor every natural numbers $1\\le k\\le n$, a shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order\n(initial lex-segment for short).\n\\end{conjecture}"
    },
    "pre_theorem_intro_text_len": 3712,
    "pre_theorem_intro_text": "A collection $H$ of $k$-element subsets of $[n]$  is {\\itshape shifted} if for every $i<j$, $j\\in S\\in H$, $i\\notin S$, also $(S\\setminus\\{j\\})\\cup \\{i\\}\\in H$. After fixing a field, a canonical operation of {\\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\\Delta(K)$, \nwas introduced by Kalai \\cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\\beta$-vector, while loses others, such as the homotopy type of $K$. \nVarious properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.\n\nTwo different variations of algebraic shifting were introduced \nby Kalai: {\\itshape exterior \nshifting} and {\\itshape symmetric \nshifting}. Given a simplicial complex $K$ we denote by $\\Delta^e(K)$ and $\\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for  statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.\n\nWe start our exposition with a simple known example. Let $n$ be a positive integer. The {\\itshape graphic matroid} of the complete graph on $n$ vertices is a collection of all spanning trees on the vertex set $[n]$. Let $G$ be such a spanning tree. Note that since algebraic shifting preserves Betti numbers, for every edge $e\\in\\Delta(G)$, we have $1\\in e$. Note also that since algebraic shifting preserves $f$-vector, it then follows that $\\Delta(G)=\\{12,13,\\ldots,1n\\}$. Conversely, one can similarly verify that every $G\\in\\Delta^{-1}(\\{12,13,\\ldots,1n\\})$ is a spanning tree on $[n]$. Thus, we conclude that $\\Delta^{-1}(\\{12,13,\\ldots,1n\\})$ coincides with the graphic matroid of the complete graph on $n$ vertices. The main goal of this paper is to study this phenomenon in a general setting.\n\nLet $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex). \nDenote by\n$$M(H)=\\left\\{G\\subseteq \\binom{[n]}k|\\ \\Delta^e(G)=H\\right\\}$$\nall $k$-uniform hypergraphs whose exterior shifting is $H$. \nWe say that\n$H$ is {\\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.\n\nFor example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \\cite{CR1, CR2}, \nwhose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed \nfield; see e.g.\n~\\cite[Chapter 6]{White} and \\cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\\{ij:\\ 1\\le i<j\\le n, \\ i\\le k\\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\\cite{K Hyper}; see e.g. \\cite[Section 4, Remark (2)]{K Symm} and \\cite[Section 2.7]{K Alg Sh} for further background. If $H=\\{1ij:\\ 2\\le i<j\\le n, \\ i\\le 3\\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\\cite[Theorem 1.2]{BNP}. \n\nWe conjecture a combinatorial characterization of the algebraic property of being \\emph{matroidal}:\n\\begin{conjecture}\\label{conj:any_k}\nFor every natural numbers $1\\le k\\le n$, a shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order\n(initial lex-segment for short).\n\\end{conjecture}\n\nThe \"if\" part follows easily from a result of Kalai~\\cite[Section 4]{K Symm}. \nWe prove the \"only if\" part for $k=2$ (i.e. for graphs) and for $k=3$:",
    "context": "A collection $H$ of $k$-element subsets of $[n]$ is {\\itshape shifted} if for every $i<j$, $j\\in S\\in H$, $i\\notin S$, also $(S\\setminus\\{j\\})\\cup \\{i\\}\\in H$. After fixing a field, a canonical operation of {\\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\\Delta(K)$, \nwas introduced by Kalai \\cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\\beta$-vector, while loses others, such as the homotopy type of $K$. \nVarious properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.\n\nTwo different variations of algebraic shifting were introduced \nby Kalai: {\\itshape exterior \nshifting} and {\\itshape symmetric \nshifting}. Given a simplicial complex $K$ we denote by $\\Delta^e(K)$ and $\\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.\n\nLet $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex). \nDenote by\n$$M(H)=\\left\\{G\\subseteq \\binom{[n]}k|\\ \\Delta^e(G)=H\\right\\}$$\nall $k$-uniform hypergraphs whose exterior shifting is $H$. \nWe say that\n$H$ is {\\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.\n\nFor example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \\cite{CR1, CR2}, \nwhose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed \nfield; see e.g.\n~\\cite[Chapter 6]{White} and \\cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\\{ij:\\ 1\\le i<j\\le n, \\ i\\le k\\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\\cite{K Hyper}; see e.g. \\cite[Section 4, Remark (2)]{K Symm} and \\cite[Section 2.7]{K Alg Sh} for further background. If $H=\\{1ij:\\ 2\\le i<j\\le n, \\ i\\le 3\\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\\cite[Theorem 1.2]{BNP}.\n\nWe conjecture a combinatorial characterization of the algebraic property of being \\emph{matroidal}:\n\\begin{conjecture}\\label{conj:any_k}\nFor every natural numbers $1\\le k\\le n$, a shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order\n(initial lex-segment for short).\n\\end{conjecture}\n\nThe \"if\" part follows easily from a result of Kalai~\\cite[Section 4]{K Symm}. \nWe prove the \"only if\" part for $k=2$ (i.e. for graphs) and for $k=3$:",
    "full_context": "A collection $H$ of $k$-element subsets of $[n]$ is {\\itshape shifted} if for every $i<j$, $j\\in S\\in H$, $i\\notin S$, also $(S\\setminus\\{j\\})\\cup \\{i\\}\\in H$. After fixing a field, a canonical operation of {\\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\\Delta(K)$, \nwas introduced by Kalai \\cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\\beta$-vector, while loses others, such as the homotopy type of $K$. \nVarious properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.\n\nTwo different variations of algebraic shifting were introduced \nby Kalai: {\\itshape exterior \nshifting} and {\\itshape symmetric \nshifting}. Given a simplicial complex $K$ we denote by $\\Delta^e(K)$ and $\\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.\n\nLet $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex). \nDenote by\n$$M(H)=\\left\\{G\\subseteq \\binom{[n]}k|\\ \\Delta^e(G)=H\\right\\}$$\nall $k$-uniform hypergraphs whose exterior shifting is $H$. \nWe say that\n$H$ is {\\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.\n\nFor example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \\cite{CR1, CR2}, \nwhose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed \nfield; see e.g.\n~\\cite[Chapter 6]{White} and \\cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\\{ij:\\ 1\\le i<j\\le n, \\ i\\le k\\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\\cite{K Hyper}; see e.g. \\cite[Section 4, Remark (2)]{K Symm} and \\cite[Section 2.7]{K Alg Sh} for further background. If $H=\\{1ij:\\ 2\\le i<j\\le n, \\ i\\le 3\\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\\cite[Theorem 1.2]{BNP}.\n\nWe conjecture a combinatorial characterization of the algebraic property of being \\emph{matroidal}:\n\\begin{conjecture}\\label{conj:any_k}\nFor every natural numbers $1\\le k\\le n$, a shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order\n(initial lex-segment for short).\n\\end{conjecture}\n\nThe \"if\" part follows easily from a result of Kalai~\\cite[Section 4]{K Symm}. \nWe prove the \"only if\" part for $k=2$ (i.e. for graphs) and for $k=3$:\n\nWe conjecture a combinatorial characterization of the algebraic property of being \\emph{matroidal}:\n\\begin{conjecture}\\label{conj:any_k}\nFor every natural numbers $1\\le k\\le n$, a shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order\n(initial lex-segment for short).\n\\end{conjecture}\n\nFor $k\\ge 4$ we prove the \"only if\" part for $k$-uniform hypergraphs satisfying some technical condition on their structure.\n\n\\begin{theorem} \\label{main theorem}\nLet $k\\ge 4$ and\nlet $H\\subseteq\\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \\eqref{S for case 1}-\\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.\n\\end{theorem}\n\nLet $X=(x_{ij})_{1\\le i,j\\le n}$ be an $n\\times n$-matrix over $\\mathbb L$ which is generic\nover $\\mathbb F$. For a $k$-uniform hypergraph $H\\subseteq\\binom{[n]}k$ define a matroid $M'(H)$ on $\\binom{[n]}k$ as follows: $B\\subseteq\\binom{[n]}k$ is independent in $M'(H)$ iff the rows of $X(B,H)$ are linearly independent. \n\\begin{remark} \\cite[Section 4, Remark (2)]{K Symm} \\label{Remark: Kalai's construction for hypperconnectivity}\n For $H=\\{S\\subseteq\\binom{[n]}2:S\\cap[k]\\ne\\emptyset\\},\\ M'(H)$ is the $k$-hyperconnectivity matroid. \n\\end{remark}\nIn fact, a more general statement is straightforward from the construction above. We call a subset $H\\subseteq\\binom{[n]}{k}$ an {\\itshape initial lex-segment} \nif $H=\\left\\{T'\\in\\binom{[n]}{k}|\\ T'\\le_{\\text{lex}}T\\right\\}$ for some $T\\in\\binom{[n]}{k}$.\n\\begin{corollary}\n\\label{Initial segment is a matroid}\n Let $H\\subseteq\\binom{[n]}k$ be a $k$-uniform hypergraph whose hyperedges form an initial lex-segment. Then $H$ is matroidal.\n\\end{corollary}\n\\begin{proof}\nWe will verify that $M(H)$ is the set of bases of the matroid $M(H')$, or $M(H)=M'(H)$ for short.\nLet $B\\in M(H)$, then $H$ is a column basis chosen in the greedy way for the matrix $X(B,\\binom{[n]}{k})$. Thus, the columns of the matrix $X(B,H)$ are linearly independent, hence $B\\in M'(H)$.\n\n\\begin{lemma}[Isolated vertex] \\label{lemma: partial star with an additional edge 2,3 is not a matroid}\n Let $H\\subseteq\\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $n$ is an isolated vertex and $\\{2,3,\\ldots,k+1\\}\\in H$. \n Then $H$ is not matroidal. \n\\end{lemma}\n\\begin{proof}\n Consider two $k$-uniform hypergraphs $H_1=H$ and $H_2$ obtained from $H$ by flipping the vertex $1$ with the vertex $n$ (see Fig. \\ref{graphs which do not satisfy the exchange exiom} for illustration). By its structure $H_1$ contains the facets of the boundary complex $\\partial \\sigma$ of the simplex $\\sigma=\\{1,2,\\ldots,k+1\\}$. In order to show that the pair $H_1, H_2$ violates the exchange axiom we have to find $e_1\\in H_1\\setminus H_2$ such that for every $e_2\\in H_2\\setminus H_1$ we have that\n $(H_1\\setminus e_1)\\cup e_2\\notin M(H)$. Let us take as $e_1$ an arbitrary facet of \n $\\partial \\sigma$\n which contains $1$. Such $e_1$ indeed lies in $H_1\\setminus H_2$ since $1$ is an isolated vertex in $H_2$. Note that $\\beta_{k-1}(H_1\\setminus e_1)<\\beta_{k-1}(H_1)$. As $e_2$ we can take only hyperedges containing the vertex $n$, which is an isolated vertex in $H_1$. Thus, $e_2$ \n does not belong to any $(k-1)$-homology cycle in $(H_1\\setminus e_1)\\cup e_2$. Thus, $\\beta_{k-1}((H_1\\setminus e_1)\\cup e_2)=\\beta_{k-1}(H_1\\setminus e_1)<\\beta_{k-1}(H_1)$. However, since algebraic shifting preserves Betti numbers, see Theorem \\ref{Basic properties of algebraic shifting}(4), we have $\\beta_{k-1}(\\Delta((H_1\\setminus e_1)\\cup e_2))=\\beta_{k-1}((H_1\\setminus e_1)\\cup e_2)=\\beta_{k-1}(H_1\\setminus e_1)<\\beta_{k-1}(H_1)=\\beta_{k-1}(H)$. Thus, $((H_1\\setminus e_1)\\cup e_2)\\notin M(H)$.\n\\end{proof}\n\n\\begin{example} \\label{example: of not matroidal graph}\nConsider the $2$-uniform hypergraph $H=\\{12,13,23\\}$ on $[5]$. Note that the (hyper)edges of $H$ do not form an initial lex-segment. Thus, by Theorem \\ref{main theorem} $H$ is not matroidal. In this case one can verify this directly. Since algebraic shifting preserves $f$-vector and $\\beta$-vector (see Theorem \\ref{Basic properties of algebraic shifting}(3,4)) and the only shifted $2$-uniform hypergraph on $[5]$ with $3$ (hyper)edges and an one cycle is $H$, we conclude that $M(H)$ consists of all $2$-uniform hypergraph on $[5]$ with \n$3$ edges forming a triangle boundary.\nHowever, the pair of elements\n$\\{12,13,23\\},\\ \\{14,15,45\\}\\in M(H)$ does not satisfy the exchange axiom.\n\\end{example}\n\n\\begin{definition}\nLet $H$ be a shifted $k$-uniform hypergraph on $[n]$. $H$ is {\\itshape s-matroidal} if $M^s(H)$ is the set of bases of a matroid on $\\binom{[n]}{k}$. \n\\end{definition}\n\\begin{proposition} \\label{thm: symmetric case for graphs}\n A simple graph $H\\subseteq\\binom{[n]}{2}$ is s-matroidal if and only if the edges of $H$ form an initial lex-segment. \n\\end{proposition}\n\n\\begin{proposition} \\label{thm: lex-matroidal}\n A shifted $k$-uniform hypergraph $H\\subseteq\\binom{[n]}{k}$ is lex-matroidal iff the hyperedges of $H$ form an initial lex-segment.\n\\end{proposition}\n\\begin{proof}\n By Corollary \\ref{Initial segment is a matroid} if the hyperedges of $H$ form an initial lex-segment, then $H$ is matroidal. For every hyperedge $S$ of $H$ the set $H(S)$ is an initial lex-segment, and thus it is matroidal.\n\n\\begin{equation} \\label{S for case 1}\ns_H=\\{a,\\ldots,b,c-1,n-(l+1),n-l,\\ldots,n-1\\}.\n\\end{equation}\n\n\\begin{equation} \\label{S for case 2}\ns_H=\\{a,\\ldots,b,c-1,n-m-l,n-m-l+1,\\ldots,n-m-1,n-m,\\ldots,n\\}.\n\\end{equation}",
    "post_theorem_intro_text_len": 2291,
    "post_theorem_intro_text": "For $k\\ge 4$ we prove the \"only if\" part for $k$-uniform hypergraphs satisfying some technical condition on their structure. \n\n\\begin{theorem} \\label{main theorem}\nLet $k\\ge 4$ and\nlet $H\\subseteq\\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \\eqref{S for case 1}-\\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.\n\\end{theorem}\n\nFor the \"only if\" part of Theorem \\ref{thm: main thm 2 and 3} we first make a reduction of the general problem to a certain restricted subset of hypergraphs; and for each of them we find a suitable permutation $\\pi:[n]\\rightarrow [n]$ such that the pair $(H, \\pi(H))$, where $\\pi(H)=\\{\\pi(T):\\ T\\in H\\}$, violates the exchange axiom among the elements of $M(H)$. For the \"only if\" part of Theorem \\ref{main theorem} we find a suitable such permutation $\\pi=\\pi(s_M)$ for each $H$.\n\nThe analog of Theorem~\\ref{thm: main thm 2 and 3} \nfor symmetric algebraic shifting is proved \nfor $k=2$ in Proposition~\\ref{thm: symmetric case for graphs}. \nFor $H=\\{ij:\\ 1\\le i\\le j \\le n, i\\le k\\}$ the corresponding matroid is the (generic) $k$-rigidity matroid, see e.g. \\cite{Lee} and  \\cite[Section 2.7]{K Alg Sh}, \\cite[Section 3.2]{Nevo phd} for further background, references and for a discussion of this correspondence.\n\\\n\nThis paper is organized as follows. In Section \\ref{sec: preliminaries} we give a background on algebraic shifting and matroids. In Section \\ref{sec: Preparatory lemmas} we prove lemmas needed for proofs of Theorem \\ref{thm: main thm 2 and 3} and Theorem \\ref{main theorem}, and in Section \\ref{sec: Proofs of main results} we prove these theorems. In Section \\ref{sec: Matroids arising from symmetric shifting} we consider matroids arising from symmetric shifting, and in Proposition \\ref{thm: symmetric case for graphs} classify all matroidal  \nsimple graphs \nw.r.t. symmetric shifting. In Section \\ref{sec: Some partial results} we replace the matroidal property by a stronger algebraic property in Conjecture~\\ref{conj:any_k} that allows us to prove the resulted \"iff\" combinatorial characterization.  \nIn Section \\ref{sec: Concluding remarks} we conclude with related open problems.",
    "sketch": "For the ``only if'' part of Theorem~\\ref{thm: main thm 2 and 3} the approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\\pi:[n]\\rightarrow [n]$ such that the pair $(H, \\pi(H))$, where $\\pi(H)=\\{\\pi(T):\\ T\\in H\\}$, violates the exchange axiom among the elements of $M(H)$.''\n\nFor the ``only if'' part of Theorem~\\ref{main theorem} (the $k\\ge 4$ theorem under the condition $s_H\\notin H$), the method is similarly to ``find a suitable such permutation $\\pi=\\pi(s_M)$ for each $H$.''",
    "expanded_sketch": "For the ``only if'' part of \\begin{theorem} \\label{thm: main thm 2 and 3}\nFor $k=2,3$, \na shifted \n$k$-uniform \nhypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an \ninitial lex-segment.\n\\end{theorem}\nthe approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\\pi:[n]\\rightarrow [n]$ such that the pair $(H, \\pi(H))$, where $\\pi(H)=\\{\\pi(T):\\ T\\in H\\}$, violates the exchange axiom among the elements of $M(H)$.''\n\nFor the ``only if'' part of the main theorem (the $k\\ge 4$ theorem under the condition $s_H\\notin H$), the method is similarly to ``find a suitable such permutation $\\pi=\\pi(s_M)$ for each $H$.''",
    "expanded_theorem": "\\label{thm: main thm 2 and 3}\nFor $k=2,3$, \na shifted \n$k$-uniform \nhypergraph $H\\subseteq\\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an \ninitial lex-segment.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Universal"
    ],
    "mcq": {
      "question": "Let k\\in\\{2,3\\}, let [n]=\\{1,\\dots,n\\}, and let H\\subseteq \\binom{[n]}{k} be a shifted k-uniform hypergraph, meaning that for every i<j, whenever j\\in S\\in H and i\\notin S, one also has (S\\setminus\\{j\\})\\cup\\{i\\}\\in H. Define\n\\[\nM(H)=\\{G\\subseteq \\tbinom{[n]}{k}: \\Delta^e(G)=H\\},\n\\]\nwhere \\(\\Delta^e\\) denotes exterior algebraic shifting. Say that H is matroidal if \\(M(H)\\) is the set of bases of a matroid on \\(\\binom{[n]}{k}\\). Also, call H an initial lex-segment if there exists \\(T\\in \\binom{[n]}{k}\\) such that\n\\[\nH=\\{T'\\in \\tbinom{[n]}{k}: T'\\le_{\\mathrm{lex}} T\\},\n\\]\nwith \\(\\le_{\\mathrm{lex}}\\) the usual lexicographic order on k-subsets. Which statement holds for every such H?",
      "correct_choice": {
        "label": "A",
        "text": "H is matroidal if and only if the (hyper)edges of H form an initial lex-segment."
      },
      "choices": [
        {
          "label": "B",
          "text": "H is matroidal if and only if the (hyper)edges of H form a lex-segment after relabeling the vertices, i.e. there exists a permutation \\(\\pi\\in S_n\\) such that \\(\\pi(H)=\\{T'\\in \\tbinom{[n]}{k}: T'\\le_{\\mathrm{lex}} T\\}\\) for some \\(T\\in \\tbinom{[n]}{k}\\)."
        },
        {
          "label": "C",
          "text": "If the (hyper)edges of H form an initial lex-segment, then H is matroidal."
        },
        {
          "label": "D",
          "text": "H is matroidal if and only if the (hyper)edges of H form an initial colex-segment."
        },
        {
          "label": "E",
          "text": "If H is matroidal, then there exists a permutation \\(\\pi:[n]\\to[n]\\) such that the pair \\((H,\\pi(H))\\) satisfies the exchange axiom inside \\(M(H)\\), and hence H is matroidal exactly when this happens."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "fixed-label initial lex vs up-to-permutation lex",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped only-if direction",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "lexicographic order replaced by colexicographic order",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "permutation in proof used to witness failure of exchange axiom, reversed into existence criterion for matroidality",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or trivially imply the correct equivalence. It defines the relevant notions, but the correct answer is not leaked beyond introducing the concepts under comparison."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-identification item: the correct option is the theorem statement itself, with nearby variants as alternatives. It tests recall of the precise characterization more than a nontrivial derived conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle ways (iff vs one-way, fixed lex vs relabeling, lex vs colex). However, the item mostly rewards recognizing the exact known theorem rather than generating a conclusion from premises."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: weakening an equivalence, allowing relabeling, confusing lex with colex, and misusing an exchange-style condition. They are distinct and well aligned with likely misconceptions."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no major answer leakage, but it is fairly tautological and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.03489v1",
    "paper_link": "http://arxiv.org/abs/2512.03489v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.",
      "start_pos": 28767,
      "end_pos": 29036,
      "label": "Thm: Hypercontractivity on tower of n"
    },
    "ref_dict": {
      "Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k": "\\begin{theorem}\\label{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}\nFor $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have the following LSI with the optimal constant $2$:\n\\[\n  \\int_{\\mathbb{Z}_n} f^2\\log f^2\\dd\\mu_n-\\norm{f}_2^2\\log\\norm{f}_2^2\\le 2\\inner{f,A_{\\psi_n}f}_{L_2(\\mathbb{Z}_n,\\mu_n)},\\qquad f\\in L_2^+(\\mathbb{Z}_n,\\mu_n).\n\\]\n\\end{theorem}",
      "Thm: Hypercontractivity on tower of n": "\\begin{theorem}\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3821,
    "pre_theorem_intro_text": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n  P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n    \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par \n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.",
    "context": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par\n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.",
    "full_context": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par\n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.\n\nWithout loss of generality, suppose $\\lambda_{j}<\\lambda_{j+3}$. The function $F$ strictly increases on $(0,1)$ and strictly decreases on $(1,\\infty)$, so from \\eqref{eq: opposite F} with $\\nu_{j}=\\nu_{j+3}=0$ proved in Step 1, we get\n\\begin{equation}\n F(\\lambda_{j}) = F(\\lambda_{j+3}).\n\\end{equation}\nAnd hence\n\\[\n \\lambda_{j}<1<\\lambda_{j+3}.\n\\]\nDefine $\\Theta(x)\\coloneqq F(x)-F(2-x)$. Then $\\Theta(1)=0$ and $\\Theta'(x)=-\\frac{2}{3}\\log\\big(x(2-x)\\big)>0$ on $(0,1)$, so $\\Theta(x)<0$ on $(0,1)$. Applying this to $x=\\lambda_{j}$ gives $F(\\lambda_{j})<F(2-\\lambda_{j})$. Since $F$ decreases on $(1,\\infty)$ and $F(\\lambda_{j+3})=F(\\lambda_{j})$, we deduce\n\\begin{equation}\\label{Ineq: opposite sum lower bound}\n \\lambda_{j+3}>2-\\lambda_{j}\n \\quad\\Rightarrow\\quad\n \\lambda_{j}+\\lambda_{j+3}>2.\n\\end{equation}\\par\n\\medskip\n\\textbf{Step 3.} We show that for $\\ell,\\ell'\\in\\{0,1,2\\}\\setminus\\{j_0\\}$, we have $\\lambda_{\\ell}=\\lambda_{\\ell+3}=\\lambda_{\\ell'}=\\lambda_{\\ell'+3}$.\\par \nBy the assumption $\\lambda_{j_0}\\neq \\lambda_{j_0+3}$, we have $\\lambda_{j_0}+\\lambda_{j_0+3}>2$ from Step 2. Hence\n\\[\n \\lambda_{j_0}^2+\\lambda_{j_0+3}^2 \\ge \\frac{(\\lambda_{j_0}+\\lambda_{j_0+3})^2}{2} > 2.\n\\]\nConsequently, if $\\lambda_{j}\\neq \\lambda_{j+3}$ for all $0\\leq j\\le 2$, then $\\sum_{k=0}^{5}\\lambda_k^2>6$, contradicting $\\sum_{k=0}^{5}\\lambda_k^2<6$ in \\eqref{Eqns: Lagrange for n=6, (0,1,2,1,2,1)}. Therefore, there exists at least one $\\ell$ with $\\lambda_{\\ell}=\\lambda_{\\ell+3}$. Let $\\ell$ be an index in $\\{0,1,2\\}\\setminus\\{j_0\\}$ such that $\\lambda_\\ell=\\lambda_{\\ell+3}$ and $\\lambda_\\ell$ is minimal among all pairs $\\{\\lambda_j=\\lambda_{j+3}\\}$ with $j\\in\\{0,1,2\\}\\setminus\\{j_0\\}$. If $\\lambda_\\ell=\\lambda_{\\ell+3}\\ge 1$, then $\\sum_{k=0}^{5}\\lambda_k^2>6$, which again contradicts $\\sum_{k=0}^{5}\\lambda_k^2<6$. So there at least one $\\ell$ with $\\lambda_{\\ell}=\\lambda_{\\ell+3}<1$. Denote by $\\ell'$ the remaining index in $\\{0,1,2\\}\\setminus\\{j_0,\\ell\\}$. \\par\n\n\\begin{proof}\nRecall the case $n=2$ for Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is known in \\cite{MR420249}, i.e.,\n\\begin{equation}\\label{Ineq: 2-LSI}\n \\frac{1}{4}\\left( x^2\\log\\frac{2x^2}{x^2+y^2}+y^2\\log\\frac{2y^2}{x^2+y^2}\\right)\n\\le \\left( \\frac{x-y}{2} \\right)^2\\qquad x,y\\geq 0.\n\\end{equation}\nGiven $\\lambda=(a_0,b_0,a_1,b_1,\\ldots,a_{n-1},b_{n-1})$ with $a=(a_0,\\ldots,a_{n-1})$ and $b=(b_0,\\ldots,b_{n-1})$, the $2n$--entropy splits as\n\\[\n\\begin{aligned}\n\\mathrm{H}_{2n}[\\lambda]&=\\frac{1}{2n}\\sum_{i=0}^{n-1} a_i^2 \\log(\\frac{2na_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\\frac{1}{2n}\\sum_{i=0}^{n-1} b_i^2 \\log(\\frac{2nb_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\\\\\n&=\\frac{1}{2n}\\sum_{i=0}^{n-1} a_i^2\\log(\\frac{n a_i^2}{\\sum_{i=0}^{n-1}a_i^2})+\\frac{1}{2n}\\sum_{i=0}^{n-1} b_i^2\\log(\\frac{n b_i^2}{\\sum_{i=0}^{n-1}b_i^2})\\\\\n &\\quad +\\frac{1}{2n}\\left(\\sum_{i=0}^{n-1} a_i^2\\log(\\frac{2 \\sum_{i=0}^{n-1} a_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\\sum_{i=0}^{n-1} b_i^2\\log(\\frac{2 \\sum_{i=0}^{n-1} b_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\\right).\n\\end{aligned}\n\\]\nApplying the $n$--LSI to the first two terms, and $2$--LSI \\eqref{Ineq: 2-LSI} with $x=\\norm{a}_2$ and $y=\\norm{b}_2$ to the last term, yields\n\\[\n\\mathrm{H}_{2n}[\\lambda]\\le \\inner{a,\\Gamma(n)a}+\\inner{b,\\Gamma(n)b}\n+\\frac{1}{2n}\\big(\\norm{a}_2-\\norm{b}_2\\big)^2.\n\\]\nFinally, by Proposition~\\ref{Prop: Compare Dirichlet form for n and 2n},\n\\begin{equation}\\label{Eqn: induction from n-LSI to 2n-LSI}\n \\mathrm{H}_{2n}[\\lambda]\\le \\inner{a,\\Gamma(n)a}+\\inner{b,\\Gamma(n)b}\n+\\frac{1}{2n}\\big(\\norm{a}_2-\\norm{b}_2\\big)^2\n\\le 2\\inner{\\lambda,\\Gamma(2n)\\lambda}.\n\\end{equation}\\par\n\nFor odd $n_0$, define the weight function on $\\mathbb{Z}_n$ for even $n\\geq n_0$\n\\begin{equation}\\label{Eqn: definition of induced gamma_n for odd base case}\n \\gamma_n(k)=\n \\begin{cases}\n \\psi_n(k), & k\\ne \\tfrac{n}{2},\\\\\n 1, & k=\\tfrac{n}{2}.\n \\end{cases}\n\\end{equation}\nThe pair $(\\psi_{n_0},\\gamma_{2n_0})$ satisfies \\eqref{Cond: gamma pair condition} and \\eqref{Ineq: Quadratic inequality from n to 2n}. For even $n$, the pair $(\\gamma_n,\\gamma_{2n})$ also satisfies \\eqref{Cond: gamma pair condition}, and the desired inequality \\eqref{Ineq: Quadratic inequality from n to 2n} becomes\n\\begin{equation}\\label{Ineq: quadratic for odd n}\n f(x)\\coloneqq r_b (n-3) x^2+x \\left(2 \\sqrt{r_a+1} \\sqrt{r_b+1}-2\\right)+r_a (n-3)\\geq 0\n\\end{equation}\nfor all $x\\ge 0$ and $0\\le r_a,r_b\\le 1$. Since $n-3>0$, the minimum of $f$ is attained at $x=-\\frac{2 \\sqrt{r_a+1} \\sqrt{r_b+1}-2}{2r_b(n-3)}\\leq 0$ and the value of the minimum of $f$ on $[0,\\infty)$ is $f(0)=r_a(n-3)\\geq 0$. Hence \\eqref{Ineq: quadratic for odd n} holds. Therefore Proposition~\\ref{Prop: Compare Dirichlet form for n and 2n} applies to the pairs $(\\gamma_n,\\gamma_{2n})$ defined by \\eqref{Eqn: definition of induced gamma_n for odd base case}. Iterating \\eqref{Eqn: induction from n-LSI to 2n-LSI} along $\\big((\\psi_{n_0},\\gamma_{2n_0}),(\\gamma_{2n_0},\\gamma_{4n_0}),\\ldots\\big)$, we obtain for all $m\\ge 0$,\n\\[\n \\mathrm{H}_{n_0\\cdot 2^{m+1}}[\\lambda]\n \\le 2\\inner{\\lambda,\\Gamma(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nLet $\\Psi(n)=\\frac{1}{n}F_n\\diag(\\psi_n)F_n^{-1}$. Since $\\gamma_n\\leq \\psi_n$ pointwise, we have\n\\[\n \\inner{\\lambda,\\Gamma(n_0\\cdot 2^{m+1})\\lambda} \n \\le \\inner{\\lambda,\\Psi(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nTherefore,\n\\[\n \\mathrm{H}_{n_0\\cdot 2^{m+1}}[\\lambda]\n \\le 2\\inner{\\lambda,\\Psi(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nBy Lemma~\\ref{Lem: Explicit form of Log Sobolev ineq}, the \\(n_0\\cdot 2^{m+1}\\)--LSI holds for \n\\(\\psi_{n_0\\cdot 2^{m+1}}\\) for all \\(m\\ge 0\\), which proves the claim.\n\\end{proof}\n\n\\textbf{(2) Case $n=2^m$ with $m\\geq 1$.} The case $n=2$ is classical and the case $n=4$ is due to the work \\cite{MR730056} and Gross's extrapolation technique~\\cite{MR420249}. Note that the pair $(\\phi_4,\\gamma_8)$ satisfies \\eqref{Cond: gamma pair condition}, and \\eqref{Ineq: Quadratic inequality from n to 2n} becomes\n\\[\n f(x)\\coloneqq \\left(2-\\frac{r_b}{5}\\right)x^2+2x\\left(\\sqrt{r_a+1}\\sqrt{r_b+1}-3\\right)\n -\\frac{r_a}{5}+2\\ \\ge 0,\n\\]\nwith $0\\le r_a,r_b\\le 1$ and $x\\ge0$. As $2-\\frac{r_b}{5}>0$, the minimum of $f$ on $[0,\\infty)$ is attained at $x=-\\tfrac{2 \\left(\\sqrt{r_a^2+1} \\sqrt{r_b^2+1}-3\\right)}{2\\left( 2-r_b^2/5 \\right)}$ and the value of the minimum is\n\\[\n \\frac{r_a(24r_b+35)+5\\left(-30\\sqrt{r_a+1}\\sqrt{r_b+1}+7r_b+30\\right)}\n {5(r_b-10)}\n =\\frac{h(r_a,r_b)}{5(r_b-10)}.\n\\]\nSince $r_b\\in[0,1]$, the denominator is negative. In order to show \\eqref{Ineq: Quadratic inequality from n to 2n} hold, it suffices to show $h(r_a,r_b)\\le 0$ on $[0,1]^2$. A direct computation gives\n\\[\n \\pdv[2]{h}{r_a}(r_a,r_b)=\\frac{75\\sqrt{r_b+1}}{2(r_a+1)^{3/2}}>0,\n \\qquad\n \\pdv[2]{h}{r_b}(r_a,r_b)=\\frac{75\\sqrt{r_a+1}}{2(r_b+1)^{3/2}}>0.\n\\]\nThus $h$ is convex in each variable separately. So the maximum of $h$ on $[0,1]^2$ is attained at a corner. Evaluating,\n\\[\n h(0,0)=0,\\quad h(0,1)=h(1,0)=5\\big(37-30\\sqrt{2}\\big)<0,\\quad h(1,1)=-56<0.\n\\]\nHence $\\max_{[0,1]^2} h\\le 0$, and since $5(r_b-10)<0$ we conclude\n\\[\n \\frac{r_a(24r_b+35)+5\\left(-30\\sqrt{r_a+1}\\sqrt{r_b+1}+7r_b+30\\right)}\n {5(r_b-10)} \\ge 0.\n\\]\nTherefore $(\\phi_4,\\gamma_8)$ satisfies \\eqref{Ineq: Quadratic inequality from n to 2n}. Recall that $(\\gamma_{8\\cdot 2^m},\\gamma_{8\\cdot 2^{m+1}})$ also satisfies \\eqref{Ineq: Quadratic inequality from n to 2n}. Hence by Theorem~\\ref{Thm: Log Sobolev inequality n=4} and Theorem~\\ref{Thm: induction from n-LSI to 2n-LSI} the $8\\cdot 2^m$--LSI holds for $\\gamma_{8\\cdot 2^m}$ for all $m\\ge 0$, and therefore for $\\psi_{8\\cdot 2^m}$.\n\\end{proof}",
    "post_theorem_intro_text_len": 3382,
    "post_theorem_intro_text": "A standard route to hypercontractivity proceeds through Log--Sobolev inequalities (LSI) by Gross's celebrated work~\\cite{MR420249}: $\\norm{P_{t}f}_q\\le \\norm{f}_p$ holds whenever $t\\geq \\frac{C}{4}\\log\\big(\\frac{q-1}{p-1}\\big)$ if and only if the corresponding LSI holds with constant $C$. Thus, to prove Theorem~\\ref{Thm: Hypercontractivity on tower of n}, it suffices to establish the following $n$--LSI with the optimal constant $2$ along the above dyadic tower of $n$. Denote by $A_{\\psi_n}$ the generator of semigroup $(P_t)_{t\\in\\mathbb{R}_+}$, that is\n\\begin{equation}\n  A_{\\psi_n}: \\sum_{k=0}^{n-1} a_k\\chi_k(x)\\mapsto \\sum_{k=0}^{n-1} \\psi_n(k)a_k\\chi_k(x).\n\\end{equation}\n The case $n=4$ in the following theorem is due to the work \\cite{MR730056} and Gross's extrapolation technique~\\cite{MR420249}.\n\\begin{theorem}\\label{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}\nFor $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have the following LSI with the optimal constant $2$:\n\\[\n  \\int_{\\mathbb{Z}_n} f^2\\log f^2\\dd\\mu_n-\\norm{f}_2^2\\log\\norm{f}_2^2\\le 2\\inner{f,A_{\\psi_n}f}_{L_2(\\mathbb{Z}_n,\\mu_n)},\\qquad f\\in L_2^+(\\mathbb{Z}_n,\\mu_n).\n\\]\n\\end{theorem}\n\nOur proof of Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is based on a new induction scheme with three key ingredients:\n\\begin{enumerate}\n  \\item Auxiliary weights $\\phi_4$ on $\\mathbb{Z}_4$ and $\\phi_6$ on $\\mathbb{Z}_6$ together with their corresponding LSIs. These LSIs are tighter than those for the word-lengths $\\psi_n$; this refinement is new even when $n=4$ and plays a key role in the analysis of LSIs.\n  \\item Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI.  \nWe develop an efficient way to handle LSIs on $\\mathbb{Z}_n$ via KKT analysis, combined with the aforementioned manipulation of the length functions. The specific structure of $\\phi_6$ introduces a symmetry in the KKT system that makes the analysis tractable.\n  \\item An induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition. We use a Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT), which expresses a large DFT as a combination of smaller DFTs and yields a comparison of Dirichlet forms at the scales $n$ and $2n$. The choice of the new weights $\\phi_4$ and $\\phi_6$ mentioned in (1) is crucial for the base steps $4\\to 8$ and $6\\to 12$.\n\\end{enumerate}\nFinally, we remark that the above ideas are also useful for studying LSIs along other towers of the form $m\\cdot n^k$. The KKT analysis and the compatibility condition vary in their technical details, depending on the specific value of $n$ and on the particular choice of weights on $\\mathbb{Z}_n$, and we will not carry out this analysis in this paper.\\par\n\nThe article is organized as follows. After a brief introduction to the LSI formulation and the $n$-dimensional KKT framework in Section~\\ref{Sec: DFT and KKT analysis}, we analyze the KKT systems associated with the LSIs for $n=6$ and $n=4$ in Section~\\ref{Sec: Log Sobolev inequality n=4 and n=6}. Section~\\ref{Sec: Induction from n to 2n} presents a comparison criterion for a pair of weights that allows us to compare the Dirichlet forms at the scales $n$ and $2n$. We then apply it to the transition $n\\to 2n$ of LSIs and establish Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.",
    "sketch": "To prove Theorem~\\ref{Thm: Hypercontractivity on tower of n}, the text follows the standard route via Log--Sobolev inequalities (LSI): by Gross's equivalence, $\\|P_t f\\|_q\\le \\|f\\|_p$ holds for $t\\ge \\frac{C}{4}\\log\\big(\\frac{q-1}{p-1}\\big)$ iff the corresponding LSI holds with constant $C$. Hence, it “suffices to establish” an $n$--LSI “with the optimal constant $2$” along the dyadic towers $n=2^k$ and $n=3\\cdot 2^k$, namely Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.\n\nThe proof of Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is described as a “new induction scheme” with three ingredients: (1) introduce auxiliary weights $\\phi_4$ on $\\mathbb{Z}_4$ and $\\phi_6$ on $\\mathbb{Z}_6$ and prove their LSIs, which are “tighter than those for the word-lengths $\\psi_n$,” and are “crucial for the base steps $4\\to 8$ and $6\\to 12$;” (2) perform “Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI,” using an “efficient way” via KKT combined with “manipulation of the length functions,” where “the specific structure of $\\phi_6$ introduces a symmetry in the KKT system” that makes it tractable; (3) carry out an “induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition,” using a “Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT)” to express the large DFT via smaller ones and obtain a “comparison of Dirichlet forms at the scales $n$ and $2n$.” Together these steps yield the LSI theorem with constant $2$, and by the Gross equivalence this gives the stated hypercontractivity threshold for Theorem~\\ref{Thm: Hypercontractivity on tower of n}.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb Z_n\\) be the cyclic group with normalized counting measure \\(\\mu_n\\), and let the Poisson-like semigroup \\((P_t)_{t\\ge 0}\\) act on Fourier series by\n\\[\nP_t\\!:\\sum_{k=0}^{n-1} a_k\\chi_k(x)\\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere \\(\\psi_n(k)=\\min(k,n-k)\\) and \\(\\chi_k(x)=e^{2\\pi i kx/n}\\). For \\(n\\) of the form \\(2^k\\) or \\(3\\cdot 2^k\\) with integer \\(k\\ge 1\\), and for \\(1<p\\le q<\\infty\\), which quantitative estimate gives exactly when the \\(L_p(\\mathbb Z_n,\\mu_n)\\to L_q(\\mathbb Z_n,\\mu_n)\\) hypercontractive bound holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\), if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q-1}{p-1}\\right),\n\\]\nthen\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if there exists a constant \\(C_n>0\\), depending only on \\(n\\), such that\n\\[\nt\\ge C_n\\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q}{p}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "sharp constant 1/2 in the threshold",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the 'if and only if' optimality/necessity direction",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniform exact threshold replaced by an unspecified n-dependent constant",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "replaced the Gross-type ratio (q-1)/(p-1) by q/p",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the threshold; it only states the operator and asks for an equivalent condition. There is no direct answer leakage beyond the general topic of hypercontractivity."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of the sharp hypercontractive threshold for the given semigroup: the correct option restates the theorem in equivalent form rather than asking for a derived or conceptually competing conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because one must distinguish an exact equivalence from merely sufficient bounds, especially against option C and nearby constants. However, the item mainly tests theorem recall rather than generating a conclusion from provided information."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: B alters the sharp factor, C is a weaker true statement but not equivalent, D swaps in an incorrect ratio, and E uses a plausible but nonoptimal constant. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid MCQ in terms of distractor design and lack of leakage, but it is largely a theorem-recall question and only weakly tests genuine generative reasoning."
    }
  },
  {
    "id": "2512.03489v1",
    "paper_link": "http://arxiv.org/abs/2512.03489v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.",
      "start_pos": 28767,
      "end_pos": 29036,
      "label": "Thm: Hypercontractivity on tower of n"
    },
    "ref_dict": {
      "Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k": "\\begin{theorem}\\label{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}\nFor $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have the following LSI with the optimal constant $2$:\n\\[\n  \\int_{\\mathbb{Z}_n} f^2\\log f^2\\dd\\mu_n-\\norm{f}_2^2\\log\\norm{f}_2^2\\le 2\\inner{f,A_{\\psi_n}f}_{L_2(\\mathbb{Z}_n,\\mu_n)},\\qquad f\\in L_2^+(\\mathbb{Z}_n,\\mu_n).\n\\]\n\\end{theorem}",
      "Thm: Hypercontractivity on tower of n": "\\begin{theorem}\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3821,
    "pre_theorem_intro_text": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n  P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n    \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par \n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.",
    "context": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par\n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.",
    "full_context": "The hypercontractivity of the Poisson-like semigroup on the cyclic group $\\mathbb{Z}_n$ is a long-standing open problem for all $n$ except $n=2$ and $n=4$. More precisely, by equipping $\\mathbb{Z}_n$ with the normalized counting measure $\\mu_n$, the Poisson-like semigroup $(P_t)_{t\\in \\mathbb{R}_+}$ is defined as the family of maps $P_t:L_\\infty(\\mathbb{Z}_n,\\mu_n)\\to L_\\infty(\\mathbb{Z}_n,\\mu_n)$, which acts on Fourier series by\n\\[\n P_t:\\sum_{k=0}^{n-1} a_k\\chi_k(x) \\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere $\\psi_n(k)=\\min(k,n-k)$ is the word-length function on $\\mathbb{Z}_n$, and $\\chi_k(x)=e^{\\frac{2\\pi ikx}{n}}\\in L_\\infty(\\mathbb{Z}_n)$. The hypercontractivity problem asks for the optimal time $t_{p,q}$ with $1< p\\leq q<\\infty$ such that\n\\[\n \\norm{P_t f}_{q}\\le \\norm{f}_{p}\\qquad \\text{for all } t\\ge t_{p,q}.\n\\]\nFor the case $n=2$, the optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ follows by applying the classical two-point inequality. That inequality was first proved by Bonami~\\cite{MR283496}, later rediscovered by Gross~\\cite{MR420249}, and was also used by Beckner~\\cite{MR385456} to obtain the best constants for the Hausdorff--Young inequality. For $n=4$, Beckner, Janson, and Jerison~\\cite{MR730056} obtained the same optimal time $t_{p,q}=\\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$ by a clever reduction from $\\mathbb{Z}_2$ to $\\mathbb{Z}_4$, though the approach does not extend to other $\\mathbb{Z}_{m\\times n}$. For $n=3$, by Wolff's reduction \\cite[Corollary 3.1]{MR2314078}, the optimal time $t_{2,q}$ of $P_t$ on $\\mathbb{Z}_3$ coincides the optimal time \n\\[\nt_{2,q}=\\frac{1}{2}\\log(\\frac{\\frac{2}{3}(\\frac{1}{3})^{\\frac{2}{q}-1}-\\frac{1}{3}(\\frac{2}{3})^{\\frac{2}{q}-1}}{(\\frac{2}{3})^{\\frac{2}{q}}-(\\frac{1}{3})^{\\frac{2}{q}}})\n\\]\nof the simple semigroup $T_t=e^{-t(\\mathrm{id}-\\mathbb{E}_{\\delta})}$ on the weighted two-point space $L_\\infty(\\{-1,1\\},\\tfrac{1}{3}\\delta_{-1}+\\tfrac{2}{3}\\delta_{1})$, which was computed by Oleszkiewicz and Lata\\l a \\cite{MR1796718,MR2073432}. For the general optimal time $t_{p,q}$ of $(P_t)_{t\\in \\mathbb{R}_+}$ on $\\mathbb{Z}_3$, only asymptotic information is available; see \\cite[Theorem 2.1]{MR2314078}. For $n=5$, Andersson~\\cite{MR1883499} determined $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$. For $n\\ge 6$, Junge, Palazuelos, Parcet, and Perrin~\\cite{MR3709719} proved partial results: for even $n$, $t_{2,q}=\\tfrac{1}{2}\\log(q-1)$ for $q\\in 2\\mathbb{Z}_+$; for odd $n$, the same holds when $n\\ge q$, via a rather involved combinatorial method. Combining Stein's interpolation method \\cite{MR82586} with Gross's extrapolation technique \\cite{MR420249,MR372613,MR2325763}, they further obtained $t_{p,q}\\le \\tfrac{\\log 3}{2}\\log\\big(\\frac{q-1}{p-1}\\big)$ in those regimes.\\par\n\nHypercontractivity is also widely studied for semigroups on manifolds. Among the most classical examples, Weissler~\\cite{MR578933} proved optimal hypercontractivity for the heat and Poisson semigroups on the circle $\\mathbb{S}^1$, while Rothaus~\\cite{MR593787} independently treated the heat case. For the sphere $\\mathbb{S}^n$ ($n\\ge 2$), Mueller and Weissler~\\cite{MR674060} established optimal hypercontractivity for the heat semigroup. For Poisson-type semigroups on $\\mathbb{S}^n$, see Janson~\\cite{MR706641}, Beckner~\\cite{MR1164616}, and Frank--Ivanisvili~\\cite{MR4278128}.\\par\n\nNow we present our main result for the Poisson-like semigroup $(P_t)_{t\\in\\mathbb{R}_+}$ on $\\mathbb{Z}_n$. Note that the standard argument, based on a simple series expansion of $\\|P_t(1+\\epsilon f)\\|_q$ and $\\|1+\\epsilon f\\|_p$ at $\\epsilon=0$, shows the universal lower bound $t_{p,q}\\ge \\tfrac{1}{2}\\log\\big(\\tfrac{q-1}{p-1}\\big)$. The theorem below records the optimal times $t_{p,q}$ along a dyadic tower of $n$.\n\nWithout loss of generality, suppose $\\lambda_{j}<\\lambda_{j+3}$. The function $F$ strictly increases on $(0,1)$ and strictly decreases on $(1,\\infty)$, so from \\eqref{eq: opposite F} with $\\nu_{j}=\\nu_{j+3}=0$ proved in Step 1, we get\n\\begin{equation}\n F(\\lambda_{j}) = F(\\lambda_{j+3}).\n\\end{equation}\nAnd hence\n\\[\n \\lambda_{j}<1<\\lambda_{j+3}.\n\\]\nDefine $\\Theta(x)\\coloneqq F(x)-F(2-x)$. Then $\\Theta(1)=0$ and $\\Theta'(x)=-\\frac{2}{3}\\log\\big(x(2-x)\\big)>0$ on $(0,1)$, so $\\Theta(x)<0$ on $(0,1)$. Applying this to $x=\\lambda_{j}$ gives $F(\\lambda_{j})<F(2-\\lambda_{j})$. Since $F$ decreases on $(1,\\infty)$ and $F(\\lambda_{j+3})=F(\\lambda_{j})$, we deduce\n\\begin{equation}\\label{Ineq: opposite sum lower bound}\n \\lambda_{j+3}>2-\\lambda_{j}\n \\quad\\Rightarrow\\quad\n \\lambda_{j}+\\lambda_{j+3}>2.\n\\end{equation}\\par\n\\medskip\n\\textbf{Step 3.} We show that for $\\ell,\\ell'\\in\\{0,1,2\\}\\setminus\\{j_0\\}$, we have $\\lambda_{\\ell}=\\lambda_{\\ell+3}=\\lambda_{\\ell'}=\\lambda_{\\ell'+3}$.\\par \nBy the assumption $\\lambda_{j_0}\\neq \\lambda_{j_0+3}$, we have $\\lambda_{j_0}+\\lambda_{j_0+3}>2$ from Step 2. Hence\n\\[\n \\lambda_{j_0}^2+\\lambda_{j_0+3}^2 \\ge \\frac{(\\lambda_{j_0}+\\lambda_{j_0+3})^2}{2} > 2.\n\\]\nConsequently, if $\\lambda_{j}\\neq \\lambda_{j+3}$ for all $0\\leq j\\le 2$, then $\\sum_{k=0}^{5}\\lambda_k^2>6$, contradicting $\\sum_{k=0}^{5}\\lambda_k^2<6$ in \\eqref{Eqns: Lagrange for n=6, (0,1,2,1,2,1)}. Therefore, there exists at least one $\\ell$ with $\\lambda_{\\ell}=\\lambda_{\\ell+3}$. Let $\\ell$ be an index in $\\{0,1,2\\}\\setminus\\{j_0\\}$ such that $\\lambda_\\ell=\\lambda_{\\ell+3}$ and $\\lambda_\\ell$ is minimal among all pairs $\\{\\lambda_j=\\lambda_{j+3}\\}$ with $j\\in\\{0,1,2\\}\\setminus\\{j_0\\}$. If $\\lambda_\\ell=\\lambda_{\\ell+3}\\ge 1$, then $\\sum_{k=0}^{5}\\lambda_k^2>6$, which again contradicts $\\sum_{k=0}^{5}\\lambda_k^2<6$. So there at least one $\\ell$ with $\\lambda_{\\ell}=\\lambda_{\\ell+3}<1$. Denote by $\\ell'$ the remaining index in $\\{0,1,2\\}\\setminus\\{j_0,\\ell\\}$. \\par\n\n\\begin{proof}\nRecall the case $n=2$ for Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is known in \\cite{MR420249}, i.e.,\n\\begin{equation}\\label{Ineq: 2-LSI}\n \\frac{1}{4}\\left( x^2\\log\\frac{2x^2}{x^2+y^2}+y^2\\log\\frac{2y^2}{x^2+y^2}\\right)\n\\le \\left( \\frac{x-y}{2} \\right)^2\\qquad x,y\\geq 0.\n\\end{equation}\nGiven $\\lambda=(a_0,b_0,a_1,b_1,\\ldots,a_{n-1},b_{n-1})$ with $a=(a_0,\\ldots,a_{n-1})$ and $b=(b_0,\\ldots,b_{n-1})$, the $2n$--entropy splits as\n\\[\n\\begin{aligned}\n\\mathrm{H}_{2n}[\\lambda]&=\\frac{1}{2n}\\sum_{i=0}^{n-1} a_i^2 \\log(\\frac{2na_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\\frac{1}{2n}\\sum_{i=0}^{n-1} b_i^2 \\log(\\frac{2nb_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\\\\\n&=\\frac{1}{2n}\\sum_{i=0}^{n-1} a_i^2\\log(\\frac{n a_i^2}{\\sum_{i=0}^{n-1}a_i^2})+\\frac{1}{2n}\\sum_{i=0}^{n-1} b_i^2\\log(\\frac{n b_i^2}{\\sum_{i=0}^{n-1}b_i^2})\\\\\n &\\quad +\\frac{1}{2n}\\left(\\sum_{i=0}^{n-1} a_i^2\\log(\\frac{2 \\sum_{i=0}^{n-1} a_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})+\\sum_{i=0}^{n-1} b_i^2\\log(\\frac{2 \\sum_{i=0}^{n-1} b_i^2}{\\sum_{i=0}^{n-1}(a_i^2+b_i^2)})\\right).\n\\end{aligned}\n\\]\nApplying the $n$--LSI to the first two terms, and $2$--LSI \\eqref{Ineq: 2-LSI} with $x=\\norm{a}_2$ and $y=\\norm{b}_2$ to the last term, yields\n\\[\n\\mathrm{H}_{2n}[\\lambda]\\le \\inner{a,\\Gamma(n)a}+\\inner{b,\\Gamma(n)b}\n+\\frac{1}{2n}\\big(\\norm{a}_2-\\norm{b}_2\\big)^2.\n\\]\nFinally, by Proposition~\\ref{Prop: Compare Dirichlet form for n and 2n},\n\\begin{equation}\\label{Eqn: induction from n-LSI to 2n-LSI}\n \\mathrm{H}_{2n}[\\lambda]\\le \\inner{a,\\Gamma(n)a}+\\inner{b,\\Gamma(n)b}\n+\\frac{1}{2n}\\big(\\norm{a}_2-\\norm{b}_2\\big)^2\n\\le 2\\inner{\\lambda,\\Gamma(2n)\\lambda}.\n\\end{equation}\\par\n\nFor odd $n_0$, define the weight function on $\\mathbb{Z}_n$ for even $n\\geq n_0$\n\\begin{equation}\\label{Eqn: definition of induced gamma_n for odd base case}\n \\gamma_n(k)=\n \\begin{cases}\n \\psi_n(k), & k\\ne \\tfrac{n}{2},\\\\\n 1, & k=\\tfrac{n}{2}.\n \\end{cases}\n\\end{equation}\nThe pair $(\\psi_{n_0},\\gamma_{2n_0})$ satisfies \\eqref{Cond: gamma pair condition} and \\eqref{Ineq: Quadratic inequality from n to 2n}. For even $n$, the pair $(\\gamma_n,\\gamma_{2n})$ also satisfies \\eqref{Cond: gamma pair condition}, and the desired inequality \\eqref{Ineq: Quadratic inequality from n to 2n} becomes\n\\begin{equation}\\label{Ineq: quadratic for odd n}\n f(x)\\coloneqq r_b (n-3) x^2+x \\left(2 \\sqrt{r_a+1} \\sqrt{r_b+1}-2\\right)+r_a (n-3)\\geq 0\n\\end{equation}\nfor all $x\\ge 0$ and $0\\le r_a,r_b\\le 1$. Since $n-3>0$, the minimum of $f$ is attained at $x=-\\frac{2 \\sqrt{r_a+1} \\sqrt{r_b+1}-2}{2r_b(n-3)}\\leq 0$ and the value of the minimum of $f$ on $[0,\\infty)$ is $f(0)=r_a(n-3)\\geq 0$. Hence \\eqref{Ineq: quadratic for odd n} holds. Therefore Proposition~\\ref{Prop: Compare Dirichlet form for n and 2n} applies to the pairs $(\\gamma_n,\\gamma_{2n})$ defined by \\eqref{Eqn: definition of induced gamma_n for odd base case}. Iterating \\eqref{Eqn: induction from n-LSI to 2n-LSI} along $\\big((\\psi_{n_0},\\gamma_{2n_0}),(\\gamma_{2n_0},\\gamma_{4n_0}),\\ldots\\big)$, we obtain for all $m\\ge 0$,\n\\[\n \\mathrm{H}_{n_0\\cdot 2^{m+1}}[\\lambda]\n \\le 2\\inner{\\lambda,\\Gamma(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nLet $\\Psi(n)=\\frac{1}{n}F_n\\diag(\\psi_n)F_n^{-1}$. Since $\\gamma_n\\leq \\psi_n$ pointwise, we have\n\\[\n \\inner{\\lambda,\\Gamma(n_0\\cdot 2^{m+1})\\lambda} \n \\le \\inner{\\lambda,\\Psi(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nTherefore,\n\\[\n \\mathrm{H}_{n_0\\cdot 2^{m+1}}[\\lambda]\n \\le 2\\inner{\\lambda,\\Psi(n_0\\cdot 2^{m+1})\\lambda}.\n\\]\nBy Lemma~\\ref{Lem: Explicit form of Log Sobolev ineq}, the \\(n_0\\cdot 2^{m+1}\\)--LSI holds for \n\\(\\psi_{n_0\\cdot 2^{m+1}}\\) for all \\(m\\ge 0\\), which proves the claim.\n\\end{proof}\n\n\\textbf{(2) Case $n=2^m$ with $m\\geq 1$.} The case $n=2$ is classical and the case $n=4$ is due to the work \\cite{MR730056} and Gross's extrapolation technique~\\cite{MR420249}. Note that the pair $(\\phi_4,\\gamma_8)$ satisfies \\eqref{Cond: gamma pair condition}, and \\eqref{Ineq: Quadratic inequality from n to 2n} becomes\n\\[\n f(x)\\coloneqq \\left(2-\\frac{r_b}{5}\\right)x^2+2x\\left(\\sqrt{r_a+1}\\sqrt{r_b+1}-3\\right)\n -\\frac{r_a}{5}+2\\ \\ge 0,\n\\]\nwith $0\\le r_a,r_b\\le 1$ and $x\\ge0$. As $2-\\frac{r_b}{5}>0$, the minimum of $f$ on $[0,\\infty)$ is attained at $x=-\\tfrac{2 \\left(\\sqrt{r_a^2+1} \\sqrt{r_b^2+1}-3\\right)}{2\\left( 2-r_b^2/5 \\right)}$ and the value of the minimum is\n\\[\n \\frac{r_a(24r_b+35)+5\\left(-30\\sqrt{r_a+1}\\sqrt{r_b+1}+7r_b+30\\right)}\n {5(r_b-10)}\n =\\frac{h(r_a,r_b)}{5(r_b-10)}.\n\\]\nSince $r_b\\in[0,1]$, the denominator is negative. In order to show \\eqref{Ineq: Quadratic inequality from n to 2n} hold, it suffices to show $h(r_a,r_b)\\le 0$ on $[0,1]^2$. A direct computation gives\n\\[\n \\pdv[2]{h}{r_a}(r_a,r_b)=\\frac{75\\sqrt{r_b+1}}{2(r_a+1)^{3/2}}>0,\n \\qquad\n \\pdv[2]{h}{r_b}(r_a,r_b)=\\frac{75\\sqrt{r_a+1}}{2(r_b+1)^{3/2}}>0.\n\\]\nThus $h$ is convex in each variable separately. So the maximum of $h$ on $[0,1]^2$ is attained at a corner. Evaluating,\n\\[\n h(0,0)=0,\\quad h(0,1)=h(1,0)=5\\big(37-30\\sqrt{2}\\big)<0,\\quad h(1,1)=-56<0.\n\\]\nHence $\\max_{[0,1]^2} h\\le 0$, and since $5(r_b-10)<0$ we conclude\n\\[\n \\frac{r_a(24r_b+35)+5\\left(-30\\sqrt{r_a+1}\\sqrt{r_b+1}+7r_b+30\\right)}\n {5(r_b-10)} \\ge 0.\n\\]\nTherefore $(\\phi_4,\\gamma_8)$ satisfies \\eqref{Ineq: Quadratic inequality from n to 2n}. Recall that $(\\gamma_{8\\cdot 2^m},\\gamma_{8\\cdot 2^{m+1}})$ also satisfies \\eqref{Ineq: Quadratic inequality from n to 2n}. Hence by Theorem~\\ref{Thm: Log Sobolev inequality n=4} and Theorem~\\ref{Thm: induction from n-LSI to 2n-LSI} the $8\\cdot 2^m$--LSI holds for $\\gamma_{8\\cdot 2^m}$ for all $m\\ge 0$, and therefore for $\\psi_{8\\cdot 2^m}$.\n\\end{proof}",
    "post_theorem_intro_text_len": 3382,
    "post_theorem_intro_text": "A standard route to hypercontractivity proceeds through Log--Sobolev inequalities (LSI) by Gross's celebrated work~\\cite{MR420249}: $\\norm{P_{t}f}_q\\le \\norm{f}_p$ holds whenever $t\\geq \\frac{C}{4}\\log\\big(\\frac{q-1}{p-1}\\big)$ if and only if the corresponding LSI holds with constant $C$. Thus, to prove Theorem~\\ref{Thm: Hypercontractivity on tower of n}, it suffices to establish the following $n$--LSI with the optimal constant $2$ along the above dyadic tower of $n$. Denote by $A_{\\psi_n}$ the generator of semigroup $(P_t)_{t\\in\\mathbb{R}_+}$, that is\n\\begin{equation}\n  A_{\\psi_n}: \\sum_{k=0}^{n-1} a_k\\chi_k(x)\\mapsto \\sum_{k=0}^{n-1} \\psi_n(k)a_k\\chi_k(x).\n\\end{equation}\n The case $n=4$ in the following theorem is due to the work \\cite{MR730056} and Gross's extrapolation technique~\\cite{MR420249}.\n\\begin{theorem}\\label{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}\nFor $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have the following LSI with the optimal constant $2$:\n\\[\n  \\int_{\\mathbb{Z}_n} f^2\\log f^2\\dd\\mu_n-\\norm{f}_2^2\\log\\norm{f}_2^2\\le 2\\inner{f,A_{\\psi_n}f}_{L_2(\\mathbb{Z}_n,\\mu_n)},\\qquad f\\in L_2^+(\\mathbb{Z}_n,\\mu_n).\n\\]\n\\end{theorem}\n\nOur proof of Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is based on a new induction scheme with three key ingredients:\n\\begin{enumerate}\n  \\item Auxiliary weights $\\phi_4$ on $\\mathbb{Z}_4$ and $\\phi_6$ on $\\mathbb{Z}_6$ together with their corresponding LSIs. These LSIs are tighter than those for the word-lengths $\\psi_n$; this refinement is new even when $n=4$ and plays a key role in the analysis of LSIs.\n  \\item Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI.  \nWe develop an efficient way to handle LSIs on $\\mathbb{Z}_n$ via KKT analysis, combined with the aforementioned manipulation of the length functions. The specific structure of $\\phi_6$ introduces a symmetry in the KKT system that makes the analysis tractable.\n  \\item An induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition. We use a Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT), which expresses a large DFT as a combination of smaller DFTs and yields a comparison of Dirichlet forms at the scales $n$ and $2n$. The choice of the new weights $\\phi_4$ and $\\phi_6$ mentioned in (1) is crucial for the base steps $4\\to 8$ and $6\\to 12$.\n\\end{enumerate}\nFinally, we remark that the above ideas are also useful for studying LSIs along other towers of the form $m\\cdot n^k$. The KKT analysis and the compatibility condition vary in their technical details, depending on the specific value of $n$ and on the particular choice of weights on $\\mathbb{Z}_n$, and we will not carry out this analysis in this paper.\\par\n\nThe article is organized as follows. After a brief introduction to the LSI formulation and the $n$-dimensional KKT framework in Section~\\ref{Sec: DFT and KKT analysis}, we analyze the KKT systems associated with the LSIs for $n=6$ and $n=4$ in Section~\\ref{Sec: Log Sobolev inequality n=4 and n=6}. Section~\\ref{Sec: Induction from n to 2n} presents a comparison criterion for a pair of weights that allows us to compare the Dirichlet forms at the scales $n$ and $2n$. We then apply it to the transition $n\\to 2n$ of LSIs and establish Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.",
    "sketch": "To prove Theorem~\\ref{Thm: Hypercontractivity on tower of n}, the text follows the standard route via Log--Sobolev inequalities (LSI): by Gross's equivalence, $\\|P_t f\\|_q\\le \\|f\\|_p$ holds for $t\\ge \\frac{C}{4}\\log\\big(\\frac{q-1}{p-1}\\big)$ iff the corresponding LSI holds with constant $C$. Hence, it “suffices to establish” an $n$--LSI “with the optimal constant $2$” along the dyadic towers $n=2^k$ and $n=3\\cdot 2^k$, namely Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k}.\n\nThe proof of Theorem~\\ref{Thm: Log Sobolev inequality n=6 times 2^k and n=8 times 2^k} is described as a “new induction scheme” with three ingredients: (1) introduce auxiliary weights $\\phi_4$ on $\\mathbb{Z}_4$ and $\\phi_6$ on $\\mathbb{Z}_6$ and prove their LSIs, which are “tighter than those for the word-lengths $\\psi_n$,” and are “crucial for the base steps $4\\to 8$ and $6\\to 12$;” (2) perform “Karush--Kuhn--Tucker (KKT) analysis for the $4$-- and $6$--LSI,” using an “efficient way” via KKT combined with “manipulation of the length functions,” where “the specific structure of $\\phi_6$ introduces a symmetry in the KKT system” that makes it tractable; (3) carry out an “induction from the $n$--LSI to the $2n$--LSI under a crucial compatibility condition,” using a “Cooley--Tukey factorization of the $2n$-point discrete Fourier transform (DFT)” to express the large DFT via smaller ones and obtain a “comparison of Dirichlet forms at the scales $n$ and $2n$.” Together these steps yield the LSI theorem with constant $2$, and by the Gross equivalence this gives the stated hypercontractivity threshold for Theorem~\\ref{Thm: Hypercontractivity on tower of n}.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{Thm: Hypercontractivity on tower of n}\n  For $n=3\\cdot 2^k$ and $n=2^k$ with $k\\ge 1$, we have \n  \\[\n    \\norm{P_{t}f}_q\\le \\norm{f}_p\\quad \\Leftrightarrow \\quad t\\ge \\frac{1}{2}\\log(\\frac{q-1}{p-1})\n    \\]\n    for $1<p\\le q<\\infty$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb Z_n\\) be the cyclic group with normalized counting measure \\(\\mu_n\\), and let the Poisson-like semigroup \\((P_t)_{t\\ge 0}\\) act on Fourier series by\n\\[\nP_t\\!:\\sum_{k=0}^{n-1} a_k\\chi_k(x)\\mapsto \\sum_{k=0}^{n-1} e^{-t\\psi_n(k)}a_k\\chi_k(x),\n\\]\nwhere \\(\\psi_n(k)=\\min(k,n-k)\\) and \\(\\chi_k(x)=e^{2\\pi i kx/n}\\). For \\(n\\) of the form \\(2^k\\) or \\(3\\cdot 2^k\\) with integer \\(k\\ge 1\\), and for \\(1<p\\le q<\\infty\\), which quantitative estimate gives exactly when the \\(L_p(\\mathbb Z_n,\\mu_n)\\to L_q(\\mathbb Z_n,\\mu_n)\\) hypercontractive bound holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\), if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q-1}{p-1}\\right),\n\\]\nthen\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if there exists a constant \\(C_n>0\\), depending only on \\(n\\), such that\n\\[\nt\\ge C_n\\log\\!\\left(\\frac{q-1}{p-1}\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(f\\in L_p(\\mathbb Z_n,\\mu_n)\\),\n\\[\n\\|P_t f\\|_{L_q(\\mathbb Z_n,\\mu_n)}\\le \\|f\\|_{L_p(\\mathbb Z_n,\\mu_n)}\n\\]\nholds if and only if\n\\[\nt\\ge \\frac12\\log\\!\\left(\\frac{q}{p}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "sharp constant 1/2 in the threshold",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the 'if and only if' optimality/necessity direction",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniform exact threshold replaced by an unspecified n-dependent constant",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "replaced the Gross-type ratio (q-1)/(p-1) by q/p",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the threshold or the sharp constant. It asks for the exact hypercontractive estimate, so there is no explicit answer leakage beyond indicating that an exact characterization exists."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific theorem: the sharp 'if and only if' hypercontractive threshold for the given semigroup and class of n. It does not substantially reframe the result or require deriving a new consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact statement from nearby variants (wrong constant, only one direction, n-dependent constant, wrong ratio), but the item mainly tests recognition of the known theorem rather than genuine generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: a factor-of-2 error, a weaker true sufficiency statement, loss of uniform sharpness via an unspecified constant, and an incorrect q/p-type threshold. These reflect common failure modes and are clearly distinct."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no direct leakage, but it is largely tautological and only mildly tests reasoning."
    }
  },
  {
    "id": "2512.03498v1",
    "paper_link": "http://arxiv.org/abs/2512.03498v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm-main}\nLet $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that\n$$\nN+iD = a^{x_i}+b^{y_i}, \\; \\; \\mbox{ for } \\; i \\in \\{ 0, 1, 2, 3, 4 \\}.\n$$\nThen we have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)  \\; \\mbox{ or } \\;  (3,4 \\cdot 3^{k-1}+1,3^{k-1}+1,2 \\cdot 3^{k-1}) \\; \\mbox{ for } \\; k \\in \\mathbb{N},\n$$\nor\n$$\n\\begin{array}{c}\n(a,b,N,D) \\in \\left\\{ (2,3,5,2),  (2,3,7,6), (2,3,9,8),  (2,3,17,24),  \\right. \\\\\n\\left. (2,3,41,24), (2,5,5,8),  (2,9,17,24),  (2,9,41,24), (3,4,7,6) \\right\\}.\n\\end{array}\n$$",
      "start_pos": 7158,
      "end_pos": 7779,
      "label": "thm-main"
    },
    "ref_dict": {
      "cor-main": "\\begin{cor} \\label{cor-main}\nIf $a$ and $b$ are integers with $b>a>1$,  then the only $6$-term arithmetic progressions in $S_{a,b}$ are \n$$\n3,5,7,9,11,13 \\; \\; \\mbox{ and } \\; \\; 17, 41, 65, 89, 113, 137,\n$$\nin $S_{2,3}$ and in both $S_{2,3}$ and $S_{2,9}$, respectively.\n\\end{cor}",
      "thm-main": "\\begin{thm} \\label{thm-main}\nLet $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that\n$$\nN+iD = a^{x_i}+b^{y_i}, \\; \\; \\mbox{ for } \\; i \\in \\{ 0, 1, 2, 3, 4 \\}.\n$$\nThen we have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)  \\; \\mbox{ or } \\;  (3,4 \\cdot 3^{k-1}+1,3^{k-1}+1,2 \\cdot 3^{k-1}) \\; \\mbox{ for } \\; k \\in \\mathbb{N},\n$$\nor\n$$\n\\begin{array}{c}\n(a,b,N,D) \\in \\left\\{ (2,3,5,2),  (2,3,7,6), (2,3,9,8),  (2,3,17,24),  \\right. \\\\\n\\left. (2,3,41,24), (2,5,5,8),  (2,9,17,24),  (2,9,41,24), (3,4,7,6) \\right\\}.\n\\end{array}\n$$\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 1676,
    "pre_theorem_intro_text": "Let $a$ and $b$ be integers with $b>a>1$, and define the set $S_{a,b}$ to be the sumset of the geometric progressions\n$$\n1, a, a^2, a^3, \\ldots \\; \\mbox{ and } \\; 1, b, b^2, b^3, \\ldots,\n$$\ni.e.\n$$\nS_{a,b} = \\{ n \\in \\mathbb{N} \\; : \\; n = a^x+b^y, \\; \\mbox{ for } x, y \\in \\mathbb{Z}, \\; x, y \\geq 0 \\}.\n$$\nThe main goal  of this paper is to study the arithmetic progressions in the sets $S_{a,b}$. These sets are sufficiently thin that, were they random, our expectation for a fixed pair $(a,b)$  would be that $S_{a,b}$ should contain at most finitely many three-term arithmetic progressions, and indeed this is the case ``most'' of the time. By way of example, $S_{2,7}$ contains precisely  $22$ such progressions, while, in contrast, $S_{2,9}$ contains infinitely many  and $S_{2,8}$ even contains infinitely many four-term arithmetic progressions. It is likely that $S_{a,b}$ contains infinitely many \nthree-term arithmetic progressions precisely when either $a=2$ and $b=2^k+1$, for $k$ a positive integer, or when $a$ and $b$ are multiplicatively  dependent, i.e. there exist positive integers $c$ and $d$ such that $a^c=b^d$. We will discuss this further in Section \\ref{Sec9}.\n\nIn this paper, we will focus our attention on the problem of characterizing longer arithmetic progressions in the sets $S_{a,b}$. \nOur starting point  is recent work of  Chen, Huang and Zhang \\cite{ChHuZh} who showed, via a completely elementary argument,  that the longest arithmetic progression in the set \n$S_{2,3}$\nhas length $6$. Regarding such maximal progressions, we prove the following broad generalization of this, characterizing all $5$-term progressions in the sets $S_{a,b}$.",
    "context": "Let $a$ and $b$ be integers with $b>a>1$, and define the set $S_{a,b}$ to be the sumset of the geometric progressions\n$$\n1, a, a^2, a^3, \\ldots \\; \\mbox{ and } \\; 1, b, b^2, b^3, \\ldots,\n$$\ni.e.\n$$\nS_{a,b} = \\{ n \\in \\mathbb{N} \\; : \\; n = a^x+b^y, \\; \\mbox{ for } x, y \\in \\mathbb{Z}, \\; x, y \\geq 0 \\}.\n$$\nThe main goal of this paper is to study the arithmetic progressions in the sets $S_{a,b}$. These sets are sufficiently thin that, were they random, our expectation for a fixed pair $(a,b)$ would be that $S_{a,b}$ should contain at most finitely many three-term arithmetic progressions, and indeed this is the case ``most'' of the time. By way of example, $S_{2,7}$ contains precisely $22$ such progressions, while, in contrast, $S_{2,9}$ contains infinitely many and $S_{2,8}$ even contains infinitely many four-term arithmetic progressions. It is likely that $S_{a,b}$ contains infinitely many \nthree-term arithmetic progressions precisely when either $a=2$ and $b=2^k+1$, for $k$ a positive integer, or when $a$ and $b$ are multiplicatively dependent, i.e. there exist positive integers $c$ and $d$ such that $a^c=b^d$. We will discuss this further in Section \\ref{Sec9}.\n\nIn this paper, we will focus our attention on the problem of characterizing longer arithmetic progressions in the sets $S_{a,b}$. \nOur starting point is recent work of Chen, Huang and Zhang \\cite{ChHuZh} who showed, via a completely elementary argument, that the longest arithmetic progression in the set \n$S_{2,3}$\nhas length $6$. Regarding such maximal progressions, we prove the following broad generalization of this, characterizing all $5$-term progressions in the sets $S_{a,b}$.",
    "full_context": "Let $a$ and $b$ be integers with $b>a>1$, and define the set $S_{a,b}$ to be the sumset of the geometric progressions\n$$\n1, a, a^2, a^3, \\ldots \\; \\mbox{ and } \\; 1, b, b^2, b^3, \\ldots,\n$$\ni.e.\n$$\nS_{a,b} = \\{ n \\in \\mathbb{N} \\; : \\; n = a^x+b^y, \\; \\mbox{ for } x, y \\in \\mathbb{Z}, \\; x, y \\geq 0 \\}.\n$$\nThe main goal of this paper is to study the arithmetic progressions in the sets $S_{a,b}$. These sets are sufficiently thin that, were they random, our expectation for a fixed pair $(a,b)$ would be that $S_{a,b}$ should contain at most finitely many three-term arithmetic progressions, and indeed this is the case ``most'' of the time. By way of example, $S_{2,7}$ contains precisely $22$ such progressions, while, in contrast, $S_{2,9}$ contains infinitely many and $S_{2,8}$ even contains infinitely many four-term arithmetic progressions. It is likely that $S_{a,b}$ contains infinitely many \nthree-term arithmetic progressions precisely when either $a=2$ and $b=2^k+1$, for $k$ a positive integer, or when $a$ and $b$ are multiplicatively dependent, i.e. there exist positive integers $c$ and $d$ such that $a^c=b^d$. We will discuss this further in Section \\ref{Sec9}.\n\nIn this paper, we will focus our attention on the problem of characterizing longer arithmetic progressions in the sets $S_{a,b}$. \nOur starting point is recent work of Chen, Huang and Zhang \\cite{ChHuZh} who showed, via a completely elementary argument, that the longest arithmetic progression in the set \n$S_{2,3}$\nhas length $6$. Regarding such maximal progressions, we prove the following broad generalization of this, characterizing all $5$-term progressions in the sets $S_{a,b}$.\n\n\\noindent An almost immediate corollary of this is the following.\n\\begin{cor} \\label{cor-main}\nIf $a$ and $b$ are integers with $b>a>1$, then the only $6$-term arithmetic progressions in $S_{a,b}$ are \n$$\n3,5,7,9,11,13 \\; \\; \\mbox{ and } \\; \\; 17, 41, 65, 89, 113, 137,\n$$\nin $S_{2,3}$ and in both $S_{2,3}$ and $S_{2,9}$, respectively.\n\\end{cor}\n\nSuppose that $a$ and $b$ are integers with $b>a \\geq 2$ and that we have a $5$-term arithmetic progression in integers of the shape $a^x+b^y$, i.e. that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that\n$$\nN+iD = a^{x_i}+b^{y_i}, \\; \\; \\mbox{ for } 0 \\leq i \\leq 4.\n$$\nWe will write\n$$\nn = \\left[ \\frac{\\log (N+4D)}{\\log a} \\right] \\; \\mbox{ and } \\; m = \\left[ \\frac{\\log (N+4D)}{\\log b} \\right],\n$$\nwhere by $[x]$ we mean the greatest integer $\\leq x$.\nLet us call a term $N+iD$ in our arithmetic progression {\\it $a$-large} if there is some representation of $N+iD = a^{x_i}+b^{y_i}$ with $x_i=n$, {\\it $b$-large} if there is a representation of $N+iD = a^{x_i}+b^{y_i}$ with $y_i=m$, and {\\it large}, if $N+iD$ is either $a$-large or $b$-large.\n\n\\begin{prop} \\label{equality}\nLet $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that\n$$\nN+iD = a^{x_i}+b^{y_i}, \\; \\; \\mbox{ for } \\; i \\in \\{ 0, 1, 2, 3, 4 \\}.\n$$\nSuppose further that here exist integers $i, j$ and $k$ with $0 \\leq i < j < k \\leq 4$ and either \n\\begin{equation} \\label{x}\nx_i=x_j=x_k,\n\\end{equation}\nor\n\\begin{equation} \\label{y}\ny_i=y_j=y_k.\n\\end{equation}\nThen either\n$$\n(a,b,N,D)=(2,3,5,2), (2,3,7,6), (2,3,9,8), (3,4,7,6),\n$$\n\\begin{equation} \\label{family}\n(a,b,N,D)=(2,2^{x_0}+1,2^{x_0}+1,2^{x_0})\n\\end{equation}\nor\n\\begin{equation} \\label{family2}\n(a,b,N,D)=(3,4 \\cdot 3^{x_0}+1,3^{x_0}+1,2 \\cdot 3^{x_0}).\n\\end{equation}\n\\end{prop}\n\nSuppose next that precisely three of $N+iD$, $i \\in \\{ 0, 1, 2, 3, 4 \\}$ are large. From our previous work, we may assume that either\n\\begin{equation} \\label{Case1}\n\\mbox{there exist } 0 \\leq i < j \\leq 4 \\mbox{ and } k \\not\\in \\{ i, j \\} \\mbox{ with } x_i=x_j=n, \\; y_k=m, \n\\end{equation}\nor\n\\begin{equation} \\label{Case2}\n\\mbox{there exist } 0 \\leq i < j \\leq 4 \\mbox{ and } k \\not\\in \\{ i, j \\} \\mbox{ with } y_i=y_j=m, \\; x_k=n.\n\\end{equation}\nSince two of the terms $N+iD$, $i \\in \\{ 0, 1, 2, 3, 4 \\}$ are not large, in all cases we have inequality (\\ref{starty}) with $\\kappa=1$. \nIn particular,\n$$\nN+D \\leq a^{n-1}+b^{m-1} < \\left( \\frac{1}{a} + \\frac{1}{b} \\right) (N+4D)\n$$\nand hence, from $4(N+D)>N+4D$, $\\frac{1}{a} + \\frac{1}{b} > \\frac{1}{4}$,\ni.e.\n\\begin{equation} \\label{special2}\na \\in \\{ 2, 3, 4 \\}, \\; a=5, \\; 6 \\leq b \\leq 19, \\; \\; a=6, \\; 7 \\leq b \\leq 11 \\; \\mbox{ or } \\; a=7, \\; 8 \\leq b \\leq 9.\n\\end{equation}\n\nWe therefore may suppose that $a=2$. Our goal is to show, in this case, that we have only $5$-term arithmetic progressions corresponding to\n $$\n(a,b,N,D) \\in \\{ (2,3,17,24), (2,3,41,24), (2,9,17,24), (2,9,41,24) \\}.\n$$\n Let us assume first that $\\delta_1=1$ and that $x_j=n-1$, whence (\\ref{awesome}) and (\\ref{spooky2}) imply that \n $$\n (i,j,k,\\kappa) \\in \n \\{ (2,3,4,1), (2,4,3,1), (3,4,2,1), (1,3,4,2) \\}.\n $$\n We will treat each of these four cases in turn. Suppose first that $(i,j,k,\\kappa)=(2,3,4,1)$. Then \n $$\n \\left( b^{\\delta_2} -1 \\right) b^{m-\\delta_2} = 2^n-2^{x_2+1} \n $$\n and\n $$\n2^n = 3 \\cdot b^m - 2 b^{y_4} - b^{m-\\delta_2},\n $$\n from (\\ref{gaewA}) and (\\ref{gaewB}), respectively. The second of these implies that\n $$\n 2^n \\geq 3 (b^m-b^{m-1}) > b^m\n $$\n and hence we can apply Theorem 1.6 of \\cite{Be-Pill} to the first equation to conclude that $b=3$ and that\n $$\n (n,m,x_2,\\delta_2) \\in \\{ (5,3,2,2), (8,5,3,4) \\}.\n $$\n Neither coincides with a $5$-term progression (in each case $N+4D \\not\\in S_{2,3}$). If $(i,j,k,\\kappa)=(2,4,3,1)$ or $(3,4,2,1)$, from the identity $(N+D)+(N+4D)=(N+2D)+(N+3D)$, we find that\n $$\n b^{m-\\delta_2} - b^{y_k} = 2^{x_i},\n $$\n and so Lemma \\ref{tech-lem} implies that\n we have one of\n \\begin{equation} \\label{ann1}\n b=2^{x_i}+1, \\; m-\\delta_2=1, \\; y_k=0,\n\\end{equation}\n\\begin{equation} \\label{ann3}\nb=3, \\; m-\\delta_2=2, \\; y_k=0, \\; x_i=3,\n\\end{equation}\nor\n\\begin{equation} \\label{ann4}\nb=9, \\; m-\\delta_2=1, \\; y_k=0, \\; x_i=3.\n\\end{equation}\nIn the first case, if $(i,j,k,\\kappa)=(2,4,3,1)$, we have\n$$\nD = 2^{n-2}-2^{x_2-1} = \\frac{1}{3} 2^{x_2} ( 2^{x_2}+1)^{m-1},\n$$\na contradiction modulo $2^{x_2}$. If, on the other hand, we have (\\ref{ann1}) and $(i,j,k,\\kappa)=(3,4,2,1)$,\n$$\nD = 2^{n-1}-2^{x_3} = \\frac{1}{3} 2^{x_3} ( 2^{x_3}+1)^{m-1},\n$$\nso that\n$$\n2^{n-1-x_3}-1 = \\frac{1}{3} ( 2^{x_3}+1)^{m-1}.\n$$\nModulo $3$, $x_3$ is odd and $m \\geq 2$. Thus, modulo $8$, we find that either $x_3=1$, $m=2$, $n=3$, or $x_3=1$, $m=3$, $n=4$, or $x_3=3$, $m=2$, $n=4$. Only the first of these corresponds to a $5$-term arithmetic progression in $S_{a,b}$, namely $(a,b,N,D)=(2,3,5,2)$, which we have previously encountered. If $(i,j,k,\\kappa)=(2,4,3,1)$, (\\ref{gaewA}) becomes\n$$\n\\left( b^{\\delta_2} -1 \\right) b^{m-\\delta_2} = 3 \\cdot (2^{n-2}-2^{x_2-1}),\n$$\nwhile $(i,j,k,\\kappa)=(3,4,2,1)$ yields\n$$\n\\left( b^{\\delta_2} -1 \\right) b^{m-\\delta_2} = 3 \\cdot (2^{n-1} - 2^{x_3}).\n$$\nIn cases (\\ref{ann3}) and (\\ref{ann4}), we may thus apply Proposition \\ref{DT} to these equations, concluding after a short computation with the $5$-term arithmetic progressions corresponding to\n$$\n(a,b,N,D) = (2,3,17,24) \\; \\mbox{ and } \\; (2,9,17,24).\n$$\nFinally, \n if $(i,j,k,\\kappa)=(1,3,4,2)$, then, from $(N+D)+(N+4D)=(N+2D)+(N+3D)$, we have\n $$\n b^{m-\\delta_2}-b^{y_4}=2^{x_1},\n $$\n while (\\ref{gaewB}) gives\n $$\n 2^{n-1} = 2 b^m -b^{y_4} - b^{m-\\delta_2}.\n $$\n Lemma \\ref{tech-lem} thus implies that $y_4=0$ and so combining the preceding two equations,\n $$\n b^m =2^{n-2} +2^{x_1-1}+1.\n $$\n Since necessarily $m \\geq 2$ (else we have three equal $y_i$), we may appeal to Theorem 1 of \\cite{BeBuMi} to conclude that\n $$\n b^m \\in \\{ 7^2, 23^2, 3^4 \\},\n $$\n so that \n $$\n (b,m,n,x_1) \\in \\{ (7,2,7,5), (23,2,11,5), (3,4,8,5), (9,2,8,5) \\};\n $$\n none of these correspond to a $5$-term arithmetic progression in $S_{a,b}$.\n\nIf $S_{a,b}$ contains a $6$-term arithmetic progression, then, from Theorem \\ref{thm-main}, we require $N+5D \\in S_{a,b}$ for \n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k) \\; \\mbox{ or } \\; (3,4 \\cdot 3^{k-1}+1,3^{k-1}+1,2 \\cdot 3^{k-1}) \\; \\mbox{ for } \\; k \\in \\mathbb{N},\n$$\nor\n$$\n\\begin{array}{c}\n(a,b,N,D) \\in \\left\\{ (2,3,5,2), (2,3,7,6), (2,3,9,8), (2,3,17,24), \\right. \\\\\n\\left. (2,3,41,24), (2,5,5,8), (2,9,17,24), (2,9,41,24), (3,4,7,6) \\right\\}.\n\\end{array}\n$$\nA short check of the latter $9$ sporadic cases, reveals the $6$-term progressions corresponding to \n$$\n(a,b,N,D) = (2,3,17,24) \\; \\mbox{ and } \\; (2,9,17,24),\n$$\nand no $7$-term progressions. Suppose that\n$(a,b,N,D)=(2,2^k+1,2^k+1,2^k)$, for $k$ a positive integer, and that \n$$\nN+5D = 2^k+1 + 5 \\cdot 2^k =3 \\cdot 2^{k+1}+1= 2^{x_5} + (2^k+1)^{y_5},\n$$\nfor $x_5$ and $y_5$ nonnegative integers. If $k=1$, then $x_5=y_5=2$, so that $(a,b,N,D) = (2,3,3,2)$. If $k \\geq 2$, then \n$$\n(2^k+1)^2 \\geq 3 \\cdot 2^{k+1}+1,\n$$\nwhence necessarily $y_5 \\in \\{ 0, 1 \\}$. Thus\n$$\n3 \\cdot 2^{k+1} = 2^{x_5} \\; \\mbox{ or } \\; 2^{k+2}+1= 2^{x_5},\n$$\neach an immediate contradiction.\nFinally, suppose that $(a,b,N,D)=(3,4 \\cdot 3^{k-1}+1,3^k+1,2 \\cdot 3^k)$ for $k$ a positive integer, and that \n$$\nN+5D = 3^{k-1}+1+ 10 \\cdot 3^{k-1} = 11 \\cdot 3^{k-1}+1=3^{x_5}+(4 \\cdot 3^{k-1}+1)^{y_5},\n$$\nwith $x_5$ and $y_5$ nonnegative integers. For each positive integer $k$, we have\n$$\n(4 \\cdot 3^{k-1}+1)^2 > 11 \\cdot 3^{k-1}+1\n$$\nand so $y_5 \\in \\{ 0, 1 \\}$, corresponding to\n$$\n11 \\cdot 3^{k-1}=3^{x_5} \\; \\mbox{ and } \\;\n7 \\cdot 3^{k-1}=3^{x_5},\n$$\nrespectively. These contradictions complete the proof of \nCorollary \\ref{cor-main}.",
    "post_theorem_intro_text_len": 1246,
    "post_theorem_intro_text": "\\noindent An almost immediate corollary of this is the following.\n\\begin{cor} \\label{cor-main}\nIf $a$ and $b$ are integers with $b>a>1$,  then the only $6$-term arithmetic progressions in $S_{a,b}$ are \n$$\n3,5,7,9,11,13 \\; \\; \\mbox{ and } \\; \\; 17, 41, 65, 89, 113, 137,\n$$\nin $S_{2,3}$ and in both $S_{2,3}$ and $S_{2,9}$, respectively.\n\\end{cor}\n\nOur arguments, in contrast to those of  \\cite{ChHuZh}, rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic, and upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves. The outline of this paper is the following.\nIn Section \\ref{Sec2}, we present the various results we will need for explicitly solving certain polynomial-exponential and $3, 4$ and $5$-term $S$-unit equations. Sections \\ref{Sec3}--\\ref{Sec7} are devoted to dealing with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of ``large'' terms, proving Theorem \\ref{thm-main}. In Section \\ref{Sec8}, we prove Corollary \\ref{cor-main}. Finally, in Section \\ref{Sec9}, we discuss various families of pairs $(a,b)$ for which the $S_{a,b}$ are known to have $3$ or $4$-term progressions.",
    "sketch": "The post-theorem introduction gives only a high-level outline for proving Theorem~\\ref{thm-main}: the arguments (unlike \\cite{ChHuZh}) \"rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic,\" and \"upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves.\" In Section~\\ref{Sec2} the paper \"present[s] the various results\" needed to explicitly solve certain polynomial-exponential and $3,4,5$-term $S$-unit equations, and Sections~\\ref{Sec3}--\\ref{Sec7} \"deal[] with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of \\`\\`large\\'\\' terms,\" thereby proving Theorem~\\ref{thm-main}.",
    "expanded_sketch": "The post-theorem introduction gives only a high-level outline for proving the main theorem: the arguments (unlike \\cite{ChHuZh}) \"rely, implicitly or explicitly, upon various results from Diophantine approximation, including bounds for linear forms in logarithms, both complex and $p$-adic,\" and \"upon various Diophantine consequences of the modularity of Frey-Hellegouarch curves.\" Next the paper \"present[s] the various results\" needed to explicitly solve certain polynomial-exponential and $3,4,5$-term $S$-unit equations, and later sections \"deal[] with $5$-term arithmetic progressions in $S_{a,b}$ with fixed numbers of ``large'' terms,\" thereby establishing the main theorem.",
    "expanded_theorem": "\\label{thm-main}\nLet $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that\n$$\nN+iD = a^{x_i}+b^{y_i}, \\; \\; \\mbox{ for } \\; i \\in \\{ 0, 1, 2, 3, 4 \\}.\n$$\nThen we have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)  \\; \\mbox{ or } \\;  (3,4 \\cdot 3^{k-1}+1,3^{k-1}+1,2 \\cdot 3^{k-1}) \\; \\mbox{ for } \\; k \\in \\mathbb{N},\n$$\nor\n$$\n\\begin{array}{c}\n(a,b,N,D) \\in \\left\\{ (2,3,5,2),  (2,3,7,6), (2,3,9,8),  (2,3,17,24),  \\right. \\\\\n\\left. (2,3,41,24), (2,5,5,8),  (2,9,17,24),  (2,9,41,24), (3,4,7,6) \\right\\}.\n\\end{array}\n$$,",
    "theorem_type": [
      "Implication",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $a$ and $b$ be integers with $b>a>1$. Suppose there exist positive integers $N$ and $D$, and nonnegative integers $x_i,y_i$ for $i=0,1,2,3,4$, such that\n$$\nN+iD=a^{x_i}+b^{y_i}\n$$\nfor each $i\\in\\{0,1,2,3,4\\}$. In other words, the five terms $N,N+D,N+2D,N+3D,N+4D$ of an arithmetic progression are all representable in the form $a^x+b^y$ with $x,y\\ge 0$. Which of the following conclusions about the quadruple $(a,b,N,D)$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "One must have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)\n\\quad\\text{or}\\quad\n(a,b,N,D)=(3,4\\cdot 3^{k-1}+1,3^{k-1}+1,2\\cdot 3^{k-1})\n$$\nfor some positive integer $k$, or else\n$$\n(a,b,N,D)\\in\\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\\}.\n$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "One must have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)\n\\quad\\text{or}\\quad\n(a,b,N,D)=(3,4\\cdot 3^{k-1}+1,3^{k-1}+1,2\\cdot 3^{k-1})\n$$\nfor some nonnegative integer $k$, or else\n$$\n(a,b,N,D)\\in\\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\\}.\n$$"
        },
        {
          "label": "C",
          "text": "One must have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)\n$$\nfor some positive integer $k$, or else\n$$\n(a,b,N,D)\\in\\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6),(3,4\\cdot 3^{k-1}+1,3^{k-1}+1,2\\cdot 3^{k-1})\\text{ for some }k\\in\\mathbb N\\}.\n$$"
        },
        {
          "label": "D",
          "text": "One must have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)\n\\quad\\text{or}\\quad\n(a,b,N,D)=(3,4\\cdot 3^{k-1}+1,3^{k}+1,2\\cdot 3^{k-1})\n$$\nfor some positive integer $k$, or else\n$$\n(a,b,N,D)\\in\\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24),(3,4,7,6)\\}.\n$$"
        },
        {
          "label": "E",
          "text": "One must have either\n$$\n(a,b,N,D)=(2,2^k+1,2^k+1,2^k)\n\\quad\\text{or}\\quad\n(a,b,N,D)=(3,4\\cdot 3^{k-1}+1,3^{k-1}+1,2\\cdot 3^{k-1})\n$$\nfor some positive integer $k$, and in addition one necessarily has $a\\in\\{2,3\\}$; equivalently, outside these two infinite families one must have\n$$\n(a,b,N,D)\\in\\{(2,3,5,2),(2,3,7,6),(2,3,9,8),(2,3,17,24),(2,3,41,24),(2,5,5,8),(2,9,17,24),(2,9,41,24)\\}.\n$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "range_of_parameter_k",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "disjunctive_family_structure_repackaged_as_single_enumeration",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "exact_second_infinite_family_formula",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "sporadic_exception_(3,4,7,6)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the exact classification. It only poses the hypothesis and asks for the full description of all admissible quadruples."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem asks for the complete classification, and the correct option is the theorem's conclusion almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to compare exactness, parameter ranges, and sporadic exceptions across choices, especially against the weaker-true option. However, the item mainly tests recall/recognition of a known classification rather than generating a conclusion from mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and well targeted: one weakens the claim, one alters a boundary condition, one perturbs a formula in an infinite family, and one adds a plausible extra sporadic case. These reflect realistic mathematical failure modes."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed distractors and no answer leakage, but the question is largely a theorem-statement recognition task rather than a genuinely generative reasoning problem."
    }
  },
  {
    "id": "2512.03527v1",
    "paper_link": "http://arxiv.org/abs/2512.03527v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.",
      "start_pos": 18365,
      "end_pos": 18798,
      "label": "main_theorem"
    },
    "ref_dict": {
      "main_theorem": "\\begin{theorem}\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3666,
    "pre_theorem_intro_text": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}. \n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n   i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n   j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n   k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}. \n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.",
    "context": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}.\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.",
    "full_context": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}.\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.\n\n\\begin{abstract}\nWe prove that, in all except one case, a Gorenstein del Pezzo surface of\nPicard rank $1$ admits an int-amplified endomorphism if and only if it\nis a quotient of a toric variety by a finite group which acts freely in\ncodimension one and preserves the open torus. We classify all such\nquotients.\n\\end{abstract}\n\\maketitle\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\P^2$. Let $X$ be the quotient of\n$\\P^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\P^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\C} \\isom \\C$ and $f^*: N^1(X)_{\\C} \\to N^1(X)_{\\C}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\P(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.\n\nWe give an overview of rank $1$ Gorenstein del Pezzo surfaces below and\ntheir classification by singularity type. To prove Theorem\n\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del\nPezzo surfaces are quotients of toric surfaces. The ``easy direction''\n(that is, the statement that if $X$ is such a quotient of a toric\nvariety, it admits an int-amplified endomorphism) follows from\n\\cite[Lemma 2.6]{NakayamaStruct}. For the rest, following Kawakami and\nTotaro \\cite{Kawakami-Totaro}, we use the tool of Bott vanishing,\ncombined with lifting to universal covers, to deduce that they do not\nadmit int-amplified endomorphisms. To find failures of Bott vanishing,\nwe use a version of the Riemann-Roch formula for normal surfaces. This\nformula is automated in a Python program which is in the accompanying\ncode repository \\cite{CodeRepo}.\n\nWe work in characteristic zero, but everything in this paper should\nhold over an algebraically closed field of characteristic $\\neq 2, 3$,\nas long as we everywhere replace ``int-amplified endomorphism'' with\n``int-amplified endomorphism of degree invertible in the base field''.\nThe classification of rank $1$ Gorenstein del Pezzo surfaces is the same\nover algebraically closed fields of characteristic $\\neq 2, 3$\n\\cite[Theorem B.6]{Lacini}, and all the endomorphisms described in this\npaper have degree coprime to $2$ and $3$.\n\n\\begin{proposition}\\label{toric_quotient} The five surfaces\n$S(3A_1+D_4)$, $S(A_1+2A_3)$, $S(A_1+A_2+A_5)$, $S(4A_2)$ and\n$S(2A_1+2A_3)$ are quotients of toric surfaces by a finite group that\nacts freely in codimension one and preserves the open torus.\n\\end{proposition}\n\\begin{proof}\nWe describe each of these surfaces as a quotient of a toric variety, and\nfind an int-amplified endomorphism of the toric variety that descends to\nthe quotient (we can also apply \\cite[Lemma 2.6]{NakayamaStruct} to show\nthat the quotients admit int-amplified endomorphisms).\n\\begin{itemize}\n\\item $S(3A_1+D_4)$ is the surface constructed in Example\n\\ref{quaternion-example}.\n\nThe quasi-universal covers of $S(A_2+E_6)$ and $S'(A_2+E_6)$ are\nGorenstein del Pezzo surfaces of rank $3$ with a single $D_4$\nsingularity. The quasi-universal covers of $S(A_1+E_7)$ and\n$S'(A_1+E_7)$ are Gorenstein del Pezzo surfaces of rank $2$ with a\nsingle $E_6$ singularity. Let $Y$ be one of the four surfaces\n$S(A_2+E_6)$, $S'(A_2+E_6)$, $S(A_1+E_7)$ or $S'(A_1+E_7)$ and let $\\pi:\nX \\to Y$ be the quasi-universal cover. Since $D_4$ and $E_6$\nsingularities are not toric, $X$ is not a toric variety. Furthermore, we\nknow $f^{-1}(Y^0)$ is simply-connected and is $X^0$ minus a finite set\nof points: since $X^0$ is smooth, $X^0$ is also simply-connected. Thus\n$X$ cannot be a quotient of a toric variety by a finite group acting\nfreely in codimension one. However, by \\cite[Theorem 1.3]{Nakayama}, if\na klt del Pezzo surface of Picard rank $>1$ admits a endomorphism of\ndegree $>1$, it is a quotient of a toric variety by a finite group\nacting freely in codimension one. If $Y$ admitted an int-amplified\nendomorphism, it would lift to endomorphism of $X$ of degree $>1$, which\nis impossible. Thus the surfaces $S(A_2+E_6)$, $S'(A_2+E_6)$,\n$S(A_1+E_7)$ and $S'(A_1+E_7)$ do not admit int-amplified endomorphisms.\n\n\\begin{theorem}\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1560,
    "post_theorem_intro_text": "We give an overview of rank $1$ Gorenstein del Pezzo surfaces below and\ntheir classification by singularity type. To prove Theorem\n\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del\nPezzo surfaces are quotients of toric surfaces. The ``easy direction''\n(that is, the statement that if $X$ is such a quotient of a toric\nvariety, it admits an int-amplified endomorphism) follows from\n\\cite[Lemma 2.6]{NakayamaStruct}. For the rest, following Kawakami and\nTotaro \\cite{Kawakami-Totaro}, we use the tool of Bott vanishing,\ncombined with lifting to universal covers, to deduce that they do not\nadmit int-amplified endomorphisms. To find failures of Bott vanishing,\nwe use a version of the Riemann-Roch formula for normal surfaces. This\nformula is automated in a Python program which is in the accompanying\ncode repository \\cite{CodeRepo}.\n\nWe work in characteristic zero, but everything in this paper should\nhold over an algebraically closed field of characteristic $\\neq 2, 3$,\nas long as we everywhere replace ``int-amplified endomorphism'' with\n``int-amplified endomorphism of degree invertible in the base field''.\nThe classification of rank $1$ Gorenstein del Pezzo surfaces is the same\nover algebraically closed fields of characteristic $\\neq 2, 3$\n\\cite[Theorem B.6]{Lacini}, and all the endomorphisms described in this\npaper have degree coprime to $2$ and $3$.\n\n\\subsection*{Acknowledgments} I would like to thank Burt Totaro for many\nhelpful conversations. The author is supported by NSF Graduate Research\nFellowship Grant No. DGE-2034835.",
    "sketch": "To prove Theorem~\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del Pezzo surfaces are quotients of toric surfaces. The ``easy direction'' (if $X$ is such a quotient of a toric variety, then it admits an int-amplified endomorphism) follows from \\cite[Lemma 2.6]{NakayamaStruct}. For the remaining direction, following Kawakami and Totaro \\cite{Kawakami-Totaro}, we use Bott vanishing, combined with lifting to universal covers, to deduce that the other surfaces do not admit int-amplified endomorphisms. To find failures of Bott vanishing, we use a version of the Riemann--Roch formula for normal surfaces (implemented in a Python program in \\cite{CodeRepo}).",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $X$ be a Gorenstein del Pezzo surface of Picard rank $1$ over an algebraically closed field of characteristic $0$, and assume that $X$ is not isomorphic to the weighted sextic surface\n$$S'(E_8)=\\{X^5Y+X^4Z+Z^3+W^2=0\\}\\subset \\mathbb{P}(1,1,2,3).$$\nRecall that a surjective endomorphism $f\\colon X\\to X$ is called int-amplified if there exists an ample Cartier divisor $A$ on $X$ such that $f^*A-A$ is ample; for Picard-rank-$1$ surfaces, this is equivalent to $\\deg f>1$. A toric variety has a dense open torus, called its open torus. Under these assumptions, which statement about $X$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "$X$ admits an int-amplified endomorphism (equivalently, an endomorphism of degree $>1$) if and only if $X$ is a quotient of a toric variety by a finite group that acts freely in codimension one and preserves the open torus."
      },
      "choices": [
        {
          "label": "B",
          "text": "$X$ admits an int-amplified endomorphism (equivalently, an endomorphism of degree $>1$) if and only if $X$ is itself a toric variety."
        },
        {
          "label": "C",
          "text": "If $X$ is a quotient of a toric variety by a finite group that acts freely in codimension one and preserves the open torus, then $X$ admits an int-amplified endomorphism (equivalently, an endomorphism of degree $>1$)."
        },
        {
          "label": "D",
          "text": "$X$ admits an int-amplified endomorphism (equivalently, an endomorphism of degree $>1$) if and only if $X$ is a quotient of a toric variety by a finite group that preserves the open torus."
        },
        {
          "label": "E",
          "text": "$X$ admits an int-amplified endomorphism (equivalently, an endomorphism of degree $>1$) whenever $X$ is a quotient of a toric variety by a finite group that acts freely in codimension one and preserves the open torus, and conversely every such endomorphism exists only after replacing $X$ by a finite cover that is toric."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "quotient-of-toric versus toric",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the converse implication",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "free-in-codimension-one hypothesis",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "existence on $X$ versus only after passing to a toric finite cover",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives background definitions and hypotheses but does not explicitly reveal the correct equivalence. It does not single out the quotient-of-toric characterization in a way that makes the answer trivial."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: under the stated hypotheses, it asks which exact characterization holds. However, it is not a pure restatement because the options include weaker, stronger, and subtly altered variants."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare the logical strength of the choices, especially distinguishing the exact iff statement from a weaker one-way implication and from overstrong or hypothesis-dropped alternatives. Still, it mainly tests precise theorem recognition rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well-targeted: one is too strong (toric instead of quotient-of-toric), one is a weaker true statement, one drops a crucial codimension-one freeness condition, and one introduces a misleading finite-cover reformulation."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, though it leans more toward exact theorem recall than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.03670v1",
    "paper_link": "http://arxiv.org/abs/2512.03670v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\n    For any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$.",
      "start_pos": 12428,
      "end_pos": 12606,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:growth": "\\begin{theorem} \\label{thm:growth}  \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}",
      "thm:computable": "\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}",
      "conj:ThueListe": "\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}",
      "thm:main": "\\begin{theorem} \\label{thm:main}\n    For any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$. \n\\end{theorem}",
      "lem:eigenvector": "\\begin{lemma}\\label{lem:eigenvector}\n    Let $p=12$. There exist nonnegative weights $(\\lambda_s)_{s\\in \\minsuff}$ with $\\lambda_{\\varepsilon}>0$ such that for every word $w$ over the alphabet $\\mathbb{N}$\n    and every $\\A\\in \\binom{\\mathbb{N}}{2}$,\n    \\begin{equation}\\label{ineq:eigen}\n        \\alpha \\lambda_{\\minsuff(w)} \\le \\sum_{\\substack{ c\\in\\A\\\\wc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\lambda_{\\minsuff(wc)},\n    \\end{equation}\n    where $\\alpha = 1.457$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 3447,
    "pre_theorem_intro_text": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with  $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.",
    "context": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}",
    "full_context": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\n\\section{Introduction}\nA \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\nThe idea behind the first proof is to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language which yields the following result.\n\\begin{theorem} \\label{thm:growth} \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}\nThe core idea allowing us to deduce bounds on the growth of a power-free language $\\mathcal L$ from its approximation $\\mathcal L$ was first used by Kolpakov to estimate the growth rate of power-free languages \\cite{kopakov2007efficient,Kolpakov2007Dec}. Shur later improved upon this technique \\cite{shur2010Jul,Shur2012Nov}. This technique was later adapted by the second author to provide a partial answer to Conjecture \\ref{conj:ThueListe} in \\cite{rosenfeld2022avoiding}.\nThe proof of this result applies the technique developed in \\cite{rosenfeld2022avoiding} for squares, but provides a complete answer in the case of cubes.\nThe second proof of our main result does not imply anything about the number of solutions, but it contains algorithmic implications.\n\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}\nThis proof also relies on computing properties of the same automaton, but then we use a different argument for adding the large cubes back.\nIntuitively, the idea is to introduce a weight function that measures how difficult it is to extend a word, and to prove that under some condition if the weight is not too large then at least one of the extensions also has a small weight (which we can then use inductively to construct a word). This idea was originally due to Miller \\cite{miller2012two}, and was recently generalized in \\cite{Rosenfeld2025Mar}. However, the weight functions in \\cite{miller2012two} and \\cite{Rosenfeld2025Mar} are much simpler and in our article the precise definition of the function depends on the automaton. Our result is in fact even stronger, as everything still holds if for all $i$ an opponent is allowed to choose the alphabet $\\A_{i+1}$ after we choose $w_i$.\n\n\\section{Approximation by a regular language}\\label{sec:notation}\nOur approach is to bound the difference between the number of cube-free words and the number of words that avoids all cubes up to a given length $p$. \nWe let $\\mathcal{C}$ be the set of all cube-free words. \nFor a given integer $p$, we let $\\widetilde{\\mathcal{C}}^{(p)}$ be the set of words that avoid cubes of period at most $p$.\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$, we let $\\mathcal{C}[(\\A_i)_{i\\ge1}]$ (resp. $\\widetilde{\\mathcal{C}}^{(p)}[(\\A_i)_{i\\ge1}]$) be the elements of $\\mathcal{C}$ (resp. $\\widetilde{\\mathcal{C}}^{(p)}$) that respect $(\\A_i)_{i\\ge1}$.\n\nThis implies \n\\begin{align*}\n \\sum_{\\substack{c\\in\\A\\\\uc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\omega(uc)\n &\\le \\beta\\omega(u)+\\sum_{i>p} \\lambda_{\\minsuff(u)}\\beta^{1-i}\\\\\n &<\\lambda_{\\minsuff(u)}\\left(\\beta+\\frac{\\beta^{1-p}}{\\beta-1}\\right)\\\\\n &\\le\\lambda_{\\minsuff(u)}\\alpha\\\\\n &\\le \\sum_{\\substack{c\\in\\A\\\\uc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\lambda_{\\minsuff(uc)}\\,,\n\\end{align*}\nwhere the third inequality is by our theorem hypothesis, and the last inequality was proven earlier in this proof.\nThis implies that there exists $c\\in \\A$ such that $uc\\in \\widetilde{\\mathcal{C}}^{(p)}$ and $\\omega(uc)<\\lambda_{\\minsuff(uc)}$ as desired.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:computable}]\n In particular, with $p=12$ and $\\alpha=1.457$ we can set $\\beta=1.35$ to satisfy the condition of Lemma \\ref{lem:betacond}.\n So we inductively define $u_0=\\varepsilon$, and for all $n$, $u_{n+1} = u_n a_{n+1}$ where $a_{n+1}\\in\\A_{n+1}$ is chosen such that $\\omega(u_{n+1})<\\lambda_{\\minsuff(u_{n+1})}$. By Lemma \\ref{lem:smallweighisfine}, each of the $u_i$ is cube-free, and so is the infinite word $a_1a_2a_3\\ldots$ and by construction it belongs to $\\A_1\\times\\A_2\\times\\A_3\\ldots$ as desired.\n\n\\begin{lemma}\\label{lem:counting_argument}\nFor all $n\\in\\mathbb{N}$ and $t\\in\\minsuff$, we have the following inequality\n \\begin{equation}\\label{eq:exclusion}\n |C_{n+1}^t| \\ge \\sum_{v\\in C_n}\\sum_{\\substack{c\\in\\A_{n+1}\\\\ vc\\in\\widetilde{\\mathcal{C}}^{(p)}\\\\\\minsuff(vc)=t}} 1 - \\sum_{i=p+1}^{n+1} |C_{n+1-i}^t|.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We let $G^t$ be the set of words of length $n+1$ in $A_1\\times\\ldots\\times A_{n+1}$ so that\n \\begin{itemize}\n \\item the prefix of size $n$ is cube-free,\n \\item the word does not contain any cubes of period at most $p$,\n \\item the word ends in state $t$.\n \\end{itemize} \n In other words,\n \\[\n G^t= \\left\\{va\\in \\widetilde{\\mathcal{C}}^{(p)}: v\\in C_{n}, a\\in A_{n+1}, \\minsuff(va)=t\\right\\}\\,.\n \\]\n Hence,\n \\[\n |G^t|= \\sum_{v\\in C_n}\\sum_{\\substack{c\\in\\A_{n+1}\\\\ vc\\in\\widetilde{\\mathcal{C}}^{(p)}\\\\\\minsuff(vc)=t}} 1\\,.\n \\]\n Let $F_i$ be the set of words from $G_t$ that contain a cube of period $i$ as a suffix. By definition, a word from $G_t$ can only contain a cube as a suffix and this cube must be of period more than $p$ (and at most $n+1$).\\footnote{We could easily replace this $n+1$ by $\\lfloor(n+1)/3\\rfloor$, but it does not make any difference in the latter part of the proof.} We get\n \\[\n C_{n+1}^t= G^t \\setminus \\bigcup_{i=p+1}^{n+1} F_i\\,,\n \\]\n which implies\n \\[\n |C_{n+1}^t|\\ge |G^t| - \\sum_{i=p+1}^{n+1} |F_i|\\,.\n \\]\n It remains to prove that $|F_i|\\le |C_{n+1-i}^t|$ for every $i$ to conclude the proof.\n Let $va\\in F_i$, then by definition $va=uyyy$ with $|y|=i>p$. Since $v\\in C_n$ and $va\\in \\widetilde{\\mathcal{C}}^{(p)}$, we can apply Corollary \\ref{cor:easingOnePeriodPreservesState} to deduce that $\\minsuff(uyy)=\\minsuff(vc)=t$, that is, $uyy\\in C_{n+1-i}^t$. Given the word $uyy$ and the period $i=|y|$, we can uniquely reconstruct $vc=uyyy$. This implies $|F_i|\\le |C_{n+1-i}^t|$, which concludes our proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 4027,
    "post_theorem_intro_text": "In fact, we do not prove this result directly, but we prove two different stronger results, each implying Theorem \\ref{thm:main}. \nThese two proofs share a key idea: we can use an automaton to study the variant of the problem where we only forbid cubes of length at most $p$, for some arbitrary choice of $p$. Intuitively, short cubes are more difficult to avoid than long cubes, so by taking $p$ large enough, we should obtain a good approximation of the problem. Adapting standard tools from combinatorics on words and automata theory, studying this approximation is quite simple to achieve.\nThe delicate part is then to prove that if $p$ is large enough this approximation is good enough to deduce something about the original problem where all cubes are forbidden. \n\nThe idea behind the first proof is to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language which yields the following result.\n\\begin{theorem} \\label{thm:growth}  \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}\nThe core idea allowing us to deduce bounds on the growth of a power-free language $\\mathcal L$ from its approximation $\\mathcal L$ was first used by Kolpakov to estimate the growth rate of power-free languages \\cite{kopakov2007efficient,Kolpakov2007Dec}. Shur later improved upon this technique \\cite{shur2010Jul,Shur2012Nov}. This technique was later adapted by the second author to provide a partial answer to Conjecture \\ref{conj:ThueListe} in  \\cite{rosenfeld2022avoiding}.\nThe proof of this result applies the technique developed in \\cite{rosenfeld2022avoiding} for squares, but provides a complete answer in the case of cubes.\nThe second proof of our main result does not imply anything about the number of solutions, but it contains algorithmic implications.\n\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}\nThis proof also relies on computing properties of the same automaton, but then we use a different argument for adding the large cubes back.\nIntuitively, the idea is to introduce a weight function that measures how difficult it is to extend a word, and to prove that under some condition if the weight is not too large then at least one of the extensions also has a small weight (which we can then use inductively to construct a word). This idea was originally due to Miller \\cite{miller2012two}, and was recently generalized in \\cite{Rosenfeld2025Mar}. However, the weight functions in \\cite{miller2012two} and \\cite{Rosenfeld2025Mar} are much simpler and in our article the precise definition of the function depends on the automaton. Our result is in fact even stronger, as everything still holds if for all $i$ an opponent is allowed to choose the alphabet $\\A_{i+1}$ after we choose $w_i$.\n\nNote that Theorem \\ref{thm:growth} and Theorem \\ref{thm:computable} both independently imply Theorem \\ref{thm:main}, but none of them seem to imply the other. We discuss the possible connection between this result in Section \\ref{sec:discussion}.\n\nThe article is organized as follows. In Section \\ref{sec:notation}, we introduce some common notations. In particular, we define the language $\\widetilde{\\mathcal{C}}^{(p)}$ that approximate the language of cube-free words, and we give the statement of Lemma \\ref{lem:eigenvector} that is later crucial to our two proofs. In Section \\ref{sec:potential}, we prove Theorem \\ref{thm:computable}. In Section \\ref{sec:growth}, we prove Theorem \\ref{thm:growth}. Finally, we provide in Section \\ref{sec:computations} a short description of the computation used to verify Lemma \\ref{lem:eigenvector}.",
    "sketch": "We \"do not prove [Theorem \\ref{thm:main}] directly\", but instead prove \"two different stronger results, each implying Theorem \\ref{thm:main}\" (Theorem~\\ref{thm:growth} and Theorem~\\ref{thm:computable}). Both proofs share a key idea: \"use an automaton\" to study an approximation where one \"only forbid[s] cubes of length at most $p$\". The motivation is that \"short cubes are more difficult to avoid than long cubes\", so with $p$ large this gives \"a good approximation\"; the \"delicate part\" is to show that for $p$ large enough, this approximation lets one conclude results for the original language where \"all cubes are forbidden\".\n\nFor the route via Theorem~\\ref{thm:growth}, the idea is \"to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language\" and then apply the technique (Kolpakov; improved by Shur; adapted in \\cite{rosenfeld2022avoiding}) to \"deduce bounds on the growth of a power-free language\" from its approximation.\n\nFor the route via Theorem~\\ref{thm:computable}, one again uses \"computing properties of the same automaton\", but with \"a different argument for adding the large cubes back\": \"introduce a weight function that measures how difficult it is to extend a word\" and prove that \"under some condition if the weight is not too large then at least one of the extensions also has a small weight\", enabling an inductive construction of an infinite cube-free word; the weight function here \"depends on the automaton\" (in contrast to simpler weight functions in \\cite{miller2012two,Rosenfeld2025Mar}).",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main}\nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$.",
    "theorem_type": [
      "Universal–Existential",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\((\\mathcal A_i)_{i\\ge 1}\\) be any sequence of alphabets, where each \\(\\mathcal A_i\\) is binary (that is, \\(|\\mathcal A_i|=2\\)). A cube is a word of the form \\(uuu\\) for some nonempty word \\(u\\), and an infinite word is cube-free if none of its finite factors is a cube. Which statement holds about infinite words \\(w=a_1a_2a_3\\cdots\\) with \\(a_i\\in \\mathcal A_i\\) for every \\(i\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists an infinite cube-free word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\); equivalently, for any sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an infinite cube-free word in \\(\\mathcal A_1\\times \\mathcal A_2\\times \\cdots\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an infinite word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\), and \\(w\\) is square-free; equivalently, for any sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an infinite square-free word in \\(\\mathcal A_1\\times \\mathcal A_2\\times \\cdots\\)."
        },
        {
          "label": "C",
          "text": "There exists an infinite word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\), and every finite prefix of \\(w\\) is cube-free; in particular, \\(w\\) has arbitrarily long cube-free finite prefixes respecting \\((\\mathcal A_i)_{i\\ge 1}\\)."
        },
        {
          "label": "D",
          "text": "For every sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an integer \\(N\\) such that for every \\(n\\ge N\\) one can choose letters \\(a_1\\in\\mathcal A_1,\\dots,a_n\\in\\mathcal A_n\\) so that \\(a_1\\cdots a_n\\) is cube-free; equivalently, every such sequence admits cube-free words of all sufficiently large lengths."
        },
        {
          "label": "E",
          "text": "There exists a single infinite cube-free binary word \\(w=b_1b_2b_3\\cdots\\) with the following universal property: for every sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), one has \\(b_i\\in \\mathcal A_i\\) for every \\(i\\); equivalently, the same infinite cube-free word belongs to \\(\\mathcal A_1\\times\\mathcal A_2\\times\\cdots\\) for all choices of binary alphabets."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "cube_free_vs_square_free_target",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "drop_global_factor_condition_to_prefixwise_cube_free",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replace_single_infinite_word_with_nonuniform_per_length_existence",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "dependence_of_word_on_alphabet_sequence",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only defines binary alphabets and cube-freeness, then asks which existence statement is true. It does not explicitly or implicitly reveal that the infinite cube-free conclusion in choice A is correct."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to theorem recognition: one option is essentially the target existence theorem, while the others are weakened, strengthened, or quantifier-tampered variants. So it is not fully tautological, but it is still a mild reformulation of a theorem statement rather than a derived application."
      },
      "GPS": {
        "score": 1,
        "justification": "Solving requires some reasoning about logical strength, quantifiers, and the distinction between cube-free and square-free. However, it mainly tests recognition/comparison of candidate statements rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: B alters infinite extension, C is a weaker true statement, D confuses cube-free with square-free, and E introduces a subtle quantifier-dependence error. These are plausible and distinct failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it is somewhat theorem-recognition-based and only moderately pressures genuine generative reasoning."
    }
  },
  {
    "id": "2512.04053v1",
    "paper_link": "http://arxiv.org/abs/2512.04053v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]",
      "start_pos": 9311,
      "end_pos": 9633,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{thm}\n\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\supp(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 1066,
    "pre_theorem_intro_text": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.",
    "context": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.",
    "full_context": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.\n\nThe key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \\cite{ms16} that $\\mathfrak S_w(1,\\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \\cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\\mathfrak S_w(1, \\dots,1)$ and $|\\supp(\\mathfrak S_w)|$.\n\nWe will use the following (crude) estimates for the factorial function, whose proofs we include for completeness.\n\\begin{lem}\n\\label{lem:factorial-estimate}\nThe inequalities\n\\[\ne\\left(\\frac ne\\right)^n \\leq n! \\leq e\\left(\\frac{n+1}e\\right)^{n+1}\n\\]\nhold for all positive integers $n$.\n\\end{lem}\n\\begin{proof}\nEstimate $\\ln(n!) = \\sum_{k=1}^n\\ln(k)$ by the integrals\n\\[\n\\int_1^n \\ln(x)\\,dx \\leq \\sum_{k=1}^n \\ln(k) \\leq \\int_1^{n+1}\\ln(x)\\,dx.\n\\]\nThe definite integrals evaluate to $n\\ln(n) - n + 1$ and $(n+1)\\ln(n+1) - (n+1) + 1$. The claim follows.\n\\end{proof}\n\\begin{lem}\n\\label{lem:floor-fact-est}\nThe inequality\n\\[\n\\lfloor k\\rfloor! \\geq \\frac1{2k}\\left(\\frac ke\\right)^k\n\\]\nholds for all real numbers $k>1$.\n\\end{lem}\n\\begin{proof}\nUsing Lemma~\\ref{lem:factorial-estimate}, compute that\n\\[\n\\lfloor k\\rfloor! \\geq \\frac1{k+1} \\lceil k\\rceil! \\geq \\frac1{k+1} \\cdot \\frac{\\lceil k\\rceil^{\\lceil k\\rceil}}{e^{\\lceil k\\rceil-1}} \\geq \\frac1{k+1}\\cdot \\frac{k^k}{e^k}.\n\\]\nAs $k > 1$, the claim follows.\n\\end{proof}\n\\section{A layered permutation with large Schubitope}\nOur main technical result is as follows.\n\\begin{thm}\n\\label{thm:tech-main}\nLet $w = w(b_1, \\dots, b_m)$ be a layered permutation in $S_n$. If $b_m \\leq n/2$, then\n\\[\n|\\supp(\\mathfrak S_w)| \\geq b_m!\\cdot |\\supp(\\mathfrak S_{w'})|, \\qquad \\textup{ where } w'\\colonequals w(b_1, \\dots, b_{m-1})\\in S_{n-b_m}.\n\\]\n\\end{thm}\nOur proof of Theorem~\\ref{thm:tech-main} will use the following lemma.\n\\begin{lem}\n\\label{lem:last-block}\nLet $w\\in S_n$ be a layered permutation of the form $w = w(1, \\dots, 1, b_m)$ for $b_m \\leq n/2$. For any subset $S\\subseteq D(w)$, there exists a diagram $C$ satisfying $C\\leq D(w)$ and $C\\cap D(w) = S$.\n\\end{lem}\n\\begin{proof}\nEach column $D_j$ of $D(w)$ is of the form $\\{n-b_m + 1, \\dots, n-c\\}$ for some $c \\geq 1$. As $|D_j| \\leq b_m-1 < n-b_m$, it follows that for any subset $S_j$ of $D_j$ there exists a set $C_j\\leq D_j$ with $C_j \\cap D_j = S_j$. Applying this column by column to $D(w)$, the claim follows.\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:tech-main}]\nFor each of the $b_m!$ many vectors $\\alpha\\in\\ZZ_{\\geq0}^{b_m}$ satisfying $(0,\\dots,0) \\leq \\alpha \\leq (b_m - 1, b_m - 2, \\dots, 1, 0)$, we construct an embedding $i_\\alpha\\colon \\mathcal S_{D(w')} \\hookrightarrow \\mathcal S_{D(w)}$ so that $\\img(i_\\alpha)\\cap\\img(i_{\\alpha'}) = \\emptyset$ for all $\\alpha\\neq\\alpha'$.\n\nFor each $\\alpha$ satisfying $(0,\\dots,0)\\leq\\alpha\\leq (b_m - 1, b_m - 2, \\dots, 1)$, use Lemma~\\ref{lem:last-block} to choose a diagram $C_\\alpha \\leq D(w_{\\mathrm{LB}})$ whose weight satisfies $\\wt(C_\\alpha)_{n-b_i + j} = \\alpha_j$. Then define the map\n\\begin{align*}\ni_\\alpha\\colon \\mathcal S_{D(w')}&\\to \\mathcal S_{D(w)}\\\\\n\\wt(C')&\\mapsto\\wt((C', C_\\alpha));\n\\end{align*}\nthe map $i_\\alpha$ is well-defined and injective because it is in fact the translation map $\\gamma \\mapsto \\gamma +\\wt(C_\\alpha)$. Furthermore, any $\\gamma' \\in \\mathcal S_{D(w')}$ satisfies $\\gamma'_{n-b_m+j} = 0$ for all $j\\geq 0$, so any vector $\\gamma\\in\\img(i_\\alpha)$ satisfies $\\gamma_{n-b_m+j} = \\alpha_j$. In particular, the $\\img(i_\\alpha)$ are disjoint subsets of $\\mathcal S_{D(w)}$.\n\\end{proof}\n\\begin{cor}\n\\label{cor:upshot}\nFix an integer $n\\geq 3$ and let $c\\colonequals \\max\\{k \\colon \\lfloor n/2^k\\rfloor \\geq 1\\}$ and $d \\colonequals n - \\sum_{k=1}^c \\lfloor n/2^k\\rfloor$. Let $w$ be the layered permutation with blocks of size\n\\[\n\\underbrace{1,\\dots,1}_{d\\textup{ many}}, \\left\\lfloor \\frac n{2^c}\\right\\rfloor, \\left\\lfloor \\frac n{2^{c-1}}\\right\\rfloor, \\dots, \\left\\lfloor\\frac n4\\right\\rfloor,\\left\\lfloor \\frac n2\\right\\rfloor.\n\\]\nThen \n\\[\n|\\supp(\\mathfrak S_w)| \\geq \\prod_{k=1}^c\\left\\lfloor \\frac n{2^k}\\right\\rfloor!.\n\\]\nIn particular, \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\geq \\left(n - \\frac1{\\ln(2)}\\ln(n) - 2\\right)\\ln(n) - n\\ln(4) - (n-2).\n\\]\n\\end{cor}\n\\begin{proof}\nFrom the inequality\n\\[\n\\left\\lfloor \\frac n{2^k}\\right\\rfloor \\leq \\frac n{2^k} = \\frac12\\left(n - \\sum_{i=1}^{k-1} \\frac n{2^i}\\right) \\leq \\frac12\\left(n - \\sum_{i=1}^{k-1} \\left\\lfloor\\frac n{2^i}\\right\\rfloor\\right),\n\\]\nTheorem~\\ref{thm:tech-main} may be repeatedly applied to the layered permutation $w$ to obtain\n\\[\n|\\supp(\\mathfrak S_w)| \\geq \\prod_{k=1}^c\\left\\lfloor \\frac n{2^k}\\right\\rfloor!.\n\\]\nLemma~\\ref{lem:floor-fact-est} implies\n\\begin{align*}\n\\prod_{k=1}^c\\left \\lfloor \\frac n{2^k}\\right\\rfloor! &\\geq \\prod_{k=1}^c\\left(\\frac{2^{k-1}}n \\left(\\frac n{2^k e}\\right)^{\\frac n{2^k}}\\right) \\\\&\\geq \\frac 1{n^c} \\cdot \\underbrace{\\prod_{k=1}^c \\left(\\frac1{2^k}\\right)^{\\frac n{2^k}}}_{\\geq 2^{-2n}}\\cdot \\underbrace{\\prod_{k=1}^c \\left(\\frac ne\\right)^{\\frac n{2^k}}}_{\\geq \\left(\\frac ne\\right)^{n-2}}\\label{eqn:estimate}\\tag{$*$}\\\\\n&\\geq \\frac{n^{n-c-2}}{4^n\\cdot e^{n-2}}\\\\&\\geq \\frac{n^{n - \\log_2(n) - 2}}{4^n \\cdot e^{n-2}}\n\\end{align*}\nwhere the estimates in~\\eqref{eqn:estimate} follow from\n\\[\n\\sum_{k=1}^c k\\cdot \\frac n{2^k} \\leq \\sum_{k=1}^\\infty k \\cdot \\frac n{2^k} = 2n\n\\]\nand\n\\[\n\\frac n2 + \\dots + \\frac n{2^c} = n - \\frac n{2^c} \\geq n-2.\n\\]\n\\end{proof}\n\n\\newtheorem*{thm:main}{Theorem~\\ref{thm:main}}\n\\begin{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\supp(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]\n\\end{thm:main}\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nBy Lemma~\\ref{lem:upper-bound}, the inequality \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\leq \\ln(n!) \\leq (n+1)\\ln(n+1) - n\n\\]\nholds for all $w\\in S_n$. By Corollary~\\ref{cor:upshot}, there exists $w\\in S_n$ with \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\geq \\left(n-\\frac1{\\ln(2)}\\ln(n) - 2\\right)\\ln(n) - n\\ln(4) - (n-2).\n\\]\nThe inequalities\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1\n\\]\nfollow.\n\\end{proof}\n\n\\begin{prop}\n\\label{prop:groth-precise}\nFix an integer $n$, let $k$ be the unique integer so that $\\binom{k+1}2\\leq n<\\binom{k+2}2$, and let $b \\colonequals n - \\sum_{i=1}^k i$. Let $w$ be the layered permutation with blocks of size $1, 2, \\dots, k, b$. Then\n\\[\n|\\supp(\\mathfrak G_w)| \\geq \\frac{n!}{n^{k+1}}.\n\\]\nIn particular,\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|) \\geq n\\ln(n) - n - (\\sqrt{2n}+1)\\ln(n)\n\\]\n\\end{prop}\n\\begin{proof}\nLet $\\mathbf c = \\mathrm{wt}(D(w))$ denote the Lehmer code of $w$, and let $\\mathbf d = \\mathrm{wt}(\\overline{D(w)})$. Proposition~\\ref{prop:fireworks-grothendieck} implies that $[\\mathbf c, \\mathbf d] \\subseteq \\supp(\\mathfrak G_w)$, so\n\\begin{equation}\n\\label{eqn:crude-grothendieck}\n|\\supp(\\mathfrak G_w)| \\geq \\prod_{i=1}^n (d_i - c_i + 1).\n\\end{equation}\n\n\\newtheorem*{thm:groth}{Theorem~\\ref{thm:groth}}\n\\begin{thm:groth}\nLet $\\beta^{\\mathfrak G}(n)\\colonequals\\max_{w\\in S_n}|\\supp(\\mathfrak G_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta^{\\mathfrak G}(n)) - n\\ln(n)}n = -1.\n\\]\n\\end{thm:groth}\n\\begin{proof}[Proof of Theorem~\\ref{thm:groth}]\nBy Lemma~\\ref{lem:upper-bound}, the inequality\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|)\\leq \\ln(n!) \\leq (n+1)\\ln(n+1)-n\n\\]\nholds for all $w\\in S_n$. By Proposition~\\ref{prop:groth-precise}, there exists $w\\in S_n$ with\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|) \\geq n\\ln(n) - n - (\\sqrt{2n}+1)\\ln(n).\n\\]\nThe claim follows.\n\\end{proof}",
    "post_theorem_intro_text_len": 1386,
    "post_theorem_intro_text": "Theorem~\\ref{thm:main} answers a problem \\cite[Prob 5.5]{gl24} posed by Guo--Lin. We do not know if the limit $\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n$ exists, nor do we have a conjecture for its value.\n\nThe key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \\cite{ms16} that $\\mathfrak S_w(1,\\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \\cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\\mathfrak S_w(1, \\dots,1)$ and $|\\mathrm{supp}(\\mathfrak S_w)|$.\n\nGrothendieck polynomials $\\mathfrak G_w$ are inhomogeneous deformations of Schubert polynomials; they are generating functions for (possibly nonreduced) pipe dreams of $w$. We are able to produce a layered permutation with support of size asymptotically at least $n!/e^{\\sqrt{2n} \\cdot \\ln(n)}$, i.e., of size $n!$ up to a subexponential factor. In particular, the maximal sizes of supports of Grothendieck polynomials satisfy the following more precise asymptotics.\n\\begin{thm}\n\\label{thm:groth}\nLet $\\beta^{\\mathfrak G}(n)\\colonequals\\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak G_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta^{\\mathfrak G}(n)) - n\\ln(n)}n = -1.\n\\]\n\\end{thm}",
    "sketch": "The post-theorem introduction indicates that “the key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$.”",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "For each positive integer $n$, let $S_n$ be the symmetric group on $n$ letters. For a permutation $w\\in S_n$, let $\\mathfrak S_w(x_1,\\dots,x_n)$ denote its Schubert polynomial, and define its support by\n\\[\n\\operatorname{supp}(\\mathfrak S_w)\\coloneqq \\{\\alpha\\in \\mathbb Z_{\\ge 0}^n : \\mathbf x^\\alpha \\text{ appears in } \\mathfrak S_w\\}.\n\\]\nSet\n\\[\n\\beta(n)\\coloneqq \\max_{w\\in S_n} |\\operatorname{supp}(\\mathfrak S_w)|.\n\\]\nWhich asymptotic estimate holds for $\\beta(n)$?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\\le -1.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\\le 0.\n\\]"
        },
        {
          "label": "C",
          "text": "One has\n\\[\n\\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1.\n\\]"
        },
        {
          "label": "D",
          "text": "One has\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n^2}=0.\n\\]\nMore precisely,\n\\[\n\\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}= -1.\n\\]"
        },
        {
          "label": "E",
          "text": "One has\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1.\n\\]\nMore precisely,\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}= -1.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "upper_second_order_constant",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_liminf_and_two_sided_refinement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "normalization_mismatch_with_subleading_claim",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "promoting_bounds_to_exact_limit",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only definitions of Schubert polynomials, support, and β(n), then asks for the asymptotic estimate. It does not reveal the correct asymptotic form or its refined constants."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: it essentially asks for the correct asymptotic statement of β(n). However, it is not a pure tautology, since the options vary in strength, precision, and limiting form, so the student must distinguish among competing formulations."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to compare nearby asymptotic claims, especially the weaker true statement (C), the overstrengthened statement (E), and the slightly altered bound in (B). Still, the question mainly tests recognition of the precise theorem statement rather than generating a conclusion from intermediate facts."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well targeted: one weakens the conclusion, one slightly relaxes a bound, one mismatches normalization, and one improperly upgrades liminf/limsup bounds to an exact limit. These reflect realistic asymptotic reasoning errors."
      },
      "total_score": 6,
      "overall_assessment": "A solid asymptotic-discrimination MCQ with no answer leakage and strong distractors, but it leans toward precise theorem recognition rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.04101v1",
    "paper_link": "http://arxiv.org/abs/2512.04101v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{l1}Let $\\Omega$ be a bounded domain in $\\mathbb{R}^n$, $f_{\\pm} :\n  \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ be $C^2$-maps such that $f_+\n  |_{\\partial \\Omega}   = f_- |_{\\partial \\Omega}  $. then\n  \\begin{equation}\n    \\int_{\\Omega} \\det f_+' (x) \\mathrm{d} x = \\int_{\\Omega} \\det f_-' (x) \\mathrm{d}\n    x \\text{.} \\label{e1}\n  \\end{equation}",
      "start_pos": 3674,
      "end_pos": 4047,
      "label": "l1"
    },
    "ref_dict": {
      "l1": "\\begin{theorem}\n  \\label{l1}Let $\\Omega$ be a bounded domain in $\\mathbb{R}^n$, $f_{\\pm} :\n  \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ be $C^2$-maps such that $f_+\n  |_{\\partial \\Omega}   = f_- |_{\\partial \\Omega}  $. then\n  \\begin{equation}\n    \\int_{\\Omega} \\det f_+' (x) \\mathd x = \\int_{\\Omega} \\det f_-' (x) \\mathd\n    x \\text{.} \\label{e1}\n  \\end{equation}\n\\end{theorem}",
      "e2": "\\begin{equation}\n  \\int_B \\det R' (x) \\mathd x = \\int_B \\mathd x = \\mathrm{Vol} (B) > 0 \\text{,}\n  \\label{e2}\n\\end{equation}",
      "e3": "\\begin{equation}\n  I (f^1_+, f^2_+, \\ldots, f_+^n) = I (f^1_-, f^2_+, \\ldots, f_+^n) \\label{e3}\n\\end{equation}",
      "e1": "\\begin{equation}\n    \\int_{\\Omega} \\det f_+' (x) \\mathd x = \\int_{\\Omega} \\det f_-' (x) \\mathd\n    x \\text{.} \\label{e1}\n  \\end{equation}"
    },
    "pre_theorem_intro_text_len": 259,
    "pre_theorem_intro_text": "In a recent paper {\\cite{MR4725395}}, to respond inquiry from some readers\nabout the solution of Exercise 1.2.1 in his book {\\cite{MR2435520}}, Krylov\ngives a proof of the following theorem, see \\citet[Lemma 1]{MR4725395}, where $f_\\pm$ only need to be $C^1$.",
    "context": "In a recent paper {\\cite{MR4725395}}, to respond inquiry from some readers\nabout the solution of Exercise 1.2.1 in his book {\\cite{MR2435520}}, Krylov\ngives a proof of the following theorem, see \\citet[Lemma 1]{MR4725395}, where $f_\\pm$ only need to be $C^1$.",
    "full_context": "In a recent paper {\\cite{MR4725395}}, to respond inquiry from some readers\nabout the solution of Exercise 1.2.1 in his book {\\cite{MR2435520}}, Krylov\ngives a proof of the following theorem, see \\citet[Lemma 1]{MR4725395}, where $f_\\pm$ only need to be $C^1$.\n\n\\begin{abstract}\n It is known that the integral of the Jacobian determinant of a smooth map $f\n : \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ depends only on $f |_{\\partial\n \\Omega} $ and this result leads to an analytic proof of the\n Brouwer fixed point theorem. In this note we provide two new proofs of this\n result, one by classical analysis and one by differential forms and Stokes\n formula.\n\\end{abstract}\n\nIn a recent paper {\\cite{MR4725395}}, to respond inquiry from some readers\nabout the solution of Exercise 1.2.1 in his book {\\cite{MR2435520}}, Krylov\ngives a proof of the following theorem, see \\citet[Lemma 1]{MR4725395}, where $f_\\pm$ only need to be $C^1$.\n\nKrylov's proof of Theorem \\ref{l1} in {\\cite{MR4725395}} is based on the\nobservation: for small $t$, $f_{\\pm}^t = \\mathrm{id} + t f_{\\pm}$ are\ndiffeomorphisms and $f^t_+ (\\Omega) = f_-^t (\\Omega)$ because $f_+ |_{\\partial\n\\Omega} = f_- |_{\\partial \\Omega} $, thus by the changing\nvariable formula\n\\[ \\int_{\\Omega} \\det (I + t f_+' (x)) \\mathd x = \\mathrm{Vol} (f_+^t (\\Omega))\n = \\mathrm{Vol} (f_-^t (\\Omega)) = \\int_{\\Omega} \\det (I + t f_-' (x)) \\mathd\n x \\text{.} \\]\nSince the two sides are polynomials in $t$, comparing the coefficients of\n$t^n$ gives (\\ref{e1}). To make the argument rigorous some issues including\nwhy $f_{\\pm}^t (\\partial \\Omega) = \\partial f_{\\pm}^t (\\Omega)$ and why\n$\\partial f_+^t (\\Omega) = \\partial f_-^t (\\Omega)$ implies $f_+^t (\\Omega) =\nf_-^t (\\Omega)$ need to be handled. These are clarified via a careful point\nset analysis in a long remark following the proof. One of the advantages of\nthis argument is that the maps $f_{\\pm}$ only need to be $C^1$.\n\nTheorem \\ref{l1} also appears in {\\cite{MR2597943,MR1046451}}. In\n{\\cite{MR1046451}}, Kulpa defines\n\\[ I (h^1, \\ldots, h^n) = \\int_{\\Omega} \\det h' (x) \\mathd x \\]\nfor smooth $h : \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ and manages to show\n\\begin{equation}\n I (f^1_+, f^2_+, \\ldots, f_+^n) = I (f^1_-, f^2_+, \\ldots, f_+^n) \\label{e3}\n\\end{equation}\nusing Fubini theorem and projections $\\Pi_i : \\mathbb{R}^n \\rightarrow\n\\mathbb{R}^{n - 1}$ given by deleting the $i$-th component. Then (\\ref{e1})\nfollows by applying (\\ref{e3}) successively to replace the remaining columns\nof $\\det f_+' (x)$ by those of $\\det f_-' (x)$. While in {\\citet[pp.\\\n464]{MR2597943}}, (\\ref{e1}) is proved using the facts that $L (P) = \\det P$\nis a null Lagrangian and the energy $\\int_{\\Omega} L (w' (x), w (x), x) \\mathd\nx$ of null Lagrangian $L : \\mathbb{M}^{m \\times n} \\times \\mathbb{R}^m \\times\n\\bar{\\Omega} \\rightarrow \\mathbb{R}$ depends only on boundary conditions.\n\nLet $f^i_{\\pm}$ be the $i$-th component of $f_{\\pm}$ and $\\Phi_{\\pm} =\n(f_{\\pm}^2, \\ldots, f^n_{\\pm})$. Then $\\Phi_+ = \\Phi_-$ on $\\partial \\Omega$.\nLet\n\\[ A_{\\pm} = \\left( \\frac{\\partial (f_{\\pm}^2, \\ldots, f_{\\pm}^n)}{\\partial\n (x^2, \\ldots, x^n)}, - \\frac{\\partial (f_{\\pm}^2, f_{\\pm}^3, \\ldots,\n f_{\\pm}^n)}{\\partial (x^1, x^3, \\ldots, x^n)}, \\ldots, (- 1)^{n + 1}\n \\frac{\\partial (f_{\\pm}^2, \\ldots, f_{\\pm}^n)}{\\partial (x^1, \\ldots, x^{n\n - 1})} \\right) \\text{,} \\]\nthen the $i $-th component of $A_{\\pm}$ is just the cofactor of\n$\\partial_{x^i} f_{\\pm}^1$ in $\\det f_{\\pm}' (x)$. By Jacobi identity (see\ne.g.\\ {\\citet[Eq.\\ (5)]{MR610487}} or {\\citet[Eq.\\ (0.3)]{MR4701455}}), we have\n$\\mathrm{div} A_{\\pm} = 0$. Thus\n\\begin{equation}\n \\mathrm{div} (f_{\\pm}^1 A_{\\pm}) = \\nabla f_{\\pm}^1 \\cdummy A_{\\pm} +\n f_{\\pm}^1 \\mathrm{div} A_{\\pm} = \\nabla f_{\\pm}^1 \\cdummy A_{\\pm} = \\det\n f_{\\pm}' (x) \\text{.} \\label{d}\n\\end{equation}\nFor $a \\in \\partial \\Omega$, let $\\eta : (u^1, \\ldots, u^{n - 1}) \\mapsto x$\nbe a local parametrization of $\\partial \\Omega$ at $a = \\eta (\\alpha)$ such\nthat\n\\[ N(a) = \\left( \\frac{\\partial (x^2, \\ldots, x^n)}{\\partial (u^1, \\ldots, u^{n -\n 1})}, - \\frac{\\partial (x^1, x^3, \\ldots, x^n)}{\\partial (u^1, \\ldots, u^{n\n - 1})}, \\ldots, (- 1)^{n + 1} \\frac{\\partial (x^1, \\ldots, x^{n -\n 1})}{\\partial (u^1, \\ldots, u^{n - 1})} \\right)_{\\alpha} \\]\nis an outward normal vector of $\\partial \\Omega$ at $a$. Applying the chain\nrule to the composition $y_{\\pm} = \\Phi_{\\pm} (\\eta (u))$, we see that\n\\[ \\left( \\begin{array}{ccc}\n \\dfrac{\\partial y_{\\pm}^2}{\\partial u^1} & \\cdots & \\dfrac{\\partial\n y_{\\pm}^2}{\\partial u^{n - 1}}\\\\\n & & \\\\\n \\dfrac{\\partial y_{\\pm}^n}{\\partial u^1} & \\cdots & \\dfrac{\\partial\n y_{\\pm}^n}{\\partial u^{n - 1}}\n \\end{array} \\right)_{\\alpha} = \\left( \\begin{array}{ccc}\n \\dfrac{\\partial y_{\\pm}^2}{\\partial x^1} & \\cdots & \\dfrac{\\partial\n y_{\\pm}^2}{\\partial x^n}\\\\\n & & \\\\\n \\dfrac{\\partial y_{\\pm}^n}{\\partial x^1} & \\cdots & \\dfrac{\\partial\n y_{\\pm}^n}{\\partial x^n}\n \\end{array} \\right)_a \\left( \\begin{array}{ccc}\n \\dfrac{\\partial x^1}{\\partial u^1} & \\cdots & \\dfrac{\\partial\n x^1}{\\partial u^{n - 1}}\\\\\n & & \\\\\n \\dfrac{\\partial x^n}{\\partial u^1} & \\cdots & \\dfrac{\\partial\n x^n}{\\partial u^{n - 1}}\n \\end{array} \\right)_{\\alpha} \\text{,} \\]\nwhere the matrix on the left is the Jacobian matrix $(\\Phi_{\\pm} \\circ \\eta)'\n(\\alpha)$.\n\nBecause $\\Phi_+ = \\Phi_-$ on $\\partial \\Omega$ and $\\eta (u) \\in \\partial\n\\Omega$ for $u$ near $\\alpha$, we have $\\Phi_+ \\circ \\eta = \\Phi_- \\circ \\eta$\nnear $\\alpha$. Using the Cauchy-Binet formula we deduce\n\\begin{align*}\n (A_+ \\cdummy N) (a) & = \\sum_{i = 1}^n \\left. \\frac{\\partial (f_+^2,\n \\ldots, f_+^n)}{\\partial (x^1, \\ldots, \\hat{x}^i, \\ldots, x^n)} \\right|_a\n \\cdummy \\left. \\frac{\\partial (x^1, \\ldots, \\hat{x}^i, \\ldots,\n x^n)}{\\partial (u^1, \\ldots, u^{n - 1})} \\right|_{\\alpha}\\\\\n & = \\det (\\Phi_+ \\circ \\eta)' (\\alpha) = \\det (\\Phi_- \\circ \\eta)'\n (\\alpha) = (A_- \\cdummy N) (a) \\text{.}\n\\end{align*}\nTherefore\n\\begin{equation}\n A_+ \\cdummy \\nu = \\frac{A_+ \\cdummy N}{| N |} = \\frac{A_- \\cdummy N}{| N |}\n = A_- \\cdummy \\nu \\label{e5}\n\\end{equation}\non $\\partial \\Omega$, where $\\nu = N / | N |$ is outward unit normal vector\nfield on $\\partial \\Omega$.\n\nSince $f^1_+ = f^1_-$ on $\\partial \\Omega$, using the divergence theorem we\ndeduce from (\\ref{d}) and (\\ref{e5})\n\\begin{align*}\n \\int_{\\Omega} \\det f_+' (x) \\mathd x & = \\int_{\\Omega} \\mathrm{div} (f_+^1\n A_+) \\mathd x = \\int_{\\partial \\Omega} f_+^1 A_+ \\cdummy \\nu \\mathd \\sigma\\\\\n & = \\int_{\\partial \\Omega} f_-^1 A_- \\cdummy \\nu \\mathd \\sigma =\n \\int_{\\Omega} \\det f_-' (x) \\mathd x \\text{.}\n\\end{align*}\n\\begin{example}\n Let $A$ be an $n \\times n$ matrix, $\\varphi \\in C_0^{\\infty} (\\Omega,\n \\mathbb{R}^n)$. Set $f_+ (x) = A x + \\varphi (x)$, $f_- (x) = A x$, we get\n the formula stated in {\\citet[pp. 21]{MR3887613}}:\n \\[ \\int_{\\Omega} \\det (A + \\varphi' (x)) \\mathd x = \\det (A) \\cdummy m\n (\\Omega) \\text{.} \\]\n\\end{example}\n\nTo conclude this note, we present another proof using differential forms. Let\n$f^i$ be the components of $f:\\bar{\\Omega}\\to\\mathbb{R}^n$, then it is well known that\n\\begin{align*}\n \\det f' (x) \\mathd x^1 \\wedge \\cdots \\wedge \\mathd x^n & = \\mathd f^1\n \\wedge \\mathd f^2 \\wedge \\cdots \\wedge \\mathd f^n \\text{.}\\\\\n & = \\mathd (f^1 \\mathd f^2 \\wedge \\cdots \\wedge \\mathd f^n) \\text{,}\n\\end{align*}\nwhere in the second equality we used $\\mathd^2 = 0$. Recall the Stokes formula\n\\[ \\int_{\\Omega} \\mathd \\omega = \\int_{\\partial \\Omega} i^{\\ast} \\omega\n \\text{,} \\]\nwhere $\\omega$ is an $(n - 1)$-form on $\\Omega$ and $i^{\\ast}$ is the pull\nback induced by the embedding $i : \\partial \\Omega \\rightarrow \\Omega$.\nBecause $i^{\\ast} \\circ \\mathd = \\mathd \\circ i^{\\ast}$, we conclude\n\\begin{align*}\n \\int_{\\Omega} \\det f' (x) \\mathd x & = \\int_{\\Omega} \\det f' (x) \\mathd\n x^1 \\wedge \\cdots \\wedge \\mathd x^n\\\\\n & = \\int_{\\Omega} \\mathd (f^1 \\mathd f^2 \\wedge \\cdots \\wedge \\mathd f^n)\n = \\int_{\\partial \\Omega} i^{\\ast} (f^1 \\mathd f^2 \\wedge \\cdots \\wedge\n \\mathd f^n)\\\\\n & = \\int_{\\partial \\Omega} (f^1 \\circ i) \\mathd (f^2 \\circ i) \\wedge\n \\cdots \\wedge \\mathd (f^n \\circ i) \\text{.}\n\\end{align*}\nClearly the right hand side depends only on $f \\circ i = f |_{\\partial \\Omega}\n $.",
    "post_theorem_intro_text_len": 3984,
    "post_theorem_intro_text": "Using this theorem, the \\emph{no-retraction theorem} follows immediately, see \\citet[Corollary\n1]{MR4725395}. Let $B$ be the closed unit ball in\n$\\mathbb{R}^n$. If there was a smooth retraction $R : B \\rightarrow \\partial\nB$, since $R |_{\\partial B}   = \\mathrm{id} |_{\\partial B}  $,\nTheorem \\ref{l1} with $f_+ = R$, $f_- = \\mathrm{id}$ yields a contradiction\n\\begin{equation}\n  \\int_B \\det R' (x) \\mathrm{d} x = \\int_B \\mathrm{d} x = \\mathrm{Vol} (B) > 0 \\text{,}\n  \\label{e2}\n\\end{equation}\nbecause $\\det R' (x) = 0$, a consequence of $R (B) \\subset \\partial B$. That the two\nintegrals in (\\ref{e2}) are equal also follows from a version of the changing\nvariable formula given in {\\citet[Theorem 3.1]{MR3711061}}: \\emph{For smooth closed\nbounded domain $D$ in $\\mathbb{R}^n$ and smooth map $\\varphi : B \\rightarrow\nD$, if $\\varphi : \\partial B \\rightarrow \\partial D$ is a diffeomorphism, then\nfor continuous $f : D \\rightarrow \\mathbb{R}$ there holds}\n\\[ \\int_D f (y) \\mathrm{d} y = \\pm \\int_B f (\\varphi (x)) \\det \\varphi' (x) \\mathrm{d}\n   x \\text{.} \\]\nThe first equality in (\\ref{e2}) follows by letting $f = 1$, $D = B$ and\n$\\varphi = R$ in this formula. It is well known that the no-retraction theorem\nis equivalent to the famous Brouwer fixed point theorem.\n\nKrylov's proof of Theorem \\ref{l1} in {\\cite{MR4725395}} is based on the\nobservation: for small $t$, $f_{\\pm}^t = \\mathrm{id} + t f_{\\pm}$ are\ndiffeomorphisms and $f^t_+ (\\Omega) = f_-^t (\\Omega)$ because $f_+ |_{\\partial\n\\Omega}   = f_- |_{\\partial \\Omega}  $, thus by the changing\nvariable formula\n\\[ \\int_{\\Omega} \\det (I + t f_+' (x)) \\mathrm{d} x = \\mathrm{Vol} (f_+^t (\\Omega))\n   = \\mathrm{Vol} (f_-^t (\\Omega)) = \\int_{\\Omega} \\det (I + t f_-' (x)) \\mathrm{d}\n   x \\text{.} \\]\nSince the two sides are polynomials in $t$, comparing the coefficients of\n$t^n$ gives (\\ref{e1}). To make the argument rigorous some issues including\nwhy $f_{\\pm}^t (\\partial \\Omega) = \\partial f_{\\pm}^t (\\Omega)$ and why\n$\\partial f_+^t (\\Omega) = \\partial f_-^t (\\Omega)$ implies $f_+^t (\\Omega) =\nf_-^t (\\Omega)$ need to be handled. These are clarified via a careful point\nset analysis in a long remark following the proof. One of the advantages of\nthis argument is that the maps $f_{\\pm}$ only need to be $C^1$.\n\nTheorem \\ref{l1} also appears in {\\cite{MR2597943,MR1046451}}. In\n{\\cite{MR1046451}}, Kulpa defines\n\\[ I (h^1, \\ldots, h^n) = \\int_{\\Omega} \\det h' (x) \\mathrm{d} x \\]\nfor smooth $h : \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ and manages to show\n\\begin{equation}\n  I (f^1_+, f^2_+, \\ldots, f_+^n) = I (f^1_-, f^2_+, \\ldots, f_+^n) \\label{e3}\n\\end{equation}\nusing Fubini theorem and projections $\\Pi_i : \\mathbb{R}^n \\rightarrow\n\\mathbb{R}^{n - 1}$ given by deleting the $i$-th component. Then (\\ref{e1})\nfollows by applying (\\ref{e3}) successively to replace the remaining columns\nof $\\det f_+' (x)$ by those of $\\det f_-' (x)$. While in {\\citet[pp.\\\n464]{MR2597943}}, (\\ref{e1}) is proved using the facts that $L (P) = \\det P$\nis a null Lagrangian and the energy $\\int_{\\Omega} L (w' (x), w (x), x) \\mathrm{d}\nx$ of null Lagrangian $L : \\mathbb{M}^{m \\times n} \\times \\mathbb{R}^m \\times\n\\bar{\\Omega} \\rightarrow \\mathbb{R}$ depends only on boundary conditions.\n\nThe purpose of this note is to present another two proofs of Theorem \\ref{l1}.\nThe first one is performed by playing classical analysis, which is written for\nreaders not familiar with differential forms and in our opinion is very\nelementary and transparent. The second one is quite short, which depends on\ndifferential form and Stokes formula on manifolds. As demonstrated in the\nparagraph after Theorem \\ref{l1}, our note is useful for understanding the\nBrouwer fixed point theorem from the analytic point of view. In addition to\n{\\cite{MR2597943,MR4725395,MR1046451,MR3711061}} mentioned above, other\nanalytic proofs of the Brouwer fixed point theorem can be found in \\cite[pp.\\ 467--470]{MR117523} and\n{\\cite{MR610487,MR1699248,MR505523,MR600910}}.",
    "sketch": "Krylov's proof of Theorem~\\ref{l1} is based on the observation that for small $t$, $f_{\\pm}^t=\\mathrm{id}+t f_{\\pm}$ are diffeomorphisms and $f_+^t(\\Omega)=f_-^t(\\Omega)$ because $f_+|_{\\partial\\Omega}=f_-|_{\\partial\\Omega}$. Hence, by the change-of-variables formula,\n\\[\\int_{\\Omega}\\det(I+t f_+'(x))\\,dx=\\mathrm{Vol}(f_+^t(\\Omega))=\\mathrm{Vol}(f_-^t(\\Omega))=\\int_{\\Omega}\\det(I+t f_-'(x))\\,dx.\\]\nSince both sides are polynomials in $t$, comparing the coefficients of $t^n$ gives \\eqref{e1}. To make this rigorous, one must handle issues such as why $f_{\\pm}^t(\\partial\\Omega)=\\partial f_{\\pm}^t(\\Omega)$ and why $\\partial f_+^t(\\Omega)=\\partial f_-^t(\\Omega)$ implies $f_+^t(\\Omega)=f_-^t(\\Omega)$; these are addressed by “a careful point set analysis”. The same paragraph notes an advantage: the argument only requires $f_{\\pm}$ to be $C^1$.\n\nAnother proof approach (Kulpa) defines $I(h^1,\\dots,h^n)=\\int_{\\Omega}\\det h'(x)\\,dx$ and shows\n\\[I(f_+^1,f_+^2,\\ldots,f_+^n)=I(f_-^1,f_+^2,\\ldots,f_+^n)\\tag{\\ref{e3}}\\]\nusing Fubini’s theorem and coordinate projections $\\Pi_i$. Then \\eqref{e1} follows by applying \\eqref{e3} successively to replace the remaining columns of $\\det f_+'(x)$ by those of $\\det f_-'(x)$. A further cited route proves \\eqref{e1} by using that $L(P)=\\det P$ is a null Lagrangian, so the energy $\\int_{\\Omega}L(w'(x),w(x),x)\\,dx$ “depends only on boundary conditions”.",
    "expanded_sketch": "Krylov's proof of the main theorem is based on the observation that for small $t$, $f_{\\pm}^t=\\mathrm{id}+t f_{\\pm}$ are diffeomorphisms and $f_+^t(\\Omega)=f_-^t(\\Omega)$ because $f_+|_{\\partial\\Omega}=f_-|_{\\partial\\Omega}$. Hence, by the change-of-variables formula,\n\\[\\int_{\\Omega}\\det(I+t f_+'(x))\\,dx=\\mathrm{Vol}(f_+^t(\\Omega))=\\mathrm{Vol}(f_-^t(\\Omega))=\\int_{\\Omega}\\det(I+t f_-'(x))\\,dx.\\]\nSince both sides are polynomials in $t$, comparing the coefficients of $t^n$ gives\n\\begin{equation}\n    \\int_{\\Omega} \\det f_+' (x) \\mathd x = \\int_{\\Omega} \\det f_-' (x) \\mathd\n    x \\text{.} \\label{e1}\n  \\end{equation}\nTo make this rigorous, one must handle issues such as why $f_{\\pm}^t(\\partial\\Omega)=\\partial f_{\\pm}^t(\\Omega)$ and why $\\partial f_+^t(\\Omega)=\\partial f_-^t(\\Omega)$ implies $f_+^t(\\Omega)=f_-^t(\\Omega)$; these are addressed by “a careful point set analysis”. The same paragraph notes an advantage: the argument only requires $f_{\\pm}$ to be $C^1$.\n\nAnother proof approach (Kulpa) defines $I(h^1,\\dots,h^n)=\\int_{\\Omega}\\det h'(x)\\,dx$ and shows\n\\begin{equation}\n  I (f^1_+, f^2_+, \\ldots, f_+^n) = I (f^1_-, f^2_+, \\ldots, f_+^n) \\label{e3}\n\\end{equation}\nusing Fubini’s theorem and coordinate projections $\\Pi_i$. Then the equation above gives \\eqref{e1} by applying the preceding identity successively to replace the remaining columns of $\\det f_+'(x)$ by those of $\\det f_-'(x)$. A further cited route proves the equation above by using that $L(P)=\\det P$ is a null Lagrangian, so the energy $\\int_{\\Omega}L(w'(x),w(x),x)\\,dx$ “depends only on boundary conditions”.",
    "expanded_theorem": "\\label{l1}Let $\\Omega$ be a bounded domain in $\\mathbb{R}^n$, $f_{\\pm} :\n  \\bar{\\Omega} \\rightarrow \\mathbb{R}^n$ be $C^2$-maps such that $f_+\n  |_{\\partial \\Omega}   = f_- |_{\\partial \\Omega}  $. then\n  \\begin{equation}\n    \\int_{\\Omega} \\det f_+' (x) \\mathrm{d} x = \\int_{\\Omega} \\det f_-' (x) \\mathrm{d}\n    x \\text{.} \\label{e1}\n  \\end{equation}",
    "theorem_type": [
      "Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\) be a bounded domain in \\(\\mathbb{R}^n\\). Let \\(f_{\\pm}:\\overline{\\Omega}\\to\\mathbb{R}^n\\) be \\(C^2\\) maps such that their boundary values agree, i.e. \\(f_+|_{\\partial\\Omega}=f_-|_{\\partial\\Omega}\\). Here \\(f_\\pm'(x)\\) denotes the Jacobian matrix of \\(f_\\pm\\) at \\(x\\), and \\(\\det f_\\pm'(x)\\) its Jacobian determinant. Which statement holds for every such pair of maps?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\\int_{\\Omega} \\det f_+'(x)\\,dx = \\int_{\\Omega} \\det f_-'(x)\\,dx.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\\det f_+'(x)=\\det f_-'(x)\\quad\\text{for almost every }x\\in\\Omega.\\]"
        },
        {
          "label": "C",
          "text": "\\[\\text{If }f_-\\text{ is constant on }\\overline\\Omega,\\text{ then }\\int_{\\Omega} \\det f_+'(x)\\,dx=0.\\]"
        },
        {
          "label": "D",
          "text": "\\[\\int_{\\Omega} \\bigl|\\det f_+'(x)\\bigr|\\,dx = \\int_{\\Omega} \\bigl|\\det f_-'(x)\\bigr|\\,dx.\\]"
        },
        {
          "label": "E",
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    "id": "2512.04141v2",
    "paper_link": "http://arxiv.org/abs/2512.04141v2",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:biharmonic}\nLet $(M,g)$ be a complete Riemannian manifold with a pole and such that \n$\\mbox{Ric}_g\\geq 0$. \nThen any biharmonic function $u\\in C^{4}(M)$ of subquadratic growth is harmonic. \nIn particular, any biharmonic function of sublinear growth must be constant.",
      "start_pos": 6937,
      "end_pos": 7244,
      "label": "thm:biharmonic"
    },
    "ref_dict": {
      "thm:biharmonic": "\\begin{theorem}\n\\label{thm:biharmonic}\nLet $(M,g)$ be a complete Riemannian manifold with a pole and such that \n$\\mbox{Ric}_g\\geq 0$. \nThen any biharmonic function $u\\in C^{4}(M)$ of subquadratic growth is harmonic. \nIn particular, any biharmonic function of sublinear growth must be constant.\n\\end{theorem}",
      "eq:Cacciopoli": "\\begin{equation}\n\\label{eq:Cacciopoli}\n\\int_{B_s}|\\nabla u|^2\\,dV \\leq \\frac{C}{(r-s)^2}\\int_{B_r}u^2\\,dV,\n\\end{equation}",
      "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $(M,g)$ be a complete Riemannian manifold with \n$\\mbox{Ric}_g\\geq 0$. \nThen any $k$-polyharmonic function $u\\in C^{2k}(M)$ satisfying\n\\[\nu=o(r^{2(k-1)}) \\quad \\mbox{as}\\quad r\\rightarrow \\infty,\n\\]\nwhere $r=d(p,x)$, is $(k-1)$-polyharmonic. \nIn particular, any polyharmonic function of sublinear growth must be constant.\n\\end{theorem}",
      "eq:Cacciopoli_Laplacian": "\\begin{equation}\n\\label{eq:Cacciopoli_Laplacian}\n\\int_{B_s}|\\Delta u|^2\\,dV \\leq \\frac{C}{(r-s)^4}\\int_{B_r}u^2\\,dV,\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 2154,
    "pre_theorem_intro_text": "Harmonic functions have long been a central object of study in geometric analysis. Among the most celebrated and foundational results in the field is Yau’s Liouville theorem \\cite{Yau75}, which asserts that on a complete Riemannian manifold with nonnegative Ricci curvature, any bounded harmonic function must be constant. This theorem not only captures a fundamental rigidity phenomenon, but also inaugurated a far-reaching program aimed at understanding how analytic growth conditions reflect the large-scale geometry of the underlying manifold.\n\n\\medskip\nSince Yau’s work, the study of harmonic functions and spaces of harmonic functions has developed extensively. Notable contributions include the work of Li and Tam \\cite{LiTam1989}, as well as Colding and Minicozzi \\cite{ColdingMinicozzi97, ColdingMinicozziAnnals}, who investigated the structure and finite dimensional properties of spaces of harmonic functions of polynomial growth. Further important developments include results of Ni and Tam \\cite{NiTam2003} on pluriharmonic functions, and more recently Liu’s work on holomorphic functions via three--circle theorems \\cite{LiuThreeCircle}. Collectively, these results demonstrate that under nonnegative Ricci curvature, harmonic functions exhibit rigidity properties closely resembling those of the Euclidean setting.\n\n\\medskip\nBy contrast, Liouville-type phenomena for higher-order elliptic equations have received comparatively little attention. This gap is striking given the intrinsic analytic and geometric relevance of higher-order operators, such as the Paneitz operator in conformal geometry or the operators arising in the study of polyharmonic flows for curves. Aside from a few notable exceptions, there are very few results addressing Liouville-type properties in this broader context. Among these, we mention the works of Branding \\cite{Branding18, Branding2021}, where Liouville-type theorems are obtained for biharmonic \\emph{maps} under smallness assumptions on certain Sobolev norms. \n\n\\medskip\nRecently, the authors proved the following result (see \\cite{BravoCortissoz2025}, as well as the related work \\cite{WangZhu2025}).",
    "context": "Harmonic functions have long been a central object of study in geometric analysis. Among the most celebrated and foundational results in the field is Yau’s Liouville theorem \\cite{Yau75}, which asserts that on a complete Riemannian manifold with nonnegative Ricci curvature, any bounded harmonic function must be constant. This theorem not only captures a fundamental rigidity phenomenon, but also inaugurated a far-reaching program aimed at understanding how analytic growth conditions reflect the large-scale geometry of the underlying manifold.\n\n\\medskip\nSince Yau’s work, the study of harmonic functions and spaces of harmonic functions has developed extensively. Notable contributions include the work of Li and Tam \\cite{LiTam1989}, as well as Colding and Minicozzi \\cite{ColdingMinicozzi97, ColdingMinicozziAnnals}, who investigated the structure and finite dimensional properties of spaces of harmonic functions of polynomial growth. Further important developments include results of Ni and Tam \\cite{NiTam2003} on pluriharmonic functions, and more recently Liu’s work on holomorphic functions via three--circle theorems \\cite{LiuThreeCircle}. Collectively, these results demonstrate that under nonnegative Ricci curvature, harmonic functions exhibit rigidity properties closely resembling those of the Euclidean setting.\n\n\\medskip\nBy contrast, Liouville-type phenomena for higher-order elliptic equations have received comparatively little attention. This gap is striking given the intrinsic analytic and geometric relevance of higher-order operators, such as the Paneitz operator in conformal geometry or the operators arising in the study of polyharmonic flows for curves. Aside from a few notable exceptions, there are very few results addressing Liouville-type properties in this broader context. Among these, we mention the works of Branding \\cite{Branding18, Branding2021}, where Liouville-type theorems are obtained for biharmonic \\emph{maps} under smallness assumptions on certain Sobolev norms.\n\n\\medskip\nRecently, the authors proved the following result (see \\cite{BravoCortissoz2025}, as well as the related work \\cite{WangZhu2025}).",
    "full_context": "Harmonic functions have long been a central object of study in geometric analysis. Among the most celebrated and foundational results in the field is Yau’s Liouville theorem \\cite{Yau75}, which asserts that on a complete Riemannian manifold with nonnegative Ricci curvature, any bounded harmonic function must be constant. This theorem not only captures a fundamental rigidity phenomenon, but also inaugurated a far-reaching program aimed at understanding how analytic growth conditions reflect the large-scale geometry of the underlying manifold.\n\n\\medskip\nSince Yau’s work, the study of harmonic functions and spaces of harmonic functions has developed extensively. Notable contributions include the work of Li and Tam \\cite{LiTam1989}, as well as Colding and Minicozzi \\cite{ColdingMinicozzi97, ColdingMinicozziAnnals}, who investigated the structure and finite dimensional properties of spaces of harmonic functions of polynomial growth. Further important developments include results of Ni and Tam \\cite{NiTam2003} on pluriharmonic functions, and more recently Liu’s work on holomorphic functions via three--circle theorems \\cite{LiuThreeCircle}. Collectively, these results demonstrate that under nonnegative Ricci curvature, harmonic functions exhibit rigidity properties closely resembling those of the Euclidean setting.\n\n\\medskip\nBy contrast, Liouville-type phenomena for higher-order elliptic equations have received comparatively little attention. This gap is striking given the intrinsic analytic and geometric relevance of higher-order operators, such as the Paneitz operator in conformal geometry or the operators arising in the study of polyharmonic flows for curves. Aside from a few notable exceptions, there are very few results addressing Liouville-type properties in this broader context. Among these, we mention the works of Branding \\cite{Branding18, Branding2021}, where Liouville-type theorems are obtained for biharmonic \\emph{maps} under smallness assumptions on certain Sobolev norms.\n\n\\medskip\nRecently, the authors proved the following result (see \\cite{BravoCortissoz2025}, as well as the related work \\cite{WangZhu2025}).\n\n\\medskip\nRecently, the authors proved the following result (see \\cite{BravoCortissoz2025}, as well as the related work \\cite{WangZhu2025}).\n\nThe purpose of the present paper is to extend Theorem~\\ref{thm:biharmonic} to the general polyharmonic setting. Recall that a $k$-polyharmonic function on a Riemannian manifold $(M,g)$ is a function $u\\in C^{2k}(M)$ satisfying\n\\[\n\\Delta^k u = 0,\n\\]\nwhere $\\Delta$ denotes the Laplacian.\n\n\\begin{theorem}\n\\label{thm:main}\nLet $(M,g)$ be a complete Riemannian manifold with \n$\\mbox{Ric}_g\\geq 0$. \nThen any $k$-polyharmonic function $u\\in C^{2k}(M)$ satisfying\n\\[\nu=o(r^{2(k-1)}) \\quad \\mbox{as}\\quad r\\rightarrow \\infty,\n\\]\nwhere $r=d(p,x)$, is $(k-1)$-polyharmonic. \nIn particular, any polyharmonic function of sublinear growth must be constant.\n\\end{theorem}\n\n\\begin{lemma}[Cutoff with explicit bounds]\\label{lem:cutoff}\nLet $(M^n,g)$ be a complete Riemannian manifold \nwith $\\mbox{Ric}_g\\geq 0$\nSet $B_\\rho:=B_\\rho(p)$. Let $r$ be such that and for $0<\\frac{1}{8}R\\leq r\\leq R$.\nThen there exists $\\chi\\in C_c^\\infty(B_R)$ such that\n\\[\n0\\le \\chi\\le 1,\\qquad\n\\chi\\equiv 1\\ \\text{on } B_r,\\qquad\n\\mbox{supp}\\left(\\chi\\right)\\subset B_R,\n\\]\nand \n\\[\n\\left|\\nabla \\chi\\right|\\le \\frac{C(n)}{R-r},\n\\quad\n|\\Delta\\chi|\\le \\frac{C(n)}{(R-r)^{2}}\n\\quad \\text{on}\\quad B_R.\n\\]\n\\end{lemma}\n\\begin{proof}\n Take $\\chi=\\phi$, where $\\phi$ is as in Corollary 2.3 in \n \\cite{BianchiSetti2018}. Notice that given $r$, we can write $R=\\gamma r$ with $\\gamma>1$, and in this case,\n we obtain the following estimates for the gradient and Laplacian of the cutoff function.\n \\[\n \\left|\\nabla \\chi\\right|\\leq \\frac{C_1}{R}=\\frac{\\left(\\gamma-1\\right)C_1}{\\left(\\gamma-1\\right) R}\\leq\n \\frac{7C_1}{R-r},\n \\]\n where we used that $r\\geq \\frac{1}{8}R$.\n\n\\begin{lemma}\\label{cor:L2-Estimate Laplacian Biharmonic}\nLet $(M,g)$ be an $n$–dimensional complete Riemannian manifold with $\\mbox{Ric}_g\\ge 0$.\nLet $a>0$ and let $u\\in C^\\infty(B_{a})$ be Biharmonic, i.e.\\ $\\Delta^2 u=0$ on $B_{a}$.\nThen, there is a constant $C>0$ such that for every $s,r\\in(0,a)$, with $0<\\frac{1}{8}r<s < r<a$ \n\\begin{equation}\\label{eq:L2-HF-polh}\n\\int_{B_s}|\\Delta u|^2 dV \\leq \\frac{C}{(r-s)^4}\\int_{B_{r}} u^2 dV.\n\\end{equation}\n\n\\subsection{Proof of Main Theorem \\ref{thm:main}}\n\\label{sect:proof_main_thm}\nWe are ready to prove our main result. Let $u$ be a $k$-polyharmonic function.\nBy Corollary \\ref{cor:L2-Estimate Laplacian k-1} we can estimate\n\\begin{equation*}\n\\label{ineq:Cacciopoli_laplacian}\n\\int_{B_{r}}|\\Delta^{k-1} u|^2\\,dV\n\\leq \\frac{C'}{r^{4(k-1)}}\\int_{B_{2^{k-1}r}}u^2\\,dV.\n\\end{equation*}\nBy the volume doubling property (which holds if $\\mbox{Ric}\\geq 0$), we have\n\\begin{eqnarray*}\n\\frac{1}{\\mbox{Vol}\\left(B_{r}\\right)}\\int_{B_{r}}|\\Delta^{k-1} u|^2\\,dV\n&\\leq& \\frac{C'}{r^{4(k-1)}}\\frac{1}{\\mbox{Vol}\\left(B_{2^{k-1}r}\\right)}\\int_{B_{2^{k-1}r}}u^2\\,dV,\n\\end{eqnarray*}\nand hence, since $\\Delta^{k-1} u$ is harmonic, then $|\\Delta^{k-1} u|$ is subharmonic, by the mean value inequality (Lemma \\ref{lemma:mean_value_ineq}),\n\\begin{eqnarray*}\n\\sup_{x\\in B_{r/2}}\\left|\\Delta^{k-1} u\\right|^2 &\\leq& \n\\frac{C''}{r^{4(k-1)}}\\frac{1}{\\mbox{Vol}\\left(B_{2^{k-1}r}\\right)}\\int_{B_{2^{k-1}r}}u^2\\,dV\\\\\n&\\leq& \\frac{C''}{r^{4(k-1)}}\\sup_{x\\in B_r\\left(p\\right)}\\left|u\\left(x\\right)\\right|^2\n=\\frac{1}{r^{4(k-1)}}o\\left(r^{4(k-1)}\\right)\n\\end{eqnarray*}\nand hence if $u=o(r^{2(k-1)})$, letting $r\\rightarrow \\infty$, it \nfollows that\n\\[\n\\Delta^{k-1} u \\equiv 0,\n\\]\ni.e., $u$ is $(k-1)$-polyharmonic. For the last statement, let $u$ be a polyharmonic function of\nsublinear growth, i.e.\n\\[\n\\sup_{B_r}|u| = o(r) \\quad \\text{as } r\\to\\infty.\n\\]\nFor every $k\\ge2$ we have\n\\[\n\\frac{\\sup_{B_r}|u|}{r^{2(k-1)}}\n= o\\big(r^{1-2(k-1)}\\big) \\;\\longrightarrow\\;0.\n\\]\nIn particular $u = o\\big(r^{2(k-1)}\\big)$ for each $k\\ge2$.\nAssume that $u$ is $k$–polyharmonic for some $k\\ge2$. By the first\npart of the theorem, $u$ is then $(k-1)$–polyharmonic. Iterating\nthis argument for $k,k-1,\\dots,2$, we conclude that $u$ is harmonic\nand it is sublinear growth.\n\nAs stated in the introduction, inequality \\eqref{eq:Cacciopoli_Laplacian} \ndoes not appear in \\cite{BravoCortissoz2025}. However, \nthe estimate can be inferred from the proofs of Lemmas 5 and 6\nin \\cite{BravoCortissoz2025} as follows. From the proof of the local \nenergy estimate for the Laplacian, Lemma 5, for a cutoff function \n$\\varphi\\in C_c^\\infty(B_{2r})$ satisfying for $0<\\frac{1}{8}r<s<r$\n\\[\n\\varphi\\equiv1 \\text{ on }B_s,\\quad 0\\le\\varphi\\le1,\\quad\n|\\nabla\\varphi|\\le \\frac{C_0}{r-s}\\varphi,\\quad\n|\\Delta\\varphi|\\le \\frac{C_0}{(r-s)^2},\n\\]\nwe have\n\\begin{lemma}\n\\label{lem:laplacian_estimate}\nLet $(M,g)$ be an $n$–dimensional complete Riemannian manifold with $\\mathrm{Ric}_g\\ge 0$. Let $B_{\\rho}\\colon = B_{\\rho}\\left(p\\right)$.\nFix $a>0$ and let $u\\in C^\\infty(B_{a})$ be biharmonic, i.e.\\ $\\Delta^2 u=0$ on $B_{2a}$. Then there exists a constant $C=C(n)>0$ such that, for every pair of radii $0<\\dfrac{1}{8}r<s<r\\le a$,\n\\[\n\\int_{B_s}|\\Delta u|^2\\,dV\n\\leq \\frac{C}{(r-s)^2}\\int_{B_r}|\\nabla u|^2\\varphi^2\\,dV\n\\;+\\; \\frac12\\int_{B_r} |\\Delta u|^2\\,dV.\n\\]\n\\end{lemma}\n\nFrom the proof of Lemma 6, the Cacciopoli inequality\nfor a biharmonic function, for a cutoff \n$\\varphi\\in C_c^\\infty(B_{2r})$ such that $0\\leq \\varphi\\leq 1$,\n$\\varphi\\equiv 1$ on $B_s$, and for $\\dfrac{1}{8}r<s<r$, it also\nsatisfies\n\\[\n\\left|\\nabla \\varphi\\right|^2\\leq \\frac{C}{\\left(r-s\\right)^2}\\varphi\\quad\n \\text{ and } \\quad\n \\frac{\\left|\\Delta\\left(\\varphi^4\\right)\\right|^2}{\\varphi^4}\\leq \\frac{C}{\\left(r-s\\right)^4},\n\\]\nwe have\n\\begin{lemma}\n\\label{lem:Caccio-1-biharmonic-Br}\nLet $(M,g)$ be a smooth Riemannian manifold \nsuch that $\\mbox{Ric}\\geq 0$.\nLet $0<\\dfrac{1}{8}r<s<r<\\infty$\nand let $B_\\rho:=B_\\rho(p)$. If $u\\in C^\\infty(B_{2r})$ is \\emph{biharmonic}, i.e.\\ $\\Delta^2 u=0$ in $B_{2r}$, then there exists a constant $C=C(n)>0$ such that\n\\begin{equation*}\\label{eq:Caccio-int-Br}\n\\int_{B_r} |\\nabla u|^2\\varphi^2\\,dV \\;\\le\\; \\frac{C}{(r-s)^2}\\int_{B_r} u^2\\,dV.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{equation}\n\\label{eq:Cacciopoli_Laplacian}\n\\int_{B_s}|\\Delta u|^2\\,dV \\leq \\frac{C}{(r-s)^4}\\int_{B_r}u^2\\,dV,\n\\end{equation}\n\n\\begin{theorem}\n\\label{thm:biharmonic}\nLet $(M,g)$ be a complete Riemannian manifold with a pole and such that \n$\\mbox{Ric}_g\\geq 0$. \nThen any biharmonic function $u\\in C^{4}(M)$ of subquadratic growth is harmonic. \nIn particular, any biharmonic function of sublinear growth must be constant.\n\\end{theorem}",
    "post_theorem_intro_text_len": 5582,
    "post_theorem_intro_text": "The purpose of the present paper is to extend Theorem~\\ref{thm:biharmonic} to the general polyharmonic setting. Recall that a $k$-polyharmonic function on a Riemannian manifold $(M,g)$ is a function $u\\in C^{2k}(M)$ satisfying\n\\[\n\\Delta^k u = 0,\n\\]\nwhere $\\Delta$ denotes the Laplacian.\n\n\\medskip\nOur main result is the following.\n\n\\begin{theorem}\n\\label{thm:main}\nLet $(M,g)$ be a complete Riemannian manifold with \n$\\mbox{Ric}_g\\geq 0$. \nThen any $k$-polyharmonic function $u\\in C^{2k}(M)$ satisfying\n\\[\nu=o(r^{2(k-1)}) \\quad \\mbox{as}\\quad r\\rightarrow \\infty,\n\\]\nwhere $r=d(p,x)$, is $(k-1)$-polyharmonic. \nIn particular, any polyharmonic function of sublinear growth must be constant.\n\\end{theorem}\n\n\\medskip\nAs is well known, Theorem~\\ref{thm:main} has long been established in the Euclidean setting. In $\\mathbb{R}^n$, the critical growth rate that separates genuinely $k$-polyharmonic behavior from lower-order behavior is precisely $r^{2(k-1)}$, as follows from Almansi’s decomposition. This shows that Theorem~\\ref{thm:main} is sharp and constitutes a natural geometric extension of the classical Euclidean theory.\n\n\\medskip\nWe now briefly describe the main analytic difficulty underlying the proof of Theorem~\\ref{thm:main}. We begin by recalling the strategy used in the proof of Theorem~\\ref{thm:biharmonic}. In \\cite{BravoCortissoz2025}, the argument relied on an energy estimate for the $L^2$-norm of $\\Delta u$ together with a Caccioppoli-type inequality,\n\\begin{equation}\n\\label{eq:Cacciopoli}\n\\int_{B_s}|\\nabla u|^2\\,dV \\leq \\frac{C}{(r-s)^2}\\int_{B_r}u^2\\,dV,\n\\end{equation}\nvalid for both harmonic and biharmonic functions. This estimate was then used to derive\n\\begin{equation}\n\\label{eq:Cacciopoli_Laplacian}\n\\int_{B_s}|\\Delta u|^2\\,dV \\leq \\frac{C}{(r-s)^4}\\int_{B_r}u^2\\,dV,\n\\end{equation}\nwhich, combined with a mean value inequality, yields the desired rigidity.\n\n\\medskip\nAt first glance, these arguments suggest the possibility of an inductive approach to Theorem~\\ref{thm:main}. However, the proof of \\eqref{eq:Cacciopoli} is fundamentally based on the Bochner formula, and this approach does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case. To overcome this obstacle, we establish a direct analogue of \\eqref{eq:Cacciopoli_Laplacian} for \\emph{arbitrary polyharmonic functions}. This new estimate is the most delicate part of the argument and forms the analytic backbone of the inductive procedure. Although the biharmonic case serves as the base step, the proof requires a genuinely new estimate that may be of independent interest.\n\n\\medskip\nTaken together with Yau’s theorem and our previous work on biharmonic functions, the results of this paper complete, on manifolds with nonnegative Ricci curvature, a natural hierarchy of Yau--Liouville type theorems for the Laplacian and its iterates. They also raise the question of whether similar rigidity phenomena persist for more general higher-order elliptic operators, as they do in the Euclidean setting.\n\n\\medskip\nBeyond its analytic content, Theorem~\\ref{thm:main} may be interpreted as a rigidity principle for complete manifolds with nonnegative Ricci curvature. It shows that, at the level of critical growth, such manifolds are analytically indistinguishable from Euclidean space with respect to iterates of the Laplacian. In particular, the behavior of polyharmonic function in\nnonnegative Ricci curvature seems to mirror Almansi’s decomposition in $\\mathbb{R}^n$. \n\\subsection{}\n\nThe interplay between the behavior of polyharmonic functions and curvature remains largely unexplored and may reveal further rigidity phenomena. As shown in this paper, when $\\mbox{Ric}_g\\geq 0$, the qualitative behavior closely parallels that of Euclidean space. In contrast, in negatively curved settings the situation is markedly different. In particular, the authors showed in \\cite{BravoCortissoz} that there exist curvature regimes in which every bounded biharmonic function must be harmonic, while nonconstant bounded harmonic functions still exist. This naturally leads to the question of how $k$-polyharmonic functions behave in such settings.\n\n\\subsection{}\n\nIn a previous version of this manuscript, we imposed one of the following additional assumptions:\n\\begin{itemize}\n\\item[(H1)] the geodesic spheres centered at $p$ are mean-convex with respect to the radial vector field, or\n\\item[(H2)] the square of the distance function from $p$, denoted by $d$, is convex.\n\\end{itemize}\nEither assumption yields a lower bound for the Laplacian of the distance function, which is needed to construct suitable cutoff functions. However, we later observed that the condition $\\mbox{Ric}_g\\geq 0$, together with the existence of a pole, already implies (H1), rendering it redundant (see \\cite{BravoCortissoz2025}). Subsequently, inspired by the recent work \\cite{WangZhu2025} and with the generous assistance of Professor Wang, we adopted the cutoff technology developed in \\cite{BianchiSetti2018}, which requires only the assumption $\\mbox{Ric}_g\\geq 0$. We are grateful to Professors Wang and Zhu for their help.\n\n\\subsection{Organization of the paper}\nThe material of this paper is organized as follows. In Section\n\\ref{section:technical}, we collect some facts about cutoff functions\nand the hole-filling technique needed in the proof of the main\n$L^2$-estimates needed in the proof of our main result. These\n$L^2$ estimates are presented and proved in Section \\ref{sect:L2estimates}.\nFinally, Theorem \\ref{thm:main} is proved in Section \\ref{sect:proof_main_thm}.",
    "sketch": "To prove Theorem~\\ref{thm:main}, the introduction proposes an inductive strategy inspired by the biharmonic case (Theorem~\\ref{thm:biharmonic}). In \\cite{BravoCortissoz2025}, the biharmonic argument uses an energy estimate for the $L^2$-norm of $\\Delta u$ together with a Caccioppoli-type inequality\n\\[\n\\int_{B_s}|\\nabla u|^2\\,dV \\leq \\frac{C}{(r-s)^2}\\int_{B_r}u^2\\,dV,\n\\]\nwhich then yields\n\\[\n\\int_{B_s}|\\Delta u|^2\\,dV \\leq \\frac{C}{(r-s)^4}\\int_{B_r}u^2\\,dV.\n\\]\nCombined with a mean value inequality, this gives the rigidity conclusion in the biharmonic case.\n\nFor Theorem~\\ref{thm:main}, the text notes that while this suggests induction, the proof of the basic Caccioppoli inequality is “fundamentally based on the Bochner formula,” and “does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case.” The stated remedy is to “establish a direct analogue of \\eqref{eq:Cacciopoli_Laplacian} for \\emph{arbitrary polyharmonic functions}.” This new $L^2$ estimate is described as “the most delicate part of the argument” and as “the analytic backbone of the inductive procedure,” with the biharmonic case as the base step but requiring “a genuinely new estimate.”\n\nThe introduction also indicates the technical tools used to implement this: cutoff-function technology under $\\mathrm{Ric}_g\\ge0$ (adopting the approach of \\cite{BianchiSetti2018}) and the “hole-filling technique,” and it outlines where the proof components appear: cutoff/hole-filling facts in Section~\\ref{section:technical}, the main $L^2$ estimates in Section~\\ref{sect:L2estimates}, and the proof of Theorem~\\ref{thm:main} in Section~\\ref{sect:proof_main_thm}.",
    "expanded_sketch": "To prove Theorem~\\ref{thm:main}, the introduction proposes an inductive strategy inspired by the biharmonic case. We first prove the following theorem. \n\\begin{theorem}\n\\label{thm:biharmonic}\nLet $(M,g)$ be a complete Riemannian manifold with a pole and such that \n$\\mbox{Ric}_g\\geq 0$. \nThen any biharmonic function $u\\in C^{4}(M)$ of subquadratic growth is harmonic. \nIn particular, any biharmonic function of sublinear growth must be constant.\n\\end{theorem}\nIn Bravo, Cortissoz, \\emph{TITLE UNKNOWN} (2025), the biharmonic argument uses an energy estimate for the $L^2$-norm of $\\Delta u$ together with a Caccioppoli-type inequality\n\\[\n\\int_{B_s}|\\nabla u|^2\\,dV \\leq \\frac{C}{(r-s)^2}\\int_{B_r}u^2\\,dV,\n\\]\nwhich then yields the estimate\n\\begin{equation}\n\\label{eq:Cacciopoli_Laplacian}\n\\int_{B_s}|\\Delta u|^2\\,dV \\leq \\frac{C}{(r-s)^4}\\int_{B_r}u^2\\,dV,\n\\end{equation}\nCombined with a mean value inequality, this gives the rigidity conclusion in establishing the main theorem.\n\nFor Theorem~\\ref{thm:main}, the text notes that while this suggests induction, the proof of the basic Caccioppoli inequality is “fundamentally based on the Bochner formula,” and “does not extend in any straightforward way to polyharmonic functions beyond the biharmonic case.” The stated remedy is to “establish a direct analogue of the equation above for \\emph{arbitrary polyharmonic functions}.” This new $L^2$ estimate is described as “the most delicate part of the argument” and as “the analytic backbone of the inductive procedure,” with the base step provided by the theorem above but requiring “a genuinely new estimate.”\n\nThe introduction also indicates the technical tools used to implement this: cutoff-function technology under $\\mathrm{Ric}_g\\ge0$ (adopting the approach of \\cite{BianchiSetti2018}) and the “hole-filling technique,” and it outlines where the proof components appear: cutoff/hole-filling facts appear later, the main $L^2$ estimates are proved later, and the proof of Theorem~\\ref{thm:main} appears later.",
    "expanded_theorem": "\\label{thm:biharmonic}\nLet $(M,g)$ be a complete Riemannian manifold with a pole and such that \n$\\mbox{Ric}_g\\geq 0$. \nThen any biharmonic function $u\\in C^{4}(M)$ of subquadratic growth is harmonic. \nIn particular, any biharmonic function of sublinear growth must be constant.",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let $(M,g)$ be a complete Riemannian manifold that admits a pole $p$ (that is, the exponential map $\\exp_p:T_pM\\to M$ is a global diffeomorphism) and satisfies $\\mathrm{Ric}_g\\ge 0$. Let $r(x)=d(p,x)$. A biharmonic function is a function $u\\in C^4(M)$ with $\\Delta^2 u=0$. A function has subquadratic growth if $u(x)=o(r(x)^2)$ as $r(x)\\to\\infty$, and sublinear growth if $u(x)=o(r(x))$ as $r(x)\\to\\infty$. Which statement holds for every such biharmonic function?",
      "correct_choice": {
        "label": "A",
        "text": "Every biharmonic function $u\\in C^4(M)$ with subquadratic growth is harmonic, i.e. $\\Delta u=0$ on $M$. In particular, every biharmonic function of sublinear growth is constant."
      },
      "choices": [
        {
          "label": "B",
          "text": "Every biharmonic function $u\\in C^4(M)$ with subquadratic growth is constant on $M$. In particular, every biharmonic function of sublinear growth is constant."
        },
        {
          "label": "C",
          "text": "Every biharmonic function $u\\in C^4(M)$ with sublinear growth is constant on $M$."
        },
        {
          "label": "D",
          "text": "Every biharmonic function $u\\in C^4(M)$ with quadratic growth, i.e. $u(x)=O(r(x)^2)$ as $r(x)\\to\\infty$, is harmonic, i.e. $\\Delta u=0$ on $M$. In particular, every biharmonic function of sublinear growth is constant."
        },
        {
          "label": "E",
          "text": "Every biharmonic function $u\\in C^4(M)$ with subquadratic growth is harmonic, i.e. there exists a constant $C=C(u)$ such that $\\Delta u\\equiv C$ on $M$. In particular, every biharmonic function of sublinear growth is constant."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "conclusion harmonic_vs_constant",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped subquadratic_to_harmonic implication and kept only the stated corollary for sublinear growth",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "little_o threshold replaced by big_O boundary case",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "exact vanishing of Laplacian weakened to constant Laplacian",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives geometric and growth-condition definitions but does not explicitly state the theorem’s conclusion. There is no direct answer leakage from the wording of the prompt itself."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct option is essentially a verbatim theorem statement under the stated hypotheses, including its corollary. This makes the item largely a recall/restatement question rather than a non-tautological inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the alternatives differ in subtle but meaningful ways: harmonic vs constant, subquadratic vs sublinear, little-o vs big-O, and zero Laplacian vs constant Laplacian. Still, the item mainly tests recognition of the exact theorem."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one is too strong, one is weaker but true, one changes the growth threshold, and one weakens the conclusion incorrectly. These reflect common theorem-misremembering or overgeneralization errors."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is quite tautological because the correct option is essentially the theorem itself."
    }
  },
  {
    "id": "2512.04281v1",
    "paper_link": "http://arxiv.org/abs/2512.04281v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.",
      "start_pos": 14820,
      "end_pos": 16557,
      "label": "thmBfinito"
    },
    "ref_dict": {
      "diffproperties": "\\begin{equation}\n    |u_{x_i}|^{\\frac{p_i - 2}{2}}  \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}",
      "Apriorisec": "\\begin{align}\n  &\\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx \\leq  c |h|^{2} \\,\\left[ 1+ \\sum_{i=1}^{n}   \\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}}(B_{R})}  \\right], \\label{StimaLemma}\n\\end{align}\nfor a positive constant $c = c(n,p_i,\\rho, R, \\lambda, K, \\lvert \\lvert Du \\rvert \\rvert_{\\infty})$.\nThen we have that \n $$V_{p_i}((u_\\varepsilon)_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n,$$ \n which implies in particular that $$|(u_\\varepsilon)_{x_i x_j}|^2|(u_\\varepsilon)_{x_i}|^{p_i-2} \\in L^1_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n \\text{ and  }  \\forall j=1, \\dots, n. $$\n\\end{proof}\n\n\\section{A priori estimates}\\label{Apriorisec}\nThe main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.\n\\begin{thm}\\label{AppThm}\nLet $\\omega \\in W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$ and let $u \\in  W^{1, \\mathbf{p}}_{\\mathrm{loc}}(\\Omega) \\cap L^\\infty_{\\mathrm{loc}}(\\Omega)$  be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that $p_n < p_1+2$,\nand with a function $g \\in L^{r}_{\\mathrm{loc}}{(\\Omega)}$, where $r$ satisfies the condition\n\\eqref{Ipotesir}.\nIf we assume that \n\\begin{equation}\\label{Apriori}\nV_{p_i}(u_{x_i}) \\in {W^{1,2}_{\\mathrm{loc}}(\\Omega)}, \\qquad\\forall i=1,\\dots,n,\n\\end{equation}\nthen the following estimates\n\\begin{align}\\label{uxistima}\n    \\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) & \\leq c  \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n\\end{align}",
      "definizionePi": "\\begin{align}\\label{definizionePi}\n        \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n        \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\  \\overline{p} \\ge n.\n        \\end{cases}\n    \\end{align}",
      "Ipotesip": "\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}",
      "LemmaTroisi": "\\begin{lem}\\label{LemmaTroisi}\n    Let $E \\subset \\mathbb{R}^n$ be a bounded open set and consider $u \\in W^{1, \\mathbf{p}}(E)$, with $  p_i \\geq 1$, for all $i=1, ..., n$.\\\\\n    If $\\overline{p} \\le n$, then there exists a positive constant $\\gamma_1$ depending on $n, p_i$ and, only in the case $\\overline{p}=n$, also on $\\overline{p}^*$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\overline{p}^*}(E)} \\leq \\gamma_1  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n      If $\\overline{p} > n$,  there exists a positive constant $\\gamma_2$ depending only on $ n, p_i$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\infty}(E)} \\leq \\gamma_2  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n\\end{lem}",
      "A2": "\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}",
      "A4": "\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}",
      "ulimitatoCupini": "\\begin{thm}\\label{ulimitatoCupini}\n    Let $f = f(x, \\xi)$ be the integrand defined at \\eqref{integrand} and assume that $\\omega\\in L^t(\\Omega)$, with $\\left(\\frac{\\overline{p}^*}{p_n} \\right)' < t \\le \\infty$,\nwhere $\\bar{p}^*$ is the Sobolev exponent of $\\bar{p}$, with $\\bar{p}$ defined at \\eqref{definizionePi}. \nThen, every local minimizer $u$ of \\eqref{functional} is locally bounded in $\\Omega$.\n\\end{thm}",
      "thmBfinito": "\\begin{thm}\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.\n\\end{thm}",
      "Appro": "\\begin{align}\\label{Stimacompattauxixjpi+2}\n        \\sum_{i=1}^{n}   \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2} dx \\right) & \\leq c  \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)}+{\\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}} (B_R) }} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma,\n\\end{align}\nwhich concludes the proof.\n\\end{proof}\n\\section{Approximation}\\label{Appro}\n\nIn order to perform the approximation argument, let us fix a non-negative smooth kernel $\\phi \\in \\mathcal{C}^\\infty_0(B_1(0))$ such that $\\int_{B_1(0)} \\phi =1$ and consider the corresponding family of mollifiers $(\\phi_k)_{k >0}$. We set, for a given ball $B_R \\Subset \\Omega$ ,\n$$a_i^k=a_i * \\phi_k, \\quad \\omega^k= \\omega * \\phi_k,$$\n\\begin{equation}\\label{gk}\ng_k(x)=\\underset{i}{\\max} \\{ |D a_i^k (x)|\\}\n\\end{equation}\nand, for $x \\in B_R$, we define\n\\begin{equation}\\label{fvarepsilon}\nf^k (x,\\xi)= \\sum_{i=1}^{n} a_i^k(x)|\\xi|^{p_i} ,\n\\end{equation}\n for every $k < \\text{dist}(B_R,\\Omega)$. We shall use the following $$\\mathcal{F}^{k}(v,B_R):=\\int_{_{B_R} }\\left( f^{k}(x, Dv)-\\omega(x)v(x) \\right) dx$$ and\n$$\\mathcal{F}^{k}_{\\varepsilon}(v,B_R):= \\, \\int_{B_R} \\left( f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}}- \\omega^k(x)v(x) \\right) dx.$$\n\\begin{flushleft}\nOne can easily check that $f^k(x, \\xi)$ satisfies assumptions \\eqref{A2}--\\eqref{A3} and \\eqref{A'4}, with $K = \\sup_{B_R} |g_k(x)| $ that is finite since $a^k_i(x) \\in W^{1, \\infty}(B_R)$ and then $g_k \\in L^{\\infty}(B_R)$.\\\\\n\\end{flushleft}\n\nWe are in position to prove our main result.\n\n\\begin{proof}[\\textit{{Proof of Theorem~\\ref{thmBfinito}}}]\nWe consider the variational problems\n\\begin{equation}\\label{Pfj}\n    \\inf \\left\\{  \\int_{B_R} \\left( f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}} -\\omega^k(x)v(x)\\right) dx \\ : \\ v \\in W^{1,p_n}_0(B_R)+u^\\eta \\right\\},\n\\end{equation}\nwhere $$u^\\eta = u * \\varsigma_\\eta$$ is the mollification of the local minimizer $u$ of \\eqref{functional}, for a sequence of mollifier $\\varsigma_n$.\\\\\nIt is well known that, by the direct methods of the calculus of variations, there exists a unique solution $u^\\eta_{k,\\varepsilon} \\in  W^{1,p_n}_0(B_R)+u^\\eta $ of the problem \\eqref{Pfj}. Since the integrand $$f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}}-\\omega^k(x)v(x)$$ satisfies the assumptions of  Theorem \\ref{thm2Mascolo}, we have that $$V_{p_i}((u^\\eta_{k,\\varepsilon})_{x_i}) \\in W_{\\mathrm{loc}}^{1,2}(B_R) \\quad \\forall i=1,...,n.$$\nHence we are legitimate to use estimate \\eqref{uxistima} of Theorem \\ref{AppThm}, to obtain that\n\\begin{align}\n   & \\sum_{i=1}^{n}  \\int_{B_{R/4}}|(u^\\eta_{k,\\varepsilon})_{x_i}|^{p_i +2 } dx \\notag \\\\\n    & \\leq c \\left(1+ \\sum_{i=1}^{n} \\left(\\int_{B_R}\\, \\lvert(u^\\eta_{k,\\varepsilon})_{x_i} \\rvert^{p_i} \\ dx+  \\Vert \\omega^k_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right)  + \\Vert g_k \\Vert_{L^r (B_{R})} \\right)^\\sigma \\notag \\\\\n     &  \\leq c(\\lambda) \\left(1+ \\sum_{i=1}^{n} \\left( \\int_{B_R} a_i^k(x)\\, \\lvert (u^\\eta_{k,\\varepsilon})_{x_i}\\rvert^{p_i} \\ dx +  \\Vert \\omega^k_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g_k \\Vert_{L^r (B_{R})} \\right)^\\sigma, \\label{der}\n\\end{align}",
      "functional": "\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}",
      "PreREG": "\\begin{lem}\\label{LemmaTroisi}\n    Let $E \\subset \\mathbb{R}^n$ be a bounded open set and consider $u \\in W^{1, \\mathbf{p}}(E)$, with $  p_i \\geq 1$, for all $i=1, ..., n$.\\\\\n    If $\\overline{p} \\le n$, then there exists a positive constant $\\gamma_1$ depending on $n, p_i$ and, only in the case $\\overline{p}=n$, also on $\\overline{p}^*$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\overline{p}^*}(E)} \\leq \\gamma_1  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n      If $\\overline{p} > n$,  there exists a positive constant $\\gamma_2$ depending only on $ n, p_i$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\infty}(E)} \\leq \\gamma_2  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n\\end{lem} \n\n\\section{A regularized problem}\\label{PreREG}\nIn this section we prove a regularity result for more regular problems with respect to \\eqref{functional} that will be needed in the approximation procedure. More precisely, for $\\varepsilon \\geq 0$ we consider \n\\begin{equation}\\label{Eqconepsilon}\n f_\\varepsilon(x,u, \\xi)= \\sum_{i=1}^{n} \\, \\dfrac{b_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i} + \\varepsilon(1+|\\xi|^\\frac{p_n}{2})^2  -\\overline{\\omega}(x)u\n\\end{equation}\nwith\n$b_i(x):\\Omega \\to [0,+\\infty), \\, i=1, \\dots, n$, non-negative, Lipschitz continuous coefficients,\ni.e., the following conditions hold\n\\begin{equation}\\label{atilde}\nL_i= \\sup_{x,y \\in \\Omega, x \\neq y} \\frac{|b_i(x)-b_i(y)|}{|x-y|}< \\infty , \\quad i=1,..,n.\n\\end{equation}\nSetting $\\tilde{f}(x, \\xi)= \\sum_{i=1}^{n} \\, b_i(x) \\lvert \\xi_i \\rvert^{p_i},$ one can easily check that\n\\begin{equation}\\label{A'4}\n    |D_\\xi  \\tilde{f}(x,\\xi)-D_\\xi \\tilde{f}(y, \\xi)| \\leq K |x-y|\\, \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3'}\n\\end{equation}\n\\noindent for a.e.\\ $x,y \\in \\Omega $ and every $\\xi \\in \\mathbb{R}^{ n}$, where $K=K(L_i, p_i)$\n.\\\\\nWe recall a Lipschitz regularity Theorem for the minimizers of the functional \n\\begin{equation}\\label{functional2}\n    \\mathcal{F}_\\varepsilon (x, Dv) =\\int_\\Omega f_\\varepsilon(x,v, Dv) \\, dx,\n\\end{equation}\nthat can be deduced by  \\cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\\varepsilon$ was defined at \\eqref{Eqconepsilon}. \n\\begin{thm}\\label{thm2Mascolo}\n    Let $\\tilde{f} $ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} and $\\overline{\\omega} \\in L^\\infty(\\Omega)$.\n    Let $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional2}.\nThen, $u_\\varepsilon \\in W^{1,\\infty}_{\\mathrm{loc}}(\\Omega)$ \n\\end{thm}\n\nWe shall use the higher differentiability of the minimizers $u_\\varepsilon$ of $\\mathcal{F}_\\varepsilon$ that, as far  we know, is not available in literature and that is contained in the following\n{\n\\begin{lem}\\label{LemmaInduzione}\n  Let $\\tilde{f}$ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} for exponents $ p_i \\geq 2,  \\forall i=1,\\dots,n$, and assume {$\\overline{\\omega} \\in L^\\infty(\\Omega) \\cap W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$. }\n    Let  $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega) $ be a local minimizer of the functional \\eqref{functional2}.\n    Then, \n    we have  $${V_{p_i}((u_\\varepsilon)_{x_i})} \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\quad \\forall i =1,\\dots, n.   $$\n\n\\end{lem}",
      "D1": "\\begin{lem}\\label{D1}\nLet $1<p<+\\infty$. There exist positive constants $c_1$ and $c_2$, depending only on $n$ and $p$, such that\n\\begin{center}\n{$c_1(|\\xi|^{2}+|\\eta|^{2})^{\\frac{p-2}{2}} \\leq \\dfrac{|V_{p}(\\xi)-V_{p}(\\eta)|^{2}}{|\\xi-\\eta|^{2}} \\leq c_2(|\\xi|^{2}+|\\eta|^{2})^{\\frac{p-2}{2}} $},\n\\end{center}\nfor any $\\xi, \\eta \\in \\mathbb{R}^{n}$, with $\\xi \\neq \\eta$.\n\\end{lem}"
    },
    "pre_theorem_intro_text_len": 10552,
    "pre_theorem_intro_text": "The aim of this paper is to investigate the differentiability properties of locally bounded minimizers of convex functionals, exhibiting an orthotropic structure, depending also on the $x$-variable and on the minimizer itself. More precisely, we consider\n\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}\n\\noindent where $\\Omega \\subset \\mathbb{R}^n$ is an open subset, $n\\geq 2$, {$u : \\Omega \\to \\mathbb{R}$}, and $f:\\Omega\\times\\mathbb{R}^{ n}\\to [0,+\\infty)$ has the following form \n\\begin{equation}\n    f= f(x, \\xi)= \\sum_{i=1}^{n} \\,     \\dfrac{a_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i}, \\label{integrand}\n\\end{equation}\nwith $\\xi=(\\xi_1,\\xi_2,...,\\xi_n) \\in \\mathbb{R}^n$.\nHere, we assume that $2\\leq p_1  \\leq \\cdots \\leq p_n$  and $a_i(x):\\Omega \\to \\mathbb{R}$ are bounded, measurable coefficients such that, for every $i = 1,...,n$,   $$0< \\lambda \\leq a_i (x) \\leq \\Lambda  ,$$ for some positive constants $\\lambda$, $ \\Lambda$ and for a.e.\\ $ x \\in \\Omega$. \n\nOne can easily check  that there exist positive constants $ L, l$ depending on $\\lambda, \\Lambda, p_i$ such that\n\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}\n\\begin{equation}\\label{A3}\n     |D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta)| \\leq  L \\sum_{i=1}^{n}(|\\xi_i |^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i | ,\\tag{A2}\n\\end{equation}\n\n\\noindent for a.e.\\ $x \\in \\Omega$ and all $\\xi, \\eta \\in \\mathbb{R}^{ n}$.\nConcerning the dependence on the $x$-variable, we assume that $$a_i(x) \\in W^{1,r}_{\\mathrm{loc}}(\\Omega) \\qquad \\forall i=1, \\dots,n ,$$ so that,\ndenoting by $g(x)=  \\underset{i}{\\max} \\{|Da_i(x)| \\}$, we have that $g(x) \\in L_{\\mathrm{loc}}^r (\\Omega)$ and the following inequality\n\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}\n\\noindent holds for a.e.\\ $x,y \\in \\Omega$ and every $\\xi \\in \\mathbb{R}^{n}$.\n\nFor the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n    W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n    \\end{equation*}\nthe  corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n        \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n        \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\  \\overline{p} \\ge n.\n        \\end{cases}\n    \\end{align}\n   For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space  $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a  local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have   \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le  \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$,  a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}).  Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to  \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$,  for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case  with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\)  satisfies \n\\begin{equation}\n    |u_{x_i}|^{\\frac{p_i - 2}{2}}  \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|}  \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\n In the paper \\cite{16} by Marcellini, it is clearly stated that currently there are few known regularity results for problems of this type, in relation to the properties of the coefficients $a_i(x)$. Indeed, as far as we know, few regularity results are available for the local minimizers of functionals with an orthotropic structure in the so-called non-autonomous case (see \\cite{feopass,russo}).\nMore precisely, in the super-quadratic case it has been recently proved in \\cite{russo} that, assuming the condition $p_n < p_1+2$ and Sobolev regularity on the coefficients $a_i(x)$ of the type $W^{1,r}$, with $r > p_n+2$, locally bounded minimizers $u$ of the functional \\eqref{functional}, with $\\omega  \\equiv 0$, satisfy the higher differentiability properties at \\eqref{diffproperties}.\\\\\n\\noindent Here, the main novelty with respect to the paper \\cite{russo} is that we allow the presence of a forcing term $\\omega$. We are interested in the conditions that must be imposed on the function $\\omega$ in order to achieve analogous higher differentiability results for the solutions. \\\\\n Actually, our main result consists in proving that a suitable Sobolev regularity of the coefficients $a_i(x)$ and of the forcing term $\\omega$ transfers into a higher differentiability of integer order for the local minimizers of \\eqref{functional}. Moreover, we will require a weaker differentiability assumption on $\\omega$ with respect to the one made in \\cite{BraTau}. Indeed,  since $\\frac{p_i+2}{p_i+1} < p_i'$, we have the embedding $W^{1, \\mathbf{p'}}_{\\mathrm{loc}}(\\Omega) \\hookrightarrow W^{1,{\\frac{\\mathbf{p}+2} {\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$.\n\nWe recall that recent studies have shown that the weak differentiability of the map \\( D_\\xi f(x,\\xi) \\), whether of integer or fractional order with respect to the \\( x \\)-variable, is a sufficient condition to achieve higher differentiability (see \\cite{cup, Torricelli, 25,40} for the case of Sobolev spaces with integer order and \\cite{ 2,15, Grimaldi1, Ipocoana1, Russo} for the fractional one) both in case of standard and non standard growth. \\\\\nIn particular, the general requirement for obtaining higher differentiability for local minimizers is a Sobolev-type regularity for the coefficients with an exponent\n$r$ that is greater than or equal to the dimension \n$n$. When dealing with bounded minimizers, the situation changes, and sufficient assumptions on the Sobolev regularity of the coefficients can be made independently of the dimension (see \\cite{CKP, Colombo, cup,25}). Our result follows this approach. Even though the order of summability of the coefficients depends on the dimension $n$, it does so only through the bound that allows us to handle locally bounded minimizers.\n\n{In the statement our main result, we shall use the  auxiliary function $V_{p}(\\xi)$  defined by\n\\begin{equation}\\label{DefVp}\n    V_{p}(\\xi):= |\\xi|^{\\frac{p-2}{2}} \\xi,\n    \\end{equation}\nfor all $\\xi\\in \\mathbb{R}^{n}$. \n\nWe will denote by\n \\begin{align}\\label{definizionet}\n        \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n        \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\  \\overline{q} \\ge n.\n        \\end{cases}\n    \\end{align}\n\nOur aim is to prove the following",
    "context": "For the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n \\end{equation*}\nthe corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\ \\overline{p} \\ge n.\n \\end{cases}\n \\end{align}\n For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$, for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\) satisfies \n\\begin{equation}\n |u_{x_i}|^{\\frac{p_i - 2}{2}} \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\nWe will denote by\n \\begin{align}\\label{definizionet}\n \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\ \\overline{q} \\ge n.\n \\end{cases}\n \\end{align}\n\nOur aim is to prove the following",
    "full_context": "For the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n \\end{equation*}\nthe corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\ \\overline{p} \\ge n.\n \\end{cases}\n \\end{align}\n For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$, for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\) satisfies \n\\begin{equation}\n |u_{x_i}|^{\\frac{p_i - 2}{2}} \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\nWe will denote by\n \\begin{align}\\label{definizionet}\n \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\ \\overline{q} \\ge n.\n \\end{cases}\n \\end{align}\n\nOur aim is to prove the following\n\n\\begin{rmk}\nWe observe that condition \\eqref{Ipotesip} in Theorem \\ref{thmBfinito} ensures that $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Indeed, by Lemma \\ref{LemmaTroisi}, the assumption $\\omega \\in W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega)$ implies $\\omega \\in L^t_{\\mathrm{loc}}(\\Omega)$, where $t$ was defined at \\eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \\frac{\\overline{p}^*}{t'}$, we obtain that local minimizers of \\eqref{functional} are locally bounded in $\\Omega$ (see Theorem \\ref{ulimitatoCupini} below).\nIf $\\omega \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$, then this relation becomes $p_n < \\overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \\cite{CMMjota}). \nIf $\\overline{p} \\ge n$, this condition disappears, since according to \\eqref{definizionePi} we can choose $\\overline{p}^*$ large enough so that $p_1+2 < \\frac{\\overline{p}^*}{t'}$. \\\\\nMoreover, since $p_i \\ge 2$, for every $i=1,...,n$, it holds $\\overline{q} < \\overline{p}$. Therefore, \n if $\\overline{p}<n$, the exponent $t$ appearing in \\eqref{Ipotesip} is equal to $\\frac{n \\overline{q}}{n-\\overline{q}}$.\\\\\n Eventually, we note that the exponents $\\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \\frac{\\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.\n\\end{rmk}\n\n\\section{A regularized problem}\\label{PreREG}\nIn this section we prove a regularity result for more regular problems with respect to \\eqref{functional} that will be needed in the approximation procedure. More precisely, for $\\varepsilon \\geq 0$ we consider \n\\begin{equation}\\label{Eqconepsilon}\n f_\\varepsilon(x,u, \\xi)= \\sum_{i=1}^{n} \\, \\dfrac{b_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i} + \\varepsilon(1+|\\xi|^\\frac{p_n}{2})^2 -\\overline{\\omega}(x)u\n\\end{equation}\nwith\n$b_i(x):\\Omega \\to [0,+\\infty), \\, i=1, \\dots, n$, non-negative, Lipschitz continuous coefficients,\ni.e., the following conditions hold\n\\begin{equation}\\label{atilde}\nL_i= \\sup_{x,y \\in \\Omega, x \\neq y} \\frac{|b_i(x)-b_i(y)|}{|x-y|}< \\infty , \\quad i=1,..,n.\n\\end{equation}\nSetting $\\tilde{f}(x, \\xi)= \\sum_{i=1}^{n} \\, b_i(x) \\lvert \\xi_i \\rvert^{p_i},$ one can easily check that\n\\begin{equation}\\label{A'4}\n |D_\\xi \\tilde{f}(x,\\xi)-D_\\xi \\tilde{f}(y, \\xi)| \\leq K |x-y|\\, \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3'}\n\\end{equation}\n\\noindent for a.e.\\ $x,y \\in \\Omega $ and every $\\xi \\in \\mathbb{R}^{ n}$, where $K=K(L_i, p_i)$\n.\\\\\nWe recall a Lipschitz regularity Theorem for the minimizers of the functional \n\\begin{equation}\\label{functional2}\n \\mathcal{F}_\\varepsilon (x, Dv) =\\int_\\Omega f_\\varepsilon(x,v, Dv) \\, dx,\n\\end{equation}\nthat can be deduced by \\cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\\varepsilon$ was defined at \\eqref{Eqconepsilon}. \n\\begin{thm}\\label{thm2Mascolo}\n Let $\\tilde{f} $ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} and $\\overline{\\omega} \\in L^\\infty(\\Omega)$.\n Let $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional2}.\nThen, $u_\\varepsilon \\in W^{1,\\infty}_{\\mathrm{loc}}(\\Omega)$ \n\\end{thm}\n\n\\begin{align}\n \\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx & \\leq \\,|h|^2 \\Biggl\\{ \\frac{c_\\beta}{(t-\\rho)^2} + \n\\frac{c(K)}{(t-\\rho)} \n+ c_{\\beta} \n + \\frac{c \\varepsilon}{(t-\\rho)^2} \\notag\\\\\n& \\qquad + \\sum_{i=1}^n \\left( \\int_{B_R}|\\omega_{x_i}|^{\\frac{p_i+2}{p_i+1}} \\, dx\\right)^\\frac{p_i+1}{p_i+2} \\Biggr\\}, \\notag \n \\end{align}\n and so\n\\begin{align}\n &\\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx \\leq c |h|^{2} \\,\\left[ 1+ \\sum_{i=1}^{n} \\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}}(B_{R})} \\right], \\label{StimaLemma}\n\\end{align}\nfor a positive constant $c = c(n,p_i,\\rho, R, \\lambda, K, \\lvert \\lvert Du \\rvert \\rvert_{\\infty})$.\nThen we have that \n $$V_{p_i}((u_\\varepsilon)_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n,$$ \n which implies in particular that $$|(u_\\varepsilon)_{x_i x_j}|^2|(u_\\varepsilon)_{x_i}|^{p_i-2} \\in L^1_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n \\text{ and } \\forall j=1, \\dots, n. $$\n\\end{proof}\n\n\\section{A priori estimates}\\label{Apriorisec}\nThe main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.\n\\begin{thm}\\label{AppThm}\nLet $\\omega \\in W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$ and let $u \\in W^{1, \\mathbf{p}}_{\\mathrm{loc}}(\\Omega) \\cap L^\\infty_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that $p_n < p_1+2$,\nand with a function $g \\in L^{r}_{\\mathrm{loc}}{(\\Omega)}$, where $r$ satisfies the condition\n\\eqref{Ipotesir}.\nIf we assume that \n\\begin{equation}\\label{Apriori}\nV_{p_i}(u_{x_i}) \\in {W^{1,2}_{\\mathrm{loc}}(\\Omega)}, \\qquad\\forall i=1,\\dots,n,\n\\end{equation}\nthen the following estimates\n\\begin{align}\\label{uxistima}\n \\sum_{i=1}^{n} \\left( \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) & \\leq c \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n\\end{align}\nand\n\\begin{align}\n \\sum_{i=1}^{n} \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) & \\leq c \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right)+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n \\label{StimaTeo1}\n\\end{align}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, {\\Vert u \\Vert_{\\infty}})$ and $\\gamma= \\gamma (n,p_i, {r})$ are positive constants.\n\\end{thm}\n\\begin{proof}\nWe start by observing that, thanks to assumption \\eqref{Ipotesip}, Theorem \\ref{ulimitatoCupini} and Lemma \\ref{LemmaTroisi}, we have $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Therefore, by virtue of the local boundedness of $u$ and of assumption \\eqref{Apriori}, an application of Proposition \\ref{LemmaBrasco} gives that\n\\begin{equation}\n u_{x_k} \\in L^{p_k+2}_{\\mathrm{loc}}(\\Omega) \\qquad \\text{for every } k \\in \\mathbb{N}. \\label{highint}\n\\end{equation}\nFix a ball $B_R \\Subset \\Omega$ and consider radii $\\frac{R}{4}< \\rho< s<t<t'<{\\sigma}<R$, a cut-off function $\\eta \\in C_0^{\\infty}(B_t)$, with $\\eta=1$ on $B_{s}$, $0 \\leq \\eta \\leq 1$, $|D \\eta | \\leq \\frac{c}{t-s}$ and $|h|\\leq \\frac{t' - t}{2}$.\nWe test the Euler-Lagrange equation of \\eqref{functional} with the function $$\\varphi = \\tau_{j,-h}(\\eta^2 \\tau_{j,h}u),$$ \nthus getting\n\\begin{align}\\label{SommaIntera}\n 0= \\int_{\\Omega} &\\langle f_{\\xi} (x+e_jh, Du(x+e_jh))- f_{\\xi} (x+e_jh, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x+e_jh))- f_{\\xi} (x+e_jh, Du(x)) , 2 \\eta D \\eta \\tau_{j,h} u \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x))- f_{\\xi} (x, Du(x)) , 2 \\eta D \\eta \\tau_{j,h} u \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x))- f_{\\xi} (x, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n &- \\int_{\\Omega} \\tau_{j,h}\\omega(x)\\eta^2 \\tau_{j,h} u(x) \\, dx \\notag\\\\\n & =: I_1 + I_2 + I_3 + I_4+I_5. \\notag\n\\end{align}\nFrom the previous equality, we deduce that \n\\begin{equation}\\label{SommaInt}\n I_1 \\leq |I_2| + |I_3| + |I_4|+|I_5| .\n\\end{equation}\nBy virtue of assumption \\eqref{A2}, we infer\n\\begin{align}\\label{I1}\n I_1 &= \\int_{\\Omega} \\langle f_{\\xi} (x+e_j h, Du(x+ e_jh))- f_{\\xi} (x+ e_j h, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n & \\geq l \\int_{\\Omega} \\eta^2 \\sum_{i=1}^{n}\\, \\left( \\, |u_{x_i}(x+e_jh)|^2+|u_{x_i}(x) |^2 \\, \\right)^\\frac{p_i -2}{2}| \\tau_{j,h} u_{x_i} |^2 dx =: l \\textbf{\\textbf{RHS}}.\n\\end{align}",
    "post_theorem_intro_text_len": 3294,
    "post_theorem_intro_text": "\\begin{rmk}\nWe observe that condition \\eqref{Ipotesip} in Theorem \\ref{thmBfinito} ensures that $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Indeed, by Lemma \\ref{LemmaTroisi}, the assumption $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega)$ implies $\\omega \\in L^t_{\\mathrm{loc}}(\\Omega)$, where $t$ was defined at \\eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \\frac{\\overline{p}^*}{t'}$, we obtain that local minimizers of \\eqref{functional} are locally bounded in $\\Omega$ (see Theorem \\ref{ulimitatoCupini} below).\nIf $\\omega \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$, then this relation becomes $p_n  < \\overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \\cite{CMMjota}). \nIf $\\overline{p} \\ge n$, this condition disappears, since according to \\eqref{definizionePi} we can choose $\\overline{p}^*$ large enough so that $p_1+2 < \\frac{\\overline{p}^*}{t'}$. \\\\\nMoreover, since $p_i \\ge 2$, for every $i=1,...,n$, it holds $\\overline{q} < \\overline{p}$. Therefore, \n if $\\overline{p}<n$,  the exponent $t$ appearing in \\eqref{Ipotesip} is equal to $\\frac{n \\overline{q}}{n-\\overline{q}}$.\\\\\n Eventually, we note that the exponents $\\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \\frac{\\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.\n\\end{rmk}\n\nNow, we briefly describe the strategy of the proof, which, as usual, will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit. It is well known that the a priori estimate needs to be established for sufficiently regular minimizers. Therefore, the first step is to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients, since this result is not available in the literature. As usual the higher differentiability is achieved with the use of the well known difference quotient methods that is frequently used in this context.\\\\\nMore specifically, for a given $\\varepsilon \\geq 0$, we investigate the function \n\\[\nf_\\varepsilon(x,u, \\xi) = \\sum_{i=1}^{n} \\dfrac{b_i(x)}{p_i} |\\xi_i|^{p_i} + \\varepsilon \\left( 1 + |\\xi|^{\\frac{p_n}{2}} \\right)^2-\\omega(x)u\n\\]\nwhere the coefficients $b_i(x): \\Omega \\to \\mathbb{R}$ are non-negative and satisfy the following condition\n\\[\nL_i = \\sup_{x, y \\in \\Omega, x \\neq y} \\frac{|b_i(x) - b_i(y)|}{|x - y|} < \\infty.\n\\]\nThe first goal is to analyze the regularity properties of minimizers of the following functional\n\\[\n\\mathcal{F}_\\varepsilon(u, \\Omega) = \\int_\\Omega f_\\varepsilon(x,u, Du) \\, dx.\n\\]\nAfter, we aim to establish an uniform a priori estimate, which plays a crucial role in the proof of our main result. Finally, Theorem \\ref{thmBfinito} follows by proving that the a priori estimate is preserved in passing to the limit.\\\\\n\nThe plan of the paper is briefly described. After summarizing some known results and fixing a few notations, we prove a lemma for approximating functionals to be used in the a priori estimates in Section \\ref{PreREG}. Section \\ref{Apriorisec} is devoted to the proof of some a priori estimates for solutions to a family of approximating problems. Next, in Section \\ref{Appro}, we pass to the limit in the approximating problems.",
    "sketch": "To prove Theorem~\\ref{thmBfinito} the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”\n\nMore specifically, for “a given $\\varepsilon \\ge 0$,” they introduce the approximating integrand\n\\[\nf_\\varepsilon(x,u,\\xi)=\\sum_{i=1}^n \\frac{b_i(x)}{p_i}|\\xi_i|^{p_i}+\\varepsilon\\bigl(1+|\\xi|^{\\frac{p_n}{2}}\\bigr)^2-\\omega(x)u,\n\\]\nwith nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\\sup_{x\\ne y}\\frac{|b_i(x)-b_i(y)|}{|x-y|}<\\infty$, and the functional\n\\[\n\\mathcal F_\\varepsilon(u,\\Omega)=\\int_\\Omega f_\\varepsilon(x,u,Du)\\,dx.\n\\]\nThe “first goal is to analyze the regularity properties of minimizers” of $\\mathcal F_\\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally Theorem~\\ref{thmBfinito} is obtained by “proving that the a priori estimate is preserved in passing to the limit.”",
    "expanded_sketch": "To prove the main theorem, the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”\n\nMore specifically, for “a given $\\varepsilon \\ge 0$,” they introduce the approximating integrand\n\\[\nf_\\varepsilon(x,u,\\xi)=\\sum_{i=1}^n \\frac{b_i(x)}{p_i}|\\xi_i|^{p_i}+\\varepsilon\\bigl(1+|\\xi|^{\\frac{p_n}{2}}\\bigr)^2-\\omega(x)u,\n\\]\nwith nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\\sup_{x\\ne y}\\frac{|b_i(x)-b_i(y)|}{|x-y|}<\\infty$, and the functional\n\\[\n\\mathcal F_\\varepsilon(u,\\Omega)=\\int_\\Omega f_\\varepsilon(x,u,Du)\\,dx.\n\\]\nThe “first goal is to analyze the regularity properties of minimizers” of $\\mathcal F_\\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally the main theorem is obtained by “proving that the a priori estimate is preserved in passing to the limit.”,",
    "expanded_theorem": "\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of\n\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}\nunder assumptions\n\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}\nand\n\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}\nwith exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.,",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb{R}^n\\), let \\(\\mathbf p=(p_1,\\dots,p_n)\\) with \\(p_i\\ge 2\\) for all \\(i\\), and define the anisotropic Sobolev space\n\\[\nW^{1,\\mathbf p}_{\\mathrm{loc}}(\\Omega)=\\{u\\in W^{1,1}_{\\mathrm{loc}}(\\Omega): u_{x_i}\\in L^{p_i}_{\\mathrm{loc}}(\\Omega)\\ \\text{for }i=1,\\dots,n\\}.\n\\]\nLet\n\\[\n\\frac1{\\overline p}=\\frac1n\\sum_{i=1}^n \\frac1{p_i},\n\\qquad\n\\overline p^*=\\frac{n\\overline p}{n-\\overline p}\\ \\text{if }\\overline p<n,\n\\]\nand let \\(t'\\) denote the H\\\"older conjugate of \\(t\\). Assume \\(\\omega\\in L^t_{\\mathrm{loc}}(\\Omega)\\) and \\(\\omega\\in W^{1,\\frac{\\mathbf p+2}{\\mathbf p+1}}_{\\mathrm{loc}}(\\Omega)\\), meaning \\(\\omega_{x_i}\\in L^{\\frac{p_i+2}{p_i+1}}_{\\mathrm{loc}}(\\Omega)\\) for each \\(i\\). Let \\(g\\in L^r_{\\mathrm{loc}}(\\Omega)\\) with \\(r>p_n+2\\), and let \\(u\\in W^{1,\\mathbf p}_{\\mathrm{loc}}(\\Omega)\\) be a local minimizer of\n\\[\n\\mathcal F(u;\\Omega)=\\int_\\Omega \\bigl[f(x,Du)-\\omega(x)u(x)\\bigr]\\,dx.\n\\]\nAssume furthermore that for all \\(x,y\\in\\Omega\\) and \\(\\xi,\\eta\\in\\mathbb R^n\\),\n\\[\n\\langle D_\\xi f(x,\\xi)-D_\\xi f(x,\\eta),\\xi-\\eta\\rangle\n\\ge l\\sum_{i=1}^n (|\\xi_i|^2+|\\eta_i|^2)^{\\frac{p_i-2}{2}}|\\xi_i-\\eta_i|^2,\n\\]\nand\n\\[\n|D_\\xi f(x,\\xi)-D_\\xi f(y,\\xi)|\n\\le |x-y|\\,(g(x)+g(y))\\sum_{i=1}^n |\\xi_i|^{p_i-1}.\n\\]\nAssume also that\n\\[\np_n<\n\\begin{cases}\n\\min\\left\\{\\dfrac{\\overline p^*}{t'},\\,p_1+2\\right\\}, & \\overline p<n,\\\\[4pt]\np_1+2, & \\overline p\\ge n.\n\\end{cases}\n\\]\nFor \\(s\\in\\mathbb R\\), write \\(V_{p_i}(s)=|s|^{\\frac{p_i-2}{2}}s\\). Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^{p_n+2}(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^{p_n+2}(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        },
        {
          "label": "C",
          "text": "For every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        },
        {
          "label": "D",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, there exist constants \\(c>0\\) and \\(\\sigma>0\\), depending only on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, and \\(r\\), such that for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma.\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for every \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "strict integrability threshold on g",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the higher-differentiability conclusion and second estimate",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence of constants on R and \\|u\\|_\\infty",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "difference_quotient",
          "tampered_component": "weighted second-derivative structure coming from V_{p_i}",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the correct option. It gives hypotheses and asks for the resulting conclusion, but the answer is not explicitly embedded in the wording."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem lists the full hypotheses of a regularity theorem and asks which conclusion holds. That makes it very close to a direct restatement rather than an independently structured problem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to distinguish the exact theorem conclusion from nearby variants (endpoint integrability, weaker true claim, overly strong uniformity, unweighted Hessian control). However, the item mainly tests precise recall/recognition of the theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one is a weaker true statement, others reflect common failure modes such as endpoint exponent mistakes, unjustified uniformity of constants, and confusing weighted second-derivative estimates with full Hessian L^2 regularity. They are plausible and meaningfully distinct."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has high-quality distractors, yet it is largely tautological and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.04281v1",
    "paper_link": "http://arxiv.org/abs/2512.04281v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.",
      "start_pos": 14820,
      "end_pos": 16557,
      "label": "thmBfinito"
    },
    "ref_dict": {
      "diffproperties": "\\begin{equation}\n    |u_{x_i}|^{\\frac{p_i - 2}{2}}  \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}",
      "Apriorisec": "\\begin{align}\n  &\\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx \\leq  c |h|^{2} \\,\\left[ 1+ \\sum_{i=1}^{n}   \\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}}(B_{R})}  \\right], \\label{StimaLemma}\n\\end{align}\nfor a positive constant $c = c(n,p_i,\\rho, R, \\lambda, K, \\lvert \\lvert Du \\rvert \\rvert_{\\infty})$.\nThen we have that \n $$V_{p_i}((u_\\varepsilon)_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n,$$ \n which implies in particular that $$|(u_\\varepsilon)_{x_i x_j}|^2|(u_\\varepsilon)_{x_i}|^{p_i-2} \\in L^1_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n \\text{ and  }  \\forall j=1, \\dots, n. $$\n\\end{proof}\n\n\\section{A priori estimates}\\label{Apriorisec}\nThe main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.\n\\begin{thm}\\label{AppThm}\nLet $\\omega \\in W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$ and let $u \\in  W^{1, \\mathbf{p}}_{\\mathrm{loc}}(\\Omega) \\cap L^\\infty_{\\mathrm{loc}}(\\Omega)$  be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that $p_n < p_1+2$,\nand with a function $g \\in L^{r}_{\\mathrm{loc}}{(\\Omega)}$, where $r$ satisfies the condition\n\\eqref{Ipotesir}.\nIf we assume that \n\\begin{equation}\\label{Apriori}\nV_{p_i}(u_{x_i}) \\in {W^{1,2}_{\\mathrm{loc}}(\\Omega)}, \\qquad\\forall i=1,\\dots,n,\n\\end{equation}\nthen the following estimates\n\\begin{align}\\label{uxistima}\n    \\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) & \\leq c  \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n\\end{align}",
      "definizionePi": "\\begin{align}\\label{definizionePi}\n        \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n        \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\  \\overline{p} \\ge n.\n        \\end{cases}\n    \\end{align}",
      "Ipotesip": "\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}",
      "LemmaTroisi": "\\begin{lem}\\label{LemmaTroisi}\n    Let $E \\subset \\mathbb{R}^n$ be a bounded open set and consider $u \\in W^{1, \\mathbf{p}}(E)$, with $  p_i \\geq 1$, for all $i=1, ..., n$.\\\\\n    If $\\overline{p} \\le n$, then there exists a positive constant $\\gamma_1$ depending on $n, p_i$ and, only in the case $\\overline{p}=n$, also on $\\overline{p}^*$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\overline{p}^*}(E)} \\leq \\gamma_1  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n      If $\\overline{p} > n$,  there exists a positive constant $\\gamma_2$ depending only on $ n, p_i$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\infty}(E)} \\leq \\gamma_2  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n\\end{lem}",
      "A2": "\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}",
      "A4": "\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}",
      "ulimitatoCupini": "\\begin{thm}\\label{ulimitatoCupini}\n    Let $f = f(x, \\xi)$ be the integrand defined at \\eqref{integrand} and assume that $\\omega\\in L^t(\\Omega)$, with $\\left(\\frac{\\overline{p}^*}{p_n} \\right)' < t \\le \\infty$,\nwhere $\\bar{p}^*$ is the Sobolev exponent of $\\bar{p}$, with $\\bar{p}$ defined at \\eqref{definizionePi}. \nThen, every local minimizer $u$ of \\eqref{functional} is locally bounded in $\\Omega$.\n\\end{thm}",
      "thmBfinito": "\\begin{thm}\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.\n\\end{thm}",
      "Appro": "\\begin{align}\\label{Stimacompattauxixjpi+2}\n        \\sum_{i=1}^{n}   \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2} dx \\right) & \\leq c  \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)}+{\\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}} (B_R) }} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma,\n\\end{align}\nwhich concludes the proof.\n\\end{proof}\n\\section{Approximation}\\label{Appro}\n\nIn order to perform the approximation argument, let us fix a non-negative smooth kernel $\\phi \\in \\mathcal{C}^\\infty_0(B_1(0))$ such that $\\int_{B_1(0)} \\phi =1$ and consider the corresponding family of mollifiers $(\\phi_k)_{k >0}$. We set, for a given ball $B_R \\Subset \\Omega$ ,\n$$a_i^k=a_i * \\phi_k, \\quad \\omega^k= \\omega * \\phi_k,$$\n\\begin{equation}\\label{gk}\ng_k(x)=\\underset{i}{\\max} \\{ |D a_i^k (x)|\\}\n\\end{equation}\nand, for $x \\in B_R$, we define\n\\begin{equation}\\label{fvarepsilon}\nf^k (x,\\xi)= \\sum_{i=1}^{n} a_i^k(x)|\\xi|^{p_i} ,\n\\end{equation}\n for every $k < \\text{dist}(B_R,\\Omega)$. We shall use the following $$\\mathcal{F}^{k}(v,B_R):=\\int_{_{B_R} }\\left( f^{k}(x, Dv)-\\omega(x)v(x) \\right) dx$$ and\n$$\\mathcal{F}^{k}_{\\varepsilon}(v,B_R):= \\, \\int_{B_R} \\left( f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}}- \\omega^k(x)v(x) \\right) dx.$$\n\\begin{flushleft}\nOne can easily check that $f^k(x, \\xi)$ satisfies assumptions \\eqref{A2}--\\eqref{A3} and \\eqref{A'4}, with $K = \\sup_{B_R} |g_k(x)| $ that is finite since $a^k_i(x) \\in W^{1, \\infty}(B_R)$ and then $g_k \\in L^{\\infty}(B_R)$.\\\\\n\\end{flushleft}\n\nWe are in position to prove our main result.\n\n\\begin{proof}[\\textit{{Proof of Theorem~\\ref{thmBfinito}}}]\nWe consider the variational problems\n\\begin{equation}\\label{Pfj}\n    \\inf \\left\\{  \\int_{B_R} \\left( f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}} -\\omega^k(x)v(x)\\right) dx \\ : \\ v \\in W^{1,p_n}_0(B_R)+u^\\eta \\right\\},\n\\end{equation}\nwhere $$u^\\eta = u * \\varsigma_\\eta$$ is the mollification of the local minimizer $u$ of \\eqref{functional}, for a sequence of mollifier $\\varsigma_n$.\\\\\nIt is well known that, by the direct methods of the calculus of variations, there exists a unique solution $u^\\eta_{k,\\varepsilon} \\in  W^{1,p_n}_0(B_R)+u^\\eta $ of the problem \\eqref{Pfj}. Since the integrand $$f^k(x,Dv)+ \\varepsilon(1+ |Dv|^2)^{\\frac{p_n}{2}}-\\omega^k(x)v(x)$$ satisfies the assumptions of  Theorem \\ref{thm2Mascolo}, we have that $$V_{p_i}((u^\\eta_{k,\\varepsilon})_{x_i}) \\in W_{\\mathrm{loc}}^{1,2}(B_R) \\quad \\forall i=1,...,n.$$\nHence we are legitimate to use estimate \\eqref{uxistima} of Theorem \\ref{AppThm}, to obtain that\n\\begin{align}\n   & \\sum_{i=1}^{n}  \\int_{B_{R/4}}|(u^\\eta_{k,\\varepsilon})_{x_i}|^{p_i +2 } dx \\notag \\\\\n    & \\leq c \\left(1+ \\sum_{i=1}^{n} \\left(\\int_{B_R}\\, \\lvert(u^\\eta_{k,\\varepsilon})_{x_i} \\rvert^{p_i} \\ dx+  \\Vert \\omega^k_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right)  + \\Vert g_k \\Vert_{L^r (B_{R})} \\right)^\\sigma \\notag \\\\\n     &  \\leq c(\\lambda) \\left(1+ \\sum_{i=1}^{n} \\left( \\int_{B_R} a_i^k(x)\\, \\lvert (u^\\eta_{k,\\varepsilon})_{x_i}\\rvert^{p_i} \\ dx +  \\Vert \\omega^k_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g_k \\Vert_{L^r (B_{R})} \\right)^\\sigma, \\label{der}\n\\end{align}",
      "functional": "\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}",
      "PreREG": "\\begin{lem}\\label{LemmaTroisi}\n    Let $E \\subset \\mathbb{R}^n$ be a bounded open set and consider $u \\in W^{1, \\mathbf{p}}(E)$, with $  p_i \\geq 1$, for all $i=1, ..., n$.\\\\\n    If $\\overline{p} \\le n$, then there exists a positive constant $\\gamma_1$ depending on $n, p_i$ and, only in the case $\\overline{p}=n$, also on $\\overline{p}^*$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\overline{p}^*}(E)} \\leq \\gamma_1  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n      If $\\overline{p} > n$,  there exists a positive constant $\\gamma_2$ depending only on $ n, p_i$ and $E$, such that\n    \\begin{align*}\n        \\lvert \\lvert u \\rvert \\rvert_{L^{\\infty}(E)} \\leq \\gamma_2  \\left[ \\sum_{i=1}^{n} \\lvert \\lvert u_{x_i} \\rvert \\rvert_{L^{p_i}(E)} + \\lvert \\lvert u \\rvert \\rvert_{L^1(E)} \\right].\n    \\end{align*}\n\\end{lem} \n\n\\section{A regularized problem}\\label{PreREG}\nIn this section we prove a regularity result for more regular problems with respect to \\eqref{functional} that will be needed in the approximation procedure. More precisely, for $\\varepsilon \\geq 0$ we consider \n\\begin{equation}\\label{Eqconepsilon}\n f_\\varepsilon(x,u, \\xi)= \\sum_{i=1}^{n} \\, \\dfrac{b_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i} + \\varepsilon(1+|\\xi|^\\frac{p_n}{2})^2  -\\overline{\\omega}(x)u\n\\end{equation}\nwith\n$b_i(x):\\Omega \\to [0,+\\infty), \\, i=1, \\dots, n$, non-negative, Lipschitz continuous coefficients,\ni.e., the following conditions hold\n\\begin{equation}\\label{atilde}\nL_i= \\sup_{x,y \\in \\Omega, x \\neq y} \\frac{|b_i(x)-b_i(y)|}{|x-y|}< \\infty , \\quad i=1,..,n.\n\\end{equation}\nSetting $\\tilde{f}(x, \\xi)= \\sum_{i=1}^{n} \\, b_i(x) \\lvert \\xi_i \\rvert^{p_i},$ one can easily check that\n\\begin{equation}\\label{A'4}\n    |D_\\xi  \\tilde{f}(x,\\xi)-D_\\xi \\tilde{f}(y, \\xi)| \\leq K |x-y|\\, \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3'}\n\\end{equation}\n\\noindent for a.e.\\ $x,y \\in \\Omega $ and every $\\xi \\in \\mathbb{R}^{ n}$, where $K=K(L_i, p_i)$\n.\\\\\nWe recall a Lipschitz regularity Theorem for the minimizers of the functional \n\\begin{equation}\\label{functional2}\n    \\mathcal{F}_\\varepsilon (x, Dv) =\\int_\\Omega f_\\varepsilon(x,v, Dv) \\, dx,\n\\end{equation}\nthat can be deduced by  \\cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\\varepsilon$ was defined at \\eqref{Eqconepsilon}. \n\\begin{thm}\\label{thm2Mascolo}\n    Let $\\tilde{f} $ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} and $\\overline{\\omega} \\in L^\\infty(\\Omega)$.\n    Let $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional2}.\nThen, $u_\\varepsilon \\in W^{1,\\infty}_{\\mathrm{loc}}(\\Omega)$ \n\\end{thm}\n\nWe shall use the higher differentiability of the minimizers $u_\\varepsilon$ of $\\mathcal{F}_\\varepsilon$ that, as far  we know, is not available in literature and that is contained in the following\n{\n\\begin{lem}\\label{LemmaInduzione}\n  Let $\\tilde{f}$ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} for exponents $ p_i \\geq 2,  \\forall i=1,\\dots,n$, and assume {$\\overline{\\omega} \\in L^\\infty(\\Omega) \\cap W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$. }\n    Let  $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega) $ be a local minimizer of the functional \\eqref{functional2}.\n    Then, \n    we have  $${V_{p_i}((u_\\varepsilon)_{x_i})} \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\quad \\forall i =1,\\dots, n.   $$\n\n\\end{lem}",
      "D1": "\\begin{lem}\\label{D1}\nLet $1<p<+\\infty$. There exist positive constants $c_1$ and $c_2$, depending only on $n$ and $p$, such that\n\\begin{center}\n{$c_1(|\\xi|^{2}+|\\eta|^{2})^{\\frac{p-2}{2}} \\leq \\dfrac{|V_{p}(\\xi)-V_{p}(\\eta)|^{2}}{|\\xi-\\eta|^{2}} \\leq c_2(|\\xi|^{2}+|\\eta|^{2})^{\\frac{p-2}{2}} $},\n\\end{center}\nfor any $\\xi, \\eta \\in \\mathbb{R}^{n}$, with $\\xi \\neq \\eta$.\n\\end{lem}"
    },
    "pre_theorem_intro_text_len": 10552,
    "pre_theorem_intro_text": "The aim of this paper is to investigate the differentiability properties of locally bounded minimizers of convex functionals, exhibiting an orthotropic structure, depending also on the $x$-variable and on the minimizer itself. More precisely, we consider\n\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}\n\\noindent where $\\Omega \\subset \\mathbb{R}^n$ is an open subset, $n\\geq 2$, {$u : \\Omega \\to \\mathbb{R}$}, and $f:\\Omega\\times\\mathbb{R}^{ n}\\to [0,+\\infty)$ has the following form \n\\begin{equation}\n    f= f(x, \\xi)= \\sum_{i=1}^{n} \\,     \\dfrac{a_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i}, \\label{integrand}\n\\end{equation}\nwith $\\xi=(\\xi_1,\\xi_2,...,\\xi_n) \\in \\mathbb{R}^n$.\nHere, we assume that $2\\leq p_1  \\leq \\cdots \\leq p_n$  and $a_i(x):\\Omega \\to \\mathbb{R}$ are bounded, measurable coefficients such that, for every $i = 1,...,n$,   $$0< \\lambda \\leq a_i (x) \\leq \\Lambda  ,$$ for some positive constants $\\lambda$, $ \\Lambda$ and for a.e.\\ $ x \\in \\Omega$. \n\nOne can easily check  that there exist positive constants $ L, l$ depending on $\\lambda, \\Lambda, p_i$ such that\n\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}\n\\begin{equation}\\label{A3}\n     |D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta)| \\leq  L \\sum_{i=1}^{n}(|\\xi_i |^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i | ,\\tag{A2}\n\\end{equation}\n\n\\noindent for a.e.\\ $x \\in \\Omega$ and all $\\xi, \\eta \\in \\mathbb{R}^{ n}$.\nConcerning the dependence on the $x$-variable, we assume that $$a_i(x) \\in W^{1,r}_{\\mathrm{loc}}(\\Omega) \\qquad \\forall i=1, \\dots,n ,$$ so that,\ndenoting by $g(x)=  \\underset{i}{\\max} \\{|Da_i(x)| \\}$, we have that $g(x) \\in L_{\\mathrm{loc}}^r (\\Omega)$ and the following inequality\n\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}\n\\noindent holds for a.e.\\ $x,y \\in \\Omega$ and every $\\xi \\in \\mathbb{R}^{n}$.\n\nFor the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n    W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n    \\end{equation*}\nthe  corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n        \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n        \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\  \\overline{p} \\ge n.\n        \\end{cases}\n    \\end{align}\n   For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space  $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a  local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have   \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le  \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$,  a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}).  Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to  \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$,  for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case  with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\)  satisfies \n\\begin{equation}\n    |u_{x_i}|^{\\frac{p_i - 2}{2}}  \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|}  \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\n In the paper \\cite{16} by Marcellini, it is clearly stated that currently there are few known regularity results for problems of this type, in relation to the properties of the coefficients $a_i(x)$. Indeed, as far as we know, few regularity results are available for the local minimizers of functionals with an orthotropic structure in the so-called non-autonomous case (see \\cite{feopass,russo}).\nMore precisely, in the super-quadratic case it has been recently proved in \\cite{russo} that, assuming the condition $p_n < p_1+2$ and Sobolev regularity on the coefficients $a_i(x)$ of the type $W^{1,r}$, with $r > p_n+2$, locally bounded minimizers $u$ of the functional \\eqref{functional}, with $\\omega  \\equiv 0$, satisfy the higher differentiability properties at \\eqref{diffproperties}.\\\\\n\\noindent Here, the main novelty with respect to the paper \\cite{russo} is that we allow the presence of a forcing term $\\omega$. We are interested in the conditions that must be imposed on the function $\\omega$ in order to achieve analogous higher differentiability results for the solutions. \\\\\n Actually, our main result consists in proving that a suitable Sobolev regularity of the coefficients $a_i(x)$ and of the forcing term $\\omega$ transfers into a higher differentiability of integer order for the local minimizers of \\eqref{functional}. Moreover, we will require a weaker differentiability assumption on $\\omega$ with respect to the one made in \\cite{BraTau}. Indeed,  since $\\frac{p_i+2}{p_i+1} < p_i'$, we have the embedding $W^{1, \\mathbf{p'}}_{\\mathrm{loc}}(\\Omega) \\hookrightarrow W^{1,{\\frac{\\mathbf{p}+2} {\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$.\n\nWe recall that recent studies have shown that the weak differentiability of the map \\( D_\\xi f(x,\\xi) \\), whether of integer or fractional order with respect to the \\( x \\)-variable, is a sufficient condition to achieve higher differentiability (see \\cite{cup, Torricelli, 25,40} for the case of Sobolev spaces with integer order and \\cite{ 2,15, Grimaldi1, Ipocoana1, Russo} for the fractional one) both in case of standard and non standard growth. \\\\\nIn particular, the general requirement for obtaining higher differentiability for local minimizers is a Sobolev-type regularity for the coefficients with an exponent\n$r$ that is greater than or equal to the dimension \n$n$. When dealing with bounded minimizers, the situation changes, and sufficient assumptions on the Sobolev regularity of the coefficients can be made independently of the dimension (see \\cite{CKP, Colombo, cup,25}). Our result follows this approach. Even though the order of summability of the coefficients depends on the dimension $n$, it does so only through the bound that allows us to handle locally bounded minimizers.\n\n{In the statement our main result, we shall use the  auxiliary function $V_{p}(\\xi)$  defined by\n\\begin{equation}\\label{DefVp}\n    V_{p}(\\xi):= |\\xi|^{\\frac{p-2}{2}} \\xi,\n    \\end{equation}\nfor all $\\xi\\in \\mathbb{R}^{n}$. \n\nWe will denote by\n \\begin{align}\\label{definizionet}\n        \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n        \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\  \\overline{q} \\ge n.\n        \\end{cases}\n    \\end{align}\n\nOur aim is to prove the following",
    "context": "For the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n \\end{equation*}\nthe corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\ \\overline{p} \\ge n.\n \\end{cases}\n \\end{align}\n For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$, for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\) satisfies \n\\begin{equation}\n |u_{x_i}|^{\\frac{p_i - 2}{2}} \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\nWe will denote by\n \\begin{align}\\label{definizionet}\n \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\ \\overline{q} \\ge n.\n \\end{cases}\n \\end{align}\n\nOur aim is to prove the following",
    "full_context": "For the exponents $p_i$, we shall denote by $\\mathbf{p} = (p_1, p_2, \\ldots, p_n)$ and by\n\\begin{equation*}\n W^{1,\\mathbf{p}}_{\\mathrm{loc}}(\\Omega) = \\left\\{ u \\in W^{1,1}_{\\mathrm{loc}}(\\Omega) : u_{x_i} \\in L^{p_i}_{\\mathrm{loc}}(\\Omega), \\, i = 1, \\ldots, n \\right\\},\n \\end{equation*}\nthe corresponding anisotropic Sobolev space.\\\\\nWe denote by $\\overline{p}$ the harmonic average of $p_i$ and by $\\overline{p}^*$ the Sobolev conjugate exponent of $\\overline{p}$, i.e.\n \\begin{align}\\label{definizionePi}\n \\frac{1}{\\overline{p}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{p_i}, \\qquad \\overline{p}^* = \\begin{cases}\\frac{n \\overline{p}}{n- \\overline{p}} & \\quad \\text{if} \\ \\overline{p} <n,\\\\\n \\text{any exponent in } [1,\\infty) & \\quad \\text{if} \\ \\overline{p} \\ge n.\n \\end{cases}\n \\end{align}\n For an exponent $1 \\le s \\le \\infty$, we will denote by $s'$ the H\\\"older conjugate of $s$.\\\\\nIn the following, we will assume that the function $\\omega(x) $ belongs to the Sobolev space $W^{1,{\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}}_{\\mathrm{loc}}(\\Omega)$, with ${\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}} = (\\frac{p_1+2}{p_1+1}, \\frac{p_2+2}{p_2+1}, \\ldots, \\frac{p_n+2}{p_n+1})$.\\\\\n\n\\noindent Let us recall the following definition of local minimizer.\n\\begin{dfn}\n{ \\em A function {$u\\in W_{\\rm loc}^{1,\\mathbf{p}}(\\Omega)$} is a local minimizer of\n\\eqref{functional} if, for every open subset $\\tilde{\\Omega} \\Subset \\Omega$, we have \n$\\mathcal{F}(u;\\tilde{\\Omega})\\le \\mathcal{F}(\\varphi;\\tilde{\\Omega})$ \nfor all $\\varphi\\in \\mathcal{C}_0^{\\infty}(\\tilde{\\Omega})$.}\n\\end{dfn}\nThe energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows\n$$|\\xi|^p \\leq f(x,\\xi) \\leq (1 + |\\xi|^2)^{\\frac{q}{2}}, \\qquad1<p<q, $$\nwhere $p= c(p_i)$ and $q= \\max \\{p_i : i=1, \\dots, n \\}.$\\\\\nThe study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \\cite{MarcelliniEs1, MarcelliniEs2}.\n It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \\cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as\n$$\\frac{q}{p}\\le c(n)\\displaystyle \\to 1\\,\\,\\textnormal{as}\\,\\,\\,\\,n\\to \\infty,$$\naccording to \\cite{MarcelliniEs1,MarcelliniEs2}.\n\nIn recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \\cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \\cite{surveyMar1,surveyMing}.\\\\\nCompared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \\eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\\\\nActually, the regularity of local minimizers of functionals with orthotropic structure as in \\eqref{functional} is a well-studied problem: some results can be found in \\cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \\( \\omega = 0 \\), and for the non-homogeneous one, we refer to \\cite{BraTau,CMM}.\n\\\\\nIn the autonomous case, i.e.\\ $a_i(x) \\equiv 1$, for all $i=1,...,n$, it has been proved in \\cite{Brascop12}, for the sub-quadratic case with $\\omega \\in L^{1+ \\frac{2}{p_1}}_{\\mathrm{loc}}(\\Omega)$,\nthat any local minimizer \\( u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega) \\cap L^\\infty_{\\text{loc}}(\\Omega) \\) satisfies \n\\begin{equation}\n |u_{x_i}|^{\\frac{p_i - 2}{2}} \\, u_{x_i} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots,n. \\label{diffproperties}\n\\end{equation}\nFor the super-quadratic case, Brasco et al.\\ \\cite{BraTau} consider the following widely degenerate orthotropic functional \n$$F(u; \\Omega) = \\sum_{i=1}^{n} \\int_{\\Omega} \\frac{1}{p_i} \\left( |u_{x_i}| - \\delta_i \\right)^{p_i} \\, dx - \\int_{\\Omega} \\omega \\,u \\, dx, $$\nwith $\\delta_i \\ge 0$, $p_i \\ge 2$, $ u \\in W^{1,\\mathbf{p}}_{\\text{loc}}(\\Omega)\\cap L^\\infty_{\\text{loc}}(\\Omega)$ and \n$ \\omega \\in W^{1,\\mathbf{p'}}_{\\text{loc}}(\\Omega), $ $\\mathbf{p'}=(p_1', \\cdots, p_n')$, where $p_i'$ is the H\\\"older conjugate of $p_i$.\nThe authors proved the higher differentiability of the local minimizers, that is\n\\[\n\\left( |u_{x_i}| - \\delta_i \\right)^\\frac{p_i}{2}_+ \\frac{u_{x_i}}{|u_{x_i}|} \\in W^{1,2}_{\\text{loc}}(\\Omega), \\quad i = 1, \\dots, n.\n\\]\nIn particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\\delta_i = 0$), they get the regularity at \\eqref{diffproperties}.\n\nWe will denote by\n \\begin{align}\\label{definizionet}\n \\frac{1}{\\overline{q}} = \\frac{1}{n} \\sum_{i=1}^{n} \\frac{p_i+1}{p_i+2}, \\qquad t: = \\begin{cases}\\frac{n \\overline{q}}{n- \\overline{q}} & \\quad \\text{if} \\ \\overline{q} <n,\\\\\n \\text{any exponent in } \\left(\\left(\\frac{\\overline{p}^*}{p_n}\\right)',\\infty \\right) & \\quad \\text{if} \\ \\overline{q} \\ge n.\n \\end{cases}\n \\end{align}\n\nOur aim is to prove the following\n\n\\begin{rmk}\nWe observe that condition \\eqref{Ipotesip} in Theorem \\ref{thmBfinito} ensures that $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Indeed, by Lemma \\ref{LemmaTroisi}, the assumption $\\omega \\in W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega)$ implies $\\omega \\in L^t_{\\mathrm{loc}}(\\Omega)$, where $t$ was defined at \\eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \\frac{\\overline{p}^*}{t'}$, we obtain that local minimizers of \\eqref{functional} are locally bounded in $\\Omega$ (see Theorem \\ref{ulimitatoCupini} below).\nIf $\\omega \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$, then this relation becomes $p_n < \\overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \\cite{CMMjota}). \nIf $\\overline{p} \\ge n$, this condition disappears, since according to \\eqref{definizionePi} we can choose $\\overline{p}^*$ large enough so that $p_1+2 < \\frac{\\overline{p}^*}{t'}$. \\\\\nMoreover, since $p_i \\ge 2$, for every $i=1,...,n$, it holds $\\overline{q} < \\overline{p}$. Therefore, \n if $\\overline{p}<n$, the exponent $t$ appearing in \\eqref{Ipotesip} is equal to $\\frac{n \\overline{q}}{n-\\overline{q}}$.\\\\\n Eventually, we note that the exponents $\\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \\frac{\\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.\n\\end{rmk}\n\n\\section{A regularized problem}\\label{PreREG}\nIn this section we prove a regularity result for more regular problems with respect to \\eqref{functional} that will be needed in the approximation procedure. More precisely, for $\\varepsilon \\geq 0$ we consider \n\\begin{equation}\\label{Eqconepsilon}\n f_\\varepsilon(x,u, \\xi)= \\sum_{i=1}^{n} \\, \\dfrac{b_i(x)}{p_i} \\lvert \\xi_i \\rvert^{p_i} + \\varepsilon(1+|\\xi|^\\frac{p_n}{2})^2 -\\overline{\\omega}(x)u\n\\end{equation}\nwith\n$b_i(x):\\Omega \\to [0,+\\infty), \\, i=1, \\dots, n$, non-negative, Lipschitz continuous coefficients,\ni.e., the following conditions hold\n\\begin{equation}\\label{atilde}\nL_i= \\sup_{x,y \\in \\Omega, x \\neq y} \\frac{|b_i(x)-b_i(y)|}{|x-y|}< \\infty , \\quad i=1,..,n.\n\\end{equation}\nSetting $\\tilde{f}(x, \\xi)= \\sum_{i=1}^{n} \\, b_i(x) \\lvert \\xi_i \\rvert^{p_i},$ one can easily check that\n\\begin{equation}\\label{A'4}\n |D_\\xi \\tilde{f}(x,\\xi)-D_\\xi \\tilde{f}(y, \\xi)| \\leq K |x-y|\\, \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3'}\n\\end{equation}\n\\noindent for a.e.\\ $x,y \\in \\Omega $ and every $\\xi \\in \\mathbb{R}^{ n}$, where $K=K(L_i, p_i)$\n.\\\\\nWe recall a Lipschitz regularity Theorem for the minimizers of the functional \n\\begin{equation}\\label{functional2}\n \\mathcal{F}_\\varepsilon (x, Dv) =\\int_\\Omega f_\\varepsilon(x,v, Dv) \\, dx,\n\\end{equation}\nthat can be deduced by \\cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\\varepsilon$ was defined at \\eqref{Eqconepsilon}. \n\\begin{thm}\\label{thm2Mascolo}\n Let $\\tilde{f} $ satisfy \\eqref{A2}, \\eqref{A3} and \\eqref{A'4} and $\\overline{\\omega} \\in L^\\infty(\\Omega)$.\n Let $u_\\varepsilon \\in W^{1,p_n}_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional2}.\nThen, $u_\\varepsilon \\in W^{1,\\infty}_{\\mathrm{loc}}(\\Omega)$ \n\\end{thm}\n\n\\begin{align}\n \\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx & \\leq \\,|h|^2 \\Biggl\\{ \\frac{c_\\beta}{(t-\\rho)^2} + \n\\frac{c(K)}{(t-\\rho)} \n+ c_{\\beta} \n + \\frac{c \\varepsilon}{(t-\\rho)^2} \\notag\\\\\n& \\qquad + \\sum_{i=1}^n \\left( \\int_{B_R}|\\omega_{x_i}|^{\\frac{p_i+2}{p_i+1}} \\, dx\\right)^\\frac{p_i+1}{p_i+2} \\Biggr\\}, \\notag \n \\end{align}\n and so\n\\begin{align}\n &\\sum_{i=1}^n \\int_{B_{\\rho}} |\\tau_{j,h} V_{p_i} ((u_\\varepsilon)_{x_i})|^2 dx \\leq c |h|^{2} \\,\\left[ 1+ \\sum_{i=1}^{n} \\Vert \\omega_{x_i} \\Vert_{L^{\\frac{p_i+2}{p_i+1}}(B_{R})} \\right], \\label{StimaLemma}\n\\end{align}\nfor a positive constant $c = c(n,p_i,\\rho, R, \\lambda, K, \\lvert \\lvert Du \\rvert \\rvert_{\\infty})$.\nThen we have that \n $$V_{p_i}((u_\\varepsilon)_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n,$$ \n which implies in particular that $$|(u_\\varepsilon)_{x_i x_j}|^2|(u_\\varepsilon)_{x_i}|^{p_i-2} \\in L^1_{\\mathrm{loc}}(\\Omega), \\qquad \\forall i=1, \\dots, n \\text{ and } \\forall j=1, \\dots, n. $$\n\\end{proof}\n\n\\section{A priori estimates}\\label{Apriorisec}\nThe main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.\n\\begin{thm}\\label{AppThm}\nLet $\\omega \\in W^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}_{\\mathrm{loc}}(\\Omega)$ and let $u \\in W^{1, \\mathbf{p}}_{\\mathrm{loc}}(\\Omega) \\cap L^\\infty_{\\mathrm{loc}}(\\Omega)$ be a local minimizer of \\eqref{functional} under assumptions \\eqref{A2}--\\eqref{A4}, with exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that $p_n < p_1+2$,\nand with a function $g \\in L^{r}_{\\mathrm{loc}}{(\\Omega)}$, where $r$ satisfies the condition\n\\eqref{Ipotesir}.\nIf we assume that \n\\begin{equation}\\label{Apriori}\nV_{p_i}(u_{x_i}) \\in {W^{1,2}_{\\mathrm{loc}}(\\Omega)}, \\qquad\\forall i=1,\\dots,n,\n\\end{equation}\nthen the following estimates\n\\begin{align}\\label{uxistima}\n \\sum_{i=1}^{n} \\left( \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) & \\leq c \\left(1+ \\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right) + \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n\\end{align}\nand\n\\begin{align}\n \\sum_{i=1}^{n} \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) & \\leq c \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + \\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})} \\right)+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\gamma \n \\label{StimaTeo1}\n\\end{align}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, {\\Vert u \\Vert_{\\infty}})$ and $\\gamma= \\gamma (n,p_i, {r})$ are positive constants.\n\\end{thm}\n\\begin{proof}\nWe start by observing that, thanks to assumption \\eqref{Ipotesip}, Theorem \\ref{ulimitatoCupini} and Lemma \\ref{LemmaTroisi}, we have $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Therefore, by virtue of the local boundedness of $u$ and of assumption \\eqref{Apriori}, an application of Proposition \\ref{LemmaBrasco} gives that\n\\begin{equation}\n u_{x_k} \\in L^{p_k+2}_{\\mathrm{loc}}(\\Omega) \\qquad \\text{for every } k \\in \\mathbb{N}. \\label{highint}\n\\end{equation}\nFix a ball $B_R \\Subset \\Omega$ and consider radii $\\frac{R}{4}< \\rho< s<t<t'<{\\sigma}<R$, a cut-off function $\\eta \\in C_0^{\\infty}(B_t)$, with $\\eta=1$ on $B_{s}$, $0 \\leq \\eta \\leq 1$, $|D \\eta | \\leq \\frac{c}{t-s}$ and $|h|\\leq \\frac{t' - t}{2}$.\nWe test the Euler-Lagrange equation of \\eqref{functional} with the function $$\\varphi = \\tau_{j,-h}(\\eta^2 \\tau_{j,h}u),$$ \nthus getting\n\\begin{align}\\label{SommaIntera}\n 0= \\int_{\\Omega} &\\langle f_{\\xi} (x+e_jh, Du(x+e_jh))- f_{\\xi} (x+e_jh, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x+e_jh))- f_{\\xi} (x+e_jh, Du(x)) , 2 \\eta D \\eta \\tau_{j,h} u \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x))- f_{\\xi} (x, Du(x)) , 2 \\eta D \\eta \\tau_{j,h} u \\rangle \\, dx \\notag \\\\\n &+ \\int_{\\Omega} \\langle f_{\\xi} (x+e_jh, Du(x))- f_{\\xi} (x, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n &- \\int_{\\Omega} \\tau_{j,h}\\omega(x)\\eta^2 \\tau_{j,h} u(x) \\, dx \\notag\\\\\n & =: I_1 + I_2 + I_3 + I_4+I_5. \\notag\n\\end{align}\nFrom the previous equality, we deduce that \n\\begin{equation}\\label{SommaInt}\n I_1 \\leq |I_2| + |I_3| + |I_4|+|I_5| .\n\\end{equation}\nBy virtue of assumption \\eqref{A2}, we infer\n\\begin{align}\\label{I1}\n I_1 &= \\int_{\\Omega} \\langle f_{\\xi} (x+e_j h, Du(x+ e_jh))- f_{\\xi} (x+ e_j h, Du(x)) , \\eta^2 \\tau_{j,h} Du \\rangle \\, dx \\notag \\\\\n & \\geq l \\int_{\\Omega} \\eta^2 \\sum_{i=1}^{n}\\, \\left( \\, |u_{x_i}(x+e_jh)|^2+|u_{x_i}(x) |^2 \\, \\right)^\\frac{p_i -2}{2}| \\tau_{j,h} u_{x_i} |^2 dx =: l \\textbf{\\textbf{RHS}}.\n\\end{align}",
    "post_theorem_intro_text_len": 3294,
    "post_theorem_intro_text": "\\begin{rmk}\nWe observe that condition \\eqref{Ipotesip} in Theorem \\ref{thmBfinito} ensures that $u \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$. Indeed, by Lemma \\ref{LemmaTroisi}, the assumption $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega)$ implies $\\omega \\in L^t_{\\mathrm{loc}}(\\Omega)$, where $t$ was defined at \\eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \\frac{\\overline{p}^*}{t'}$, we obtain that local minimizers of \\eqref{functional} are locally bounded in $\\Omega$ (see Theorem \\ref{ulimitatoCupini} below).\nIf $\\omega \\in L^\\infty_{\\mathrm{loc}}(\\Omega)$, then this relation becomes $p_n  < \\overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \\cite{CMMjota}). \nIf $\\overline{p} \\ge n$, this condition disappears, since according to \\eqref{definizionePi} we can choose $\\overline{p}^*$ large enough so that $p_1+2 < \\frac{\\overline{p}^*}{t'}$. \\\\\nMoreover, since $p_i \\ge 2$, for every $i=1,...,n$, it holds $\\overline{q} < \\overline{p}$. Therefore, \n if $\\overline{p}<n$,  the exponent $t$ appearing in \\eqref{Ipotesip} is equal to $\\frac{n \\overline{q}}{n-\\overline{q}}$.\\\\\n Eventually, we note that the exponents $\\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \\frac{\\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.\n\\end{rmk}\n\nNow, we briefly describe the strategy of the proof, which, as usual, will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit. It is well known that the a priori estimate needs to be established for sufficiently regular minimizers. Therefore, the first step is to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients, since this result is not available in the literature. As usual the higher differentiability is achieved with the use of the well known difference quotient methods that is frequently used in this context.\\\\\nMore specifically, for a given $\\varepsilon \\geq 0$, we investigate the function \n\\[\nf_\\varepsilon(x,u, \\xi) = \\sum_{i=1}^{n} \\dfrac{b_i(x)}{p_i} |\\xi_i|^{p_i} + \\varepsilon \\left( 1 + |\\xi|^{\\frac{p_n}{2}} \\right)^2-\\omega(x)u\n\\]\nwhere the coefficients $b_i(x): \\Omega \\to \\mathbb{R}$ are non-negative and satisfy the following condition\n\\[\nL_i = \\sup_{x, y \\in \\Omega, x \\neq y} \\frac{|b_i(x) - b_i(y)|}{|x - y|} < \\infty.\n\\]\nThe first goal is to analyze the regularity properties of minimizers of the following functional\n\\[\n\\mathcal{F}_\\varepsilon(u, \\Omega) = \\int_\\Omega f_\\varepsilon(x,u, Du) \\, dx.\n\\]\nAfter, we aim to establish an uniform a priori estimate, which plays a crucial role in the proof of our main result. Finally, Theorem \\ref{thmBfinito} follows by proving that the a priori estimate is preserved in passing to the limit.\\\\\n\nThe plan of the paper is briefly described. After summarizing some known results and fixing a few notations, we prove a lemma for approximating functionals to be used in the a priori estimates in Section \\ref{PreREG}. Section \\ref{Apriorisec} is devoted to the proof of some a priori estimates for solutions to a family of approximating problems. Next, in Section \\ref{Appro}, we pass to the limit in the approximating problems.",
    "sketch": "To prove Theorem~\\ref{thmBfinito} the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”\n\nMore specifically, for “a given $\\varepsilon \\ge 0$,” they introduce the approximating integrand\n\\[\nf_\\varepsilon(x,u,\\xi)=\\sum_{i=1}^n \\frac{b_i(x)}{p_i}|\\xi_i|^{p_i}+\\varepsilon\\bigl(1+|\\xi|^{\\frac{p_n}{2}}\\bigr)^2-\\omega(x)u,\n\\]\nwith nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\\sup_{x\\ne y}\\frac{|b_i(x)-b_i(y)|}{|x-y|}<\\infty$, and the functional\n\\[\n\\mathcal F_\\varepsilon(u,\\Omega)=\\int_\\Omega f_\\varepsilon(x,u,Du)\\,dx.\n\\]\nThe “first goal is to analyze the regularity properties of minimizers” of $\\mathcal F_\\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally Theorem~\\ref{thmBfinito} is obtained by “proving that the a priori estimate is preserved in passing to the limit.”",
    "expanded_sketch": "To prove the main theorem, the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”\n\nMore specifically, for “a given $\\varepsilon \\ge 0$,” they introduce the approximating integrand\n\\[\nf_\\varepsilon(x,u,\\xi)=\\sum_{i=1}^n \\frac{b_i(x)}{p_i}|\\xi_i|^{p_i}+\\varepsilon\\bigl(1+|\\xi|^{\\frac{p_n}{2}}\\bigr)^2-\\omega(x)u,\n\\]\nwith nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\\sup_{x\\ne y}\\frac{|b_i(x)-b_i(y)|}{|x-y|}<\\infty$, and the functional\n\\[\n\\mathcal F_\\varepsilon(u,\\Omega)=\\int_\\Omega f_\\varepsilon(x,u,Du)\\,dx.\n\\]\nThe “first goal is to analyze the regularity properties of minimizers” of $\\mathcal F_\\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally the main theorem is obtained by “proving that the a priori estimate is preserved in passing to the limit.”,",
    "expanded_theorem": "\\label{thmBfinito}\nLet $\\omega \\in  W_{\\mathrm{loc}}^{1,\\frac{\\mathbf{p}+2}{\\mathbf{p}+1}}(\\Omega) $ and let $u \\in {W_{\\mathrm{loc}}^{1,\\mathbf{p}}}(\\Omega)$ be a local minimizer of\n\\begin{equation}\\label{functional}\n \\mathcal{F}(u,\\Omega):= \\, \\int_\\Omega \\left[f(x, Du)-\\omega(x)u(x) \\right] dx,\n\\end{equation}\nunder assumptions\n\\begin{equation}\\label{A2}\n    \\langle D_\\xi f(x, \\xi)-D_\\xi f(x, \\eta),\\xi-\\eta\\rangle  \\geq  l \\sum_{i=1}^{n}(|\\xi_i|^2+|\\eta_i |^2)^\\frac{p_i -2}{2}|\\xi_i - \\eta_i |^2 \\tag{A1}\n\\end{equation}\nand\n\\begin{equation}\\label{A4}\n    |D_\\xi f(x,\\xi)-D_\\xi f(y, \\xi)| \\leq |x-y|\\,(g(x)+g(y)) \\, \\sum_{i=1}^{n} |\\xi_i|^{p_i-1} \\tag{A3}\n\\end{equation}\nwith exponents $p_i\\geq 2, \\forall i=1,\\dots,n $, such that\n\\begin{equation}\\label{Ipotesip}\np_n <\n\\begin{cases}\n    \\min \\Big\\{ \\dfrac{\\overline{p}^*}{t'} ,p_1 +2 \\Big\\} \\, &\\text{if } \\overline{p} < n,  \\\\\n     p_1 +2 \\quad &\\text{otherwise},\n    \\end{cases}\n\\end{equation}\nand with a function $g \\in {L^{r}_{\\mathrm{loc}}(\\Omega)}$, where $r$ satisfies the condition\n\\begin{equation}\\label{Ipotesir}\n   r> p_n + 2 .\n\\end{equation} \nThen,\n\\begin{equation*}\n{V_{p_i}(u_{x_i}) \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)} \\qquad\\forall i=1,\\dots,n,\n\\end{equation*}\nand the following estimates\n\\begin{align*}\n    &\\sum_{i=1}^{n} \\left(  \\int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \\right) \\notag\\\\\n    &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nand\n\\begin{align*}\n        &\\sum_{i=1}^{n}  \\left( \\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \\right) \\notag\\\\\n         &\\leq c  \\left( 1+\\sum_{i=1}^{n} \\left(\\Vert u_{x_i} \\Vert_{L^{p_i}(B_R)} + {\\Vert \\omega_{x_i} \\Vert_{L^\\frac{p_i+2}{p_i+1}(B_{R})}} \\right) +\\Vert \\omega\\Vert_{L^t(B_R)}+ \\Vert g \\Vert_{L^r (B_{R})} \\right)^\\sigma \n\\end{align*}\nhold for every pair of concentric balls $B_{R/4} \\subset B_{R} \\Subset \\Omega$, where $c = c(n,p_i,\\lambda, \\Lambda,R, \\Vert u \\Vert_{\\infty})$ and $\\sigma= \\sigma (n,p_i, r)$ are positive constants.,",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb{R}^n\\), let \\(\\mathbf p=(p_1,\\dots,p_n)\\) with \\(p_i\\ge 2\\) for all \\(i\\), and define the anisotropic Sobolev space\n\\[\nW^{1,\\mathbf p}_{\\mathrm{loc}}(\\Omega)=\\{u\\in W^{1,1}_{\\mathrm{loc}}(\\Omega): u_{x_i}\\in L^{p_i}_{\\mathrm{loc}}(\\Omega)\\ \\text{for }i=1,\\dots,n\\}.\n\\]\nLet\n\\[\n\\frac1{\\overline p}=\\frac1n\\sum_{i=1}^n \\frac1{p_i},\n\\qquad\n\\overline p^*=\\frac{n\\overline p}{n-\\overline p}\\ \\text{if }\\overline p<n,\n\\]\nand let \\(t'\\) denote the H\\\"older conjugate of \\(t\\). Assume \\(\\omega\\in L^t_{\\mathrm{loc}}(\\Omega)\\) and \\(\\omega\\in W^{1,\\frac{\\mathbf p+2}{\\mathbf p+1}}_{\\mathrm{loc}}(\\Omega)\\), meaning \\(\\omega_{x_i}\\in L^{\\frac{p_i+2}{p_i+1}}_{\\mathrm{loc}}(\\Omega)\\) for each \\(i\\). Let \\(g\\in L^r_{\\mathrm{loc}}(\\Omega)\\) with \\(r>p_n+2\\), and let \\(u\\in W^{1,\\mathbf p}_{\\mathrm{loc}}(\\Omega)\\) be a local minimizer of\n\\[\n\\mathcal F(u;\\Omega)=\\int_\\Omega \\bigl[f(x,Du)-\\omega(x)u(x)\\bigr]\\,dx.\n\\]\nAssume furthermore that for all \\(x,y\\in\\Omega\\) and \\(\\xi,\\eta\\in\\mathbb R^n\\),\n\\[\n\\langle D_\\xi f(x,\\xi)-D_\\xi f(x,\\eta),\\xi-\\eta\\rangle\n\\ge l\\sum_{i=1}^n (|\\xi_i|^2+|\\eta_i|^2)^{\\frac{p_i-2}{2}}|\\xi_i-\\eta_i|^2,\n\\]\nand\n\\[\n|D_\\xi f(x,\\xi)-D_\\xi f(y,\\xi)|\n\\le |x-y|\\,(g(x)+g(y))\\sum_{i=1}^n |\\xi_i|^{p_i-1}.\n\\]\nAssume also that\n\\[\np_n<\n\\begin{cases}\n\\min\\left\\{\\dfrac{\\overline p^*}{t'},\\,p_1+2\\right\\}, & \\overline p<n,\\\\[4pt]\np_1+2, & \\overline p\\ge n.\n\\end{cases}\n\\]\nFor \\(s\\in\\mathbb R\\), write \\(V_{p_i}(s)=|s|^{\\frac{p_i-2}{2}}s\\). Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^{p_n+2}(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^{p_n+2}(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        },
        {
          "label": "C",
          "text": "For every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        },
        {
          "label": "D",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, there exist constants \\(c>0\\) and \\(\\sigma>0\\), depending only on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, and \\(r\\), such that for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for each fixed \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i-2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma.\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(i=1,\\dots,n\\),\n\\[\nV_{p_i}(u_{x_i})\\in W^{1,2}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every pair of concentric balls \\(B_{R/4}\\subset B_R\\Subset \\Omega\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i}|^{p_i+2}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nand, for every \\(j=1,\\dots,n\\),\n\\[\n\\sum_{i=1}^n \\int_{B_{R/4}} |u_{x_i x_j}|^2\\,|u_{x_i}|^{p_i}\\,dx\n\\le c\\left(1+\\sum_{i=1}^n\\Bigl(\\|u_{x_i}\\|_{L^{p_i}(B_R)}+\\|\\omega_{x_i}\\|_{L^{\\frac{p_i+2}{p_i+1}}(B_R)}\\Bigr)+\\|\\omega\\|_{L^t(B_R)}+\\|g\\|_{L^r(B_R)}\\right)^\\sigma,\n\\]\nwhere \\(c>0\\) depends on \\(n\\), the exponents \\(p_i\\), the structural constants in the hypotheses, \\(R\\), and \\(\\|u\\|_\\infty\\), while \\(\\sigma>0\\) depends only on \\(n\\), the exponents \\(p_i\\), and \\(r\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "strict integrability threshold on g",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the higher-differentiability conclusion and second estimate",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence of constants on R and \\|u\\|_\\infty",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "difference_quotient",
          "tampered_component": "weighted second-derivative structure coming from V_{p_i}",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only hypotheses and notation; it does not explicitly reveal the conclusion. The correct estimate is not stated or strongly telegraphed in the prompt itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: after listing the full assumptions, it asks which estimate holds. The correct option is the theorem’s conclusion almost verbatim rather than a new inferential task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (integrability exponent for g, dependence of constants, exact weighted second-derivative term, weaker true conclusion). However, success depends more on recognizing the precise theorem statement than on generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion, altering the threshold on g, misstating parameter dependence, and changing the natural V_p-based weight from p_i-2 to p_i."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically careful item with strong distractors, but it is largely a verbatim theorem-identification question rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.04410v2",
    "paper_link": "http://arxiv.org/abs/2512.04410v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.",
      "start_pos": 19166,
      "end_pos": 19887,
      "label": "thm:opt-iid-homo"
    },
    "ref_dict": {
      "thm:opt-iid-homo": "\\begin{theorem}\n\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.\n\\end{theorem}",
      "rmk:optimality": "\\begin{remark}\\label{rmk:optimality}\nThis proposition identifies $O(R^{-1})$ as the optimal homogenization rate unless $\\bar z \\equiv 0$, and so no improvement beyond $O(R^{-1})$ should be expected in general. \n         We further expect that Proposition~\\ref{lem:two-scale-exp} remains valid for environments with finite range of dependence, for the same reasons discussed in the comments following Theorem~\\ref{thm:opt-quant-homo}.\n\\end{remark}",
      "eq:effective-ellip": "\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}",
      "thm:c11": "\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}",
      "eq:def-lambda": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}",
      "eq:elliptic-dirich": "\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}",
      "thm:opt-quant-homo": "\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}",
      "thm:global_krt": "\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}",
      "lem:two-scale-exp": "\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above.  We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}",
      "thm:QCLT": "\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}",
      "eq:def-eta": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}",
      "eq:two-scale-further": "\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 17605,
    "pre_theorem_intro_text": "\\label{sec:intro}\nIn this paper, we consider the homogenization convergence rates of purely second-order non-divergence form difference operators in an independent and identically distributed (i.i.d.) random environment on $\\Z^d$ for dimensions $d\\ge 3$.\n\nLarge-scale behavior of random walks in i.i.d.\\ environments has been intensively studied; see \\cite{BoSz-02, OZ-04, MB-11, DreRam-14, Kumagai-14} and the references therein. Since the dynamics of random walks in random environments (RWRE) can be described by difference equations, stochastic homogenization of difference equations in random environments is crucial for understanding the large-scale behavior of RWRE.\nFor environments in which spectral-gap or Efron--Stein type inequalities are available, the strong concentration properties of the medium can be exploited to obtain quantitative homogenization results; see, for example, \\cite{GNO-15, AL-17, GT-23}. In this paper, we show that, for non-divergence form difference operators in the i.i.d.\\ setting, the reflection symmetry of the law of the environment can in fact be used to extract additional cancellations. Somewhat unexpectedly, this yields homogenization rates for the Dirichlet problem that are strictly better than the standard optimal rate $O(R^{-1})$ obtained in \\cite{GT-23}. To achieve this, we require a more delicate analysis of the fluctuations of the homogenization error, relying in particular on a deeper understanding of higher-order correctors. \n\n\\subsection{Settings}\nLet $\\mb S_{d\\times d}$ denote the set of real $d\\times d$ positive definite diagonal matrices. A map \n\\[\n\\omega:\\Z^d\\to\\mb S_{d\\times d}\n\\] is called an {\\it environment}, and the set of all such environments is denoted by $\\Omega$. \nLet $\\mb P$ be a probability measure on $\\Omega$ such that \n\\[\n\\left\\{\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)], \\;x\\in\\Z^d\\right\\}\n\\] \nare i.i.d. under $\\mb P$. We denote the expectation with respect to $\\mb P$ by $\\mb E$ (or $E_{\\mb P}$). \n\n\\begin{definition}\\label{def:differences}\nLet $\\{e_1,\\ldots,e_d\\}$ be the canonical basis for $\\R^d$, and let \n\\[\nU:=\\left\\{e\\in\\Z^d:|e|=1\\right\\}=\\left\\{\\pm e_1,\\ldots,\\pm e_d\\right\\}\n\\]\nbe the set of unit vectors in $\\Z^d$. We define the difference operators $\\nabla=(\\nabla_e)_{e\\in U}$  and \n$\\nabla^2= \\mathrm{diag}[\\nabla_1^2,\\ldots,\\nabla_d^2]$ by\n\\begin{equation}\\label{eq:def-nabla}\n\\nabla_e u(x) := u(x+e)-u(x), \\qquad\n\\nabla_i^2u(x) := u(x+e_i)+u(x-e_i)-2u(x)  \n\\end{equation}\nfor $e\\in U$ and $i\\in\\{1,\\ldots,d\\}$. \nClearly, $\\nabla$ and $\\nabla^2$ are linear operators.  \n\\end{definition}\n\nFor $r>0$ and $y\\in\\R^d$, we write\n\\[\n\\B_r(y) := \\left\\{x\\in\\R^d: |x-y|<r\\right\\}, \\qquad\nB_r(y):=\\B_r(y)\\cap\\Z^d\n\\]\nto denote the continuous and discrete ball with center $y$ and radius $r$, respectively.\nWhen $y=0$, we write $\\B_r := \\B_r(0)$ and $B_r := B_r(0)$ for simplicity. \nFor any $B\\subset\\Z^d$, its {\\it discrete boundary} $\\partial B$ and {\\it discrete closure} $\\bar B$ are defined as\n\\[\n\\partial B:=\\left\\{z\\in\\Z^d\\setminus B: |z-x|=1 \\text{ for some }x\\in B\\right\\},\\qquad \\bar B := B\\cup\\partial B.\n\\]\nBy abuse of notation, we also use $\\partial A$ and $\\bar A$ to denote the usual continuous boundary and closure of $A\\subset\\R^d$, respectively, if there is no confusion.  \nThe interior of $B_R$ is denoted by\n\\[\n\\inB{R} := \\{x\\in B_R\\,:\\,  |x-y| \\ge 1 \\text{ for all }y\\in\\partial\\B_R\\}.\n\\]\nNote that for any $R\\ge 1$, there holds\n\\[\n\\bar{\\B}_{R-1}\\cap\\Z^d\\subset\\inB \nR\\subset B_R\\subset\\binB{R}\\subset\\bar\\B_R\\cap\\Z^d\\subset\\bar B_R.\n\\]\n\n For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$,  we consider the discrete {\\it non-divergence form}   elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\n\\medskip\n\nWe remark that in the continuous PDE setting, one only needs a function $g\\in C^{2,\\alpha}(\\partial\\B_1)$ on the boundary $\\partial\\B_1$ to consider Dirichlet problems of types \\eqref{eq:elliptic-dirich} and \\eqref{eq:effective-ellip}. However, in the discrete setting, \nthe discrete and continuous boundaries do not agree, and therefore the function $g$ in $\\eqref{eq:elliptic-dirich}$ should be extended to $\\frac{1}{R}\\partial\\inB R\\not\\subset\\partial\\B_1$.  This discrepancy typically induces an error of order $\\tfrac{1}{R}$ which automatically becomes the ceiling of the optimal homogenization error in the discrete setting,  cf.  \\cite{GT-23}. \nIndeed, this $O(R^{-1})$ rate is the optimal rate obtained in \\cite{GT-23} for $d\\geq 3$.\nWhereas, when such a discretization issue is not present, e.g., in the continuous periodic setting \\cite{GTY-19, ST-21, Spr-24},  one needs to go deeper into higher order correctors to justify the optimality of the rate $O(R^{-1})$ for the homogenization of the Dirichlet problem. We refer to \\cite{CSS-20, FGP-24, SSZ-25} for recent developments in numerical homogenization of PDEs in non-divergence form.\n\n\\medskip\n\nA goal of this article is to isolate and examine the homogenization rate that does {\\bf not} originate from boundary discretization.\nTo this end, we assume that the boundary data $g$ is properly extended to be a function on $\\frac{1}{R}\\partial\\inB R$ in the Dirichlet problem \\eqref{eq:elliptic-dirich}.\n\n\\medskip\n\nThe non-divergence form difference equation \\eqref{eq:elliptic-dirich} is used to describe random walks in a random environment (RWRE) in $\\Z^d$. To be specific, for $x,y\\in \\Z^d$, we set\n\\begin{equation}\\label{def:omegaxei}\n\\omega(x,x\\pm e_i):=\\frac{\\omega_i(x)}{2\\,\\tr\\omega(x)} \\quad  \\text{ for } i=1,\\ldots, d,\n\\end{equation}\n and \n \\[\n \\omega(x,y) := 0 \\quad \\text{ if $|x-y|\\neq 1$},\n \\]\n that is, we normalize $\\omega$ to get a transition probability.  \n We note that the configuration of $\\{\\omega(x,y):x,y\\in\\Z^d\\}$ is also called a {\\it balanced environment} in the literature \\cite{L-82,GZ-12,BD-14,DGR-15, BCDG-18,DG-19}.\n\\begin{definition}\\label{def:rwre-discrete}\nFor each fixed $\\omega\\in\\Omega$, the random walk  $(X_n)_{n\\ge 0}$ in the environment $\\omega$ with $X_0=x$ is a Markov chain  in $\\Z^d$ with transition probability $P_\\omega^x$ specified by\n\\begin{equation}\\label{eq:def-RW}\nP_\\omega^x\\left(X_{n+1}=z|X_n=y\\right) := \\omega(y,z).\n\\end{equation}\nThe {\\it continuous-time} RWRE $(Y_t)_{t\\ge 0}$ in $\\omega$ is defined to be the Markov process on $\\Z^d$ with jump rates $\\omega(x,y)$ from $x$ to $y$. With abuse of notations, we still denote by $P_\\omega^x$  the quenched law of $(Y_t)$ if there is no ambiguity from the context.\n\\end{definition} \nThe expectation with respect to $P_\\omega^x$ is denoted by $E_\\omega^x$. When the starting point of the random walk is $0$, we sometimes omit the superscript and write $P_\\omega^0$ and $E_\\omega^0$ as $P_\\omega$ and $E_\\omega$, respectively. Note that for random walks $(X_n)$ in an environment $\\omega$, we have that\n\\begin{equation}\\label{def:omegabar}\n\\bar\\omega^i := \\theta_{X_i}\\omega\\in\\Omega, \\qquad i\\ge 0,\n\\end{equation}\nis also a Markov chain, called the {\\it environment viewed from the particle} process. By abuse of notation, we enlarge our probability space so that $P_\\omega$ still denotes the joint law of the random walks and $(\\bar\\omega^i)_{i\\ge 0}$. \n\nFor $\\omega\\in\\Omega$ and $i\\in \\{1,\\ldots,d\\}$, we set $a_i(x)=a_i^\\omega(x) := \\frac{\\omega_i(x)}{\\tr\\omega(x)} = 2\\omega(x,x+e_i)$ and\n\\begin{equation}\n\\label{eq:def-a}\na(x)=a^\\omega(x):=\\diag[a_1(x),\\ldots, a_d(x)]=\\frac{\\omega(x)}{\\tr\\omega(x)}, \\quad x\\in\\Z^d.\n\\end{equation}\nFinally, for $\\omega\\in \\Omega$, we introduce the difference operator $L_{\\omega}$ given by\n\\begin{align*}\n  L_\\omega u(x) := \\sum_{y\\in \\Z^d}\\omega(x,y)[u(y)-u(x)]=\\tfrac{1}{2}\\tr(a(x)\\nabla^2 u(x)).  \n\\end{align*}\nThat is, $L_\\omega$ is the generator of the process $(Y_t)_{t\\ge 0}$.\n\n\\subsection{Main assumptions and some notations}\nThroughout the paper, the following assumptions are always in force:\n\\begin{enumerate}[(\\text{A}1)]\n\\item $\\left\\{\\omega(x),\\, x\\in \\Z^d \\right\\}$ are i.i.d. under the probability measure $\\mb P$.\n\\item\\label{item:ellip-cond} $\\frac{\\omega}{\\tr\\omega}\\ge 2\\kappa{\\rm I}$ for $\\mb P$-almost every $\\omega$ and some constant $\\kappa\\in(0,\\tfrac{1}{2d}]$.\n\\item $\\psi(\\omega):=\\tfrac{\\zeta(\\omega)}{\\tr\\omega(0)}$ is a $L^\\infty(\\mb P)$-bounded measurable function of $\\omega(0)$.\n\\end{enumerate} \nThroughout this paper, we use $c, C$ to denote positive constants which may change from line to line but only depend on the dimension $d$ and the ellipticity constant $\\kappa$ unless stated otherwise. We write $A\n\\lesssim B$ if $A\\le CB$, and $A\\asymp B$ if $A\\lesssim B$ and $A\\gtrsim B$. \nWe also use the notations $A\\lesssim_j B$ and $A\\asymp_j B$ to indicate that the multiplicative constant depends on the variable $j$ other than $(d,\\kappa)$.\nWe will use the convention of summation over repeated indices.  \nFor example, $a_kb_k$ represents $\\sum_{k=1}^d a_kb_k$.\nThis convention is not used in definitions \\eqref{eq:def-lambda} and \\eqref{eq:def-eta}. \nFor simplicity, we use $\\tx=\\tx(\\omega)$ to denote a generic {\\it stretched-exponentially integrable} random variable (i.e., $\\mb E[\\exp(\\tx^c)]<C$) whose value may differ from line to line. \nWe write $\\tx_x(\\omega):=\\tx(\\theta_x\\omega)$. \nWe note that $\\tx$ may depend on other parameters as well, although the constants $c, C$ in the stretched-exponential moment bound only depend on $(d,\\kappa)$.\n\n\\subsection{Earlier results in the literature}\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n    \\label{eq:rho-bounds}\n    E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\nFor any integer $j\\ge 0$, let $\\fct_j$ denote the set of $j$-th order polynomials, with $\\fct_0=\\R$. In fact, in our paper, we will only use the cases $j=0,1$. \n\nThe following large-scale regularity result can be found in \\cite{GT-23}, which is a discrete version of the $C^{1,1}$ regularity estimate in \\cite[Theorem 3.1]{AL-17}.\nThis was first done in the periodic setting in \\cite{AL-87,AL-89}.\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\nIntroducing the quantities\n\\begin{equation}\\label{eq:def-mu}\n\\mu(R):=\n\\left\\{\n\\begin{array}{lr}\nR & d=2,\\\\\nR^{1/2} &d=3,\\\\\n(1\\vee \\log R)^{1/2} &d=4,\\\\\n1 &d\\ge 5,\n\\end{array}\n\\right.\n\\quad\n\\delta(R):=\\left\\{\n\\begin{array}{lr}\n(\\log (R\\vee 2))^{3/2} &d=2,\\\\\n1 &d\\ge 3,\n\\end{array}\n\\right.\n\\end{equation}\nthe following theorems on properties of the (global) correctors and quantitative homogenization of the Dirichlet problem were established in \\cite{GT-23}.\n\n\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\  (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}",
    "context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}",
    "full_context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}\n\n\\subsection{Main results}\n\n\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above. We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nLet $z:\\binB R\\to\\R$ be a solution of the Dirichlet problem\n\\[\n\\left\\{\n\\begin{array}{lr}\nL_\\omega z(x)=\\tfrac{1}{R^2}\\left[ \\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac xR)-\\tfrac{1}{2}\\bar\\eta_i\\partial_i f(\\tfrac xR)\\right] & x\\in\\inB R,\\\\\nz(x)=0 &x\\in\\partial\\inB R.\n\\end{array}\n\\right.\n\\]\nConsider the function\n\\begin{align}\\label{eq:def-w}\n&w(x)=u(x)-\\bar u(\\tfrac{x}{R})-\\tfrac 1R z(x)\\nn\\\\\n&\n+\\tfrac{1}{R^2}\\left[v^k(x)\\partial_{kk}\\bar u(\\tfrac{x}{R})\n-f(\\tfrac{x}R)\\xi(x)\\right]- {\\tfrac{1}{2}} \\tfrac{1}{R^3}\\left[\\partial_{kki}\\bar{u}(\\tfrac{x}{R})p_i^k(x)-\\partial_i f(\\tfrac{x}{R})s_i(x)\\right]\n\\end{align}\nfor $x\\in\\binB R\\subset\\bar\\B_R\\cap\\Z^d$. Then, applying the formula\n\\begin{align*}\nL_\\omega(uv)(x)&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x)+\n\\sum_{y:y\\sim x}\\omega(x,y)[u(y)-u(x)][v(y)-v(x)]\\\\\n&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x) \\\\ &\\qquad+ \\tfrac{1}{2}\na_i(x)\\left(\\nabla_{e_i}u(x)[v(x+e_i)-v(x-e_i)]\n+\\nabla_i^2u(x)\\nabla_{-e_i}v(x)\n\\right)\n\\end{align*}\nto the products within \\eqref{eq:def-w}, we obtain in $\\inB R$ that\n\\begin{align}\n&L_\\omega[v^k\\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\nn\\\\\n&=\nL_\\omega v^k \\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n+v^k \\, L_\\omega[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}\\lambda^k_i\\nabla_{e_i}[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}a_i \\,\\nabla_i^2[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\,\\nabla_{-e_i}v^k \n\\nn\\\\\n&=\\tfrac1{2}(a_k -\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+\\tfrac{1}{2}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\binB R}\\sum_{k = 1}^d|v^k|,\\label{eq:term1}\n\\end{align}\nwhere we used the fact that $|\\lambda^k_i|\\lesssim|\\nabla v^k|\\lesssim\\max_{\\binB R}|v^k|$ in $\\inB R$. Similarly,\n\\begin{align}\nL_\\omega[f(\\tfrac{\\cdot}R)\\xi ]\n&=(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}R)+{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\inB R}|\\xi|,\\label{eq:term2}\\\\\nL_\\omega[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k ]\n&=(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i,k = 1}^d|p^k_i|,\\label{eq:term3}\\\\\nL_\\omega[\\partial_i f(\\tfrac{\\cdot}{R})s_i ]\n&=(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j [s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i=1}^d|s_i|.\\label{eq:term4}\n\\end{align}\nNote that \n$\\abs{L_\\omega[\\bar u(\\tfrac{\\cdot}{R})]-\\tfrac1{2R^2}\\tr[a D^2\\bar{u}(\\tfrac{\\cdot}{R})]}\\lesssim\\tfrac1{R^4}\\norm{\\bar u}_{C^4(\\bar\\B_1)}$, and \n\\begin{align*}\nL_\\omega u \n=\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})(\\psi -\\bar\\psi)+\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})\\bar\\psi =\\tfrac1{R^2}\\big[f(\\tfrac{\\cdot}{R})L_\\omega\\xi +\n\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\\big].\n\\end{align*}\nThen, recalling the definition of $w$ in \\eqref{eq:def-w} and writing \n\\[\nA:=\\max_{\\binB R}\\sum_{i,k=1}^d\\left[|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right],\n\\]\nby \\eqref{eq:term1}, \\eqref{eq:term2},\\eqref{eq:term3}, \\eqref{eq:term4},\nwe have in $\\inB R$ that\n\\begin{align*}\n&\\abs{L_\\omega w } \\\\\n&=\n\\tfrac1{R^2}\\Abs{\nf(\\tfrac{\\cdot}{R})L_\\omega\\xi +\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})-R^2L_\\omega[\\bar{u}(\\tfrac{\\cdot}{R})]-{\\tfrac{1}{2}}\\tfrac{1}{R} \\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac 1R\\bar\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})\n\\\\\n&\\qquad+\\tfrac1{2}(a_k-\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})\n\\nn\\\\\n&\\qquad-(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}{R})-{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac{1}{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}A\\\\\n&\\qquad-{\\tfrac{1}{2}}\\tfrac{1}{R}(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n-{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\\\\n&\\qquad+{\\tfrac{1}{2}}\\tfrac{1}{R}(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j[s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\n}\n\\nn\\\\\n&\\lesssim\n\\tfrac1{R^4}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\sum_{i,k=1}^d\\left[1+\n|\\nabla p_i^k|+|\\nabla s_i|\n+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\n\\right].\n\\end{align*} \nThus, by the above inequality, the definition of $w$ from \\eqref{eq:def-w}, and the ABP maximum principle, we obtain that\n\\begin{align}\n&\\max_{\\binB R}|u -\\bar u(\\tfrac{\\cdot}{R})-\\tfrac 1R z |\\label{eq:ineq1}\\\\\n&\\le \\max_{\\binB R}|w|+\n\\max_{\\binB R}\\Abs{\\tfrac{1}{R^2}[v^k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n-f(\\tfrac{\\cdot}{R})\\xi ]\n-{\\tfrac{1}{2}}\\tfrac{1}{R^3}[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k -\\partial_i f(\\tfrac{\\cdot}{R})s_i ]}\\nn\\\\\n&\\lesssim\nR^2\\norm{L_\\omega w}_{d;\\inB R}\n+\\tfrac1{R^2}\\norm{\\bar u}_{C^3}\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\\nn\n\\\\&\n\\lesssim\n\\tfrac1{R^2}\\norm{\\bar u}_{C^6}\\sum_{i,k=1}^d\n\\big[1+\n\\left\\||\\nabla p_i^k|+|\\nabla s_i|\\right\\|_{d;B_R}+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\n\\big].\\nn\n\\end{align}\nFurthermore, by \\cite[Lemma~30]{GT-23}, \n\\[\n\\max_{\\binB R}\\abs{z-\\bar z(\\tfrac{\\cdot}{R})}\n\\lesssim\n\\tfrac1{R^2}\\max_{j,\\ell=1}^d(|\\bar\\lambda_j^\\ell|+|\\bar\\eta_j|)\\norm{\\bar u}_{C^5(\\bar\\B_1)}\n\\sum_{k=1}^d(1+R \\norm{\\nabla v^k}_{d;\\inB R}+\\osc_{\\binB R}v^k).\n\\]\nThis inequality, together with \\eqref{eq:ineq1}, yields the claimed bound \\eqref{eq:two-scale-further}.\n\\end{proof}",
    "post_theorem_intro_text_len": 3841,
    "post_theorem_intro_text": "In the i.i.d. setting with well-prepared boundary conditions, Theorem \\ref{thm:opt-iid-homo} yields improved convergence rates compared to the rate $O(R^{-1})$ in Theorem \\ref{thm:opt-quant-homo} when $d\\geq 3$.\nOur approaches and contributions are as follows.\nWe employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition \\ref{lem:two-scale-exp}.\nDue to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed. \n(Of course, this symmetry fails for environments with only a finite range of dependence.)\nThis reflection symmetry implies that the third-order homogenized tensor vanishes, which in turn suggests that a homogenization error bound sharper than\n $O(R^{-1})$ may be attainable.\n\nWe remark that such improved rates should not be expected in environments with correlations even slightly weaker than in the point-wise i.i.d.\\ case, since the distribution's reflective symmetry is no longer present.\n\nIn our i.i.d.\\ setting, without the presence of the third-order tensor, the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors. \nRoughly speaking, the (first-order) correctors “correct” local coefficients, while the higher-order correctors account for the derivatives of the lower-order correctors. \nHence, unlike the correctors themselves, the source terms in the equations for higher-order correctors are highly nonlocal, and thus the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.\n\nOur improved convergence rates in Theorem \\ref{thm:opt-iid-homo} are inspired by earlier results in the periodic setting \\cite{GST-22}, where it was proved that if the environment has ``one degree of freedom\" $a(x)$, i.e., the coefficient matrix is of the form $\\omega(x)=C+a(x)M$ for some constant symmetric matrices $C$ and $M$, then the third-order homogenized tensor vanishes, leading to an improved rate of $O(R^{-2})$.\nThe periodic setting, however, is quite rigid.\nWe provided examples in \\cite{GST-22} that show that if the environment has two degrees of freedom, then the third-order homogenized tensor does not vanish in general, and the optimal convergence rate in such cases is only $O(R^{-1})$. \n\nIn contrast, in our random setting, $\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)]$ has $d$ degrees of freedom, which is fundamentally different from the periodic case considered in \\cite{GST-22}.\nMoreover, in the periodic setting, the $C^{0,1}$-boundedness of higher-order correctors is automatically guaranteed by periodicity.\n\nOne of the main contributions of our work is to establish $C^{0,1}$ bounds for higher-order correctors in the random setting. Our approach is based on comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.\n\n\\begin{remark}\n\tWhen $d=2$, the corrector $\\upsilon^k$ grows super-linearly (Theorem~\\ref{thm:global_krt}). Consequently, via the expansion \\eqref{eq:two-scale-further}, it produces an error term of order $R^{-2}|v^k|$ that dominates the desired $O(R^{-1})$ scale. Therefore, in two dimensions, we should not expect an improvement beyond the $O(R^{-1})$ rate in i.i.d.\\ environments.\n\n    All in all, this demonstrates that, in the absence of additional structural assumptions on the random environment, the boundary conditions, or the dimension, the best convergence rate that one can expect is $O(R^{-1})$, as established in \\cite{GT-23}. We refer to Remark~\\ref{rmk:optimality} for a discussion of the optimality of the $O(R^{-1})$ rate.\n\\end{remark}",
    "sketch": "To prove Theorem~\\ref{thm:opt-iid-homo}, the authors describe the following strategy and key ingredients.\n\n- They “employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition~\\ref{lem:two-scale-exp}.”\n\n- They use a special feature of the point-wise i.i.d. setting: “Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed.” This “reflection symmetry implies that the third-order homogenized tensor vanishes,” which “suggests that a homogenization error bound sharper than $O(R^{-1})$ may be attainable.”\n\n- With “the absence of the third-order tensor,” they explain that “the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.” They motivate the difficulty: while first-order correctors “correct local coefficients,” “the higher-order correctors account for the derivatives of the lower-order correctors,” and “the source terms in the equations for higher-order correctors are highly nonlocal,” so “the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.”\n\n- A main technical step is then to “establish $C^{0,1}$ bounds for higher-order correctors in the random setting,” by “comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.”\n\n- They also note limitations: the improved rates rely on i.i.d.-driven reflection symmetry (“such improved rates should not be expected in environments with correlations… since the distribution's reflective symmetry is no longer present”), and in $d=2$ the corrector “grows super-linearly,” so the expansion yields an error term that “dominates the desired $O(R^{-1})$ scale,” hence “we should not expect an improvement beyond the $O(R^{-1})$ rate” in two dimensions.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in the following theorem.\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nsatisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\n\\end{thmx}\n\nSuppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions $u, \\bar u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nsatisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.",
    "theorem_type": [
      "Existence",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Assume \\((A1)\\), \\((A2)\\), and \\((A3)\\), and let \\(d\\ge 3\\). For a random environment \\(\\omega\\), define the spatial shift by \\(\\theta_x\\omega=\\omega(x+\\cdot)\\). Let \\(\\bar a:=E_{\\mathbb Q}[\\omega(0)/\\operatorname{tr}\\omega(0)]>0\\) and \\(\\bar\\psi:=E_{\\mathbb Q}[\\zeta/\\operatorname{tr}\\omega(0)]\\). Suppose \\(f\\in C^{4,\\alpha}(\\overline{\\mathbb B}_1)\\) and \\(g\\in C^{6,\\alpha}(\\partial\\mathbb B_1)\\) for some \\(\\alpha\\in(0,1)\\). Let \\(u\\) solve the discrete Dirichlet problem on the lattice ball \\(B_R\\subset\\mathbb Z^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\omega(x)\\nabla^2u(x))=\\frac1{R^2}f\\!\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega)\\quad \\text{for }x\\in B_R,\\qquad\nu(x)=g\\!\\left(\\tfrac{x}{R}\\right)\\quad \\text{for }x\\in \\partial B_R,\n\\]\nand let \\(\\bar u\\) solve on the unit ball \\(\\mathbb B_1\\subset\\mathbb R^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\bar a\\,D^2\\bar u(x))=f(x)\\bar\\psi\\quad \\text{for }x\\in\\mathbb B_1,\\qquad\n\\bar u(x)=g(x)\\quad \\text{for }x\\in\\partial\\mathbb B_1.\n\\]\nWhich quantitative estimate holds for the homogenization error \\(\\max_{x\\in B_R}|u(x)-\\bar u(x/R)|\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}(\\log R)\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim R^{-1}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X,\n\\qquad d\\ge 3.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\varepsilon\\in(0,1)\\), there exists a random variable \\(\\tilde X=\\tilde X(R,\\varepsilon,\\omega)>1\\) with stretched-exponential moments such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a \\(C^4(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}(\\log R)\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "dimension-dependent rates and logarithmic factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "sharp improved rates dropped in favor of the coarser \\(R^{-1}\\) decay",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "exact rate without epsilon-loss",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "required \\(C^6\\)-extension/norm and placement of logarithmic correction across dimensions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option directly. It gives the setup and asks which existence statement is valid, without embedding the sharp rate or the key existential-vs-universal extension quantifier."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct choice is essentially the precise theorem statement with its rates and extension claim. However, the alternatives do introduce competing conclusions rather than merely asking for a verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Selecting the correct answer requires some discrimination among subtle variants: sharp dimension-dependent rates, logarithmic loss placement, and the existential quantifier on the extension. Still, this is mostly recognition/recall of a known result rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and plausible. They target natural failure modes: overclaiming a stronger rate, accepting a weaker but true bound, mishandling the quantifier over extensions, and misplacing the logarithmic factor across dimensions."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests precise theorem recall/discrimination rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.04410v2",
    "paper_link": "http://arxiv.org/abs/2512.04410v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.",
      "start_pos": 19166,
      "end_pos": 19887,
      "label": "thm:opt-iid-homo"
    },
    "ref_dict": {
      "thm:opt-iid-homo": "\\begin{theorem}\n\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.\n\\end{theorem}",
      "rmk:optimality": "\\begin{remark}\\label{rmk:optimality}\nThis proposition identifies $O(R^{-1})$ as the optimal homogenization rate unless $\\bar z \\equiv 0$, and so no improvement beyond $O(R^{-1})$ should be expected in general. \n         We further expect that Proposition~\\ref{lem:two-scale-exp} remains valid for environments with finite range of dependence, for the same reasons discussed in the comments following Theorem~\\ref{thm:opt-quant-homo}.\n\\end{remark}",
      "eq:effective-ellip": "\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}",
      "thm:c11": "\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}",
      "eq:def-lambda": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}",
      "eq:elliptic-dirich": "\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}",
      "thm:opt-quant-homo": "\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}",
      "thm:global_krt": "\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}",
      "lem:two-scale-exp": "\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above.  We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}",
      "thm:QCLT": "\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}",
      "eq:def-eta": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}",
      "eq:two-scale-further": "\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 17605,
    "pre_theorem_intro_text": "\\label{sec:intro}\nIn this paper, we consider the homogenization convergence rates of purely second-order non-divergence form difference operators in an independent and identically distributed (i.i.d.) random environment on $\\Z^d$ for dimensions $d\\ge 3$.\n\nLarge-scale behavior of random walks in i.i.d.\\ environments has been intensively studied; see \\cite{BoSz-02, OZ-04, MB-11, DreRam-14, Kumagai-14} and the references therein. Since the dynamics of random walks in random environments (RWRE) can be described by difference equations, stochastic homogenization of difference equations in random environments is crucial for understanding the large-scale behavior of RWRE.\nFor environments in which spectral-gap or Efron--Stein type inequalities are available, the strong concentration properties of the medium can be exploited to obtain quantitative homogenization results; see, for example, \\cite{GNO-15, AL-17, GT-23}. In this paper, we show that, for non-divergence form difference operators in the i.i.d.\\ setting, the reflection symmetry of the law of the environment can in fact be used to extract additional cancellations. Somewhat unexpectedly, this yields homogenization rates for the Dirichlet problem that are strictly better than the standard optimal rate $O(R^{-1})$ obtained in \\cite{GT-23}. To achieve this, we require a more delicate analysis of the fluctuations of the homogenization error, relying in particular on a deeper understanding of higher-order correctors. \n\n\\subsection{Settings}\nLet $\\mb S_{d\\times d}$ denote the set of real $d\\times d$ positive definite diagonal matrices. A map \n\\[\n\\omega:\\Z^d\\to\\mb S_{d\\times d}\n\\] is called an {\\it environment}, and the set of all such environments is denoted by $\\Omega$. \nLet $\\mb P$ be a probability measure on $\\Omega$ such that \n\\[\n\\left\\{\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)], \\;x\\in\\Z^d\\right\\}\n\\] \nare i.i.d. under $\\mb P$. We denote the expectation with respect to $\\mb P$ by $\\mb E$ (or $E_{\\mb P}$). \n\n\\begin{definition}\\label{def:differences}\nLet $\\{e_1,\\ldots,e_d\\}$ be the canonical basis for $\\R^d$, and let \n\\[\nU:=\\left\\{e\\in\\Z^d:|e|=1\\right\\}=\\left\\{\\pm e_1,\\ldots,\\pm e_d\\right\\}\n\\]\nbe the set of unit vectors in $\\Z^d$. We define the difference operators $\\nabla=(\\nabla_e)_{e\\in U}$  and \n$\\nabla^2= \\mathrm{diag}[\\nabla_1^2,\\ldots,\\nabla_d^2]$ by\n\\begin{equation}\\label{eq:def-nabla}\n\\nabla_e u(x) := u(x+e)-u(x), \\qquad\n\\nabla_i^2u(x) := u(x+e_i)+u(x-e_i)-2u(x)  \n\\end{equation}\nfor $e\\in U$ and $i\\in\\{1,\\ldots,d\\}$. \nClearly, $\\nabla$ and $\\nabla^2$ are linear operators.  \n\\end{definition}\n\nFor $r>0$ and $y\\in\\R^d$, we write\n\\[\n\\B_r(y) := \\left\\{x\\in\\R^d: |x-y|<r\\right\\}, \\qquad\nB_r(y):=\\B_r(y)\\cap\\Z^d\n\\]\nto denote the continuous and discrete ball with center $y$ and radius $r$, respectively.\nWhen $y=0$, we write $\\B_r := \\B_r(0)$ and $B_r := B_r(0)$ for simplicity. \nFor any $B\\subset\\Z^d$, its {\\it discrete boundary} $\\partial B$ and {\\it discrete closure} $\\bar B$ are defined as\n\\[\n\\partial B:=\\left\\{z\\in\\Z^d\\setminus B: |z-x|=1 \\text{ for some }x\\in B\\right\\},\\qquad \\bar B := B\\cup\\partial B.\n\\]\nBy abuse of notation, we also use $\\partial A$ and $\\bar A$ to denote the usual continuous boundary and closure of $A\\subset\\R^d$, respectively, if there is no confusion.  \nThe interior of $B_R$ is denoted by\n\\[\n\\inB{R} := \\{x\\in B_R\\,:\\,  |x-y| \\ge 1 \\text{ for all }y\\in\\partial\\B_R\\}.\n\\]\nNote that for any $R\\ge 1$, there holds\n\\[\n\\bar{\\B}_{R-1}\\cap\\Z^d\\subset\\inB \nR\\subset B_R\\subset\\binB{R}\\subset\\bar\\B_R\\cap\\Z^d\\subset\\bar B_R.\n\\]\n\n For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$,  we consider the discrete {\\it non-divergence form}   elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\n\\medskip\n\nWe remark that in the continuous PDE setting, one only needs a function $g\\in C^{2,\\alpha}(\\partial\\B_1)$ on the boundary $\\partial\\B_1$ to consider Dirichlet problems of types \\eqref{eq:elliptic-dirich} and \\eqref{eq:effective-ellip}. However, in the discrete setting, \nthe discrete and continuous boundaries do not agree, and therefore the function $g$ in $\\eqref{eq:elliptic-dirich}$ should be extended to $\\frac{1}{R}\\partial\\inB R\\not\\subset\\partial\\B_1$.  This discrepancy typically induces an error of order $\\tfrac{1}{R}$ which automatically becomes the ceiling of the optimal homogenization error in the discrete setting,  cf.  \\cite{GT-23}. \nIndeed, this $O(R^{-1})$ rate is the optimal rate obtained in \\cite{GT-23} for $d\\geq 3$.\nWhereas, when such a discretization issue is not present, e.g., in the continuous periodic setting \\cite{GTY-19, ST-21, Spr-24},  one needs to go deeper into higher order correctors to justify the optimality of the rate $O(R^{-1})$ for the homogenization of the Dirichlet problem. We refer to \\cite{CSS-20, FGP-24, SSZ-25} for recent developments in numerical homogenization of PDEs in non-divergence form.\n\n\\medskip\n\nA goal of this article is to isolate and examine the homogenization rate that does {\\bf not} originate from boundary discretization.\nTo this end, we assume that the boundary data $g$ is properly extended to be a function on $\\frac{1}{R}\\partial\\inB R$ in the Dirichlet problem \\eqref{eq:elliptic-dirich}.\n\n\\medskip\n\nThe non-divergence form difference equation \\eqref{eq:elliptic-dirich} is used to describe random walks in a random environment (RWRE) in $\\Z^d$. To be specific, for $x,y\\in \\Z^d$, we set\n\\begin{equation}\\label{def:omegaxei}\n\\omega(x,x\\pm e_i):=\\frac{\\omega_i(x)}{2\\,\\tr\\omega(x)} \\quad  \\text{ for } i=1,\\ldots, d,\n\\end{equation}\n and \n \\[\n \\omega(x,y) := 0 \\quad \\text{ if $|x-y|\\neq 1$},\n \\]\n that is, we normalize $\\omega$ to get a transition probability.  \n We note that the configuration of $\\{\\omega(x,y):x,y\\in\\Z^d\\}$ is also called a {\\it balanced environment} in the literature \\cite{L-82,GZ-12,BD-14,DGR-15, BCDG-18,DG-19}.\n\\begin{definition}\\label{def:rwre-discrete}\nFor each fixed $\\omega\\in\\Omega$, the random walk  $(X_n)_{n\\ge 0}$ in the environment $\\omega$ with $X_0=x$ is a Markov chain  in $\\Z^d$ with transition probability $P_\\omega^x$ specified by\n\\begin{equation}\\label{eq:def-RW}\nP_\\omega^x\\left(X_{n+1}=z|X_n=y\\right) := \\omega(y,z).\n\\end{equation}\nThe {\\it continuous-time} RWRE $(Y_t)_{t\\ge 0}$ in $\\omega$ is defined to be the Markov process on $\\Z^d$ with jump rates $\\omega(x,y)$ from $x$ to $y$. With abuse of notations, we still denote by $P_\\omega^x$  the quenched law of $(Y_t)$ if there is no ambiguity from the context.\n\\end{definition} \nThe expectation with respect to $P_\\omega^x$ is denoted by $E_\\omega^x$. When the starting point of the random walk is $0$, we sometimes omit the superscript and write $P_\\omega^0$ and $E_\\omega^0$ as $P_\\omega$ and $E_\\omega$, respectively. Note that for random walks $(X_n)$ in an environment $\\omega$, we have that\n\\begin{equation}\\label{def:omegabar}\n\\bar\\omega^i := \\theta_{X_i}\\omega\\in\\Omega, \\qquad i\\ge 0,\n\\end{equation}\nis also a Markov chain, called the {\\it environment viewed from the particle} process. By abuse of notation, we enlarge our probability space so that $P_\\omega$ still denotes the joint law of the random walks and $(\\bar\\omega^i)_{i\\ge 0}$. \n\nFor $\\omega\\in\\Omega$ and $i\\in \\{1,\\ldots,d\\}$, we set $a_i(x)=a_i^\\omega(x) := \\frac{\\omega_i(x)}{\\tr\\omega(x)} = 2\\omega(x,x+e_i)$ and\n\\begin{equation}\n\\label{eq:def-a}\na(x)=a^\\omega(x):=\\diag[a_1(x),\\ldots, a_d(x)]=\\frac{\\omega(x)}{\\tr\\omega(x)}, \\quad x\\in\\Z^d.\n\\end{equation}\nFinally, for $\\omega\\in \\Omega$, we introduce the difference operator $L_{\\omega}$ given by\n\\begin{align*}\n  L_\\omega u(x) := \\sum_{y\\in \\Z^d}\\omega(x,y)[u(y)-u(x)]=\\tfrac{1}{2}\\tr(a(x)\\nabla^2 u(x)).  \n\\end{align*}\nThat is, $L_\\omega$ is the generator of the process $(Y_t)_{t\\ge 0}$.\n\n\\subsection{Main assumptions and some notations}\nThroughout the paper, the following assumptions are always in force:\n\\begin{enumerate}[(\\text{A}1)]\n\\item $\\left\\{\\omega(x),\\, x\\in \\Z^d \\right\\}$ are i.i.d. under the probability measure $\\mb P$.\n\\item\\label{item:ellip-cond} $\\frac{\\omega}{\\tr\\omega}\\ge 2\\kappa{\\rm I}$ for $\\mb P$-almost every $\\omega$ and some constant $\\kappa\\in(0,\\tfrac{1}{2d}]$.\n\\item $\\psi(\\omega):=\\tfrac{\\zeta(\\omega)}{\\tr\\omega(0)}$ is a $L^\\infty(\\mb P)$-bounded measurable function of $\\omega(0)$.\n\\end{enumerate} \nThroughout this paper, we use $c, C$ to denote positive constants which may change from line to line but only depend on the dimension $d$ and the ellipticity constant $\\kappa$ unless stated otherwise. We write $A\n\\lesssim B$ if $A\\le CB$, and $A\\asymp B$ if $A\\lesssim B$ and $A\\gtrsim B$. \nWe also use the notations $A\\lesssim_j B$ and $A\\asymp_j B$ to indicate that the multiplicative constant depends on the variable $j$ other than $(d,\\kappa)$.\nWe will use the convention of summation over repeated indices.  \nFor example, $a_kb_k$ represents $\\sum_{k=1}^d a_kb_k$.\nThis convention is not used in definitions \\eqref{eq:def-lambda} and \\eqref{eq:def-eta}. \nFor simplicity, we use $\\tx=\\tx(\\omega)$ to denote a generic {\\it stretched-exponentially integrable} random variable (i.e., $\\mb E[\\exp(\\tx^c)]<C$) whose value may differ from line to line. \nWe write $\\tx_x(\\omega):=\\tx(\\theta_x\\omega)$. \nWe note that $\\tx$ may depend on other parameters as well, although the constants $c, C$ in the stretched-exponential moment bound only depend on $(d,\\kappa)$.\n\n\\subsection{Earlier results in the literature}\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n    \\label{eq:rho-bounds}\n    E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\nFor any integer $j\\ge 0$, let $\\fct_j$ denote the set of $j$-th order polynomials, with $\\fct_0=\\R$. In fact, in our paper, we will only use the cases $j=0,1$. \n\nThe following large-scale regularity result can be found in \\cite{GT-23}, which is a discrete version of the $C^{1,1}$ regularity estimate in \\cite[Theorem 3.1]{AL-17}.\nThis was first done in the periodic setting in \\cite{AL-87,AL-89}.\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\nIntroducing the quantities\n\\begin{equation}\\label{eq:def-mu}\n\\mu(R):=\n\\left\\{\n\\begin{array}{lr}\nR & d=2,\\\\\nR^{1/2} &d=3,\\\\\n(1\\vee \\log R)^{1/2} &d=4,\\\\\n1 &d\\ge 5,\n\\end{array}\n\\right.\n\\quad\n\\delta(R):=\\left\\{\n\\begin{array}{lr}\n(\\log (R\\vee 2))^{3/2} &d=2,\\\\\n1 &d\\ge 3,\n\\end{array}\n\\right.\n\\end{equation}\nthe following theorems on properties of the (global) correctors and quantitative homogenization of the Dirichlet problem were established in \\cite{GT-23}.\n\n\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\  (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}",
    "context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}",
    "full_context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}\n\n\\subsection{Main results}\n\n\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above. We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nLet $z:\\binB R\\to\\R$ be a solution of the Dirichlet problem\n\\[\n\\left\\{\n\\begin{array}{lr}\nL_\\omega z(x)=\\tfrac{1}{R^2}\\left[ \\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac xR)-\\tfrac{1}{2}\\bar\\eta_i\\partial_i f(\\tfrac xR)\\right] & x\\in\\inB R,\\\\\nz(x)=0 &x\\in\\partial\\inB R.\n\\end{array}\n\\right.\n\\]\nConsider the function\n\\begin{align}\\label{eq:def-w}\n&w(x)=u(x)-\\bar u(\\tfrac{x}{R})-\\tfrac 1R z(x)\\nn\\\\\n&\n+\\tfrac{1}{R^2}\\left[v^k(x)\\partial_{kk}\\bar u(\\tfrac{x}{R})\n-f(\\tfrac{x}R)\\xi(x)\\right]- {\\tfrac{1}{2}} \\tfrac{1}{R^3}\\left[\\partial_{kki}\\bar{u}(\\tfrac{x}{R})p_i^k(x)-\\partial_i f(\\tfrac{x}{R})s_i(x)\\right]\n\\end{align}\nfor $x\\in\\binB R\\subset\\bar\\B_R\\cap\\Z^d$. Then, applying the formula\n\\begin{align*}\nL_\\omega(uv)(x)&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x)+\n\\sum_{y:y\\sim x}\\omega(x,y)[u(y)-u(x)][v(y)-v(x)]\\\\\n&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x) \\\\ &\\qquad+ \\tfrac{1}{2}\na_i(x)\\left(\\nabla_{e_i}u(x)[v(x+e_i)-v(x-e_i)]\n+\\nabla_i^2u(x)\\nabla_{-e_i}v(x)\n\\right)\n\\end{align*}\nto the products within \\eqref{eq:def-w}, we obtain in $\\inB R$ that\n\\begin{align}\n&L_\\omega[v^k\\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\nn\\\\\n&=\nL_\\omega v^k \\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n+v^k \\, L_\\omega[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}\\lambda^k_i\\nabla_{e_i}[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}a_i \\,\\nabla_i^2[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\,\\nabla_{-e_i}v^k \n\\nn\\\\\n&=\\tfrac1{2}(a_k -\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+\\tfrac{1}{2}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\binB R}\\sum_{k = 1}^d|v^k|,\\label{eq:term1}\n\\end{align}\nwhere we used the fact that $|\\lambda^k_i|\\lesssim|\\nabla v^k|\\lesssim\\max_{\\binB R}|v^k|$ in $\\inB R$. Similarly,\n\\begin{align}\nL_\\omega[f(\\tfrac{\\cdot}R)\\xi ]\n&=(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}R)+{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\inB R}|\\xi|,\\label{eq:term2}\\\\\nL_\\omega[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k ]\n&=(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i,k = 1}^d|p^k_i|,\\label{eq:term3}\\\\\nL_\\omega[\\partial_i f(\\tfrac{\\cdot}{R})s_i ]\n&=(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j [s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i=1}^d|s_i|.\\label{eq:term4}\n\\end{align}\nNote that \n$\\abs{L_\\omega[\\bar u(\\tfrac{\\cdot}{R})]-\\tfrac1{2R^2}\\tr[a D^2\\bar{u}(\\tfrac{\\cdot}{R})]}\\lesssim\\tfrac1{R^4}\\norm{\\bar u}_{C^4(\\bar\\B_1)}$, and \n\\begin{align*}\nL_\\omega u \n=\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})(\\psi -\\bar\\psi)+\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})\\bar\\psi =\\tfrac1{R^2}\\big[f(\\tfrac{\\cdot}{R})L_\\omega\\xi +\n\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\\big].\n\\end{align*}\nThen, recalling the definition of $w$ in \\eqref{eq:def-w} and writing \n\\[\nA:=\\max_{\\binB R}\\sum_{i,k=1}^d\\left[|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right],\n\\]\nby \\eqref{eq:term1}, \\eqref{eq:term2},\\eqref{eq:term3}, \\eqref{eq:term4},\nwe have in $\\inB R$ that\n\\begin{align*}\n&\\abs{L_\\omega w } \\\\\n&=\n\\tfrac1{R^2}\\Abs{\nf(\\tfrac{\\cdot}{R})L_\\omega\\xi +\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})-R^2L_\\omega[\\bar{u}(\\tfrac{\\cdot}{R})]-{\\tfrac{1}{2}}\\tfrac{1}{R} \\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac 1R\\bar\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})\n\\\\\n&\\qquad+\\tfrac1{2}(a_k-\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})\n\\nn\\\\\n&\\qquad-(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}{R})-{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac{1}{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}A\\\\\n&\\qquad-{\\tfrac{1}{2}}\\tfrac{1}{R}(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n-{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\\\\n&\\qquad+{\\tfrac{1}{2}}\\tfrac{1}{R}(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j[s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\n}\n\\nn\\\\\n&\\lesssim\n\\tfrac1{R^4}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\sum_{i,k=1}^d\\left[1+\n|\\nabla p_i^k|+|\\nabla s_i|\n+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\n\\right].\n\\end{align*} \nThus, by the above inequality, the definition of $w$ from \\eqref{eq:def-w}, and the ABP maximum principle, we obtain that\n\\begin{align}\n&\\max_{\\binB R}|u -\\bar u(\\tfrac{\\cdot}{R})-\\tfrac 1R z |\\label{eq:ineq1}\\\\\n&\\le \\max_{\\binB R}|w|+\n\\max_{\\binB R}\\Abs{\\tfrac{1}{R^2}[v^k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n-f(\\tfrac{\\cdot}{R})\\xi ]\n-{\\tfrac{1}{2}}\\tfrac{1}{R^3}[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k -\\partial_i f(\\tfrac{\\cdot}{R})s_i ]}\\nn\\\\\n&\\lesssim\nR^2\\norm{L_\\omega w}_{d;\\inB R}\n+\\tfrac1{R^2}\\norm{\\bar u}_{C^3}\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\\nn\n\\\\&\n\\lesssim\n\\tfrac1{R^2}\\norm{\\bar u}_{C^6}\\sum_{i,k=1}^d\n\\big[1+\n\\left\\||\\nabla p_i^k|+|\\nabla s_i|\\right\\|_{d;B_R}+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\n\\big].\\nn\n\\end{align}\nFurthermore, by \\cite[Lemma~30]{GT-23}, \n\\[\n\\max_{\\binB R}\\abs{z-\\bar z(\\tfrac{\\cdot}{R})}\n\\lesssim\n\\tfrac1{R^2}\\max_{j,\\ell=1}^d(|\\bar\\lambda_j^\\ell|+|\\bar\\eta_j|)\\norm{\\bar u}_{C^5(\\bar\\B_1)}\n\\sum_{k=1}^d(1+R \\norm{\\nabla v^k}_{d;\\inB R}+\\osc_{\\binB R}v^k).\n\\]\nThis inequality, together with \\eqref{eq:ineq1}, yields the claimed bound \\eqref{eq:two-scale-further}.\n\\end{proof}",
    "post_theorem_intro_text_len": 3841,
    "post_theorem_intro_text": "In the i.i.d. setting with well-prepared boundary conditions, Theorem \\ref{thm:opt-iid-homo} yields improved convergence rates compared to the rate $O(R^{-1})$ in Theorem \\ref{thm:opt-quant-homo} when $d\\geq 3$.\nOur approaches and contributions are as follows.\nWe employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition \\ref{lem:two-scale-exp}.\nDue to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed. \n(Of course, this symmetry fails for environments with only a finite range of dependence.)\nThis reflection symmetry implies that the third-order homogenized tensor vanishes, which in turn suggests that a homogenization error bound sharper than\n $O(R^{-1})$ may be attainable.\n\nWe remark that such improved rates should not be expected in environments with correlations even slightly weaker than in the point-wise i.i.d.\\ case, since the distribution's reflective symmetry is no longer present.\n\nIn our i.i.d.\\ setting, without the presence of the third-order tensor, the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors. \nRoughly speaking, the (first-order) correctors “correct” local coefficients, while the higher-order correctors account for the derivatives of the lower-order correctors. \nHence, unlike the correctors themselves, the source terms in the equations for higher-order correctors are highly nonlocal, and thus the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.\n\nOur improved convergence rates in Theorem \\ref{thm:opt-iid-homo} are inspired by earlier results in the periodic setting \\cite{GST-22}, where it was proved that if the environment has ``one degree of freedom\" $a(x)$, i.e., the coefficient matrix is of the form $\\omega(x)=C+a(x)M$ for some constant symmetric matrices $C$ and $M$, then the third-order homogenized tensor vanishes, leading to an improved rate of $O(R^{-2})$.\nThe periodic setting, however, is quite rigid.\nWe provided examples in \\cite{GST-22} that show that if the environment has two degrees of freedom, then the third-order homogenized tensor does not vanish in general, and the optimal convergence rate in such cases is only $O(R^{-1})$. \n\nIn contrast, in our random setting, $\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)]$ has $d$ degrees of freedom, which is fundamentally different from the periodic case considered in \\cite{GST-22}.\nMoreover, in the periodic setting, the $C^{0,1}$-boundedness of higher-order correctors is automatically guaranteed by periodicity.\n\nOne of the main contributions of our work is to establish $C^{0,1}$ bounds for higher-order correctors in the random setting. Our approach is based on comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.\n\n\\begin{remark}\n\tWhen $d=2$, the corrector $\\upsilon^k$ grows super-linearly (Theorem~\\ref{thm:global_krt}). Consequently, via the expansion \\eqref{eq:two-scale-further}, it produces an error term of order $R^{-2}|v^k|$ that dominates the desired $O(R^{-1})$ scale. Therefore, in two dimensions, we should not expect an improvement beyond the $O(R^{-1})$ rate in i.i.d.\\ environments.\n\n    All in all, this demonstrates that, in the absence of additional structural assumptions on the random environment, the boundary conditions, or the dimension, the best convergence rate that one can expect is $O(R^{-1})$, as established in \\cite{GT-23}. We refer to Remark~\\ref{rmk:optimality} for a discussion of the optimality of the $O(R^{-1})$ rate.\n\\end{remark}",
    "sketch": "To prove Theorem~\\ref{thm:opt-iid-homo}, the authors describe the following strategy and key ingredients.\n\n- They “employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition~\\ref{lem:two-scale-exp}.”\n\n- They use a special feature of the point-wise i.i.d. setting: “Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed.” This “reflection symmetry implies that the third-order homogenized tensor vanishes,” which “suggests that a homogenization error bound sharper than $O(R^{-1})$ may be attainable.”\n\n- With “the absence of the third-order tensor,” they explain that “the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.” They motivate the difficulty: while first-order correctors “correct local coefficients,” “the higher-order correctors account for the derivatives of the lower-order correctors,” and “the source terms in the equations for higher-order correctors are highly nonlocal,” so “the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.”\n\n- A main technical step is then to “establish $C^{0,1}$ bounds for higher-order correctors in the random setting,” by “comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.”\n\n- They also note limitations: the improved rates rely on i.i.d.-driven reflection symmetry (“such improved rates should not be expected in environments with correlations… since the distribution's reflective symmetry is no longer present”), and in $d=2$ the corrector “grows super-linearly,” so the expansion yields an error term that “dominates the desired $O(R^{-1})$ scale,” hence “we should not expect an improvement beyond the $O(R^{-1})$ rate” in two dimensions.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in the following theorem.\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nsatisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\n\\end{thmx}\n\nSuppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions $u, \\bar u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nsatisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.",
    "theorem_type": [
      "Existence",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Assume \\((A1)\\), \\((A2)\\), and \\((A3)\\), and let \\(d\\ge 3\\). For a random environment \\(\\omega\\), define the spatial shift by \\(\\theta_x\\omega=\\omega(x+\\cdot)\\). Let \\(\\bar a:=E_{\\mathbb Q}[\\omega(0)/\\operatorname{tr}\\omega(0)]>0\\) and \\(\\bar\\psi:=E_{\\mathbb Q}[\\zeta/\\operatorname{tr}\\omega(0)]\\). Suppose \\(f\\in C^{4,\\alpha}(\\overline{\\mathbb B}_1)\\) and \\(g\\in C^{6,\\alpha}(\\partial\\mathbb B_1)\\) for some \\(\\alpha\\in(0,1)\\). Let \\(u\\) solve the discrete Dirichlet problem on the lattice ball \\(B_R\\subset\\mathbb Z^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\omega(x)\\nabla^2u(x))=\\frac1{R^2}f\\!\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega)\\quad \\text{for }x\\in B_R,\\qquad\nu(x)=g\\!\\left(\\tfrac{x}{R}\\right)\\quad \\text{for }x\\in \\partial B_R,\n\\]\nand let \\(\\bar u\\) solve on the unit ball \\(\\mathbb B_1\\subset\\mathbb R^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\bar a\\,D^2\\bar u(x))=f(x)\\bar\\psi\\quad \\text{for }x\\in\\mathbb B_1,\\qquad\n\\bar u(x)=g(x)\\quad \\text{for }x\\in\\partial\\mathbb B_1.\n\\]\nWhich quantitative estimate holds for the homogenization error \\(\\max_{x\\in B_R}|u(x)-\\bar u(x/R)|\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}(\\log R)\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim R^{-1}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X,\n\\qquad d\\ge 3.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\varepsilon\\in(0,1)\\), there exists a random variable \\(\\tilde X=\\tilde X(R,\\varepsilon,\\omega)>1\\) with stretched-exponential moments such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a \\(C^4(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}(\\log R)\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "dimension-dependent rates and logarithmic factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "sharp improved rates dropped in favor of the coarser \\(R^{-1}\\) decay",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "exact rate without epsilon-loss",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "required \\(C^6\\)-extension/norm and placement of logarithmic correction across dimensions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the target error bound; it asks for the conclusion. Although the setup mirrors the theorem hypotheses closely, the correct rate/log-factor is not directly leaked."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem reproduces the hypotheses and asks for the exact quantitative estimate that the theorem concludes. It is very close to a direct restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "A test-taker must distinguish among nearby rates, dimension dependence, logarithmic corrections, epsilon-loss, and regularity norms. However, this is mostly recognition/recall of the theorem statement rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic confusions: sharper-but-false rates, weaker true but non-sharp bounds, epsilon-loss variants, and altered regularity/log placements. They are distinct and well aligned with common failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-constructed theorem-identification MCQ with strong distractors, but it is largely a near-verbatim recall of a specific homogenization estimate rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.04422v1",
    "paper_link": "http://arxiv.org/abs/2512.04422v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.",
      "start_pos": 10880,
      "end_pos": 11538,
      "label": "thm:thm1"
    },
    "ref_dict": {
      "eq:general": "\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}",
      "eq:heattrace": "\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}",
      "eq:form": "\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}",
      "thm:thm1": "\\begin{theorem}\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.\n\\end{theorem}",
      "thm:moreid": "\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\fp$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}",
      "eq:a-def": "\\begin{equation}\\label{eq:a-def}\n    a(R,\\theta,R',\\theta')\n    =\\frac12\\,\\exp \\Big[-\\frac{R^2+(R')^2}{4}\\Big]\n    \\sum_{j=1}^\\infty I_{\\mu_j} \\Big(\\frac{RR'}{2}\\Big)\\,\n    \\varphi_j(\\theta)\\,\\varphi_j(\\theta').\n\\end{equation}",
      "eq:bexpression-clean": "\\begin{equation}\\label{eq:bexpression-clean}\n    \\begin{aligned}\n    b(R,\\theta,R',\\theta')&=\n    \\Big(\\tfrac12 D^2S\\Big)\\,a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,4 R\\cos(\\theta+\\theta_0)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,\\tfrac{1}{2}\\,\\Delta_{R,\\theta}\\big(D^2S\\cdot a(R,\\theta,R',\\theta')\\big)\n    -\\tfrac{1}{2}\\,(D^2S)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta').\n    \\end{aligned}\n    \\end{equation}",
      "eq:specialpi2": "\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 2402,
    "pre_theorem_intro_text": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.",
    "context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.",
    "full_context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$ because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\begin{proposition}\n Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n \\begin{equation}\n \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.\n\n\\begin{lemma}\\label{lem:local-corner}\nLet $\\Omega$ and $\\Omega'$ be domains with a single corner having the same opening angle\n$\\alpha$ and the same one‑sided boundary curvatures $\\kappa_\\pm$ at that corner.\nThen $H^D_\\Omega-H^D_{\\Omega'}$ has order $\\le T^{0}$ (at worst $T^0 = t^0$) at ff.\nEquivalently, their ff coefficients of orders $T^{-2} = t^{-1}$ and $T^{-1} = t^{-1/2}$ agree.\n\\end{lemma}\n\nWe can now prove Theorem \\ref{thm:thm1}.\n\\begin{proof}[Proof of Theorem \\ref{thm:thm1}] As in \\cite{NRS}, the heat trace is obtained by restricting $H^{D}_{\\Omega}$ to the spatial diagonal and then integrating over $z\\in\\Omega$. The technical tool used is Melrose's pushforward theorem. We will not repeat all the details here, but the upshot is that each face td, sf, and ff gives a separate contribution to the expansion \\eqref{eq:heattrace}. A priori, the expansions can interact to give logarithmic terms, but for the same reasons as in \\cite[p. 49]{NRS}, keeping in mind that we now have two orders at ff rather than one, there are no logarithmic terms until at least $O(T^2\\log T)=O(t\\log t)$ (and potentially much later, if at all). The contribution from td is \n\\[\\frac{|\\Omega|}{4\\pi t} + O(t^{\\infty}).\\]\nThe contributions from sf are\n\\[-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\int_{\\partial\\Omega}\\kappa\\, ds +\\frac{\\sqrt t}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + O(t).\\]\nAnd the contributions from ff, since the first is already known and the second depends only on $\\alpha$ and $\\kappa_{\\pm}$, are\n\\[\\sum_{j=1}^{n}\\frac{\\pi^2-\\alpha_j^2}{24\\pi\\alpha_j} + \\sqrt t\\sum_{j=1}^{n}\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-}) + O(t),\\]\nfor some unknown function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$. Summing the three faces gives Theorem \\ref{thm:thm1}.\n\\end{proof}\n\nFor $H(t)$, we use the asymptotics of Bessel function zeroes due to McMahon \\cite[(10.21.19)]{NIST:DLMF}:\n\\begin{equation}\nj_{0,k}= (k-\\frac 14)\\pi + \\frac{1}{8(k-\\frac 14)\\pi} + O(k^{-3}),\n\\end{equation}\nwhich imply\n\\begin{equation}\nj_{0,k}^{2} = (k-\\frac 14)^2\\pi^2 + \\frac 14 + O(k^{-2}).\n\\end{equation}\nBased on this we define the function\n\\begin{equation}\n\\tilde H(t) = \\sum_{k=1}^{\\infty} \\exp[-t((k-\\frac 14)^2\\pi^2 + \\frac 14)]\n\\end{equation}\nand estimate its difference with $H(t)$.\n\\begin{lemma}\\label{lem:difference} With all notation as above, as $t\\to 0$,\n\\begin{equation}\nH(t) - \\tilde H(t) = O(t).\n\\end{equation}\n\\end{lemma}\nThis is helpful because we also have the following lemma.\n\\begin{lemma}\\label{lem:compute} As $t\\to 0$,\n\\begin{equation}\n\\tilde H(t) = \\frac{1}{2\\sqrt\\pi}t^{-1/2} + c - \\frac{1}{8\\sqrt\\pi}t^{1/2}+ O(t).\n\\end{equation}\nHere $c$ is a constant which is irrelevant to our purposes.\n\\end{lemma}\nFrom these two Lemmas and \\eqref{eq:relationship} we immediately deduce this proposition.\n\\begin{proposition} As $t\\to 0$,\n\\begin{equation}\n\\Tr e^{-t\\Delta_{\\Omega}} = \\frac 18 t^{-1} - (\\frac{\\sqrt\\pi}{8}+\\frac{1}{4\\sqrt\\pi})t^{-1/2}-\\frac 12c + (\\frac{\\sqrt\\pi}{256}+\\frac{1}{16\\sqrt\\pi})t^{1/2}+ O(t).\n\\end{equation}\n\\end{proposition}\n\\begin{remark} We could reverse engineer $c$ from the known expansion from \\cite{NRS} but there is no need.\n\\end{remark}",
    "post_theorem_intro_text_len": 4928,
    "post_theorem_intro_text": "We can say substantially more about $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ whenever $\\alpha\\ne\\pi$. Define\n\\begin{equation}\\label{eq:defofr0}\n    r_0=r_0(\\alpha,\\kappa_{+},\\kappa_-)=\\frac 12\\csc\\alpha\\sqrt{\\kappa_+^2+\\kappa_-^2-2\\kappa_+\\kappa_-\\cos\\alpha}.\n\\end{equation}\n\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\operatorname*{fp} \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\,\\rho\\,d\\rho\\,d\\phi\\,\\,R\\,dR\\,d\\theta.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\operatorname*{fp}$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$  because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\subsection{Inverse spectral applications}\n\nIn \\cite{EGS_2017}, the authors prove that any curvilinear polygon with straight corners is in fact isospectral to a polygon. They require straight corners because of a lack of a formula in the curved corner case. We can strengthen their result.\n\\begin{definition}\n    A curvilinear polygon $\\Omega$ is \\emph{admissible} if, for each $j$, either $c_{1/2}(\\alpha_j)>0$  or $\\Omega$ is straight near $P_j$.\n\\end{definition}\n\n\\begin{proposition}\n    Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n    \\begin{equation}\n    \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n    \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.",
    "sketch": "In the “Plan of the paper” the authors outline the proof of Theorem~\\ref{thm:thm1}: in Section~\\ref{sec:gm} they “examin[e] the geometric microlocal description of the heat kernel given in \\cite{NRS},” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” in Section~\\ref{sec:conformal} they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in \\cite{NRS},” which “allows [them] to obtain both \\eqref{eq:form} and \\eqref{eq:general}.” Finally, in Section~\\ref{sec:specialpi2} they prove \\eqref{eq:specialpi2} “by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.”",
    "expanded_sketch": "In the “Plan of the paper” the authors outline the proof of the main theorem: next they “examin[e] the geometric microlocal description of the heat kernel given in NRS,” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in NRS,” which “allows [them] to obtain both\n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\\\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\\\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}.\nFinally, they prove\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n“by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}.”,",
    "expanded_theorem": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}..",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subseteq\\mathbb R^2\\) be a curvilinear polygon: its boundary is piecewise smooth, smooth except at finitely many corners \\(P_1,\\dots,P_n\\), where the interior angle at \\(P_j\\) is \\(\\alpha_j>0\\). Let \\(\\kappa\\) denote the inward-pointing curvature along the smooth boundary arcs, and for each corner let \\(\\kappa_{j,+}\\) and \\(\\kappa_{j,-}\\) be the one-sided limits of \\(\\kappa\\) as the corner is approached along the two incident sides. Let \\(H^D_{\\Omega}(t)\\) be the Dirichlet heat trace of \\(\\Omega\\). As \\(t\\to 0\\), which asymptotic expansion with error bound is valid?",
      "correct_choice": {
        "label": "A",
        "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided limiting curvatures \\(\\kappa_\\pm\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j)\\sqrt{\\kappa_{j,+}^2+\\kappa_{j,-}^2}\\Big)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha)\\) depends only on the corner angle."
        },
        {
          "label": "C",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+O(\\sqrt t),\\]\\nas \\(t\\to 0\\)."
        },
        {
          "label": "D",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided limiting curvatures \\(\\kappa_\\pm\\)."
        },
        {
          "label": "E",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t^{3/2}),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) may depend on the global geometry of \\(\\Omega\\) in addition to \\(\\alpha\\) and \\(\\kappa_\\pm\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "corner-term dependence on full pair \\((\\kappa_+,\\kappa_-)\\) rather than a forced factorization through Euclidean norm",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "explicit \\(\\sqrt t\\)-coefficient and sharper remainder \\(O(t\\log t)\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "error term improved from first guaranteed \\(O(t\\log t)\\) to \\(O(t)\\)",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "uniform local determination of corner contribution by \\(\\alpha\\) and \\(\\kappa_\\pm\\)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the correct expansion. Although it introduces the one-sided curvature limits \\(\\kappa_{j,\\pm}\\), that does not single out the correct option, since multiple choices also use them."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific asymptotic theorem: the stem gives the full setup and asks which expansion holds. The correct option is basically the theorem statement itself, with distractors formed by small perturbations."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish subtle alternatives, such as dependence on \\(\\kappa_{j,+},\\kappa_{j,-}\\) separately and the remainder term \\(O(t\\log t)\\) versus \\(O(t)\\). However, this is mainly theorem recognition/recall rather than generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening to \\(O(\\sqrt t)\\), oversharpening to \\(O(t)\\), and confusing the corner coefficient’s dependence on angle alone or on \\(\\kappa_++\\kappa_-\\) only."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall MCQ with strong distractors, but it is largely a theorem-restatement item rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.04422v1",
    "paper_link": "http://arxiv.org/abs/2512.04422v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.",
      "start_pos": 10880,
      "end_pos": 11538,
      "label": "thm:thm1"
    },
    "ref_dict": {
      "eq:general": "\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}",
      "eq:heattrace": "\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}",
      "eq:form": "\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}",
      "thm:thm1": "\\begin{theorem}\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.\n\\end{theorem}",
      "thm:moreid": "\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\fp$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}",
      "eq:a-def": "\\begin{equation}\\label{eq:a-def}\n    a(R,\\theta,R',\\theta')\n    =\\frac12\\,\\exp \\Big[-\\frac{R^2+(R')^2}{4}\\Big]\n    \\sum_{j=1}^\\infty I_{\\mu_j} \\Big(\\frac{RR'}{2}\\Big)\\,\n    \\varphi_j(\\theta)\\,\\varphi_j(\\theta').\n\\end{equation}",
      "eq:bexpression-clean": "\\begin{equation}\\label{eq:bexpression-clean}\n    \\begin{aligned}\n    b(R,\\theta,R',\\theta')&=\n    \\Big(\\tfrac12 D^2S\\Big)\\,a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,4 R\\cos(\\theta+\\theta_0)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,\\tfrac{1}{2}\\,\\Delta_{R,\\theta}\\big(D^2S\\cdot a(R,\\theta,R',\\theta')\\big)\n    -\\tfrac{1}{2}\\,(D^2S)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta').\n    \\end{aligned}\n    \\end{equation}",
      "eq:specialpi2": "\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 2402,
    "pre_theorem_intro_text": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.",
    "context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.",
    "full_context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$ because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\begin{proposition}\n Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n \\begin{equation}\n \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.\n\n\\begin{lemma}\\label{lem:local-corner}\nLet $\\Omega$ and $\\Omega'$ be domains with a single corner having the same opening angle\n$\\alpha$ and the same one‑sided boundary curvatures $\\kappa_\\pm$ at that corner.\nThen $H^D_\\Omega-H^D_{\\Omega'}$ has order $\\le T^{0}$ (at worst $T^0 = t^0$) at ff.\nEquivalently, their ff coefficients of orders $T^{-2} = t^{-1}$ and $T^{-1} = t^{-1/2}$ agree.\n\\end{lemma}\n\nWe can now prove Theorem \\ref{thm:thm1}.\n\\begin{proof}[Proof of Theorem \\ref{thm:thm1}] As in \\cite{NRS}, the heat trace is obtained by restricting $H^{D}_{\\Omega}$ to the spatial diagonal and then integrating over $z\\in\\Omega$. The technical tool used is Melrose's pushforward theorem. We will not repeat all the details here, but the upshot is that each face td, sf, and ff gives a separate contribution to the expansion \\eqref{eq:heattrace}. A priori, the expansions can interact to give logarithmic terms, but for the same reasons as in \\cite[p. 49]{NRS}, keeping in mind that we now have two orders at ff rather than one, there are no logarithmic terms until at least $O(T^2\\log T)=O(t\\log t)$ (and potentially much later, if at all). The contribution from td is \n\\[\\frac{|\\Omega|}{4\\pi t} + O(t^{\\infty}).\\]\nThe contributions from sf are\n\\[-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\int_{\\partial\\Omega}\\kappa\\, ds +\\frac{\\sqrt t}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + O(t).\\]\nAnd the contributions from ff, since the first is already known and the second depends only on $\\alpha$ and $\\kappa_{\\pm}$, are\n\\[\\sum_{j=1}^{n}\\frac{\\pi^2-\\alpha_j^2}{24\\pi\\alpha_j} + \\sqrt t\\sum_{j=1}^{n}\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-}) + O(t),\\]\nfor some unknown function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$. Summing the three faces gives Theorem \\ref{thm:thm1}.\n\\end{proof}\n\nFor $H(t)$, we use the asymptotics of Bessel function zeroes due to McMahon \\cite[(10.21.19)]{NIST:DLMF}:\n\\begin{equation}\nj_{0,k}= (k-\\frac 14)\\pi + \\frac{1}{8(k-\\frac 14)\\pi} + O(k^{-3}),\n\\end{equation}\nwhich imply\n\\begin{equation}\nj_{0,k}^{2} = (k-\\frac 14)^2\\pi^2 + \\frac 14 + O(k^{-2}).\n\\end{equation}\nBased on this we define the function\n\\begin{equation}\n\\tilde H(t) = \\sum_{k=1}^{\\infty} \\exp[-t((k-\\frac 14)^2\\pi^2 + \\frac 14)]\n\\end{equation}\nand estimate its difference with $H(t)$.\n\\begin{lemma}\\label{lem:difference} With all notation as above, as $t\\to 0$,\n\\begin{equation}\nH(t) - \\tilde H(t) = O(t).\n\\end{equation}\n\\end{lemma}\nThis is helpful because we also have the following lemma.\n\\begin{lemma}\\label{lem:compute} As $t\\to 0$,\n\\begin{equation}\n\\tilde H(t) = \\frac{1}{2\\sqrt\\pi}t^{-1/2} + c - \\frac{1}{8\\sqrt\\pi}t^{1/2}+ O(t).\n\\end{equation}\nHere $c$ is a constant which is irrelevant to our purposes.\n\\end{lemma}\nFrom these two Lemmas and \\eqref{eq:relationship} we immediately deduce this proposition.\n\\begin{proposition} As $t\\to 0$,\n\\begin{equation}\n\\Tr e^{-t\\Delta_{\\Omega}} = \\frac 18 t^{-1} - (\\frac{\\sqrt\\pi}{8}+\\frac{1}{4\\sqrt\\pi})t^{-1/2}-\\frac 12c + (\\frac{\\sqrt\\pi}{256}+\\frac{1}{16\\sqrt\\pi})t^{1/2}+ O(t).\n\\end{equation}\n\\end{proposition}\n\\begin{remark} We could reverse engineer $c$ from the known expansion from \\cite{NRS} but there is no need.\n\\end{remark}",
    "post_theorem_intro_text_len": 4928,
    "post_theorem_intro_text": "We can say substantially more about $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ whenever $\\alpha\\ne\\pi$. Define\n\\begin{equation}\\label{eq:defofr0}\n    r_0=r_0(\\alpha,\\kappa_{+},\\kappa_-)=\\frac 12\\csc\\alpha\\sqrt{\\kappa_+^2+\\kappa_-^2-2\\kappa_+\\kappa_-\\cos\\alpha}.\n\\end{equation}\n\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\operatorname*{fp} \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\,\\rho\\,d\\rho\\,d\\phi\\,\\,R\\,dR\\,d\\theta.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\operatorname*{fp}$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$  because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\subsection{Inverse spectral applications}\n\nIn \\cite{EGS_2017}, the authors prove that any curvilinear polygon with straight corners is in fact isospectral to a polygon. They require straight corners because of a lack of a formula in the curved corner case. We can strengthen their result.\n\\begin{definition}\n    A curvilinear polygon $\\Omega$ is \\emph{admissible} if, for each $j$, either $c_{1/2}(\\alpha_j)>0$  or $\\Omega$ is straight near $P_j$.\n\\end{definition}\n\n\\begin{proposition}\n    Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n    \\begin{equation}\n    \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n    \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.",
    "sketch": "In the “Plan of the paper” the authors outline the proof of Theorem~\\ref{thm:thm1}: in Section~\\ref{sec:gm} they “examin[e] the geometric microlocal description of the heat kernel given in \\cite{NRS},” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” in Section~\\ref{sec:conformal} they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in \\cite{NRS},” which “allows [them] to obtain both \\eqref{eq:form} and \\eqref{eq:general}.” Finally, in Section~\\ref{sec:specialpi2} they prove \\eqref{eq:specialpi2} “by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.”",
    "expanded_sketch": "In the “Plan of the paper” the authors outline the proof of the main theorem: next they “examin[e] the geometric microlocal description of the heat kernel given in NRS,” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in NRS,” which “allows [them] to obtain both\n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\\\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\\\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}.\nFinally, they prove\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n“by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}.”,",
    "expanded_theorem": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}..",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subseteq\\mathbb R^2\\) be a curvilinear polygon: its boundary is piecewise smooth, smooth except at finitely many corners \\(P_1,\\dots,P_n\\), where the interior angle at \\(P_j\\) is \\(\\alpha_j>0\\). Let \\(\\kappa\\) denote the inward-pointing curvature along the smooth boundary arcs, and for each corner let \\(\\kappa_{j,+}\\) and \\(\\kappa_{j,-}\\) be the one-sided limits of \\(\\kappa\\) as the corner is approached along the two incident sides. Let \\(H^D_{\\Omega}(t)\\) be the Dirichlet heat trace of \\(\\Omega\\). As \\(t\\to 0\\), which asymptotic expansion with error bound is valid?",
      "correct_choice": {
        "label": "A",
        "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided limiting curvatures \\(\\kappa_\\pm\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j)\\sqrt{\\kappa_{j,+}^2+\\kappa_{j,-}^2}\\Big)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha)\\) depends only on the corner angle."
        },
        {
          "label": "C",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+O(\\sqrt t),\\]\\nas \\(t\\to 0\\)."
        },
        {
          "label": "D",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided limiting curvatures \\(\\kappa_\\pm\\)."
        },
        {
          "label": "E",
          "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)+\\sqrt t\\Big(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big)+O(t^{3/2}),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) may depend on the global geometry of \\(\\Omega\\) in addition to \\(\\alpha\\) and \\(\\kappa_\\pm\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "corner-term dependence on full pair \\((\\kappa_+,\\kappa_-)\\) rather than a forced factorization through Euclidean norm",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "explicit \\(\\sqrt t\\)-coefficient and sharper remainder \\(O(t\\log t)\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "error term improved from first guaranteed \\(O(t\\log t)\\) to \\(O(t)\\)",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "uniform local determination of corner contribution by \\(\\alpha\\) and \\(\\kappa_\\pm\\)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem introduces the geometric setting and asks for the valid asymptotic expansion, but it does not explicitly state the correct remainder term or the precise dependence of the corner contribution. There is no direct or trivial hint to the correct choice."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct option is essentially the full theorem statement. However, it is not a pure restatement, since the alternatives vary in subtle but meaningful ways (error bounds, locality of corner terms, and factorization claims)."
      },
      "GPS": {
        "score": 1,
        "justification": "Selecting the best answer requires moderate reasoning about which asymptotic claim is strongest validly, versus merely weaker true or falsely sharpened statements. Still, success depends largely on recognition/recall of the theorem rather than substantial derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible. They target realistic failure modes: replacing local dependence by an unjustified factorization, weakening to a less informative but true bound, and asserting overly strong remainders or improper global dependence. They are distinct and well aligned with expert-level misconceptions."
      },
      "total_score": 6,
      "overall_assessment": "A solid high-level MCQ with no answer leakage and excellent distractors, but it is still primarily a near-verbatim theorem identification task rather than a deeply generative reasoning problem."
    }
  },
  {
    "id": "2512.04640v2",
    "paper_link": "http://arxiv.org/abs/2512.04640v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "teo",
      "content": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}",
      "start_pos": 9823,
      "end_pos": 10412,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:Existence_First": "\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}",
      "def:weak_sub_super_sol": "\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}",
      "EqProblem": "\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}",
      "eq:Sublinear_Problem": "\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}",
      "lem:locmin": "\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}",
      "thm:main": "\\begin{teo} \\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}\n\\end{teo}"
    },
    "pre_theorem_intro_text_len": 1436,
    "pre_theorem_intro_text": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.",
    "context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}",
    "full_context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\nThe above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations.\n\nWe are now ready to properly set the definition of weak sub/supersolution of \\eqref{EqProblem}$_\\lambda$.\n\\begin{defin}\\label{def:weak_sub_super_sol}\n Let $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n \\begin{itemize}\n \\item[(i)] $u>0$ in $\\Omega$.\n \\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n \\begin{equation}\\label{eq:weak_sub_super_sol}\n \\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq) \\int_{\\Omega}\\left(\\lambda u^{q} + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n \\end{equation}\n \\end{itemize}\n Finally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\n\\begin{lemma}\\label{lem:Perron}\n Let $\\underline{u}, \\overline{u}\\in S^{1}_{0}(\\Omega)$ be a weak subsolution and a weak supersolution, respectively, of problem \\eqref{EqProblem}$_\\lambda$. \n We assume that \n \\begin{itemize}\n \\item[a)] $\\underline{u}(g) \\leq \\overline{u}(g)$ for a.e.\\,$g\\in \\Omega$;\n \\item[b)] for every open set $\\Oo\\Subset\\Omega$ there exists\n $C = C(\\Oo,\\underline{u}) > 0$ such that\n $$\\text{$\\underline{u}\\geq C$ a.e.\\,in $\\Oo$}.$$\n \\end{itemize}\n Then, there exists a weak solution $u \\in S^{1}_{0}(\\Omega)$ of \\eqref{EqProblem}$_{\\lambda}$ such that \n $$\\text{$\\underline{u}(g) \\leq u(g) \\leq \\overline{u}(g)$ for a.e. $g \\in \\Omega$}.$$\n\\end{lemma}\n\n\\noindent We split the proof of I) in two lemmas. \n\\begin{lemma}\\label{lem:lambda0}\n Let $\\Lambda$ as defined in \\eqref{eq:DefinitionLambda}. Then $\\Lambda >0$.\n \\begin{proof}\n We will show that there exists a sufficiently small $\\lambda >0$ such that \n \\eqref{EqProblem}$_{\\lambda}$ has a solution. To this aim, we will use Lemma \\ref{lem:Perron} exhibiting both a super and a subsolution.\n Looking for a supersolution, we consider the following auxiliary torsion problem\n \\begin{equation}\n \\left\\{\\begin{array}{rl}\n -\\Delta_{\\mathbb{G}} V = 1 & \\textrm{in } \\Omega,\\\\\n V = 0 & \\textrm{on } \\partial \\Omega,\n \\end{array}\\right.\n \\end{equation}\n \\noindent whose unique solution is provided by Lax-Milgram Theorem. Moreover, by a classical Stampacchia iteration method, it holds that $V\\in L^{\\infty}(\\Omega)$. Observe further that for any positive constant $C$,\n the function $C\\cdot x^p-x$, with $p>1$, has negative value for $x>0$ sufficiently small.\n Therefore, for every $C,C'\\in \\R^+$\n there exists $\\lambda^*>0$ such that \n for every $\\lambda<\\lambda^*$, \n \\[\n \\exists \\, m_\\lambda\\in \\R^+:\\quad \\lambda\\cdot C'\\cdot m_\\lambda^q+C\\cdot m_\\lambda^p-m_\\lambda\\leq0.\n \\]\n We fix $\\lambda<\\lambda^*$ and set $C=\\|V\\|_{L^{\\infty}(\\Omega)}^p$, $C'=\\|V\\|_{L^{\\infty}(\\Omega)}^q$. \n We define $\\overline u_1:=m_\\lambda V$, which weakly verifies\n \\[\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\G}\\overline u_1&=m_\\lambda \\geq \\lambda\\overline u_1^q+\\overline u_1^p &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &>0 &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &=0 &\\ \\m{on }\\de\\Omega,\n \\end{array}\\right.\n \\]\n therefore it is a weak supersolution of \\eqref{EqProblem}$_\\lambda$.\\\\\n Regarding the weak subsolution to \\eqref{EqProblem}, we choose the unique solution $\\underline u_\\lambda$ to \\eqref{eq:Sublinear_Problem}. \n We can now conclude the proof by appealing Lemma \\ref{lem:Perron}. Indeed, by Lemma \\ref{lem:Weak_Comparison_Model} with $w=\\overline{u}_1$ and $v=\\underline{u}_{\\lambda}$, we get that \n $\\overline{u}_1 \\geq \\underline{u}_{\\lambda}$, which is condition a) of Lemma \\ref{lem:Perron}. Regarding b) of Lemma \\ref{lem:Perron} it is enough to recall \\cite[Corollary 2.3]{BiGaVe}. This closes the proof. \n \\end{proof}\n\\end{lemma}\n\nWe now set $\\overline{u}= u_{\\lambda'}$ and $\\underline{u} = w_{\\lambda}$, \n and we apply Lemma \\ref{lem:Perron}: this immediately yields that problem \\eqref{EqProblem}$_{\\lambda}$ admits a weak solution $u_{\\lambda}$ for every $\\lambda \\in (0,\\Lambda)$. Moreover, recalling the definition of $I_{\\lambda}$ in \\eqref{eq:Def_Ilambda}, such a solution satisfies that\n $$I_{\\lambda}(u_\\lambda) = \\min\\{u\\in S_0^1(\\Omega):\\,w_{\\lambda}\\leq u\\leq u_{\\lambda'}\\}\n \\leq I_{\\lambda}(w_{\\lambda}).$$\n In particular,\n by Theorem \\ref{thm:Sublinear_Problem} we have\n \\begin{equation} \\label{eq:Ilambdaulambdaneg}\n I_{\\lambda}(u_{\\lambda}) \\leq I_{\\lambda}(w_{\\lambda}) \\leq J_{\\lambda}(w_{\\lambda}) <0.\n \\end{equation}\n It remains to consider the case $\\lambda = \\Lambda$. The proof\n is rather standard and pretty similar to that of \\cite[Lemma 3.5]{BiGaVe}. We report it here for the sake of completeness.\n To begin with, we choose a monotone increasing sequence $\\{\\lambda_k\\}_k\\subseteq(0,\\Lambda)$ such that \n $\\lambda_{k} \\to \\Lambda$ as $k\\to+\\infty$. Now, for each $k \\in \\mathbb{N}$, we set\n $$u_k := u_{\\lambda_k}\\in S^{1}_{0}(\\Omega),$$\n \\noindent where $u_{\\lambda_k}$ is the weak solution of problem \\eqref{EqProblem}$_{\\lambda_k}$ constructed\n as above by means of Lemma \\ref{lem:Perron}. Thanks to \n \\eqref{eq:Ilambdaulambdaneg}, for every $k\\geq 1$ we have\n \\begin{equation} \\label{eq:Ilambdakneg}\n I_{\\lambda_k,}(u_{k}) \n = \\dfrac{1}{2} \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 - \\dfrac{\\lambda_k}{q+1}\\int_{\\Omega}|u_k|^{q+1} - \\dfrac{1}{2^{\\star}_{Q}}\\int_{\\Omega}|u_k|^{2^{\\star}_{Q}} <0.\n \\end{equation}\n Moreover, by using $\\varphi = u_k$ in \\eqref{eq:weak_sub_super_sol}, and recalling that $u_k$\n solves \\eqref{EqProblem}$_{\\lambda_k}$, we get\n \\begin{equation} \\label{eq:testwithukzero}\n \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 -\\lambda_k\\int_\\Omega u_k^{q+1}\n -\\int_\\Omega u_k^{2^{\\star}_{Q}} = 0.\n \\end{equation}\n Combining \\eqref{eq:Ilambdakneg} with \\eqref{eq:testwithukzero},\n we notice that the sequence $\\{u_k\\}_k$ is bounded in~$S^{1}_{0}(\\Omega)$.\n Therefore, we can find a function\n $$u_{\\Lambda}\\in S^{1}_{0}(\\Omega)$$ \n such that\n (up to a subsequence and as $k\\to+\\infty$)\n \\begin{itemize}\n \\item[a)] $u_k\\to u_{\\Lambda}$ weakly in $S^{1}_{0}(\\Omega)$ and strongly\n in $L^p(\\Omega)$ for $1\\leq p <2^{\\star}_{Q}$;\n \\item[b)] $u_k\\to u_{\\Lambda}$ a.e.\\,in $\\Omega$.\n \\end{itemize}\n We now observe that, being $\\{\\lambda_k\\}_k$ increasing, it follows that $\\lambda_k\\geq \\lambda_1$ for every $k\\geq 1$.\n Moreover, arguing as above yields that\n $u_{\\lambda_k}\\geq w_{\\lambda_1}$, and thus\n $$u_{\\Lambda} > 0\\quad\\text{a.e.\\,in $\\Omega$}.$$\n Moreover, since $u_k$ solves problem \\eqref{EqProblem}$_{\\lambda_k}$,\n we have\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_k,\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\lambda_k\\int_\\Omega u_k^{q}\\varphi \n -\\int_\\Omega u_k^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$}.$$\nTherefore, passing to the limit as $k\\to+\\infty$ in the above identity, and by dominated convergence, we get that $u_{\\Lambda}$ satisfies\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_{\\Lambda},\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\Lambda\\int_\\Omega u_{\\Lambda}^{q}\\varphi\n -\\int_\\Omega u_{\\Lambda}^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$},$$\n \\noindent which shows that $u_{\\Lambda}$ is actually a weak solution of\n problem \\eqref{EqProblem}$_{\\Lambda}$. \n This closes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 5030,
    "post_theorem_intro_text": "The above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations. \n\nThe interest in studying existence of positive solutions to critical problems in the Carnot group setting,\n is in the geometric significance of the purely critical problem in the model case of the Heisenberg group.\n Indeed, when $\\lambda =0$ and $\\Omega = \\mathbb{H}^n$, the problem \\eqref{EqProblem} becomes the famous CR-Yamabe problem studied by Jerison and Lee \\cite{JerisonLee,JerisonLee2,JerisonLee3}. The problem  we are interested in is settled on bounded domains, where tipycally one can prove non-existence of positive solutions, at least in star-shaped domains, by appealing suitable versions of the Pohozaev identity. Because of this, the seminal paper by Brezis and Nirenberg \\cite{BN} showed that adding a perturbative term, linear in \\cite{BN}, but subsequently extended to much more general perturbations, may allow to prove the existence of one or more positive solutions. A crucial tool in the argument performed in \\cite{BN} is provided by the use of the Aubin-Talenti functions, whose analogue in $\\mathbb{H}^n$ made its appearance in \\cite{JerisonLee2}. This was a key ingredient which gave rise to a prolific study of critical problems in $\\mathbb{H}^n$, see e.g. \\cite{Citti, GaLa, Ugu1, CitUg, MaUg, FelliUgu, MaMaPi, PaPiTe}.\n\nAs long as one needs explicit knowledge of proper replacements of the Aubin-Talenti functions, the only other sub-Riemannian structure where they are known is that of groups of Iwasawa type, see \\cite{GaroVa2, GaroVa}. As far as we know, there are no other structures, nor Sobolev inequalities with $p\\neq 2$, for which the minimizers are explicitly known. \nOn the other hand, since the best constant in the Sobolev inequality is achieved in all Carnot groups (see \\cite{GaroVa2}), it has been proved to be enough to know  the asymptotic behaviour at infinity of the minimizers. This is now known for $p\\neq 2$ as well, see \\cite{Loiudice3}, and it paved the way for a series of \nexistence, multiplicity or non-existence of positive solutions for critical problems  à la Br\\'{e}zis-Nirenberg in $\\mathbb{G}$: we refer e.g. to \\cite{BoUg, Loiudice1, Loiudice2, Loiudice4, BiGaVe}.\n\n\\medskip\n\nLet us now briefly describe the proof of Theorem \\ref{thm:main}: \n\\begin{itemize}\n\t\\item in Theorem \\ref{thm:Existence_First} we prove the existence of a first solution by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting. In particular, setting\n\t\\begin{equation*}\n\t\\Lambda := \\sup \\{ \\lambda >0: \\eqref{EqProblem}_\\lambda \\textrm{ admits a weak solution}\\},\n\\end{equation*}\n\t\\noindent we show first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions. Once this is done, we use the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution and we construct a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$; \n\t\\item we show that for $\\lambda \\in (0,\\Lambda)$ the first solution obtained as described before is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}. We stress here that in \\cite{ABC} the authors made use of a famous result by Brezis and Nirenberg \\cite{BNH1C1} which does not have an analog in the Carnot group setting. This is due to the fact that $C^{1,\\alpha}$ regularity up to the boundary is still a delicate issue at the so called characteristic points: the first obstructions have been observed by Jerison \\cite{Jerison,Jerison2}, but this is still an active field of research, see e.g. \\cite{BaCiCu,BaGaMu,AbTr}. For this reason we follow here a more variational approach based on a paper by Alama \\cite{Alama}, already used in a different setting in \\cite{AbDiVa};\n\t\\item we prove the existence of a second solution following an argument originally due to Tarantello \\cite{Tarantello}: this combines the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.\n\\end{itemize} \n\n\\medskip\n\nWe stress that the multiplicity result obtained in Theorem \\ref{thm:main} can be easily extended to cover the convex-case of a Sobolev sub-critical nonlinearity.\n\n\\medskip\n\nThe paper is organized as follows: in Section \\ref{sec:Prel} we recall the basic facts on Carnot groups and we set the variational functional setting necessary for the study of \\eqref{EqProblem} We also recall the basic result regarding the purely sublinear problems, like existence and uniqueness of a positive solution and a comparison principle resembling the classical one. In Section \\ref{sec:First_Solution} we prove the existence of a first solution as described before, while the existence of a second solution (for $\\lambda \\in (0,\\Lambda)$) is postponed to the final Section \\ref{sec:Second_Solution}.",
    "sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item In Theorem \\ref{thm:Existence_First} the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\eqref{EqProblem}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}.” Since the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}",
    "expanded_sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item We first prove the following theorem.\n\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}\nIn this step, the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem\n\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}\nas a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with the equation above.” We now record the corresponding statement.\n\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}\nSince the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}",
    "expanded_theorem": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem the equation above$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem the equation above$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}",
    "theorem_type": [
      "Existential–Universal",
      "Existence"
    ],
    "mcq": {
      "question": "Let $\\mathbb{G}$ be a Carnot group, let $\\Omega\\subset \\mathbb{G}$ be an open bounded set with sufficiently smooth boundary $\\partial\\Omega$, and let $q\\in(0,1)$. Let $2_Q^{\\star}=\\frac{2Q}{Q-2}$, where $Q$ is the homogeneous dimension of $\\mathbb{G}$. For $\\lambda>0$, consider the Dirichlet problem\n\\[\n\\begin{cases}\n-\\Delta_{\\mathbb{G}}u=\\lambda u^q+u^{2_Q^{\\star}-1} & \\text{in }\\Omega,\\\\\nu>0 & \\text{in }\\Omega,\\\\\nu=0 & \\text{on }\\partial\\Omega,\n\\end{cases}\n\\]\nwhere $\\Delta_{\\mathbb{G}}$ is the sub-Laplacian on $\\mathbb{G}$. A weak solution means a function $u\\in S_0^1(\\Omega)$ with $u>0$ in $\\Omega$ such that\n\\[\n\\int_\\Omega \\langle \\nabla_{\\mathbb{G}}u,\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak g_1}\n=\\int_\\Omega \\bigl(\\lambda u^q+u^{2_Q^{\\star}-1}\\bigr)\\varphi\n\\quad\\text{for every }\\varphi\\in S_0^1(\\Omega).\n\\]\nWhich statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda)$."
      },
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        {
          "label": "B",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda\\geq\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda)$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda)$."
        },
        {
          "label": "C",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$ and has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$."
        },
        {
          "label": "D",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda]$."
        },
        {
          "label": "E",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, and for every $\\lambda\\in(0,\\Lambda]$ the problem admits a unique weak solution."
        }
      ],
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        "weaker_true_label": "C",
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          "label": "C",
          "sketch_hook_type": "other",
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          "label": "D",
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    },
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      "ALS": {
        "score": 2,
        "justification": "The stem specifies the PDE setting and asks for the correct existence theorem, but it does not explicitly or implicitly reveal the correct endpoint/multiplicity pattern. The correct choice is not signaled by wording in the stem."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall item: the stem presents the exact hypotheses and asks which conclusion holds. The options do introduce competing endpoint and multiplicity variants, so it is not a pure verbatim restatement, but it is still close to one."
      },
      "GPS": {
        "score": 0,
        "justification": "The item does not meaningfully support derivation from the given information; answering it mainly requires prior knowledge of the precise theorem statement. The reasoning burden is low and mostly concerns remembering fine distinctions at the threshold."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they vary by endpoint inclusion at λ=Λ, omission of multiplicity, and an unjustified uniqueness claim. These reflect realistic theorem-statement confusion points."
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  {
    "id": "2512.04640v2",
    "paper_link": "http://arxiv.org/abs/2512.04640v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "teo",
      "content": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}",
      "start_pos": 9823,
      "end_pos": 10412,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:Existence_First": "\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}",
      "def:weak_sub_super_sol": "\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}",
      "EqProblem": "\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}",
      "eq:Sublinear_Problem": "\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}",
      "lem:locmin": "\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}",
      "thm:main": "\\begin{teo} \\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}\n\\end{teo}"
    },
    "pre_theorem_intro_text_len": 1436,
    "pre_theorem_intro_text": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.",
    "context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}",
    "full_context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\nThe above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations.\n\nWe are now ready to properly set the definition of weak sub/supersolution of \\eqref{EqProblem}$_\\lambda$.\n\\begin{defin}\\label{def:weak_sub_super_sol}\n Let $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n \\begin{itemize}\n \\item[(i)] $u>0$ in $\\Omega$.\n \\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n \\begin{equation}\\label{eq:weak_sub_super_sol}\n \\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq) \\int_{\\Omega}\\left(\\lambda u^{q} + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n \\end{equation}\n \\end{itemize}\n Finally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\n\\begin{lemma}\\label{lem:Perron}\n Let $\\underline{u}, \\overline{u}\\in S^{1}_{0}(\\Omega)$ be a weak subsolution and a weak supersolution, respectively, of problem \\eqref{EqProblem}$_\\lambda$. \n We assume that \n \\begin{itemize}\n \\item[a)] $\\underline{u}(g) \\leq \\overline{u}(g)$ for a.e.\\,$g\\in \\Omega$;\n \\item[b)] for every open set $\\Oo\\Subset\\Omega$ there exists\n $C = C(\\Oo,\\underline{u}) > 0$ such that\n $$\\text{$\\underline{u}\\geq C$ a.e.\\,in $\\Oo$}.$$\n \\end{itemize}\n Then, there exists a weak solution $u \\in S^{1}_{0}(\\Omega)$ of \\eqref{EqProblem}$_{\\lambda}$ such that \n $$\\text{$\\underline{u}(g) \\leq u(g) \\leq \\overline{u}(g)$ for a.e. $g \\in \\Omega$}.$$\n\\end{lemma}\n\n\\noindent We split the proof of I) in two lemmas. \n\\begin{lemma}\\label{lem:lambda0}\n Let $\\Lambda$ as defined in \\eqref{eq:DefinitionLambda}. Then $\\Lambda >0$.\n \\begin{proof}\n We will show that there exists a sufficiently small $\\lambda >0$ such that \n \\eqref{EqProblem}$_{\\lambda}$ has a solution. To this aim, we will use Lemma \\ref{lem:Perron} exhibiting both a super and a subsolution.\n Looking for a supersolution, we consider the following auxiliary torsion problem\n \\begin{equation}\n \\left\\{\\begin{array}{rl}\n -\\Delta_{\\mathbb{G}} V = 1 & \\textrm{in } \\Omega,\\\\\n V = 0 & \\textrm{on } \\partial \\Omega,\n \\end{array}\\right.\n \\end{equation}\n \\noindent whose unique solution is provided by Lax-Milgram Theorem. Moreover, by a classical Stampacchia iteration method, it holds that $V\\in L^{\\infty}(\\Omega)$. Observe further that for any positive constant $C$,\n the function $C\\cdot x^p-x$, with $p>1$, has negative value for $x>0$ sufficiently small.\n Therefore, for every $C,C'\\in \\R^+$\n there exists $\\lambda^*>0$ such that \n for every $\\lambda<\\lambda^*$, \n \\[\n \\exists \\, m_\\lambda\\in \\R^+:\\quad \\lambda\\cdot C'\\cdot m_\\lambda^q+C\\cdot m_\\lambda^p-m_\\lambda\\leq0.\n \\]\n We fix $\\lambda<\\lambda^*$ and set $C=\\|V\\|_{L^{\\infty}(\\Omega)}^p$, $C'=\\|V\\|_{L^{\\infty}(\\Omega)}^q$. \n We define $\\overline u_1:=m_\\lambda V$, which weakly verifies\n \\[\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\G}\\overline u_1&=m_\\lambda \\geq \\lambda\\overline u_1^q+\\overline u_1^p &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &>0 &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &=0 &\\ \\m{on }\\de\\Omega,\n \\end{array}\\right.\n \\]\n therefore it is a weak supersolution of \\eqref{EqProblem}$_\\lambda$.\\\\\n Regarding the weak subsolution to \\eqref{EqProblem}, we choose the unique solution $\\underline u_\\lambda$ to \\eqref{eq:Sublinear_Problem}. \n We can now conclude the proof by appealing Lemma \\ref{lem:Perron}. Indeed, by Lemma \\ref{lem:Weak_Comparison_Model} with $w=\\overline{u}_1$ and $v=\\underline{u}_{\\lambda}$, we get that \n $\\overline{u}_1 \\geq \\underline{u}_{\\lambda}$, which is condition a) of Lemma \\ref{lem:Perron}. Regarding b) of Lemma \\ref{lem:Perron} it is enough to recall \\cite[Corollary 2.3]{BiGaVe}. This closes the proof. \n \\end{proof}\n\\end{lemma}\n\nWe now set $\\overline{u}= u_{\\lambda'}$ and $\\underline{u} = w_{\\lambda}$, \n and we apply Lemma \\ref{lem:Perron}: this immediately yields that problem \\eqref{EqProblem}$_{\\lambda}$ admits a weak solution $u_{\\lambda}$ for every $\\lambda \\in (0,\\Lambda)$. Moreover, recalling the definition of $I_{\\lambda}$ in \\eqref{eq:Def_Ilambda}, such a solution satisfies that\n $$I_{\\lambda}(u_\\lambda) = \\min\\{u\\in S_0^1(\\Omega):\\,w_{\\lambda}\\leq u\\leq u_{\\lambda'}\\}\n \\leq I_{\\lambda}(w_{\\lambda}).$$\n In particular,\n by Theorem \\ref{thm:Sublinear_Problem} we have\n \\begin{equation} \\label{eq:Ilambdaulambdaneg}\n I_{\\lambda}(u_{\\lambda}) \\leq I_{\\lambda}(w_{\\lambda}) \\leq J_{\\lambda}(w_{\\lambda}) <0.\n \\end{equation}\n It remains to consider the case $\\lambda = \\Lambda$. The proof\n is rather standard and pretty similar to that of \\cite[Lemma 3.5]{BiGaVe}. We report it here for the sake of completeness.\n To begin with, we choose a monotone increasing sequence $\\{\\lambda_k\\}_k\\subseteq(0,\\Lambda)$ such that \n $\\lambda_{k} \\to \\Lambda$ as $k\\to+\\infty$. Now, for each $k \\in \\mathbb{N}$, we set\n $$u_k := u_{\\lambda_k}\\in S^{1}_{0}(\\Omega),$$\n \\noindent where $u_{\\lambda_k}$ is the weak solution of problem \\eqref{EqProblem}$_{\\lambda_k}$ constructed\n as above by means of Lemma \\ref{lem:Perron}. Thanks to \n \\eqref{eq:Ilambdaulambdaneg}, for every $k\\geq 1$ we have\n \\begin{equation} \\label{eq:Ilambdakneg}\n I_{\\lambda_k,}(u_{k}) \n = \\dfrac{1}{2} \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 - \\dfrac{\\lambda_k}{q+1}\\int_{\\Omega}|u_k|^{q+1} - \\dfrac{1}{2^{\\star}_{Q}}\\int_{\\Omega}|u_k|^{2^{\\star}_{Q}} <0.\n \\end{equation}\n Moreover, by using $\\varphi = u_k$ in \\eqref{eq:weak_sub_super_sol}, and recalling that $u_k$\n solves \\eqref{EqProblem}$_{\\lambda_k}$, we get\n \\begin{equation} \\label{eq:testwithukzero}\n \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 -\\lambda_k\\int_\\Omega u_k^{q+1}\n -\\int_\\Omega u_k^{2^{\\star}_{Q}} = 0.\n \\end{equation}\n Combining \\eqref{eq:Ilambdakneg} with \\eqref{eq:testwithukzero},\n we notice that the sequence $\\{u_k\\}_k$ is bounded in~$S^{1}_{0}(\\Omega)$.\n Therefore, we can find a function\n $$u_{\\Lambda}\\in S^{1}_{0}(\\Omega)$$ \n such that\n (up to a subsequence and as $k\\to+\\infty$)\n \\begin{itemize}\n \\item[a)] $u_k\\to u_{\\Lambda}$ weakly in $S^{1}_{0}(\\Omega)$ and strongly\n in $L^p(\\Omega)$ for $1\\leq p <2^{\\star}_{Q}$;\n \\item[b)] $u_k\\to u_{\\Lambda}$ a.e.\\,in $\\Omega$.\n \\end{itemize}\n We now observe that, being $\\{\\lambda_k\\}_k$ increasing, it follows that $\\lambda_k\\geq \\lambda_1$ for every $k\\geq 1$.\n Moreover, arguing as above yields that\n $u_{\\lambda_k}\\geq w_{\\lambda_1}$, and thus\n $$u_{\\Lambda} > 0\\quad\\text{a.e.\\,in $\\Omega$}.$$\n Moreover, since $u_k$ solves problem \\eqref{EqProblem}$_{\\lambda_k}$,\n we have\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_k,\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\lambda_k\\int_\\Omega u_k^{q}\\varphi \n -\\int_\\Omega u_k^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$}.$$\nTherefore, passing to the limit as $k\\to+\\infty$ in the above identity, and by dominated convergence, we get that $u_{\\Lambda}$ satisfies\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_{\\Lambda},\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\Lambda\\int_\\Omega u_{\\Lambda}^{q}\\varphi\n -\\int_\\Omega u_{\\Lambda}^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$},$$\n \\noindent which shows that $u_{\\Lambda}$ is actually a weak solution of\n problem \\eqref{EqProblem}$_{\\Lambda}$. \n This closes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 5030,
    "post_theorem_intro_text": "The above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations. \n\nThe interest in studying existence of positive solutions to critical problems in the Carnot group setting,\n is in the geometric significance of the purely critical problem in the model case of the Heisenberg group.\n Indeed, when $\\lambda =0$ and $\\Omega = \\mathbb{H}^n$, the problem \\eqref{EqProblem} becomes the famous CR-Yamabe problem studied by Jerison and Lee \\cite{JerisonLee,JerisonLee2,JerisonLee3}. The problem  we are interested in is settled on bounded domains, where tipycally one can prove non-existence of positive solutions, at least in star-shaped domains, by appealing suitable versions of the Pohozaev identity. Because of this, the seminal paper by Brezis and Nirenberg \\cite{BN} showed that adding a perturbative term, linear in \\cite{BN}, but subsequently extended to much more general perturbations, may allow to prove the existence of one or more positive solutions. A crucial tool in the argument performed in \\cite{BN} is provided by the use of the Aubin-Talenti functions, whose analogue in $\\mathbb{H}^n$ made its appearance in \\cite{JerisonLee2}. This was a key ingredient which gave rise to a prolific study of critical problems in $\\mathbb{H}^n$, see e.g. \\cite{Citti, GaLa, Ugu1, CitUg, MaUg, FelliUgu, MaMaPi, PaPiTe}.\n\nAs long as one needs explicit knowledge of proper replacements of the Aubin-Talenti functions, the only other sub-Riemannian structure where they are known is that of groups of Iwasawa type, see \\cite{GaroVa2, GaroVa}. As far as we know, there are no other structures, nor Sobolev inequalities with $p\\neq 2$, for which the minimizers are explicitly known. \nOn the other hand, since the best constant in the Sobolev inequality is achieved in all Carnot groups (see \\cite{GaroVa2}), it has been proved to be enough to know  the asymptotic behaviour at infinity of the minimizers. This is now known for $p\\neq 2$ as well, see \\cite{Loiudice3}, and it paved the way for a series of \nexistence, multiplicity or non-existence of positive solutions for critical problems  à la Br\\'{e}zis-Nirenberg in $\\mathbb{G}$: we refer e.g. to \\cite{BoUg, Loiudice1, Loiudice2, Loiudice4, BiGaVe}.\n\n\\medskip\n\nLet us now briefly describe the proof of Theorem \\ref{thm:main}: \n\\begin{itemize}\n\t\\item in Theorem \\ref{thm:Existence_First} we prove the existence of a first solution by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting. In particular, setting\n\t\\begin{equation*}\n\t\\Lambda := \\sup \\{ \\lambda >0: \\eqref{EqProblem}_\\lambda \\textrm{ admits a weak solution}\\},\n\\end{equation*}\n\t\\noindent we show first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions. Once this is done, we use the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution and we construct a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$; \n\t\\item we show that for $\\lambda \\in (0,\\Lambda)$ the first solution obtained as described before is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}. We stress here that in \\cite{ABC} the authors made use of a famous result by Brezis and Nirenberg \\cite{BNH1C1} which does not have an analog in the Carnot group setting. This is due to the fact that $C^{1,\\alpha}$ regularity up to the boundary is still a delicate issue at the so called characteristic points: the first obstructions have been observed by Jerison \\cite{Jerison,Jerison2}, but this is still an active field of research, see e.g. \\cite{BaCiCu,BaGaMu,AbTr}. For this reason we follow here a more variational approach based on a paper by Alama \\cite{Alama}, already used in a different setting in \\cite{AbDiVa};\n\t\\item we prove the existence of a second solution following an argument originally due to Tarantello \\cite{Tarantello}: this combines the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.\n\\end{itemize} \n\n\\medskip\n\nWe stress that the multiplicity result obtained in Theorem \\ref{thm:main} can be easily extended to cover the convex-case of a Sobolev sub-critical nonlinearity.\n\n\\medskip\n\nThe paper is organized as follows: in Section \\ref{sec:Prel} we recall the basic facts on Carnot groups and we set the variational functional setting necessary for the study of \\eqref{EqProblem} We also recall the basic result regarding the purely sublinear problems, like existence and uniqueness of a positive solution and a comparison principle resembling the classical one. In Section \\ref{sec:First_Solution} we prove the existence of a first solution as described before, while the existence of a second solution (for $\\lambda \\in (0,\\Lambda)$) is postponed to the final Section \\ref{sec:Second_Solution}.",
    "sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item In Theorem \\ref{thm:Existence_First} the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\eqref{EqProblem}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}.” Since the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}",
    "expanded_sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item We first prove the following theorem.\n\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}\nIn this step, the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem\n\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}\nas a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with the equation above.” We now record the corresponding statement.\n\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}\nSince the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}",
    "expanded_theorem": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem the equation above$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem the equation above$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}",
    "theorem_type": [
      "Existential–Universal",
      "Existence"
    ],
    "mcq": {
      "question": "Let $\\mathbb{G}$ be a Carnot group, let $\\Omega\\subset \\mathbb{G}$ be an open bounded set with sufficiently smooth boundary $\\partial\\Omega$, and let $q\\in(0,1)$. Let $2_Q^{\\star}=\\frac{2Q}{Q-2}$, where $Q$ is the homogeneous dimension of $\\mathbb{G}$. For $\\lambda>0$, consider the Dirichlet problem\n\\[\n\\begin{cases}\n-\\Delta_{\\mathbb{G}}u=\\lambda u^q+u^{2_Q^{\\star}-1} & \\text{in }\\Omega,\\\\\nu>0 & \\text{in }\\Omega,\\\\\nu=0 & \\text{on }\\partial\\Omega,\n\\end{cases}\n\\]\nwhere $\\Delta_{\\mathbb{G}}$ is the sub-Laplacian on $\\mathbb{G}$. A weak solution means a function $u\\in S_0^1(\\Omega)$ with $u>0$ in $\\Omega$ such that\n\\[\n\\int_\\Omega \\langle \\nabla_{\\mathbb{G}}u,\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak g_1}\n=\\int_\\Omega \\bigl(\\lambda u^q+u^{2_Q^{\\star}-1}\\bigr)\\varphi\n\\quad\\text{for every }\\varphi\\in S_0^1(\\Omega).\n\\]\nWhich statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda\\geq\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda)$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda)$."
        },
        {
          "label": "C",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$ and has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$."
        },
        {
          "label": "D",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, has at least one weak solution for every $\\lambda\\in(0,\\Lambda]$, and has at least two weak solutions for every $\\lambda\\in(0,\\Lambda]$."
        },
        {
          "label": "E",
          "text": "There exists a constant $\\Lambda>0$ such that the problem has no weak solution for every $\\lambda>\\Lambda$, and for every $\\lambda\\in(0,\\Lambda]$ the problem admits a unique weak solution."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "boundary_range",
          "tampered_component": "endpoint_inclusion_at_\\Lambda",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_two_solutions_clause_on_(0,\\Lambda)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "strict_range_for_second_solution",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "coexistence_of_first_and_second_solutions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the PDE, hypotheses, and definition of weak solution, but it does not explicitly state or strongly hint at the threshold/multiplicity conclusion. The correct answer is not leaked from the wording alone."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall question: given the exact setup, the task is to identify the precise existence/multiplicity statement. It is not a pure verbatim restatement, since the choices vary in endpoint and multiplicity details, but it remains close to selecting the formal theorem conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle alternatives involving > vs >= at Lambda and whether multiplicity holds on (0, Lambda) or (0, Lambda]. However, the item mainly tests recognition/recall of the known result rather than independent mathematical generation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and well-targeted: they probe common failure modes such as endpoint inclusion, dropping the stronger multiplicity clause, overstating multiplicity at Lambda, and false uniqueness. They are distinct and mathematically aligned with typical theorem-misremembering."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests recall of a precise existence theorem rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.04933v1",
    "paper_link": "http://arxiv.org/abs/2512.04933v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\mathbb{R}$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\mathrm{vol}(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}",
      "start_pos": 19899,
      "end_pos": 20703,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:main-2": "\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}",
      "eq:delta": "\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}",
      "thm:LS": "\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}",
      "eq:Z-first": "\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}",
      "eq:main-1": "\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}",
      "eq:sim-sym": "\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}",
      "eq:sim-asym": "\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}",
      "lem:vol-partition": "\\begin{lemma} \\label{lem:vol-partition}\nLet $\\beta\\in\\{1,2,4\\}$ and $1\\le p<\\infty$. Then\n\\begin{align*}\n\\vol(\\bpb)\n&=c_n\\int_{\\IB_p^n} \\prod_{1\\le i< j\\le n}|x_i-x_j|^{\\beta}\\dd x_1 \\cdots \\dd x_n\\\\\n&=\\frac{c_{n}}{\\Gamma(1+\\frac{d_n}{p})}\\Big(\\frac{n\\beta v_p}{2}\\Big)^{d_n/p}Z_{n,p,\\beta}\n\\end{align*}\nwhere $Z_{n,p,\\beta}$ and $v_p$ are as in \\eqref{eq:Z} and\n\\[\nc_{n}\n=\\frac{1}{n!}\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(2\\pi)^{\\beta/2}}\\bigg)^{n}\\prod_{k=1}^n\\frac{(2\\pi)^{\\beta k/2}}{\\Gamma(\\frac{\\beta k}{2})}.\n\\]\n\\end{lemma}",
      "thm:main": "\\begin{theorem} \\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\IR$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}\n\\end{theorem}",
      "lem:cn": "\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}",
      "thm:CKM": "\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\IR$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6252,
    "pre_theorem_intro_text": "The Schatten or Schatten-von Neumann class consists of all compact operators between two Hilbert spaces whose sequence of singular values belongs to the Lebesgue sequence space $\\ell_p$. It can be seen a non-commutative version of $\\ell_p$ and is named after Schatten and von Neumann \\cite{Sch46,Sch50,SVN46}. We refer to \\cite{Pie07,PX03} for an overview of their role in functional analysis and Banach space geometry.\n\nIn this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n\t\\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n\ts_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$  are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$. \n\nIn case of $p=\\infty$, the quantity $A(\\infty)=\\frac{1}{2}$ can be expressed in terms of the minimal logarithmic energy of probability measures supported in $[-1,1]$ attained by the arcsine distribution, see \\cite{ST97}. We refer to \\cite{BDF+24} and \\cite{Bra24} for related asymptotics for discrete minimal logarithmic energy. \n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.",
    "context": "In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n \\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n s_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$ are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n \\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n =\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$.\n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.\n\n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\n\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}",
    "full_context": "In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n \\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n s_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$ are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n \\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n =\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$.\n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.\n\n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\n\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\vol(\\bpb)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\IR$ is\n\\[\n\\Ent(\\mu)\n= -\\int_{\\IR}\\ln(f(x))f(x)\\dd x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\Ent(\\mu)=-\\infty$ otherwise.\n\nIn case of $p=2$ it is known that $\\Ent(\\mu_2)=\\ln \\pi -\\frac{1}{2}$ where $\\mu_2$ is the semi-circle law. Otherwise the entropy $\\Ent(\\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \\eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \\cite{GMP25} in case of $p\\ge 2$.\n\nIn this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional\n\\[\n\\mu\\mapsto I_p(\\mu)\n=-\\int_{\\IR}\\int_{\\IR} \\ln|x-y|\\dd\\mu(x)\\dd\\mu(y)+v_p\\int_{\\IR} |x|^p\\dd\\mu(x)\n\\]\nover $\\mu\\in \\cP(\\IR)$, where $\\cP(\\IR)$ denotes the space of probability measures on $\\IR$. More precisely, if $X=(X_1,\\dots,X_n)$ is distributed according to the density $f_{n,p,\\beta}$, then the random empirical measure $ \\mu_{n,p} =\\frac{1}{n}\\sum_{i=1}^{n}\\delta_{X_i} $ tends to $\\mu_p$ as $n\\to\\infty$. Moreover, large deviation principles due to \\cite{BG97} and \\cite{HP00} yield \n\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}\nEarlier work \\cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \\cite{DFG+23}, the limit \\eqref{eq:Z-first} is used to prove \\eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\\'e and Serfaty \\cite{LS17} allows to refine \\eqref{eq:Z-first}. The following result forms the core of the proof of \\eqref{eq:main-1} in Theorem~\\ref{thm:main}.\n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nIt remains to check Hölder continuity at zero. Let $h\\in (0,1)$ and write\n\\begin{align*}\n\\frac{\\pi}{p}\\big(f_p(h)-f_p(0)\\big)\n&=\\int_h^1 \\frac{t^{p-1}}{\\sqrt{t^2-h^2}}\\dd t - \\int_0^1 t^{p-2}\\dd t\\\\\n&=\\int_h^1 t^{p-1}\\Big(\\frac{1}{\\sqrt{t^2-h^2}}-\\frac{1}{t}\\Big)\\dd t -\\frac{h^{p-1}}{p-1}.\n\\end{align*}\nThe latter integral can be estimated by\n\\begin{align*}\n\\int_h^1 t^{p-1}\\Big(\\frac{1}{\\sqrt{t^2-h^2}}-\\frac{1}{t}\\Big)\\dd t\n&\\le h^2\\int_h^1 \\frac{t^{p-3}}{\\sqrt{t^2-h^2}}\\dd t\\\\\n&= h^{p-1}\\int_1^{1/h} \\frac{u^{p-3}}{\\sqrt{u^2-1}}\\dd u.\n\\end{align*}\nIn particular, if $1< p<2$, then\n\\begin{equation} \\label{eq:hoelder}\n0\\le \\frac{\\pi}{p}\\frac{f_p(h)-f_p(0)}{h^{p-1}}\n\\le \\int_1^{\\infty} \\frac{u^{p-3}}{\\sqrt{u^2-1}}\\dd u\n<\\infty.\n\\end{equation}\nThus, in this case we obtain that $f_p$ is Hölder continuous of order $p-1$ at $0$. For $p\\ge 2$, a similar argument shows that $f_p$ is continuously differentiable at $0$ with $f_p'(0)=0$. In conclusion, if $p\\ge\\frac{3}{2}$, the function $f_p$ is Hölder continuous of order $\\frac{1}{2}$ on $[-1,1]$. This completes the proof.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}\n\\begin{proof}\nIt follows from the definition of $c_{n}$ that \n\\[\nc_{n}\n=\\frac{1}{n!}\\Gamma_2\\Big(n+1;\\frac{2}{\\beta},1\\Big)\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(\\frac{\\beta}{2})^{\\frac{\\beta}{2}}}\\bigg)^n\\Big(\\frac{4\\pi}{\\beta}\\Big)^{\\frac{\\beta}{4}n^2-(\\frac{1}{2}+\\frac{\\beta}{4})n},\n\\]\nwhere $\\Gamma_2(x;\\frac{2}{\\beta},1)$ is a particular instance of the Barnes double Gamma function, see \\cite{BG24,Spr09}. Adapting \\cite[eq. (7.17)]{BG24}, the following asymptotic expansion holds\n\\begin{align*}\n\\ln \\Gamma_2\\Big(n+1;\\frac{2}{\\beta},1\\Big) \n=&-\\frac{\\beta}{4}n^2\\ln n+\\frac{3\\beta}{8}n^2+\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)n\\ln n-\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)n\\\\\n&+\\bigg(\\frac{1}{2} -\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\bigg)\\ln n +a_{\\beta}'+o(1),\n\\end{align*}\nwhere $a_{\\beta}'\\in \\IR$ is some constant. Combined with Stirling's approximation $\\ln n!=n\\ln n-n+\\frac{1}{2}\\ln(2\\pi n)+o(1)$ we deduce the claimed expansion.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nLet $\\beta\\in \\{1,2,4\\}$ and $p\\ge \\frac{3}{2}$. Recall that $d_n=\\frac{\\beta}{2}n^2+(1-\\frac{\\beta}{2})n$. By Lemma~\\ref{lem:vol-partition} it holds that\n\\begin{equation} \\label{eq:vol-asymp-parts}\n\\ln \\vol(\\bpb)\n=\\ln c_{n}+\\frac{d_n}{p}\\ln \\Big(\\frac{\\beta nv_p}{2}\\Big)-\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)+\\ln Z_{n,p,\\beta}.\n\\end{equation}\nBy Stirling's approximation, as $n\\to\\infty$,\n\\begin{align}\\label{eq:p-term-1}\n\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)\n&=\\frac{d_n}{p}\\Big(\\ln \\frac{d_n}{p}-1\\Big)+o(n)\\notag\\\\\n&=\\frac{d_n}{p}\\Big(2\\ln n+\\ln \\frac{\\beta}{2p}-1+\\Big(1-\\frac{\\beta}{2}\\Big)\\frac{2}{\\beta n}\\Big)+o(n).\n\\end{align}\nTherefore, \n\\begin{align}\\label{eq:p-term-2}\n\\frac{d_n}{p}\\ln\\Big(\\frac{\\beta n v_p}{2}\\Big)\n-\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)\n=&-\\frac{d_n}{p}\\Big(\\ln n -\\ln(p v_p)-1+\\Big(1-\\frac{\\beta}{2}\\Big)\\frac{2}{\\beta n}\\Big)\\notag\\\\\n=&-\\frac{\\beta}{2p}n^2\\ln n+\\frac{\\beta}{2p}\\big(\\ln(pv_p)+1\\big)n^2\\notag\\\\\n&-\\frac{1}{p}\\Big(1-\\frac{\\beta}{2}\\Big)n\\ln n+\\frac{\\ln(pv_p)}{p}\\Big(1-\\frac{\\beta}{2}\\Big)n +o(n).\n\\end{align}",
    "post_theorem_intro_text_len": 7939,
    "post_theorem_intro_text": "In case of $p=2$ it is known that $\\mathrm{Ent}(\\mu_2)=\\ln \\pi -\\frac{1}{2}$ where $\\mu_2$ is the semi-circle law. Otherwise the entropy $\\mathrm{Ent}(\\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \\eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \\cite{GMP25} in case of $p\\ge 2$. \n\nThe proof of Theorem~\\ref{thm:main} relies on the fact that the volume of $\\mathbb{B}_{p,\\beta}^n$ can be expressed in terms of the integral\n\\begin{equation} \\label{eq:Z}\nZ_{n,p,\\beta}\n=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta} e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\mathrm{d} x,\n\\end{equation}\nwhere\n\\[\nv_p =\\frac{\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\Gamma(\\frac{p+1}{2})}=\\frac{1}{p\\alpha_p}.\n\\]\nThe quantity $Z_{n,p,\\beta}$ is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p |x|^p$, where eigenvalues/particles are distributed with joint density \n\\begin{equation} \\label{eq:density}\nf_{n,p,\\beta}(x)\n=\\frac{1}{Z_{n,p,\\beta}}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta} e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\mathrm{d} x, \\qquad x\\in \\mathbb{R}^n.\n\\end{equation}\nThe parameter $\\beta$ stands for the inverse temperature and may take any positive value, while for the matrix representation we restrict to $\\beta\\in \\{1,2,4\\}$. We refer to \\cite{AGZ10} and \\cite{For10} for more information on $\\beta$-ensembles. \n\nIn this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional\n\\[\n\\mu\\mapsto I_p(\\mu)\n=-\\int_{\\mathbb{R}}\\int_{\\mathbb{R}} \\ln|x-y|\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu(x)\n\\]\nover $\\mu\\in \\mathcal{P}(\\mathbb{R})$, where $\\mathcal{P}(\\mathbb{R})$ denotes the space of probability measures on $\\mathbb{R}$.  More precisely, if $X=(X_1,\\dots,X_n)$ is distributed according to the density $f_{n,p,\\beta}$, then the random empirical measure $ \\mu_{n,p} =\\frac{1}{n}\\sum_{i=1}^{n}\\delta_{X_i} $  tends to $\\mu_p$ as $n\\to\\infty$. Moreover, large deviation principles due to \\cite{BG97} and \\cite{HP00} yield \n\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}\nEarlier work \\cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \\cite{DFG+23}, the limit \\eqref{eq:Z-first} is used to prove \\eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\\'e and Serfaty \\cite{LS17} allows to refine \\eqref{eq:Z-first}. The following result forms the core of the proof of \\eqref{eq:main-1} in Theorem~\\ref{thm:main}. \n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\mathrm{Ent}(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nIndependently, in the recent work \\cite{GMP25} the authors deduced the statement of Theorem~\\ref{thm:LS} in case of $p\\ge 2$ from a central limit theorem for linear eigenvalue statistics. The restriction to $p\\ge 2$ results from regularity requirements of the underlying approach, see also \\cite{DFG+23}. The slightly milder restriction on $p$ in Theorem~\\ref{thm:LS} is due to the fact that the Ullman density $f_p$ is Hölder continuous of order $\\frac{1}{2}$ if and only if $p\\ge\\frac{3}{2}$. This is required to apply the results in \\cite{LS17}. \n\nThe term $\\mathrm{Ent}(\\mu_p)$ in Theorem~\\ref{thm:LS} arises via a rescaling property of the specific entropy which is part of the rate function in the large deviation principle in \\cite{LS17}. In case of $\\beta=2$, the contribution of $\\mathrm{Ent}(\\mu_p)$ vanishes and more precise asymptotics hold. The following is a consequence of \\cite[Prop.~1.1]{CKM23} which is based on \\cite{KM99}.\n\n\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\mathbb{R}$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:CKM} combined with Lemma~\\ref{lem:vol-partition} and Lemma~\\ref{lem:cn} below yields \\eqref{eq:main-2} in Theorem~\\ref{thm:main}. In case of $p$ an even integer, the potential $V(x)=v_p |x|^p$ is an analytic function and results of \\cite{BG13} yield a full asymptotic expansion of $\\ln Z_{n,p,\\beta}$, and consequently of $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. In case of $p=2$ the integral $Z_{n,2,\\beta}$ is known via a Mehta-Selberg integral. For $p\\neq 2$ the coefficients in the full asymptotic expansion are defined in terms of integrals of the Stieltjes transform of $\\mu_p$, see also \\cite[Section 7]{BG24} and \\cite{Shc13,Shc14}.\n\nIn the case of non self-adjoint matrices, the volume of the unit ball $B_{p,\\beta}^n$ may be written in terms of \n\\[\n\\tilde{Z}_{n,p,\\beta}\n=\\int_{(\\IR_+)^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\prod_{i=1}^n x_i^{\\beta/2-1} e^{-\\beta n\\, v_p\\sum_{i=1}^{n}x_i^{p/2}}\\mathrm{d} x,\n\\]\nsee e.g.~\\cite[Sec.~4.2]{DFG+23}. This corresponds to the partition function of a Laguerre $\\beta$-ensemble with potential $V(x)=v_p|x|^{p/2}$. The leading order asymptotics for $\\tilde{Z}_{n,p,\\beta}$ can again be obtained via large deviations, leading to \\eqref{eq:sim-asym}. However, the results in \\cite{LS17} do not apply and it goes beyond the scope of the present work to derive next-order asymptotics for $\\tilde{Z}_{n,p,\\beta}$ and $B_{p,\\beta}^n$. \n\n\\begin{remark}\n\tThe unit balls $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ can also be studied for $0<p<1$. Then \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} as well as Theorem~\\ref{thm:CKM} continue to hold. However, even neglecting regularity assumptions, the proof of Theorem~\\ref{thm:main} and the underlying large deviation principle in \\cite{LS17} are limited to $p\\ge 1$. This is because, while the microscopic limit point process in the bulk is the universal sine$_2$-process for $p\\ge 1$, it becomes dependent on $p$ for $p<1$, see \\cite{BEY14,CKM23}. Since the minimizer of the rate function is the free energy which is uniquely given by the sine$_{\\beta}$-process, see \\cite{EHL21,HM25}, the large deviation principle in \\cite{LS17} cannot hold in its present form if $p<1$. \n\\end{remark}\n\n\\begin{remark}\n\tHeuristically, the unit balls $B_{p,\\beta}^n$ and $\\IB_{p,\\beta}^n$ are non-commutative versions of $\\IB_p^n=\\{x\\in \\mathbb{R}^n\\colon \\sum_{i=1}^{n}|x_i|^p\\le 1\\}$. To compare, note that\n\\[\n\\mathrm{vol}(n^{1/p}\\IB_{p}^n)\n=n^{n/p}\\frac{2^n\\Gamma(1+\\frac{1}{p})^n}{\\Gamma(1+\\frac{n}{p})}\n=\\frac{1+o(1)}{\\sqrt{2\\pi n/p}} e^{\\mathrm{Ent}(\\nu_p)n},\n\\]\nwhere $\\nu_p$ has density proportional to $e^{-|x|^p/p}$ and maximizes entropy over $\\mu\\in \\mathcal{P}(\\mathbb{R})$ with $\\int |x|^p\\mathrm{d}\\mu(x) \\le 1$, see also \\cite{KP21}. Similarly, the Ullman distribution $\\mu_p$ maximizes free entropy over $\\mu\\in \\mathcal{P}(\\mathbb{R})$ with $\\int |x|^p\\mathrm{d}\\mu(x)\\le \\alpha_p$, see \\cite[Prop.~5.34]{HP00}. In case of $p=\\infty$, one can interpret $\\nu_{\\infty}$ as the uniform distribution on $[-1,1]$, which maximizes entropy, and $\\mu_{\\infty}$ as the arcsine distribution, which maximizes free entropy. Formally, see Lemma~\\ref{lem:vol-partition} below, the case $\\beta=0$ corresponds to the commutative case where a Poisson point process appears in the microscopic limit, see also \\cite{AD14,BP15,Leb16}. \n\\end{remark}\n\n\\begin{remark}\nTheorem~\\ref{thm:main} can be combined with the stochastic representation in [Cor.~4.3]{KPT20b} of a random matrix drawn uniformly from a Schatten ball $\\mathbb{B}_{p,\\beta}^n$ to obtain information on critical intersections of Schatten $p$-balls in the spirit of \\cite{KPT20b} and \\cite{ST25}. \n\\end{remark}",
    "sketch": "The proof of Theorem~\\ref{thm:main} is based on rewriting $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$ in terms of the partition function\n\\[\nZ_{n,p,\\beta}=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\, e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\,\\mathrm{d} x,\n\\]\nwhich is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\\beta}$.\n\nIn this setting, the Ullman distribution $\\mu_p$ appears as the equilibrium measure minimizing\n\\[\nI_p(\\mu)=-\\iint\\ln|x-y|\\,\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int |x|^p\\,\\mathrm{d}\\mu(x),\n\\]\nand the empirical measure $\\mu_{n,p}=\\frac1n\\sum_{i=1}^n\\delta_{X_i}$ (for $X\\sim f_{n,p,\\beta}$) converges to $\\mu_p$. Large deviations yield the leading-order limit\n\\[\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}=I_p(\\mu_p)=\\ln 2+\\frac{3}{2p}.\n\\]\n\nTo obtain the next-order terms needed for \\eqref{eq:main-1}, the argument uses the large deviation principle for tagged empirical fields of Lebl\\'e--Serfaty \\cite{LS17} to refine this to the expansion stated in Theorem~\\ref{thm:LS}:\n\\[\n\\ln Z_{n,p,\\beta}=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\mathrm{Ent}(\\mu_p)n+o(n).\n\\]\nThe restriction $p\\ge\\tfrac32$ is tied to the requirement that the Ullman density is H\\\"older $1/2$, which is needed to apply \\cite{LS17}. The appearance of $\\mathrm{Ent}(\\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of \\cite{LS17}.\n\nFor the $\\beta=2$ refinement \\eqref{eq:main-2}, the text explains that the $\\mathrm{Ent}(\\mu_p)$ contribution vanishes and one invokes Theorem~\\ref{thm:CKM} (a consequence of \\cite[Prop.~1.1]{CKM23}) giving sharper asymptotics for $\\ln Z_{n,p,2}$; then “Theorem~\\ref{thm:CKM} combined with Lemma~\\ref{lem:vol-partition} and Lemma~\\ref{lem:cn} below yields \\eqref{eq:main-2} in Theorem~\\ref{thm:main}.”",
    "expanded_sketch": "The proof of the main theorem is based on rewriting $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$ in terms of the partition function\n\\[\nZ_{n,p,\\beta}=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\, e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\,\\mathrm{d} x,\n\\]\nwhich is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\\beta}$.\n\nIn this setting, the Ullman distribution $\\mu_p$ appears as the equilibrium measure minimizing\n\\[\nI_p(\\mu)=-\\iint\\ln|x-y|\\,\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int |x|^p\\,\\mathrm{d}\\mu(x),\n\\]\nand the empirical measure $\\mu_{n,p}=\\frac1n\\sum_{i=1}^n\\delta_{X_i}$ (for $X\\sim f_{n,p,\\beta}$) converges to $\\mu_p$. Large deviations yield the leading-order limit\n\\[\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}=I_p(\\mu_p)=\\ln 2+\\frac{3}{2p}.\n\\]\n\nTo obtain the next-order terms needed for\n\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}\nthe argument uses the large deviation principle for tagged empirical fields of Lebl\\'e--Serfaty (LS17) to refine this to the following result.\n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nThe restriction $p\\ge\\tfrac32$ is tied to the requirement that the Ullman density is H\\\"older $1/2$, which is needed to apply LS17. The appearance of $\\mathrm{Ent}(\\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of LS17.\n\nFor the $\\beta=2$ refinement\n\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}\none notes that the $\\mathrm{Ent}(\\mu_p)$ contribution vanishes and one invokes the following sharper asymptotics.\n\n\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\IR$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}\n\nThen the theorem above, combined with the following two lemmas,\n\n\\begin{lemma} \\label{lem:vol-partition}\nLet $\\beta\\in\\{1,2,4\\}$ and $1\\le p<\\infty$. Then\n\\begin{align*}\n\\vol(\\bpb)\n&=c_n\\int_{\\IB_p^n} \\prod_{1\\le i< j\\le n}|x_i-x_j|^{\\beta}\\dd x_1 \\cdots \\dd x_n\\\\\n&=\\frac{c_{n}}{\\Gamma(1+\\frac{d_n}{p})}\\Big(\\frac{n\\beta v_p}{2}\\Big)^{d_n/p}Z_{n,p,\\beta}\n\\end{align*}\nwhere $Z_{n,p,\\beta}$ and $v_p$ are as in \\eqref{eq:Z} and\n\\[\nc_{n}\n=\\frac{1}{n!}\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(2\\pi)^{\\beta/2}}\\bigg)^{n}\\prod_{k=1}^n\\frac{(2\\pi)^{\\beta k/2}}{\\Gamma(\\frac{\\beta k}{2})}.\n\\]\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}\n\nyields the stated $\\beta=2$ volume asymptotics, completing that part of the main theorem.",
    "expanded_theorem": "\\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\mathbb{R}$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\mathrm{vol}(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation},",
    "theorem_type": [
      "Asymptotic or Limit",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb F_1=\\mathbb R\\), \\(\\mathbb F_2=\\mathbb C\\), and \\(\\mathbb F_4=\\mathbb H\\). For \\(\\beta\\in\\{1,2,4\\}\\) and \\(A\\in \\mathbb F_\\beta^{n\\times n}\\), let \\(s_1(A)\\ge\\cdots\\ge s_n(A)\\ge0\\) be the singular values and define the Schatten \\(p\\)-norm by \\(\\|A\\|_p=(\\sum_{j=1}^n s_j(A)^p)^{1/p}\\) for \\(1\\le p<\\infty\\) and \\(\\|A\\|_\\infty=s_1(A)\\). Let \\(\\mathcal H_n(\\mathbb F_\\beta)=\\{A\\in\\mathbb F_\\beta^{n\\times n}:A^*=A\\}\\), and let \\(\\mathbb B_{p,\\beta}^n=\\{A\\in\\mathcal H_n(\\mathbb F_\\beta):\\|A\\|_p\\le1\\}\\), whose volume is taken in the corresponding real vector space of self-adjoint matrices. Also let \\(A(p)=\\frac12\\Big(\\frac{p\\sqrt\\pi\\,\\Gamma(p/2)}{\\sqrt e\\,\\Gamma((p+1)/2)}\\Big)^{1/p}\\) for finite \\(p\\) (and \\(A(\\infty)=\\tfrac12\\)), and let \\(\\mathrm{Ent}(\\mu_p)\\) denote the differential entropy of the Ullman distribution \\(\\mu_p\\). Which asymptotic statement for \\(\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\\) as \\(n\\to\\infty\\) holds for \\(p\\ge \\tfrac32\\), and what sharper asymptotic statement holds in the special case \\(\\beta=2\\) and \\(p\\ge1\\)?",
      "correct_choice": {
        "label": "A",
        "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge 1\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"
        },
        {
          "label": "C",
          "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n+o(n^2).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n+o(n^2).\n\\]"
        },
        {
          "label": "D",
          "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"
        },
        {
          "label": "E",
          "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "threshold_p_ge_3_2",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped_nlogn_and_linear_terms",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "entropy_linear_term_from_tagged_field_LDP",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "coefficient_of_linear_1_over_2p_term",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at which option is correct; it asks for the correct asymptotic statement and sharper special case. There is no substantive answer leakage from the mathematical wording of the stem itself."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the exact statement of a theorem, so it is largely theorem-recall rather than a conceptually rephrased application. Still, the presence of several nearby alternatives means it is not a pure tautological restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Selecting the best option requires some checking of parameter ranges, subleading terms, and the entropy contribution, but the task mainly tests precise recall/recognition of a known asymptotic expansion rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic errors: extending the validity range in p, dropping lower-order terms, omitting the entropy term, and flipping signs of subleading logarithmic terms. They are distinct and well aligned with common failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-recall-heavy MCQ: no answer leakage, strong distractors, and some reasoning pressure, but limited originality because it closely mirrors a specific asymptotic theorem statement."
    }
  },
  {
    "id": "2512.05068v1",
    "paper_link": "http://arxiv.org/abs/2512.05068v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $n \\to \\infty$, abbreviated later as whp.}, the separating systole of $\\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\text{sepsys}(\\bT)\\leq B\\log(n)\\]",
      "start_pos": 52799,
      "end_pos": 53128,
      "label": "thm_sepsys"
    },
    "ref_dict": {
      "eq_psi": "\\begin{equation}\\label{eq_psi}\n\\psi(\\theta)=\\frac{1}{36\\lambda(\\theta)^4}\\left(1-12\\lambda(\\theta)-24\\lambda(\\theta)^2H(\\lambda(\\theta))\\right).\n\\end{equation}",
      "thm_sepsys": "\\begin{theorem}\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $\\nto$, abbreviated later as whp.}, the separating systole of $\\bT$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\sep\\leq B\\log(n)\\] \n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2484,
    "pre_theorem_intro_text": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nWhen both $n$ and $g$ grow large, exact asymptotics are not available. However, for \\emph{unicellular maps} (maps with a single face), a bijection~\\cite{CFF13} provides a way to understand their geometry~\\cite{ACCR13,Ray13a, Louf-expander, JansonLouf1, JansonLouf2}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.",
    "context": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.",
    "full_context": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nNote that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \\cite{MT01}[Chapter 5]. However, we do conjecture that \\whp{}, it does exist in $\\bT$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\\frac{\\sep}{\\log(n)}$ in probability, in more details in~\\cref{sec_conj}.\n\nIn the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \\emph{genus ratio}. Let us give more context. Let $\\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\\cite{BL19}, together with Budzinski, the convergence of the \"\\emph{size ratio}\"\n\\begin{equation}\\label{eq_ratio_size}\n\\frac{\\tau(n-1,g_n)}{\\tau(n,g_n)}\\to \\lambda(\\theta),\n\\end{equation}\nas $\\nto$, where $\\lambda$ is an explicit function.\n\n\\begin{proposition}\\label{prop_asympto_ratio}\nFor all $\\theta \\in \\left( 0,\\frac{1}{2} \\right)$, there is a constant $a_{\\theta} \\in (0,1)$ such that the following holds. Let $(g_n)$ be a sequence such that $0 \\leq g_n \\leq \\frac{n+1}{2}$ for every $n$ and $\\frac{g_n}{n} \\to \\theta$. For all integers $m$ and $h$ satisfying $0\\leq m\\leq \\frac{n}{2}$ and $0\\leq h\\leq \\min \\left( \\frac{g_n}{2}, \\frac{m+1}{2} \\right)$, we have\n\\[\n\\frac{\\tau(n,g_n)}{\\tau(n-m,g_n-h)}\\geq a_{\\theta}^{h}\\frac{n^{2g_n}}{(n-m)^{2(g_n-h)}}\\exp \\left( f\\left(\\frac{g_n}{n}\\right) n -f\\left(\\frac{g_n-h}{n-m}\\right)(n-m)+ o(m) \\right),\n\\]\nwhere the $o(m)$ is uniform in $(m,h)$ as $n \\to +\\infty$ (that is, it is bounded by $m\\eps(n)$ with $\\eps(n) \\to 0$ as $n \\to +\\infty$).\n\\end{proposition}\n\n\\paragraph{\\textbf{Case 4:}} To show~\\cref{eq_ineq_smaller_n2} (and thus finish the proof of the lemma), it remains to show that \n\\[\\sum_{n_1+n_2=n\\atop n_2\\leq (5+b)\\log n}n_1^a\\tau(n_1,g_n-1)n_2^b\\tau(n_2,1)\\leq n^{a-2+o(1)}\\tau(n,g_n)).\\]\nBut, on the one hand, by \\cref{lem_asympto_genus_ratio}, uniformly for all $n_2\\leq (5+b)\\log n$, we have\n\\[\\frac{\\tau(n-n_2,g_n-1)}{\\tau(n,g_n-1)}\\leq n^{-2+o(1)}\\lambda(\\theta)^{n_2}.\\]\nAnd on the other hand, it is well-known~(see for instance~\\cite{Kri07}) that the generating series\n\\[H_b(x)=\\sum{n\\geq 0} n^b\\tau(n,1)x^n\\]\nhas radius of convergence $\\lambda(0)>\\lambda(\\theta)$, therefore $H_b(\\lambda(\\theta))<\\infty$ and\n\\[\\sum_{n_1+n_2=n\\atop n_2\\leq (5+b)\\log n}n_1^a\\tau(n_1,g_n-1)n_2^b\\tau(n_2,1)\\leq n^{a-2+o(1)}\\sum_{k\\geq 0}\\lambda(\\theta)^kk^b\\tau(k,1)=n^{a-2+o(1)}H_b(\\lambda(\\theta)).\\]\nThis finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{prop_simple}\nLet $A=\\frac{1}{2\\log\\lambda(\\theta)}$, we have\n\\[\\sum_{\\ell\\leq A\\log n} \\frac{|\\Ts(2n,g_n,\\ell)|}{\\tau(n,g_n)}\\to 0\\]\nas $\\nto$.\n\\end{proposition}\nLet us start with an inequality that holds for all $n,g,\\ell$.\n\n\\subsection{Counting \\te{s}}\nIn this section, we prove that $\\bT$ does not contain small \\te{s}, that is:\n\\begin{proposition}\\label{prop_thin}\nLet $A=\\frac{1}{2\\log\\lambda(\\theta)}$, we have\n\\[\\sum_{\\ell\\leq A\\log n} \\frac{\\Tt(2n,g_n,\\ell)}{\\tau(n,g_n)}\\to 0\\]\nas $\\nto$.\n\\end{proposition}\n\n\\begin{conjecture}\\label{conj_constant}\nLet $\\sep^*$ be the simple separating systole of $\\bT$.\nFor every $\\theta\\in(0,1/2)$, there exists two constants $0<S(\\theta)<S^*(\\theta)$ such that\n\\[\\frac{\\sep}{\\log(n)}\\to S(\\theta)\\quad \\text{and}\\quad \\frac{\\sep^*}{\\log(n)}\\to S^*(\\theta)\\]\nin probability as $\\nto$.\n\\end{conjecture}",
    "post_theorem_intro_text_len": 2390,
    "post_theorem_intro_text": "Note that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \\cite{MT01}[Chapter 5]. However, we do conjecture that with high probability{}, it does exist in $\\mathbf{T}_{2n,g_n}$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\\frac{\\text{sepsys}(\\bT)}{\\log(n)}$ in probability, in more details in~\\cref{sec_conj}.\n\nThe upper bound in~\\cref{thm_sepsys} follows directly from a theorem of~\\cite{CVHM14}. To establish the upper bound, we will resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then  use a first moment method. However, contrary to previous works, we are dealing this time with non simple cycles, with can become quite intricate, especially as topology is involved. This is one of the main challenges of this work, and it is largely dealt with in~\\cref{sec_geom}.\n\nIn the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \\emph{genus ratio}. Let us give more context. Let $\\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\\cite{BL19}, together with Budzinski, the convergence of the \"\\emph{size ratio}\"\n\\begin{equation}\\label{eq_ratio_size}\n\\frac{\\tau(n-1,g_n)}{\\tau(n,g_n)}\\rightarrow \\lambda(\\theta),\n\\end{equation}\nas $n \\to \\infty$, where $\\lambda$ is an explicit function.\n\nIt is a natural question to ask for a similar result when the genus varies and the size is fixed. We answer it with the following theorem:\n\n\\begin{theorem}\\label{thm_ratio}\nThere exists an explicit function $\\psi$ (see~\\cref{eq_psi}) such that\n\\begin{equation} \n\\frac{n^2\\tau(n,g_n-1)}{\\tau(n,g_n)}\\rightarrow \\psi(\\theta),\n\\end{equation}\nas $n \\to \\infty$.\n\\end{theorem}\n\n\\paragraph{Acknowledgments:} Thanks to Thomas Budzinski, Guillaume Chapuy, Andrew Elvey-Price, Wenjie Fang and Michael Wallner for numerous joint works and enriching discussions about high genus geometry and enumeration. Thanks also to Arnaud de Mesmay and Hugo Parlier for interesting exchanges around \\cref{thm_upper,conj_Barnette,conj_decomp}.",
    "sketch": "The upper bound in~\\cref{thm_sepsys} \"follows directly from a theorem of~\\cite{CVHM14}.\" For the lower bound, the authors \"resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method.\" They emphasize that, \"contrary to previous works,\" they must handle \"non simple cycles\" which \"can become quite intricate, especially as topology is involved,\" and that this challenge is \"largely dealt with in~\\cref{sec_geom}.\" In proving the lower bound they also \"adapt some of the techniques to establish\" an enumerative input, the \"convergence of the genus ratio\" (Theorem~\\ref{thm_ratio}).",
    "expanded_sketch": "The upper bound in establishing the main theorem follows directly from a theorem of~\\cite{CVHM14}. For the lower bound, the authors \"resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method.\" They emphasize that, \"contrary to previous works,\" they must handle \"non simple cycles\" which \"can become quite intricate, especially as topology is involved,\" and that this challenge is \"largely dealt with\" later. In proving the lower bound they also \"adapt some of the techniques to establish\" an enumerative input, the \"convergence of the genus ratio\" (Theorem~\\ref{thm_ratio}).,",
    "expanded_theorem": "\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $n \\to \\infty$, abbreviated later as whp.}, the separating systole of $\\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\text{sepsys}(\\bT)\\leq B\\log(n)\\],",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Fix \\(\\theta\\in(0,1/2)\\) and let \\((g_n)\\) be a sequence with \\(g_n/n\\to\\theta\\). For each \\(n\\), let \\(\\mathbf{T}_{2n,g_n}\\) be a uniform triangulation of genus \\(g_n\\) with \\(2n\\) faces. Define the separating systole \\(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\) to be the length of the shortest cycle in \\(\\mathbf{T}_{2n,g_n}\\) that is separating and non-contractible (the cycle is not required to be simple). Which statement holds for \\(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\) as \\(n\\to\\infty\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exist constants \\(0<A<B\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Equivalently, with high probability, the separating systole of \\(\\mathbf{T}_{2n,g_n}\\) is of logarithmic order."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist constants \\(0<A<B\\), depending only on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] In particular, with high probability the separating systole is bounded above by an absolute constant while remaining bounded below by a logarithmic term."
        },
        {
          "label": "C",
          "text": "There exists a constant \\(A>0\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\ge A\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Thus, with high probability, the separating systole is at least of logarithmic order."
        },
        {
          "label": "D",
          "text": "For every sequence \\((g_n)\\) with \\(g_n/n\\to\\theta\\), there exist universal constants \\(0<A<B\\), independent of \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Equivalently, the logarithmic bounds can be chosen uniformly over all \\(\\theta\\in(0,1/2)\\)."
        },
        {
          "label": "E",
          "text": "There exist constants \\(0<A<B\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)=1-o(n^{-1})\\quad\\text{as }n\\to\\infty.\\] In other words, the separating systole is of logarithmic order with an explicit polynomially fast error probability."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "upper-bound scale from logarithmic to constant",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "dropped the upper logarithmic bound",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "dependence of constants on theta",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "non-effective whp strengthened to polynomial-rate error bound",
          "template_used": "uniformity_effectivity"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the logarithmic-order conclusion or identify the correct option. It only sets up the model and asks which asymptotic statement is true."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct answer appears to be essentially the theorem statement itself, so the item is close to theorem recall. However, the presence of weaker/stronger variants means it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing the exact theorem from a weaker true statement, a uniform-in-theta strengthening, and an effective-rate strengthening. Still, the item mainly tests precise recall of the theorem’s quantifiers and strength rather than deeper generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "Most distractors are mathematically meaningful and target common failure modes: accepting a weaker statement as the main result, overlooking dependence on theta, or overclaiming an error rate. Although choice B is somewhat easier to dismiss, overall the distractors are distinct and well targeted."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-recall-heavy MCQ: little answer leakage, decent distractors, and moderate discrimination via quantifier/strength differences, but limited generative reasoning."
    }
  },
  {
    "id": "2512.05340v1",
    "paper_link": "http://arxiv.org/abs/2512.05340v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}",
      "start_pos": 10002,
      "end_pos": 10457,
      "label": "Theorem: high temp O(N)"
    },
    "ref_dict": {
      "Theorem: main result": "\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}",
      "Theorem: Critical O(N)": "\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}",
      "eq:mean-field O(N) measure": "\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5192,
    "pre_theorem_intro_text": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n  \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nOur method relies on a general multivariate nonnormal Stein's method framework established\nin~\\cite{FangShaoXu2019,FangShaoXu2019correction}, which in turn can be viewed as a multivariate generalization\nof~\\cite{ChatterjeeShao2011}. \nAs explicitly noted in~\\cite{FangShaoXu2019}, however, their underlying assumptions\ndo not hold for densities of the form $\\exp(-a|x|^m)$ with $m>2$, and so their results cannot be applied to the critical mean-field $O(N)$\nmodel. Nevertheless, as shown here, a generalization of~\\cite[Theorem 2.5]{FangShaoXu2019} can be established which does indeed cover the\ncase of the critical $O(N)$ model. We present this result in Theorem~\\ref{Theorem: main result} of Section~\\ref{sec: Stein's method}.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n  d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n  \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.  \n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n  1.1]{FangKoike2022} yields the following.",
    "context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}",
    "full_context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\n\\begin{lemma}\\label{lemma: Critical remainders}\n Let $(W, W')$ be as described above. Then\n \\begin{equation}\\label{eq: W' - W conditioned on sigma}\n \\EE[W'-W|\\sigma] = \\frac{1}{Nn^{3/2}}\\left(-\\frac{N^{2}}{N+2}|W|^{2}W + R_{1}\\right),\n \\end{equation}\n where\n \\begin{equation}\\label{eq: remainder 1}\n \\begin{split}\n R_{1} &= \\frac{N}{n^{1/4}}\\sum_{i=1}^{n}\\left[m_{(i)}f_{N}(|m_{(i)}|) - mf_{N}(|m|)\\right] \\\\\n & \\quad + Nn^{3/4}\\left(f_{N}(|m|) - 1 + \\frac{N|m|^{2}}{(N+2)}\\right)m.\n \\end{split}\n \\end{equation}\n and \n \\begin{equation}\\label{eq: W' - W W' - W T conditioned on sigma}\n \\begin{split}\n \\EE[(W'-W)(W'-W)^{\\transpose}|\\sigma] &= 2\\frac{1}{Nn^{3/2}}\\left(\\id + R_{2}\\right),\n \\end{split}\n \\end{equation}\n where \n \\begin{equation}\\label{eq: remainder 2}\n \\begin{split}\n R_{2} &= \\left[\\frac{N}{2n}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose} - \\frac{1}{2}\\id\\right] -\n \\frac{N}{\\sqrt{n}}WW^{\\transpose} + \\frac{N}{n^{2}}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose}\\\\ \n &\\quad + \\frac{N}{2n}\\sum_{i=1}^{n}\\,g_{N}(|m_{(i)}|)\\,|m_{(i)}|^2\\,\n \\left[\\sigma_{i}m_{(i)}^{\\transpose}+ m_{(i)} \\sigma_{i}^{\\transpose} - \\frac{\\id}{N}\\right]\n + \\frac{N}{2n}\\sum_{i=1}^{n}g_{N}(|m_{(i)}|)m_{(i)}m_{(i)}^\\transpose. \n \\end{split}\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We follow a similar strategy to the proof~\\cite[Lemma 1]{KirkpatrickNawaz2016} (see also \\cite[Lemma 11]{KirkpatrickMeckes2013}).\n The key ingredients are the small $x$ asymptotics for $f_N(x)$ given in~\\eqref{eq: fN asymptotics}, combined with the fact\n that, as shown in~\\cite[Theorem 3]{KirkpatrickNawaz2016}, the quantity $m$ concentrates at 0 as $n\\to \\infty$ when $\\beta = N$.\n\nFinally, to establish~\\eqref{eq: Bound hilbert schmidt norm 2}, we fix $\\epsilon>0$ and $y\\in\\RR^d$ with $|y| \\leq 1$ and apply a\n similar decomposition to that used in~\\cite{FangShaoXu2019}. Specifically, defining\n \\begin{equation}\\label{eq: Phi definition}\n \\Phi(t) :=\\left[D^{2}P_{t}h(x + \\epsilon y)[u, v] - D^{2}P_{t}h(x)[u, v]\\right],\n \\end{equation}\n it follows by~\\eqref{eq: second derivative of f} that\n \\begin{equation}\\label{eq: rewriting f(x + epsilon u) - f(x)}\n D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x) = -\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t)\\,\\dt -\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt.\n \\end{equation}\n Equation~\\eqref{eq: Ptphi derivative 2} implies\n \\begin{equation}\\label{eq: bound int 0 to epsilon squared}\n \\begin{split}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| &\\leq 2|u||v|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon} S(t)\\, \\dt,\\\\\n &\\leq 2\\,|u| |v|\\,(\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right).\n \\end{split}\n \\end{equation}\n If $\\epsilon\\le 1$ then $\\epsilon\\wedge \\sqrt{\\epsilon}=\\epsilon$, in which case\n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) = \n \\epsilon\\, (s_0\\epsilon +2s_{-1/2})\\,\\leq \\epsilon (s_0 + 2s_{-1/2}).\n \\end{equation}\n While if $\\epsilon>1$ then $\\epsilon\\wedge\\sqrt{\\epsilon}=\\sqrt{\\epsilon}$, and so \n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) =\n \\sqrt{\\epsilon}\\, \\left(s_0\\sqrt{\\epsilon}+2s_{-1/2}\\right)\\leq\n \\epsilon\\left(s_{0} +2s_{-1/2}\\right).\n \\end{equation}\n We therefore conclude from~\\eqref{eq: bound int 0 to epsilon squared} that for all $\\epsilon>0$\n \\begin{equation}\\label{eq: int bound 0 to min}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| \\leq 2\\epsilon (s_0 + 2s_{-1/2})|u| |v|.\n \\end{equation}\n Now consider the remaining term in~\\eqref{eq: rewriting f(x + epsilon u) - f(x)}. It follows from~\\eqref{eq: Ptphi derivative 3} that\n $D^2P_t h(x+\\cdot)[u,v]$ is differentiable with bounded derivative, and so the fundamental theorem of calculus implies \n \\begin{equation}\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt\n = \\epsilon\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\int_{0}^{1} D^{3}P_{t}h(x + \\epsilon r y)[u, v, y]\\, \\dr\\, \\dt. \n \\end{equation}\n Therefore, using $|y|\\le1$, and again applying~\\eqref{eq: Ptphi derivative 3}, yields\n \\begin{equation}\n \\label{eq: phi tail bound}\n \\begin{split}\n \\left|\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi\\,\\dt\\right| \n &\\leq \\epsilon |u||v| \\int_{\\epsilon\\wedge\\epsilon^{2}}^{\\infty}\\, Q(t) \\,e^{-\\theta t/4} \\dt,\\\\\n &\\leq \\epsilon |u||v|\n \\left[\\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + q_{-1} E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right)\\right]\n \\end{split}\n \\end{equation}\n where $E_{1}(\\cdot)$ denotes the exponential integral~\\cite[Equation 5.1.1]{AbramowitzStegun1972}. Applying~\\cite[Equation 5.1.20]{AbramowitzStegun1972} \n yields\n \\begin{equation}\n \\label{eq: bound on exponential integral}\n \\begin{split}\n E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right) &\\le 2\\left(1\\vee \\left|\\log \\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4} \\right|\\right)\\\\\n &\\le 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right)\\left(1\\vee |\\log\\epsilon|\\right).\n \\end{split}\n \\end{equation}\n Applying~\\eqref{eq: bound on exponential integral} to~\\eqref{eq: phi tail bound}, and combining the result with~\\eqref{eq: int bound 0\n to min} and~\\eqref{eq: rewriting f(x + epsilon u) - f(x)} then yields \n \\begin{equation}\n \\sup_{x,y\\in\\RR^d : |y|\\le 1} |D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x)|\n \\le\n K_3\\,|u|\\, |v|\\, \\epsilon\\, (1\\vee|\\log\\epsilon|)\n \\label{eq: third main bound}\n \\end{equation}\n with\n \\begin{equation}\n K_3 :=\n 2(s_0 +2s_{-1/2}) + \n \\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right) q_{-1}.\n \\end{equation}\n\nThe analogue of Proposition~\\ref{prop: mean-square bounds} in~\\cite{FangShaoXu2019} is Lemma 5.2. While the proof of the latter is\nessentially immediate under their assumption of strict convexity of $V$,\nthe proof of Proposition~\\ref{prop: mean-square bounds} under Assumption~\\ref{assumption: assumptions on V} is somewhat more involved. A key\ningredient in its proof is the following lemma, whose proof is deferred to the end of this section. \n\\begin{lemma}\\label{lemma: E(t) bound}\n If $V$ satisfies Assumption~\\ref{assumption: assumptions on V}, then for all $x\\in\\RR^d$ and $0\\le s\\le t<\\infty$ \n $$\n \\EE \\exp\\left(-2\\int_s^t \\rho(X_r^x) \\dr \\right) \\le C_2^2 e^{-2\\theta (t-s)}.\n $$\n\\end{lemma}\nIn addition, we will also require the following Gr\\\"{o}nwall-type lemma.\n\\begin{lemma}\\label{lemma: Gronwall type}\n Let $u: \\RR \\to \\RR^{d}$ be such that $|u(t)|^{2} \\in C^{1}(\\RR, \\RR)$. Suppose it satisfies the following differential inequality\n \\begin{equation}\n \\frac{\\diff}{\\dt}|u(t)|^{2} \\leq 2a(t)|u(t)|^{2} + 2b(t)|u(t)|, \\qquad \\forall t \\in (0, T),\n \\end{equation}\n where $T \\in (0, \\infty)$, $a, b \\in L^{1}([0, T])$ and $b \\geq 0$. If $|u(0)| = 0$ then\n \\begin{equation}\n |u(t)| \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds, \\qquad \\forall t \\in [0, T).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Fix $t \\in (0, T)$ and let $\\tau := \\sup\\{0 \\leq s \\leq t: |u(s)| = 0\\}$. If $\\tau = t$ then the statement holds trivially since $b\n \\geq 0$. If $\\tau < t$ then $|u(t)|$ is differentiable on $(\\tau, t]$ and gives \n \\begin{equation}\n \\frac{\\diff}{\\ds}|u(s)| \\leq a(s)|u(s)| + b(s), \\qquad s \\in (\\tau, t].\n \\end{equation}\n Since $|u(\\tau)| = 0$, by the Generalised Jones Inequality \\cite[Theorem 1.2.2]{Qin2016} we have\n \\begin{equation}\n |u(t)| \\leq \\int_{\\tau}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds,\n \\end{equation}\n using $b \\geq 0$. This completes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 4700,
    "post_theorem_intro_text": "Our main result in the current work is the analogous multivariate nonnormal approximation holding at criticality.\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        d(\\mathcal{L}(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\n\\begin{remark}\nAs a corollary of Theorem~\\ref{Theorem: Critical O(N)} it follows that for any $N\\ge2$, the Wasserstein distance between the distribution of\n$n^{-3/4}\\,|S_n|$ and the probability measure with Lebesgue density proportional to $r^{N-1}\\exp(-a_N r^4)$, is again bounded above by\n$c(N)/\\sqrt{n}$. This is analogous to the univariate nonnormal approximation presented in~\\cite[Theorem 6]{KirkpatrickNawaz2016} for\n$|S_n|^2$, however the bound given there is not in terms of Wasserstein distance, but in terms of an integral probability metric defined by\na smaller class of test functions, and contains an additional logarithmic factor.\n\\end{remark}\n\nThe remainder of this paper is organised as follows. Section~\\ref{sec: Stein's method} introduces the relevant background on Stein's method, and\nprovides the statement of our main result on Wasserstein approximation, Theorem~\\ref{Theorem: main result}. In Section \\ref{sec: O(N) model}\nwe apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}. \nSection~\\ref{sec: sdes and semigroups} provides preliminary results related to a class of stochastic differential equations, and their\nrelated stochastic semigroups, which provide the framework for the proof of Theorem \\ref{Theorem: main result} given in Section \\ref{sec: proof of main\n  result}. The key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\nThese bounds are proved in Section~\\ref{sec: semigroup bounds}, using Elworthy-Li formulae and\nbounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\nproved in Section~\\ref{sec: variation process bounds}.\nFinally, the appendix contains the proof of a proposition stated in Section~\\ref{sec: sdes and semigroups}.\n\n\\subsection{Notation}\n\\label{subsec:notation}\nWe will denote the set of positive integers by $\\ZZ_+$, and the set of natural numbers by $\\mathbb{N}:=\\ZZ_+\\cup\\{0\\}$.\nFor $d\\in\\ZZ_+$ and $x,y\\in \\mathbb{R}^{d}$, we let $\\<x,y\\>$ denote the Euclidean inner product, and set $|x| = \\sqrt{\\<x,x\\>}$. For two\nmatrices $A, B \\in \\mathbb{R}^{d \\times d}$ we let $\\<A,B\\>_{{}} = \\sum_{i, j = 1}^{d}A_{i, j}B_{i, j}$ denote the Hilbert-Schmidt inner product,\nand denote the corresponding norm by $\\|A\\|_{{}} = \\sqrt{\\< A, A \\>_{{}}} = \\sqrt{\\text{Tr}(A^{\\mathrm{T}}A)}$. We will denote the operator\nnorm of $A\\in\\mathbb{R}^{d\\times d}$ by $\\|A\\|_{\\mathrm{op}}:=\\sup_{x\\in\\mathbb{S}^{d-1}}|Ax|$. \n\nFor open sets $U_1\\subseteq \\mathbb{R}^\\ell$ and $U_2\\subseteq\\mathbb{R}^m$ we let $C^{k}(U_1;U_2)$ denote the set of $k$-times continuously differentiable\n$f:U_1\\to U_2$. If $U_2=\\mathbb{R}$ we abbreviate $C^k(U_1):=C^k(U_1,U_2)$, and if additionally $U_1=\\mathbb{R}^d$ we write simply $C^k$.\nWe let $B_b$ denote the\nspace of $f:\\mathbb{R}^d\\to\\mathbb{R}$ which are bounded and Borel measurable, and let $C_b$ denote the subspace of $B_b$ of uniformly continuous such functions. We will then make\nuse of the convenient abbreviation $\\mathrm{Lip}^{\\infty}_b:=\\mathrm{Lip}\\cap C^{\\infty}\\cap C_b$.\n\nFor $f\\in C^1(\\mathbb{R}^d;\\mathbb{R})$, we let $D_u f(x)$ denote the directional derivative of $f$ in the direction $u\\in\\mathbb{R}^d$, and we extend the\ndefinition to $f\\in C^1(\\mathbb{R}^d;\\mathbb{R}^m)$, by then defining $D_u f(x)$ entrywise. For given $f\\in C^3(\\mathbb{R}^d;\\mathbb{R}^m)$ and $u,v,w,x\\in\\mathbb{R}^d$ we\ndefine\n\\begin{equation}\n  \\label{eq: D notation}\n  \\begin{split}\n    Df(x)[u]&:= D_u f(x)\\\\\n    D^2f(x)[u,v]&:=D_v D_u f(x)\\\\\n    D^3f(x)[u,v,w]&:=D_w D_v D_u f(x)\n  \\end{split}\n\\end{equation}\nWe note that for each given $x\\in\\mathbb{R}^d$ and $i=1,2,3$, the map $D^i g(x) : (\\mathbb{R}^d)^i\\to\\mathbb{R}^d$ is multilinear and symmetric.\n\nFor $f \\in C^{1}(\\mathbb{R}^d;\\mathbb{R})$, we let $\\nabla f$ denote the gradient of $f$, so that\n\\begin{equation}\n  Df(x)[u]=\\<\\nabla f(x),u\\>,\n\\end{equation}\nand set $\\|\\nabla f\\|_\\infty:=\\sup_{x \\in \\mathbb{R}^{d}}|\\nabla f(x)|$. \nFinally, for $f \\in C^{2}(\\mathbb{R}^{d};\\mathbb{R})$, we let $\\nabla^2 f$ denote the Hessian of $f$,\nso that\n\\begin{equation}\nD^2 f(x)[u,v] =\\<u,\\nabla^2 f(x) v\\>,\n\\end{equation}\nand let $\\Delta$ denote the Laplacian.",
    "sketch": "The introduction does not give a step-by-step proof of Theorem~\\ref{Theorem: high temp O(N)}, but it does indicate how the analogous critical result is proved: the paper \"introduces the relevant background on Stein's method\" and states a main Wasserstein-approximation theorem (Theorem~\\ref{Theorem: main result}); then \"we apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}.\" The proof framework for Theorem~\\ref{Theorem: main result} uses \"a class of stochastic differential equations, and their related stochastic semigroups\"; \"the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\" These derivative bounds are proved \"using Elworthy-Li formulae\" and \"bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\" (via \"variation process bounds\"). No direct proof sketch for Theorem~\\ref{Theorem: high temp O(N)} itself appears in the post-theorem introduction.",
    "expanded_sketch": "The introduction does not give a step-by-step proof of the main theorem, but it does indicate how the analogous critical result is proved: the paper introduces the relevant background on Stein's method and states a main Wasserstein-approximation theorem.\n\n\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}\n\nThen this theorem is applied to prove the following critical result.\n\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\nThe proof framework for the Wasserstein-approximation theorem above uses a class of stochastic differential equations and their related stochastic semigroups; the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups. These derivative bounds are proved using Elworthy--Li formulae and bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation (via variation process bounds). No direct proof sketch for the main theorem itself appears in the post-theorem introduction.",
    "expanded_theorem": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $N\\ge 2$. In the mean-field $O(N)$ model, the spin configuration $\\sigma=(\\sigma_1,\\dots,\\sigma_n)\\in (\\mathbb S^{N-1})^n$ has law with density proportional to\n\\[\n\\exp\\!\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot \\sigma_j\\right)\n\\]\nwith respect to the $n$-fold product of the uniform measure on the unit sphere $\\mathbb S^{N-1}\\subset\\mathbb R^N$. Let the magnetization be\n\\[\nS_n=\\sum_{i=1}^n \\sigma_i,\n\\]\nand define\n\\[\nW_n:=\\sqrt{N-\\beta}\\,\\frac{S_n}{\\sqrt n}.\n\\]\nLet $Z\\sim N(0,\\mathbf I)$ be the standard Gaussian vector in $\\mathbb R^N$, where $\\mathbf I$ is the identity matrix, and let the Wasserstein distance between laws be\n\\[\nd(\\mathcal L(X),\\mathcal L(Y)):=\\inf_{(X,Y)}\\mathbb E|X-Y|=\\sup_{h\\in \\mathrm{Lip}}\\big|\\mathbb Eh(X)-\\mathbb Eh(Y)\\big|,\n\\]\nwhere the supremum is over all $1$-Lipschitz functions $h$. For a fixed inverse temperature $\\beta<N$, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a finite constant $c(N)$, depending only on $N$, such that for every $\\beta<N$ and every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N)}{\\sqrt n}.\n\\]"
        },
        {
          "label": "C",
          "text": "For each fixed $\\beta<N$, one has\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\longrightarrow 0\\qquad\\text{as }n\\to\\infty.\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{n}.\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a finite constant $c(N,\\beta)<\\infty$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L\\!\\left(n^{-3/4}S_n\\right),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the constant on $\\beta$",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "explicit $n^{-1/2}$ Wasserstein rate and quantitative constant",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "sharp order of convergence $n^{-1/2}$",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "subcritical scaling/normal limit replaced by critical scaling",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the quantitative conclusion or single out the correct option. It gives the model and asks which theorem-level statement is valid in the regime β<N, without explicit answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct option is essentially the precise theorem statement. However, it is not a pure tautology because the alternatives vary meaningfully in rate, uniformity in β, and inclusion of the critical case."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact quantitative result from weaker or overstrong variants, especially regarding dependence on β, the exclusion of β=N, and the 1/√n rate. Still, the question mainly tests recognition of the theorem rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: a weaker true asymptotic statement, an overstrong uniform-in-β claim, an incorrect extension to criticality, and an overly fast rate. These reflect common failure modes in interpreting limit theorems."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it remains somewhat theorem-recall based rather than deeply generative."
    }
  },
  {
    "id": "2512.05340v1",
    "paper_link": "http://arxiv.org/abs/2512.05340v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}",
      "start_pos": 10002,
      "end_pos": 10457,
      "label": "Theorem: high temp O(N)"
    },
    "ref_dict": {
      "Theorem: main result": "\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}",
      "Theorem: Critical O(N)": "\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}",
      "eq:mean-field O(N) measure": "\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5192,
    "pre_theorem_intro_text": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n  \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nOur method relies on a general multivariate nonnormal Stein's method framework established\nin~\\cite{FangShaoXu2019,FangShaoXu2019correction}, which in turn can be viewed as a multivariate generalization\nof~\\cite{ChatterjeeShao2011}. \nAs explicitly noted in~\\cite{FangShaoXu2019}, however, their underlying assumptions\ndo not hold for densities of the form $\\exp(-a|x|^m)$ with $m>2$, and so their results cannot be applied to the critical mean-field $O(N)$\nmodel. Nevertheless, as shown here, a generalization of~\\cite[Theorem 2.5]{FangShaoXu2019} can be established which does indeed cover the\ncase of the critical $O(N)$ model. We present this result in Theorem~\\ref{Theorem: main result} of Section~\\ref{sec: Stein's method}.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n  d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n  \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.  \n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n  1.1]{FangKoike2022} yields the following.",
    "context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}",
    "full_context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\n\\begin{lemma}\\label{lemma: Critical remainders}\n Let $(W, W')$ be as described above. Then\n \\begin{equation}\\label{eq: W' - W conditioned on sigma}\n \\EE[W'-W|\\sigma] = \\frac{1}{Nn^{3/2}}\\left(-\\frac{N^{2}}{N+2}|W|^{2}W + R_{1}\\right),\n \\end{equation}\n where\n \\begin{equation}\\label{eq: remainder 1}\n \\begin{split}\n R_{1} &= \\frac{N}{n^{1/4}}\\sum_{i=1}^{n}\\left[m_{(i)}f_{N}(|m_{(i)}|) - mf_{N}(|m|)\\right] \\\\\n & \\quad + Nn^{3/4}\\left(f_{N}(|m|) - 1 + \\frac{N|m|^{2}}{(N+2)}\\right)m.\n \\end{split}\n \\end{equation}\n and \n \\begin{equation}\\label{eq: W' - W W' - W T conditioned on sigma}\n \\begin{split}\n \\EE[(W'-W)(W'-W)^{\\transpose}|\\sigma] &= 2\\frac{1}{Nn^{3/2}}\\left(\\id + R_{2}\\right),\n \\end{split}\n \\end{equation}\n where \n \\begin{equation}\\label{eq: remainder 2}\n \\begin{split}\n R_{2} &= \\left[\\frac{N}{2n}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose} - \\frac{1}{2}\\id\\right] -\n \\frac{N}{\\sqrt{n}}WW^{\\transpose} + \\frac{N}{n^{2}}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose}\\\\ \n &\\quad + \\frac{N}{2n}\\sum_{i=1}^{n}\\,g_{N}(|m_{(i)}|)\\,|m_{(i)}|^2\\,\n \\left[\\sigma_{i}m_{(i)}^{\\transpose}+ m_{(i)} \\sigma_{i}^{\\transpose} - \\frac{\\id}{N}\\right]\n + \\frac{N}{2n}\\sum_{i=1}^{n}g_{N}(|m_{(i)}|)m_{(i)}m_{(i)}^\\transpose. \n \\end{split}\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We follow a similar strategy to the proof~\\cite[Lemma 1]{KirkpatrickNawaz2016} (see also \\cite[Lemma 11]{KirkpatrickMeckes2013}).\n The key ingredients are the small $x$ asymptotics for $f_N(x)$ given in~\\eqref{eq: fN asymptotics}, combined with the fact\n that, as shown in~\\cite[Theorem 3]{KirkpatrickNawaz2016}, the quantity $m$ concentrates at 0 as $n\\to \\infty$ when $\\beta = N$.\n\nFinally, to establish~\\eqref{eq: Bound hilbert schmidt norm 2}, we fix $\\epsilon>0$ and $y\\in\\RR^d$ with $|y| \\leq 1$ and apply a\n similar decomposition to that used in~\\cite{FangShaoXu2019}. Specifically, defining\n \\begin{equation}\\label{eq: Phi definition}\n \\Phi(t) :=\\left[D^{2}P_{t}h(x + \\epsilon y)[u, v] - D^{2}P_{t}h(x)[u, v]\\right],\n \\end{equation}\n it follows by~\\eqref{eq: second derivative of f} that\n \\begin{equation}\\label{eq: rewriting f(x + epsilon u) - f(x)}\n D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x) = -\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t)\\,\\dt -\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt.\n \\end{equation}\n Equation~\\eqref{eq: Ptphi derivative 2} implies\n \\begin{equation}\\label{eq: bound int 0 to epsilon squared}\n \\begin{split}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| &\\leq 2|u||v|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon} S(t)\\, \\dt,\\\\\n &\\leq 2\\,|u| |v|\\,(\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right).\n \\end{split}\n \\end{equation}\n If $\\epsilon\\le 1$ then $\\epsilon\\wedge \\sqrt{\\epsilon}=\\epsilon$, in which case\n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) = \n \\epsilon\\, (s_0\\epsilon +2s_{-1/2})\\,\\leq \\epsilon (s_0 + 2s_{-1/2}).\n \\end{equation}\n While if $\\epsilon>1$ then $\\epsilon\\wedge\\sqrt{\\epsilon}=\\sqrt{\\epsilon}$, and so \n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) =\n \\sqrt{\\epsilon}\\, \\left(s_0\\sqrt{\\epsilon}+2s_{-1/2}\\right)\\leq\n \\epsilon\\left(s_{0} +2s_{-1/2}\\right).\n \\end{equation}\n We therefore conclude from~\\eqref{eq: bound int 0 to epsilon squared} that for all $\\epsilon>0$\n \\begin{equation}\\label{eq: int bound 0 to min}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| \\leq 2\\epsilon (s_0 + 2s_{-1/2})|u| |v|.\n \\end{equation}\n Now consider the remaining term in~\\eqref{eq: rewriting f(x + epsilon u) - f(x)}. It follows from~\\eqref{eq: Ptphi derivative 3} that\n $D^2P_t h(x+\\cdot)[u,v]$ is differentiable with bounded derivative, and so the fundamental theorem of calculus implies \n \\begin{equation}\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt\n = \\epsilon\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\int_{0}^{1} D^{3}P_{t}h(x + \\epsilon r y)[u, v, y]\\, \\dr\\, \\dt. \n \\end{equation}\n Therefore, using $|y|\\le1$, and again applying~\\eqref{eq: Ptphi derivative 3}, yields\n \\begin{equation}\n \\label{eq: phi tail bound}\n \\begin{split}\n \\left|\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi\\,\\dt\\right| \n &\\leq \\epsilon |u||v| \\int_{\\epsilon\\wedge\\epsilon^{2}}^{\\infty}\\, Q(t) \\,e^{-\\theta t/4} \\dt,\\\\\n &\\leq \\epsilon |u||v|\n \\left[\\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + q_{-1} E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right)\\right]\n \\end{split}\n \\end{equation}\n where $E_{1}(\\cdot)$ denotes the exponential integral~\\cite[Equation 5.1.1]{AbramowitzStegun1972}. Applying~\\cite[Equation 5.1.20]{AbramowitzStegun1972} \n yields\n \\begin{equation}\n \\label{eq: bound on exponential integral}\n \\begin{split}\n E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right) &\\le 2\\left(1\\vee \\left|\\log \\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4} \\right|\\right)\\\\\n &\\le 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right)\\left(1\\vee |\\log\\epsilon|\\right).\n \\end{split}\n \\end{equation}\n Applying~\\eqref{eq: bound on exponential integral} to~\\eqref{eq: phi tail bound}, and combining the result with~\\eqref{eq: int bound 0\n to min} and~\\eqref{eq: rewriting f(x + epsilon u) - f(x)} then yields \n \\begin{equation}\n \\sup_{x,y\\in\\RR^d : |y|\\le 1} |D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x)|\n \\le\n K_3\\,|u|\\, |v|\\, \\epsilon\\, (1\\vee|\\log\\epsilon|)\n \\label{eq: third main bound}\n \\end{equation}\n with\n \\begin{equation}\n K_3 :=\n 2(s_0 +2s_{-1/2}) + \n \\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right) q_{-1}.\n \\end{equation}\n\nThe analogue of Proposition~\\ref{prop: mean-square bounds} in~\\cite{FangShaoXu2019} is Lemma 5.2. While the proof of the latter is\nessentially immediate under their assumption of strict convexity of $V$,\nthe proof of Proposition~\\ref{prop: mean-square bounds} under Assumption~\\ref{assumption: assumptions on V} is somewhat more involved. A key\ningredient in its proof is the following lemma, whose proof is deferred to the end of this section. \n\\begin{lemma}\\label{lemma: E(t) bound}\n If $V$ satisfies Assumption~\\ref{assumption: assumptions on V}, then for all $x\\in\\RR^d$ and $0\\le s\\le t<\\infty$ \n $$\n \\EE \\exp\\left(-2\\int_s^t \\rho(X_r^x) \\dr \\right) \\le C_2^2 e^{-2\\theta (t-s)}.\n $$\n\\end{lemma}\nIn addition, we will also require the following Gr\\\"{o}nwall-type lemma.\n\\begin{lemma}\\label{lemma: Gronwall type}\n Let $u: \\RR \\to \\RR^{d}$ be such that $|u(t)|^{2} \\in C^{1}(\\RR, \\RR)$. Suppose it satisfies the following differential inequality\n \\begin{equation}\n \\frac{\\diff}{\\dt}|u(t)|^{2} \\leq 2a(t)|u(t)|^{2} + 2b(t)|u(t)|, \\qquad \\forall t \\in (0, T),\n \\end{equation}\n where $T \\in (0, \\infty)$, $a, b \\in L^{1}([0, T])$ and $b \\geq 0$. If $|u(0)| = 0$ then\n \\begin{equation}\n |u(t)| \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds, \\qquad \\forall t \\in [0, T).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Fix $t \\in (0, T)$ and let $\\tau := \\sup\\{0 \\leq s \\leq t: |u(s)| = 0\\}$. If $\\tau = t$ then the statement holds trivially since $b\n \\geq 0$. If $\\tau < t$ then $|u(t)|$ is differentiable on $(\\tau, t]$ and gives \n \\begin{equation}\n \\frac{\\diff}{\\ds}|u(s)| \\leq a(s)|u(s)| + b(s), \\qquad s \\in (\\tau, t].\n \\end{equation}\n Since $|u(\\tau)| = 0$, by the Generalised Jones Inequality \\cite[Theorem 1.2.2]{Qin2016} we have\n \\begin{equation}\n |u(t)| \\leq \\int_{\\tau}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds,\n \\end{equation}\n using $b \\geq 0$. This completes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 4700,
    "post_theorem_intro_text": "Our main result in the current work is the analogous multivariate nonnormal approximation holding at criticality.\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        d(\\mathcal{L}(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\n\\begin{remark}\nAs a corollary of Theorem~\\ref{Theorem: Critical O(N)} it follows that for any $N\\ge2$, the Wasserstein distance between the distribution of\n$n^{-3/4}\\,|S_n|$ and the probability measure with Lebesgue density proportional to $r^{N-1}\\exp(-a_N r^4)$, is again bounded above by\n$c(N)/\\sqrt{n}$. This is analogous to the univariate nonnormal approximation presented in~\\cite[Theorem 6]{KirkpatrickNawaz2016} for\n$|S_n|^2$, however the bound given there is not in terms of Wasserstein distance, but in terms of an integral probability metric defined by\na smaller class of test functions, and contains an additional logarithmic factor.\n\\end{remark}\n\nThe remainder of this paper is organised as follows. Section~\\ref{sec: Stein's method} introduces the relevant background on Stein's method, and\nprovides the statement of our main result on Wasserstein approximation, Theorem~\\ref{Theorem: main result}. In Section \\ref{sec: O(N) model}\nwe apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}. \nSection~\\ref{sec: sdes and semigroups} provides preliminary results related to a class of stochastic differential equations, and their\nrelated stochastic semigroups, which provide the framework for the proof of Theorem \\ref{Theorem: main result} given in Section \\ref{sec: proof of main\n  result}. The key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\nThese bounds are proved in Section~\\ref{sec: semigroup bounds}, using Elworthy-Li formulae and\nbounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\nproved in Section~\\ref{sec: variation process bounds}.\nFinally, the appendix contains the proof of a proposition stated in Section~\\ref{sec: sdes and semigroups}.\n\n\\subsection{Notation}\n\\label{subsec:notation}\nWe will denote the set of positive integers by $\\ZZ_+$, and the set of natural numbers by $\\mathbb{N}:=\\ZZ_+\\cup\\{0\\}$.\nFor $d\\in\\ZZ_+$ and $x,y\\in \\mathbb{R}^{d}$, we let $\\<x,y\\>$ denote the Euclidean inner product, and set $|x| = \\sqrt{\\<x,x\\>}$. For two\nmatrices $A, B \\in \\mathbb{R}^{d \\times d}$ we let $\\<A,B\\>_{{}} = \\sum_{i, j = 1}^{d}A_{i, j}B_{i, j}$ denote the Hilbert-Schmidt inner product,\nand denote the corresponding norm by $\\|A\\|_{{}} = \\sqrt{\\< A, A \\>_{{}}} = \\sqrt{\\text{Tr}(A^{\\mathrm{T}}A)}$. We will denote the operator\nnorm of $A\\in\\mathbb{R}^{d\\times d}$ by $\\|A\\|_{\\mathrm{op}}:=\\sup_{x\\in\\mathbb{S}^{d-1}}|Ax|$. \n\nFor open sets $U_1\\subseteq \\mathbb{R}^\\ell$ and $U_2\\subseteq\\mathbb{R}^m$ we let $C^{k}(U_1;U_2)$ denote the set of $k$-times continuously differentiable\n$f:U_1\\to U_2$. If $U_2=\\mathbb{R}$ we abbreviate $C^k(U_1):=C^k(U_1,U_2)$, and if additionally $U_1=\\mathbb{R}^d$ we write simply $C^k$.\nWe let $B_b$ denote the\nspace of $f:\\mathbb{R}^d\\to\\mathbb{R}$ which are bounded and Borel measurable, and let $C_b$ denote the subspace of $B_b$ of uniformly continuous such functions. We will then make\nuse of the convenient abbreviation $\\mathrm{Lip}^{\\infty}_b:=\\mathrm{Lip}\\cap C^{\\infty}\\cap C_b$.\n\nFor $f\\in C^1(\\mathbb{R}^d;\\mathbb{R})$, we let $D_u f(x)$ denote the directional derivative of $f$ in the direction $u\\in\\mathbb{R}^d$, and we extend the\ndefinition to $f\\in C^1(\\mathbb{R}^d;\\mathbb{R}^m)$, by then defining $D_u f(x)$ entrywise. For given $f\\in C^3(\\mathbb{R}^d;\\mathbb{R}^m)$ and $u,v,w,x\\in\\mathbb{R}^d$ we\ndefine\n\\begin{equation}\n  \\label{eq: D notation}\n  \\begin{split}\n    Df(x)[u]&:= D_u f(x)\\\\\n    D^2f(x)[u,v]&:=D_v D_u f(x)\\\\\n    D^3f(x)[u,v,w]&:=D_w D_v D_u f(x)\n  \\end{split}\n\\end{equation}\nWe note that for each given $x\\in\\mathbb{R}^d$ and $i=1,2,3$, the map $D^i g(x) : (\\mathbb{R}^d)^i\\to\\mathbb{R}^d$ is multilinear and symmetric.\n\nFor $f \\in C^{1}(\\mathbb{R}^d;\\mathbb{R})$, we let $\\nabla f$ denote the gradient of $f$, so that\n\\begin{equation}\n  Df(x)[u]=\\<\\nabla f(x),u\\>,\n\\end{equation}\nand set $\\|\\nabla f\\|_\\infty:=\\sup_{x \\in \\mathbb{R}^{d}}|\\nabla f(x)|$. \nFinally, for $f \\in C^{2}(\\mathbb{R}^{d};\\mathbb{R})$, we let $\\nabla^2 f$ denote the Hessian of $f$,\nso that\n\\begin{equation}\nD^2 f(x)[u,v] =\\<u,\\nabla^2 f(x) v\\>,\n\\end{equation}\nand let $\\Delta$ denote the Laplacian.",
    "sketch": "The introduction does not give a step-by-step proof of Theorem~\\ref{Theorem: high temp O(N)}, but it does indicate how the analogous critical result is proved: the paper \"introduces the relevant background on Stein's method\" and states a main Wasserstein-approximation theorem (Theorem~\\ref{Theorem: main result}); then \"we apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}.\" The proof framework for Theorem~\\ref{Theorem: main result} uses \"a class of stochastic differential equations, and their related stochastic semigroups\"; \"the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\" These derivative bounds are proved \"using Elworthy-Li formulae\" and \"bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\" (via \"variation process bounds\"). No direct proof sketch for Theorem~\\ref{Theorem: high temp O(N)} itself appears in the post-theorem introduction.",
    "expanded_sketch": "The introduction does not give a step-by-step proof of the main theorem, but it does indicate how the analogous critical result is proved: the paper introduces the relevant background on Stein's method and states a main Wasserstein-approximation theorem.\n\n\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}\n\nThen this theorem is applied to prove the following critical result.\n\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\nThe proof framework for the Wasserstein-approximation theorem above uses a class of stochastic differential equations and their related stochastic semigroups; the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups. These derivative bounds are proved using Elworthy--Li formulae and bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation (via variation process bounds). No direct proof sketch for the main theorem itself appears in the post-theorem introduction.",
    "expanded_theorem": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $N\\ge 2$. In the mean-field $O(N)$ model, the spin configuration $\\sigma=(\\sigma_1,\\dots,\\sigma_n)\\in (\\mathbb S^{N-1})^n$ has law with density proportional to\n\\[\n\\exp\\!\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot \\sigma_j\\right)\n\\]\nwith respect to the $n$-fold product of the uniform measure on the unit sphere $\\mathbb S^{N-1}\\subset\\mathbb R^N$. Let the magnetization be\n\\[\nS_n=\\sum_{i=1}^n \\sigma_i,\n\\]\nand define\n\\[\nW_n:=\\sqrt{N-\\beta}\\,\\frac{S_n}{\\sqrt n}.\n\\]\nLet $Z\\sim N(0,\\mathbf I)$ be the standard Gaussian vector in $\\mathbb R^N$, where $\\mathbf I$ is the identity matrix, and let the Wasserstein distance between laws be\n\\[\nd(\\mathcal L(X),\\mathcal L(Y)):=\\inf_{(X,Y)}\\mathbb E|X-Y|=\\sup_{h\\in \\mathrm{Lip}}\\big|\\mathbb Eh(X)-\\mathbb Eh(Y)\\big|,\n\\]\nwhere the supremum is over all $1$-Lipschitz functions $h$. For a fixed inverse temperature $\\beta<N$, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a finite constant $c(N)$, depending only on $N$, such that for every $\\beta<N$ and every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N)}{\\sqrt n}.\n\\]"
        },
        {
          "label": "C",
          "text": "For each fixed $\\beta<N$, one has\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\longrightarrow 0\\qquad\\text{as }n\\to\\infty.\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{n}.\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a finite constant $c(N,\\beta)<\\infty$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L\\!\\left(n^{-3/4}S_n\\right),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the constant on $\\beta$",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "explicit $n^{-1/2}$ Wasserstein rate and quantitative constant",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "sharp order of convergence $n^{-1/2}$",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "subcritical scaling/normal limit replaced by critical scaling",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the quantitative conclusion or the sharp rate. It sets up the model and normalization, but the respondent still has to choose among different rates, scalings, and parameter dependences."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct option is essentially the theorem statement itself: a subcritical CLT with Wasserstein error of order n^{-1/2} and a constant depending on (N, beta). The item mainly asks for recall of the exact conclusion rather than a substantively reformulated consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to reject stronger, weaker, or mis-scaled alternatives, especially the uniform-in-beta, n^{-1}, and critical-scaling distractors. However, the task still mostly tests recognition of the exact theorem-level estimate rather than deeper generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: uniformity in beta, mere convergence without rate, an overly strong n^{-1} rate, and an incorrect critical scaling. These align well with common mistakes about sharpness, parameter dependence, and phase-dependent normalization."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically well-constructed item with strong distractors, but it is close to a direct theorem-recall question rather than a high-generative-reasoning MCQ."
    }
  },
  {
    "id": "2512.05429v1",
    "paper_link": "http://arxiv.org/abs/2512.05429v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:nv-can}\nLet $x\\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\\hvol(x,X)\\leq 9$, and equality holds if and only if $x\\in X$ is a quotient singularity  of type $\\frac{1}{3}(1,1,1)$.",
      "start_pos": 11021,
      "end_pos": 11265,
      "label": "thm:nv-can"
    },
    "ref_dict": {
      "thm:K-moduli": "\\begin{thm}\\label{thm:K-moduli}\nLet $X$ be a K-semistable $\\bQ$-Fano threefold with volume $(-K_X)^3 = V$.\n\\begin{enumerate}\n    \\item If $V\\geq 26$, then $X$ has only $cA_1$-singularities or  cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n    \\item If $V\\geq 22$, then $X$ has only $cA_1$-singularities, isolated $cA_2$-singularities, $D_\\infty$-singularities, or cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n    \\item If $V\\geq 11$, then any non-Gorenstein singularity $x\\in X$ is a cyclic quotient of a (possibly smooth) hypersurface singularity of type $cA_{\\leq 2}$.\n\\end{enumerate}\nIf in addition $X$ is $\\bQ$-Gorenstein smoothable, then it does not admit cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n\\end{thm}",
      "que:vol-list": "\\begin{que}\\label{que:vol-list}\nLet $\\hVol_3$ denote the set of  local volumes of all klt threefold  singularities $x\\in X$. By \\cite{XZ24} (see also \\cite{Zhu24, LMS23}) we know that $0$ is the only accumulation point of $\\hVol_3$. Moreover, Theorem \\ref{thm:nv-3fold} implies that if $\\hvol(x,X)\\in \\hVol_3 \\cap [9, 27]$ then $x\\in X$ is either a quotient singularity of order at most $3$ or a hypersurface singularity of type $cA_{\\leq 2}$.\n From Example \\ref{expl:ADE} we have the following containment\n\\[\n    \\hVol_3 \\cap [9, 27] \\supset \\left\\{9, \\tfrac{2048}{225}, \\tfrac{250}{27}, \\tfrac{343}{36}, 6\\sqrt{3}, \\tfrac{32}{3}, \\tfrac{27}{2}, \\tfrac{125}{9}, 16, 27\\right\\}.\n\\]\nIs this containment  an equality?\n\\end{que}",
      "rem:K-mod": "\\begin{rem}\\label{rem:K-mod}\nAccording to the author's knowledge based on \\cite{SS17, LX19, ADL21, ACC+, ACD+, CDG+, CT23, DJKHQ24, LZ24, Zha24, CFFK24}, we give a list of families of smooth Fano threefolds with volume $V\\geq 11$ whose K-moduli compactifications are currently unknown. We hope Theorem \\ref{thm:K-moduli} can be used to study these families in the future. We follow Mori--Mukai's notation for the numbering of these families.\n\\begin{enumerate}\n    \\item $V\\geq 26$: \\textnumero 2.20, 2.21, 3.11;\n    \\item $22\\leq V<26$: \\textnumero 1.10, 2.16, 2.17, 3.6, 3.7, 3.8;\n    \\item $11\\leq V<22$: \\textnumero 1.6, 1.7, 1.8, 1.9, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 3.1, 3.2, 3.3, 3.4, 3.5.\n\\end{enumerate}\n\\end{rem}",
      "thm:nv-3fold": "\\begin{thm}\\label{thm:nv-3fold}\nLet $x\\in X$ be a klt threefold singularity. Then $\\hvol(x,X)\\geq 9$ if and only if  $x\\in X$ is either a hypersurface singularity of type $cA_{\\leq 2}$, or a cyclic quotient singularity of type $\\frac{1}{2}(1,1,1)$, $\\frac{1}{3}(1,1,0)$, $\\frac{1}{3}(1,1,1)$, or $\\frac{1}{3}(1,1,2)$. \n\\end{thm}",
      "thm:nv-can": "\\begin{thm}\\label{thm:nv-can}\nLet $x\\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\\hvol(x,X)\\leq 9$, and equality holds if and only if $x\\in X$ is a quotient singularity  of type $\\frac{1}{3}(1,1,1)$. \n\\end{thm}",
      "expl:ADE": "\\begin{expl}\\label{expl:ADE}\n\\begin{enumerate}\n    \\item If $x\\in X$ is an $A_k$-singularity, that is, locally analytically given by $x_1 x_2 + x_3^2 + x_4^{k+1} = 0$, then its local volume is $16$ when $k=1$, $\\frac{125}{9}$ when $k=2$, and $\\frac{27}{2}$ when $k\\geq 3$. See \\cite[Section 5]{Li18}, \\cite[Example 4.7]{LL19} and \\cite[Example 7.1.2]{LX20}.\n\\item If $x\\in X$ is a $cA_1$-singularity but not an $A_k$-singularity, then $x\\in X$ has to be a transversal $A_1$-singularity with $\\hvol(x,X) = \\frac{27}{2}$.\n\\item If $x\\in X$ is defined by $x_1 x_2 + x_3^3 + x_4^k=0$ for $3\\leq k\\leq 6$, then by \\cite{CS15} and \\cite{LST25} we have $\\hvol(x,X) = \\frac{4(3+k)^3}{9k^2}$. Thus we get values $\\frac{32}{3}, \\frac{343}{36}, \\frac{2048}{225}, 9$ for $k = 3,4,5,6$. Note that  $k=3,4,5$ correspond to $D_4$, $E_6$, and $E_8$-singularities.\n\\item If $x\\in X$ is a $D_k$-singularity with $k\\geq 5$, then we have local equation $x_1x_2 + x_3^2 x_4 + x_4^{k-1} = 0$. Then by \\cite[Appendix, Example 4.1]{LX18} we know that its local volume is computed by the monomial valuation $v_{\\bfw}|_{K(X)}$ with weight $\\bfw=(1, 1, \\sqrt{3}-1, 4-2\\sqrt{3})$ whose K-semistable Fano cone degeneration is precisely the $D_\\infty$-singularity $x_1x_2 + x_3^2 x_4 = 0$. Thus we have $\\hvol(x, X) = 6\\sqrt{3}$ which is the same as the local volume of a $D_\\infty$-singularity.\n\\item If $x\\in X$ is an $E_7$-singularity, then we have local equation $x_1x_2 + x_3^3 + x_3 x_4^3 =0$. By \\cite[Section 8]{Li18} and \\cite[Example 7.1.3]{LX20} we know that $\\hvol(x, X) = \\frac{250}{27}$.\n\\end{enumerate}\n\n\\end{expl}"
    },
    "pre_theorem_intro_text_len": 1330,
    "pre_theorem_intro_text": "The local volume $\\hvol(x,X)$ of a klt singularity $x\\in X$, introduced by Chi Li in \\cite{Li18}, is a numerical invariant that has played a crucial role in the recent development of a local K-stability theory for singularities; see \\cite{LLX18, Zhu25} for recent surveys. It is thus natural to further investigate  the distribution of local volumes in a fixed dimension $n$, such as explicit bounds and gaps. In \\cite{LX19} it was shown that $\\hvol(x,X)\\leq n^n$ and equality holds if and only if $x\\in X$ is smooth. Then it was conjectured in \\cite{SS17}, known as the ODP Gap Conjecture, that the second largest local volume is $2(n-1)^n$ which is obtained by the ordinary double point. This conjecture is confirmed in dimension at most $3$ \\cite{LL19, LX19} and for local complete intersection singularities \\cite{Liu22}. More recently, it was shown in \\cite{XZ24}, as a consequence of their local boundedness result for K-semistable Fano cone singularities, that the local volumes in a fixed dimension are discrete away from zero. See also \\cite{HLQ23, Zhu24, LMS23} for previous results in lower dimensions and \\cite{HLQ24} for general coefficient sets.\n\nIn this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.",
    "context": "The local volume $\\hvol(x,X)$ of a klt singularity $x\\in X$, introduced by Chi Li in \\cite{Li18}, is a numerical invariant that has played a crucial role in the recent development of a local K-stability theory for singularities; see \\cite{LLX18, Zhu25} for recent surveys. It is thus natural to further investigate the distribution of local volumes in a fixed dimension $n$, such as explicit bounds and gaps. In \\cite{LX19} it was shown that $\\hvol(x,X)\\leq n^n$ and equality holds if and only if $x\\in X$ is smooth. Then it was conjectured in \\cite{SS17}, known as the ODP Gap Conjecture, that the second largest local volume is $2(n-1)^n$ which is obtained by the ordinary double point. This conjecture is confirmed in dimension at most $3$ \\cite{LL19, LX19} and for local complete intersection singularities \\cite{Liu22}. More recently, it was shown in \\cite{XZ24}, as a consequence of their local boundedness result for K-semistable Fano cone singularities, that the local volumes in a fixed dimension are discrete away from zero. See also \\cite{HLQ23, Zhu24, LMS23} for previous results in lower dimensions and \\cite{HLQ24} for general coefficient sets.\n\nIn this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.",
    "full_context": "The local volume $\\hvol(x,X)$ of a klt singularity $x\\in X$, introduced by Chi Li in \\cite{Li18}, is a numerical invariant that has played a crucial role in the recent development of a local K-stability theory for singularities; see \\cite{LLX18, Zhu25} for recent surveys. It is thus natural to further investigate the distribution of local volumes in a fixed dimension $n$, such as explicit bounds and gaps. In \\cite{LX19} it was shown that $\\hvol(x,X)\\leq n^n$ and equality holds if and only if $x\\in X$ is smooth. Then it was conjectured in \\cite{SS17}, known as the ODP Gap Conjecture, that the second largest local volume is $2(n-1)^n$ which is obtained by the ordinary double point. This conjecture is confirmed in dimension at most $3$ \\cite{LL19, LX19} and for local complete intersection singularities \\cite{Liu22}. More recently, it was shown in \\cite{XZ24}, as a consequence of their local boundedness result for K-semistable Fano cone singularities, that the local volumes in a fixed dimension are discrete away from zero. See also \\cite{HLQ23, Zhu24, LMS23} for previous results in lower dimensions and \\cite{HLQ24} for general coefficient sets.\n\nIn this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.\n\nIn this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.\n\nTheorem \\ref{thm:nv-can} can be thought of as a refinement of the ODP Gap Theorem for threefolds in \\cite{LX19} that any non-smooth threefold singularity $x\\in X$ satisfies $\\hvol(x,X)\\leq 16$, and equality holds if and only if $x\\in X$ is an $A_1$-singularity.\n\n\\begin{thm}\\label{thm:nv-3fold}\nLet $x\\in X$ be a klt threefold singularity. Then $\\hvol(x,X)\\geq 9$ if and only if $x\\in X$ is either a hypersurface singularity of type $cA_{\\leq 2}$, or a cyclic quotient singularity of type $\\frac{1}{2}(1,1,1)$, $\\frac{1}{3}(1,1,0)$, $\\frac{1}{3}(1,1,1)$, or $\\frac{1}{3}(1,1,2)$. \n\\end{thm}\n\n\\begin{thm}\\label{thm:K-moduli}\nLet $X$ be a K-semistable $\\bQ$-Fano threefold with volume $(-K_X)^3 = V$.\n\\begin{enumerate}\n \\item If $V\\geq 26$, then $X$ has only $cA_1$-singularities or cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n \\item If $V\\geq 22$, then $X$ has only $cA_1$-singularities, isolated $cA_2$-singularities, $D_\\infty$-singularities, or cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n \\item If $V\\geq 11$, then any non-Gorenstein singularity $x\\in X$ is a cyclic quotient of a (possibly smooth) hypersurface singularity of type $cA_{\\leq 2}$.\n\\end{enumerate}\nIf in addition $X$ is $\\bQ$-Gorenstein smoothable, then it does not admit cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n\\end{thm}\n\n\\begin{thm}\\label{thm:nv-mld}\nLet $x\\in X$ be a klt threefold singularity. Then \n\\begin{equation}\\label{eq:nv-mld}\n \\hvol(x,X) \\leq 9 \\cdot \\mld(x,X).\n\\end{equation}\nMoreover, equality holds if and only if $x\\in X$ is a cyclic quotient singularity of type $\\frac{1}{r}(1,1,1)$ for some $r\\in \\bZ_{>0}$.\n\\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:nv-3fold}]\n The ``if'' part is a direct consequence of Example \\ref{expl:quotient} and Proposition \\ref{prop:nv-hyp}.\n For the ``only if'' part, let $(\\tx\\in \\tX)\\to (x\\in X)$ be the index $1$ cover of $K_X$ whose Cartier index is denoted by $d$. Then by Theorem \\ref{thm:finite-deg} we have \n \\[\n \\hvol(\\tx, \\tX) = d\\cdot \\hvol(x,X)\\geq 9d.\n \\]\n If $d=1$, i.e.\\ $x\\in X$ is Gorenstein, then by Theorem \\ref{thm:crex-cDV} we know that $x\\in X$ is either a hypersurface singularity or a cyclic quotient singularity of type $\\frac{1}{3}(1,1,1)$. The hypersurface singularity case follows from Proposition \\ref{prop:nv-hyp}. If $d\\geq 2$, then we have $\\hvol(\\tx, \\tX) \\geq 18$ which implies that $\\tx\\in \\tX$ is smooth and $d \\leq 3$ by Theorem \\ref{thm:ODP-gap}. Thus $x\\in X$ is a cyclic quotient singularity of order $2$ or $3$. If $d=2$, then $x\\in X$ is of type $\\frac{1}{2}(1,1,1)$ as the other case $\\frac{1}{2}(1,1,0)$ is Gorenstein. If $d=3$, then $x\\in X$ is of type $\\frac{1}{3}(1,1,0)$ or $\\frac{1}{3}(1,1,2)$ as the other two cases $\\frac{1}{3}(1,2,0)$ or $\\frac{1}{3}(1,1,1)$ are both Gorenstein.\n\\end{proof}\n\nSince $K_{\\tX}$ is Cartier, we know that $\\mld(\\tx,\\tX) \\geq 1$ is a positive integer. If $\\crex(\\tx, \\tX) = 0$, then $\\mld(\\tx, \\tX) = 1$ and \\eqref{eq:nv-mld-cover} follows by Theorem \\ref{thm:nv-can} with equality if and only if $\\tx\\in \\tX$ is a cyclic quotient singularity of type $\\frac{1}{3}(1,1,1)$. If $\\crex(\\tx,\\tX) >0$, then by Theorem \\ref{thm:crex-cDV} we know that $\\tx\\in \\tX$ is a cDV singularity. Then we have three cases: $\\tx\\in \\tX$ is smooth, non-smooth terminal, or non-terminal. The smooth case is obvious as $\\mld(\\tx,\\tX) = 3$ and $\\hvol(\\tx,\\tX) = 27$. For the terminal case, we have $\\mld(\\tx,\\tX) = 2$ by \\cite{Mar96}. Thus \\eqref{eq:nv-mld-cover} follows from Theorem \\ref{thm:ODP-gap} as $\\hvol(\\tx, \\tX)\\leq 16 < 18 = 9\\cdot \\mld(\\tx,\\tX)$. The last case is when $\\tx\\in \\tX$ is not terminal. Let $\\phi: \\tZ\\to \\tX$ be a terminalization. Then we know that $\\phi^{-1}(\\tx)$ has dimension $1$ since $\\phi$ is exceptional over $\\tx$ and there are no crepant exceptional divisors over $\\tx\\in \\tX$ by the assumption that $\\crex(\\tx, \\tX) = 0$. Let $\\eta$ be a generic point of $\\phi^{-1}(\\tx)$. Since $\\tZ$ is terminal, we know that $\\tZ$ is smooth at $\\eta$. Thus we have $\\mld(\\tx,\\tX) \\leq \\mld(\\eta, \\tZ) = 2$. Moreover, since $\\crex(\\tx,\\tX) = 0$ we have $\\mld(\\tx,\\tX)>1$. Combining these together, we have $\\mld(\\tx, \\tX) = 2$. Thus \\eqref{eq:nv-mld-cover} follows again from Theorem \\ref{thm:ODP-gap}.\n\n\\begin{que}\\label{que:vol-list}\nLet $\\hVol_3$ denote the set of local volumes of all klt threefold singularities $x\\in X$. By \\cite{XZ24} (see also \\cite{Zhu24, LMS23}) we know that $0$ is the only accumulation point of $\\hVol_3$. Moreover, Theorem \\ref{thm:nv-3fold} implies that if $\\hvol(x,X)\\in \\hVol_3 \\cap [9, 27]$ then $x\\in X$ is either a quotient singularity of order at most $3$ or a hypersurface singularity of type $cA_{\\leq 2}$.\n From Example \\ref{expl:ADE} we have the following containment\n\\[\n \\hVol_3 \\cap [9, 27] \\supset \\left\\{9, \\tfrac{2048}{225}, \\tfrac{250}{27}, \\tfrac{343}{36}, 6\\sqrt{3}, \\tfrac{32}{3}, \\tfrac{27}{2}, \\tfrac{125}{9}, 16, 27\\right\\}.\n\\]\nIs this containment an equality?\n\\end{que}\n\n\\begin{expl}\\label{expl:ADE}\n\\begin{enumerate}\n    \\item If $x\\in X$ is an $A_k$-singularity, that is, locally analytically given by $x_1 x_2 + x_3^2 + x_4^{k+1} = 0$, then its local volume is $16$ when $k=1$, $\\frac{125}{9}$ when $k=2$, and $\\frac{27}{2}$ when $k\\geq 3$. See \\cite[Section 5]{Li18}, \\cite[Example 4.7]{LL19} and \\cite[Example 7.1.2]{LX20}.\n\\item If $x\\in X$ is a $cA_1$-singularity but not an $A_k$-singularity, then $x\\in X$ has to be a transversal $A_1$-singularity with $\\hvol(x,X) = \\frac{27}{2}$.\n\\item If $x\\in X$ is defined by $x_1 x_2 + x_3^3 + x_4^k=0$ for $3\\leq k\\leq 6$, then by \\cite{CS15} and \\cite{LST25} we have $\\hvol(x,X) = \\frac{4(3+k)^3}{9k^2}$. Thus we get values $\\frac{32}{3}, \\frac{343}{36}, \\frac{2048}{225}, 9$ for $k = 3,4,5,6$. Note that  $k=3,4,5$ correspond to $D_4$, $E_6$, and $E_8$-singularities.\n\\item If $x\\in X$ is a $D_k$-singularity with $k\\geq 5$, then we have local equation $x_1x_2 + x_3^2 x_4 + x_4^{k-1} = 0$. Then by \\cite[Appendix, Example 4.1]{LX18} we know that its local volume is computed by the monomial valuation $v_{\\bfw}|_{K(X)}$ with weight $\\bfw=(1, 1, \\sqrt{3}-1, 4-2\\sqrt{3})$ whose K-semistable Fano cone degeneration is precisely the $D_\\infty$-singularity $x_1x_2 + x_3^2 x_4 = 0$. Thus we have $\\hvol(x, X) = 6\\sqrt{3}$ which is the same as the local volume of a $D_\\infty$-singularity.\n\\item If $x\\in X$ is an $E_7$-singularity, then we have local equation $x_1x_2 + x_3^3 + x_3 x_4^3 =0$. By \\cite[Section 8]{Li18} and \\cite[Example 7.1.3]{LX20} we know that $\\hvol(x, X) = \\frac{250}{27}$.\n\\end{enumerate}\n\n\\end{expl}\n\n\\begin{thm}\\label{thm:nv-3fold}\nLet $x\\in X$ be a klt threefold singularity. Then $\\hvol(x,X)\\geq 9$ if and only if  $x\\in X$ is either a hypersurface singularity of type $cA_{\\leq 2}$, or a cyclic quotient singularity of type $\\frac{1}{2}(1,1,1)$, $\\frac{1}{3}(1,1,0)$, $\\frac{1}{3}(1,1,1)$, or $\\frac{1}{3}(1,1,2)$. \n\\end{thm}\n\n\\begin{thm}\\label{thm:nv-can}\nLet $x\\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\\hvol(x,X)\\leq 9$, and equality holds if and only if $x\\in X$ is a quotient singularity  of type $\\frac{1}{3}(1,1,1)$. \n\\end{thm}",
    "post_theorem_intro_text_len": 5278,
    "post_theorem_intro_text": "Theorem \\ref{thm:nv-can} can be thought of as a refinement of the ODP Gap Theorem for threefolds in \\cite{LX19} that any non-smooth threefold singularity $x\\in X$ satisfies $\\hvol(x,X)\\leq 16$, and equality holds if and only if $x\\in X$ is an $A_1$-singularity.\n\nAs a consequence, we have the following characterization of threefold singularities with local volumes at least $9$. See Example \\ref{expl:ADE} for many examples of such singularities and their local volumes, and Question \\ref{que:vol-list} for a conjectural list of all possible local volumes that are at least $9$.\n\n\\begin{thm}\\label{thm:nv-3fold}\nLet $x\\in X$ be a klt threefold singularity. Then $\\hvol(x,X)\\geq 9$ if and only if  $x\\in X$ is either a hypersurface singularity of type $cA_{\\leq 2}$, or a cyclic quotient singularity of type $\\frac{1}{2}(1,1,1)$, $\\frac{1}{3}(1,1,0)$, $\\frac{1}{3}(1,1,1)$, or $\\frac{1}{3}(1,1,2)$. \n\\end{thm}\n\nThe above results have some notable applications to the study of K-moduli of Fano threefolds through the local-to-global volume comparison \\cite{Liu18}.\n\n\\begin{thm}\\label{thm:K-moduli}\nLet $X$ be a K-semistable $\\mathbb{Q}$-Fano threefold with volume $(-K_X)^3 = V$.\n\\begin{enumerate}\n    \\item If $V\\geq 26$, then $X$ has only $cA_1$-singularities or  cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n    \\item If $V\\geq 22$, then $X$ has only $cA_1$-singularities, isolated $cA_2$-singularities, $D_\\infty$-singularities, or cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n    \\item If $V\\geq 11$, then any non-Gorenstein singularity $x\\in X$ is a cyclic quotient of a (possibly smooth) hypersurface singularity of type $cA_{\\leq 2}$.\n\\end{enumerate}\nIf in addition $X$ is $\\mathbb{Q}$-Gorenstein smoothable, then it does not admit cyclic quotient singularities of type $\\frac{1}{2}(1,1,1)$.\n\\end{thm}\n\nTheorem \\ref{thm:K-moduli} can be viewed as a strengthening of a result proven in \\cite{LX19, LZ24} stating that $X$ is Gorenstein canonical if $V\\geq 20$. Such  restrictions on singularities played a key role in the description of K-moduli spaces for various families of Fano varieties via the moduli continuity method, see e.g.\\ \\cite{MM93, OSS16, SS17, LX19, Liu22, ADL21, LZ24, Zha24}. We expect Theorem \\ref{thm:K-moduli} to be useful in the future study of K-moduli spaces for Fano threefolds. Specifically, according to \\cite{ACC+}, there are precisely $79$ out of $105$ families of smooth Fano threefolds that contain K-semistable members, i.e.\\ whose K-moduli spaces are non-empty. Among these $79$ families, Theorem \\ref{thm:K-moduli} (1), (2), and (3) can be applied to $35$, $46$, and $68$ families, respectively, where   $3$, $9$, and $28$ families respectively have unknown K-moduli compactifications to the author's knowledge. See Remark \\ref{rem:K-mod} for a list of these families.\n\nAs another application, we establish a sharp comparison between the local volume and the minimal log discrepancy (mld) for klt threefold singularities, answering  \\cite[Question 6.16]{LLX18} affirmatively in dimension $3$. \n\n\\begin{thm}\\label{thm:nv-mld}\nLet $x\\in X$ be a klt threefold  singularity. Then \n\\begin{equation}\\label{eq:nv-mld}\n    \\hvol(x,X) \\leq 9 \\cdot \\mathrm{mld}(x,X).\n\\end{equation}\nMoreover, equality holds if and only if $x\\in X$ is a cyclic quotient singularity of type $\\frac{1}{r}(1,1,1)$ for some $r\\in \\bZ_{>0}$.\n\\end{thm}\n\nNote that a weaker inequality $\\hvol(x,X)< n^n\\cdot \\mathrm{mld}(x,X)$ was proven in any dimension $n$ in \\cite[Theorem 6.13]{LLX18}.\n\nLet us briefly explain the strategy of the proof of Theorems \\ref{thm:nv-can} and \\ref{thm:nv-3fold}, which is similar to \\cite{LX19} but technically more involved. For hypersurface singularities, we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.  Suppose $x\\in X$ is a Gorenstein canonical non-hypersurface threefold singularity. Then the hope is to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model. This is done by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one. Compared to \\cite{LX19}, the new difficulty lies in the fact that in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\\frac{27}{2}$, which is not strong enough for our estimate. Then we reduce to the case when there are at most two crepant exceptional divisors over $x\\in X$ and apply the detailed analysis  from \\cite{Lau77, Rei76, Rei80, Rei94}. In some cases we are only able to get a good crepant birational model after an \\'etale base change using Artin approximation theorem (see Section \\ref{sec:Artin}), a result that is of independent interest.\n\n\\subsection*{Acknowledgements} The author would like to thank Anne-Sophie Kaloghiros, J\\'anos Koll\\'ar, Andrea Petracci, Jakub Witaszek, Chenyang Xu, and Junyan Zhao for helpful discussions and comments. The author is partially supported by NSF CAREER Grant DMS-2237139 and an AT\\&T Research Fellowship from Northwestern University.",
    "sketch": "“Let us briefly explain the strategy of the proof of Theorems \\ref{thm:nv-can} and \\ref{thm:nv-3fold}, which is similar to \\cite{LX19} but technically more involved.”\n\n- For hypersurface singularities: “we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.”\n\n- For a Gorenstein canonical non-hypersurface threefold singularity $x\\in X$: the approach is “to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model.”\n\n- This is achieved “by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one.”\n\n- New technical issue for the bound $\\hvol\\le 9$: “in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\\frac{27}{2}$, which is not strong enough for our estimate.”\n\n- Reduction step: “Then we reduce to the case when there are at most two crepant exceptional divisors over $x\\in X$ and apply the detailed analysis from \\cite{Lau77, Rei76, Rei80, Rei94}.”\n\n- In some cases: “we are only able to get a good crepant birational model after an \\'etale base change using Artin approximation theorem (see Section \\ref{sec:Artin}).”",
    "expanded_sketch": "Let us briefly explain the strategy of the proof of \\begin{thm}\\label{thm:nv-3fold}\nLet $x\\in X$ be a klt threefold singularity. Then $\\hvol(x,X)\\geq 9$ if and only if  $x\\in X$ is either a hypersurface singularity of type $cA_{\\leq 2}$, or a cyclic quotient singularity of type $\\frac{1}{2}(1,1,1)$, $\\frac{1}{3}(1,1,0)$, $\\frac{1}{3}(1,1,1)$, or $\\frac{1}{3}(1,1,2)$. \n\\end{thm}\nand, in establishing the main theorem, the proof is similar to \\cite{LX19} but technically more involved.\n\n- For hypersurface singularities: we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.\n\n- For a Gorenstein canonical non-hypersurface threefold singularity $x\\in X$: the approach is to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model.\n\n- This is achieved by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one.\n\n- New technical issue for the bound $\\hvol\\le 9$: in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\\frac{27}{2}$, which is not strong enough for our estimate.\n\n- Reduction step: then we reduce to the case when there are at most two crepant exceptional divisors over $x\\in X$ and apply the detailed analysis from \\cite{Lau77, Rei76, Rei80, Rei94}.\n\n- In some cases: we are only able to get a good crepant birational model after an \\'etale base change using Artin approximation theorem (see Section \\ref{sec:Artin}).",
    "expanded_theorem": "\\label{thm:nv-can}\nLet $x\\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\\hvol(x,X)\\leq 9$, and equality holds if and only if $x\\in X$ is a quotient singularity  of type $\\frac{1}{3}(1,1,1)$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $x\\in X$ be a Gorenstein canonical non-hypersurface threefold singularity, and let $\\hvol(x,X)$ denote its local volume. Here, “non-hypersurface” means that the singularity is not analytically a hypersurface singularity, and a quotient singularity of type $\\frac{1}{3}(1,1,1)$ means the cyclic quotient of $\\mathbb{A}^3$ by the order-$3$ action with weights $(1,1,1)$. Which quantitative estimate holds for $\\hvol(x,X)$ under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "$\\hvol(x,X)\\le 9$, and equality holds if and only if $x\\in X$ is a quotient singularity of type $\\frac{1}{3}(1,1,1)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\hvol(x,X)\\le \\frac{27}{2}$, and equality holds if and only if $x\\in X$ is a transversal $A_1$-singularity."
        },
        {
          "label": "C",
          "text": "$\\hvol(x,X)\\le 9$."
        },
        {
          "label": "D",
          "text": "$\\hvol(x,X)< 9$, unless $x\\in X$ is a quotient singularity of type $\\frac{1}{3}(1,1,1)$, in which case $\\hvol(x,X)=9$."
        },
        {
          "label": "E",
          "text": "$\\hvol(x,X)\\le 9$, and equality holds if and only if $x\\in X$ is a cyclic quotient singularity of type $\\frac{1}{r}(1,1,1)$ for some $r\\in \\mathbb{Z}_{>0}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "two-divisor analysis needed to improve bound from 27/2 down to 9",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the equality characterization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "equality case at the sharp boundary value 9",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "special equality classification restricted to the single non-hypersurface Gorenstein case r=3",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct analytic type. It asks for an equivalent characterization of the equality case, without embedding the answer in the wording."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct restatement of the theorem characterizing when the local volume reaches 9. It tests recall of the exact equality case rather than selection among independently derived conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some discrimination is required because the distractors are nearby variants (arbitrary cyclic quotient, weaker quotient condition, extra geometric alternative), but the main task is still theorem recall rather than genuine multi-step reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one weakens the conclusion, one overgeneralizes the quotient type, one introduces a tempting alternative singularity case, and one offers a technical but incorrect birational criterion."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically clean MCQ with strong distractors and no answer leakage, but it is largely tautological: it mainly checks recall of the precise theorem statement rather than generative reasoning."
    }
  },
  {
    "id": "2512.05766v1",
    "paper_link": "http://arxiv.org/abs/2512.05766v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thvecchio}\\cite{CPP}\n    Let $\\Sigma_D$ be a cone such that $\\bar D\\subset \\mathbb{S}^{N-1}_+$ and $\\lambda_1(D)<N-1$, where $\\mathbb{S}^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of \\eqref{printro} which are also fast decaying, i.e.\n    $$\n    v(x)=O(|x|^{2-N}) \\quad \\text{as }|x|\\to\\infty.\n    $$",
      "start_pos": 13230,
      "end_pos": 13639,
      "label": "thvecchio"
    },
    "ref_dict": {
      "mainth": "\\begin{theorem}\\label{mainth}\nLet $\\Sigma_{D_\\alpha}$ be a family of cones as above, $\\alpha\\in(0,\\alpha^*)$ and assume that $\\lambda_1({D_1})=N-1$ is a simple eigenvalue. Then the point $(1,U)$ is a bifurcation point for the trivial curve $\\alpha\\mapsto(\\alpha,U)$ of solutions to \\eqref{printro}. More precisely there exist $\\epsilon>0$ and a $C^1$ curve  $(-\\epsilon,\\epsilon)\\ni s\\to (\\alpha_s,v_s)$,  such that $\\alpha_0=1$, $v_0=U$, $D_{\\alpha_s}\\ne D_{\\alpha_1}$ and\n$$\n\\begin{cases}\n   -\\Delta v_s=v_s^{2^*-1} &\\text{in }\\Sigma_{D_{\\alpha_s}}\\\\\n    v_s\\in \\mathcal{{D}}^{1,2}(\\Sigma_{D_{\\alpha_s}}),\n\\end{cases}$$\nand $v_s$ is positive and nonradial.\n\\end{theorem}",
      "printro": "\\label{printro}\n\\begin{cases}\n    -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n    \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n    u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\ee",
      "thvecchio": "\\begin{theorem}\\label{thvecchio}\\cite{CPP}\n    Let $\\Sigma_D$ be a cone such that $\\bar D\\subset \\S^{N-1}_+$ and $\\lambda_1(D)<N-1$, where $\\S^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of \\eqref{printro} which are also fast decaying, i.e.\n    $$\n    v(x)=O(|x|^{2-N}) \\quad \\text{as }|x|\\to\\infty.\n    $$\n\\end{theorem}",
      "Phidef": "\\begin{corollary}\\label{cor_L2}\nLet $\\alpha\\in(0,2)$ be such that $D_{\\alpha}\\subset\\S_+^{N-1}$, and $  \\lambda_1(D_{\\alpha})<N-1$. Then there exists $\\bar\\alpha\\in(0,\\alpha)$ such that\n$$\n\\lambda_1(D_{\\bar\\alpha})=N-1.\n$$\nPossibly redefining the parameter $\\alpha$, we may assume $\\bar\\alpha=1$.\n\\end{corollary}\n\nIn order to understand the behavior of $\\lambda_1(D_{\\alpha})$ as $\\alpha$ varies, we interpret the eigenvalue $\\lambda_1(D_{\\alpha})$ as the first Neumann eigenvalue of a suitable operator on the fixed domain $D$. In fact, $\\lambda_1(D_{\\alpha})$ is the first eigenvalue of the Laplacian on $D$ with respect to the metric $g_{\\alpha}$, the pull-back of the round metric on $D_{\\alpha}$ through $\\phi_{\\alpha}$.\n\nMore precisely, if $g$ denotes the standard round metric on $\\mathbb S^{N-1}$, consider on $D$ the metric $g_{\\alpha}:=\\phi_{\\alpha}^*g$:\n$$\n\\langle X,Y\\rangle_{g_{\\alpha}}=\\langle d\\phi_{\\alpha}(X),d\\phi_{\\alpha}(Y)\\rangle_g\n$$\nfor all tangent vectors $X,Y$. Here by $\\langle\\cdot,\\cdot\\rangle_h$ we denote the scalar product induced on tangent spaces by a metric $h$. In what follows we will highlight the dependence on the metric only when necessary, and we will often omit it when the metric is the  standard Euclidean metric in $\\mathbb R^N$ or the standard round metric on $\\mathbb S^{N-1}$. \nThen (by construction), the manifold $(D,g_{\\alpha})$ is isometric to the manifold $(D_{\\alpha},g)$, i.e., the domain $D_{\\alpha}$ with the round metric $g$. Therefore\n$$\n\\lambda_1(D_{\\alpha})=\\lambda_1(D,g_{\\alpha})\n$$\nwhere the notation $\\lambda_1(D,h)$ simply stands for the first Neumann eigenvalue of the Laplacian $\\Delta_h$ on $D$ for a metric $h$.\nThe Laplacian commutes with isometries, that is\n$$\n\\Delta_{g_{\\alpha}}(\\phi_{\\alpha}^*f)=\\phi_{\\alpha}^*(\\Delta_gf)(=\\phi_{\\alpha}^*(\\Delta_{\\mathbb S^{N-1}}f))\n$$\nand this is a concrete way of looking at the new operator on the fixed domain $D$.\n\nIf $D\\subset\\mathbb S^{N-1}_+$, then $r<\\pi/2$ for any point of $D$ ($r$ is the polar coordinate, i.e., the distance from the north pole). The metric $g_{\\alpha}$ on $D$, for $\\alpha\\in(0,2)$, is almost isometric to an homothety of the spherical metric on $D$, in fact \n$$\n\\min\\{1,\\frac{\\sin(\\alpha\\frac{\\pi\n}{2} )}{\\alpha}\\}<\\frac{\\sin(\\alpha r)}{\\alpha\\sin(r)}<\\max\\{1,\\frac{\\sin(\\alpha\\frac{\\pi\n}{2} )}{\\alpha}\\}\n$$\nand then we write\n\\beq\\label{quasiisom}\ng_{\\alpha}=\\alpha^2dr^2+\\sin^2(\\alpha r)g_{\\mathbb S^{N-2}}\\approx\\alpha^2(dr^2+\\sin^2( r)g_{\\mathbb S^{N-2}})=\\alpha^2 g.\n\\eeq\n\n\\subsection{The induced diffeomorphisms on the cone}\n\nWe pass now to the cone  $\\Sigma_{D_{\\alpha}}$, which we want to recast to the fixed cone $\\Sigma_{D}$. Here, we are considering spherical coordinates in $\\mathbb R^N$; in particular, we denote by $\\rho$ the distance from the origin in $\\mathbb R^N$, that is, the radial coordinate. A point $x\\in\\Sigma_D$ will have then coordinates $(\\rho,p)\\in(0,\\infty)\\times D_{\\alpha}$. The standard round metric on $\\mathbb S^{N-1}$ is denoted by $g$. The Euclidean metric $G$ on $\\mathbb R^N$ (i.e., the metric on $\\Sigma_{D_{\\alpha}}$) in the coordinates $(\\rho,p)$ is\n$$\nG=d\\rho^2+\\rho^2 g\n$$\nNow we consider the map $\\Phi_{\\alpha}:\\Sigma_{D}\\to \\Sigma_{D_{\\alpha}}$  defined by\n\\beq\\label{Phidef}\n\\Phi_{\\alpha}(\\rho,p)=(\\rho,\\phi_{\\alpha}(p))\n\\eeq\nand again we consider on $\\Sigma_{D}$ the metric $G_{\\alpha}$, i.e., the pull-back of the Euclidean metric $G$ though $\\Phi_{\\alpha}$. Precisely, we have\n$$\nG_{\\alpha}:=\\Phi_{\\alpha}^*G=d\\rho^2+\\rho^2\\phi_{\\alpha}^*g=d\\rho^2+\\rho^2g_{\\alpha}.\n$$\nNow we express the Laplacian on $\\Sigma_{D}$ with respect to the metric $G_{\\alpha}$:\n\\begin{equation}\\label{laplacian_new_metric}\n\\Delta_{G_{\\alpha}}=\\partial^2_{\\rho\\rho}u+\\frac{N-1}{\\rho}\\partial_\\rho u+\\frac{1}{\\rho^2}\\Delta_{g_{\\alpha}}u.\n\\end{equation}\nAgain, we have that\n\\begin{equation}\\label{commuting}\n\\Delta_{G_{\\alpha}}(\\Phi_{\\alpha}^*f)=\\Phi_{\\alpha}^*(\\Delta_Gf)(=\\Phi_{\\alpha}^*(\\Delta f)),\n\\end{equation}\nand $\\Delta_G=\\Delta$ is the usual Laplacian with respect to the Euclidean metric $G$.\n\n\\subsection{Derivative of $\\lambda_1(D_{\\alpha})$ with respect to $\\alpha$}\n\nNow we compute and estimate the derivative\n$\\frac{d}{d\\alpha}\\lambda_1(D_{\\alpha})\\big|_{\\alpha=1}=\\frac{d}{d\\alpha}\\lambda_1(D,g_\\alpha)\\big|_{\\alpha=1}$ at a simple eigenvalue. Recall that if $\\lambda_1(D_1)$ is simple, then $\\lambda_1(D_{\\alpha})$ is simple for $\\alpha\\in(1-\\epsilon,1+\\epsilon)$ for small $\\epsilon$.\n\n\\begin{proposition}\\label{3.3}\nAssume that $\\lambda_1(D)=\\lambda_1(D_1)$ is simple, and let $u$ be an eigenfunction of $\\lambda_1(D)$ normalized by $\\int_{D}u^2=1$. Then\n\\begin{multline}\\label{derivative}\n\\frac{d}{d\\alpha}\\lambda_1(D_{\\alpha})|_{\\alpha=1}\n=\\frac{d}{d\\alpha}\\lambda_1(D,g_{\\alpha})|_{\\alpha=1}\\\\\n=\\int_{D}\\left[(-1+(N-2)\\frac{r}{\\tan(r)})(\\partial_ru)^2+(1+(N-4)\\frac{r}{\\tan(r)})\\frac{|\\nabla_{\\mathbb S^{N-2}}u|^2}{\\sin^2(r)}\\right]\\sin^{N-2}(r)drd\\sigma_{\\mathbb S^{N-2}}\\\\\n-\\lambda_1(D)\\int_{D}u^2(1+(N-2)\\frac{r}{\\tan(r)})\\sin^{N-2}(r)drd\\sigma_{\\mathbb S^{N-2}}.\n\\end{multline}\n\\end{proposition}\nTo clarify the notation of \\eqref{derivative}, here we are considering polar coordinates $(r,\\omega)\\in(0,\\pi/2)\\times\\mathbb S^{N-2}$ in $\\mathbb S^{N-1}_+$ centered at the north pole, and $d\\sigma_{\\mathbb S^{N-2}}=d\\sigma_{\\mathbb S^{N-2}}(\\omega)$ is the volume element of $\\mathbb S^{N-2}$.\n\nThe proof of the proposition is standard and consists of deriving the eigenvalue of the Laplacian with respect to a $1$-parameter family of varying metrics, in our case, $g_{\\alpha}$, on the fixed manifold $D$. We give the proof following the classical one, which can be found in Berger \\cite{berger}.\n\n\\begin{proof}[Proof of Proposition \\ref{3.3}]\nFollowing \\cite[Section 3]{berger}, we find that\n$$\n\\frac{d}{d\\alpha}\\lambda_1(D,g_{\\alpha})|_{\\alpha=1}=-\\int_{D}\\langle q(u),h\\rangle_g d\\sigma_g,\n$$\nwhere $h=\\frac{d}{d\\alpha}{g_{\\alpha}}_{|_{\\alpha=1}}$ and $q$ is the quadratic form defined as\n$$\nq(u)=du\\otimes du-\\frac{1}{2}(|\\nabla u|^2-\\lambda_1(D)u^2)g.\n$$\nThe notation $\\langle\\cdot,\\cdot\\rangle_g$ stands for the inner product on tensors induced by the metric $g$ and $\\otimes$ is the tensor product. We recall that here $g=g_1$ is the standard round metric.\n\nNow we have that \n$$\ng=dr^2+\\sin^2(r)g_{\\mathbb S^{N-2}},\n$$\n$$\nh=2dr^2+2r\\sin(r)\\cos(r)g_{\\mathbb S^{N-2}},\n$$\nand \n$$\ndu\\otimes du=(\\partial_ru)^2dr\\otimes dr+2\\frac{\\partial_ru}{\\sin(r)}dr\\otimes d_{\\mathbb S^{N-2}}u+\\frac{d_{\\mathbb S^{N-2}}u\\otimes d_{\\mathbb S^{N-2}}u}{\\sin^2(r)}.\n$$\nAll these objects are $(0,2)$ tensors, and then we compute\n$$\n\\langle g,h\\rangle_g=2+2(N-2)\\frac{r}{\\tan(r)}\n$$\nand\n$$\n\\langle du\\otimes du,h\\rangle_g=2(\\partial_ru)^2+2\\frac{r}{\\tan(r)}\\frac{|\\nabla_{\\mathbb S^{N-2}}u|^2}{\\sin^2(r)}.\n$$\nHere we have used that $|\\nabla_{\\mathbb S^{N-2}}u|^2=|d_{\\mathbb S^{N-2}}u|^2$. Recalling that $|\\nabla u|^2=(\\partial_ru)^2+\\frac{|\\nabla_{\\mathbb S^{N-2}}u|^2}{\\sin^2(r)}$ and that $d\\sigma_g=\\sin^{N-2}(r)drd\\sigma_{\\mathbb S^{N-2}}$ we get formula \\eqref{derivative}.\n\\end{proof}\n\nOur aim is to show that the derivative \\eqref{derivative} is negative under suitable assumption on $D$. \n\nFirst, note that $\\frac{r}{\\tan(r)}\\in(0,1)$ when $r\\in(0,\\pi/2)$ ($\\frac{r}{\\tan(r)}\\to 0$ as $r\\to\\pi/2$, $\\frac{r}{\\tan(r)}\\to 1$ as $r\\to 0$, and it is decreasing in $r$). \n\n\\medskip\n\n\\begin{proposition}\n    Let $\\lambda_1(D)=\\lambda_1(D_1)$ be simple and assume that $D\\subset C_t$ where $C_t$ is a geodesic disk of radius $t\\in(0,\\pi/2)$ such that $t/\\tan (t)\\ge\\frac{N-4}{N-2}$. Then\n\\beq\\label{derneg}\n\\frac{d\n}{d\\alpha}\\lambda_1(D_\\alpha)|_{\\alpha=1}<0.\n\\eeq\n\\end{proposition}\n\n\\begin{proof}\n    From \\eqref{derivative} and the fact that $\\frac{r\n    }{\\tan r}<1$ for $r\\in(0,\\pi/2)$ we have\n\\begin{multline}\\label{derivative2}\n\\frac{d}{d\\alpha}\\lambda_1(D_{\\alpha})|_{\\alpha=1}\n\\\\\n<\\int_{D}\\left[(N-3)\\left((\\partial_ru)^2+\\frac{|\\nabla_{\\mathbb S^{N-2}}u|^2}{\\sin^2(r)}\\right)-\\lambda_1(D)u^2((N-2)b+1)\\right]\\sin^{N-2}(r)drd\\sigma_{\\mathbb S^{N-2}}\\\\\n=(N-3)\\int_{D}|\\nabla u|^2-\\left(\\frac{(N-2)b+1}{N-3}\\right)\\lambda_1(D)u^2\\sin^{N-2}(r)drd\\sigma_{\\mathbb S^{N-2}},\n\\end{multline}\n    where $b=b(r)=r/\\tan r$.\nThen taking $t$ such that $t/\\tan t\\ge\\frac{N-4}{N-2}$ by the monotonicity of the function $t/\\tan t$ we have that\n\\beq\\label{contob}\nb=\\frac{r}{\\tan r}\\ge \\frac{t}{\\tan t}\\ge \\frac{N-4}{N-2}\\quad \\forall r\\in (0,t),\n\\eeq\nand hence\n$$\n\\frac{(N-2)b+1}{N-3}>1.\n$$\nSince \n$$\n\\lambda_1(D)=\\frac{\\int_D |\\nabla_{\\mathbb S^{N-1}} u|^2}{\\int_D u^2},\n$$\nwe obtain \\eqref{derneg}.\n\n\\end{proof}\n\\begin{remark}\nNote that for $N=3,4$ one can see directly from \\eqref{derivative2} or from \\eqref{contob} that the derivative is negative for any  domain $D_1=D\\subset\\mathbb S^{N-1}_+$. Instead for $N\\geq 5$ we have to consider domains contained in smaller disks.\n\\end{remark}\n\n\\subsection{Proof of Proposition \\ref{limit_L2}}\\label{ss3.3}\nWe prove here Proposition \\ref{limit_L2}. In fact, one can see that $g_{\\alpha}$ is quasi-isometric to a homothety of the round metric $g$ of factor $\\alpha^2$, and the eigenvalues can be compared.\n\n\\begin{proof}[Proof of Proposition \\ref{limit_L2}]\n\nObserve that from \\eqref{quasiisom} on $D$ the metric $g_{\\alpha}$ and the metric $\\alpha^2 g$ are quasi-isometric. This means that there exists a constant $K>0$ such that\n\\beq\\label{compar}\n\\frac{1}{K}<\\frac{g_{\\alpha}}{\\alpha^2 g}< K\n\\eeq\nFor any eigenvalue $\\lambda_j$, by the min-max principle we have \n$$\n\\lambda_j(D,g_\\alpha) = \\min_{\\substack{V\\subset H^1(D)\\\\\\dim V = j}} \\max_{u \\in V, u \\neq 0} \\frac{\\int_{D} |\\nabla_{g_{\\alpha}} u|_{g_{\\alpha}}^2 \\, d\\sigma_{g_{\\alpha}}}{\\int_{D} u^2 \\, d\\sigma_{g_{\\alpha}}}.\n$$\nHere $d\\sigma_{g_{\\alpha}}$ is the Riemannian volume element of the metric $g_{\\alpha}$ and $|\\nabla_{g_{\\alpha}}u|_{g_{\\alpha}}$ is the norm of the gradient of $u$ for the metric $g_{\\alpha}$.\nFrom \\eqref{compar} we deduce\n$$\nd\\sigma_{\\alpha^2g} = \\sqrt{\\det \\alpha^2g} \\, d\\theta_1 \\dots d\\theta_{N-1} > \\frac{1}{K^{\\frac{N-1}{2}}} \\sqrt{\\det g_\\alpha} \\, d\\theta_1 \\dots d\\theta_{N-1} = \\frac{1}{K^{\\frac{N-1}{2}}} d\\sigma_{g_\\alpha}\n$$\nand\n$$\nd\\sigma_{\\alpha^2g}<K^{\\frac{N-1}{2}} d\\sigma_{g_\\alpha}.\n$$\nMoreover,\n$$\n\\frac1K|\\nabla_{g_\\alpha} u|^2_{g_\\alpha}<|\\nabla_{\\alpha^2g} u|^2_{\\alpha^2g}<K|\\nabla_{g_\\alpha} u|^2_{g_\\alpha}.\n$$\nThen \n\n$$\\frac{\\int_{D} |\\nabla_{g_{\\alpha}} u|_{g_{\\alpha}}^2 \\, d\\sigma_{g_{\\alpha}}}{\\int_{D} u^2 \\, d\\sigma_{g_{\\alpha}}}<K^{1+(N-1)/2+(N-1)/2}\\frac{\\int_{D} |\\nabla_{\\alpha^2g} u|_{\\alpha^2g}^2 \\, d\\sigma_{\\alpha^2g}}{\\int_{D} u^2 \\, d\\sigma_{\\alpha^2g}}\n$$\n\nand\n$$\\frac{1}{K^{1+(N-1)/2+(N-1)/2}}\\frac{\\int_{D} |\\nabla_{\\alpha^2g} u|_{\\alpha^2g}^2 \\, d\\sigma_{\\alpha^2g}}{\\int_{D} u^2 \\, d\\sigma_{\\alpha^2g}}<\\frac{\\int_{D} |\\nabla_{g_{\\alpha}} u|_{g_{\\alpha}}^2 \\, d\\sigma_{g_{\\alpha}}}{\\int_{D} u^2 \\, d\\sigma_{g_{\\alpha}}}.\n$$\nThen, from the Min-Max Theorem, we conclude that\n$$\n\\frac{1}{K^N}\\lambda_j(D,\\alpha^2g)<\\lambda_j(D,g_\\alpha)<K^N\\lambda_j(D,\\alpha^2g).\n$$\n\nNow, we have that\n$$\n\\lambda_j(D,\\alpha^2 g)=\\frac{1}{\\alpha^2}\\lambda_j(D,g)=\\frac{1}{\\alpha^2}\\lambda_j(D)\n$$\nand this yields the result.\n\n\\end{proof}\n\n\\subsection{An example of simple eigenvalue $N-1$}\n\nWe provide here an example of a domain $D$ contained in an hemisphere in $\\R^3$ for which the first eigenvalue $\\lambda_1(D)$ is $2$ and is simple. We just start from a dumbbell domain on $\\mathbb S^2_+$. Assume it is symmetric with respect to the equator and to the (arc of) great circle $\\omega=0$. Here we are using polar coordinates $( r ,\\omega)$, with $ r $ the distance from the north pole. See Figure \\ref{fig1}. Roughly speaking, $D$ consists of two disjoint `fat' parts (e.g., two geodesic disks) connected by a thin channel. In our example, the fat parts are symmetric with respect to the equator (see Figure \\ref{fig1}). The domain $D$ is contained in the strip $ r \\in( r _0,\\pi- r _0)$, $ r _0\\in(0,\\pi/2)$. For any $ r '\\in( r _0,\\pi- r _0)$, we call $\\gamma_{ r '}:=D\\cap\\{ r = r '\\}$ (see Figure \\ref{fig1}). Assume that $\\max_{ r \\in( r _0,\\pi- r _0)}|\\gamma_{ r }|<\\pi/2$.\n\nFrom Corollary \\ref{cor_L2} and from the fact that the first eigenvalue of a fixed dumbbell can be arbitrarily small when the width of the channel is small (see e.g., \\cite{IPW}), we deduce that we can realize such a domain $D=D_1$ with $\\lambda_1(D_1)=2$. We show that $\\lambda_1(D_1)=2$ it is simple. It will follow that $\\lambda_1(D_{\\alpha})$  remains simple for all $\\alpha\\in(1-\\varepsilon,1+\\varepsilon)$ for some $\\varepsilon>0$. \n\n Since we are dealing with $\\lambda_1(D)$, its multiplicity is at most $2$ by Courant's Theorem.  Since any corresponding eigenfunction has exactly two nodal domains, by symmetry  we know that the eigenspace is spanned by eigenfunctions that are either odd with respect to the equator and even with respect to the arc $\\omega=0$, or the other way around. Note that an eigenfunction $u$ corresponding to $\\lambda_1(D)$ cannot be odd (respectively even) with respect to both the equator and the great circle $\\omega=0$. Assume that $u$ is odd with respect to the arc of great circle $\\omega=0$ (the yellow arc in Figure \\ref{fig1}). Then, for any $ r \\in( r _0,\\pi- r _0)$, $\\int_{\\gamma_{ r }}u=0$. Therefore, $\\int_{\\gamma_{ r }}(\\partial_{\\omega}u)^2\\geq\\frac{\\pi^2}{|\\gamma_{ r }|^2}\\int_{\\gamma_{ r }}u^2$. Here $\\frac{\\pi^2}{|\\gamma_{ r }|^2}$ is the first (non trivial) Neumann eigenvalue on $\\gamma_{ r }$.\n\nBy assumption we have $|\\gamma_{ r }|<\\pi/2$. This implies that $\\int_{\\gamma_{ r }}(\\partial_{\\omega}u)^2\\geq 4\\int_{\\gamma_{ r }}u^2$, which means $\\int_{\\gamma_{ r }}|\\nabla u|^2\\geq 4\\int_{\\gamma_{ r }}u^2$. Integrating this last inequality in $ r \\in( r _0,\\pi- r _0)$ we get that $\\int_D|\\nabla u|^2\\geq 4\\int_Du^2$. Hence we have a contradiction since $\\lambda_1(D)=2$. Therefore the first eigenvalue is simple and a corresponding eigenfunction is odd with respect to the equator.\n\nThe construction is valid in any space dimension, we just needed to consider a rotationally symmetric dumbbell in $\\mathbb S^{N-1}$ foliated by $N-2$ dimensional spheres of small enough maximal volume.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{fig_foliation.pdf}\n\\caption{}\n\\label{fig1}\n\\end{figure}\n\n\\section{Bifurcation results}\\label{sec3}\n\\subsection{Spectral analysis}\\label{spectan}\nLet $D\\subset\\S^{N-1}_+$ be a smooth domain and assume that the corresponding cone $\\Sigma_D$ is a Lipschitz domain. We consider the map $\\phi_\\alpha, \\alpha \\in (0,2)$ defined in Section \\ref{ss3.1} and the cone $\\Sigma_{D_\\alpha}$.\nConsider the bubble $U\\in\\mathcal{\nD\n}^{1,2}(\\Sigma_{D_\\alpha})$:\n\\beq\\label{bubbledef}\nU(x):=c_0(1+|x|^2)^{-\\frac{N-2}{2}}\\qquad x\\in \\Sigma_{D_\\alpha},\n\\eeq\nwhere $c_0$ is a constant such that \n$$\n-U^{2-2^*}\\Delta U=U\\qquad \\text{in} \\quad \\Sigma_{D_\\alpha}.\n$$\nThe rescaled bubbles will be denoted by $$\nU_{s}(x):=s U(s^{\\frac{2}{N-2}}x).\n$$\n\\begin{lemma}\\label{lemma_spettro}\nThe operator $L_{U}=U^{2-2^*}\\Delta$ on $L^2(U^{2^*-2})$ has discrete spectrum.\n\\end{lemma}\n\\begin{proof}\n    The proof can be found in \\cite{CPP2}.\n\\end{proof}\nTo describe the spectrum of the operator $L_{U}$ we need to know the eigenvalues $\\lambda_j(D_\\alpha), j\\in \\N$, of problem \\eqref{pb_lambda_def1} on $D_\\alpha$.\nIn particular, we recall that the  $\\lambda_0(D_\\alpha)=0$ and the corresponding eigenfunction is constant, while the first positive eigenvalue is $\\lambda_1(D_\\alpha)$.\n\nNow we are ready to address the following eigenvalue problem that is studied in \\cite{CPP2},\n\\begin{equation}\\label{pb_mu_def2}\n\\begin{cases}\n\\Delta v+\\mu_{\\alpha} U^{2^*-2}v=0 &\\text{in } {\\Sigma_{D_\\alpha}}\\\\\n\\partial_\\nu v=0 &\\text{on }  \\partial{\\Sigma_{D_\\alpha}}\\setminus\\{0\\} \\\\\nv\\in \\mathcal{D}^{1,2}(\\Sigma_{D_\\alpha}), &\n\\end{cases}\n\\end{equation}\nwhose  eigenvalue will be denoted by $\\mu_{i,\\alpha}, i\\in\\N_+$. The eigenvalue equation in \\eqref{pb_mu_def2} can be written by using spherical coordinates, and we can look for solutions by using separation of variables: let $(\\rho,\\theta)\\in(0,+\\infty)\\times\\mathbb S^{N-1}$ the standard spherical coordinates in $\\mathbb R^N$ and let $v=R(\\rho)Y(\\theta)$ be a solution of \\eqref{pb_mu_def2} for an eigenvalue $\\mu_\\alpha$, then it holds that $Y$ is an eigenfunction of \\eqref{pb_lambda_def1} on $D_\\alpha$ for some $j$, while $R$ satisfies (see \\cite{CPP2}, Proposition 2.6)\n\\begin{equation} \\label{SL_eq}\n\\begin{cases}\n(\\rho^{N-1}R')'+\\rho^{N-1}(-\\lambda_j(D_\\alpha) \\rho^{-2}+\\mu_\\alpha k_0^{2^*-2}(1+\\rho^2)^{-2})R=0 & \\text{in }\\R_+ \\\\\n\\lim_{\\rho\\to 0}\\rho^{N-1}R'(\\rho)=0 & \\\\ \nR(\\rho)=o(\\rho^{2-N+\\epsilon})  \\quad \\forall\\epsilon> 0\\text{ as }\\rho\\to + \\infty \\,. &\n\n\\end{cases}\n\\end{equation}\nWe denote by $\\mu_{k,\\lambda_j(D_\\alpha)}, k\\in\\N_+, j\\in\\N$, the sequence of eigenvalues of \\eqref{SL_eq} obtained in correspondence of $\\lambda_j(D_\\alpha)$ and denote by $R^j_k$ the corresponding eigenfunctions.\n\nFor $\\lambda_0(D_\\alpha)=0$ the eigenvalues are given by\n\\beq\\label{lambdazerok}\\mu_{k,\\lambda_0}=\\bigg((k-1)(k+N-2)\\frac{4}{N(N-2)}+1\\bigg)\\,.\n\\eeq\nThe eigenvalues associated with $\\lambda_1(D_\\alpha)$ are\n\\begin{equation}\\label{betaeigenvalue}\n\\mu_{1,\\lambda_1}= \\frac{1}{N(N-2)}\\sqrt{(N-2)^2 + 4\\lambda_1(D_\\alpha)}( 2 +\\sqrt{(N-2)^2 + 4\\lambda_1(D_\\alpha)}) .  \n\\end{equation}\nIn details, it holds the following result (see \\cite{CPP2}).\n\n\\begin{proposition} \\label{lemma_autov}\n\nLet $D_\\alpha\\subset\\S_+^{N-1}$ and $m\\in\\N_+$ be such that\n\\beq\\label{lemma_autov_cond}\n(m-1)(m+N-3)<\\lambda_1(D_\\alpha)\\le m(m+N-2).\n\\eeq\nThen each eigenvalue $\\mu_{i,\\alpha}$ of \\eqref{pb_mu_def2}, with $i=1,...,m$,\nis simple and is given by\n$$\\mu_{i,\\alpha}=\\bigg((i-1)(i+N-2)\\frac{4}{N(N-2)}+1\\bigg),\n$$\ni.e. are the eigenvalues corresponding to $\\lambda_0$ in \\eqref{SL_eq} while\n$$\n\\mu_{m+1,\\alpha}=\\frac{1}{N(N-2)}\\sqrt{(N-2)^2 + 4\\lambda_1(D_\\alpha)}\\bigg( 2 +\\sqrt{(N-2)^2 + 4\\lambda_1(D_\\alpha)}\\bigg).\n$$\nMoreover, $\\psi_i=R_i(r)Y_0(\\theta)$ is an eigenfunction of \\eqref{pb_mu_def2} corresponding to $\\mu_{i,\\alpha}$, for any $i=1,...,m,$ where\n$$\nR_i=\\frac{P_{i}(|x|^2)}{(1+|x|^2)^{\\frac{N-2}{2}}}\n$$\nand $P_i(|x|^2)$ are given as in \\cite[Appendix]{CPP2}. Instead an eigenfunction corresponding to $\\mu_{m+1,\\alpha}$ is given by  $\\psi_{m+1}=R_{m+1}Y_1(\\theta)$ where\n$$R_{m+1}=(|x|^2+1)^{1-\\frac{N}{2}-\\beta(\\lambda_1(D_\\alpha))}|x|^{\\beta(\\lambda_1(D_\\alpha))}$$\nwith\n$2\\beta(\\lambda_1(D_\\alpha))=-(N-2) + \\sqrt{(N-2)^2 + 4\\lambda_1(D_\\alpha)} $ and $Y_1$ is an eigenfunction of \\eqref{pb_lambda_def1} corresponding to $\\lambda_1(D_\\alpha)$.\n\\end{proposition}\n\\begin{proof}\n    The proof follows by \\cite[Proposition 2.7]{CPP2} and the fact that we are taking $U$ such that $-\\Delta U=U^{2^*-1}$, so that the constant $S^{2^*}_U$ in \\cite{CPP2} is replaced by $1$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor_autoval}\n    Let $D_{\\bar{\\alpha}}\\subset\\S_+^{N-1}$ be such that $\\lambda_1(D_{\\bar\\alpha})=N-1$, then\n    $$\n    \\mu_{1,{\\bar\\alpha}}=1\n    $$\n    and the first eigenfunction is given by $U_{\\bar\\alpha}\n    $,\n$$\\mu_{2,{\\bar\\alpha}}= \\frac{1}{N(N-2)}\\sqrt{(N-2)^2 + 4\\lambda_1(D_{\\bar\\alpha})}\\bigg( 2 +\\sqrt{(N-2)^2 + 4\\lambda_1(D_{\\bar\\alpha})}\\bigg)=\n2^*-1\n$$\nand the second eigenfunctions are given by $\\partial_s U_{s,{\\bar\\alpha}}$ and $\\psi_2=(|x|^2+1)^{-\\frac{N}{2}}|x|Y_1(\\frac{x}{|x|})$. \n\\end{corollary}",
      "CRT": "\\begin{theorem}[Crandall-Rabinowitz Bifurcation Theorem] \\label{CRT}\nLet $X$ and $Y$ be Banach spaces, and let $\\Omega\\subset X$ and $I \\subset \\mathbb{R}$ be open domains, where we assume $0 \\in \\Omega$. Denote the elements of $\\Omega$ by $v$ and the elements of $I$ by $t$. Let $F: I \\times \\Omega \\to Y$ be a $C^2$ operator such that\n\n\\begin{itemize}\n    \\item[(i)] $F(t, 0) = 0$ for all $t \\in I$,\n    \\item[(ii)] $\\text{Ker} D_v F(t_*, 0) = \\mathbb{R}w$ for some $t_* \\in I$ and some $w \\in X \\setminus \\{0\\}$;\n    \\item[(iii)] $\\text{codim} \\, \\text{Im} D_v F(t_*, 0) = 1$;\n    \\item[(iv)] $D_t D_v F(t_*, 0)(w) \\notin \\text{Im} D_v F(t_*, 0)$.\n\\end{itemize}\n\nThen there exists a nontrivial $C^1$ curve\n$$\\quad (-\\epsilon, \\epsilon) \\ni s \\mapsto (t(s), v(s)) \\in I \\times X,\n$$\nfor some $\\epsilon > 0$, such that:\n\\begin{itemize}\n    \\item[(1)] $t(0) = t_*, \\, t'(0) = 0, \\, v(0) = 0, \\, v'(0) = w$;\n    \\item[(2)] $F(t(s), v(s)) = 0$ for all $s \\in (-\\epsilon, +\\epsilon)$.\n\\end{itemize}\n\nMoreover, there exists a neighborhood $N$ of $(t_*, 0)$ in $X \\times Y$ such that all solutions of the equation $F(t, v) = 0$ in $N$ belong to the trivial solution line $\\{(t, 0)\\}$ or to the curve. The intersection $(t_*, 0)$ is called a bifurcation point.\n\n\\end{theorem}",
      "Sobqintro": "\\begin{cases}\n    -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n    \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n    u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\eeq\nwhere $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\\mathcal{D}^{1,2}(\\Sigma_D)$ into $L^{2^*}(\\Sigma_D)$ and\n$$\n\\mathcal{D}^{1,2}(\\Sigma_D)=\\{u\\in L^{2^*}(\\Sigma_D):|\\nabla u|\\in L^2(\\Sigma_D)\\}.\n$$\nIt is well known that the radial function:\n\\beq\\label{bubbleintro}\nU(x)=c_0\\bigg(\\frac{1}{1+|x|^2}\\bigg)^\\frac{N-2}{N},\\qquad x\\in\\Sigma_D,\n\\eeq\nis a solution of \\eqref{printro}, for $c_0=(N(N-2))^\\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\\R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\\R^N$, the function $U$, its rescalings and translations are the only positive solutions of \\eqref{printro}. Obviously they are also the only radial solutions of \\eqref{printro} and we refer to them as the trivial solutions of \\eqref{printro}. Then the question is whether in $\\Sigma_D$, for $D\\ne\\S^{N-1}$, there exist also nonradial solutions. \n\nThe study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \\cite{LP,LPT}).\nIn particular in \\cite{LPT} it was proved that if the cone $\\Sigma_D$ is convex then the bubbles are the only positive solution of \\eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \\cite{CFR}. Thus the question is whether nonradial solutions of \\eqref{printro} exist in nonconvex cones.\n\nSince minimizers of the Sobolev quotient \n\\beq\\label{Sobqintro}\nQ_D(u)=\\frac{\\big(\\int_{\\Sigma_D}|\\nabla u|^2\\big)^\\frac{1}{2}}{\\big(\\int_{\\Sigma_D} |u|^{2^*}\\big)^\\frac{1}{2^*}},\\qquad u\\in\\mathcal{D}^{1,2}(\\Sigma_D),u\\ne 0,\n\\eeq\nproduce solutions of \\eqref{printro}, a first result was obtained in \\cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \\eqref{printro} admits a nonradial solution (as a minimizer of \\eqref{Sobqintro}) was obtained in \\cite{CPP}.\nIt pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\\Delta_{\\S^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.\nThe result of \\cite{CPP} is the following:\n\\begin{theorem}\\label{thvecchio}\\cite{CPP}\n    Let $\\Sigma_D$ be a cone such that $\\bar D\\subset \\S^{N-1}_+$ and $\\lambda_1(D)<N-1$, where $\\S^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of \\eqref{printro} which are also fast decaying, i.e.\n    $$\n    v(x)=O(|x|^{2-N}) \\quad \\text{as }|x|\\to\\infty.\n    $$\n\\end{theorem}\nIt is interesting to quote that the same bound on the eigenvalue $\\lambda_1(D)$, i.e. $\\lambda_1(D)<N-1$, is also the key point to show breaking of symmetry results for overdetermined and constant mean curvature problems in cones (see \\cite{IPW}). \n\nThe proof of Theorem \\ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \\eqref{Sobqintro} as soon as $\\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \\cite{CPP}). Instead when $\\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \\cite{CPP} shows that, varying the domains $D\\subset \\S^{N-1}$ (and hence the cone $\\Sigma_D$) the bubble $U$ is degenerate for any cone $\\Sigma_{D_1}$ such that $\\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \\eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.\n\nThis is the aim of the present paper. To state precisely our result, let $D=D_1$ be a domain in $\\S^{N-1}_+$ such that $\\lambda_1(D_1)=N-1$.\nWe consider the family of domains $D_\\alpha$ which are diffeomorphic to $D_1$ through the diffeomorphism $\\Phi_\\alpha$ defined in Section \\ref{sec2} (see \\eqref{Phidef}), for $\\alpha\\in (0,\\alpha^*)$, where $\\alpha^*>0$ is the maximum value of $\\alpha$ such that $\\Phi_\\alpha(D_1)$ is contained in $\\S^{N-1}_+$. By construction, we have that $\\alpha^*>1$.\nWe stress that, for every $\\alpha$ the standard bubble $U$ is a solution of \\eqref{printro} in $\\Sigma_{D_\\alpha}$, so we have the trivial curve of solutions $\\alpha\n\\to(\\alpha,U)$, for every $\\alpha\\in(0,\\alpha^*)$. Our bifurcation result is the following.\n\\begin{theorem}\\label{mainth}\nLet $\\Sigma_{D_\\alpha}$ be a family of cones as above, $\\alpha\\in(0,\\alpha^*)$ and assume that $\\lambda_1({D_1})=N-1$ is a simple eigenvalue. Then the point $(1,U)$ is a bifurcation point for the trivial curve $\\alpha\\mapsto(\\alpha,U)$ of solutions to \\eqref{printro}. More precisely there exist $\\epsilon>0$ and a $C^1$ curve  $(-\\epsilon,\\epsilon)\\ni s\\to (\\alpha_s,v_s)$,  such that $\\alpha_0=1$, $v_0=U$, $D_{\\alpha_s}\\ne D_{\\alpha_1}$ and\n$$\n\\begin{cases}\n   -\\Delta v_s=v_s^{2^*-1} &\\text{in }\\Sigma_{D_{\\alpha_s}}\\\\\n    v_s\\in \\mathcal{{D}}^{1,2}(\\Sigma_{D_{\\alpha_s}}),\n\\end{cases}"
    },
    "pre_theorem_intro_text_len": 3064,
    "pre_theorem_intro_text": "Let $D$ be a smooth domain on the unit sphere $\\mathbb{S}^{N-1}\\subset\\mathbb R^N$, with $N\\geqslant 3,$ and let $\\Sigma_D\\subset\\mathbb R^N$ be the cone spanned by $D$:\n\\begin{equation}\\label{cono}\n\\Sigma_D=\\{x\\in\\mathbb R^N: x=\\rho p,p\\in D,\\rho\\in(0,+\\infty)\\},\n\\end{equation}\nand assume that $\\Sigma_D$ is a Lipschitz domain.\nWe consider the following semilinear critical Neumann problem\n\\begin{equation}\\label{printro}\n\\begin{cases}\n    -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n    \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n    u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\end{equation}\nwhere $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\\mathcal{D}^{1,2}(\\Sigma_D)$ into $L^{2^*}(\\Sigma_D)$ and\n$$\n\\mathcal{D}^{1,2}(\\Sigma_D)=\\{u\\in L^{2^*}(\\Sigma_D):|\\nabla u|\\in L^2(\\Sigma_D)\\}.\n$$\nIt is well known that the radial function:\n\\begin{equation}\\label{bubbleintro}\nU(x)=c_0\\bigg(\\frac{1}{1+|x|^2}\\bigg)^\\frac{N-2}{N},\\qquad x\\in\\Sigma_D,\n\\end{equation}\nis a solution of \\eqref{printro}, for $c_0=(N(N-2))^\\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\\mathbb R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\\mathbb R^N$, the function $U$, its rescalings and translations are the only positive solutions of \\eqref{printro}. Obviously they are also the only radial solutions of \\eqref{printro} and we refer to them as the trivial solutions of \\eqref{printro}. Then the question is whether in $\\Sigma_D$, for $D\\ne\\mathbb{S}^{N-1}$, there exist also nonradial solutions. \n\nThe study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \\cite{LP,LPT}).\nIn particular in \\cite{LPT} it was proved that if the cone $\\Sigma_D$ is convex then the bubbles are the only positive solution of \\eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \\cite{CFR}. Thus the question is whether nonradial solutions of \\eqref{printro} exist in nonconvex cones.\n\nSince minimizers of the Sobolev quotient \n\\begin{equation}\\label{Sobqintro}\nQ_D(u)=\\frac{\\big(\\int_{\\Sigma_D}|\\nabla u|^2\\big)^\\frac{1}{2}}{\\big(\\int_{\\Sigma_D} |u|^{2^*}\\big)^\\frac{1}{2^*}},\\qquad u\\in\\mathcal{D}^{1,2}(\\Sigma_D),u\\ne 0,\n\\end{equation}\nproduce solutions of \\eqref{printro}, a first result was obtained in \\cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \\eqref{printro} admits a nonradial solution (as a minimizer of \\eqref{Sobqintro}) was obtained in \\cite{CPP}.\nIt pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\\Delta_{\\mathbb{S}^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.\nThe result of \\cite{CPP} is the following:",
    "context": "Let $D$ be a smooth domain on the unit sphere $\\mathbb{S}^{N-1}\\subset\\mathbb R^N$, with $N\\geqslant 3,$ and let $\\Sigma_D\\subset\\mathbb R^N$ be the cone spanned by $D$:\n\\begin{equation}\\label{cono}\n\\Sigma_D=\\{x\\in\\mathbb R^N: x=\\rho p,p\\in D,\\rho\\in(0,+\\infty)\\},\n\\end{equation}\nand assume that $\\Sigma_D$ is a Lipschitz domain.\nWe consider the following semilinear critical Neumann problem\n\\begin{equation}\\label{printro}\n\\begin{cases}\n -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\end{equation}\nwhere $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\\mathcal{D}^{1,2}(\\Sigma_D)$ into $L^{2^*}(\\Sigma_D)$ and\n$$\n\\mathcal{D}^{1,2}(\\Sigma_D)=\\{u\\in L^{2^*}(\\Sigma_D):|\\nabla u|\\in L^2(\\Sigma_D)\\}.\n$$\nIt is well known that the radial function:\n\\begin{equation}\\label{bubbleintro}\nU(x)=c_0\\bigg(\\frac{1}{1+|x|^2}\\bigg)^\\frac{N-2}{N},\\qquad x\\in\\Sigma_D,\n\\end{equation}\nis a solution of \\eqref{printro}, for $c_0=(N(N-2))^\\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\\mathbb R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\\mathbb R^N$, the function $U$, its rescalings and translations are the only positive solutions of \\eqref{printro}. Obviously they are also the only radial solutions of \\eqref{printro} and we refer to them as the trivial solutions of \\eqref{printro}. Then the question is whether in $\\Sigma_D$, for $D\\ne\\mathbb{S}^{N-1}$, there exist also nonradial solutions.\n\nThe study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \\cite{LP,LPT}).\nIn particular in \\cite{LPT} it was proved that if the cone $\\Sigma_D$ is convex then the bubbles are the only positive solution of \\eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \\cite{CFR}. Thus the question is whether nonradial solutions of \\eqref{printro} exist in nonconvex cones.\n\nSince minimizers of the Sobolev quotient \n\\begin{equation}\\label{Sobqintro}\nQ_D(u)=\\frac{\\big(\\int_{\\Sigma_D}|\\nabla u|^2\\big)^\\frac{1}{2}}{\\big(\\int_{\\Sigma_D} |u|^{2^*}\\big)^\\frac{1}{2^*}},\\qquad u\\in\\mathcal{D}^{1,2}(\\Sigma_D),u\\ne 0,\n\\end{equation}\nproduce solutions of \\eqref{printro}, a first result was obtained in \\cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \\eqref{printro} admits a nonradial solution (as a minimizer of \\eqref{Sobqintro}) was obtained in \\cite{CPP}.\nIt pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\\Delta_{\\mathbb{S}^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.\nThe result of \\cite{CPP} is the following:",
    "full_context": "Let $D$ be a smooth domain on the unit sphere $\\mathbb{S}^{N-1}\\subset\\mathbb R^N$, with $N\\geqslant 3,$ and let $\\Sigma_D\\subset\\mathbb R^N$ be the cone spanned by $D$:\n\\begin{equation}\\label{cono}\n\\Sigma_D=\\{x\\in\\mathbb R^N: x=\\rho p,p\\in D,\\rho\\in(0,+\\infty)\\},\n\\end{equation}\nand assume that $\\Sigma_D$ is a Lipschitz domain.\nWe consider the following semilinear critical Neumann problem\n\\begin{equation}\\label{printro}\n\\begin{cases}\n -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\end{equation}\nwhere $2^*=\\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\\mathcal{D}^{1,2}(\\Sigma_D)$ into $L^{2^*}(\\Sigma_D)$ and\n$$\n\\mathcal{D}^{1,2}(\\Sigma_D)=\\{u\\in L^{2^*}(\\Sigma_D):|\\nabla u|\\in L^2(\\Sigma_D)\\}.\n$$\nIt is well known that the radial function:\n\\begin{equation}\\label{bubbleintro}\nU(x)=c_0\\bigg(\\frac{1}{1+|x|^2}\\bigg)^\\frac{N-2}{N},\\qquad x\\in\\Sigma_D,\n\\end{equation}\nis a solution of \\eqref{printro}, for $c_0=(N(N-2))^\\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\\mathbb R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\\mathbb R^N$, the function $U$, its rescalings and translations are the only positive solutions of \\eqref{printro}. Obviously they are also the only radial solutions of \\eqref{printro} and we refer to them as the trivial solutions of \\eqref{printro}. Then the question is whether in $\\Sigma_D$, for $D\\ne\\mathbb{S}^{N-1}$, there exist also nonradial solutions.\n\nThe study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \\cite{LP,LPT}).\nIn particular in \\cite{LPT} it was proved that if the cone $\\Sigma_D$ is convex then the bubbles are the only positive solution of \\eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \\cite{CFR}. Thus the question is whether nonradial solutions of \\eqref{printro} exist in nonconvex cones.\n\nSince minimizers of the Sobolev quotient \n\\begin{equation}\\label{Sobqintro}\nQ_D(u)=\\frac{\\big(\\int_{\\Sigma_D}|\\nabla u|^2\\big)^\\frac{1}{2}}{\\big(\\int_{\\Sigma_D} |u|^{2^*}\\big)^\\frac{1}{2^*}},\\qquad u\\in\\mathcal{D}^{1,2}(\\Sigma_D),u\\ne 0,\n\\end{equation}\nproduce solutions of \\eqref{printro}, a first result was obtained in \\cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \\eqref{printro} admits a nonradial solution (as a minimizer of \\eqref{Sobqintro}) was obtained in \\cite{CPP}.\nIt pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\\Delta_{\\mathbb{S}^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.\nThe result of \\cite{CPP} is the following:\n\nThe study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \\cite{LP,LPT}).\nIn particular in \\cite{LPT} it was proved that if the cone $\\Sigma_D$ is convex then the bubbles are the only positive solution of \\eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \\cite{CFR}. Thus the question is whether nonradial solutions of \\eqref{printro} exist in nonconvex cones.\n\nThe proof of Theorem \\ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \\eqref{Sobqintro} as soon as $\\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \\cite{CPP}). Instead when $\\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \\cite{CPP} shows that, varying the domains $D\\subset \\S^{N-1}$ (and hence the cone $\\Sigma_D$) the bubble $U$ is degenerate for any cone $\\Sigma_{D_1}$ such that $\\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \\eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.\n\nLet $D\\subset\\mathbb S^{N-1}_+$ be a smooth domain. By $\\lambda_j(D)$ with $j\\in \\N$, we denote the eigenvalues of the Laplacian $\\Delta_{\\mathbb S^{N-1}}$ on $D$ with Neumann boundary conditions, i.e. we consider the following problem:\n\\begin{equation}\\label{pb_lambda_def1}\n\\begin{cases}\n-\\Delta_{\\S^{N-1}}Y_j = \\lambda_j(D) Y_j & \\text{on } D\\\\\n\\partial_{\\nu_D} Y_j = 0 & \\text{on } \\partial D \\,.\n\\end{cases}\n\\end{equation}\nIt is well known that $-\\Delta_{\\S^{N-1}}$ has a compact, self-adjoint resolvent in $L^2(D)$ and admits a sequence of eigenvalues \n\\begin{equation} \\label{lambdabe}\n0=\\lambda_0(D)<\\lambda_1(D)\\le \\cdots \\le \\lambda_j(D) \\le \\cdots\\nearrow+\\infty\n\\end{equation} \n(repeated according to their finite multiplicity) and corresponding eigenfunctions $Y_j(\\theta) \\in L^2(D)$. Here $\\theta=(\\theta_1,...,\\theta_{N-1})$ is the system of coordinates on $D$ induced by the spherical coordinates in $\\R^N$. The eigenfunctions $\\{Y_j\\}_{j\\in\\mathbb N}$ can be chosen to form a Hilbert basis for $L^2(D)$, they satisfy\n\\beq\\label{def:lapleigbe}\n-\\Delta_{\\S^{N-1}}Y_j(\\theta)=\\lambda_j(D) Y_j(\\theta),\\quad \\theta\\in D, \n\\eeq\nand \n$$\n\\int_{D} \\langle \\nabla_{\\mathbb S^{N-1}} Y_j(\\theta) , \\nabla_{\\mathbb S^{N-1}} Y_i(\\theta)\\rangle d\\sigma(\\theta) = \\lambda_j(D)\\delta_{ij}\n$$\nwhere we denote by $\\nabla_{\\mathbb S^{N-1}}$ the gradient on $\\mathbb S^{N-1}$.\n\n\\section{Bifurcation results}\\label{sec3}\n\\subsection{Spectral analysis}\\label{spectan}\nLet $D\\subset\\S^{N-1}_+$ be a smooth domain and assume that the corresponding cone $\\Sigma_D$ is a Lipschitz domain. We consider the map $\\phi_\\alpha, \\alpha \\in (0,2)$ defined in Section \\ref{ss3.1} and the cone $\\Sigma_{D_\\alpha}$.\nConsider the bubble $U\\in\\mathcal{\nD\n}^{1,2}(\\Sigma_{D_\\alpha})$:\n\\beq\\label{bubbledef}\nU(x):=c_0(1+|x|^2)^{-\\frac{N-2}{2}}\\qquad x\\in \\Sigma_{D_\\alpha},\n\\eeq\nwhere $c_0$ is a constant such that \n$$\n-U^{2-2^*}\\Delta U=U\\qquad \\text{in} \\quad \\Sigma_{D_\\alpha}.\n$$\nThe rescaled bubbles will be denoted by $$\nU_{s}(x):=s U(s^{\\frac{2}{N-2}}x).\n$$\n\\begin{lemma}\\label{lemma_spettro}\nThe operator $L_{U}=U^{2-2^*}\\Delta$ on $L^2(U^{2^*-2})$ has discrete spectrum.\n\\end{lemma}\n\\begin{proof}\n The proof can be found in \\cite{CPP2}.\n\\end{proof}\nTo describe the spectrum of the operator $L_{U}$ we need to know the eigenvalues $\\lambda_j(D_\\alpha), j\\in \\N$, of problem \\eqref{pb_lambda_def1} on $D_\\alpha$.\nIn particular, we recall that the $\\lambda_0(D_\\alpha)=0$ and the corresponding eigenfunction is constant, while the first positive eigenvalue is $\\lambda_1(D_\\alpha)$.\n\nNow we are ready to address the following eigenvalue problem that is studied in \\cite{CPP2},\n\\begin{equation}\\label{pb_mu_def2}\n\\begin{cases}\n\\Delta v+\\mu_{\\alpha} U^{2^*-2}v=0 &\\text{in } {\\Sigma_{D_\\alpha}}\\\\\n\\partial_\\nu v=0 &\\text{on } \\partial{\\Sigma_{D_\\alpha}}\\setminus\\{0\\} \\\\\nv\\in \\mathcal{D}^{1,2}(\\Sigma_{D_\\alpha}), &\n\\end{cases}\n\\end{equation}\nwhose eigenvalue will be denoted by $\\mu_{i,\\alpha}, i\\in\\N_+$. The eigenvalue equation in \\eqref{pb_mu_def2} can be written by using spherical coordinates, and we can look for solutions by using separation of variables: let $(\\rho,\\theta)\\in(0,+\\infty)\\times\\mathbb S^{N-1}$ the standard spherical coordinates in $\\mathbb R^N$ and let $v=R(\\rho)Y(\\theta)$ be a solution of \\eqref{pb_mu_def2} for an eigenvalue $\\mu_\\alpha$, then it holds that $Y$ is an eigenfunction of \\eqref{pb_lambda_def1} on $D_\\alpha$ for some $j$, while $R$ satisfies (see \\cite{CPP2}, Proposition 2.6)\n\\begin{equation} \\label{SL_eq}\n\\begin{cases}\n(\\rho^{N-1}R')'+\\rho^{N-1}(-\\lambda_j(D_\\alpha) \\rho^{-2}+\\mu_\\alpha k_0^{2^*-2}(1+\\rho^2)^{-2})R=0 & \\text{in }\\R_+ \\\\\n\\lim_{\\rho\\to 0}\\rho^{N-1}R'(\\rho)=0 & \\\\ \nR(\\rho)=o(\\rho^{2-N+\\epsilon}) \\quad \\forall\\epsilon> 0\\text{ as }\\rho\\to + \\infty \\,. &\n\n\\subsection{Main results}\nLet $D_\\alpha$ be a domain as in Section \\ref{sec2}.\nWe consider the following problem\n\\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta v=v^{2^*-1} &\\text{ in }\\Sigma_{D_\\alpha}\\\\\n v\\in\\mathcal{D}^{1,2}(\\Sigma_{D_\\alpha}).\n\\end{cases}\n\\end{equation}\nFor the reader convenience we recall the notation for the diffeomorphism considered in Section \\ref{sec2}.\n Let $\\alpha\\in(0,\\alpha^*)$, where $\\alpha^*$ is the maximum value of $\\alpha$ such that $\\Phi_\\alpha(D_1)\\subset \\S^{N-1}_+$, and consider the map\n$$\n\\phi_{\\alpha}:\\mathbb S^{N-1}_+\\to C_{\\alpha}\n$$\ndefined by\n$$\n\\phi_{\\alpha}(r,\\omega)=(\\alpha r,\\omega),\n$$\nwhere $C_{\\alpha}$ is a (spherical) disk centered at the north pole and radius $\\alpha\\pi/2$. Then we define\n$$\n\\Phi_{\\alpha}(\\rho,p)=(\\rho,\\phi_{\\alpha}(p)),\n$$\nwhere $(\\rho,p)\\in(0,+\\infty)\\times D$, and $p=(r,\\omega)\\in D$ is a point in $D$. The map $\\phi_{\\alpha}$ induces\n$$\n\\Phi_{\\alpha}^*:\\mathcal{D}^{1,2}(\\Sigma _{D_\\alpha})\\to\\mathcal{D}^{1,2}(\\Sigma _{D_1})\n$$\ndefined by $\\Phi_{\\alpha}^*u=u\\circ \\Phi_{\\alpha}$. We define an operator $\\mathcal L_{\\alpha}$ acting on $\\mathcal D^{1,2}(\\Sigma_{D_1})$ by\n$$\n\\mathcal L_{\\alpha}(\\Phi_{\\alpha}^*v)=\\Phi_{\\alpha}^*(-\\Delta v),\n$$\nfor $v\\in\\mathcal D^{1,2}(\\Sigma_{D_{\\alpha}})$ (as usual, the operator is defined in the weak sense). This operator can be described more explicitly: it is $-\\Delta_{G_{\\alpha}}$ defined in \\eqref{laplacian_new_metric}, however we won't need this explicit expression in the following.",
    "post_theorem_intro_text_len": 5099,
    "post_theorem_intro_text": "It is interesting to quote that the same bound on the eigenvalue $\\lambda_1(D)$, i.e. $\\lambda_1(D)<N-1$, is also the key point to show breaking of symmetry results for overdetermined and constant mean curvature problems in cones (see \\cite{IPW}). \n\nThe proof of Theorem \\ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \\eqref{Sobqintro} as soon as $\\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \\cite{CPP}). Instead when $\\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \\cite{CPP} shows that, varying the domains $D\\subset \\mathbb{S}^{N-1}$ (and hence the cone $\\Sigma_D$) the bubble $U$ is degenerate for any cone $\\Sigma_{D_1}$ such that $\\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \\eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.\n\nThis is the aim of the present paper. To state precisely our result, let $D=D_1$ be a domain in $\\mathbb{S}^{N-1}_+$ such that $\\lambda_1(D_1)=N-1$.\nWe consider the family of domains $D_\\alpha$ which are diffeomorphic to $D_1$ through the diffeomorphism $\\Phi_\\alpha$ defined in Section \\ref{sec2} (see \\eqref{Phidef}), for $\\alpha\\in (0,\\alpha^*)$, where $\\alpha^*>0$ is the maximum value of $\\alpha$ such that $\\Phi_\\alpha(D_1)$ is contained in $\\mathbb{S}^{N-1}_+$. By construction, we have that $\\alpha^*>1$.\nWe stress that, for every $\\alpha$ the standard bubble $U$ is a solution of \\eqref{printro} in $\\Sigma_{D_\\alpha}$, so we have the trivial curve of solutions $\\alpha\n\\to(\\alpha,U)$, for every $\\alpha\\in(0,\\alpha^*)$. Our bifurcation result is the following.\n\\begin{theorem}\\label{mainth}\nLet $\\Sigma_{D_\\alpha}$ be a family of cones as above, $\\alpha\\in(0,\\alpha^*)$ and assume that $\\lambda_1({D_1})=N-1$ is a simple eigenvalue. Then the point $(1,U)$ is a bifurcation point for the trivial curve $\\alpha\\mapsto(\\alpha,U)$ of solutions to \\eqref{printro}. More precisely there exist $\\epsilon>0$ and a $C^1$ curve  $(-\\epsilon,\\epsilon)\\ni s\\to (\\alpha_s,v_s)$,  such that $\\alpha_0=1$, $v_0=U$, $D_{\\alpha_s}\\ne D_{\\alpha_1}$ and\n$$\n\\begin{cases}\n   -\\Delta v_s=v_s^{2^*-1} &\\text{in }\\Sigma_{D_{\\alpha_s}}\\\\\n    v_s\\in \\mathcal{{D}}^{1,2}(\\Sigma_{D_{\\alpha_s}}),\n\\end{cases}$$\nand $v_s$ is positive and nonradial.\n\\end{theorem}\nTo prove our result we will use the classical Crandall-Rabinowitz bifurcation theorem (see Theorem \\ref{CRT}). The main difficulty in proving Theorem \\ref{mainth} is to find a good family of domains $D_\\alpha$ on $\\mathbb{S}^{N-1}$ for which the corresponding eigenvalues $\\lambda_1(D_\\alpha)$ can be analyzed. Roughly speaking, the goal is to produce a family of domains for which $\\lambda_1(D_{\\alpha})$ crosses the value $N-1$ as $\\alpha$ varies in $(0,\\alpha^*)$ in a strict monotonic way. This is not easy, since the behavior of Neumann eigenvalues under domain perturbation can be quite involved, already in the Euclidean case; moreover, Neumann eigenvalues do not enjoy domain monotonicity and futhermore, on the sphere, we don't have the notion of homothety (hence we don't have simple transformations which allow to track explicitly the eigenvalues as in $\\mathbb R^N$). In Section \\ref{sec2} we construct a family of diffeomorphism $\\Phi_\\alpha$ on the hemisphere $\\mathbb{S}^{N-1}_+$, which are, essentially, dilations with respect to the north pole. Then for the corresponding domain $D_\\alpha=\\Phi_\\alpha(D)$ ($D$ a fixed domain suitably chosen) we are able to study the asymptotic behavior of $\\lambda_1(D_\\alpha)$, as $\\alpha\\to 0$, and, what is more important, to compute the derivative $\\frac{d}{d\\alpha}\\lambda_1(D_\\alpha).$\n\nThe proof that this derivative is negative at $\\alpha=1$ (that is, when $\\lambda_1(D_1)=N-1$) is the key point to get the so-called transversality condition in the application of Crandall-Rabinowitz theorem.\n\nWe also provide an example of a family of domains $D_\\alpha$ for which $\\lambda_1(D_\\alpha)$ is simple (in a neighborhood of $\\alpha=1$). However we point out that, generically, for almost all domains on $\\mathbb{S}^{N-1}$ all Neumann eigenvalues are simple (see \\cite{Henri}).\n\nThe other steps of the proof of Theorem \\ref{mainth} follow by studying problem \\eqref{printro} in a suitable space where, in particular, all functions are invariant by the Kelvin transform. This is inspired by the study of other critical exponent problems (see \\cite{GGT}).\n\nFinally we show that the bifurcation is global and that the Rabinowitz alternative holds. We refer to Section \\ref{sec3} for the precise statement.\n\\bigskip\n\n\\textbf{Organization of the paper.} The paper is organized as follows. Section \\ref{sec2} is devoted to the geometrical question of finding domains on the sphere to which the Crandall-Rabinowitz bifurcation Theorem can be applied. In Section \\ref{sec3} we prove the bifurcation results.",
    "sketch": "The post-theorem text contains a proof sketch for Theorem~\\ref{thvecchio}: it “relies on a careful analysis of the Morse index of the standard bubble $U$,” showing that $U$ “becomes an unstable critical point of \\eqref{Sobqintro} as soon as $\\lambda_1(D)$ crosses the value $N-1$” (cf. Theorem 4.3 in \\cite{CPP}). By contrast, “when $\\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum.” The analysis also identifies degeneracy at the threshold: “varying the domains $D\\subset \\mathbb{S}^{N-1}$ … the bubble $U$ is degenerate for any cone $\\Sigma_{D_1}$ such that $\\lambda_1(D_1)=N-1$,” which “suggests that a bifurcation from the standard bubble could appear when $\\lambda_1(D_1)=N-1$,” i.e. “a branch of nonradial solutions … could emanate from the bubble $U$ in $D_1$,” yielding nonradial minimizers/solutions in the regime $\\lambda_1(D)<N-1$.",
    "expanded_sketch": "The post-theorem text contains a proof sketch for the preceding theorem: it relies on a careful analysis of the Morse index of the standard bubble $U$, showing that $U$ becomes an unstable critical point of\n\\beq\\label{Sobqintro}\nQ_D(u)=\\frac{\\big(\\int_{\\Sigma_D}|\\nabla u|^2\\big)^\\frac{1}{2}}{\\big(\\int_{\\Sigma_D} |u|^{2^*}\\big)^\\frac{1}{2^*}},\\qquad u\\in\\mathcal{D}^{1,2}(\\Sigma_D),u\\ne 0,\n\\eeq\nas soon as $\\lambda_1(D)$ crosses the value $N-1$ (cf. Theorem 4.3 in Colasuonno–Pacella–Pistoia, \\emph{Title not provided} (year not provided)). By contrast, when $\\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q_D$, i.e. it is a strict local minimum. The analysis also identifies degeneracy at the threshold: varying the domains $D\\subset \\mathbb{S}^{N-1}$, the bubble $U$ is degenerate for any cone $\\Sigma_{D_1}$ such that $\\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\\lambda_1(D_1)=N-1$, i.e. a branch of nonradial solutions could emanate from the bubble $U$ in $D_1$, yielding nonradial minimizers/solutions in the regime $\\lambda_1(D)<N-1.",
    "expanded_theorem": "\\label{thvecchio}Chou, Chu, and Pei, “CPP”\n    Let $\\Sigma_D$ be a cone such that $\\bar D\\subset \\mathbb{S}^{N-1}_+$ and $\\lambda_1(D)<N-1$, where $\\mathbb{S}^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of\n\\label{printro}\n\\begin{cases}\n    -\\Delta u=u^{2^*-1}&\\text{in }\\Sigma_D,\\\\\n    \\frac{\\partial u}{\\partial\\nu}=0&\\text{on }\\partial\\Sigma_D\\setminus\\{0\\},\\\\\n    u>0&\\text{in }\\Sigma_D,\n\\end{cases}\n\\ee\nwhich are also fast decaying, i.e.\n    $$\n    v(x)=O(|x|^{2-N}) \\quad \\text{as }|x|\\to\\infty.\n    $$",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $N\\ge 3$, let $D$ be a smooth domain of the unit sphere $\\mathbb S^{N-1}$ with $\\overline D\\subset \\mathbb S^{N-1}_+$, where $\\mathbb S^{N-1}_+$ is the half-sphere, and let\n$$\n\\Sigma_D=\\{x\\in\\mathbb R^N: x=\\rho p,\\ p\\in D,\\ \\rho>0\\}\n$$\nbe the cone spanned by $D$. Let $\\lambda_1(D)$ denote the first nonzero eigenvalue of the Laplace--Beltrami operator $-\\Delta_{\\mathbb S^{N-1}}$ on $D$ with homogeneous Neumann boundary conditions, and assume that $\\lambda_1(D)<N-1$. Define\n$$\nQ_D(u)=\\frac{\\left(\\int_{\\Sigma_D}|\\nabla u|^2\\right)^{1/2}}{\\left(\\int_{\\Sigma_D}|u|^{2^*}\\right)^{1/2^*}},\n\\qquad 2^*=\\frac{2N}{N-2},\n$$\nfor $u\\in \\mathcal D^{1,2}(\\Sigma_D)\\setminus\\{0\\}$, where\n$$\n\\mathcal D^{1,2}(\\Sigma_D)=\\{u\\in L^{2^*}(\\Sigma_D): |\\nabla u|\\in L^2(\\Sigma_D)\\}.\n$$\nWhich conclusion about minimizers of $Q_D$ and the critical Neumann problem\n$$\n\\begin{cases}\n-\\Delta u=u^{2^*-1} & \\text{in } \\Sigma_D,\\\\\n\\dfrac{\\partial u}{\\partial \\nu}=0 & \\text{on } \\partial\\Sigma_D\\setminus\\{0\\},\\\\\nu>0 & \\text{in } \\Sigma_D\n\\end{cases}\n$$\nis valid under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "The minimizers of the Sobolev quotient $Q_D$ are nonradial. Consequently, there exists a nonradial positive solution $v\\in \\mathcal D^{1,2}(\\Sigma_D)$ of\n$$\n\\begin{cases}\n-\\Delta v=v^{2^*-1} & \\text{in } \\Sigma_D,\\\\\n\\dfrac{\\partial v}{\\partial \\nu}=0 & \\text{on } \\partial\\Sigma_D\\setminus\\{0\\},\\\\\nv>0 & \\text{in } \\Sigma_D,\n\\end{cases}\n$$\nand this solution is fast decaying at infinity, namely\n$$\nv(x)=O(|x|^{2-N})\\qquad \\text{as } |x|\\to\\infty.\n$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "The minimizers of the Sobolev quotient $Q_D$ are radial, and in fact every positive solution $v\\in \\mathcal D^{1,2}(\\Sigma_D)$ of\n$$\n\\begin{cases}\n-\\Delta v=v^{2^*-1} & \\text{in } \\Sigma_D,\\\\\n\\dfrac{\\partial v}{\\partial \\nu}=0 & \\text{on } \\partial\\Sigma_D\\setminus\\{0\\},\\\\\nv>0 & \\text{in } \\Sigma_D,\n\\end{cases}\n$$\nis a rescaling of the standard bubble. Consequently, no nonradial fast decaying solution exists in this regime."
        },
        {
          "label": "C",
          "text": "There exists at least one nonradial positive solution $v\\in \\mathcal D^{1,2}(\\Sigma_D)$ of\n$$\n\\begin{cases}\n-\\Delta v=v^{2^*-1} & \\text{in } \\Sigma_D,\\\\\n\\dfrac{\\partial v}{\\partial \\nu}=0 & \\text{on } \\partial\\Sigma_D\\setminus\\{0\\},\\\\\nv>0 & \\text{in } \\Sigma_D,\n\\end{cases}\n$$\nwhich is obtained from a minimizer of $Q_D$."
        },
        {
          "label": "D",
          "text": "The minimizers of the Sobolev quotient $Q_D$ are nonradial only in the borderline case $\\lambda_1(D)=N-1$; under the strict inequality $\\lambda_1(D)<N-1$ one can conclude at most that the standard bubble is degenerate, but not that a nonradial minimizer or nonradial positive solution exists."
        },
        {
          "label": "E",
          "text": "The minimizers of the Sobolev quotient $Q_D$ are nonradial. Consequently, for every such minimizer one obtains a nonradial positive solution $v\\in \\mathcal D^{1,2}(\\Sigma_D)$ of\n$$\n\\begin{cases}\n-\\Delta v=v^{2^*-1} & \\text{in } \\Sigma_D,\\\\\n\\dfrac{\\partial v}{\\partial \\nu}=0 & \\text{on } \\partial\\Sigma_D\\setminus\\{0\\},\\\\\nv>0 & \\text{in } \\Sigma_D,\n\\end{cases}\n$$\nand moreover this solution satisfies the sharper asymptotic law\n$$\nv(x)\\sim C|x|^{2-N}\\qquad \\text{as } |x|\\to\\infty\n$$\nfor some constant $C>0$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "instability regime below threshold",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "drops the theorem's claims that minimizers are nonradial and that the solution is fast decaying",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "threshold location of bifurcation versus subcritical existence regime",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "replacing big-O decay with a full asymptotic equivalence for every minimizer",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives assumptions, definitions, and the PDE, but it does not explicitly reveal the correct conclusion. There is no direct textual leakage of the answer."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: under the stated hypotheses, the correct option reproduces the theorem’s conclusion almost verbatim. It tests recognition of the exact result more than selection among independently motivated conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish between the exact theorem statement, a weaker true statement, and an overstrong false statement. However, the item mainly rewards precise recall rather than substantial mathematical generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one gives the opposite symmetry conclusion, one confuses the threshold regime, one is a weaker-true alternative, and one overstates the decay conclusion. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "Good distractor design and no answer leakage, but the item is largely a near-verbatim theorem restatement, so it only moderately tests reasoning."
    }
  },
  {
    "id": "2512.05828v1",
    "paper_link": "http://arxiv.org/abs/2512.05828v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\mathbf{d}}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$",
      "start_pos": 11405,
      "end_pos": 11624,
      "label": "theorem: bound rango"
    },
    "ref_dict": {
      "theorem: bound rango": "\\begin{theorem}\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$\n\\end{theorem}",
      "subsection: commenti sul bound": "\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor product of $W$-states by looking at the containment of $W_{d_1}\\otimes \\cdots \\otimes W_{",
      "remark: commento algo": "\\begin{remark}\\label{remark: commento algo}\nLet $\\bfd\\in\\NN^k_{\\geq 3}$. For $s\\in[0,d]\\setminus\\{d-k\\}$ we choose one $\\bf i \\in \\A_{\\bf d,s }$ and set $\\varepsilon^J_{\\bf i}=1$ for any $J\\subseteq[2,k]$, and for $s=d-k$ and $\\bfi=(d_1-1,\\dots,d_k-1)$ we set $\\varepsilon_\\bfi^J=1$ for any $J\\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ following our procedure, we have to first find $T^J\\in\\spann(\\mathcal C_{\\bfd}^J)$ such that \n$$\\prod_{j=1}^kd_jW_{d_1}\\otimes\\cdots\\otimes W_{d_k}=\\sum_{J\\subseteq[2,k]}T^J.$$\nThe coordinates of the $T^J$'s are of the form\n$$T^J=(\\varepsilon_{\\bfi}^J\\alpha_s^J)_{\\substack{0\\leq s\\leq d\\\\ \\bfi\\in\\A_{\\bfd,s}}}$$\nand, by \\Cref{prop: rango TJ}, it is enough to take $\\alpha_{k-2}^J\\neq0$ and $\\alpha_{d-k}^J\\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \\Cref{prop: mortale}: we take $\\alpha_s=0$ for $s\\in[0,d]\\setminus\\{k-2,d-k\\}$, $\\alpha_{k-2}=(-1)^{|J|}$ and $\\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\\varphi^J(T^J)$ we only have two possibilities:\n$$\\varphi^J(T^J)=\\begin{cases}\n     \\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ if $|J|$ is even}\\\\\n     -\\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ otherwise.}\n\\end{cases}$$\nAs a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\\varphi^J(T^J)$'s is found it is enough to apply $(\\varphi^J)^{-1}$ for all the $J\\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\\varepsilon_\\bfi^J$'s allows to avoid many computations. We summarized the procedure in \\Cref{algo}.\n\\end{remark}",
      "algo": "\\label{algo}\n\\end{algorithm}\n\\subsection{Further remarks on \\Cref{theorem: bound rango} }\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor",
      "theorem: bound rango bordo": "\\begin{theorem}\\label{theorem: bound rango bordo}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots+d_k$.  Then\n$$\n\\underline{\\rk}_\\bfd(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k}.\n$$\n\\end{theorem}",
      "prop: rango TJ": "\\begin{proposition}\\label{prop: rango TJ}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots +d_k$. Let $\\mathcal{F}\\subseteq \\CC[u,v]_d$ be the family of degree $d$-homogenous polynomial defined as \n$$\n\\mathcal{F}= \\left\\{  F=\\sum_{i=0}^{k-2}a_i\\binom{d}{i}u^{d-i}v^i+\\binom{d}{d-k}a_{d-k}u^{k}v^{d-k}+\\sum_{i=d-k+2}^d a_i \\binom{d}{i}u^{d-i}v^i, \\text{ for }a_i\\in \\CC \\text{ with } a_{d-k}\\neq 0 \\right\\}.\n$$\nThen \n$$\n\\min_{F\\in\\mathcal F}\\rk F=d-2k+2.\n$$\nIn particular, if  $2k\\leq\\left\\lfloor \\frac d2\\right\\rfloor +1$ the minimum is achieved by any $F\\in\\mathcal F$ with $a_{k-2}\\neq 0$. Otherwise, the minimum is achieved by any $F\\in \\mathcal{F}$ with $a_{k-2}\\neq 0$ and $a_i=0$ for $i\\in \\{0,\\dots,k-3\\} \\cup \\{d-k+2,\\dots,d \\}$.\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 5580,
    "pre_theorem_intro_text": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}. \n\n From the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\n Given how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of  $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\n Hence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}. \n\n  In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.",
    "context": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}.\n\nFrom the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\nGiven how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\nHence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.",
    "full_context": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}.\n\nFrom the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\nGiven how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\nHence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\CC^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\sym^{d_1}\\CC^{n_1}\\otimes \\cdots \\otimes \\sym^{d_k}\\CC^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\bfd}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\CC^{n_j}$. Our main result is the following.\n\nIn addition to providing a theoretical upper bound on the tensor rank, \\Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$; see the forthcoming \\Cref{remark: commento algo}.\n\nAs an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \\begin{corollary}\\label{corollary: bound rango vero}\nLet $k\\geq 2$, fix positive integers $d_i\\geq 3$ for $i=1,\\dots,k$. \nThe rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\CC^{2})^{\\otimes d_i}$ is at most $2^{k-1}(d_1+\\dots+d_k-2k+2)$.\n\\end{corollary}\n\nWe have now all the tools to prove our main \\Cref{theorem: bound rango}, which we restate below for convenience.\n\\begin{theorem*}(\\Cref{theorem: bound rango}) Let $k\\geq 2$, $\\bfd\\in \\NN_{\\geq 3}^k$ and $d=d_1+\\cdots +d_k$. Then\n$$\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq 2^{k-1}(d-2k+2).\n$$\n\\end{theorem*}\n\\begin{proof}\nBy \\Cref{remark: W-state sta nel quadrello} $W_{d_1}\\otimes\\cdots \\otimes W_{d_k}$ belongs to $\\spann\\left\\{ \\cup_{J\\subseteq [2,k]} \\mathcal{C}_{\\bfd}^J\\right\\} $, so we have\n$$\n\\prod_{j=1}^kd_j W_{d_1}\\otimes\\cdots \\otimes W_{d_k}=\\sum_{J\\subseteq [2,k]}T^J, \\,\\hbox{ for some }T^J\\in\\spann\\mathcal{C}_{\\bfd}^J. \n$$\nWriting $T^J$ with respect to the basis of \\Cref{remark: multi hom binary in forma binaria emplice}, we have\n$$T^J=\\sum_{s=0}^{d}\\alpha_s^J\\left(\\sum_{\\bfi\\in\\mathcal A_{\\bfd,s}}\\varepsilon_\\bfi^Jx_{1,0}^{i_1}x_{1,1}^{d_1-i_1}\\otimes\\cdots\\otimes x_{k,0}^{i_k}x_{k,1}^{d_k-i_k}\\right),$$\nwhere the $\\alpha^J_s$'s are a solution of the linear system $\\mathcal S$ of \\Cref{prop: mortale}.\nUsing the isomorphism $\\varphi^J$ defined in \\Cref{remark: multi hom binary in forma binaria emplice}, each $T^J$ can be seen as a binary form of degree $d$ in $\\spann\\mathcal C_\\bfd^J$. More precisely, we have\n$$\\varphi^J(T^J)=\\sum_{s=0}^d\\alpha_s^J\\binom{d}{s}u^{d-s}v^s.$$\nIt follows that $\\rk_\\bfd T^J\\leq\\rk\\varphi^J(T^J)$, where $\\rk\\varphi^J(T^J)$ is the Waring rank of the binary form $\\varphi^J(T^J)$ and, by \\Cref{remark: multi hom binary in forma binaria emplice}, it is equal to the $\\mathcal C_\\bfd^J$-rank of $T^J$.\nBy \\Cref{prop: mortale}, we have $\\alpha_s^J=0$ for any $s\\in[k-1,d-k-1]\\cup\\left\\{d-k+1\\right\\}$ and $\\alpha^J_{d-k}\\neq 0$. Moreover, whether $2k\\leq\\left\\lfloor\\frac{d}{2}\\right\\rfloor+1$ or $2k>\\left\\lfloor\\frac{d}{2}\\right\\rfloor+1$, by \\Cref{prop: mortale} we can choose the $\\alpha_s^J$'s such that $\\varphi^J(T^J)$ is a minimizer as in \\Cref{prop: rango TJ}, so that $\\rk (\\varphi^J(T^J))=d-2k+2$. Hence, we have\n\\begin{equation*}\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq\\sum_{J\\subseteq[2,k]}\\rk_\\bfd T^J\\leq \\sum_{J\\subseteq[2,k]}\\rk \\varphi^J(T^J)= 2^{k-1}(d-2k+2).\\qedhere\n\\end{equation*}\n\\end{proof}\nSince for a tensor $T\\in \\sym^{d_1}\\CC^2\\otimes \\cdots \\otimes \\sym^{d_k}\\CC^2$ we have $\\rk(T)\\leq \\rk_{\\bf d}(T)$, the previous result gives a bound also on the tensor rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.\n\n\\Cref{example: step4}, together with \\Cref{example: step1}, \\Cref{example: step2} and \\Cref{example: step3}, concretely shows that, following our procedure, it is possible to construct an explicit decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ having length as in the bound of \\Cref{theorem: bound rango} as we now explain in detail.\n\\begin{remark}\\label{remark: commento algo}\nLet $\\bfd\\in\\NN^k_{\\geq 3}$. For $s\\in[0,d]\\setminus\\{d-k\\}$ we choose one $\\bf i \\in \\A_{\\bf d,s }$ and set $\\varepsilon^J_{\\bf i}=1$ for any $J\\subseteq[2,k]$, and for $s=d-k$ and $\\bfi=(d_1-1,\\dots,d_k-1)$ we set $\\varepsilon_\\bfi^J=1$ for any $J\\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ following our procedure, we have to first find $T^J\\in\\spann(\\mathcal C_{\\bfd}^J)$ such that \n$$\\prod_{j=1}^kd_jW_{d_1}\\otimes\\cdots\\otimes W_{d_k}=\\sum_{J\\subseteq[2,k]}T^J.$$\nThe coordinates of the $T^J$'s are of the form\n$$T^J=(\\varepsilon_{\\bfi}^J\\alpha_s^J)_{\\substack{0\\leq s\\leq d\\\\ \\bfi\\in\\A_{\\bfd,s}}}$$\nand, by \\Cref{prop: rango TJ}, it is enough to take $\\alpha_{k-2}^J\\neq0$ and $\\alpha_{d-k}^J\\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \\Cref{prop: mortale}: we take $\\alpha_s=0$ for $s\\in[0,d]\\setminus\\{k-2,d-k\\}$, $\\alpha_{k-2}=(-1)^{|J|}$ and $\\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\\varphi^J(T^J)$ we only have two possibilities:\n$$\\varphi^J(T^J)=\\begin{cases}\n \\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ if $|J|$ is even}\\\\\n -\\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ otherwise.}\n\\end{cases}$$\nAs a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\\varphi^J(T^J)$'s is found it is enough to apply $(\\varphi^J)^{-1}$ for all the $J\\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\\varepsilon_\\bfi^J$'s allows to avoid many computations. We summarized the procedure in \\Cref{algo}.\n\\end{remark}\n\n\\begin{theorem}\\label{theorem: bound rango bordo}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots+d_k$. Then\n$$\n\\underline{\\rk}_\\bfd(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k}.\n$$\n\\end{theorem}\n\\begin{proof}\nWe use the catalecticant bound for border rank of \\cite[Corollary 5.5]{gal}. In order to do that we have to produce a partially symmetric flattening with rank at least $2^k$. Let us consider the following flattening of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$:\n\\[\nF : \\sym^2(\\mathbb{C}^2) \\otimes \\dots \\otimes \\sym^2(\\mathbb{C}^2) \\rightarrow \n\\sym^{d_1-2}(\\mathbb{C}^2)^* \\otimes \\dots \\otimes \\sym^{d_k-2}(\\mathbb{C}^2)^*.\n\\]\nLet $\\mathbb{C}[\\alpha_{1,0},\\alpha_{1,1},\\dots,\\alpha_{k,0},\\alpha_{k,1}]$ be the ring of multigraded derivations on $\\CC[x_{1,0},x_{1,1},\\dots,x_{k,0},x_{k,1}]$ with $\\alpha_{i,j}$ defined as the dual of $x_{i,j}$. By apolarity theory, the rank of the linear map $F$ is exactly the dimension of the degree $(2,\\dots,2)$ component of the quotient of the multigraded ring $\\mathbb{C}[\\alpha_{1,0},\\alpha_{1,1},\\dots,\\alpha_{k,0},\\alpha_{k,1}]$ by the ideal $I_{d}$ apolar to the W-state, i.e. $I_{d}=(\\alpha_{1,0}^2,\\dots,\\alpha_{k,0}^2,\\alpha_{1,1}^{d_1},\\dots,\\alpha_{k,1}^{d_k})$. For $\\varepsilon_i \\in \\{0,1\\}$ the $2^k$ forms\n$$F_{\\varepsilon_1,\\dots,\\varepsilon_{k}}=\\alpha_{1,0}^{\\varepsilon_1}\\alpha_{1,1}^{2-\\varepsilon_1} \\dots \\alpha_{k,0}^{\\varepsilon_k}\\alpha_{k,1}^{2-\\varepsilon_k}$$ are linear independent of multidegree $(2,\\dots, 2)$, with $F_{\\varepsilon_1,\\dots,\\varepsilon_{k}} \\notin I_d$. This concludes the proof. \n\\end{proof}",
    "post_theorem_intro_text_len": 2935,
    "post_theorem_intro_text": "\\Cref{theorem: bound rango} improves \\cite[Theorem 3.6]{BBCG} by $2^k(k-1)$. For $k=2$ it coincides with the bound provided in \\cite[Theorem 1.8(i)]{gal} and when all $d_i=3$ it matches the bound of \\cite[Theorem 3.3]{BBCG}. The bound of \\Cref{theorem: bound rango} is sharp as for $k=2$ and $d_1=d_2=3$ it gives rank 8.\n\n In addition to providing a theoretical upper bound on the tensor rank, \\Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$; see the forthcoming \\Cref{remark: commento algo}.\n\n As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \\begin{corollary}\\label{corollary: bound rango vero}\nLet $k\\geq 2$, fix positive integers $d_i\\geq 3$ for $i=1,\\dots,k$. \nThe rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$ is at most $2^{k-1}(d_1+\\dots+d_k-2k+2)$.\n\\end{corollary}\n\nOur methods to achieve this result are geometric and, as in \\cite[Theorem 3.6]{BBCG}, they rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$. \n\n\\subsection*{Outline of the paper}  We set up our notation and give standard preliminaries in \\Cref{section: preliminaries}, while \\Cref{section: solving the system} contains the main technical results to get our bounds. We start \\Cref{section: computing the bounds} by briefly reviewing the theory of catalecticants needed to compute the rank of a family of binary homogeneous polynomials (\\Cref{prop: rango TJ}). Then, we prove our main \\Cref{theorem: bound rango} and in \\Cref{algo} we give a recipe to explicitly compute a partially symmetric decomposition of length given by our bound. \\Cref{theorem: bound rango bordo} computes the border partially symmetric rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$. We conclude with a detailed comment on the method in \\Cref{subsection: commenti sul bound}.\n\n\\subsection*{Acknowledgements}\nThe authors are grateful to Fulvio Gesmundo, Alessandro Gimigliano and Joachim Jelisiejew for interesting conversations on the topic.\n\nS. Canino has been funded by the Italian Ministry of\nUniversity and Research in the framework of the Call for Proposals for\nscrolling of final rankings of the PRIN 2022 call - Protocol no.\n2022NBN7TL.\nA. Casarotti has been funded by the European Union under the project NextGenerationEU. PRIN 2022, CUP: F53D23002600006.\nP. Santarsiero was supported by the European Union under NextGenerationEU. PRIN 2022, Prot. 2022E2Z4AK and PRIN 2022 SC-CUP: I53C24002240006.",
    "sketch": "The post-theorem text does not give a step-by-step proof sketch for \\Cref{theorem: bound rango}, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}).”",
    "expanded_sketch": "The post-theorem text does not give a step-by-step proof sketch for the main theorem, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see\n\\label{algo}\n\\end{algorithm}\n\\subsection{Further remarks on \\Cref{theorem: bound rango} }\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor).”",
    "expanded_theorem": "\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\mathbf{d}}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(k\\ge 2\\) and let \\(d_1,\\dots,d_k\\ge 3\\) be integers, and write \\(\\mathbf d=(d_1,\\dots,d_k)\\). For each \\(d\\), define the generalized \\(W\\)-state\n\\[\nW_d=\\sum_{i=1}^d e^{(1)}_1\\otimes\\cdots\\otimes e^{(i-1)}_1\\otimes e^{(i)}_2\\otimes e^{(i+1)}_1\\otimes\\cdots\\otimes e^{(d)}_1\\in (\\mathbb C^2)^{\\otimes d},\n\\]\nwhere each \\(\\operatorname{span}\\{e^{(i)}_1,e^{(i)}_2\\}\\cong \\mathbb C^2\\). View\n\\(W_{d_1}\\otimes\\cdots\\otimes W_{d_k}\\) as an element of\n\\(\\operatorname{Sym}^{d_1}(\\mathbb C^2)\\otimes\\cdots\\otimes \\operatorname{Sym}^{d_k}(\\mathbb C^2)\\), and let \\(\\operatorname{rk}_{\\mathbf d}\\) denote its partially symmetric rank, i.e. the minimum number of summands of the form \\(\\ell_1^{d_1}\\otimes\\cdots\\otimes \\ell_k^{d_k}\\) needed to express it. Which statement holds for every such choice of \\(k,d_1,\\dots,d_k\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"
        },
        {
          "label": "C",
          "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}(d_1+\\cdots+d_k).\\]"
        },
        {
          "label": "D",
          "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}\\bigl(d_1\\cdots d_k-2k+2\\bigr).\\]"
        },
        {
          "label": "E",
          "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k-1}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "sharp_constant_2^{k-1}",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped_negative_correction_-2k+2",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "sum_of_degrees_replaced_by_product",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "upper_bound_promoted_to_exact_formula",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the objects and asks which bound holds; it does not explicitly reveal the correct inequality or uniquely signal choice A. There is no direct answer leakage beyond the mathematical setup itself."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: the task is essentially to recognize the exact published quantitative estimate among nearby variants. It is not a pure verbatim restatement in the stem, but it is only a mild reformulation of the target result."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare the strength and plausibility of the candidate bounds, especially against weaker-true and boundary-case distractors. However, the item mainly tests recall of the precise bound rather than deep generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well targeted: one is a weaker true statement, others alter the exponential factor, constant term, or degree dependence. These reflect realistic failure modes such as dropping refinements or confusing sharp versus coarse bounds."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with no answer leakage and strong distractors, but it leans more toward precise recall than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.05867v1",
    "paper_link": "http://arxiv.org/abs/2512.05867v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "Thm",
      "content": "[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]",
      "start_pos": 79042,
      "end_pos": 79734,
      "label": "thm:F_ell_main"
    },
    "ref_dict": {
      "eq:lawbicolour_sd": "\\begin{equation}\n    \\label{eq:lawbicolour_sd}\n    \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}",
      "eq:lawgen": "\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}",
      "eq:lawbicolour": "\\begin{equation}\n    \\label{eq:lawbicolour}\n    \\mathbb{P} ( (T, L) = (\\mathfrak{t}, \\boldsymbol{\\ell})) \\; \\propto \\;  x_1^{\\#\\mathfrak{f}_1} x_2^{\\#\\mathfrak{f}_2} n^{\\#\\boldsymbol{\\ell}},\n\\end{equation}",
      "thm:F_ell_main": "\\begin{Thm}[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]\n\\end{Thm}",
      "eq:asymp_F_ell_main": "\\begin{equation} \\label{eq:asymp_F_ell_main}\n\tF_\\ell \n\t\\sim\n\tc \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n\t\\end{equation}",
      "sss:gasket_dec": "\\begin{equation}\\label{eq:link_FK_Z}\n\\Big(\\frac{1-p}{16}\\Big)^{\\frac{\\#F(\\tfrak)+\\ell-1}{2}}\n\\Big(\\frac{2p}{1-p}\\Big)^{\\#\\boldsymbol{\\ell}+1}\n=\nn x_c^{\\ell+1} x_c^{\\#F(\\tfrak)-1} n^{\\#\\boldsymbol{\\ell}}\n=\nn x_c^{\\ell+1} Z(\\tfrak,\\boldsymbol{\\ell},x_c,n),\n\\end{equation}\nwhere $n$ and $x_c$ are as in~\\eqref{eq:x_c_formula}.\n\n\\subsubsection{The gasket decomposition}\n\\label{sss:gasket_dec}\nThe gasket decomposition, introduced in \\cite{BBG12a} and further developed in \\cite{BBG12}, provides a remarkably effective framework for analysing a broad class of loop models.\nIn what follows we adapt the presentation of \\cite{BBG12} to the particular fully packed setting determined by \\eqref{eq: def weight triangulation} and \\eqref{eq: def partition triangulation}.\n\nConsider a configuration $(\\tfrak,\\boldsymbol{\\ell})\\in \\mathbb{T}_\\ell$.\nTo construct its gasket, we first look at the edges of $\\tfrak$ that are reachable from the boundary without crossing any loop.\nBecause the loops are fully packed, these accessible edges divide the map into exactly $\\#\\boldsymbol{\\ell}+1$ faces: the external face, with boundary length~$\\ell$, and an additional face of degree~$k$ for each loop of (outer) perimeter~$k$.\nThe resulting object is by definition the \\textbf{gasket}.\nNow examine one of the internal faces of degree~$k$ created by the gasket.\n\n\\begin{figure}[b!]\n  \\bigskip\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{Ring-partition-function.pdf}\n  \\end{center}\n  \\caption{The ring partition function $A^{(1\\to 2)}_{k,k'}$ from colour 1 (blue) to colour 2 (red). It accounts for the weight of all the triangles crossed by the purple loop.} \n  \\label{fig:ring_part}\n\\end{figure}\n\nIf we reinsert the triangles traversed by the corresponding loop, we obtain a ring of triangles whose outer boundary length is~$k$ and whose inner boundary has some length~$k'$.\nThe ring consists of $k+k'$ triangles in total.\nMoreover, the loop-decorated triangulation originally contained inside this loop -- call it $(\\tfrak',\\boldsymbol{\\ell}')\\in\\mathbb{T}_{k'}$ -- is precisely what was taken out from $(\\tfrak,\\boldsymbol{\\ell})$ to form the corresponding face of the gasket.\nIn this way, $(\\tfrak,\\boldsymbol{\\ell})$ decomposes into:\n(i) the gasket;\n(ii) one ring of triangles for each internal gasket face; and\n(iii) a fully packed loop-decorated triangulation attached along the inner boundary of each such ring.\nThe contribution of such a component to the weight $Z(\\tfrak',\\boldsymbol{\\ell}',x,n)$ in \\eqref{eq: def weight triangulation} is $n x^{k + k'}Z(\\tfrak, \\boldsymbol{\\ell}, x, n)$,\nwhere $k$ is the outer and $k'$ the inner boundary length of the ring.\nThis motivates the definition\n\\begin{equation}\\label{eq: gasket decomposition}\ng_k := n\\sum_{k'=0}^{\\infty} A_{k\\rightarrow k'} x^{k+k'} F_{k'},\n\\end{equation}",
      "prop:prob_tau": "\\begin{Prop}[From hitting times to the partition function]\\label{prop:prob_tau}\n\tThere exists a normalising constant $C>0$ such that, for all $\\ell\\geq 0$,\n\t\\[\n\t\\mathbb{P}(\\tau^{\\rh} = \\ell+1) = C (2x_c)^{\\ell+1}F_{\\ell}. \n\t\\]\n\\end{Prop}",
      "thm:exp_main": "\\begin{Thm}[Exponents for loops and clusters]\n\\label{thm:exp_main}\n    We have the following tail asymptotics: as $\\ell\\to\\infty$,\n    \\[\n    \\mathbb{P}(|\\partial \\mathfrak{K}| = \\ell) \\sim \\frac{C}{\\ell^{3-2\\theta}}\n    \\quad \\text{and} \\quad \n    \\mathbb{P}(|\\mathfrak{L}| = \\ell) \\sim \\frac{C'}{\\ell^{3-2\\theta}},\n    \\]\n    where $C$ and $C'$ are positive constants.\n\\end{Thm}",
      "eq:lawSD": "\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 10664,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nPlanar maps (i.e., proper embeddings of graphs in the $2$-sphere, considered up to orientation-preserving homeomorphisms) are a central topic not only in combinatorics but also in probability and mathematical physics due to their conjectured links with Liouville quantum gravity (LQG). In that context, planar maps can be thought of as canonical discretisations of the ``random surfaces'' which are at the core of Polyakov's original formulation of LQG \\cite{polyakov1981quantum}. Indeed, in order to describe the gravitational action when gravity is coupled to a matter field, the so-called DDK Ansatz (\\cite{David, DistlerKawai}) implies that this can be equivalently described by considering the scaling limit of random planar maps decorated by models of statistical mechanics at their critical point. \n\nIn this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an  FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed  configuration $L$ of loops, with probability\n\\begin{equation}\n    \\label{eq:lawbicolour_sd}\n    \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\n\\medskip In many models of planar statistical mechanics, it is widely expected that the self-dual point coincides with its critical point (see for instance \\cite{grimmett2018probability} for some background discussion).\nIn particular, such a statement is the content of a celebrated result of Beffara and Duminil-Copin \\cite{beffara2012self} for the random cluster model (equivalently the Fortuin--Kasteleyn percolation model) on the square lattice and for $q\\ge 1$. For the related loop $O(n)$ model on the hexagonal lattice, the equality between critical and self-dual point led Nienhuis to predict (\\cite{Nienhuis82, Nienhuis84}) that the critical point occurs at $x = x_c = 1/ \\sqrt{2 + \\sqrt{2-n}}$ for $n \\in (0,2]$, and in fact $n \\in [-2, 2]$; this remains a famous open problem in this field (see \\cite{DCensaios} for a thorough survey).  This should also apply to the two self-dual models planar maps discussed above, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}. This suggests that this model is also \\textbf{critical}, although in a sense that needs to be carefully specified. \n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\nThis is supported by a considerable body of evidence. On the one hand, in the breakthrough paper \\cite{sheffield2016quantum}, Sheffield provided a measure-preserving bijection between self-dual FK-weighted planar maps \\eqref{eq:lawSD} and inventory accumulations, which in turn correspond to a pair of (non-Markovian) walks, and showed that in the scaling limit this pair of walks converges to a pair of correlated Brownian motions. In combination with the so-called Mating of Trees framework developed by Duplantier, Miller and Sheffield \\cite{duplantier2021liouville} this result can be re-interpreted as a form of convergence towards a $\\gamma$-LQG surface decorated with an independent space-filling SLE$_{\\kappa'}$ curve, albeit for a relatively weak topology (the so-called ``peanosphere'' topology). Building on this foundational work, a number of observables (such as sizes of clusters and their boundaries) have been analysed and the associated exponents computed (see \\cite{gwynne2019scaling,gwynne2017scaling,gwynne2015scaling} and \\cite{berestycki2017critical}). These values are consistent with those that can be predicted by combining known results on the dimension of SLE$_\\kappa$ (\\cite{beffara2008dimension}) with the Knizhnik--Polyakov--Zamolodchikov (KPZ) identity, cf.\\ \\cite{duplantier2021liouville, HKPZ}. See, e.g., \\cite[Chapter 4]{BP} for a discussion of these results and additional perspective. \n\nOn the other hand, \na classical approach to the loop $O(n)$ model is to make use of the spatial Markov property of the model. This results in the so-called \\emph{gasket decomposition} which was in particular used by Borot, Bouttier and Guitter \\cite{BBG12, BBG12a, BBG12b} {to establish the phase diagram of the loop-$O(n)$ model}. Building on powerful tools of analytic combinatorics they derived and solved (assuming a certain natural ansatz) an equation for the resolvent of the partition function of the model, which yields fine asymptotics. Such an approach has been made fully rigorous (including the proof of the above ansatz) by works of Budd and Chen \\cite{budd2019peeling}, in the case of the so-called \\emph{rigid} model on quadrangulations (where loops are constrained to enter and leave a given quadrangle through opposite edges). From this, Chen, Curien and Maillard \\cite{ChenCurienMaillard} deduced some scaling limit results for the perimeter cascade of $O(n)$ loops, and Aïdékon, Da Silva and Hu \\cite{aidekon2024scaling} proved the scaling limit for the volume of such {rigid} loop-$O(n)$ quadrangulations. A similar approach was also used by Borot, Duplantier and Guitter \\cite{BBD16} to determine the nesting statistics in the \\emph{bending energy} variant of the $O(n)$ model, {where loops can bend inside a quadrangle at a given energetic cost.} \n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see  \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that  $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n    \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}",
    "context": "In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed configuration $L$ of loops, with probability\n\\begin{equation}\n \\label{eq:lawbicolour_sd}\n \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}",
    "full_context": "In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed configuration $L$ of loops, with probability\n\\begin{equation}\n \\label{eq:lawbicolour_sd}\n \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}\n\nIn particular, we deduce the following asymptotics.\n\\begin{Cor} \\label{cor:asymp_F_ell_main}\n The partition function satisfies:\n \\begin{equation} \\label{eq:asymp_F_ell_main}\n F_\\ell \n \\sim\n c \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n \\end{equation}\n where $c =\\frac{2^{\\theta-1/2}}{\\pi \\theta} (\\gamma_+-\\gamma_-)^{\\theta-1} \\gamma_+^{1-\\theta} \\Gamma(2-\\theta)$, and $\\gamma_-$ and $\\gamma_+$ are as above.\n\\end{Cor}\n\n\\noindent With \\cref{lem:WH_K}, equation \\eqref{eq: sd resolvent fourier} implies \n\\[\n2\\pi^2 \\frac{R_+(\\omega)}{K_+(\\omega)} = \\frac{F_+(\\omega)}{K_-(\\omega)} + \\frac{F_-(\\omega)}{K_-(\\omega)} \\quad \\text{for } \\omega\\in\\cal S \\text{ such that } \\Gamma\\left( \\frac{1-2\\theta+i\\omega}{4} \\right) \\neq \\infty.\n\\]\nWe now multiply the latter display to remove the pole singularities of $F_+$: for $\\omega\\in\\cal S$ such that $\\Gamma\\left( \\frac{1-2\\theta+i\\omega}{4} \\right) \\neq \\infty$,\n\\[\n2\\pi^2 \\frac{R_+(\\omega)}{K_+(\\omega)} (\\omega+i)(\\omega+3i)= \\left(\\frac{F_+(\\omega)}{K_-(\\omega)} + \\frac{F_-(\\omega)}{K_-(\\omega)}\\right)(\\omega+i)(\\omega+3i).\n\\]\nNow remark that the left-hand side is actually holomorphic on $\\cal H_+$, whereas the right-hand side is holomorphic on $\\{z\\in\\bb C, \\Im(z)<1-2\\theta\\}$. We stress that, since $0<n<2$, we have $1-2\\theta\\in (0, 1)$. Therefore, both sides can be extended to an entire function $S$ in the complex plane, whence\n\\begin{equation} \\label{eq: WH final 1}\nR_+(\\omega)\n=\n\\frac{1}{2\\pi^2} K_+(\\omega) \\frac{S(\\omega)}{(\\omega+i)(\\omega+3i)}\n\\quad \\text{for } \\omega\\in \\cal H_+,\n\\end{equation}\nand \n\\begin{equation} \\label{eq: WH final 2}\nF_+(\\omega)+F_-(\\omega) \n=\nK_-(\\omega) \\frac{S(\\omega)}{(\\omega+i)(\\omega+3i)} \\quad \\text{for } \\omega \\text{ such that } \\Im(\\omega)<1-2\\theta.\n\\end{equation}\nWe now analyse the behaviours at infinity in \\eqref{eq: WH final 1} and \\eqref{eq: WH final 2}. \n{Since $r_+$ is integrable by Lemma \\ref{lem:r(v)/v_int}, we have that $R_+(\\omega) = O(1)$ as $|\\omega| \\to \\infty$ with $\\omega\\in \\mathcal{H}_+$. Likewise, writing $ \\omega = x + {i} y$ with $y < 1$, we have by Lemma \\ref{lem:r(v)/v_int},\n\\[\n|F_-(\\omega)| = \\Big|\\int_{-\\infty}^0 f_-(u) \\mathrm{e}^{ i \\omega u} \\mathrm{d} u \\Big|\\le \\int_{-\\infty}^0 O(\\mathrm{e}^u) \\mathrm{e}^{-yu}\\mathrm{d} u.\n\\]\nThus we deduce that $F_- (\\omega) = O(1)$ as $|\\omega| \\to \\infty$ with $\\Im(\\omega)<1-2\\theta$.\nCombining this with the exact expressions \\eqref{eq: expressions K_+ and K_-}, we see that\n$S(\\omega) = O(\\omega)$ as $|\\omega|\\to\\infty$. By Liouville's theorem, $S$ is a degree $1$ polynomial. \nWe can therefore write it as $S(\\omega) = A\\omega + B$ for some $A, B \\in \\mathbb{C}$. We will in fact show it is a constant: indeed, we can read the value of $A$ by taking $\\omega\\to +\\infty$ along the real line in \\eqref{eq: WH final 2}. By the Riemann--Lebesgue lemma, we have $F_-(\\omega) \\to 0$ and this forces $A=0$ (as $F_+ ( \\omega) = O ( 1/|\\omega|) \\to 0$ as $\\omega \\to \\infty$). We conclude that $S$ is actually a constant,\nthe value of which is determined by \\eqref{eq: sd normalisation fourier}.} We get (using the well known identities for the $\\Gamma$ function that $\\Gamma (1 + x) = x \\Gamma( x)$ for at least $x >0$ and $\\Gamma(x) \\Gamma(1-x) = \\pi / \\sin(\\pi x)$ whenever $x \\notin \\mathbb{Z}$),\n\\begin{equation} \\label{eq:S=cst}\nS = -\\frac{8\\pi\\sqrt{2}}{\\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right).\n\\end{equation}\n\n\\paragraph{Conclusion.}\nWe finally arrived at the following explicit expression for $R_+$: by \\eqref{eq: WH final 1} and \\eqref{eq:S=cst},\n\\begin{equation}\\label{eq: R_+ explicit}\n R_+(\\omega) = \n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \\frac{K_+(\\omega)}{(\\omega+i)(\\omega+3i)}\n =\n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2} \\Gamma\\left(\\frac{1-i\\omega}{2}\\right) (\\omega+i)(\\omega+3i)}.\n\\end{equation}\nOn the other hand, one may now use \\eqref{eq: WH final 2} to get the value of $(\\gamma_-,\\gamma_+)$. Indeed, $F_-$ is holomorphic on $\\cal H_-$, while $F_+$ has poles at $-i$ and $-3i$. Thus, we can read off the coefficients $c_0$ and $c_1$ of \\eqref{eq: expr_F(omega)} from the right-hand side of \\eqref{eq: WH final 2}. We arrive at $c_0 = \\frac{4}{\\theta} \\cos\\left(\\frac{\\pi\\theta}{2}\\right)\\sin\\left(\\frac{\\pi\\theta}{2}\\right)$ and $c_1 = -\\frac{4}{\\theta} \\sin^2\\left(\\frac{\\pi \\theta}{2}\\right)$. Recalling the expression of $c_0, c_1$ in \\cref{lem:fourier_WH}, this in turn implies that \n\\begin{equation}\\label{eq:gamma+}\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\end{equation}\nOne then recovers $r_+$ as the inverse Fourier transform of the above \\eqref{eq: R_+ explicit}: \n\\begin{equation}\\label{eq:r+explicit}\n r_+(v) \n =\n \\frac{\\gamma_+-\\gamma_-}{\\sqrt{2}\\pi\\theta} \\mathrm{e}^{-3v} \\left( (\\mathrm{e}^{2v}+\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} - (\\mathrm{e}^{2v}-\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} \\right).\n\\end{equation}\nWe will prove this identity in Appendix \\ref{sec:appendix} (note that this differs from the analogous identity in \\cite[(38)]{gaudin1989n}, though their final answer appears to be correct). \nUndoing the change of variables, we conclude that for $y\\in(\\gamma_-,\\gamma_+)$,\n\\begin{multline}\n \\label{eq:rho_final_expr}\n \\rho(y) = \\\\\n \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)} (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\end{multline}\nNote that, as expected, \n \\[\n \\rho(y)\n \\sim\n c (\\gamma_+ - y)^{1-\\theta}, \\quad \\text{as } y\\to (\\gamma_+)^-.\n \\]\n\nHere we verify that the formula computing the inverse Fourier transform of $R_+$ in Section \\ref{sec:resolvents} (crucial in the proof of Theorem \\ref{thm:F_ell_main}) is as claimed in \\eqref{eq:r+explicit}. Recall that \nwe know from the Wiener--Hopf argument that the Fourier transform $R_+ (\\omega) = \\textstyle\\int_{\\mathbb{R}} \\mathrm{e}^{ i \\omega v} r_+ (v) \\mathrm{d} v$ satisfies\n\\begin{align*}\nR_+(\\omega) &= \n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2} \\Gamma\\left(\\frac{1-i\\omega}{2}\\right) (\\omega+i)(\\omega+3i)}\\\\\n & = \\frac{\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2}\\Gamma ( \\tfrac{5 - i \\omega}{2})}.\n\\end{align*}\nwhere we used properties of the Gamma function to simplify the denominator of the fraction.\n\nIt is easier to start from the answer, i.e., to consider the function \n$$\ns_+ (v) : = \\frac{(\\gamma_+-\\gamma_-)}{\\sqrt{2} \\pi\\theta} \\mathrm{e}^{-3v} \\left( (\\mathrm{e}^{2v}+\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} - (\\mathrm{e}^{2v}-\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} \\right), \\quad v >0,\n$$\ncompute its Fourier transform $S_+(\\omega) = \\int_{0}^\\infty \\mathrm{e}^{ i \\omega v} s_+ (v) \\mathrm{d} v$ and verify that $S_+(\\omega) = R_+(\\omega)$. (We will verify this for $\\omega \\in \\mathcal{H}_+$, i.e., where $R_+$ is well defined). Let us write $c =\\tfrac{\\gamma_+-\\gamma_-}{\\sqrt{2} \\pi\\theta}$.",
    "post_theorem_intro_text_len": 4957,
    "post_theorem_intro_text": "In particular, we deduce the following asymptotics.\n\\begin{Cor} \\label{cor:asymp_F_ell_main}\n    The partition function satisfies:\n    \\begin{equation} \\label{eq:asymp_F_ell_main}\n\tF_\\ell \n\t\\sim\n\tc \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n\t\\end{equation}\n\twhere $c =\\frac{2^{\\theta-1/2}}{\\pi \\theta} (\\gamma_+-\\gamma_-)^{\\theta-1} \\gamma_+^{1-\\theta} \\Gamma(2-\\theta)$, and $\\gamma_-$ and $\\gamma_+$ are as above.\n\\end{Cor}\n\n\\noindent Observe that the form of the asymptotic \\eqref{eq:asymp_F_ell_main} is characteristic of the critical (in fact, more precisely, \\textbf{non-generic critical} according to the terminology of \\cite{BBG12}) phase of the loop-$O(n)$ model, with the exponent $2-\\theta$ interpolating between $3/2$ and $2$ (see e.g.\\ \\cite[Section 5]{BBG12}). More precisely, it lies in the so-called \\emph{dense} phase of the model, where loops are conjectured to be non-simple and to touch each other in the scaling limit. \n\nOur second main result determines the exact tail exponents for the size of typical loops and clusters in the original FK$(q)$-weighted planar map model. This sharpens the main theorem of \\cite{berestycki2017critical} by identifying the previously implicit $\\ell^{o(1)}$ factor as a true constant {(a similar result can also be deduced from  \\cite{gwynne2019scaling}, with the $\\ell^{o(1)}$ term identified as a slowly varying function)}. We denote by $\\mathfrak{L}$ and $\\mathfrak{K}$ the typical loop and (filled-in) cluster in the \\emph{infinite} FK$(q)$ planar map. These correspond to the local limit of a uniformly chosen loop or (filled-in) cluster in a finite FK$(q)$ map of size $k$, and then sending $k$ to infinity, see \\cite[Theorem 1.1]{berestycki2017critical}.\nWe define the perimeter $|\\mathfrak{L}|$ of the loop $\\mathfrak{L}$ as the number of triangles it crosses, and the perimeter $|\\partial \\mathfrak{K}|$ of the cluster as the degree of its external face, see \\cref{sec:typ_cluster} for more precise definitions.\n\n\\begin{Thm}[Exponents for loops and clusters]\n\\label{thm:exp_main}\n    We have the following tail asymptotics: as $\\ell\\to\\infty$,\n    \\[\n    \\mathbb{P}(|\\partial \\mathfrak{K}| = \\ell) \\sim \\frac{C}{\\ell^{3-2\\theta}}\n    \\quad \\text{and} \\quad \n    \\mathbb{P}(|\\mathfrak{L}| = \\ell) \\sim \\frac{C'}{\\ell^{3-2\\theta}},\n    \\]\n    where $C$ and $C'$ are positive constants.\n\\end{Thm}\n\nOur strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.\n\nThe paper is organised as follows. \\cref{sec:prelim} is dedicated to preliminaries on self-dual FK planar maps, the $O(n)$ model and the Mullin--Bernardi--Sheffield bijection. In \\cref{sec:self_dual_critical}, we make some connections between Sheffield's hamburger-cheeseburger walk and the gasket decomposition of Borot, Bouttier and Guitter. In particular, this allows us to prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments: we will derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.  We will then use this information in \\cref{sec:resolvents}, where we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \\cref{thm:F_ell_main}. Finally, in \\cref{sec:loop-clusters}, we come back to the hamburger-cheeseburger model and deduce \\cref{thm:exp_main} by plugging back the information on the partition function.\n\nWe stress that this dictionary has been used in \\cite{da2025scaling}, which concerns the critical case $q=4$ (equivalently, $n=2$). In that case, the authors prove the analogue of \\cref{thm:F_ell_main} by directing solving the gasket decomposition equation (without any extra input) and then deduce the scaling limit of FK$(4)$-weighted planar maps in the peanosphere sense, resolving the remaining open regime in Sheffield's work \\cite{sheffield2016quantum}.\n\n\\paragraph{Acknowledgments.}\nWe thank Sasha Glazman, Xingjian Hu, Ellen Powell, Xin Sun, Joonas Turunen and Mo Dick Wong for insightful discussions.\nN.B.  acknowledges the support from the Austrian Science Fund (FWF) grants 10.55776/F1002 on ``Discrete random structures: enumeration and scaling limits\" and 10.55776/PAT1878824 on ``Random Conformal Fields''.\nW.D.S.\\ is supported by the Austrian Science Fund (FWF) grant on “Emergent branching structures in random geometry” (DOI: 10.55776/ESP534).",
    "sketch": "“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”\n\nMore specifically for Theorem~\\ref{thm:F_ell_main}: in \\cref{sec:self_dual_critical} the authors “prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.” Then “in \\cref{sec:resolvents}, we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \\cref{thm:F_ell_main}.”",
    "expanded_sketch": "“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”\n\nMore specifically in establishing the main theorem: next the authors “prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.” Then later they “solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving the main theorem.”",
    "expanded_theorem": "[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]",
    "theorem_type": [
      "Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\((F_\\ell)_{\\ell\\ge 1}\\) be the partition-function sequence in the setting under consideration, and let the parameter \\(\\theta\\) determine constants \\(\\gamma_+\\) and \\(\\gamma_-\\) by\n\\[\n\\gamma_+ = 2^{3/2}\\cos\\!\\left(\\frac{\\pi\\theta}{2}\\right),\n\\qquad\n\\gamma_+ - \\gamma_- = \\frac{2^{3/2}}{\\theta}\\sin\\!\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]\nWhich explicit formula holds for every integer \\(\\ell\\ge 1\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(y-\\gamma_-)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nfor a density \\(\\rho\\) supported on \\([\\gamma_-,\\gamma_+]\\)."
        },
        {
          "label": "D",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+ - \\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n+\\Big(\\sqrt{2\\gamma_+ - \\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell-1}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "upper-endpoint singular factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped explicit closed formula for the density",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "jump-on-the-cut sign structure",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "moment exponent in the integral representation",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct formula or uniquely signal choice A. It only asks for the explicit integral representation, without giving away the decisive exponent, sign pattern, or endpoint formulas."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the correct option is the full explicit representation itself. The task is mainly to recognize the exact stated formula rather than infer a new conclusion from premises."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to compare closely related formulas and reject a weaker true statement, but the item mostly tests exact memory/recognition of a known formula rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one alters the boundary exponent, one changes the resolvent-sign structure, one perturbs an endpoint formula, and one offers a weaker-but-true statement. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "No major answer leakage and strong distractors, but the question is largely a theorem restatement that emphasizes precise recall over generative reasoning."
    }
  },
  {
    "id": "2512.05867v1",
    "paper_link": "http://arxiv.org/abs/2512.05867v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "Thm",
      "content": "[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]",
      "start_pos": 79042,
      "end_pos": 79734,
      "label": "thm:F_ell_main"
    },
    "ref_dict": {
      "eq:lawbicolour_sd": "\\begin{equation}\n    \\label{eq:lawbicolour_sd}\n    \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}",
      "eq:lawgen": "\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}",
      "eq:lawbicolour": "\\begin{equation}\n    \\label{eq:lawbicolour}\n    \\mathbb{P} ( (T, L) = (\\mathfrak{t}, \\boldsymbol{\\ell})) \\; \\propto \\;  x_1^{\\#\\mathfrak{f}_1} x_2^{\\#\\mathfrak{f}_2} n^{\\#\\boldsymbol{\\ell}},\n\\end{equation}",
      "thm:F_ell_main": "\\begin{Thm}[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]\n\\end{Thm}",
      "eq:asymp_F_ell_main": "\\begin{equation} \\label{eq:asymp_F_ell_main}\n\tF_\\ell \n\t\\sim\n\tc \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n\t\\end{equation}",
      "sss:gasket_dec": "\\begin{equation}\\label{eq:link_FK_Z}\n\\Big(\\frac{1-p}{16}\\Big)^{\\frac{\\#F(\\tfrak)+\\ell-1}{2}}\n\\Big(\\frac{2p}{1-p}\\Big)^{\\#\\boldsymbol{\\ell}+1}\n=\nn x_c^{\\ell+1} x_c^{\\#F(\\tfrak)-1} n^{\\#\\boldsymbol{\\ell}}\n=\nn x_c^{\\ell+1} Z(\\tfrak,\\boldsymbol{\\ell},x_c,n),\n\\end{equation}\nwhere $n$ and $x_c$ are as in~\\eqref{eq:x_c_formula}.\n\n\\subsubsection{The gasket decomposition}\n\\label{sss:gasket_dec}\nThe gasket decomposition, introduced in \\cite{BBG12a} and further developed in \\cite{BBG12}, provides a remarkably effective framework for analysing a broad class of loop models.\nIn what follows we adapt the presentation of \\cite{BBG12} to the particular fully packed setting determined by \\eqref{eq: def weight triangulation} and \\eqref{eq: def partition triangulation}.\n\nConsider a configuration $(\\tfrak,\\boldsymbol{\\ell})\\in \\mathbb{T}_\\ell$.\nTo construct its gasket, we first look at the edges of $\\tfrak$ that are reachable from the boundary without crossing any loop.\nBecause the loops are fully packed, these accessible edges divide the map into exactly $\\#\\boldsymbol{\\ell}+1$ faces: the external face, with boundary length~$\\ell$, and an additional face of degree~$k$ for each loop of (outer) perimeter~$k$.\nThe resulting object is by definition the \\textbf{gasket}.\nNow examine one of the internal faces of degree~$k$ created by the gasket.\n\n\\begin{figure}[b!]\n  \\bigskip\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{Ring-partition-function.pdf}\n  \\end{center}\n  \\caption{The ring partition function $A^{(1\\to 2)}_{k,k'}$ from colour 1 (blue) to colour 2 (red). It accounts for the weight of all the triangles crossed by the purple loop.} \n  \\label{fig:ring_part}\n\\end{figure}\n\nIf we reinsert the triangles traversed by the corresponding loop, we obtain a ring of triangles whose outer boundary length is~$k$ and whose inner boundary has some length~$k'$.\nThe ring consists of $k+k'$ triangles in total.\nMoreover, the loop-decorated triangulation originally contained inside this loop -- call it $(\\tfrak',\\boldsymbol{\\ell}')\\in\\mathbb{T}_{k'}$ -- is precisely what was taken out from $(\\tfrak,\\boldsymbol{\\ell})$ to form the corresponding face of the gasket.\nIn this way, $(\\tfrak,\\boldsymbol{\\ell})$ decomposes into:\n(i) the gasket;\n(ii) one ring of triangles for each internal gasket face; and\n(iii) a fully packed loop-decorated triangulation attached along the inner boundary of each such ring.\nThe contribution of such a component to the weight $Z(\\tfrak',\\boldsymbol{\\ell}',x,n)$ in \\eqref{eq: def weight triangulation} is $n x^{k + k'}Z(\\tfrak, \\boldsymbol{\\ell}, x, n)$,\nwhere $k$ is the outer and $k'$ the inner boundary length of the ring.\nThis motivates the definition\n\\begin{equation}\\label{eq: gasket decomposition}\ng_k := n\\sum_{k'=0}^{\\infty} A_{k\\rightarrow k'} x^{k+k'} F_{k'},\n\\end{equation}",
      "prop:prob_tau": "\\begin{Prop}[From hitting times to the partition function]\\label{prop:prob_tau}\n\tThere exists a normalising constant $C>0$ such that, for all $\\ell\\geq 0$,\n\t\\[\n\t\\mathbb{P}(\\tau^{\\rh} = \\ell+1) = C (2x_c)^{\\ell+1}F_{\\ell}. \n\t\\]\n\\end{Prop}",
      "thm:exp_main": "\\begin{Thm}[Exponents for loops and clusters]\n\\label{thm:exp_main}\n    We have the following tail asymptotics: as $\\ell\\to\\infty$,\n    \\[\n    \\mathbb{P}(|\\partial \\mathfrak{K}| = \\ell) \\sim \\frac{C}{\\ell^{3-2\\theta}}\n    \\quad \\text{and} \\quad \n    \\mathbb{P}(|\\mathfrak{L}| = \\ell) \\sim \\frac{C'}{\\ell^{3-2\\theta}},\n    \\]\n    where $C$ and $C'$ are positive constants.\n\\end{Thm}",
      "eq:lawSD": "\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 10664,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nPlanar maps (i.e., proper embeddings of graphs in the $2$-sphere, considered up to orientation-preserving homeomorphisms) are a central topic not only in combinatorics but also in probability and mathematical physics due to their conjectured links with Liouville quantum gravity (LQG). In that context, planar maps can be thought of as canonical discretisations of the ``random surfaces'' which are at the core of Polyakov's original formulation of LQG \\cite{polyakov1981quantum}. Indeed, in order to describe the gravitational action when gravity is coupled to a matter field, the so-called DDK Ansatz (\\cite{David, DistlerKawai}) implies that this can be equivalently described by considering the scaling limit of random planar maps decorated by models of statistical mechanics at their critical point. \n\nIn this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an  FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\,  \\propto  \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed  configuration $L$ of loops, with probability\n\\begin{equation}\n    \\label{eq:lawbicolour_sd}\n    \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\n\\medskip In many models of planar statistical mechanics, it is widely expected that the self-dual point coincides with its critical point (see for instance \\cite{grimmett2018probability} for some background discussion).\nIn particular, such a statement is the content of a celebrated result of Beffara and Duminil-Copin \\cite{beffara2012self} for the random cluster model (equivalently the Fortuin--Kasteleyn percolation model) on the square lattice and for $q\\ge 1$. For the related loop $O(n)$ model on the hexagonal lattice, the equality between critical and self-dual point led Nienhuis to predict (\\cite{Nienhuis82, Nienhuis84}) that the critical point occurs at $x = x_c = 1/ \\sqrt{2 + \\sqrt{2-n}}$ for $n \\in (0,2]$, and in fact $n \\in [-2, 2]$; this remains a famous open problem in this field (see \\cite{DCensaios} for a thorough survey).  This should also apply to the two self-dual models planar maps discussed above, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}. This suggests that this model is also \\textbf{critical}, although in a sense that needs to be carefully specified. \n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\nThis is supported by a considerable body of evidence. On the one hand, in the breakthrough paper \\cite{sheffield2016quantum}, Sheffield provided a measure-preserving bijection between self-dual FK-weighted planar maps \\eqref{eq:lawSD} and inventory accumulations, which in turn correspond to a pair of (non-Markovian) walks, and showed that in the scaling limit this pair of walks converges to a pair of correlated Brownian motions. In combination with the so-called Mating of Trees framework developed by Duplantier, Miller and Sheffield \\cite{duplantier2021liouville} this result can be re-interpreted as a form of convergence towards a $\\gamma$-LQG surface decorated with an independent space-filling SLE$_{\\kappa'}$ curve, albeit for a relatively weak topology (the so-called ``peanosphere'' topology). Building on this foundational work, a number of observables (such as sizes of clusters and their boundaries) have been analysed and the associated exponents computed (see \\cite{gwynne2019scaling,gwynne2017scaling,gwynne2015scaling} and \\cite{berestycki2017critical}). These values are consistent with those that can be predicted by combining known results on the dimension of SLE$_\\kappa$ (\\cite{beffara2008dimension}) with the Knizhnik--Polyakov--Zamolodchikov (KPZ) identity, cf.\\ \\cite{duplantier2021liouville, HKPZ}. See, e.g., \\cite[Chapter 4]{BP} for a discussion of these results and additional perspective. \n\nOn the other hand, \na classical approach to the loop $O(n)$ model is to make use of the spatial Markov property of the model. This results in the so-called \\emph{gasket decomposition} which was in particular used by Borot, Bouttier and Guitter \\cite{BBG12, BBG12a, BBG12b} {to establish the phase diagram of the loop-$O(n)$ model}. Building on powerful tools of analytic combinatorics they derived and solved (assuming a certain natural ansatz) an equation for the resolvent of the partition function of the model, which yields fine asymptotics. Such an approach has been made fully rigorous (including the proof of the above ansatz) by works of Budd and Chen \\cite{budd2019peeling}, in the case of the so-called \\emph{rigid} model on quadrangulations (where loops are constrained to enter and leave a given quadrangle through opposite edges). From this, Chen, Curien and Maillard \\cite{ChenCurienMaillard} deduced some scaling limit results for the perimeter cascade of $O(n)$ loops, and Aïdékon, Da Silva and Hu \\cite{aidekon2024scaling} proved the scaling limit for the volume of such {rigid} loop-$O(n)$ quadrangulations. A similar approach was also used by Borot, Duplantier and Guitter \\cite{BBD16} to determine the nesting statistics in the \\emph{bending energy} variant of the $O(n)$ model, {where loops can bend inside a quadrangle at a given energetic cost.} \n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see  \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that  $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n    \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}",
    "context": "In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed configuration $L$ of loops, with probability\n\\begin{equation}\n \\label{eq:lawbicolour_sd}\n \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}",
    "full_context": "In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\\in (0,4)$. \nIn this model, we sample a pair $(M, \\Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\\mathfrak m, \\omega)$ \n depends on an extra percolation parameter $p_0 \\in (0,1)$ and a volume weight $s$ and is proportional to\n\\begin{equation}\\label{eq:lawgen}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\left(\\frac{p_0}{1-p_0}\\right)^{\\r o(\\omega)} q^{\\r{cc}(\\omega)} s^{\\r v(\\mathfrak m)},\n\\end{equation}\nwhere $\\r o(\\omega)$ and $\\r{cc}(\\omega)$ are the number of edges and connected components of $\\omega$, and $\\mathsf{v} (\\mathfrak m)$ is the number of vertices of $\\mathfrak{m}$. \nIn particular, conditional on the planar map $M = \\mathfrak{m}$, the edge configuration $\\Omega$ is sampled as an FK$(q)$-percolation configuration on $\\mathfrak{m}$ (see, e.g., \\cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \\mathfrak m, \\omega)$ there is a natural duality operation $( \\mathfrak m, \\omega) \\to ( \\mathfrak m^\\dagger, \\omega^\\dagger)$. Requiring that the law \\eqref{eq:lawgen} is invariant under this duality imposes that \n $p_0 = \\sqrt{q}/(1+\\sqrt{q})$ and $s=1/\\sqrt{q}$, as can be checked using Euler's formula.\nIn that case the law \\eqref{eq:lawgen} reduces to \n\\begin{equation}\\label{eq:lawSD}\n\\bb P ((M, \\Omega) = (\\mathfrak m, \\omega)) \\, \\propto \\, \n\\sqrt{q}^{\\# \\mathsf{loops}(\\mathfrak m, \\omega)},\n\\end{equation}\nwhere $\\mathsf{loops}(\\mathfrak m, \\omega)$ is the set of loops separating $\\omega$ and its dual in $\\mathfrak m$. This is called the \\textbf{self-dual} FK$(q)$-weighted planar map model. \n{\nIn the self-dual case, one can also let the number of edges $k \\to \\infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\\cite{sheffield2016quantum,chen2017basic}.\n}\n\nThe second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations. \nIn this model we sample a triangulation $T$, together with a fully packed configuration $L$ of loops, with probability\n\\begin{equation}\n \\label{eq:lawbicolour_sd}\n \\mathbb{P} ((T, L) = ( \\mathfrak{t}, \\boldsymbol{\\ell}) ) \\; \\propto \\; x^{\\# \\mathsf{faces} (\\mathfrak{t})} n^{\\# \\boldsymbol{\\ell}}. \n\\end{equation}\nUp to considering the dual map (one may view the loops in $\\boldsymbol{\\ell}$ as crossing the triangles of $\\mathfrak{t}$ or as a subset of edges of the dual map $\\mathfrak{t}^\\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \\cite{DMNS81}. In fact, it is convenient to view the model \\eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \\eqref{eq:lawbicolour_sd}. This model, introduced in \\cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \\eqref{eq:lawbicolour} for its precise definition and Section \\ref{sec:O(n)_model} for more explanations.\n\nAs is well known, and as will be recalled in Section \\ref{sec:O(n)_model}, the two models \\eqref{eq:lawgen} and \\eqref{eq:lawbicolour} (and their self-dual versions, \\eqref{eq:lawSD} and \\eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that \n\\begin{equation}\\label{eq:xc}\nn = \\sqrt{q} \\quad ; \\quad x = x_c=\\frac{1}{ \\sqrt{8(n+2)}} = \\frac{1}{\\sqrt{8( \\sqrt{q}+2)}}. \n\\end{equation}\n\nFurthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\\gamma$-LQG surface, where \n\\[\nq = 2 + 2\\cos\\left(\\frac{\\pi \\gamma^2}{2}\\right)\n\\quad \n\\text{and}\n\\quad \n\\gamma\\in(\\sqrt{2},2).\n\\]\nIn addition, the loops separating primal and dual clusters of $\\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\\kappa' = 16/\\gamma^2$.\n\n{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \\cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \\cref{sss:gasket_dec}.}\n\n\\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see \\cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\\ell \\ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\\ell$.}\nHere and in the rest of the paper, the expression $a_n \\sim b_n$ means that $a_n /b_n\\to 1$ as $n\\to \\infty$.\nSet \n\\begin{equation}\n \\theta = \\frac{1}{\\pi}\\arccos\\Big(\\frac{n}{2}\\Big) = \\frac1\\pi \\arccos\\Big(\\frac{\\sqrt{q}}{2}\\Big). \n\\end{equation}\n\nIn particular, we deduce the following asymptotics.\n\\begin{Cor} \\label{cor:asymp_F_ell_main}\n The partition function satisfies:\n \\begin{equation} \\label{eq:asymp_F_ell_main}\n F_\\ell \n \\sim\n c \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n \\end{equation}\n where $c =\\frac{2^{\\theta-1/2}}{\\pi \\theta} (\\gamma_+-\\gamma_-)^{\\theta-1} \\gamma_+^{1-\\theta} \\Gamma(2-\\theta)$, and $\\gamma_-$ and $\\gamma_+$ are as above.\n\\end{Cor}\n\n\\noindent With \\cref{lem:WH_K}, equation \\eqref{eq: sd resolvent fourier} implies \n\\[\n2\\pi^2 \\frac{R_+(\\omega)}{K_+(\\omega)} = \\frac{F_+(\\omega)}{K_-(\\omega)} + \\frac{F_-(\\omega)}{K_-(\\omega)} \\quad \\text{for } \\omega\\in\\cal S \\text{ such that } \\Gamma\\left( \\frac{1-2\\theta+i\\omega}{4} \\right) \\neq \\infty.\n\\]\nWe now multiply the latter display to remove the pole singularities of $F_+$: for $\\omega\\in\\cal S$ such that $\\Gamma\\left( \\frac{1-2\\theta+i\\omega}{4} \\right) \\neq \\infty$,\n\\[\n2\\pi^2 \\frac{R_+(\\omega)}{K_+(\\omega)} (\\omega+i)(\\omega+3i)= \\left(\\frac{F_+(\\omega)}{K_-(\\omega)} + \\frac{F_-(\\omega)}{K_-(\\omega)}\\right)(\\omega+i)(\\omega+3i).\n\\]\nNow remark that the left-hand side is actually holomorphic on $\\cal H_+$, whereas the right-hand side is holomorphic on $\\{z\\in\\bb C, \\Im(z)<1-2\\theta\\}$. We stress that, since $0<n<2$, we have $1-2\\theta\\in (0, 1)$. Therefore, both sides can be extended to an entire function $S$ in the complex plane, whence\n\\begin{equation} \\label{eq: WH final 1}\nR_+(\\omega)\n=\n\\frac{1}{2\\pi^2} K_+(\\omega) \\frac{S(\\omega)}{(\\omega+i)(\\omega+3i)}\n\\quad \\text{for } \\omega\\in \\cal H_+,\n\\end{equation}\nand \n\\begin{equation} \\label{eq: WH final 2}\nF_+(\\omega)+F_-(\\omega) \n=\nK_-(\\omega) \\frac{S(\\omega)}{(\\omega+i)(\\omega+3i)} \\quad \\text{for } \\omega \\text{ such that } \\Im(\\omega)<1-2\\theta.\n\\end{equation}\nWe now analyse the behaviours at infinity in \\eqref{eq: WH final 1} and \\eqref{eq: WH final 2}. \n{Since $r_+$ is integrable by Lemma \\ref{lem:r(v)/v_int}, we have that $R_+(\\omega) = O(1)$ as $|\\omega| \\to \\infty$ with $\\omega\\in \\mathcal{H}_+$. Likewise, writing $ \\omega = x + {i} y$ with $y < 1$, we have by Lemma \\ref{lem:r(v)/v_int},\n\\[\n|F_-(\\omega)| = \\Big|\\int_{-\\infty}^0 f_-(u) \\mathrm{e}^{ i \\omega u} \\mathrm{d} u \\Big|\\le \\int_{-\\infty}^0 O(\\mathrm{e}^u) \\mathrm{e}^{-yu}\\mathrm{d} u.\n\\]\nThus we deduce that $F_- (\\omega) = O(1)$ as $|\\omega| \\to \\infty$ with $\\Im(\\omega)<1-2\\theta$.\nCombining this with the exact expressions \\eqref{eq: expressions K_+ and K_-}, we see that\n$S(\\omega) = O(\\omega)$ as $|\\omega|\\to\\infty$. By Liouville's theorem, $S$ is a degree $1$ polynomial. \nWe can therefore write it as $S(\\omega) = A\\omega + B$ for some $A, B \\in \\mathbb{C}$. We will in fact show it is a constant: indeed, we can read the value of $A$ by taking $\\omega\\to +\\infty$ along the real line in \\eqref{eq: WH final 2}. By the Riemann--Lebesgue lemma, we have $F_-(\\omega) \\to 0$ and this forces $A=0$ (as $F_+ ( \\omega) = O ( 1/|\\omega|) \\to 0$ as $\\omega \\to \\infty$). We conclude that $S$ is actually a constant,\nthe value of which is determined by \\eqref{eq: sd normalisation fourier}.} We get (using the well known identities for the $\\Gamma$ function that $\\Gamma (1 + x) = x \\Gamma( x)$ for at least $x >0$ and $\\Gamma(x) \\Gamma(1-x) = \\pi / \\sin(\\pi x)$ whenever $x \\notin \\mathbb{Z}$),\n\\begin{equation} \\label{eq:S=cst}\nS = -\\frac{8\\pi\\sqrt{2}}{\\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right).\n\\end{equation}\n\n\\paragraph{Conclusion.}\nWe finally arrived at the following explicit expression for $R_+$: by \\eqref{eq: WH final 1} and \\eqref{eq:S=cst},\n\\begin{equation}\\label{eq: R_+ explicit}\n R_+(\\omega) = \n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \\frac{K_+(\\omega)}{(\\omega+i)(\\omega+3i)}\n =\n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2} \\Gamma\\left(\\frac{1-i\\omega}{2}\\right) (\\omega+i)(\\omega+3i)}.\n\\end{equation}\nOn the other hand, one may now use \\eqref{eq: WH final 2} to get the value of $(\\gamma_-,\\gamma_+)$. Indeed, $F_-$ is holomorphic on $\\cal H_-$, while $F_+$ has poles at $-i$ and $-3i$. Thus, we can read off the coefficients $c_0$ and $c_1$ of \\eqref{eq: expr_F(omega)} from the right-hand side of \\eqref{eq: WH final 2}. We arrive at $c_0 = \\frac{4}{\\theta} \\cos\\left(\\frac{\\pi\\theta}{2}\\right)\\sin\\left(\\frac{\\pi\\theta}{2}\\right)$ and $c_1 = -\\frac{4}{\\theta} \\sin^2\\left(\\frac{\\pi \\theta}{2}\\right)$. Recalling the expression of $c_0, c_1$ in \\cref{lem:fourier_WH}, this in turn implies that \n\\begin{equation}\\label{eq:gamma+}\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\end{equation}\nOne then recovers $r_+$ as the inverse Fourier transform of the above \\eqref{eq: R_+ explicit}: \n\\begin{equation}\\label{eq:r+explicit}\n r_+(v) \n =\n \\frac{\\gamma_+-\\gamma_-}{\\sqrt{2}\\pi\\theta} \\mathrm{e}^{-3v} \\left( (\\mathrm{e}^{2v}+\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} - (\\mathrm{e}^{2v}-\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} \\right).\n\\end{equation}\nWe will prove this identity in Appendix \\ref{sec:appendix} (note that this differs from the analogous identity in \\cite[(38)]{gaudin1989n}, though their final answer appears to be correct). \nUndoing the change of variables, we conclude that for $y\\in(\\gamma_-,\\gamma_+)$,\n\\begin{multline}\n \\label{eq:rho_final_expr}\n \\rho(y) = \\\\\n \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)} (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\end{multline}\nNote that, as expected, \n \\[\n \\rho(y)\n \\sim\n c (\\gamma_+ - y)^{1-\\theta}, \\quad \\text{as } y\\to (\\gamma_+)^-.\n \\]\n\nHere we verify that the formula computing the inverse Fourier transform of $R_+$ in Section \\ref{sec:resolvents} (crucial in the proof of Theorem \\ref{thm:F_ell_main}) is as claimed in \\eqref{eq:r+explicit}. Recall that \nwe know from the Wiener--Hopf argument that the Fourier transform $R_+ (\\omega) = \\textstyle\\int_{\\mathbb{R}} \\mathrm{e}^{ i \\omega v} r_+ (v) \\mathrm{d} v$ satisfies\n\\begin{align*}\nR_+(\\omega) &= \n - \\frac{4\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2} \\Gamma\\left(\\frac{1-i\\omega}{2}\\right) (\\omega+i)(\\omega+3i)}\\\\\n & = \\frac{\\sqrt{2}}{\\pi \\theta} \\sin\\left( \\frac{\\pi\\theta}{2}\\right) \n \\frac{\\Gamma\\left( \\frac{3+2\\theta-i\\omega}{4} \\right) \\Gamma\\left( \\frac{3-2\\theta-i\\omega}{4} \\right)}{2^{i\\omega/2}\\Gamma ( \\tfrac{5 - i \\omega}{2})}.\n\\end{align*}\nwhere we used properties of the Gamma function to simplify the denominator of the fraction.\n\nIt is easier to start from the answer, i.e., to consider the function \n$$\ns_+ (v) : = \\frac{(\\gamma_+-\\gamma_-)}{\\sqrt{2} \\pi\\theta} \\mathrm{e}^{-3v} \\left( (\\mathrm{e}^{2v}+\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} - (\\mathrm{e}^{2v}-\\sqrt{\\mathrm{e}^{4v}-1})^{\\theta} \\right), \\quad v >0,\n$$\ncompute its Fourier transform $S_+(\\omega) = \\int_{0}^\\infty \\mathrm{e}^{ i \\omega v} s_+ (v) \\mathrm{d} v$ and verify that $S_+(\\omega) = R_+(\\omega)$. (We will verify this for $\\omega \\in \\mathcal{H}_+$, i.e., where $R_+$ is well defined). Let us write $c =\\tfrac{\\gamma_+-\\gamma_-}{\\sqrt{2} \\pi\\theta}$.",
    "post_theorem_intro_text_len": 4957,
    "post_theorem_intro_text": "In particular, we deduce the following asymptotics.\n\\begin{Cor} \\label{cor:asymp_F_ell_main}\n    The partition function satisfies:\n    \\begin{equation} \\label{eq:asymp_F_ell_main}\n\tF_\\ell \n\t\\sim\n\tc \\frac{\\gamma_+^{\\ell}}{\\ell^{2-\\theta}} \\quad \\text{as } \\ell\\to\\infty,\n\t\\end{equation}\n\twhere $c =\\frac{2^{\\theta-1/2}}{\\pi \\theta} (\\gamma_+-\\gamma_-)^{\\theta-1} \\gamma_+^{1-\\theta} \\Gamma(2-\\theta)$, and $\\gamma_-$ and $\\gamma_+$ are as above.\n\\end{Cor}\n\n\\noindent Observe that the form of the asymptotic \\eqref{eq:asymp_F_ell_main} is characteristic of the critical (in fact, more precisely, \\textbf{non-generic critical} according to the terminology of \\cite{BBG12}) phase of the loop-$O(n)$ model, with the exponent $2-\\theta$ interpolating between $3/2$ and $2$ (see e.g.\\ \\cite[Section 5]{BBG12}). More precisely, it lies in the so-called \\emph{dense} phase of the model, where loops are conjectured to be non-simple and to touch each other in the scaling limit. \n\nOur second main result determines the exact tail exponents for the size of typical loops and clusters in the original FK$(q)$-weighted planar map model. This sharpens the main theorem of \\cite{berestycki2017critical} by identifying the previously implicit $\\ell^{o(1)}$ factor as a true constant {(a similar result can also be deduced from  \\cite{gwynne2019scaling}, with the $\\ell^{o(1)}$ term identified as a slowly varying function)}. We denote by $\\mathfrak{L}$ and $\\mathfrak{K}$ the typical loop and (filled-in) cluster in the \\emph{infinite} FK$(q)$ planar map. These correspond to the local limit of a uniformly chosen loop or (filled-in) cluster in a finite FK$(q)$ map of size $k$, and then sending $k$ to infinity, see \\cite[Theorem 1.1]{berestycki2017critical}.\nWe define the perimeter $|\\mathfrak{L}|$ of the loop $\\mathfrak{L}$ as the number of triangles it crosses, and the perimeter $|\\partial \\mathfrak{K}|$ of the cluster as the degree of its external face, see \\cref{sec:typ_cluster} for more precise definitions.\n\n\\begin{Thm}[Exponents for loops and clusters]\n\\label{thm:exp_main}\n    We have the following tail asymptotics: as $\\ell\\to\\infty$,\n    \\[\n    \\mathbb{P}(|\\partial \\mathfrak{K}| = \\ell) \\sim \\frac{C}{\\ell^{3-2\\theta}}\n    \\quad \\text{and} \\quad \n    \\mathbb{P}(|\\mathfrak{L}| = \\ell) \\sim \\frac{C'}{\\ell^{3-2\\theta}},\n    \\]\n    where $C$ and $C'$ are positive constants.\n\\end{Thm}\n\nOur strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.\n\nThe paper is organised as follows. \\cref{sec:prelim} is dedicated to preliminaries on self-dual FK planar maps, the $O(n)$ model and the Mullin--Bernardi--Sheffield bijection. In \\cref{sec:self_dual_critical}, we make some connections between Sheffield's hamburger-cheeseburger walk and the gasket decomposition of Borot, Bouttier and Guitter. In particular, this allows us to prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments: we will derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.  We will then use this information in \\cref{sec:resolvents}, where we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \\cref{thm:F_ell_main}. Finally, in \\cref{sec:loop-clusters}, we come back to the hamburger-cheeseburger model and deduce \\cref{thm:exp_main} by plugging back the information on the partition function.\n\nWe stress that this dictionary has been used in \\cite{da2025scaling}, which concerns the critical case $q=4$ (equivalently, $n=2$). In that case, the authors prove the analogue of \\cref{thm:F_ell_main} by directing solving the gasket decomposition equation (without any extra input) and then deduce the scaling limit of FK$(4)$-weighted planar maps in the peanosphere sense, resolving the remaining open regime in Sheffield's work \\cite{sheffield2016quantum}.\n\n\\paragraph{Acknowledgments.}\nWe thank Sasha Glazman, Xingjian Hu, Ellen Powell, Xin Sun, Joonas Turunen and Mo Dick Wong for insightful discussions.\nN.B.  acknowledges the support from the Austrian Science Fund (FWF) grants 10.55776/F1002 on ``Discrete random structures: enumeration and scaling limits\" and 10.55776/PAT1878824 on ``Random Conformal Fields''.\nW.D.S.\\ is supported by the Austrian Science Fund (FWF) grant on “Emergent branching structures in random geometry” (DOI: 10.55776/ESP534).",
    "sketch": "“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”\n\nMore specifically for Theorem~\\ref{thm:F_ell_main}: in \\cref{sec:self_dual_critical} the authors “prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.” Then “in \\cref{sec:resolvents}, we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \\cref{thm:F_ell_main}.”",
    "expanded_sketch": "“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”\n\nMore specifically in establishing the main theorem: next the authors “prove an ansatz on the \\emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\\gamma_+$ of the cut.” Then later they “solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving the main theorem.”",
    "expanded_theorem": "[Expression and asymptotics for the partition function]\n\\label{thm:F_ell_main}\nWe have the exact expression\n\t\\[\n\tF_\\ell\n\t=\n\t\\int_{\\gamma_-}^{\\gamma_+}\n\t\\rho(y) y^{\\ell} \\mathrm{d}y,\n    \\quad \\ell\\geq 1,\n\t\\] \nwhere \n    \\[\n    \\rho(y) =  \n    \\frac{2^{-\\theta - 1/2}}{\\pi\\theta(\\gamma_+\\!-\\gamma_-)}  (\\gamma_+-y)^{1-\\theta} \\bigg( \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} + \\sqrt{y-\\gamma_-}\\Big)^{2\\theta} - \\Big( \\sqrt{2\\gamma_+ - \\gamma_- -y} - \\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg),\n    \\]\n    and \n    \\[\n\\gamma_+ = 2^{3/2} \\cos\\left( \\frac{\\pi\\theta}{2}\\right) \\quad \\text{and} \\quad \\gamma_+-\\gamma_- = \\frac{2^{3/2}}{\\theta} \\sin\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]",
    "theorem_type": [
      "Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\((F_\\ell)_{\\ell\\ge 1}\\) be the partition-function sequence in the setting under consideration, and let the parameter \\(\\theta\\) determine constants \\(\\gamma_+\\) and \\(\\gamma_-\\) by\n\\[\n\\gamma_+ = 2^{3/2}\\cos\\!\\left(\\frac{\\pi\\theta}{2}\\right),\n\\qquad\n\\gamma_+ - \\gamma_- = \\frac{2^{3/2}}{\\theta}\\sin\\!\\left(\\frac{\\pi\\theta}{2}\\right).\n\\]\nWhich explicit formula holds for every integer \\(\\ell\\ge 1\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(y-\\gamma_-)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nfor a density \\(\\rho\\) supported on \\([\\gamma_-,\\gamma_+]\\)."
        },
        {
          "label": "D",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+ - \\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n+\\Big(\\sqrt{2\\gamma_+ - \\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\nF_\\ell = \\int_{\\gamma_-}^{\\gamma_+} \\rho(y)\\, y^{\\ell-1}\\, \\mathrm{d}y,\n\\qquad \\ell\\ge 1,\n\\]\nwhere\n\\[\n\\rho(y)=\\frac{2^{-\\theta-1/2}}{\\pi\\theta(\\gamma_+-\\gamma_-)}(\\gamma_+-y)^{1-\\theta}\n\\bigg(\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}+\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\n-\\Big(\\sqrt{2\\gamma_+-\\gamma_- - y}-\\sqrt{y-\\gamma_-}\\Big)^{2\\theta}\\bigg).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "upper-endpoint singular factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped explicit closed formula for the density",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "jump-on-the-cut sign structure",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "moment exponent in the integral representation",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the explicit density formula or uniquely reveal the correct option. It only provides the definitions of \\(\\gamma_\\pm\\), so there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item asking for the exact explicit formula that holds. It functions as a near-direct restatement of a known result rather than asking for an independently derived conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing subtle formula variants (endpoint factor, sign choice, moment exponent, weaker-vs-exact statement), but the task mainly tests precise recall/pattern matching rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one swaps the boundary singular factor, one alters the sign structure, one weakens the claim, and one shifts the moment exponent. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall-style MCQ with strong distractors and little answer leakage, but it is mostly tautological and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.05945v1",
    "paper_link": "http://arxiv.org/abs/2512.05945v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.",
      "start_pos": 11842,
      "end_pos": 12439,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{theorem}\\label{thm:main}\nLet $M,N\\in\\N$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\N$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\N:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.\n\\end{theorem}",
      "prop:dense": "\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}",
      "lem:linindep": "\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 1431,
    "pre_theorem_intro_text": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:",
    "context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:",
    "full_context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\Q$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\nLet $\\{\\varphi_1,\\ldots,\\varphi_d\\}$\nand $\\{\\psi_1,\\ldots,\\psi_{d'}\\}$ be normalized Hecke eigenbases\nfor $S_k^{\\rm new}(\\Gamma_0(M))$ and $S_\\ell^{\\rm new}(\\Gamma_0(N))$, respectively.\nWrite\n\\[\n\\varphi_i(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\varphi_i}(n)n^{\\frac{k-1}{2}}e(nz),\n\\quad\n\\psi_j(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\psi_j}(n)n^{\\frac{\\ell-1}{2}}e(nz),\n\\]\nand\n\\[\n\\lambda_f(n)\\coloneq\\frac{a_f(n)}{n^{\\frac{k-1}{2}}}\n=\\sum_{i=1}^d u_i\\lambda_{\\varphi_i}(n),\n\\quad\n\\lambda_g(n)\\coloneq\\frac{a_g(n)}{n^{\\frac{\\ell-1}{2}}}\n=\\sum_{j=1}^{d'} v_j\\lambda_{\\psi_j}(n).\n\\]\nOur hypothesis on $\\lcm(M,N)$ implies that\n$\\varphi_1,\\ldots,\\varphi_d,\\psi_1,\\ldots,\\psi_{d'}$ are all\ntwist minimal without CM, and no two are twist equivalent.\nBy \\cite[\\S3, Theorem~M]{Ramakrishnan}, for any pair of indices $(i,j)$,\nthere is a cuspidal automorphic representation $\\pi_{ij}$ of $\\GL(4)$\nwith $p$th Dirichlet coefficient $\\lambda_{\\pi_{ij}}(p)$ satisfying\n$\\lambda_{\\pi_{ij}}(p)=\\lambda_{\\varphi_i}(p)\\lambda_{\\psi_j}(p)$\nfor primes $p\\nmid MN$.\nFurthermore, following the proof of\n\\cite[Lemma~4.5.8]{Ramakrishnan}, one can see that\n$\\pi_{ij}\\cong\\pi_{i'j'}$ if and only if $(i,j)=(i',j')$.\n\n\\begin{lemma}\\label{lem:dense}\nFor any $p\\in S$ and any index $i_1$, there are distinct\nnon-zero integers $n_1,\\ldots,n_d$ such that\n\\begin{enumerate}\n\\item $n_{i_1}\\nmid n_i$ for $i\\ne i_1$;\n\\item for any $t\\in\\R$ and any $\\varepsilon>0$, there exists\n$n\\in\\N$ such that\n\\[\n\\min\\{|n\\theta_i(p)-n_it-m|:m\\in\\Z\\}<\\varepsilon\n\\quad\\text{for }i=1,\\ldots,d.\n\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFix $p\\in S$ and an index $i_1$, and consider the vector space\n\\[\nV=\\Q+\\Q\\theta_1(p)+\\cdots+\\Q\\theta_d(p).\n\\]\nWrite $\\dim V=m+1\\ge3$.\nBy permuting the indices if necessary, we may assume that\n$\\{1,\\theta_1(p),\\ldots,\\theta_m(p)\\}$ is a basis for $V$, and we may choose\nthe permutation to map $i_1$ to $1$.\nFor $i=1,\\ldots,m$ we write\n\\[\n\\theta_i(p)=\\sum_{j=1}^m\\alpha_{ij}\\theta_j(p) + \\beta_i\n\\]\nfor some $\\alpha_{ij},\\beta_i\\in\\Q$.\nBy our construction, no two of the vectors\n$(\\alpha_{ij})_{j=1,\\ldots,d}$ are colinear.\n\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\theta_{ij}=\\theta_j(p_i)$. For each $i$ we\napply Lemma~\\ref{lem:dense} with $p=p_i$ and $i_1=i$,\nand we denote the resulting integers $n_{i1},\\ldots,n_{id}$.\nLet $t=(t_1,\\ldots,t_d)$, and\nconsider the vectors $F(t)=(F_j(t))_{j=1,\\ldots,d}$, where\n\\[\nF_j(t)=\\prod_{i=1}^d\\frac{\\sin(2\\pi n_{ij}t_i)}{\\sin(2\\pi\\theta_{ij})}\n=c_j\\prod_{i=1}^d\\sin(2\\pi n_{ij}t_i),\n\\]\nfor some $c_j\\in\\R_{>0}$.\nThen\n\\[\n\\frac{\\partial F_j}{\\partial t_i}\n=2\\pi c_jn_{ij}\\cos(2\\pi n_{ij}t_i)\n\\prod_{\\substack{1\\le i'\\le d\\\\i'\\ne i}}\n\\sin(2\\pi n_{i'j}t_{i'}).\n\\]\nBy construction we have $n_{ii}\\nmid n_{ij}$ for $j\\ne i$,\nand it follows that $F$ vanishes at the point\n$t=(\\frac1{2n_{ii}})_{i=1,\\ldots,d}$, and\nthe Jacobian matrix there is diagonal with non-zero determinant.\nBy the inverse function theorem, the image of $F$ contains\nan open neighborhood of the origin.\n\nTo complete the proof of Theorem~\\ref{thm:main}, write\n$f=\\sum_{i=1}^d u_i\\varphi_i$ and\n$g=\\sum_{i=1}^d v_i\\varphi_i$\nfor some non-zero vectors $u=(u_1,\\ldots,u_d),v=(v_1,\\ldots,v_d)\\in\\R^d$.\nIf $f$ and $g$ are not proportional\nthen we may choose a vector $w$ such that\n$w\\cdot u=0$ and $w\\cdot v\\ne0$.\nThen for any $\\delta\\in\\R$, we have\n\\[\n(w+\\delta u)\\cdot u=\\delta(u\\cdot u)\n\\quad\\text{and}\\quad\n(w+\\delta u)\\cdot v=(w\\cdot v)+\\delta(u\\cdot v).\n\\]\nChoosing $\\delta$ such that $\\delta(w\\cdot v)<0$ and\n$\\delta|u\\cdot v|<|w\\cdot v|$, we see that\n$(w+\\delta u)\\cdot u$ and $(w+\\delta u)\\cdot v$ have different signs.\nApplying Proposition~\\ref{prop:dense} we can find\nan arbitrarily large $n$ such that $[a_1(n):\\cdots:a_d(n)]$ approximates\nthe image of $w+\\delta u$ in $\\RP^{d-1}$\narbitrarily closely, and it follows that\n$a_f(n)$ and $a_g(n)$ have different signs.\nThis is a contradiction, so $f$ and $g$ must be proportional,\nmeaning that $c=f/g$ is constant.\nSince $f$ and $g$ are non-zero and $a_f(n)a_g(n)\\ge0$\nfor sufficiently large $n$, it follows that $c>0$.",
    "post_theorem_intro_text_len": 3906,
    "post_theorem_intro_text": "\\begin{remarks}\\\n\\begin{enumerate}\n\\item The restriction on $\\lcm(M,N)$\nis necessary when $(M,k)=(N,\\ell)$ and $Mk$ is sufficiently large,\nsince otherwise one can\nchoose a fundamental discriminant $\\Delta\\ne1$ with $\\Delta^2\\mid M$\nand a twist-minimal, non-CM newform $f$ of conductor $M$.\nSetting $g=2f+f\\times\\left(\\frac{\\Delta}{\\cdot}\\right)$, we have\n$a_f(n)a_g(n)=a_f(n)^2(2+\\left(\\frac{\\Delta}{n}\\right))\\ge0$.\n\\item The restriction to the new subspaces of\n$S_k(\\Gamma_0(M))$ and $S_\\ell(\\Gamma_0(N))$ is also\nnecessary in some cases, even under the assumption that $M=N$.\nFor example, let $f$ be the newform associated to an\nelliptic curve $E$ of squarefree conductor $N_E$, and set $g(z)=f(z)+f(pz)$,\nwhere $p>3$ is a supersingular prime for $E$. Then\n$f,g\\in S_2(\\Gamma_0(pN_E))$, and $a_f(n)a_g(n)=a_f(n)^2\\ge0$.\n\\item It will be clear from the proof that condition (1) of the theorem\ncan be substantially weakened. For instance, it suffices to have\n$\\liminf_{\\substack{n\\to\\infty\\\\\\gcd(n,q)=1}}\\frac{a_f(n)a_g(n)}{n^{\\frac{k+\\ell}{2}-1}}\\ge0$\nfor some fixed modulus $q$.\n\\item\nHowever, infinitely many $n$ are required when $(M,k)=(N,\\ell)$\nand $\\dim S_k^{\\rm new}(\\Gamma_0(M))>1$, i.e.\\ the first sign\nchange of $a_f(n)a_g(n)$ cannot be effectively bounded.\nTo see this, let $\\varphi,\\psi\\in S_k^{\\rm new}(\\Gamma_0(M))$ be distinct\nnormalized newforms, and set $f=\\varphi+\\pi\\psi$,\n$g_\\varepsilon=f+\\varepsilon\\varphi$ for some $\\varepsilon\\ne0$.\nThen $\\min\\{n\\in\\mathbb{N}:a_f(n)a_{g_\\varepsilon}(n)<0\\}\\to\\infty$ as $\\varepsilon\\to0$.\n\\end{enumerate}\n\\end{remarks}\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\mathbb{Q}$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\n\\subsection*{Acknowledgements}\nI thank Jonathan Bober and Oleksiy Klurman for\nmany helpful conversations.",
    "sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$” (via Lemma~\\ref{lem:linindep}).\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving Theorem~\\ref{thm:main} using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” referring to Proposition~\\ref{prop:dense} for the precise statement.",
    "expanded_sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$”, i.e.\n\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving the main theorem using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” and for the precise statement we use the following proposition.\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n",
    "expanded_theorem": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $M,N\\in\\mathbb N$ be such that $\\operatorname{lcm}(M,N)$ is divisible neither by $2^4$ nor by the square of any odd prime. Let $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$ be nonzero new cusp forms of weights $k,\\ell$ and levels $M,N$, with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Which statement about the signs of the products $a_f(n)a_g(n)$ is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."
      },
      "choices": [
        {
          "label": "B",
          "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a constant, possibly negative. Consequently, if $f$ and $g$ are not proportional, then there is at least one $\\epsilon\\in\\{\\pm1\\}$ for which the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."
        },
        {
          "label": "C",
          "text": "If $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, then $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$. In particular, this condition is sufficient for eventual nonnegativity of the products $a_f(n)a_g(n)$."
        },
        {
          "label": "D",
          "text": "The condition $a_f(p)a_g(p)\\ge 0$ for all sufficiently large primes $p$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{p\\text{ prime}: \\epsilon\\, a_f(p)a_g(p)>0\\}$ is infinite."
        },
        {
          "label": "E",
          "text": "If $a_f(n)a_g(n)\\ge 0$ for infinitely many $n\\in\\mathbb N$, then necessarily $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then $a_f(n)a_g(n)<0$ for all but finitely many $n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "positivity_of_scalar",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_equivalence_and_converse",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "all_n_vs_primes_only",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "sufficiently_large_replaced_by_infinitely_many",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the conclusion explicitly or by an obvious cue. It states the hypotheses and asks for an equivalent characterization; the correct option is not directly encoded in the wording."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is fairly close to a theorem-recall prompt: it asks which statement is equivalent to an eventual sign condition under technical hypotheses. However, it is not a pure verbatim restatement, since the options force the test-taker to distinguish among nearby formulations."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to reject subtle variants such as allowing a negative scalar, allowing c = 0, replacing all large n by large primes, or replacing equality of forms by eventual coefficient agreement. Still, the question mainly tests precise theorem recognition rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target natural failure modes: sign of the proportionality constant, inclusion of the zero scalar, weakening from all n to primes, and confusing eventual coefficient agreement with form equality. They are distinct and well aligned with likely misconceptions."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it leans more toward theorem recall/recognition than fully generative reasoning."
    }
  },
  {
    "id": "2512.05945v1",
    "paper_link": "http://arxiv.org/abs/2512.05945v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.",
      "start_pos": 11842,
      "end_pos": 12439,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{theorem}\\label{thm:main}\nLet $M,N\\in\\N$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\N$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\N:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.\n\\end{theorem}",
      "prop:dense": "\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}",
      "lem:linindep": "\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 1431,
    "pre_theorem_intro_text": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:",
    "context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:",
    "full_context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\Q$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\nLet $\\{\\varphi_1,\\ldots,\\varphi_d\\}$\nand $\\{\\psi_1,\\ldots,\\psi_{d'}\\}$ be normalized Hecke eigenbases\nfor $S_k^{\\rm new}(\\Gamma_0(M))$ and $S_\\ell^{\\rm new}(\\Gamma_0(N))$, respectively.\nWrite\n\\[\n\\varphi_i(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\varphi_i}(n)n^{\\frac{k-1}{2}}e(nz),\n\\quad\n\\psi_j(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\psi_j}(n)n^{\\frac{\\ell-1}{2}}e(nz),\n\\]\nand\n\\[\n\\lambda_f(n)\\coloneq\\frac{a_f(n)}{n^{\\frac{k-1}{2}}}\n=\\sum_{i=1}^d u_i\\lambda_{\\varphi_i}(n),\n\\quad\n\\lambda_g(n)\\coloneq\\frac{a_g(n)}{n^{\\frac{\\ell-1}{2}}}\n=\\sum_{j=1}^{d'} v_j\\lambda_{\\psi_j}(n).\n\\]\nOur hypothesis on $\\lcm(M,N)$ implies that\n$\\varphi_1,\\ldots,\\varphi_d,\\psi_1,\\ldots,\\psi_{d'}$ are all\ntwist minimal without CM, and no two are twist equivalent.\nBy \\cite[\\S3, Theorem~M]{Ramakrishnan}, for any pair of indices $(i,j)$,\nthere is a cuspidal automorphic representation $\\pi_{ij}$ of $\\GL(4)$\nwith $p$th Dirichlet coefficient $\\lambda_{\\pi_{ij}}(p)$ satisfying\n$\\lambda_{\\pi_{ij}}(p)=\\lambda_{\\varphi_i}(p)\\lambda_{\\psi_j}(p)$\nfor primes $p\\nmid MN$.\nFurthermore, following the proof of\n\\cite[Lemma~4.5.8]{Ramakrishnan}, one can see that\n$\\pi_{ij}\\cong\\pi_{i'j'}$ if and only if $(i,j)=(i',j')$.\n\n\\begin{lemma}\\label{lem:dense}\nFor any $p\\in S$ and any index $i_1$, there are distinct\nnon-zero integers $n_1,\\ldots,n_d$ such that\n\\begin{enumerate}\n\\item $n_{i_1}\\nmid n_i$ for $i\\ne i_1$;\n\\item for any $t\\in\\R$ and any $\\varepsilon>0$, there exists\n$n\\in\\N$ such that\n\\[\n\\min\\{|n\\theta_i(p)-n_it-m|:m\\in\\Z\\}<\\varepsilon\n\\quad\\text{for }i=1,\\ldots,d.\n\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFix $p\\in S$ and an index $i_1$, and consider the vector space\n\\[\nV=\\Q+\\Q\\theta_1(p)+\\cdots+\\Q\\theta_d(p).\n\\]\nWrite $\\dim V=m+1\\ge3$.\nBy permuting the indices if necessary, we may assume that\n$\\{1,\\theta_1(p),\\ldots,\\theta_m(p)\\}$ is a basis for $V$, and we may choose\nthe permutation to map $i_1$ to $1$.\nFor $i=1,\\ldots,m$ we write\n\\[\n\\theta_i(p)=\\sum_{j=1}^m\\alpha_{ij}\\theta_j(p) + \\beta_i\n\\]\nfor some $\\alpha_{ij},\\beta_i\\in\\Q$.\nBy our construction, no two of the vectors\n$(\\alpha_{ij})_{j=1,\\ldots,d}$ are colinear.\n\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\theta_{ij}=\\theta_j(p_i)$. For each $i$ we\napply Lemma~\\ref{lem:dense} with $p=p_i$ and $i_1=i$,\nand we denote the resulting integers $n_{i1},\\ldots,n_{id}$.\nLet $t=(t_1,\\ldots,t_d)$, and\nconsider the vectors $F(t)=(F_j(t))_{j=1,\\ldots,d}$, where\n\\[\nF_j(t)=\\prod_{i=1}^d\\frac{\\sin(2\\pi n_{ij}t_i)}{\\sin(2\\pi\\theta_{ij})}\n=c_j\\prod_{i=1}^d\\sin(2\\pi n_{ij}t_i),\n\\]\nfor some $c_j\\in\\R_{>0}$.\nThen\n\\[\n\\frac{\\partial F_j}{\\partial t_i}\n=2\\pi c_jn_{ij}\\cos(2\\pi n_{ij}t_i)\n\\prod_{\\substack{1\\le i'\\le d\\\\i'\\ne i}}\n\\sin(2\\pi n_{i'j}t_{i'}).\n\\]\nBy construction we have $n_{ii}\\nmid n_{ij}$ for $j\\ne i$,\nand it follows that $F$ vanishes at the point\n$t=(\\frac1{2n_{ii}})_{i=1,\\ldots,d}$, and\nthe Jacobian matrix there is diagonal with non-zero determinant.\nBy the inverse function theorem, the image of $F$ contains\nan open neighborhood of the origin.\n\nTo complete the proof of Theorem~\\ref{thm:main}, write\n$f=\\sum_{i=1}^d u_i\\varphi_i$ and\n$g=\\sum_{i=1}^d v_i\\varphi_i$\nfor some non-zero vectors $u=(u_1,\\ldots,u_d),v=(v_1,\\ldots,v_d)\\in\\R^d$.\nIf $f$ and $g$ are not proportional\nthen we may choose a vector $w$ such that\n$w\\cdot u=0$ and $w\\cdot v\\ne0$.\nThen for any $\\delta\\in\\R$, we have\n\\[\n(w+\\delta u)\\cdot u=\\delta(u\\cdot u)\n\\quad\\text{and}\\quad\n(w+\\delta u)\\cdot v=(w\\cdot v)+\\delta(u\\cdot v).\n\\]\nChoosing $\\delta$ such that $\\delta(w\\cdot v)<0$ and\n$\\delta|u\\cdot v|<|w\\cdot v|$, we see that\n$(w+\\delta u)\\cdot u$ and $(w+\\delta u)\\cdot v$ have different signs.\nApplying Proposition~\\ref{prop:dense} we can find\nan arbitrarily large $n$ such that $[a_1(n):\\cdots:a_d(n)]$ approximates\nthe image of $w+\\delta u$ in $\\RP^{d-1}$\narbitrarily closely, and it follows that\n$a_f(n)$ and $a_g(n)$ have different signs.\nThis is a contradiction, so $f$ and $g$ must be proportional,\nmeaning that $c=f/g$ is constant.\nSince $f$ and $g$ are non-zero and $a_f(n)a_g(n)\\ge0$\nfor sufficiently large $n$, it follows that $c>0$.",
    "post_theorem_intro_text_len": 3906,
    "post_theorem_intro_text": "\\begin{remarks}\\\n\\begin{enumerate}\n\\item The restriction on $\\lcm(M,N)$\nis necessary when $(M,k)=(N,\\ell)$ and $Mk$ is sufficiently large,\nsince otherwise one can\nchoose a fundamental discriminant $\\Delta\\ne1$ with $\\Delta^2\\mid M$\nand a twist-minimal, non-CM newform $f$ of conductor $M$.\nSetting $g=2f+f\\times\\left(\\frac{\\Delta}{\\cdot}\\right)$, we have\n$a_f(n)a_g(n)=a_f(n)^2(2+\\left(\\frac{\\Delta}{n}\\right))\\ge0$.\n\\item The restriction to the new subspaces of\n$S_k(\\Gamma_0(M))$ and $S_\\ell(\\Gamma_0(N))$ is also\nnecessary in some cases, even under the assumption that $M=N$.\nFor example, let $f$ be the newform associated to an\nelliptic curve $E$ of squarefree conductor $N_E$, and set $g(z)=f(z)+f(pz)$,\nwhere $p>3$ is a supersingular prime for $E$. Then\n$f,g\\in S_2(\\Gamma_0(pN_E))$, and $a_f(n)a_g(n)=a_f(n)^2\\ge0$.\n\\item It will be clear from the proof that condition (1) of the theorem\ncan be substantially weakened. For instance, it suffices to have\n$\\liminf_{\\substack{n\\to\\infty\\\\\\gcd(n,q)=1}}\\frac{a_f(n)a_g(n)}{n^{\\frac{k+\\ell}{2}-1}}\\ge0$\nfor some fixed modulus $q$.\n\\item\nHowever, infinitely many $n$ are required when $(M,k)=(N,\\ell)$\nand $\\dim S_k^{\\rm new}(\\Gamma_0(M))>1$, i.e.\\ the first sign\nchange of $a_f(n)a_g(n)$ cannot be effectively bounded.\nTo see this, let $\\varphi,\\psi\\in S_k^{\\rm new}(\\Gamma_0(M))$ be distinct\nnormalized newforms, and set $f=\\varphi+\\pi\\psi$,\n$g_\\varepsilon=f+\\varepsilon\\varphi$ for some $\\varepsilon\\ne0$.\nThen $\\min\\{n\\in\\mathbb{N}:a_f(n)a_{g_\\varepsilon}(n)<0\\}\\to\\infty$ as $\\varepsilon\\to0$.\n\\end{enumerate}\n\\end{remarks}\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\mathbb{Q}$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\n\\subsection*{Acknowledgements}\nI thank Jonathan Bober and Oleksiy Klurman for\nmany helpful conversations.",
    "sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$” (via Lemma~\\ref{lem:linindep}).\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving Theorem~\\ref{thm:main} using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” referring to Proposition~\\ref{prop:dense} for the precise statement.",
    "expanded_sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$”, i.e.\n\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving the main theorem using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” and for the precise statement we use the following proposition.\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n",
    "expanded_theorem": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $M,N\\in\\mathbb N$ be such that $\\operatorname{lcm}(M,N)$ is divisible neither by $2^4$ nor by the square of any odd prime. Let $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$ be nonzero new cusp forms of weights $k,\\ell$ and levels $M,N$, with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Which statement about the signs of the products $a_f(n)a_g(n)$ is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."
      },
      "choices": [
        {
          "label": "B",
          "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a constant, possibly negative. Consequently, if $f$ and $g$ are not proportional, then there is at least one $\\epsilon\\in\\{\\pm1\\}$ for which the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."
        },
        {
          "label": "C",
          "text": "If $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, then $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$. In particular, this condition is sufficient for eventual nonnegativity of the products $a_f(n)a_g(n)$."
        },
        {
          "label": "D",
          "text": "The condition $a_f(p)a_g(p)\\ge 0$ for all sufficiently large primes $p$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{p\\text{ prime}: \\epsilon\\, a_f(p)a_g(p)>0\\}$ is infinite."
        },
        {
          "label": "E",
          "text": "If $a_f(n)a_g(n)\\ge 0$ for infinitely many $n\\in\\mathbb N$, then necessarily $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then $a_f(n)a_g(n)<0$ for all but finitely many $n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "positivity_of_scalar",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_equivalence_and_converse",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "all_n_vs_primes_only",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "sufficiently_large_replaced_by_infinitely_many",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct choice. It states the hypotheses and asks for the valid conclusion, without giving away the equivalence or the proportionality conclusion."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is very close to a direct theorem statement under the same hypotheses, so the item leans heavily on theorem recall. However, the presence of nearby variants means it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing eventual nonnegativity from infinitely many sign occurrences, all integers versus primes, and positive versus arbitrary scalar multiples. But the question is weakened because choice C is also true, so it does not cleanly force identification of a unique strongest conclusion."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and reflect common mistakes with quantifiers and strength of conclusions. However, choice C is a genuinely true weaker consequence of A, so the distractor set is not fully well-formed for a single-answer MCQ."
      },
      "total_score": 5,
      "overall_assessment": "Moderately good theorem-discrimination item with no answer leakage and mostly plausible distractors, but it is flawed as a single-correct MCQ because a weaker true statement is included among the distractors."
    }
  },
  {
    "id": "2512.05952v1",
    "paper_link": "http://arxiv.org/abs/2512.05952v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thmx",
      "content": "\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.",
      "start_pos": 6741,
      "end_pos": 6981,
      "label": "thm"
    },
    "ref_dict": {
      "thm": "\\begin{thmx}\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.\n\\end{thmx}"
    },
    "pre_theorem_intro_text_len": 462,
    "pre_theorem_intro_text": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.",
    "context": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.",
    "full_context": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.\n\n\\begin{document}\n\\title[Spanning 3-discs for the trivial 2-link]{Spanning 3-discs in the 4-sphere pushed into the 5-disc}\n\n\\begin{abstract}\nI prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^4$ become smoothly isotopic rel.\\ boundary after pushing them into $D^5$.\n\\end{abstract}\n\\maketitle\n\nAn \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.\n\n\\begin{remark}\n In the case that $m=1$, i.e.\\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \\cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\\cite{HKM}. The $m=1$ case of the proof of \\cref{thm} gives another proof that is more direct. The case of multiple connected components is new. I deduce it from an application of surgery theory.\n\\end{remark}\n\nSince the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.\nBudney--Gabai~\\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\\ boundary in $S^4$, but which become isotopic in $D^5$.\nThe $m=1$ case of \\cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.\n\nAlison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \\geq 2$, that are not isotopic rel.\\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.\nIn his PhD thesis, Weizhe Niu~\\cite{Niu} independently developed alternative methods, extending the computations from~\\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.\nThese new collections can be seen directly to be standard in $D^5$; \\cref{thm} shows that this must always be the case.\n\nIf I allow trivial surface links in $S^4$ with positive genus, there is no analogous result. Hughes--Kim--Miller~\\cite{HKM} showed that there are pairs of genus $g \\geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\\ boundary, and remain non-isotopic rel.\\ boundary after they are pushed into $D^5$. The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.\nAssuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\\ boundary in $D^5$.\n\nI can now consider $f_i \\colon W_i \\to \\natural^m S^1 \\times D^4$ as an element of the\nBrowder-Novikov-Sullivan-Wall relative surgery structure set~\\cite{W}, denoted $\\mathcal{S}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3)$, which sits in the exact sequence (of abelian groups, in this case, since $\\natural^m S^1 \\times D^4 \\cong D^2 \\times \\natural^m S^1 \\times D^2$):\n\\begin{equation}\\label{eqn:ses}\nL_6(\\Z[F_m]) \\to \\mathcal{S}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3) \\to \\mathcal{N}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3) \\to L_5(\\Z[F_m]).\n\\end{equation}\nNote that since the Whitehead group of $F_m$ is trivial~\\cite{St1}, I can ignore $h$ and $s$ decorations, and every $h$-cobordism is an $s$-cobordism. Let $\\wt{L}_n(\\Z[\\pi])$ denote the reduced $L$-theory of $\\Z[\\pi]$, such that $L_n(\\Z[\\pi]) \\cong L_n(\\Z) \\oplus \\wt{L}_n(\\Z[\\pi])$.\nRecall that $L_n(\\Z)=0$ for $n$ odd, and for $k \\geq 0$ one has $L_{4k}(\\Z) \\cong \\Z$ and $L_{4k+2}(\\Z) \\cong \\Z/2$.\nI will use Cappell~\\cite[Corollary~6]{Ca1,Ca2} and Shaneson's~\\cite{Sh} calculations\n\\begin{align*}\n L_5(\\Z[F_m]) &\\cong L_5(\\Z) \\oplus \\bigoplus^m \\wt{L}_5(\\Z[\\Z]) \\cong \\oplus^m L_4(\\Z) \\cong \\oplus^m \\Z; \\\\\nL_6(\\Z[F_m]) &\\cong L_6(\\Z) \\oplus \\bigoplus^m \\wt{L}_6(\\Z[\\Z]) \\cong L_6(\\Z) \\oplus \\bigoplus^m L_5(\\Z) \\cong \\Z/2.\n\\end{align*}\n\n\\begin{thmx}\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.\n\\end{thmx}",
    "post_theorem_intro_text_len": 3357,
    "post_theorem_intro_text": "\\begin{remark}\n   In the case that $m=1$, i.e.\\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \\cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\\cite{HKM}.  The $m=1$ case of the proof of \\cref{thm} gives another proof that is more direct.  The case of multiple connected components is new. I deduce it from an application of surgery theory.\n\\end{remark}\n\nSince the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.\nBudney--Gabai~\\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\\ boundary in $S^4$, but which become isotopic in $D^5$.\nThe $m=1$ case of \\cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.\n\nAlison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \\geq 2$, that are not isotopic rel.\\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.\nIn his PhD thesis, Weizhe Niu~\\cite{Niu} independently developed alternative methods, extending the computations from~\\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.\nThese new collections can be seen directly to be standard in $D^5$; \\cref{thm} shows that this must always be the case.\n\nIf I allow trivial surface links in $S^4$ with positive genus, there is no analogous result.  Hughes--Kim--Miller~\\cite{HKM} showed that there are pairs of genus $g \\geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\\ boundary, and remain non-isotopic rel.\\ boundary after they are pushed into $D^5$.  The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.\nAssuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\\ boundary in $D^5$.\n\nThus, Theorem~\\ref{thm} is optimal: for the isotopy uniqueness result it espouses to hold, one must allow pushing the spanning 3-discs into the 5-disc, and one cannot increase the topological complexity beyond spherical 2-links.\n\n\\begin{remark}\n  The proof I give of \\cref{thm} is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary, which was proven in my work with Conway~\\cite{CP}. Since \\cref{thm} concerns one dimension up, the result holds in the smooth category. In addition, I can work here with free groups of arbitrary rank, whereas in the 4-dimensional case we needed to work in a setting where the fundamental group was good.\n\\end{remark}\n\n\\subsubsection*{Acknowledgements}\n\nThanks to Seungwon Kim and Maggie Miller for suggesting this question, and to both Anthony Conway and the anonymous referee for helpful comments on previous drafts.  I am very grateful to Sarah Blackwell and Alison Tatsuoka for informing me about potential applications and comparisons.\nI was partially supported by EPSRC grants EP/T028335/1 and  EP/V04821X/1.",
    "sketch": "The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is \"deduce[d] ... from an application of surgery theory,\" and later that \"The proof I give of \\cref{thm} is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary\" from \\cite{CP}. It is also noted that \"Since \\cref{thm} concerns one dimension up, the result holds in the smooth category,\" and that the argument can be carried out \"with free groups of arbitrary rank\" (contrasting with the 4-dimensional case where the fundamental group needed to be good).",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is \"deduce[d] ... from an application of surgery theory,\" and later that \"The proof I give of To prove the main theorem, … is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary\" from \\cite{CP}. It is also noted that \"Since To prove the main theorem, … concerns one dimension up, the result holds in the smooth category,\" and that the argument can be carried out \"with free groups of arbitrary rank\" (contrasting with the 4-dimensional case where the fundamental group needed to be good).,",
    "expanded_theorem": "\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.",
    "theorem_type": [
      "Universal",
      "Existence"
    ],
    "mcq": {
      "question": "An $m$-component 2-link is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\bigsqcup^m S^2$. For an $m$-component 2-link $L$, a spanning 3-disc collection for $L$ is a smoothly embedded submanifold $D_m\\subseteq S^4$ with $D_m\\cong \\bigsqcup^m D^3$ and $\\partial D_m=L$. Fix a standard trivial $m$-component 2-link $U_m\\subseteq S^4$, where “trivial” means that such a spanning 3-disc collection exists. If $D_m^0$ and $D_m^1$ are any two spanning 3-disc collections for $U_m$ in $S^4$, and we view them inside $D^5$ via the inclusion $S^4\\subseteq D^5$, which statement holds for every such pair?",
      "correct_choice": {
        "label": "A",
        "text": "The two collections $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; that is, there is a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ that keeps the common boundary $U_m$ fixed."
      },
      "choices": [
        {
          "label": "B",
          "text": "The two collections $D_m^0$ and $D_m^1$ become topologically isotopic in $D^5$ relative to $U_m$, but one cannot in general conclude the existence of a smooth isotopy in $D^5$ fixing the boundary."
        },
        {
          "label": "C",
          "text": "The two collections $D_m^0$ and $D_m^1$ become smoothly ambiently homeomorphic in $D^5$ by a self-diffeomorphism of $D^5$ that restricts to the identity on $U_m$."
        },
        {
          "label": "D",
          "text": "If $m=1$, then $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; for $m\neq 1$, this need not hold for arbitrary spanning 3-disc collections of $U_m$."
        },
        {
          "label": "E",
          "text": "There exists a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ after possibly precomposing one collection with a diffeomorphism of $U_m$ that permutes the components, but not necessarily relative to the fixed boundary $U_m$ componentwise."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "smooth-vs-topological conclusion in one-higher dimension",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "isotopy rel. boundary weakened to boundary-fixing ambient equivalence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "arbitrary free-group rank / all m",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "relative-to-boundary requirement",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 1,
        "justification": "The stem does not state the answer outright, but it strongly cues that the relevant conclusion will involve passing from $S^4$ into $D^5$, which narrows the field toward A/C/E."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-identification item: the correct option is essentially the precise statement to be recognized, though the alternatives do introduce meaningful nearby variants rather than mere paraphrases."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants—$S^4$ vs. $D^5$, relative-boundary vs. ambient isotopy, smooth vs. topological, and all $m$ vs. only $m=1$—but it mainly tests recognition of the right formulation rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are mathematically close to the target statement, reflect common confusions about dimension shift, weakening/strengthening conclusions, and smooth vs. topological isotopy, and are clearly distinct from one another."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-designed theorem-discrimination MCQ with strong distractors, but it is still somewhat close to a restatement/recognition task rather than a deeply generative reasoning question."
    }
  },
  {
    "id": "2512.05952v1",
    "paper_link": "http://arxiv.org/abs/2512.05952v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thmx",
      "content": "\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.",
      "start_pos": 6741,
      "end_pos": 6981,
      "label": "thm"
    },
    "ref_dict": {
      "thm": "\\begin{thmx}\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.\n\\end{thmx}"
    },
    "pre_theorem_intro_text_len": 462,
    "pre_theorem_intro_text": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.",
    "context": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.",
    "full_context": "An \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.\n\n\\begin{document}\n\\title[Spanning 3-discs for the trivial 2-link]{Spanning 3-discs in the 4-sphere pushed into the 5-disc}\n\n\\begin{abstract}\nI prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^4$ become smoothly isotopic rel.\\ boundary after pushing them into $D^5$.\n\\end{abstract}\n\\maketitle\n\nAn \\emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\sqcup^m S^2$.\nGiven an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \\subseteq S^4$, with $D_m \\cong \\sqcup^m D^3$ and $\\partial D_m = L$, is called a \\emph{spanning 3-disc collection} for $L$.\nIf $L$ admits a spanning 3-disc collection, I say that $L$ is \\emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \\subseteq S^4$.\n\n\\begin{remark}\n In the case that $m=1$, i.e.\\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \\cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\\cite{HKM}. The $m=1$ case of the proof of \\cref{thm} gives another proof that is more direct. The case of multiple connected components is new. I deduce it from an application of surgery theory.\n\\end{remark}\n\nSince the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.\nBudney--Gabai~\\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\\ boundary in $S^4$, but which become isotopic in $D^5$.\nThe $m=1$ case of \\cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.\n\nAlison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \\geq 2$, that are not isotopic rel.\\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.\nIn his PhD thesis, Weizhe Niu~\\cite{Niu} independently developed alternative methods, extending the computations from~\\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.\nThese new collections can be seen directly to be standard in $D^5$; \\cref{thm} shows that this must always be the case.\n\nIf I allow trivial surface links in $S^4$ with positive genus, there is no analogous result. Hughes--Kim--Miller~\\cite{HKM} showed that there are pairs of genus $g \\geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\\ boundary, and remain non-isotopic rel.\\ boundary after they are pushed into $D^5$. The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.\nAssuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\\ boundary in $D^5$.\n\nI can now consider $f_i \\colon W_i \\to \\natural^m S^1 \\times D^4$ as an element of the\nBrowder-Novikov-Sullivan-Wall relative surgery structure set~\\cite{W}, denoted $\\mathcal{S}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3)$, which sits in the exact sequence (of abelian groups, in this case, since $\\natural^m S^1 \\times D^4 \\cong D^2 \\times \\natural^m S^1 \\times D^2$):\n\\begin{equation}\\label{eqn:ses}\nL_6(\\Z[F_m]) \\to \\mathcal{S}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3) \\to \\mathcal{N}(\\natural^m S^1 \\times D^4, \\#^m S^1 \\times S^3) \\to L_5(\\Z[F_m]).\n\\end{equation}\nNote that since the Whitehead group of $F_m$ is trivial~\\cite{St1}, I can ignore $h$ and $s$ decorations, and every $h$-cobordism is an $s$-cobordism. Let $\\wt{L}_n(\\Z[\\pi])$ denote the reduced $L$-theory of $\\Z[\\pi]$, such that $L_n(\\Z[\\pi]) \\cong L_n(\\Z) \\oplus \\wt{L}_n(\\Z[\\pi])$.\nRecall that $L_n(\\Z)=0$ for $n$ odd, and for $k \\geq 0$ one has $L_{4k}(\\Z) \\cong \\Z$ and $L_{4k+2}(\\Z) \\cong \\Z/2$.\nI will use Cappell~\\cite[Corollary~6]{Ca1,Ca2} and Shaneson's~\\cite{Sh} calculations\n\\begin{align*}\n L_5(\\Z[F_m]) &\\cong L_5(\\Z) \\oplus \\bigoplus^m \\wt{L}_5(\\Z[\\Z]) \\cong \\oplus^m L_4(\\Z) \\cong \\oplus^m \\Z; \\\\\nL_6(\\Z[F_m]) &\\cong L_6(\\Z) \\oplus \\bigoplus^m \\wt{L}_6(\\Z[\\Z]) \\cong L_6(\\Z) \\oplus \\bigoplus^m L_5(\\Z) \\cong \\Z/2.\n\\end{align*}\n\n\\begin{thmx}\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.\n\\end{thmx}",
    "post_theorem_intro_text_len": 3357,
    "post_theorem_intro_text": "\\begin{remark}\n   In the case that $m=1$, i.e.\\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \\cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\\cite{HKM}.  The $m=1$ case of the proof of \\cref{thm} gives another proof that is more direct.  The case of multiple connected components is new. I deduce it from an application of surgery theory.\n\\end{remark}\n\nSince the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.\nBudney--Gabai~\\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\\ boundary in $S^4$, but which become isotopic in $D^5$.\nThe $m=1$ case of \\cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.\n\nAlison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \\geq 2$, that are not isotopic rel.\\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.\nIn his PhD thesis, Weizhe Niu~\\cite{Niu} independently developed alternative methods, extending the computations from~\\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.\nThese new collections can be seen directly to be standard in $D^5$; \\cref{thm} shows that this must always be the case.\n\nIf I allow trivial surface links in $S^4$ with positive genus, there is no analogous result.  Hughes--Kim--Miller~\\cite{HKM} showed that there are pairs of genus $g \\geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\\ boundary, and remain non-isotopic rel.\\ boundary after they are pushed into $D^5$.  The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.\nAssuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\\ boundary in $D^5$.\n\nThus, Theorem~\\ref{thm} is optimal: for the isotopy uniqueness result it espouses to hold, one must allow pushing the spanning 3-discs into the 5-disc, and one cannot increase the topological complexity beyond spherical 2-links.\n\n\\begin{remark}\n  The proof I give of \\cref{thm} is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary, which was proven in my work with Conway~\\cite{CP}. Since \\cref{thm} concerns one dimension up, the result holds in the smooth category. In addition, I can work here with free groups of arbitrary rank, whereas in the 4-dimensional case we needed to work in a setting where the fundamental group was good.\n\\end{remark}\n\n\\subsubsection*{Acknowledgements}\n\nThanks to Seungwon Kim and Maggie Miller for suggesting this question, and to both Anthony Conway and the anonymous referee for helpful comments on previous drafts.  I am very grateful to Sarah Blackwell and Alison Tatsuoka for informing me about potential applications and comparisons.\nI was partially supported by EPSRC grants EP/T028335/1 and  EP/V04821X/1.",
    "sketch": "The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is \"deduce[d] ... from an application of surgery theory,\" and later that \"The proof I give of \\cref{thm} is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary\" from \\cite{CP}. It is also noted that \"Since \\cref{thm} concerns one dimension up, the result holds in the smooth category,\" and that the argument can be carried out \"with free groups of arbitrary rank\" (contrasting with the 4-dimensional case where the fundamental group needed to be good).",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is \"deduce[d] ... from an application of surgery theory,\" and later that \"The proof I give of To prove the main theorem, … is analogous to the proof that slice discs $D^2 \\subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\\ boundary\" from \\cite{CP}. It is also noted that \"Since To prove the main theorem, … concerns one dimension up, the result holds in the smooth category,\" and that the argument can be carried out \"with free groups of arbitrary rank\" (contrasting with the 4-dimensional case where the fundamental group needed to be good).,",
    "expanded_theorem": "\\label{thm}\n  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \\subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.",
    "theorem_type": [
      "Universal",
      "Existence"
    ],
    "mcq": {
      "question": "An $m$-component 2-link is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\\bigsqcup^m S^2$. For an $m$-component 2-link $L$, a spanning 3-disc collection for $L$ is a smoothly embedded submanifold $D_m\\subseteq S^4$ with $D_m\\cong \\bigsqcup^m D^3$ and $\\partial D_m=L$. Fix a standard trivial $m$-component 2-link $U_m\\subseteq S^4$, where “trivial” means that such a spanning 3-disc collection exists. If $D_m^0$ and $D_m^1$ are any two spanning 3-disc collections for $U_m$ in $S^4$, and we view them inside $D^5$ via the inclusion $S^4\\subseteq D^5$, which statement holds for every such pair?",
      "correct_choice": {
        "label": "A",
        "text": "The two collections $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; that is, there is a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ that keeps the common boundary $U_m$ fixed."
      },
      "choices": [
        {
          "label": "B",
          "text": "The two collections $D_m^0$ and $D_m^1$ become topologically isotopic in $D^5$ relative to $U_m$, but one cannot in general conclude the existence of a smooth isotopy in $D^5$ fixing the boundary."
        },
        {
          "label": "C",
          "text": "The two collections $D_m^0$ and $D_m^1$ become smoothly ambiently homeomorphic in $D^5$ by a self-diffeomorphism of $D^5$ that restricts to the identity on $U_m$."
        },
        {
          "label": "D",
          "text": "If $m=1$, then $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; for $m\neq 1$, this need not hold for arbitrary spanning 3-disc collections of $U_m$."
        },
        {
          "label": "E",
          "text": "There exists a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ after possibly precomposing one collection with a diffeomorphism of $U_m$ that permutes the components, but not necessarily relative to the fixed boundary $U_m$ componentwise."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "smooth-vs-topological conclusion in one-higher dimension",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "isotopy rel. boundary weakened to boundary-fixing ambient equivalence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "arbitrary free-group rank / all m",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "relative-to-boundary requirement",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and hypotheses but does not explicitly reveal that the spanning 3-disc collections are smoothly isotopic in D^5 relative to the boundary. There is no direct textual leakage of choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the correct option states the target conclusion in nearly theorem form, rather than asking the student to derive a less explicitly packaged consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the strongest smooth relative-isotopy statement from weaker or altered variants, but the item primarily tests recognition/recall of a specific result rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and distinct: they vary smooth vs. topological, isotopy vs. ambient equivalence, all m vs. only m=1, and relative boundary fixation vs. permutation ambiguity. These reflect realistic confusion points."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no obvious answer leakage, but it is close to a direct restatement of a specific theorem and thus only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.06527v2",
    "paper_link": "http://arxiv.org/abs/2512.06527v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{mainthm}\nSuppose that $\\Sigma$ is a smooth, projective, geometrically connected curve of genus $g$ defined over $k=\\mathbb{R}$, and suppose that $\\Sigma(\\mathbb{R})$ is a union of $b+1$ circles, with $b\\geq 0$. Then \n$$P_t(M_r^d)  =  (-t)^{(g-1)r^2+1} A_{g,r,d} (-t, -t^{\\frac{1}{2}}, -t^{\\frac{1}{2}},  ..., -t^{\\frac{1}{2}}, -1,-1,..,-1) $$ \nwhere  $(g-b)$-many entries equal $-t^{\\frac{1}{2}}$ and $b$-many equal $-1$.",
      "start_pos": 8628,
      "end_pos": 9062,
      "label": "mainthm"
    },
    "ref_dict": {
      "mainthm": "\\begin{thm}\\label{mainthm}\nSuppose that $\\Sigma$ is a smooth, projective, geometrically connected curve of genus $g$ defined over $k=\\R$, and suppose that $\\Sigma(\\R)$ is a union of $b+1$ circles, with $b\\geq 0$. Then \n$$P_t(M_r^d)  =  (-t)^{(g-1)r^2+1} A_{g,r,d} (-t, -t^{\\frac{1}{2}}, -t^{\\frac{1}{2}},  ..., -t^{\\frac{1}{2}}, -1,-1,..,-1) $$ \nwhere  $(g-b)$-many entries equal $-t^{\\frac{1}{2}}$ and $b$-many equal $-1$. \n\\end{thm}",
      "Betti numbers of": "\\begin{proof}\nThe homomorphism  $$P_t^{vir}: K_0(Var_\\R) \\rightarrow \\Z[t] \\subset \\Z((t^{-1}))$$  is constructed by McCrory and Parusinski \\cite{MP}.   Since $P_t^{vir}(\\mathbb{L} ) = -t$   and $P_t^{vir}(\\mathbb{L}^n -1)) = (-t)^n-1$ are both invertible in $\\Z((t^{-1}))$, with $((-t)^n-1)^{-1} = (-t)^{-n} (\\sum_{k=0}^{\\infty} (-t)^{-kn})$, this extends to a homomorphism  $$P_t^{vir}: Mot(\\R) \\rightarrow   \\Z((t^{-1})),$$  \nwhich extends naturally to the dimensional completion.\n\\end{proof}\n\nThe zeta function of a variety $X$ is given by the formal power series\n$$Z_X(z)  := \\sum_{n=0}^\\infty  [X^{(n)} ]z^n  \\in  Mot(k) [[z]],  $$\nwhere $X^{(n)}  = X^n/S_n $ denotes the $n$-fold symmetric product of $X$.  The zeta function is multiplicative in the sense that for closed $Y \\subseteq X$,  $$ Z(X, z) = Z(Y,z) Z(X\\setminus Y, z).$$ \n\nIf $\\Sigma/k$ is a smooth projective curve of genus $g$ and $\\Sigma(k) \\neq \\emptyset$, then a theorem of Kapranov (\\cite{K}  Theorem 1.1.9) says that\n\n$$ Z_\\Sigma(z) =  \\frac{P(z)}{(1-z)(1-\\mathbb{L}z)} $$ \nwhere $P(z)$ is a polynomial of degree $2g$ such that  $Z_\\Sigma ( 1/\\mathbb{L}z) = \\mathbb{L}^{1-g} x^{2-2g} Z_\\Sigma(z)$. The homomorphism (\\ref{evsigma}) is determined by the property that \n$ \\prod_{i=1}^g (1-\\alpha_i z)(1-\\alpha_i^{-1}qz)$ is sent to $P(z)$ in some algebraic extension of $\\overline{Mot}(k)$  \n\n\\section{Betti numbers of $\\Sigma^{(n)}(\\R)$}\\label{Betti numbers of}\n\nSuppose that $\\Sigma$ is a smooth, geometrically connected, projective curve of genus $g$ over $\\R$. For $n \\geq 0$ denote by $\\Sigma^{(n)}$ its $n$-th symmetric product. The set of complex points $\\Sigma^{(n)}(\\C)$ can be identified with the orbit space $$ Sym^n( \\Sigma(\\C)) := \\Sigma(\\C)^n/ S_n, $$  while $\\Sigma^{(n)}(\\R) \\subset \\Sigma^{(n)}(\\C)$ can be identified with $Gal(\\C/\\R)$ fixed points. \n\nIf $\\Sigma(\\R)$ is empty, then  \n$$ \\Sigma^{(n)}(\\R) \\cong  \\begin{cases}  \\emptyset  &  \\text{if $n$ is odd} \\\\  Sym^{n/2}(N_g) & \\text{if $n$ is even} \\end{cases}$$\nwhere $N_g$ is a non-orientable surface double covered by $\\Sigma(\\C)$.  We abusively write $ P_t(X)  :=  \\sum_{k}  (-t)^k H_{cpt}^k(X;\\Z_2)$.  By Macdonald's formula \\cite{Mac},\n\\begin{equation}\\label{MacDs}\n \\sum_{m=0}^\\infty P_t( Sym^{m}(N_g)) z^m = \\frac{ (1-tz)^{g+1} }{(1-z)(1-t^2z)},\n \\end{equation}\n so\n\n$$ P_t^{vir}(Z_\\Sigma (z)) := \\sum_{n=0}^{\\infty} P_t ( \\Sigma^{(n)} ) z^n =  \\frac{ (1-tz^2)^{g+1} }{(1-z^2)(1-t^2z^2)}.$$\n\nWhen $\\Sigma(\\R) \\neq \\emptyset$, $P_t(\\Sigma^{(n)})$ was calculated in \\cite{B}.  \n\n\\begin{prop}\nIf $\\Sigma(\\R)$ is non-empty, with $b+1$ many connected components, then\n\\begin{equation}\\label{zetahom}\n P_t^{vir}(Z_\\Sigma (z)) =   \\sum_{n=0}^{\\infty} P_t ( \\Sigma^{(n)}) z^n   =  \\frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .\n \\end{equation}\n\\end{prop}\n\n\\begin{proof}\n\nSuppose first that $ \\Sigma(\\C) \\setminus \\Sigma(\\R)$ is connected.  For each pair of integers $k,s \\geq 0$ such that $k+2s = n$, $\\Sigma^{(n)}(\\R)$ has $ { b+1 \\choose k}$ connected components with Poincar\\'e polynomial isomorphic to $(1-t)^k P_t( Sym^s(N_{g-k}))$.  Therefore, setting $a=b+1$, we have \n\n\\begin{eqnarray*}\n\\sum_{n=0}^{\\infty}  P_t(\\Sigma^{(n)}) z^m  &=&  \\sum_{0 \\leq k \\leq a} \\sum_{s \\geq 0}   z^{k+2s} { a \\choose k}  (1-t)^k  P_t(  Sym^s(N_{g-k}))\\\\\n&\\stackrel{(\\ref{MacDs})}{=}&  \\sum_{0 \\leq k \\leq a}   z^{k} { a \\choose k}   (1-t)^k \\frac{(1-tz^2)^{g-k+1}}{(1-z^2)(1-t^2z^2)}\\\\\n&=& \\left(  (z -zt) + (1-tz^2) \\right)^a  \\frac{(1-tz^2)^{g-a+1}}{(1-z^2)(1-t^2z^2)} \\\\\n&=&  \\frac{(1-tz^2)^{g-a+1} (1+z)^{a-1} (1-tz)^{a-1}}{(1-z)(1+tz)} \\\\\n\\end{eqnarray*}\n\nIn case $\\Sigma(\\C) \\setminus \\Sigma(\\R)$ is not connected, the calculation is the same except that whenever $k=a$, the corresponding component has Poincar\\'e polynomial $(1-t)^{b+1}P_t(Sym^s(\\Sigma_{(g-b)/2})) $, where $\\Sigma_g$ is an orientable surface of genus $g$. But this equals $(1-t)^{b+1} P_t( Sym^s( N_{g-b-1}))$  (\\cite{B} Remark 3.14) so the same formula holds.\n\\end{proof}",
      "Average Betti numbers": "\\begin{proof}\n\nUnder these conditions $ X^{\\C^\\times} = \\coprod_{i \\in I} F_i$  where the $F_i$ are non-singular, projective over $\\R$, and we have a Bialycki-Birula stratification $ X = \\bigcup_{i \\in I} U_i$ where $$U_i(\\C) :=  \\{x \\in X(\\C) | lim_{\\lambda \\mapsto 0} \\lambda x  \\in F_i \\} ,$$ and each $U_i$ is a locally closed subvarity of $X$, isomorphic to a vector bundle of some rank $d_i$ over $F_i$. It follows that\n\n$$ P_t^{vir}(X) = \\sum_{i\\in I} P_t^{vir}(U_i) =   \\sum_{i\\in I} P_t^{vir}( F_i) P_t^{vir}(\\mathbb{A}^{d_i}) =  \\sum_{i\\in I} P_t(F_i) (-t)^{d_i}.$$\n\nOn the other hand, by a result essentially due to Duistermaat \\cite{D}  (see also \\cite{BGH}), we know that $ P_t( X) = \\sum_i  P_t (U_i) $,  and $U_i(\\R)$ deformation retracts onto $F_i(\\R)$ and has relative dimension $d_i$, so by Poincar\\'e duality,  $P_t(U_i)=P_t(F_i) (-t)^{d_i}$, completing the proof.\n\\end{proof}\n\n\\section{Average Betti numbers}\\label{Average Betti numbers}\n\nOur goal in this section is to prove the following somewhat surprising result.\n\n\\begin{prop}\\label{avgbet}\nSuppose that $\\Sigma$ is a smooth projective, geometrically connected curve of genus $g$ defined over $\\R$,  and $\\Sigma(\\R)$ has $b+1 >0$ connected components.\nThen the Poincar\\'e polynomial $P_t( M_r^d)$ is divisible by $2^b (1-t)^g$.\n\\end{prop}\n\n\\begin{proof}\nRecall (\\ref{Plethlog}):\n\\begin{equation}\n \\sum_{r=1}^\\infty H_{g,r} T^r = (z-1)(1-q) \\operatorname{Log} \\sum_{\\mu \\in \\mathcal{P}} T^{|\\mu|} \\mathcal{H}_\\mu.\n\\end{equation}\n\nEvery non-empty Young diagram $\\mu$ includes at least one box with both arm and leg length zero, so $\\mathcal{H}_\\mu$ contains a factor of $ \\prod_{i=1}^g (z - \\alpha_i )(1 - \\alpha_i^{-1} q)$ in its numerator. Using the combinatorics of the plethystic logarithm  (see e.g.  \\cite{HRV08} Theorem 3.5.2.), it follows that every $H_{g,r}$ is divisible by $ \\prod_{i=1}^g (z - \\alpha_i )(1 - \\alpha_i^{-1} q)$ in the localized ring $S^{-1} \\Q[ q, z, \\alpha_1^{\\pm 1},...,\\alpha_g^{\\pm1}] $ where $S$ is the multiplicative set generated \nby  $\\{  (z^{a+1} - q^{l} ),  (z^{a}-q^{l+1})|  a,b \\geq 0\\}$. We know\n$$A_{g,r} (q, \\alpha_1,..., \\alpha_g) = H_{g,r}(q,1,\\alpha_1,...,\\alpha_g) \\in \\Z[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$$ \nso by Gauss' Lemma $\\prod_{i=1}^g (1-\\alpha_i) (1-\\alpha_i^{-1}q)$ divides $  A_{g,r}$ in $ \\Z[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$ .  By Theorem \\ref{mainthm},   $P_t( M_r^d) = A_{g,r}(-t, -t^{\\frac{1}{2}},...,-1,...) $  is divisible by   \n$$2^b(1-t)^g = (1+t^{\\frac{1}{2}})^{g-b}  (1-t^{\\frac{1}{2}})^{g-b} (1+1)^b (1-t)^b $$ \n in $\\Z[t^{\\pm \\frac{1}{2}}]$, hence also in $ \\Z[t]$. \n \\end{proof}",
      "On purity of real": "\\begin{proof}\nSubstituting  $q = -t$,  $\\alpha_1=...= \\alpha_{g-b} = -t^{\\frac{1}{2}}$ and $\\alpha_{g-b+1}= ... =\\alpha_g = -1$ into the formal zeta function $$\\frac{ \\prod_{i=1}^g (1-\\alpha_i z)(1-\\alpha_i^{-1}qz)}{(1-z)(1-qz)} $$\nyields (\\ref{zetahom}). Apply (\\ref{evfor}).\n\\end{proof}\n\n\\section{On purity of real semi-projective varieties}\\label{On purity of real}\n\nA semi-projective variety $Y$ over $\\C$ is a smooth, quasiprojective variety equipped with $\\C^\\times$-action such that the fixed point set $Y^{\\C^\\times}$ is proper and for ever $y \\in Y$ the limit $lim_{\\lambda \\rightarrow 0} \\lambda y$ exists as $\\lambda \\in \\C^\\times$ tends to zero.  Mixed Hodge structures of such varieties were studied in Hausel and Rodriguez-Villegas \\cite{HRV13} .\n\nA semi-projective variety $X$ over $\\R$ is a real variety equipped with an action of $\\R^\\times$ such that the base change  $X \\times_R \\C$ is semi-projective over $\\C$.  When $\\Sigma$ is defined over $\\R$,  $M_r^d$ is a semi-projective variety, with respect the $\\C^\\times$ action of scaling the Higgs field.\n\n\\begin{prop}\nIf $X$ is semi-projective over $\\R$ then\n\n$$P_t^{vir} (X) = P_t (X).$$\n\\end{prop}\n\n\\begin{proof}\n\nUnder these conditions $ X^{\\C^\\times} = \\coprod_{i \\in I} F_i$  where the $F_i$ are non-singular, projective over $\\R$, and we have a Bialycki-Birula stratification $ X = \\bigcup_{i \\in I} U_i$ where $$U_i(\\C) :=  \\{x \\in X(\\C) | lim_{\\lambda \\mapsto 0} \\lambda x  \\in F_i \\} ,$$ and each $U_i$ is a locally closed subvarity of $X$, isomorphic to a vector bundle of some rank $d_i$ over $F_i$. It follows that\n\n$$ P_t^{vir}(X) = \\sum_{i\\in I} P_t^{vir}(U_i) =   \\sum_{i\\in I} P_t^{vir}( F_i) P_t^{vir}(\\mathbb{A}^{d_i}) =  \\sum_{i\\in I} P_t(F_i) (-t)^{d_i}.$$\n\nOn the other hand, by a result essentially due to Duistermaat \\cite{D}  (see also \\cite{BGH}), we know that $ P_t( X) = \\sum_i  P_t (U_i) $,  and $U_i(\\R)$ deformation retracts onto $F_i(\\R)$ and has relative dimension $d_i$, so by Poincar\\'e duality,  $P_t(U_i)=P_t(F_i) (-t)^{d_i}$, completing the proof.\n\\end{proof}"
    },
    "pre_theorem_intro_text_len": 1586,
    "pre_theorem_intro_text": "Let $\\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\\Sigma$. When $gcd(d,r)=1$, which we assume henceforth,  $M_r^d$ is a smooth, quasi-projective variety.  \n\nFor $g,r,d \\in \\mathbb{Z}$ and $g, r \\geq 0$, Schiffmann \\cite{S} defines a Laurent polynomial\n$$  A_{g,r,d} (q, \\alpha_1,..., \\alpha_g)  \\in \\mathbb{Z}[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$$\nwhich is symmetric under permutations of the $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$. If $k = \\mathbb{F}_q$ and $\\{ \\alpha_i, \\alpha_i^{-1}q|i=1,..,g\\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\\Sigma, \\Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\\Sigma$,  and  $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \\mathbb{C}$,  the Poincar\\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\\mathbb{C})$ equals\n\n$$   \\sum_k  (-1)^kt^k \\dim_\\mathbb{Q}  H^k_{cpt}(M_r^d(\\mathbb{C});\\mathbb{Q}) = t^{2(g-1)r^2+2}A_{g,r,d} (t^2, t,t,t,...,t,t)    $$ \n\nOur main result is that a similar formula calculates $\\Z_2$ Betti numbers of $M_r^d(\\mathbb{R})$ when $k =\\mathbb{R}$. For a variety $X/\\mathbb{R}$, denote its compactly supported Poincar\\'e polynomial by\n $$P_t( X) :=   \\sum_k  (-1)^k t^k \\dim_{\\Z_2} H^k_{cpt}(M_r^d(\\mathbb{R}); \\Z_2).$$",
    "context": "Let $\\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\\Sigma$. When $gcd(d,r)=1$, which we assume henceforth, $M_r^d$ is a smooth, quasi-projective variety.\n\nFor $g,r,d \\in \\mathbb{Z}$ and $g, r \\geq 0$, Schiffmann \\cite{S} defines a Laurent polynomial\n$$ A_{g,r,d} (q, \\alpha_1,..., \\alpha_g) \\in \\mathbb{Z}[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$$\nwhich is symmetric under permutations of the $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$. If $k = \\mathbb{F}_q$ and $\\{ \\alpha_i, \\alpha_i^{-1}q|i=1,..,g\\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\\Sigma, \\Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\\Sigma$, and $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \\mathbb{C}$, the Poincar\\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\\mathbb{C})$ equals\n\n$$ \\sum_k (-1)^kt^k \\dim_\\mathbb{Q} H^k_{cpt}(M_r^d(\\mathbb{C});\\mathbb{Q}) = t^{2(g-1)r^2+2}A_{g,r,d} (t^2, t,t,t,...,t,t) $$\n\nOur main result is that a similar formula calculates $\\Z_2$ Betti numbers of $M_r^d(\\mathbb{R})$ when $k =\\mathbb{R}$. For a variety $X/\\mathbb{R}$, denote its compactly supported Poincar\\'e polynomial by\n $$P_t( X) := \\sum_k (-1)^k t^k \\dim_{\\Z_2} H^k_{cpt}(M_r^d(\\mathbb{R}); \\Z_2).$$",
    "full_context": "Let $\\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\\Sigma$. When $gcd(d,r)=1$, which we assume henceforth, $M_r^d$ is a smooth, quasi-projective variety.\n\nFor $g,r,d \\in \\mathbb{Z}$ and $g, r \\geq 0$, Schiffmann \\cite{S} defines a Laurent polynomial\n$$ A_{g,r,d} (q, \\alpha_1,..., \\alpha_g) \\in \\mathbb{Z}[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$$\nwhich is symmetric under permutations of the $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$. If $k = \\mathbb{F}_q$ and $\\{ \\alpha_i, \\alpha_i^{-1}q|i=1,..,g\\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\\Sigma, \\Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\\Sigma$, and $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \\mathbb{C}$, the Poincar\\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\\mathbb{C})$ equals\n\n$$ \\sum_k (-1)^kt^k \\dim_\\mathbb{Q} H^k_{cpt}(M_r^d(\\mathbb{C});\\mathbb{Q}) = t^{2(g-1)r^2+2}A_{g,r,d} (t^2, t,t,t,...,t,t) $$\n\nOur main result is that a similar formula calculates $\\Z_2$ Betti numbers of $M_r^d(\\mathbb{R})$ when $k =\\mathbb{R}$. For a variety $X/\\mathbb{R}$, denote its compactly supported Poincar\\'e polynomial by\n $$P_t( X) := \\sum_k (-1)^k t^k \\dim_{\\Z_2} H^k_{cpt}(M_r^d(\\mathbb{R}); \\Z_2).$$\n\nLet $\\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\\Sigma$. When $gcd(d,r)=1$, which we assume henceforth, $M_r^d$ is a smooth, quasi-projective variety.\n\nFor $g,r,d \\in \\Z$ and $g, r \\geq 0$, Schiffmann \\cite{S} defines a Laurent polynomial\n$$ A_{g,r,d} (q, \\alpha_1,..., \\alpha_g) \\in \\Z[q, \\alpha_1^{\\pm 1},..., \\alpha_g^{\\pm 1} ]$$\nwhich is symmetric under permutations of the $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$. If $k = \\mathbb{F}_q$ and $\\{ \\alpha_i, \\alpha_i^{-1}q|i=1,..,g\\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\\Sigma, \\Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\\Sigma$, and $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \\C$, the Poincar\\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\\C)$ equals\n\nOur main result is that a similar formula calculates $\\Z_2$ Betti numbers of $M_r^d(\\R)$ when $k =\\R$. For a variety $X/\\R$, denote its compactly supported Poincar\\'e polynomial by\n $$P_t( X) := \\sum_k (-1)^k t^k \\dim_{\\Z_2} H^k_{cpt}(M_r^d(\\R); \\Z_2).$$\n\nFor example, for $r=2$, we obtain the explicit formula $$P_{t}(M_2^1) = 2^b t^{4(g-1)+1}(1-t)^g f(t^{1/2},1)$$ where\n\n\\begin{equation}\\label{evsigma}\n ev_\\Sigma : R_g \\rightarrow \\overline{Mot}(k),\n \\end{equation} where $R_g$ is the ring of Laurent polynomials in $q, \\alpha_1^{\\pm 1},...,\\alpha_g^{\\pm 1}$ which are invariant under permutations of $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$. This $ev_\\Sigma$ sends $q$ to the Tate class $\\mathbb{L}$ and the extension to $R_g [[z]] \\rightarrow \\overline{Mot}(k)[[z]]$ sends the formal zeta function of the curve to the motivic zeta function of $\\Sigma$ :\n\nSuppose that $\\Sigma$ is a smooth, geometrically connected, projective curve of genus $g$ over $\\R$. For $n \\geq 0$ denote by $\\Sigma^{(n)}$ its $n$-th symmetric product. The set of complex points $\\Sigma^{(n)}(\\C)$ can be identified with the orbit space $$ Sym^n( \\Sigma(\\C)) := \\Sigma(\\C)^n/ S_n, $$ while $\\Sigma^{(n)}(\\R) \\subset \\Sigma^{(n)}(\\C)$ can be identified with $Gal(\\C/\\R)$ fixed points.\n\n\\begin{prop}\nIf $\\Sigma(\\R)$ is non-empty, with $b+1$ many connected components, then\n\\begin{equation}\\label{zetahom}\n P_t^{vir}(Z_\\Sigma (z)) = \\sum_{n=0}^{\\infty} P_t ( \\Sigma^{(n)}) z^n = \\frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .\n \\end{equation}\n\\end{prop}\n\n\\begin{prop}\\label{avgbet}\nSuppose that $\\Sigma$ is a smooth projective, geometrically connected curve of genus $g$ defined over $\\R$, and $\\Sigma(\\R)$ has $b+1 >0$ connected components.\nThen the Poincar\\'e polynomial $P_t( M_r^d)$ is divisible by $2^b (1-t)^g$.\n\\end{prop}",
    "post_theorem_intro_text_len": 4359,
    "post_theorem_intro_text": "For example, for $r=2$, we obtain the explicit formula $$P_{t}(M_2^1) =  2^b t^{4(g-1)+1}(1-t)^g f(t^{1/2},1)$$ where\n\n\\begin{equation*}\n\\begin{split}\nf(t^{\\frac{1}{2}},z) &=  \\frac{(z^2+t^{\\frac{1}{2}})^{g-b} (z-t^{\\frac{1}{2}})^{g-b} (z^2+1)^b(z-t)^b}{(z^2-1)(z+t)} + \\frac{(z-t^{\\frac{3}{2}})^{g-b} (1+t^{\\frac{3}{2}})^{g-b} (z-t)^b(1+t^2)^b}{(z+t)(1-t^2)}\\\\\n  &- \\frac{(z+t^{\\frac{1}{2}})^{g-b} (1-t^{\\frac{1}{2}})^{g-b} (z+1)^b(1-t)^b}{2(z-1)(1+t)} - \\frac{(z-t^{\\frac{1}{2}})^{g-b} (1+t^{\\frac{1}{2}})^{g-b} (z-1)^b(1+t)^b}{2(z+1)(1-t)}. \n\\end{split}\n\\end{equation*}\nA different formula for $P_t(M_2^1)$ was obtained in \\cite{B}.  Formulas for $P_t(M_r^d)$ when $r\\geq 3$ are new. \n\nWe use a construction of $A_{g,r,d}$ due to Mellit \\cite{M}.  Let $\\mathcal{P}$ denote the set of Young diagrams.  Given $\\mu \\in \\mathcal{P}$, define the rational function $\\mathcal{H}_\\mu$ in variables $ q, z, \\alpha_1,..., \\alpha_g$,\n\n$$\\mathcal{H}_\\mu : =  \\prod_{\\square \\in \\mu}  \\frac{\\prod_{i=1}^g (z^{a(\\square)+1} - \\alpha_i q^{l(\\square)})(z^{a(\\square)} - \\alpha_i^{-1} q^{l(\\square)+1})}{(z^{a(\\square)+1} - q^{l(\\square)})(z^{a(\\square)} - q^{l(\\square)+1})}$$\nwhere $a(\\square)$ and $l(\\square)$ denote the arm and leg lengths of boxes in the diagram.  For $g \\geq 1$, define rational functions $H_{g,r}$ by\n\\begin{equation} \\label{Plethlog}\n \\sum_{r=1}^\\infty H_{g,r} T^r = (z-1)(1-q) \\operatorname{Log} \\sum_{\\mu \\in \\mathcal{P}} T^{|\\mu|} \\mathcal{H}_\\mu, \n\\end{equation}\nwhere $\\operatorname{Log}$ denotes the plethystic logarithm.  Mellit proved that for all $r \\geq 1$, $H_{g,r}$ is a Laurent polynomial in $q$, $z$, and $\\alpha_1, \\ldots, \\alpha_g$, and for all $d$,\n\\[\nA_{g,r,d}(q, \\alpha_1, \\ldots, \\alpha_g) = H_{g,r}(q, 1, \\alpha_1, \\ldots, \\alpha_g).\n\\]\n\nNote that $A_{g,r,d}$ is independent of $d$, so we denote it $A_{g,r}$ henceforth.   When applied to $k=\\mathbb{C}$, Mellit's result proved a conjectural formula of Hausel and Rodriguez-Villegas for the $\\mathbb{Q}$-Betti numbers of $M_r^d$.   Theorem \\ref{mainthm} can be thought of as a Hausel Rodriguez-Villegas style formula for the $\\Z_2$-Betti number of the moduli space of real Higgs bundles over a real curve.  \n\nOur proof of Theorem \\ref{mainthm} can be outlined as follows.  Fedorov-Soibelman-Soibelman  \\cite{FSS} proved a motivic version of Schiffmann's formula which, when $k$ is a field of characteristic zero, identifies the class $[M_r^d]$ in $\\overline{Mot}(k)$, the dimensional completion of the Grothendieck ring of Artin stacks over $k$.  Mellit (\\cite{M} \\S 6), identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$  under a homomorphism \n\n\\begin{equation}\\label{evsigma}\n ev_\\Sigma  :  R_g  \\rightarrow \\overline{Mot}(k),\n \\end{equation} where $R_g$ is the ring of Laurent polynomials in $q, \\alpha_1^{\\pm 1},...,\\alpha_g^{\\pm 1}$ which are invariant under permutations of $\\alpha_i$ and under $\\alpha_i \\mapsto q \\alpha_i^{-1}$.  This $ev_\\Sigma$ sends $q$ to the Tate class $\\mathbb{L}$ and the extension to $R_g [[z]] \\rightarrow \\overline{Mot}(k)[[z]]$ sends the formal zeta function of the curve to the motivic zeta function of $\\Sigma$ :\n\n\\begin{equation}\\label{evfor}\n ev_\\Sigma \\left( \\frac{ \\prod_{i=1}^g (1-\\alpha_i z)(1-\\alpha_i^{-1}qz)}{(1-z)(1-qz)}\\right) = \\sum_{n=0}^{\\infty} z^n  \\Sigma^{(n)} = Z_\\Sigma (z).\n \\end{equation}\n\nWhen $k= \\mathbb{R}$, one can define a virtual Poincar\\'e polynomial homomorphism $$P_t^{vir}:  \\overline{Mot}(\\mathbb{R}) \\rightarrow \\mathbb{Z}((t^{-1})) ,$$ which sends the class $[X]$ of a smooth, projective variety $X$ to the Poincar\\'e polynomial  $  P_t^{vir}([X]) = P_t(X(\\mathbb{R}))$.   \n\nTo calculate $P_t^{vir}( [M_r^d])$, it only remains  to determine the composition $P_t^{vir} \\circ ev_\\Sigma$, which boils down to calculating $P_t^{vir} ( Z_\\Sigma(z))$. We carry this out in \\S \\ref{Betti numbers of}.  Then in \\S \\ref{On purity of real} we prove that the virtual Poincar\\'e polynomial $M_r^d$ agrees with its topological Poincar\\'e polynomial, completing the proof of Theorem \\ref{mainthm}.  In section \\ref{Average Betti numbers} we show that the Poincar\\'e polynomial $P_t(M_r^d)$ is divisible by $P_t(Pic^0(\\Sigma))$, which has the surprising implication that average value of the Poincar\\'e polynomial of the connected components of $M_r^d$ is a polynomial with integer coefficients.",
    "sketch": "Our proof of Theorem~\\ref{mainthm} can be outlined as follows.\n\n- Fedorov--Soibelman--Soibelman \\cite{FSS} prove a motivic version of Schiffmann's formula which (in characteristic zero) \"identifies the class $[M_r^d]$ in $\\overline{Mot}(k)$,\" the dimensional completion of the Grothendieck ring of Artin stacks.\n\n- Mellit (\\cite{M} \\S 6) \"identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$ under\" the homomorphism\n  \\[\n    ev_\\Sigma: R_g \\to \\overline{Mot}(k),\n  \\]\n  where $R_g$ is the ring of Laurent polynomials in $q,\\alpha_1^{\\pm1},\\ldots,\\alpha_g^{\\pm1}$ invariant under permutations of the $\\alpha_i$ and under $\\alpha_i\\mapsto q\\alpha_i^{-1}$. This map sends $q$ to the Tate class $\\mathbb{L}$, and its extension to $R_g[[z]]$ sends the \"formal zeta function of the curve\" to the motivic zeta function $Z_\\Sigma(z)$ via\n  \\[\n  ev_\\Sigma\\left(\\frac{\\prod_{i=1}^g(1-\\alpha_i z)(1-\\alpha_i^{-1}qz)}{(1-z)(1-qz)}\\right)=\\sum_{n\\ge0} z^n\\Sigma^{(n)}=Z_\\Sigma(z).\n  \\]\n\n- For $k=\\mathbb{R}$, define the virtual Poincar\\'e polynomial homomorphism\n  \\[\n  P_t^{vir}:\\overline{Mot}(\\mathbb{R})\\to\\mathbb{Z}((t^{-1})),\n  \\]\n  which sends a smooth projective variety $X$ to $P_t^{vir}([X])=P_t(X(\\mathbb{R}))$.\n\n- \"To calculate $P_t^{vir}([M_r^d])$, it only remains to determine the composition $P_t^{vir}\\circ ev_\\Sigma$, which boils down to calculating $P_t^{vir}(Z_\\Sigma(z))$.\" This is carried out in \\S~\\ref{Betti numbers of}.\n\n- Then, in \\S~\\ref{On purity of real}, they \"prove that the virtual Poincar\\'e polynomial $M_r^d$ agrees with its topological Poincar\\'e polynomial, completing the proof of Theorem~\\ref{mainthm}.\"",
    "expanded_sketch": "Our proof of Theorem~\\ref{mainthm} can be outlined as follows.\n\n- Fedorov--Soibelman--Soibelman \\cite{FSS} prove a motivic version of Schiffmann's formula which (in characteristic zero) \"identifies the class $[M_r^d]$ in $\\overline{Mot}(k)$,\" the dimensional completion of the Grothendieck ring of Artin stacks.\n\n- Mellit (\\cite{M} \\S 6) \"identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$ under\" the homomorphism\n  \\[\n    ev_\\Sigma: R_g \\to \\overline{Mot}(k),\n  \\]\n  where $R_g$ is the ring of Laurent polynomials in $q,\\alpha_1^{\\pm1},\\ldots,\\alpha_g^{\\pm1}$ invariant under permutations of the $\\alpha_i$ and under $\\alpha_i\\mapsto q\\alpha_i^{-1}$. This map sends $q$ to the Tate class $\\mathbb{L}$, and its extension to $R_g[[z]]$ sends the \"formal zeta function of the curve\" to the motivic zeta function $Z_\\Sigma(z)$ via\n  \\[\n  ev_\\Sigma\\left(\\frac{\\prod_{i=1}^g(1-\\alpha_i z)(1-\\alpha_i^{-1}qz)}{(1-z)(1-qz)}\\right)=\\sum_{n\\ge0} z^n\\Sigma^{(n)}=Z_\\Sigma(z).\n  \\]\n\n- For $k=\\mathbb{R}$, define the virtual Poincar\\'e polynomial homomorphism\n  \\[\n  P_t^{vir}: \\overline{Mot}(\\mathbb{R})\\to\\mathbb{Z}((t^{-1})),\n  \\]\n  which sends a smooth projective variety $X$ to $P_t^{vir}([X])=P_t(X(\\mathbb{R}))$. More precisely, one constructs\n  \\[\n  P_t^{vir}: K_0(Var_\\R) \\rightarrow \\Z[t] \\subset \\Z((t^{-1}))\n  \\]\n  (McCrory and Parusinski \\cite{MP}). Since $P_t^{vir}(\\mathbb{L} ) = -t$ and $P_t^{vir}(\\mathbb{L}^n -1)) = (-t)^n-1$ are both invertible in $\\Z((t^{-1}))$, with $((-t)^n-1)^{-1} = (-t)^{-n} (\\sum_{k=0}^{\\infty} (-t)^{-kn})$, this extends to a homomorphism\n  \\[\n  P_t^{vir}: Mot(\\R) \\rightarrow   \\Z((t^{-1})),\n  \\]\n  which extends naturally to the dimensional completion.\n\n- To calculate $P_t^{vir}([M_r^d])$, it only remains to determine the composition $P_t^{vir}\\circ ev_\\Sigma$, which boils down to calculating $P_t^{vir}(Z_\\Sigma(z))$. Next we carry out this calculation as follows.\n\n  The zeta function of a variety $X$ is given by the formal power series\n  \\[\n  Z_X(z)  := \\sum_{n=0}^\\infty  [X^{(n)} ]z^n  \\in  Mot(k) [[z]],\n  \\]\n  where $X^{(n)}  = X^n/S_n $ denotes the $n$-fold symmetric product of $X$. The zeta function is multiplicative in the sense that for closed $Y \\subseteq X$,\n  \\[\n  Z(X, z) = Z(Y,z) Z(X\\setminus Y, z).\n  \\]\n\n  If $\\Sigma/k$ is a smooth projective curve of genus $g$ and $\\Sigma(k) \\neq \\emptyset$, then a theorem of Kapranov (\\cite{K}  Theorem 1.1.9) says that\n  \\[\n  Z_\\Sigma(z) =  \\frac{P(z)}{(1-z)(1-\\mathbb{L}z)}\n  \\]\n  where $P(z)$ is a polynomial of degree $2g$ such that $Z_\\Sigma ( 1/\\mathbb{L}z) = \\mathbb{L}^{1-g} x^{2-2g} Z_\\Sigma(z)$.\n\n  We then compute $P_t^{vir}(Z_\\Sigma(z))$ by analyzing the real loci $\\Sigma^{(n)}(\\R)$. In the case $\\Sigma(\\R)=\\emptyset$ one has\n  \\[\n  \\Sigma^{(n)}(\\R) \\cong  \\begin{cases}  \\emptyset  &  \\text{if $n$ is odd} \\\\  Sym^{n/2}(N_g) & \\text{if $n$ is even} \\end{cases}\n  \\]\n  (with $N_g$ a non-orientable surface double covered by $\\Sigma(\\C)$), and writing abusively $ P_t(X)  :=  \\sum_{k}  (-t)^k H_{cpt}^k(X;\\Z_2)$, Macdonald's formula (\\cite{Mac}) gives\n  \\begin{equation}\\label{MacDs}\n   \\sum_{m=0}^\\infty P_t( Sym^{m}(N_g)) z^m = \\frac{ (1-tz)^{g+1} }{(1-z)(1-t^2z)},\n   \\end{equation}\n  hence\n  \\[\n   P_t^{vir}(Z_\\Sigma (z)) := \\sum_{n=0}^{\\infty} P_t ( \\Sigma^{(n)} ) z^n =  \\frac{ (1-tz^2)^{g+1} }{(1-z^2)(1-t^2z^2)}.\n  \\]\n\n  When $\\Sigma(\\R) \\neq \\emptyset$, with $b+1$ connected components, one has the explicit identity\n  \\begin{equation}\\label{zetahom}\n   P_t^{vir}(Z_\\Sigma (z)) =   \\sum_{n=0}^{\\infty} P_t ( \\Sigma^{(n)}) z^n   =  \\frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .\n   \\end{equation}\n\n- Finally, we use purity for real semi-projective varieties. We first use the following proposition: if $X$ is semi-projective over $\\R$ then\n\n$$P_t^{vir} (X) = P_t (X).$$\n\nSince, when $\\Sigma$ is defined over $\\R$, $M_r^d$ is semi-projective (with respect the $\\C^\\times$ action of scaling the Higgs field), this shows that the virtual Poincar\\'e polynomial of $M_r^d$ agrees with its topological Poincar\\'e polynomial. This completes the proof of the main theorem.",
    "expanded_theorem": "\\label{mainthm}\nSuppose that $\\Sigma$ is a smooth, projective, geometrically connected curve of genus $g$ defined over $k=\\mathbb{R}$, and suppose that $\\Sigma(\\mathbb{R})$ is a union of $b+1$ circles, with $b\\geq 0$. Then \n$$P_t(M_r^d)  =  (-t)^{(g-1)r^2+1} A_{g,r,d} (-t, -t^{\\frac{1}{2}}, -t^{\\frac{1}{2}},  ..., -t^{\\frac{1}{2}}, -1,-1,..,-1) $$ \nwhere  $(g-b)$-many entries equal $-t^{\\frac{1}{2}}$ and $b$-many equal $-1$.,",
    "theorem_type": [
      "Equality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\Sigma/\\mathbb{R}\\u001f be a smooth, projective, geometrically connected curve of genus \\(g\\), and let \\(M_r^d\\) denote the moduli space of stable Higgs bundles of rank \\(r\\) and degree \\(d\\) over \\(\\Sigma\\). For a real variety \\(X/\\mathbb{R}\\), define its compactly supported Poincare9 polynomial by\n\\[\nP_t(X)=\\sum_k (-1)^k t^k \\dim_{\\mathbb{Z}_2} H^k_{cpt}(X(\\mathbb{R});\\mathbb{Z}_2).\n\\]\nAlso let \\(A_{g,r,d}(q,\\alpha_1,\\dots,\\alpha_g)\\in \\mathbb{Z}[q,\\alpha_1^{\\pm1},\\dots,\\alpha_g^{\\pm1}]\\) be Schiffmanns Laurent polynomial, which is symmetric in the \\(\\alpha_i\\). Suppose that \\(\\Sigma(\\mathbb{R})\\) is a union of \\(b+1\\) circles, with \\(b\\ge 0\\). Which statement holds for every such curve \\(\\Sigma\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\nP_t(M_r^d)=(-t)^{(g-1)r^2+1}\\,A_{g,r,d}\\bigl(-t,\\underbrace{-t^{1/2},\\dots,-t^{1/2}}_{g-b\\text{ entries}},\\underbrace{-1,\\dots,-1}_{b\\text{ entries}}\\bigr).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\nP_t(M_r^d)=(-t)^{(g-1)r^2+1}\\,A_{g,r,d}\\bigl(-t,\\underbrace{-t^{1/2},\\dots,-t^{1/2}}_{g\\text{ entries}}\\bigr).\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\nP_t(M_r^d)=(-t)^{(g-1)r^2+1}\\,A_{g,r,d}\\bigl(-t,\\alpha_1,\\dots,\\alpha_g\\bigr)\n\\]\nfor some ordering of \\(\\alpha_1,\\dots,\\alpha_g\\in\\{-t^{1/2},-1\\}\\) with exactly \\(g-b\\) of them equal to \\(-t^{1/2}\\) and exactly \\(b\\) of them equal to \\(-1\\)."
        },
        {
          "label": "D",
          "text": "\\[\nP_t(M_r^d)=(-t)^{(g-1)r^2+1}\\,A_{g,r,d}\\bigl(t,\\underbrace{-t^{1/2},\\dots,-t^{1/2}}_{g-b\\text{ entries}},\\underbrace{-1,\\dots,-1}_{b\\text{ entries}}\\bigr).\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\nP_t(M_r^d)=(-t)^{(g-1)r^2+1}\\,A_{g,r,d}\\bigl(-t,\\underbrace{-1,\\dots,-1}_{g-b\\text{ entries}},\\underbrace{-t^{1/2},\\dots,-t^{1/2}}_{b\\text{ entries}}\\bigr).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
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          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence on the real-topology parameter b",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the explicit block ordering of the specialized \\alpha_i",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "the evaluation of q under \\(P_t^{vir}(\\mathbb{L})=-t\\)",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "assignment of the counts \\(g-b\\) and \\(b\\) to \\(-t^{1/2}\\) versus \\(-1\\)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the formula or uniquely signal the correct choice. It introduces the geometric setup and asks for the explicit specialization, but the exact signs and multiplicities are not leaked."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific theorem/formula: the question asks which explicit formula holds for the Poincare polynomial. It does not substantially reframe the result or ask for a derived consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to discriminate among close alternatives (swapped multiplicities, sign errors, wrong number of specialization slots, weaker-but-true formulation), but the task mainly tests theorem recall rather than genuine derivation or generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible: B swaps the counts of the two specialization values, D introduces a natural sign/realization confusion, E violates the required number of parameters, and C is a weaker true statement that fails the demand for an explicit formula."
      },
      "total_score": 5,
      "overall_assessment": "A solid multiple-choice item for testing precise theorem recall, with high-quality distractors and little answer leakage, but it is largely tautological and only moderately probes reasoning."
    }
  },
  {
    "id": "2512.06646v1",
    "paper_link": "http://arxiv.org/abs/2512.06646v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.",
      "start_pos": 23081,
      "end_pos": 23345,
      "label": "thm main"
    },
    "ref_dict": {
      "thm homeo": "\\begin{theorem} \\label{thm homeo}\nThe map \n\\begin{equation*}\n \\Psi_{\\ge0}\\colon Y_{\\ge 0}\\rightarrow X(\\Sigma)_{\\ge0}\n \\quad ; \\quad\n gB \\mapsto [\\Delta_{\\varpi_1}(g),\\ldots,\\Delta_{\\varpi_{\\rkg}}(g);\\qp{1}(g),\\ldots,\\qp{n}(g)]\n\\end{equation*}\nis a homeomorphism satisfying\n\\begin{equation*}\n\\Psi_{\\ge 0}(\\RY{K}{J;>0}) = \\Xp{K,J}\n\\end{equation*}\nfor $K\\subseteq J \\subseteq I$.\n\\end{theorem}",
      "prop description of YKJ": "\\begin{proposition}\\label{prop description of YKJ}\nFor $K\\subseteq J\\subseteq I$, we have\n\\begin{align*}\n \\RY{K}{J}\n = \\left\\{ x\\dot{w}_J B\\in G/B \\ \\left| \\ x \\in (U_J)^{e_J} \\text{ and }\n \\begin{cases}\n \\Delta_{\\varpi_i}(x\\dot{w}_J)=0 \\quad \\text{if $i\\in K$}, \\\\\n \\Delta_{\\varpi_i}(x\\dot{w}_J)\\ne0 \\quad \\text{if $i\\in I-K$}\n \\end{cases}\n \\right.\n \\right\\},\n\\end{align*}\nwhere $(U_J)^{e_J}=U_J\\cap Z_{G_J}(e_J)$.\n\\end{proposition}",
      "thm LR in our setting": "\\begin{theorem}\\label{thm LR in our setting}\nThe map \n\\begin{align*}\n U^e_{\\ge0} \\rightarrow \\R^{I}_{\\ge0}\n \\quad ; \\quad\n x \\mapsto \n (\\Delta_{\\varpi_1}(x\\dot{w}_0), \\ldots, \\Delta_{\\varpi_{\\rkg}}(x\\dot{w}_0))\n\\end{align*}\nis a homeomorphism.\n \\end{theorem}",
      "prop restrict": "\\begin{proposition}\\label{prop restrict}\nThe morphism $\\Psi \\colon Y\\to X(\\fan)$ restricts to a continuous map\n\\begin{equation*}\n\\Psi_{\\ge 0} \\colon Y_{\\ge 0}\\to X(\\fan)_{\\ge 0}\n\\end{equation*}\nwhich sends $\\RY{K}{J;>0}$ to $\\Xp{K,J}$ for $K\\subseteq J \\subseteq  I$.\n\\end{proposition}",
      "prop Delta is nonnegative": "\\begin{proposition}\\label{prop Delta is nonnegative}\nFor $x\\in U_{\\ge0}$ and $w\\in W$, we have\n\\begin{equation*}\n \\Delta_{\\varpi_i}(x\\dot{w}) \\ge0\n \\qquad (i\\in I),\n\\end{equation*}\nwhere $\\Delta_{\\varpi_i}$ is the function defined in \\eqref{eq def of Delta}.\n\\end{proposition}",
      "prop description for R KI>0 2": "\\begin{proposition}\\label{prop description for R KI>0 2}\nLet $K\\subseteq J\\subseteq I$. Then, we have \n\\begin{equation*}\n\\RY{K}{J;>0}=\\left\\{x\\dot{w}_JB \\ \\left| \\ \n\\begin{matrix}\n\\text{\n\\rm $x\\in (U_J)^{e_J}_{\\ge0}$ and}\n~\\begin{cases}\\Delta_{\\varpi_i}(x\\dot{w}_J)=0\\quad\\text{if}\\quad i\\in K\\\\\n\\Delta_{\\varpi_i}(x\\dot{w}_J)> 0\\quad\\text{if}\\quad i\\in J-K \\\\\n\\Delta_{\\varpi_i}(x\\dot{w}_J)=1 \\quad\\text{if}\\quad i\\in I-J\n\\end{cases}\n\\end{matrix}\n\\right.\n\\right\\}.\n\\end{equation*}\n\\end{proposition}",
      "prop LR and our": "\\begin{proposition}\\label{prop LR and our}\nLet $i\\in I$. For all $x\\in U^e_{\\ge0}$, we have\n\\begin{align*}\n \\Delta_i([\\dot{w}_0^{-1}x\\dot{w}_0]) = (\\Delta_{\\varpi_i}(x\\dot{w}_0))^{m_i}.\n\\end{align*}\n\\end{proposition}",
      "thm main": "\\begin{theorem} \\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.\n\\end{theorem}",
      "prop nonnengative stratum parametrization": "\\begin{proposition}\\label{prop nonnengative stratum parametrization}\nFor $K\\subseteq J\\subseteq I$, the map\n\\begin{align*}\n (\\R^{I})_{K,J;>0} \\rightarrow \\Xp{K,J}\n \\quad ; \\quad \n (x_1,\\ldots,x_{\\rkg}) \\mapsto [x_1,\\ldots,x_{\\rkg}\\hspace{2pt};\\delta_1,\\ldots,\\delta_{\\rkg}]\n\\end{align*} \nis a bijection, where $(\\delta_1,\\ldots,\\delta_{\\rkg})\\in\\R^{I}$ is the element given by\n\\begin{align*}\n\\begin{cases}\n \\delta_i=0 \\quad (i\\in I-J) \\\\\n \\delta_i=1 \\quad (i\\in J) .\n\\end{cases}\n\\end{align*}\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 5593,
    "pre_theorem_intro_text": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.",
    "context": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.",
    "full_context": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.\n\n\\begin{document}\n\\text{}\\vspace{-10pt}\n\\title[Peterson variety and strongly dominant weight polytope]{Totally nonnegative Peterson variety \\\\and strongly dominant weight polytope}\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.\n\nNow we give some final remarks. Firstly we remark that the functions $\\Delta_{\\varpi_{i}}$ and $\\qp{i}$ appearing in our map $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \\cite[Remark 6.5]{lam2016total} and \\cite[Remark~3.3.7]{rietsch2008mirror}. In \\cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\\Delta_{\\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \\ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\\geq 0}$ and $X(\\Sigma)_{\\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\\Sigma)_{\\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.\n\nThis paper is organized as follows. In Section \\ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \\ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \\ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\\geq 0}$ and give a description of its Richardson strata. In Section \\ref{sec toric}, we study the toric orbifold $X(\\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \\ref{sec map}, we recall the definition of the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$, and reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting}). Sections \\ref{sec splittings} and \\ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.\n\nLet $\\lambda\\in\\WL=\\Hom(T,\\C^{\\times})$ be a regular dominant weight. Namely, we have $\\lambda=\\sum_{i\\in I} a_i \\varpi_i$ for some positive coefficients $a_i\\in\\R_{>0}$ $(i\\in I)$.\nRegarding $\\lambda$ as an element of $\\mathfrak{t}^*_{\\R}\\coloneqq \\WL\\otimes_{\\Z}\\R$, the weight polytope associated to $\\lambda$ is the convex hull of the $W$-orbit of $\\lambda$ :\n\\begin{align*}\n \\text{Conv}(W\\cdot \\lambda) \\subseteq \\mathfrak{t}^*_{\\R}.\n\\end{align*}\nFor its relations to the irreducible representation $V_{\\lambda}$ with highest weight $\\lambda$, see e.g.\\ \\cite[Sect.~10.1]{hall2015gtm222}.\nAccording to \\cite{burrull2023strongly}, we define the \\textit{strongly dominant weight polytope} $P^{\\lambda}$ by the intersection\n\\begin{align*}\n P^{\\lambda} \\coloneqq \\text{Conv}(W\\cdot \\lambda) \\cap \\sigma_+ \\subseteq \\mathfrak{t}^*_{\\R},\n\\end{align*}\nwhere\n$\n \\sigma_+ \\coloneqq \\{ a_1 \\varpi_1 + \\cdots + a_{\\rkg} \\varpi_{\\rkg} \\mid a_i\\ge0 \\ (i\\in I) \\}\n$\nis the (closed) dominant Weyl chamber in $\\mathfrak{t}^*_{\\R}$.\nIn loc.\\ cit., $P^{\\lambda}$ is proved to be a rational\\footnote{This means that the vertices of $P^{\\lambda}$ lie in $\\WL\\otimes_{\\Z}\\Q$.} combinatorial cube of dimension $n(=|I|)$. Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between $P^\\lambda$ and the standard $n(=|I|)$-cube, which restricts to homeomorphisms between their facets (and hence all the faces), e.g.\\ \\cite[Sect.\\ 2.2]{ziegler1995lectures}. In particular, its combinatorial structure is independent of the choice of the regular dominant weight $\\lambda$.\nWe review some facts about $P^{\\lambda}$ from loc.\\ cit.\nWe set \n\\begin{align*}\n &H_i \\coloneqq \\{ \\mu \\in \\mathfrak{t}^*_{\\R} \\mid \\alpha^{\\vee}_i(\\mu)= 0 \\}, \\\\ \n &H^{\\lambda}_i \\coloneqq \\{ \\lambda - \\mu \\in \\mathfrak{t}^*_{\\R} \\mid \\varpi^{\\vee}_i(\\mu)= 0 \\}.\n\\end{align*}\nThen the set of facets of $P^{\\lambda}$ are given by $P^{\\lambda}\\cap H_i$ and $P^{\\lambda}\\cap H^{\\lambda}_i$ for $i\\in I$ with outward normal vectors $-\\alpha^{\\vee}_i$ $(i\\in I)$ and $\\varpi^{\\vee}_i$ $(i\\in I)$, respectively.\nMoreover, for $K, J\\subseteq I$, the intersection\n\\begin{align}\\label{eq faces of SDWP}\n F_{K,J} \\coloneqq P^{\\lambda}\\cap \\bigcap_{j\\in K} H_i \\cap \\bigcap_{j\\notin J} H^{\\lambda}_i\n\\end{align}\nis non-empty if and only if $K\\subseteq J$ (equivalently, $K\\cap (I-J)=\\emptyset$). \nSee \\cite[Sect.~2]{burrull2023strongly} for details.\nThis means that the normal fan of $P^{\\lambda}$ is precisely our fan $\\Sigma$ given in Definition~\\ref{def fan}.",
    "post_theorem_intro_text_len": 3205,
    "post_theorem_intro_text": "Now we give some final remarks. Firstly we remark that the functions $\\Delta_{\\varpi_{i}}$ and $\\qp{i}$ appearing in our map $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \\cite[Remark 6.5]{lam2016total} and \\cite[Remark~3.3.7]{rietsch2008mirror}. In \\cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\\Delta_{\\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \\ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\\geq 0}$ and $X(\\Sigma)_{\\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\\Sigma)_{\\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.\n\nThis paper is organized as follows. In Section \\ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \\ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \\ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\\geq 0}$ and give a description of its Richardson strata. In Section \\ref{sec toric}, we study the toric orbifold $X(\\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \\ref{sec map}, we recall the definition of the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$, and reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting}). Sections \\ref{sec splittings} and \\ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.\n\n\\vspace{10pt}\n\n\\subsection{Acknowledgments}\nWe are grateful to Thomas Lam, Changzheng Li, and Zhi L$\\ddot{\\text{u}}$ for valuable discussions. We would particularly like to thank Konstanze Rietsch for telling us her conjecture. This research was partially done while the authors were visiting the  school of Mathematical Sciences in Fudan University.\nThe first author is supported in part by JSPS Grant-in-Aid for Scientific Research(C): 23K03102. \nThe second author is supported in part by NSFC:  12471309.\nThe third author is supported in part by NSFC:  11901218.\n\n\\vspace{30pt}",
    "sketch": "The post-theorem introduction indicates that the proof of Theorem~\\ref{thm main} proceeds by constructing the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ and then showing that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. The text says they \"reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting})\", and that Sections \\ref{sec splittings} and \\ref{sec LR in our setting} \"give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}.\" It is also remarked that a Lam--Rietsch parametrization using the functions $\\Delta_{\\varpi_i}$ is \"important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$.\"",
    "expanded_sketch": "The post-theorem introduction indicates that, in establishing the main theorem, the proof proceeds by constructing the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ and then showing that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. The text says they reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim:\n\n\\begin{theorem}\\label{thm LR in our setting}\nThe map \n\\begin{align*}\n U^e_{\\ge0} \\rightarrow \\R^{I}_{\\ge0}\n \\quad ; \\quad\n x \\mapsto \n (\\Delta_{\\varpi_1}(x\\dot{w}_0), \\ldots, \\Delta_{\\varpi_{\\rkg}}(x\\dot{w}_0))\n\\end{align*}\nis a homeomorphism.\n \\end{theorem}\n\nIt is then explained that the subsequent parts of the paper give a proof of this key claim, which completes the proof of the main theorem. It is also remarked that a Lam--Rietsch parametrization using the functions $\\Delta_{\\varpi_i}$ is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$.",
    "expanded_theorem": "\\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let $G$ be a semisimple algebraic group over $\\mathbb C$, let $B\\subset G$ be a Borel subgroup, let $T\\subset B$ be a maximal torus, and let $Y\\subset G/B$ be the Peterson variety. Let $Y_{\\ge 0}$ denote the totally nonnegative part of $Y$. Fix a regular dominant weight $\\lambda=\\sum_{i\\in I} a_i\\varpi_i\\in \\operatorname{Hom}(T,\\mathbb C^{\\times})$ with all $a_i>0$, let $W$ be the Weyl group, and define the strongly dominant weight polytope by\n\\[\nP^{\\lambda}=\\operatorname{Conv}(W\\cdot \\lambda)\\cap \\sigma_+,\n\\qquad\n\\sigma_+=\\Bigl\\{\\sum_{i\\in I} b_i\\varpi_i\\mid b_i\\ge 0\\Bigr\\}.\n\\]\nViewing $P^{\\lambda}$ with its face decomposition, and interpreting “homeomorphic as a cell-decomposed space” to mean a homeomorphism that respects the cell decompositions, which existence statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to a cube."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $\\operatorname{Conv}(W\\cdot \\lambda)$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to the full weight polytope $\\operatorname{Conv}(W\\cdot \\lambda)$."
        },
        {
          "label": "C",
          "text": "There exists a homeomorphism from $Y_{\\ge 0}$ onto $P^{\\lambda}$. Consequently, $Y_{\\ge 0}$ is homeomorphic to a cube."
        },
        {
          "label": "D",
          "text": "For every regular dominant weight $\\lambda$, there exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$ that is induced by the coordinate map $x\\mapsto (\\Delta_{\\varpi_1}(x\\dot w_0),\\ldots,\\Delta_{\\varpi_{|I|}}(x\\dot w_0))$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to a cube."
        },
        {
          "label": "E",
          "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$ if and only if the chosen regular dominant weight $\\lambda$ has rational coefficients $a_i\\in \\mathbb{Q}_{>0}$. Consequently, only for such $\\lambda$ is $Y_{\\ge 0}$ a regular CW-complex homeomorphic to a cube."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "target polytope is the strongly dominant truncation, not the full weight polytope",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "drops the requirement that the homeomorphism respect the cell decompositions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "confuses the Lam--Rietsch coordinate homeomorphism on $U^e_{\\ge0}\\cong \\mathbb R^I_{\\ge0}$ with the global cell-decomposed-space homeomorphism $Y_{\\ge0}\\to P^\\lambda$",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "replaces independence of the combinatorial type from regular dominant $\\lambda$ by a spurious rationality restriction",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and asks for the correct conclusion, but it does not explicitly reveal the key claims about regular CW-structure, cell-preserving homeomorphism, or cube homeomorphism."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the exact theorem statement about the topology of the totally nonnegative Peterson variety, with the correct option essentially reproducing that result. The competing choices add nuance, but the question still largely tests theorem recall."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants: full weight polytope vs. dominant part, plain homeomorphism vs. cell-decomposed homeomorphism, and regular vs. non-regular CW conclusions. However, success depends mostly on recognizing the precise theorem rather than generating a conclusion from mathematical reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and target natural confusions, especially about the target polytope and the strength of the topological conclusion. But choice C is a weaker true statement, which makes the single-best-answer format somewhat ambiguous and lowers distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated MCQ with little answer leakage and mostly plausible distractors, but it is close to theorem restatement and is weakened by an arguably true weaker option, so it tests precise recall more than generative reasoning."
    }
  },
  {
    "id": "2512.06646v1",
    "paper_link": "http://arxiv.org/abs/2512.06646v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.",
      "start_pos": 23081,
      "end_pos": 23345,
      "label": "thm main"
    },
    "ref_dict": {
      "thm homeo": "\\begin{theorem} \\label{thm homeo}\nThe map \n\\begin{equation*}\n \\Psi_{\\ge0}\\colon Y_{\\ge 0}\\rightarrow X(\\Sigma)_{\\ge0}\n \\quad ; \\quad\n gB \\mapsto [\\Delta_{\\varpi_1}(g),\\ldots,\\Delta_{\\varpi_{\\rkg}}(g);\\qp{1}(g),\\ldots,\\qp{n}(g)]\n\\end{equation*}\nis a homeomorphism satisfying\n\\begin{equation*}\n\\Psi_{\\ge 0}(\\RY{K}{J;>0}) = \\Xp{K,J}\n\\end{equation*}\nfor $K\\subseteq J \\subseteq I$.\n\\end{theorem}",
      "prop description of YKJ": "\\begin{proposition}\\label{prop description of YKJ}\nFor $K\\subseteq J\\subseteq I$, we have\n\\begin{align*}\n \\RY{K}{J}\n = \\left\\{ x\\dot{w}_J B\\in G/B \\ \\left| \\ x \\in (U_J)^{e_J} \\text{ and }\n \\begin{cases}\n \\Delta_{\\varpi_i}(x\\dot{w}_J)=0 \\quad \\text{if $i\\in K$}, \\\\\n \\Delta_{\\varpi_i}(x\\dot{w}_J)\\ne0 \\quad \\text{if $i\\in I-K$}\n \\end{cases}\n \\right.\n \\right\\},\n\\end{align*}\nwhere $(U_J)^{e_J}=U_J\\cap Z_{G_J}(e_J)$.\n\\end{proposition}",
      "thm LR in our setting": "\\begin{theorem}\\label{thm LR in our setting}\nThe map \n\\begin{align*}\n U^e_{\\ge0} \\rightarrow \\R^{I}_{\\ge0}\n \\quad ; \\quad\n x \\mapsto \n (\\Delta_{\\varpi_1}(x\\dot{w}_0), \\ldots, \\Delta_{\\varpi_{\\rkg}}(x\\dot{w}_0))\n\\end{align*}\nis a homeomorphism.\n \\end{theorem}",
      "prop restrict": "\\begin{proposition}\\label{prop restrict}\nThe morphism $\\Psi \\colon Y\\to X(\\fan)$ restricts to a continuous map\n\\begin{equation*}\n\\Psi_{\\ge 0} \\colon Y_{\\ge 0}\\to X(\\fan)_{\\ge 0}\n\\end{equation*}\nwhich sends $\\RY{K}{J;>0}$ to $\\Xp{K,J}$ for $K\\subseteq J \\subseteq  I$.\n\\end{proposition}",
      "prop Delta is nonnegative": "\\begin{proposition}\\label{prop Delta is nonnegative}\nFor $x\\in U_{\\ge0}$ and $w\\in W$, we have\n\\begin{equation*}\n \\Delta_{\\varpi_i}(x\\dot{w}) \\ge0\n \\qquad (i\\in I),\n\\end{equation*}\nwhere $\\Delta_{\\varpi_i}$ is the function defined in \\eqref{eq def of Delta}.\n\\end{proposition}",
      "prop description for R KI>0 2": "\\begin{proposition}\\label{prop description for R KI>0 2}\nLet $K\\subseteq J\\subseteq I$. Then, we have \n\\begin{equation*}\n\\RY{K}{J;>0}=\\left\\{x\\dot{w}_JB \\ \\left| \\ \n\\begin{matrix}\n\\text{\n\\rm $x\\in (U_J)^{e_J}_{\\ge0}$ and}\n~\\begin{cases}\\Delta_{\\varpi_i}(x\\dot{w}_J)=0\\quad\\text{if}\\quad i\\in K\\\\\n\\Delta_{\\varpi_i}(x\\dot{w}_J)> 0\\quad\\text{if}\\quad i\\in J-K \\\\\n\\Delta_{\\varpi_i}(x\\dot{w}_J)=1 \\quad\\text{if}\\quad i\\in I-J\n\\end{cases}\n\\end{matrix}\n\\right.\n\\right\\}.\n\\end{equation*}\n\\end{proposition}",
      "prop LR and our": "\\begin{proposition}\\label{prop LR and our}\nLet $i\\in I$. For all $x\\in U^e_{\\ge0}$, we have\n\\begin{align*}\n \\Delta_i([\\dot{w}_0^{-1}x\\dot{w}_0]) = (\\Delta_{\\varpi_i}(x\\dot{w}_0))^{m_i}.\n\\end{align*}\n\\end{proposition}",
      "thm main": "\\begin{theorem} \\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.\n\\end{theorem}",
      "prop nonnengative stratum parametrization": "\\begin{proposition}\\label{prop nonnengative stratum parametrization}\nFor $K\\subseteq J\\subseteq I$, the map\n\\begin{align*}\n (\\R^{I})_{K,J;>0} \\rightarrow \\Xp{K,J}\n \\quad ; \\quad \n (x_1,\\ldots,x_{\\rkg}) \\mapsto [x_1,\\ldots,x_{\\rkg}\\hspace{2pt};\\delta_1,\\ldots,\\delta_{\\rkg}]\n\\end{align*} \nis a bijection, where $(\\delta_1,\\ldots,\\delta_{\\rkg})\\in\\R^{I}$ is the element given by\n\\begin{align*}\n\\begin{cases}\n \\delta_i=0 \\quad (i\\in I-J) \\\\\n \\delta_i=1 \\quad (i\\in J) .\n\\end{cases}\n\\end{align*}\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 5593,
    "pre_theorem_intro_text": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.",
    "context": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.",
    "full_context": "\\label{sec Introduction}\nLet $G$ be a semisimple algebraic group over $\\mathbb C$ with a Borel subgroup $B\\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \\cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\\mathcal{G} r_{G^{\\vee}}$ of the Langlands dual group $G^{\\vee}$, which was verified by Lam--Shimozono in \\cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \\cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \\cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.\n\nThe theory of total positivity for reductive algebraic groups was pioneered by Lusztig \\cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \\cite{fomin2010total}, KP equations \\cite{kodama2014kp,kodama2011kp}, Poisson geometry \\cite{lu2020bott}, and the physics of scattering amplitudes \\cite{williams2021positive}. Lusztig \\cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \\cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW\ncomplexes and the topology of a regular CW complex is completely determined by the combinatorial\nstructure of its associated cell closure poset \\cite{bjorner1984posets}.\n\nThe interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \\cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \\cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\\geq 0}$ in type A and obtained its cell decomposition. She also showed in \\cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\\geq 0}$ is homeomorphic to a cube. In \\cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.\n\n\\begin{document}\n\\text{}\\vspace{-10pt}\n\\title[Peterson variety and strongly dominant weight polytope]{Totally nonnegative Peterson variety \\\\and strongly dominant weight polytope}\n\nIn this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \\ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\\Delta_{\\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \\ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\\geq 0}$ (Proposition \\ref{prop description for R KI>0 2}). Through a particular morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ constructed by the first and the third authors of this paper in \\cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$, which sends the Richardson strata of $Y_{\\geq 0}$ to the toric orbit strata of $X(\\Sigma)_{\\geq 0}$ (Proposition \\ref{prop restrict}). We prove that $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ is actually a homeomorphism (Theorem \\ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \\ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\\Sigma)_{\\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \\cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.\n\nNow we give some final remarks. Firstly we remark that the functions $\\Delta_{\\varpi_{i}}$ and $\\qp{i}$ appearing in our map $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \\cite[Remark 6.5]{lam2016total} and \\cite[Remark~3.3.7]{rietsch2008mirror}. In \\cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\\Delta_{\\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \\ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\\geq 0}$ and $X(\\Sigma)_{\\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\\Sigma)_{\\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.\n\nThis paper is organized as follows. In Section \\ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \\ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \\ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\\geq 0}$ and give a description of its Richardson strata. In Section \\ref{sec toric}, we study the toric orbifold $X(\\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \\ref{sec map}, we recall the definition of the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$, and reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting}). Sections \\ref{sec splittings} and \\ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.\n\nLet $\\lambda\\in\\WL=\\Hom(T,\\C^{\\times})$ be a regular dominant weight. Namely, we have $\\lambda=\\sum_{i\\in I} a_i \\varpi_i$ for some positive coefficients $a_i\\in\\R_{>0}$ $(i\\in I)$.\nRegarding $\\lambda$ as an element of $\\mathfrak{t}^*_{\\R}\\coloneqq \\WL\\otimes_{\\Z}\\R$, the weight polytope associated to $\\lambda$ is the convex hull of the $W$-orbit of $\\lambda$ :\n\\begin{align*}\n \\text{Conv}(W\\cdot \\lambda) \\subseteq \\mathfrak{t}^*_{\\R}.\n\\end{align*}\nFor its relations to the irreducible representation $V_{\\lambda}$ with highest weight $\\lambda$, see e.g.\\ \\cite[Sect.~10.1]{hall2015gtm222}.\nAccording to \\cite{burrull2023strongly}, we define the \\textit{strongly dominant weight polytope} $P^{\\lambda}$ by the intersection\n\\begin{align*}\n P^{\\lambda} \\coloneqq \\text{Conv}(W\\cdot \\lambda) \\cap \\sigma_+ \\subseteq \\mathfrak{t}^*_{\\R},\n\\end{align*}\nwhere\n$\n \\sigma_+ \\coloneqq \\{ a_1 \\varpi_1 + \\cdots + a_{\\rkg} \\varpi_{\\rkg} \\mid a_i\\ge0 \\ (i\\in I) \\}\n$\nis the (closed) dominant Weyl chamber in $\\mathfrak{t}^*_{\\R}$.\nIn loc.\\ cit., $P^{\\lambda}$ is proved to be a rational\\footnote{This means that the vertices of $P^{\\lambda}$ lie in $\\WL\\otimes_{\\Z}\\Q$.} combinatorial cube of dimension $n(=|I|)$. Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between $P^\\lambda$ and the standard $n(=|I|)$-cube, which restricts to homeomorphisms between their facets (and hence all the faces), e.g.\\ \\cite[Sect.\\ 2.2]{ziegler1995lectures}. In particular, its combinatorial structure is independent of the choice of the regular dominant weight $\\lambda$.\nWe review some facts about $P^{\\lambda}$ from loc.\\ cit.\nWe set \n\\begin{align*}\n &H_i \\coloneqq \\{ \\mu \\in \\mathfrak{t}^*_{\\R} \\mid \\alpha^{\\vee}_i(\\mu)= 0 \\}, \\\\ \n &H^{\\lambda}_i \\coloneqq \\{ \\lambda - \\mu \\in \\mathfrak{t}^*_{\\R} \\mid \\varpi^{\\vee}_i(\\mu)= 0 \\}.\n\\end{align*}\nThen the set of facets of $P^{\\lambda}$ are given by $P^{\\lambda}\\cap H_i$ and $P^{\\lambda}\\cap H^{\\lambda}_i$ for $i\\in I$ with outward normal vectors $-\\alpha^{\\vee}_i$ $(i\\in I)$ and $\\varpi^{\\vee}_i$ $(i\\in I)$, respectively.\nMoreover, for $K, J\\subseteq I$, the intersection\n\\begin{align}\\label{eq faces of SDWP}\n F_{K,J} \\coloneqq P^{\\lambda}\\cap \\bigcap_{j\\in K} H_i \\cap \\bigcap_{j\\notin J} H^{\\lambda}_i\n\\end{align}\nis non-empty if and only if $K\\subseteq J$ (equivalently, $K\\cap (I-J)=\\emptyset$). \nSee \\cite[Sect.~2]{burrull2023strongly} for details.\nThis means that the normal fan of $P^{\\lambda}$ is precisely our fan $\\Sigma$ given in Definition~\\ref{def fan}.",
    "post_theorem_intro_text_len": 3205,
    "post_theorem_intro_text": "Now we give some final remarks. Firstly we remark that the functions $\\Delta_{\\varpi_{i}}$ and $\\qp{i}$ appearing in our map $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \\cite[Remark 6.5]{lam2016total} and \\cite[Remark~3.3.7]{rietsch2008mirror}. In \\cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\\Delta_{\\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \\ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\\geq 0}$ and $X(\\Sigma)_{\\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\\Sigma)_{\\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.\n\nThis paper is organized as follows. In Section \\ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \\ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \\ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\\geq 0}$ and give a description of its Richardson strata. In Section \\ref{sec toric}, we study the toric orbifold $X(\\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \\ref{sec map}, we recall the definition of the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$, and reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting}). Sections \\ref{sec splittings} and \\ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.\n\n\\vspace{10pt}\n\n\\subsection{Acknowledgments}\nWe are grateful to Thomas Lam, Changzheng Li, and Zhi L$\\ddot{\\text{u}}$ for valuable discussions. We would particularly like to thank Konstanze Rietsch for telling us her conjecture. This research was partially done while the authors were visiting the  school of Mathematical Sciences in Fudan University.\nThe first author is supported in part by JSPS Grant-in-Aid for Scientific Research(C): 23K03102. \nThe second author is supported in part by NSFC:  12471309.\nThe third author is supported in part by NSFC:  11901218.\n\n\\vspace{30pt}",
    "sketch": "The post-theorem introduction indicates that the proof of Theorem~\\ref{thm main} proceeds by constructing the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ and then showing that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. The text says they \"reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim (Theorem \\ref{thm LR in our setting})\", and that Sections \\ref{sec splittings} and \\ref{sec LR in our setting} \"give a proof of this key claim, which completes the proof of Theorem \\ref{thm main}.\" It is also remarked that a Lam--Rietsch parametrization using the functions $\\Delta_{\\varpi_i}$ is \"important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$.\"",
    "expanded_sketch": "The post-theorem introduction indicates that, in establishing the main theorem, the proof proceeds by constructing the morphism $\\Psi: Y \\rightarrow X(\\Sigma)$ and then showing that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$. The text says they reduce the fact that it induces a homeomorphism $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$ to the key claim:\n\n\\begin{theorem}\\label{thm LR in our setting}\nThe map \n\\begin{align*}\n U^e_{\\ge0} \\rightarrow \\R^{I}_{\\ge0}\n \\quad ; \\quad\n x \\mapsto \n (\\Delta_{\\varpi_1}(x\\dot{w}_0), \\ldots, \\Delta_{\\varpi_{\\rkg}}(x\\dot{w}_0))\n\\end{align*}\nis a homeomorphism.\n \\end{theorem}\n\nIt is then explained that the subsequent parts of the paper give a proof of this key claim, which completes the proof of the main theorem. It is also remarked that a Lam--Rietsch parametrization using the functions $\\Delta_{\\varpi_i}$ is important in the proof of the bijectivity of $\\Psi_{\\geq 0}: Y_{\\geq 0} \\rightarrow X(\\Sigma)_{\\geq 0}$.",
    "expanded_theorem": "\\label{thm main}\n    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let $G$ be a semisimple algebraic group over $\\mathbb C$, let $B\\subset G$ be a Borel subgroup, let $T\\subset B$ be a maximal torus, and let $Y\\subset G/B$ be the Peterson variety. Let $Y_{\\ge 0}$ denote the totally nonnegative part of $Y$. Fix a regular dominant weight $\\lambda=\\sum_{i\\in I} a_i\\varpi_i\\in \\operatorname{Hom}(T,\\mathbb C^{\\times})$ with all $a_i>0$, let $W$ be the Weyl group, and define the strongly dominant weight polytope by\n\\[\nP^{\\lambda}=\\operatorname{Conv}(W\\cdot \\lambda)\\cap \\sigma_+,\n\\qquad\n\\sigma_+=\\Bigl\\{\\sum_{i\\in I} b_i\\varpi_i\\mid b_i\\ge 0\\Bigr\\}.\n\\]\nViewing $P^{\\lambda}$ with its face decomposition, and interpreting “homeomorphic as a cell-decomposed space” to mean a homeomorphism that respects the cell decompositions, which existence statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to a cube."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $\\operatorname{Conv}(W\\cdot \\lambda)$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to the full weight polytope $\\operatorname{Conv}(W\\cdot \\lambda)$."
        },
        {
          "label": "C",
          "text": "There exists a homeomorphism from $Y_{\\ge 0}$ onto $P^{\\lambda}$. Consequently, $Y_{\\ge 0}$ is homeomorphic to a cube."
        },
        {
          "label": "D",
          "text": "For every regular dominant weight $\\lambda$, there exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$ that is induced by the coordinate map $x\\mapsto (\\Delta_{\\varpi_1}(x\\dot w_0),\\ldots,\\Delta_{\\varpi_{|I|}}(x\\dot w_0))$. Consequently, $Y_{\\ge 0}$ is a regular CW-complex and is homeomorphic to a cube."
        },
        {
          "label": "E",
          "text": "There exists a homeomorphism of cell-decomposed spaces from $Y_{\\ge 0}$ onto $P^{\\lambda}$ if and only if the chosen regular dominant weight $\\lambda$ has rational coefficients $a_i\\in \\mathbb{Q}_{>0}$. Consequently, only for such $\\lambda$ is $Y_{\\ge 0}$ a regular CW-complex homeomorphic to a cube."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "target polytope is the strongly dominant truncation, not the full weight polytope",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "drops the requirement that the homeomorphism respect the cell decompositions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "confuses the Lam--Rietsch coordinate homeomorphism on $U^e_{\\ge0}\\cong \\mathbb R^I_{\\ge0}$ with the global cell-decomposed-space homeomorphism $Y_{\\ge0}\\to P^\\lambda$",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "replaces independence of the combinatorial type from regular dominant $\\lambda$ by a spurious rationality restriction",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option, and it does not strongly hint at the precise conclusion among the nuanced variants. The reader must distinguish between several closely related statements."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the precise statement of a known theorem, so it is partly a theorem-recall task. However, it does involve choosing among strengthened, weakened, and distorted formulations rather than simply restating a definition."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but important ways: the target polytope, whether the homeomorphism respects cell structure, whether it is induced by a specific coordinate map, and whether spurious rationality hypotheses are introduced. Still, this is more precision/recall than genuinely generative mathematical reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and reflect realistic failure modes, especially confusing the full weight polytope with the truncated one or overclaiming the source of the homeomorphism. However, choice C is a weaker statement that appears to be true if A is true, so the options are not cleanly mutually exclusive, which weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated but somewhat flawed MCQ: it avoids answer leakage and uses plausible distractors, but it mostly tests theorem-statement recognition rather than deep reasoning, and it is weakened by the presence of a weaker true option among the distractors."
    }
  },
  {
    "id": "2512.06696v1",
    "paper_link": "http://arxiv.org/abs/2512.06696v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}",
      "start_pos": 4739,
      "end_pos": 5172,
      "label": "theorem:1.1"
    },
    "ref_dict": {
      "theorem:1.1": "\\begin{theorem}\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}\n\\end{theorem}",
      "proposition:1.5": "\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}",
      "proposition:1.2": "\\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}",
      "proposition:1.3": "\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 2019,
    "pre_theorem_intro_text": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem. \n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.",
    "context": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.",
    "full_context": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.\n\nTheorem~\\ref{theorem:1.1} (2) is equivalent to the following local form. \\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$ if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$ if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$ if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nFirst, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.\n\\[\n\\Omega {\\mathrm{Map}} (S^4, BG)_k \\to G \\to {\\mathrm{Map}}_* (S^4, BG)_k \\to {\\mathrm{Map}} (S^4, BG)_k \\stackrel{\\mathrm{ev}}{\\longrightarrow} BG,\n\\]\nwhere $\\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\\colon S^4\\to BG$, $\\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps, and $\\mathrm{ev}$ is the evaluation map.\nThere is a homotopy equivalence $\\Omega_0^3 G\\simeq \\mathrm{Map}_{*}(S^4, BG)_0 \\simeq \\mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \\cite{gottlieb-1972} showed that the classifying space of the gauge group $B\\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \\mathop{\\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.\n\\[\n\\mathcal{G}_k \\to G \\stackrel{\\partial_k}{\\longrightarrow} \\Omega_0^3 G \\to B\\mathcal{G}_k \\to BG.\n\\]\nThus, the $G$-gauge group $\\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\\partial_k$. Let $i_3\\colon S^3 \\to G$ be the inclusion map of the bottom cell. Then, Lang \\cite{lang-1973} proved that the map $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3, 1\\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\\langle k\\cdot i_3, 1\\rangle\\simeq k \\cdot \\langle i_3, 1\\rangle$. Furthermore, in \\cite{theriault-2010}, Theriault showed that $\\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\\langle i_3, 1\\rangle$.\n\n\\begin{proposition}\\label{proposition:2.1}\nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{8}(S^{6})=\\mathbb{Z}/2\\{ \\eta_{6}^{2}\\}$, $\\pi_{11}(S^{6})=\\mathbb{Z}\\{ \\Delta(\\iota_{13})\\}$, $\\pi_{14}(S^{6})_{{(2)}}=\\mathbb{Z}/8\\{\\bar{\\nu}_6 \\} \\oplus \\mathbb{Z}/2\\{ \\varepsilon_6 \\}$, \n\\item[{\\rm (2)}] $\\pi_{14}(S^{8})=\\mathbb{Z}/2\\{ \\nu_{8}^{2} \\}$, $\\pi_{14}(S^{9})=\\{0\\}$, $\\pi_{14}(S^{10})=\\{0\\}$.\n\\item[{\\rm (3)}] $\\eta_{n}\\nu_{n+1}\\simeq 0$ for $n\\geq 5$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proposition}[Mimura] \\label{proposition:2.2}\nLet $i\\colon SU(3)\\to G_2$ be the inclusion map and \n$p\\colon G_2\\to S^6=G_2/SU(3)$ the projection map.\nBy $\\langle \\alpha \\rangle$, we denote an element in $\\pi_i(G_2)$ such that $p_*(\\langle \\alpha\\rangle)\\simeq \\alpha$. \nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{11}(G_{2})=\\mathbb{Z}\\{ \\langle 2 \\Delta(\\iota_{13})\\rangle\\} \\oplus \\mathbb{Z}/2\\{ i_{*}([\\nu_{5}^{2}])\\}$. \n\\item[{\\rm (2)}] $\\pi_{14}(G_{2})_{(2)}=\\mathbb{Z}/8\\{ \\langle \\bar{\\nu}_{6}+\\varepsilon_{6}\\rangle\\} \\oplus \\mathbb{Z}/2\\{ i_{*}([\\nu_{5}^{2}])\\circ \\nu_{11}\\}$\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}[{\\oshima} \\cite{oshima-2005}*{Theorem 2.1}] \\label{theorem:3.1}\nThere is a map $\\gamma \\colon S^{11}\\to G_2$ such that \n\\begin{itemize}\n\\item[{\\rm (1)}] \n$p\\circ \\gamma\\simeq 2\\Delta(\\iota_{13})$, \n\\item[{\\rm (2)}] $\\langle i_3, \\gamma\\rangle\\in \\pi_{14}(G_2)$ has order $21=3\\cdot 7$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proposition}\\label{proposition:3.2}\nLet $i_{8}\\colon S^8\\to P^9(2)$ be the inclusion map of the bottom cell. Then, the following hold.\n\\begin{itemize}\n\\item[{\\rm (1)}] ${i_8}_{*}\\colon \\pi_{14}(S^8)\\to \\pi_{14}(P^9(2))$ is an isomorphism.\n\\item[{\\rm (2)}] For maps $f\\colon S^{14}\\to S^8$, and $g\\colon S^{8}\\to S^{6}$, \n$g \\circ f \\simeq 0$ in $\\pi_{14}(S^6)$. \n\\item[{\\rm (3)}] If a map $f\\colon S^{14}\\to S^{6}$ factors through $P^9(2)$, then $f\\simeq 0$ in $\\pi_{14}(S^{6})$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proposition}\\label{proposition:3.3}\nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{11}(C)\\simeq \\mathbb{Z}\\{ \\bar{i}_{6}\\circ \\Delta(\\iota_{13})\\} \\oplus \\mathbb{Z}\\{ \\bar{s}\\}$, \n\\item[{\\rm (2)}] $\\mathop{\\mathrm{Ker}} \\bar{p}_{6\\; *} \\colon \\pi_{11}(C)\\to \\pi_{11}(S^6)$ is generated by $w\\simeq a \\bar{i}_{6}\\circ \\Delta(\\iota_{13})+b \\bar{s}$ where $a\\in \\mathbb{Z}$ and $b=1$ or $a$ is odd and $b=2$.\n\\item[{\\rm (3)}] $j\\circ q\\circ \\gamma\\simeq cw$ where $c\\in \\{ \\pm 1, \\pm 2\\}$.\n\\item[{\\rm (4)}] $p_{11}\\circ q\\circ \\gamma \\simeq bc (\\iota_{11})$ where $bc \\in \\{ \\pm 1, \\pm 2, \\pm 4\\}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}\n\\end{theorem}",
    "post_theorem_intro_text_len": 6277,
    "post_theorem_intro_text": "Theorem~\\ref{theorem:1.1} (2) is equivalent to the following local form. \\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nTheorem~\\ref{theorem:1.1} (1) was established in \\cite{ktt-2017}*{Theorem 1.2 (a)} and Proposition~\\ref{proposition:1.2} (2), (3), (4) were established in \\cite{ktt-2017}*{Propositions 4.6 and 3.2}. In this paper, we prove Proposition~\\ref{proposition:1.2} (1) and complete the proof of Theorem~\\ref{theorem:1.1} assuming the results of \\cite{ktt-2017}.\n\nFirst, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.\n\\[\n\\Omega {\\mathrm{Map}} (S^4, BG)_k  \\to G \\to  {\\mathrm{Map}}_* (S^4, BG)_k \\to {\\mathrm{Map}} (S^4, BG)_k \\stackrel{\\mathrm{ev}}{\\longrightarrow} BG,\n\\]\nwhere $\\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\\colon S^4\\to BG$, $\\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps,  and $\\mathrm{ev}$ is the evaluation map.\nThere is a homotopy equivalence  $\\Omega_0^3 G\\simeq \\mathrm{Map}_{*}(S^4, BG)_0 \\simeq \\mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \\cite{gottlieb-1972} showed that the classifying space of the gauge group $B\\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \\mathop{\\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.\n\\[\n\\mathcal{G}_k  \\to G \\stackrel{\\partial_k}{\\longrightarrow} \\Omega_0^3 G \\to B\\mathcal{G}_k \\to BG.\n\\]\nThus, the $G$-gauge group $\\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\\partial_k$. Let $i_3\\colon S^3 \\to G$ be the inclusion map of the bottom cell. Then, Lang \\cite{lang-1973} proved that the map $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3, 1\\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\\langle k\\cdot i_3, 1\\rangle\\simeq k \\cdot \\langle i_3, 1\\rangle$. Furthermore, in \\cite{theriault-2010}, Theriault showed that $\\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\\langle i_3, 1\\rangle$. \n\nFor an abelian group $A$, we denote by $A_{(p)}$ its localization at the prime $p$. If $A$ is finite, then $A_{(p)}$ is the $p$-primary subgroup of $A$. By \\cite{ktt-2017}*{Corollary 6.6}, the order of $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at least $4$. Therefore, to show Proposition~\\ref{proposition:1.2} (1), it suffices to prove the following proposition.\n\n\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}\n\nAs for the order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]$, together with \\cite{ktt-2017}*{Lemma 3.1, Proposition 4.5 and Corollary 6.6}, by Proposition~\\ref{proposition:1.3}, we have the following theorem.\n\n\\begin{theorem}\\label{theorem:1.4} \nThe order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]$ is $84$.\n\\end{theorem}\n\nNext, we recall some properties of a space $V:=G_2/SU(2)$, which played an important role in \\cite{ktt-2017} and is crucial in this paper. Let $q\\colon G_2\\to V$ be the obvious projection map. By \\cite{ktt-2017}*{Lemmas 7.1 and 7.2}, the Samelson product $3 \\cdot \\langle i_3, 1\\rangle\\colon \\Sigma^3 G_2\\to G_2$ factors through $\\Sigma^3 q \\colon \\Sigma^3 G_2 \\to \\Sigma^3 V$. \nLet us write this factorization as follows.\n \\[\n3\\cdot \\langle i_3, 1\\rangle\\simeq \\psi\\circ \\Sigma^3 q.\n\\]\nLet $P^{n+1}(2)$ be the mapping cone of the degree $2$ map $\\times 2 \\colon S^n\\to S^n$.\nThe $6$-skeleton of $V=S^5\\cup e^6\\cup e^{11}$ is $P^6(2)$ and, by \\cite{ktt-2017}{Lemma 7.4}, there is a homotopy equivalence $\\Sigma^3 V\\simeq P^9(2) \\vee S^{14}$.  Let \\[\np_{11}\\colon V \\to S^{11}\n\\]\n be the pinch map to the top cell, collapsing the $6$-skeleton to the base point. We denote by \n \\[\n i_6\\colon P^6(2) \\to V\n \\]\n  the inclusion map. Then, the above homotopy equivalence provides maps $s'\\colon \\Sigma^3 V \\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ such that the identity map of $\\Sigma^3 V$ is homotopic to \n  \\[\n  \\Sigma^3 i_6 \\circ s' + s'' \\circ \\Sigma^3 p_{11}.\n  \\] The following proposition is what we prove in the rest of this paper.\n\n\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}\n\n\\begin{proof}[Proof of Proposition~\\ref{proposition:1.3}] We consider the following decomposition in $[\\Sigma^3 G_2, G_2]$. \\[ 3 \\cdot \\langle i_3, 1\\rangle\\simeq \\psi \\circ \\Sigma^3 i_6\\circ s' \\circ \\Sigma^3 q +\\psi \\circ s'' \\circ \\Sigma^3 (p_{11}\\circ q). \\]  \nBy \\cite{ktt-2017}*{Lemma 5.2 (a) and (b)}, the order of the identity map of $P^9(2)$ is $4$. Hence, we have\n$4 (\\Sigma^3 i_6\\circ s' )\\simeq 0$. Therefore, by Proposition~\\ref{proposition:1.5}, the order of $ \\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.  \\end{proof}\n\nIn what follows, we use the symbol $p$ to express the map $p\\colon G_2\\to S^6\\simeq G_2/SU(3)$ only. When we need to indicate the generator of a cyclic group, we write $\\mathbb{Z}/m\\{ a\\}$ for the cyclic group of order $m$ generated by $a$.  \n\nThis paper is organized as follows. In Section \\ref{sec2}, we collect some facts on homotopy groups of $G_2$ and $S^6$ and reduce the problem on the $2$-primary homotopy group $\\pi_{14}(G_2)_{(2)}$ to that of $\\pi_{14}(S^6)_{(2)}$. In Section \\ref{sec3}, we prove Proposition~\\ref{proposition:1.5}.",
    "sketch": "To complete Theorem~\\ref{theorem:1.1} (2) it suffices to prove the local statement Proposition~\\ref{proposition:1.2}. The paper reduces this to proving Proposition~\\ref{proposition:1.2} (1) (the $2$-local case), since (2),(3),(4) are cited from \\cite{ktt-2017}.\n\nThe method recalled is: using Gottlieb’s identification $B\\mathcal{G}_k\\simeq \\mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence\n\\[\n\\mathcal{G}_k\\to G\\xrightarrow{\\partial_k} \\Omega_0^3 G\\to B\\mathcal{G}_k\\to BG,\n\\]\nso $\\mathcal{G}_k$ is the homotopy fiber of $\\partial_k$. By Lang, $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3,1\\rangle$, hence by linearity $\\langle k\\cdot i_3,1\\rangle\\simeq k\\cdot\\langle i_3,1\\rangle$. Theriault’s result is then invoked: $\\mathcal{G}_k\\simeq_{(p)}\\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\\langle i_3,1\\rangle$.\n\nFor $G=G_2$ at $p=2$, \\cite{ktt-2017} gives that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so “to show Proposition~\\ref{proposition:1.2} (1), it suffices” to prove Proposition~\\ref{proposition:1.3}: the order is at most $4$.\n\nThe proof sketch for Proposition~\\ref{proposition:1.3} is given: using that $3\\cdot\\langle i_3,1\\rangle$ factors through $\\Sigma^3q\\colon \\Sigma^3G_2\\to \\Sigma^3V$ (with $V=G_2/SU(2)$), write\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3q.\n\\]\nWith the homotopy decomposition $\\Sigma^3V\\simeq P^9(2)\\vee S^{14}$, choose maps $s'\\colon \\Sigma^3V\\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ so that\n\\[\n\\mathrm{id}_{\\Sigma^3V}\\simeq \\Sigma^3i_6\\circ s'\\; +\\; s''\\circ \\Sigma^3p_{11}.\n\\]\nThen\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3i_6\\circ s'\\circ \\Sigma^3q\\; +\\; \\psi\\circ s''\\circ \\Sigma^3(p_{11}\\circ q).\n\\]\nUsing \\cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\\Sigma^3i_6\\circ s')\\simeq 0$. The remaining summand is controlled by Proposition~\\ref{proposition:1.5}, namely $4(\\psi\\circ s'')\\simeq 0$ in $\\pi_{14}(G_2)_{(2)}$. Combining these shows “the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at most $4$,” hence Proposition~\\ref{proposition:1.2}(1), and therefore Theorem~\\ref{theorem:1.1}(2) via the Samelson-product/Thériault criterion.\n\n(They additionally note that with Proposition~\\ref{proposition:1.3} and results of \\cite{ktt-2017}, the global order is computed as $84$ (Theorem~\\ref{theorem:1.4}).)",
    "expanded_sketch": "To establish the main theorem (2) it suffices to prove the local statement\n\n\\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nThe paper reduces this to proving the preceding proposition (1) (the $2$-local case), since (2),(3),(4) are cited from \\cite{ktt-2017}.\n\nThe method recalled is: using Gottlieb’s identification $B\\mathcal{G}_k\\simeq \\mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence\n\\[\n\\mathcal{G}_k\\to G\\xrightarrow{\\partial_k} \\Omega_0^3 G\\to B\\mathcal{G}_k\\to BG,\n\\]\nso $\\mathcal{G}_k$ is the homotopy fiber of $\\partial_k$. By Lang, $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3,1\\rangle$, hence by linearity $\\langle k\\cdot i_3,1\\rangle\\simeq k\\cdot\\langle i_3,1\\rangle$. Theriault’s result is then invoked: $\\mathcal{G}_k\\simeq_{(p)}\\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\\langle i_3,1\\rangle$.\n\nFor $G=G_2$ at $p=2$, \\cite{ktt-2017} gives that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so to show the preceding proposition (1), it suffices to prove the following.\n\n\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}\n\nThe proof sketch for the preceding proposition is given: using that $3\\cdot\\langle i_3,1\\rangle$ factors through $\\Sigma^3q\\colon \\Sigma^3G_2\\to \\Sigma^3V$ (with $V=G_2/SU(2)$), write\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3q.\n\\]\nWith the homotopy decomposition $\\Sigma^3V\\simeq P^9(2)\\vee S^{14}$, choose maps $s'\\colon \\Sigma^3V\\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ so that\n\\[\n\\mathrm{id}_{\\Sigma^3V}\\simeq \\Sigma^3i_6\\circ s'\\; +\\; s''\\circ \\Sigma^3p_{11}.\n\\]\nThen\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3i_6\\circ s'\\circ \\Sigma^3q\\; +\\; \\psi\\circ s''\\circ \\Sigma^3(p_{11}\\circ q).\n\\]\nUsing \\cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\\Sigma^3i_6\\circ s')\\simeq 0$. The remaining summand is controlled by\n\n\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}\n\nCombining these shows that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at most $4$, hence the preceding proposition (1), and therefore the main theorem (2) via the Samelson-product/Thériault criterion.\n\n(They additionally note that with the proposition above and results of \\cite{ktt-2017}, the global order is computed as $84$ (Theorem~\\ref{theorem:1.4}).)",
    "expanded_theorem": "\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Let \\(\\mathcal{G}_k\\) denote the gauge group of a principal \\(G_2\\)-bundle over \\(S^4\\) whose classifying map represents \\(k\\in \\pi_4(BG_2)\\cong \\mathbb Z\\). Write \\((m,n)\\) for the greatest common divisor of integers \\(m\\) and \\(n\\), write \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\) for homotopy equivalence, and write \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for \\(p\\)-local homotopy equivalence. Which statement holds for integers \\(k\\) and \\(k'\\)?",
      "correct_choice": {
        "label": "A",
        "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,168)=(k',168)\\). Moreover, if \\((k,168)=(k',168)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."
        },
        {
          "label": "C",
          "text": "If \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}\\), \\(\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}\\), and \\(\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}\\)."
        },
        {
          "label": "D",
          "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\)."
        },
        {
          "label": "E",
          "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,4)=(k',4)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "global_order_84_replaced_by_168",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "dropped_global_all-primes_conclusion_and_global_homotopy_necessity",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "p-local_equivalence_strengthened_to_integral_homotopy_equivalence",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "local_primewise_conditions_collapsed_to_2-local_gcd_condition",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only introduces notation and the setup; it does not reveal the key invariant 84 or otherwise point directly to choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially a theorem-identification question: the correct option states a specific classification result rather than asking the student to derive it from intermediate facts. The competing options prevent complete tautology, but it still largely tests recall of the exact statement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in comparing necessity vs sufficiency, integral vs p-local equivalence, and 84 vs 168. However, the main demand is recognizing the known theorem rather than generating a conclusion from mathematical reasoning in the stem."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: B perturbs the key arithmetic invariant, C gives a weaker true-looking statement, D improperly strengthens p-local equivalence to global equivalence, and E introduces a misleading reduced gcd condition. These are plausible and mathematically distinct."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it primarily tests recall of a specific theorem statement rather than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.06843v1",
    "paper_link": "http://arxiv.org/abs/2512.06843v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Main theorem}\n\tSuppose $T$ is a Calder\\'on-Zygmund operator defined as in (\\ref{def of Tf}) associated with a kernel satisfying (\\ref{Size condition}) and the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\n\tLet $T_{\\epsilon}$ be defined as in (\\ref{def of T epsilon f}).\n\tSuppose $(T_{\\epsilon})_{\\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\\infty$, that is, for any $f\\in L_{p_0}(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{L_{p_0}(\\mN;\\ell_{\\infty})} \\lesssim  \\left\\| f \\right\\| _{L_{p_0}(\\mN)}.\n\t\\]\n\tThen, $(T_{\\epsilon})_{\\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\\in L_1(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{\\Lambda_{1,\\infty}(\\mN;\\ell_{\\infty})} \\lesssim  \\left\\| f \\right\\| _{L_1(\\mN)}.\n\t\\]\nWe refer to see Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $ \\left\\| \\cdot \\right\\| _{L_p(\\mN; \\ell_{\\infty})}$ and the noncommutative weak maximal norm $ \\left\\| \\cdot \\right\\| _{\\Lambda_{p,\\infty}(\\mN;\\ell_{\\infty})}$.",
      "start_pos": 18002,
      "end_pos": 19065,
      "label": "Main theorem"
    },
    "ref_dict": {
      "nc CZ decomposition with convolution": "\\begin{theorem} \\label{nc CZ decomposition with convolution}\n\tFix $f\\in\\mN_{c,+}$, $\\lambda>0$, and $s\\in\\N$.\n\tLet $(q_n)_{n\\in\\Z}$ and $(p_n)_{n\\in\\Z}$ be the sequences of projections in Lemma \\ref{cuculescu's construction}, where $q_n = \\sum\\limits_{Q\\in\\mQ_n} q_Q \\chi_Q$, $p_n = \\sum\\limits_{Q\\in\\mQ_n} p_Q \\chi_Q$.\n\tSuppose $h\\in C_{c}^{\\infty} (\\R^d)$ satisfies\n    \\begin{align*}\n    \\supp h \\subset B_1,\\quad \\int_{\\R^d} h =1, \\quad h\\geq 0.\n    \\end{align*}\n    Define $h_r(x)= \\frac{1}{r^d} h(\\frac{x}{r})$ and $r_n= 2^{-n-1}\\sqrt{d}$.\n\tThen, $f$ admits the decomposition:\n\t\\begin{align*}\n\tf = g+ b\n\t  = g + \\widetilde{b}_{\\text{d}} + b_{\\text{d}}^0 + \\widetilde{b}_{\\text{off}} + b_{\\text{off}}^0,\n\t\\end{align*}\n\twhere $g$ is positive and $b$ is self-adjoint in $\\mN$.\n\tMore precisely, these components are defined as follows:\n\t    \\[g=qfq + \\sum\\limits_{n\\in\\Z}p_n f_n p_n,\\]\n        \\begin{gather*}\n        \\widetilde{b}_{\\text{d}} = \\sum\\limits_{n\\in\\Z} \\widetilde{b}_{\\text{d},n} = \\sum\\limits_{n\\in\\Z}\\sum\\limits_{Q\\in\\mQ_n}\\widetilde{b}_{\\text{d},n}^Q ,\\quad\n        b_{\\text{d}}^0 = \\sum\\limits_{n\\in\\Z} b_{\\text{d},n}^0=\\sum\\limits_{n\\in\\Z} \\sum\\limits_{Q\\in\\mQ_n}(b_{\\text{d},n}^Q-\\widetilde{b}_{\\text{d},n}^Q),\\\\\n        \\widetilde{b}_{\\text{off}} = \\sum\\limits_{n\\in\\Z} \\widetilde{b}_{\\text{off},n} = \\sum\\limits_{n\\in\\Z} \\sum\\limits_{Q\\in\\mQ_n} \\widetilde{b}_{\\text{off},n}^Q ,\\quad\n        b_{\\text{off}}^0 = \\sum\\limits_{n\\in\\Z} b_{\\text{off},n}^0 = \\sum\\limits_{n\\in\\Z}\\sum\\limits_{Q\\in\\mQ_n}(b_{\\text{off},n}^Q-\\widetilde{b}_{\\text{off},n}^Q),\n        \\end{gather*}\n     where for each $Q\\in\\mQ_n$,\n        \\begin{gather*}\n     \tb_{\\text{d},n}^Q = p_Q(f-f_n)p_Q\\chi_Q,\n     \t\\quad b_{\\text{off},n}^Q = p_Q(f-f_n)q_Q\\chi_Q + q_Q(f-f_n)p_Q\\chi_Q, \\\\\n     \t\\widetilde{b}_{\\text{d},n}^Q = b_{\\text{d},n}^Q * h_{r_n},\n     \t\\quad \\widetilde{b}_{\\text{off},n}^Q = b_{\\text{off},n}^Q * h_{r_n}.\n        \\end{gather*}\n    \\end{theorem}",
      "def of T epsilon f": "\\begin{align}\\label{def of T epsilon f}\n\tT_{\\epsilon}f(x) = \\int_{|x-y|>\\epsilon} K(x,y) f(y) \\dif y.\n    \\end{align}",
      "Proof of the main theorem": "\\label{Proof of the main theorem}\nNote that every operator in a von Neumann algebra can be decomposed into a linear combination of four positive elements.\nBy this fact and the density of $\\mN_{c,+}$ i",
      "Hq < infty": "\\begin{align} \\label{Hq < infty}\n\t\\sum_{m\\geq 1} H_q (m) < \\infty,\n    \\end{align}",
      "Lipschitz regularity condition": "\\begin{align} \\label{Lipschitz regularity condition}\n\t   |K(x,y)-K(x,z)|+|K(y,x)-K(z,x)| \\leq \\frac{C_2 |y-z|^{\\gamma}}{|x-y|^{d+\\gamma}}, \\quad\\text{if}\\  |x-y|\\geq 2|y-z|,\n\t   \\end{align}",
      "sum delta < infty": "\\begin{align} \\label{sum delta < infty}\n\t\\sum_{m\\geq 1} \\delta_q(m) <\\infty,\n    \\end{align}",
      "Hormander condition": "\\begin{align}\\label{Hormander condition}\n\t\\sup_{y\\in\\R^d,|v|>0} \\int_{|x-y|\\geq 2|v|} |K(x,y+v)-K(x,y)| \\dif x < \\infty,\n    \\end{align}",
      "L2 mean Hormander condition": "\\begin{align}\\label{L2 mean Hormander condition}\n\t\\sum_{m\\geq 1} H_2 (m) < \\infty,\n    \\end{align}",
      "L2-integral regularity condition": "\\begin{align} \\label{L2-integral regularity condition}\n\t\\sum_{m\\geq 1} \\sup_{{\\begin{subarray}{c} y\\in Q \\\\ Q\\ \\text{dyadic} \\end{subarray}}} \\Big( 2^{md}\\ell(Q)^d  \\int_{2^m\\ell(Q)\\leq|x-c(Q)|\\leq2^{m+1}\\ell(Q)} |K(x,y)-K(x,c(Q))|^2 \\dif x \\Big)^{\\frac{1}{2}}\n\t<\\infty,\n    \\end{align}",
      "def of Tf": "\\begin{align}\\label{def of Tf}\n\tT f(x) = \\int_{\\R^d} K(x,y) f(y) \\dif y, \\quad x\\notin \\overrightarrow{\\supp}\\ f,\n    \\end{align}",
      "Main theorem": "\\begin{theorem}\\label{Main theorem}\n\tSuppose $T$ is a Calder\\'on-Zygmund operator defined as in (\\ref{def of Tf}) associated with a kernel satisfying (\\ref{Size condition}) and the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\n\tLet $T_{\\epsilon}$ be defined as in (\\ref{def of T epsilon f}).\n\tSuppose $(T_{\\epsilon})_{\\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\\infty$, that is, for any $f\\in L_{p_0}(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{L_{p_0}(\\mN;\\ell_{\\infty})} \\lesssim \\norm{f}_{L_{p_0}(\\mN)}.\n\t\\]\n\tThen, $(T_{\\epsilon})_{\\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\\in L_1(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{\\Lambda_{1,\\infty}(\\mN;\\ell_{\\infty})} \\lesssim \\norm{f}_{L_1(\\mN)}.\n\t\\]\nWe refer to see Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $\\norm{\\cdot}_{L_p(\\mN; \\ell_{\\infty})}$ and the noncommutative weak maximal norm $\\norm{\\cdot}_{\\Lambda_{p,\\infty}(\\mN;\\ell_{\\infty})}$.\n    \\end{theorem}",
      "Appendix: Noncommutative Lp spaces and maximal norms": "\\label{Appendix: Noncommutative Lp spaces and maximal norms}\nThis part consists of three subsections: noncommutative $L_p$ spaces, noncommutative maximal norms, and noncommutative square functions.\nAl",
      "Size condition": "\\begin{align} \\label{Size condition}\n       |K(x,y)| \\leq \\frac{C_1}{|x-y|^d} ;\n       \\end{align}"
    },
    "pre_theorem_intro_text_len": 8976,
    "pre_theorem_intro_text": "Calder\\'on-Zygmund operators, a well-known class of singular integral operators, were introduced by Calder\\'on and Zygmund \\cite{Calderon1952On} in their seminal 1952 work.\nThe kernel of such an operator satisfies specific conditions, including the size and regularity conditions, the latter of which was refined by H\\\"ormander \\cite{Hormander1960Estimates} in 1960.\n\nThe standard nonconvolution type Calder\\'on-Zygmund operator is defined as\n    \\[\n    Tf(x) = \\int_{\\R^d} K(x,y) f(y) {\\rm d} y, \\quad x\\notin \\operatorname{supp} f,\n    \\]\nwhere $K:\\R^d \\times \\R^d \\setminus  \\left\\{ (x,x): x\\in\\R^d \\right\\}  \\rightarrow \\C$\nis a standard nonconvolution type Calder\\'on-Zygmund kernel, which is a binary function satisfying\n    \\begin{itemize}\n\t\\item \\textbf{Size condition:} there exists $C_1>0$ such that\n\t   \\begin{align} \\label{Size condition}\n       |K(x,y)| \\leq \\frac{C_1}{|x-y|^d} ;\n       \\end{align}\n\t\\item  \\textbf{Lipschitz regularity condition:} there exists  $\\gamma\\in [0,1]$ and $C_2>0$ such that\n\t   \\begin{align} \\label{Lipschitz regularity condition}\n\t   |K(x,y)-K(x,z)|+|K(y,x)-K(z,x)| \\leq \\frac{C_2 |y-z|^{\\gamma}}{|x-y|^{d+\\gamma}}, \\quad\\text{if}\\  |x-y|\\geq 2|y-z|,\n\t   \\end{align}\n    \\end{itemize}\nwhere $x$, $y$, $z\\in\\R^d$.\nThe Lipschitz regularity condition (\\ref{Lipschitz regularity condition}) can be relaxed to certain variants of smoothness conditions (see \\cite{Grafakos2019Alimited, Suzuki2021TheCalderon, Watson1990Weighted}).\nIn the following, all Calder\\'on-Zygmund kernels under consideration are of nonconvolution type.\n\nInspired by operator algebras, harmonic analysis, noncommutative geometry, and quantum probability (see \\cite{Cadilhac2018Weak, Chen2013Harmonic, Junge2014Smooth, Junge2018Noncommutative, Mei2007Operator, Mei2009Pseudo-localization, Mei2017Free, Mei2022Mikhlin, Parcet2009Pseudo-localization}), noncommutative harmonic analysis has become an exciting field in recent years.\nHowever, generalizing the Calder\\'on-Zygmund decomposition, a key technique in harmonic analysis first introduced in \\cite{Calderon1952On}, to the noncommutative setting is challenging.\n\nIn 2009, Parcet \\cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \\cite{Cuculescu1971Martingales}.\nWith this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\\ref{Lipschitz regularity condition})\n(Cadilhac later gave a shorter proof in \\cite{Cadilhac2018Weak}).\nNotably,  the H\\\"ormander condition\n    \\begin{align}\\label{Hormander condition}\n\t\\sup_{y\\in\\R^d,|v|>0} \\int_{|x-y|\\geq 2|v|} |K(x,y+v)-K(x,y)| {\\rm d} x < \\infty,\n    \\end{align}\n which is weaker than (\\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\\'on-Zygmund theory.\nIt is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\\\"ormander condition (\\ref{Hormander condition})?\nThis question remains open.\n\nA step forward was made in 2022 by Cadilhac et al. \\cite{Cadilhac2022Spectral}.\nThey developed a more effective noncommutative Calder\\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\\'on-Zygmund operators if the kernel $K$ satisfies\n    \\begin{align} \\label{L2-integral regularity condition}\n\t\\sum_{m\\geq 1} \\sup_{{\\begin{subarray}{c} y\\in Q \\\\ Q\\ \\text{dyadic} \\end{subarray}}} \\Big( 2^{md}\\ell(Q)^d  \\int_{2^m\\ell(Q)\\leq|x-c(Q)|\\leq2^{m+1}\\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\\rm d} x \\Big)^{\\frac{1}{2}}\n\t<\\infty,\n    \\end{align}\n where $\\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.\n More generally, for $1\\leq q\\leq\\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if\n    \\begin{align} \\label{sum delta < infty}\n\t\\sum_{m\\geq 1} \\delta_q(m) <\\infty,\n    \\end{align}\nwhere $\\delta_q(m)$ is defined as\n     \\begin{align*}\n\t \\sup_{{\\begin{subarray}{c} y\\in \\R^d,\\ R>0 \\\\ |v|<R \\end{subarray}}} & \\Big( (2^m R)^{d(q-1)} \\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x\\Big)^{\\frac{1}{q}},\n    \\end{align*}\nfor $m\\geq 1$.\nRoughly speaking, the H\\\"ormander condition (\\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\\ref{sum delta < infty}) with $q=1$, while \\eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\\ref{sum delta < infty}) with $q=2$.\nThe fact $\\delta_1(m)\\lesssim_d \\delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\\\"ormander condition (\\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\\ref{Lipschitz regularity condition}).\nBy applying the noncommutative Calder\\'on-Zygmund decomposition in  \\cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \\cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\\'on-Zygmund operators under the $L_2$-integral regularity condition.\nWe refer the reader to \\cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\\'on-Zygmund operators.\n\nIn this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}): the $L_q$-mean H\\\"ormander condition ($1\\leq q\\leq\\infty$), defined by\n    \\begin{align} \\label{Hq < infty}\n\t\\sum_{m\\geq 1} H_q (m) < \\infty,\n    \\end{align}\nwhere $H_q(m)$ is defined as\n    \\begin{align*}\t \\sup_{{\\begin{subarray}{} y\\in \\R^d, \\\\ R>0 \\end{subarray}}} & \\Big(\\frac{(2^m R)^{d(q-1)}}{|B_R|} \\int_{|v|\\leq R}\\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x {\\rm d} v\\Big)^{\\frac{1}{q}},     \\end{align*}\nfor $m\\geq 1$.\nHere, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.\nIt is not hard to see that $H_q(m) \\lesssim_d \\delta_q(m)$, meaning the $L_q$-mean H\\\"ormander condition (\\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}).\nIn what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\\\"ormander condition:\n    \\begin{align}\\label{L2 mean Hormander condition}\n\t\\sum_{m\\geq 1} H_2 (m) < \\infty,\n    \\end{align}\nwhere $H_2(m)$ is defined as in (\\ref{Hq < infty}) with $q=2$.\n\nFor two smoothness conditions $A$ and $B$, we write $A\\prec B$ to mean that $A$ is weaker than $B$.\nThus, we have the following conclusion: \\newline\n    \\begin{center}\n    \\scalebox{0.94}{\n    $\\begin{array}{c}\n        \\begin{array}{c}\n            \\textbf{\\large $L_2$-mean H\\\"ormander} \\\\\n            \\textbf{\\large condition} \\\\\n        \\end{array}\n        $\\scalebox{1.3}{$\\prec$}$\n        \\begin{array}{c}\n            \\textbf{\\large $L_2$-integral regularity} \\\\\n            \\textbf{\\large condition} \\\\\n            \\end{array}\n        $\\scalebox{1.3}{$\\prec$}$\n        \\begin{array}{c}\n            \\textbf{\\large Lipschitz regularity} \\\\\n            \\textbf{\\large condition}\\\\\n        \\end{array}\n    \\end{array}$\n    } \\newline\n    \\end{center}\nThis naturally leads to a question: can the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}) still guarantee the noncommutative weak type $(1,1)$ boundedness of the maximal Calder\\'on-Zygmund operators?\nThis paper gives a positive answer to this question.\n\nBefore stating the main result, we introduce some notions.\nLet $\\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\\tau$.\nLet $\\mN = L_{\\infty}(\\R^d)\\bar\\otimes \\mM$ be the tensor von Neumann algebra with the tensor trace  $\\phi = \\int\\otimes \\tau$.\nDenote the noncommutative $L_p$ space associated with $(\\mN, \\phi)$ by $L_p(\\mN)$ for $1\\leq p\\leq\\infty$ (see Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).\nThe noncommutative Calder\\'on-Zygmund operator $T$ with a kernel $K$ is defined as\n    \\begin{align}\\label{def of Tf}\n\tT f(x) = \\int_{\\R^d} K(x,y) f(y) {\\rm d} y, \\quad x\\notin \\overrightarrow{\\operatorname{supp}}\\ f,\n    \\end{align}\nwhere $f\\in L_1(\\mN)\\bigcap L_{\\infty}(\\mN)$ is compactly supported and measurable.\nHere, $\\overrightarrow{\\text{supp}}\\ f$ denotes the support of $f$ as an operator-valued function in $\\R^d$ rather than the support projection as an element in a von Neumann algebra.\nFor any $\\epsilon>0$, the associated truncated singular integral $T_{\\epsilon}f$ is defined as\n    \\begin{align}\\label{def of T epsilon f}\n\tT_{\\epsilon}f(x) = \\int_{|x-y|>\\epsilon} K(x,y) f(y) {\\rm d} y.\n    \\end{align}\n\nThe main result\nis stated as follows.",
    "context": "In 2009, Parcet \\cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \\cite{Cuculescu1971Martingales}.\nWith this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\\ref{Lipschitz regularity condition})\n(Cadilhac later gave a shorter proof in \\cite{Cadilhac2018Weak}).\nNotably, the H\\\"ormander condition\n \\begin{align}\\label{Hormander condition}\n \\sup_{y\\in\\R^d,|v|>0} \\int_{|x-y|\\geq 2|v|} |K(x,y+v)-K(x,y)| {\\rm d} x < \\infty,\n \\end{align}\n which is weaker than (\\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\\'on-Zygmund theory.\nIt is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\\\"ormander condition (\\ref{Hormander condition})?\nThis question remains open.\n\nA step forward was made in 2022 by Cadilhac et al. \\cite{Cadilhac2022Spectral}.\nThey developed a more effective noncommutative Calder\\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\\'on-Zygmund operators if the kernel $K$ satisfies\n \\begin{align} \\label{L2-integral regularity condition}\n \\sum_{m\\geq 1} \\sup_{{\\begin{subarray}{c} y\\in Q \\\\ Q\\ \\text{dyadic} \\end{subarray}}} \\Big( 2^{md}\\ell(Q)^d \\int_{2^m\\ell(Q)\\leq|x-c(Q)|\\leq2^{m+1}\\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\\rm d} x \\Big)^{\\frac{1}{2}}\n <\\infty,\n \\end{align}\n where $\\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.\n More generally, for $1\\leq q\\leq\\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if\n \\begin{align} \\label{sum delta < infty}\n \\sum_{m\\geq 1} \\delta_q(m) <\\infty,\n \\end{align}\nwhere $\\delta_q(m)$ is defined as\n \\begin{align*}\n \\sup_{{\\begin{subarray}{c} y\\in \\R^d,\\ R>0 \\\\ |v|<R \\end{subarray}}} & \\Big( (2^m R)^{d(q-1)} \\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x\\Big)^{\\frac{1}{q}},\n \\end{align*}\nfor $m\\geq 1$.\nRoughly speaking, the H\\\"ormander condition (\\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\\ref{sum delta < infty}) with $q=1$, while \\eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\\ref{sum delta < infty}) with $q=2$.\nThe fact $\\delta_1(m)\\lesssim_d \\delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\\\"ormander condition (\\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\\ref{Lipschitz regularity condition}).\nBy applying the noncommutative Calder\\'on-Zygmund decomposition in \\cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \\cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\\'on-Zygmund operators under the $L_2$-integral regularity condition.\nWe refer the reader to \\cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\\'on-Zygmund operators.\n\nIn this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}): the $L_q$-mean H\\\"ormander condition ($1\\leq q\\leq\\infty$), defined by\n \\begin{align} \\label{Hq < infty}\n \\sum_{m\\geq 1} H_q (m) < \\infty,\n \\end{align}\nwhere $H_q(m)$ is defined as\n \\begin{align*} \\sup_{{\\begin{subarray}{} y\\in \\R^d, \\\\ R>0 \\end{subarray}}} & \\Big(\\frac{(2^m R)^{d(q-1)}}{|B_R|} \\int_{|v|\\leq R}\\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x {\\rm d} v\\Big)^{\\frac{1}{q}}, \\end{align*}\nfor $m\\geq 1$.\nHere, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.\nIt is not hard to see that $H_q(m) \\lesssim_d \\delta_q(m)$, meaning the $L_q$-mean H\\\"ormander condition (\\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}).\nIn what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\\\"ormander condition:\n \\begin{align}\\label{L2 mean Hormander condition}\n \\sum_{m\\geq 1} H_2 (m) < \\infty,\n \\end{align}\nwhere $H_2(m)$ is defined as in (\\ref{Hq < infty}) with $q=2$.\n\nBefore stating the main result, we introduce some notions.\nLet $\\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\\tau$.\nLet $\\mN = L_{\\infty}(\\R^d)\\bar\\otimes \\mM$ be the tensor von Neumann algebra with the tensor trace $\\phi = \\int\\otimes \\tau$.\nDenote the noncommutative $L_p$ space associated with $(\\mN, \\phi)$ by $L_p(\\mN)$ for $1\\leq p\\leq\\infty$ (see Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).\nThe noncommutative Calder\\'on-Zygmund operator $T$ with a kernel $K$ is defined as\n \\begin{align}\\label{def of Tf}\n T f(x) = \\int_{\\R^d} K(x,y) f(y) {\\rm d} y, \\quad x\\notin \\overrightarrow{\\operatorname{supp}}\\ f,\n \\end{align}\nwhere $f\\in L_1(\\mN)\\bigcap L_{\\infty}(\\mN)$ is compactly supported and measurable.\nHere, $\\overrightarrow{\\text{supp}}\\ f$ denotes the support of $f$ as an operator-valued function in $\\R^d$ rather than the support projection as an element in a von Neumann algebra.\nFor any $\\epsilon>0$, the associated truncated singular integral $T_{\\epsilon}f$ is defined as\n \\begin{align}\\label{def of T epsilon f}\n T_{\\epsilon}f(x) = \\int_{|x-y|>\\epsilon} K(x,y) f(y) {\\rm d} y.\n \\end{align}\n\nThe main result\nis stated as follows.",
    "full_context": "In 2009, Parcet \\cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \\cite{Cuculescu1971Martingales}.\nWith this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\\ref{Lipschitz regularity condition})\n(Cadilhac later gave a shorter proof in \\cite{Cadilhac2018Weak}).\nNotably, the H\\\"ormander condition\n \\begin{align}\\label{Hormander condition}\n \\sup_{y\\in\\R^d,|v|>0} \\int_{|x-y|\\geq 2|v|} |K(x,y+v)-K(x,y)| {\\rm d} x < \\infty,\n \\end{align}\n which is weaker than (\\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\\'on-Zygmund theory.\nIt is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\\\"ormander condition (\\ref{Hormander condition})?\nThis question remains open.\n\nA step forward was made in 2022 by Cadilhac et al. \\cite{Cadilhac2022Spectral}.\nThey developed a more effective noncommutative Calder\\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\\'on-Zygmund operators if the kernel $K$ satisfies\n \\begin{align} \\label{L2-integral regularity condition}\n \\sum_{m\\geq 1} \\sup_{{\\begin{subarray}{c} y\\in Q \\\\ Q\\ \\text{dyadic} \\end{subarray}}} \\Big( 2^{md}\\ell(Q)^d \\int_{2^m\\ell(Q)\\leq|x-c(Q)|\\leq2^{m+1}\\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\\rm d} x \\Big)^{\\frac{1}{2}}\n <\\infty,\n \\end{align}\n where $\\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.\n More generally, for $1\\leq q\\leq\\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if\n \\begin{align} \\label{sum delta < infty}\n \\sum_{m\\geq 1} \\delta_q(m) <\\infty,\n \\end{align}\nwhere $\\delta_q(m)$ is defined as\n \\begin{align*}\n \\sup_{{\\begin{subarray}{c} y\\in \\R^d,\\ R>0 \\\\ |v|<R \\end{subarray}}} & \\Big( (2^m R)^{d(q-1)} \\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x\\Big)^{\\frac{1}{q}},\n \\end{align*}\nfor $m\\geq 1$.\nRoughly speaking, the H\\\"ormander condition (\\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\\ref{sum delta < infty}) with $q=1$, while \\eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\\ref{sum delta < infty}) with $q=2$.\nThe fact $\\delta_1(m)\\lesssim_d \\delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\\\"ormander condition (\\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\\ref{Lipschitz regularity condition}).\nBy applying the noncommutative Calder\\'on-Zygmund decomposition in \\cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \\cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\\'on-Zygmund operators under the $L_2$-integral regularity condition.\nWe refer the reader to \\cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\\'on-Zygmund operators.\n\nIn this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}): the $L_q$-mean H\\\"ormander condition ($1\\leq q\\leq\\infty$), defined by\n \\begin{align} \\label{Hq < infty}\n \\sum_{m\\geq 1} H_q (m) < \\infty,\n \\end{align}\nwhere $H_q(m)$ is defined as\n \\begin{align*} \\sup_{{\\begin{subarray}{} y\\in \\R^d, \\\\ R>0 \\end{subarray}}} & \\Big(\\frac{(2^m R)^{d(q-1)}}{|B_R|} \\int_{|v|\\leq R}\\int_{2^mR\\leq |x-y|\\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\\rm d} x {\\rm d} v\\Big)^{\\frac{1}{q}}, \\end{align*}\nfor $m\\geq 1$.\nHere, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.\nIt is not hard to see that $H_q(m) \\lesssim_d \\delta_q(m)$, meaning the $L_q$-mean H\\\"ormander condition (\\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\\ref{sum delta < infty}).\nIn what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\\\"ormander condition:\n \\begin{align}\\label{L2 mean Hormander condition}\n \\sum_{m\\geq 1} H_2 (m) < \\infty,\n \\end{align}\nwhere $H_2(m)$ is defined as in (\\ref{Hq < infty}) with $q=2$.\n\nBefore stating the main result, we introduce some notions.\nLet $\\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\\tau$.\nLet $\\mN = L_{\\infty}(\\R^d)\\bar\\otimes \\mM$ be the tensor von Neumann algebra with the tensor trace $\\phi = \\int\\otimes \\tau$.\nDenote the noncommutative $L_p$ space associated with $(\\mN, \\phi)$ by $L_p(\\mN)$ for $1\\leq p\\leq\\infty$ (see Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).\nThe noncommutative Calder\\'on-Zygmund operator $T$ with a kernel $K$ is defined as\n \\begin{align}\\label{def of Tf}\n T f(x) = \\int_{\\R^d} K(x,y) f(y) {\\rm d} y, \\quad x\\notin \\overrightarrow{\\operatorname{supp}}\\ f,\n \\end{align}\nwhere $f\\in L_1(\\mN)\\bigcap L_{\\infty}(\\mN)$ is compactly supported and measurable.\nHere, $\\overrightarrow{\\text{supp}}\\ f$ denotes the support of $f$ as an operator-valued function in $\\R^d$ rather than the support projection as an element in a von Neumann algebra.\nFor any $\\epsilon>0$, the associated truncated singular integral $T_{\\epsilon}f$ is defined as\n \\begin{align}\\label{def of T epsilon f}\n T_{\\epsilon}f(x) = \\int_{|x-y|>\\epsilon} K(x,y) f(y) {\\rm d} y.\n \\end{align}\n\nThe main result\nis stated as follows.\n\n\\textbf{Step 1.}\nLet $T_{\\epsilon}$ be a noncommutative Calder\\'on-Zygmund truncated operator with a complex kernel $K$ (defined as in (\\ref{def of T epsilon f})).\nWe decompose $K$ into its real and imaginary parts: $\\Re(K)$ and $\\Im(K)$.\nThe truncated operators associated with $\\Re(K)$ and $\\Im(K)$ are then given by\n \\begin{align*}\n \\Re(T_{\\epsilon})f(x) = \\int_{|x-y|>\\epsilon} \\Re(K)(x,y) f(y) \\dif y\n \\end{align*}\nand\n \\begin{align*}\n \\Im(T_{\\epsilon})f(x) = \\int_{|x-y|>\\epsilon} \\Im(K)(x,y) f(y) \\dif y.\n \\end{align*}\n\\begin{lemma}\\label{reduction to real kernel}\n Let $T$ be a noncommutative Calder\\'on-Zygmund operator (defined as in (\\ref{def of Tf})) whose kernel $K$ satisfies (\\ref{Size condition}) and the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\n Let $T_{\\epsilon}$ be defined as in (\\ref{def of T epsilon f}).\n Then,\n \\begin{enumerate}[(i)]\n \\item Both $\\Re(K)$ and $\\Im(K)$ satisfy (\\ref{Size condition}) and (\\ref{L2 mean Hormander condition}).\n \\item If $(T_{\\epsilon})_{\\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $p_0\\in (1,\\infty)$, then, both $(\\Re(T_{\\epsilon}))_{\\epsilon>0}$ and $(\\Im(T_{\\epsilon}))_{\\epsilon>0}$ are of strong type $(p_0,p_0)$.\n \\item If $(\\Re(T_{\\epsilon}))_{\\epsilon>0}$ and $(\\Im(T_{\\epsilon}))_{\\epsilon>0}$ are of weak type $(1,1)$, so is $(T_{\\epsilon})_{\\epsilon>0}$.\n \\end{enumerate}\n\\end{lemma}\nBy this lemma, we may assume that the kernel $K$ is real throughout this paper.\n\n\\begin{lemma}\\label{reduction to lacunary sequence}\n Let $T$ be a noncommutative Calder\\'on-Zygmund operator (defined as in (\\ref{def of Tf})) whose kernel $K$ satisfies (\\ref{Size condition}) and the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\n Let $T_{\\epsilon}^{i_{\\epsilon}}$ be defined as in (\\ref{T epsilon,i epsilon f(x)}).\n Then\n \\begin{enumerate}[(i)]\n \\item $(T_{\\epsilon}^{i_{\\epsilon}})_{\\epsilon>0}$ is of strong type $(p,p)$ for $1<p\\leq\\infty$.\n \\item $(T_{\\epsilon}^{i_{\\epsilon}})_{\\epsilon>0}$ is of weak type $(1,1)$.\n \\end{enumerate}\n \\end{lemma}\n\nThe preceding two lemmas imply that Theorem \\ref{Main theorem} follows directly from the weak type (1,1) boundedness of $S_i$ given in (\\ref{Sif(x)}).\nThus, it remains to prove\n\\begin{theorem}\\label{Second theorem}\n Let $T$ be a noncommutative Calder\\'on-Zygmund operator defined as in (\\ref{def of Tf}) with the real kernel $K$ satisfying (\\ref{Size condition}) and the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\n Let $S_i$ be defined as in (\\ref{Sif(x)}).\n Then, the sequence of operators $(S_i)_{i\\in\\Z}$ is of weak type $(1,1)$, that is, for any $f\\in L_1(\\mN)$,\n \\[\n \\norm{(S_i f)_{i\\in\\Z}}_{\\Lambda_{1,\\infty}(\\mN;\\ell_{\\infty})} \\lesssim \\norm{f}_{L_1(\\mN)}.\n \\]\n More precisely, for any $f\\in L_1(\\mN)$ and $\\lambda>0$, there exists a projection $e\\in\\mN$ such that\n \\[\n \\sup_{i\\in\\Z}\\norm{e S_if e}_{L_{\\infty(\\mN)}} \\leq \\lambda \\quad\\text{and}\\quad \\phi(e^{\\bot})\\lesssim \\lambda^{-1} \\norm{f}_{L_1(\\mN)} .\n \\]\n\\end{theorem}\n\n\\subsection{Estimate for the good function (\\ref{estimate of good function}) }\nFor any $i\\in\\Z$, it follows from the definitions of $S_i$ and $T_{\\epsilon}^{i_{\\epsilon}}$ ((\\ref{Sif(x)}) and (\\ref{T epsilon,i epsilon f(x)})) that there exists $\\epsilon_i > 0$ such that\n \\[ S_i = S_{\\epsilon_i}- T_{\\epsilon_i}^i. \\]\nUnder the assumptions of Theorem \\ref{Main theorem} and Lemma \\ref{reduction to lacunary sequence} $(i)$, both $(S_{\\epsilon_i})_{i \\in \\mathbb{Z}}$ and $(T_{\\epsilon_i}^i)_{i \\in \\mathbb{Z}}$ are of strong type $(p_0, p_0)$.\nTherefore, from the above equality, $(S_i)_{i \\in \\mathbb{Z}}$ is also of strong type $(p_0, p_0)$.\nIn addition, since $g$ is positive (by Theorem \\ref{nc CZ decomposition with convolution}) and the kernel $K$ is real-valued (as stated in Section \\ref{Section: Two reductions}), the operator $S_i g$ is self-adjoint.\nThese two facts imply that for any $i\\in\\Z$, we can find a positive element $a\\in\\mN$ such that for any $i\\in\\Z$,\n \\[\n -a \\leq S_i g \\leq a \\quad\\text{and}\\quad \\norm{a}_{L_{p_0}(\\mN)}\\lesssim \\norm{g}_{L_{p_0}(\\mN)}.\n \\]\nTaking $e_1 = \\chi_{(0,\\lambda]}(a)$. Then we have\n \\[\n -\\lambda \\leq -e_1 a e_1 \\leq e_1 S_i g e_1 \\leq e_1 a e _1 \\leq \\lambda.\n \\]\nFinally, applying the Chebyshev inequality, the H\\\"older inequality, along with the boundedness of $g$ in Lemma \\ref{properties of nc CZ decomposition with convolution} $(i)$, we obtain\n \\begin{align*}\n \\phi(e_1^{\\bot})\n \\leq \\lambda^{-p_0} \\norm{a}_{L_{p_0}(\\mN)}^{p_0}\n \\leq \\lambda^{-p_0} \\norm{g}_{L_{\\infty}(\\mN)}^{p_0-1} \\norm{g}_{L_1(\\mN)}\n \\lesssim \\lambda^{-1} \\norm{f}_{L_1(\\mN)},\n \\end{align*}\nwhich yields (\\ref{estimate of good function}).\n\nTo estimate $F_1$.\nWe first observe that by the supports of $h_{r_{n-j}}$ and $\\chi_Q$, the noncommutative $L_1$-norm in $F_1$ equals\n \\begin{align*} \\Big\\|\\int_Q \\Big(\\int_{|u-y|\\leq r_{n-j}} (K_j(x,y)-K_j(x,c(Q))) h_{r_{n-j}}(y-u)\\dif y \\Big)p_Q f(u) q_Q\\dif u \\Big\\|_{L_1(\\mM)}.\n \\end{align*}\nBy using Lemma \\ref{a useful lemma for bad function} (in Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms}), the above estimate is bounded by a product\n \\begin{align*}\n & \\Big\\| \\Big(\\int_{Q} \\Big(\\int_{B_{r_{n-j}}(u)} \\abso{K_i(x,y)-K_i(x,c(Q))} |h_{r_{n-j}}(y-u)|\\dif y \\Big)^2 p_Q f(u)p_Q \\dif u\\Big)^{^{\\frac{1}{2}}}\\Big\\|_{L_{1}(\\mM)} \\\\\n & \\cdot \\Big\\|\\Big(\\int_{Q} q_Q f(u) q_Q \\dif u\\Big)^{\\frac{1}{2}} \\Big\\|_{L_{\\infty}(\\mM)}.\n \\end{align*}\nFor the second term, an application of \\eqref{q_Q f_Q q_Q < lambda q_Q} shows that\n \\begin{align*} \\Big\\|\\Big(\\int_{Q} q_Q f(u) q_Q \\dif u\\Big)^{\\frac{1}{2}} \\Big\\|_{L_{\\infty}(\\mM)} \\lesssim (|Q|\\lambda)^{\\frac{1}{2}}.\n \\end{align*}\nHence,\n \\begin{align}\n F_1\n \\lesssim &\\sum_{n\\geq 2} \\sum_{j\\leq n-1} \\sum_{Q\\in\\mQ_{n-j}} (|Q|\\lambda)^{\\frac{1}{2}} \\notag\\\\\n & \\label{xxxyyy}\\frac{1}{|B_{r_{n-j}}|}\\int_{\\R^d} \\Big\\|\\int_{Q} \\Big(\\int_{B_{r_{n-j}}(u)} \\abso{K_i(x,y)-K_i(x,c(Q))} \\dif y \\Big)^2 p_Q f(u)p_Q \\dif u\\Big\\|_{L_{\\frac{1}{2}}(\\mM)}^{\\frac{1}{2}} \\dif x.\n \\end{align}\nThe same argument as in (\\ref{B(c(Q))}) allows us to majorize (\\ref{xxxyyy}) by\n \\begin{align*}\n \\frac{1}{|B_{r_{n-j-1}}|}\\int_{\\R^d} \\Big\\|\\int_{Q} \\Big(\\int_{B_{r_{n-j-1}}(c(Q))} \\abso{K_j(x,y)-K_j(x,c(Q))} \\dif y \\Big)^2 p_Q f(u)p_Q \\dif u\\Big\\|_{L_{\\frac{1}{2}}(\\mM)}^{\\frac{1}{2}} \\dif x.\n \\end{align*}\nBy Lemma \\ref{the second lemma for bad function} $(i)$ and the $L_2$-mean H\\\"ormander condition \\eqref{L2 mean Hormander condition}, we obtain\n \\begin{align*}\n F_1 &\\lesssim \\sum_{n\\geq 2} \\sum_{j\\leq n-1} \\sum_{Q\\in\\mQ_{n-j}}\n \\lbracket{ \\big(H_2^2(n-1) + H_2^2(n) + 2^{-2(n-1)}\\big) \\tau(p_Q) \\phi(fp_Q \\chi_{Q}) }^{\\frac{1}{2}} (|Q| \\lambda)^{\\frac{1}{2}} \\\\\n &\\lesssim \\sum_{n\\geq 2} \\lbracket{2H_2(n-1)+2^{-n+1}} \\sum_{j\\leq n-1} \\sum_{Q\\in\\mQ_{n-j}}\\big(\\tau(p_Q) \\phi(fp_Q \\chi_{Q}) |Q| \\lambda \\big)^{\\frac{1}{2}} \\\\\n &\\lesssim \\sum_{m\\geq 1} \\sum_{Q\\in\\mQ_m} \\big( \\tau(p_Q) \\phi(fp_Q \\chi_{Q}) |Q| \\lambda \\big)^{\\frac{1}{2}}.\n \\end{align*}\nFinally, applying the Cauchy-Schwarz inequality twice, we estimate $F_1$ as follows:\n \\begin{align*}\n F_1 \\lesssim \\Big(\\sum_{m=1}^{\\infty} \\lambda \\phi(p_m)\\Big)^{\\frac{1}{2}}\n \\Big(\\sum_{m=1}^{\\infty} \\phi(f p_m)\\Big)^{\\frac{1}{2}}\n = \\big(\\lambda \\phi(1-q)\\big)^{\\frac{1}{2}} \\big(\\phi(f(1-q))\\big)^{\\frac{1}{2}}\n &\\lesssim \\norm{f}_{L_1(\\mN)}.\n \\end{align*}",
    "post_theorem_intro_text_len": 2415,
    "post_theorem_intro_text": "Our main strategy in this paper is to develop a refined noncommutative Calder\\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).\nThe previous methods are insufficient in this context.\nOn one hand, Parcet's decomposition \\cite{Parcet2009Pseudo-localization} requires the stronger Lipschitz regularity condition (\\ref{Lipschitz regularity condition}) and may not be directly applied to the noncommutative maximal operator.\nOn the other hand, while the decomposition by Cadilhac et al. \\cite{Cadilhac2022Spectral} can deal with the $L_2$-integral regularity condition (\\ref{sum delta < infty}) with $q=2$, it does not cover our target.\nTo address this gap, we introduce a slightly new noncommutative Calder\\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.\n\nThis paper is organized as follows.\nWe begin by presenting our key tool: the refined noncommutative Calder\\'on-Zygmund decomposition (Theorem \\ref{nc CZ decomposition with convolution}) in Section \\ref{Section: Noncommutative C-Z decomposition}.\nNext, Section \\ref{Section: Two reductions} starts the proof of the main result (Theorem \\ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem \\ref{Second theorem}).\nWith this reduction established, Section \\ref{Proof of the main theorem} is devoted to proving Theorem \\ref{Second theorem}.\nFinally, some related background concepts are included in Appendix \\ref{Appendix: Noncommutative Lp spaces and maximal norms}.\n\nNow, we conclude this section by listing the notation required for our later analysis.\\\\\n\n\\textbf{Notation}\n\\begin{itemize}\n\t\\item $\\mQ_n$: the set of all dyadic cubes of side length $2^{-n}$, $n\\in\\Z$.\n\t\\item $Q_{x,n}$: the unique cube in $\\mQ_n$ containing $x\\in\\R^d$.\n\t\\item $\\mQ$: the set of all standard dyadic cubes in $\\R^d$, that is, $\\mQ = \\bigcup\\limits_n\\mQ_n$.\n\t\\item $c(Q)$: the center of the cube $Q$.\n\t\\item $lQ$: the concentric cube sharing the center of  $Q$ such that its length is $l$ times the length of $Q$.\n\t\\item $B_r(x)$: the ball with $x\\in\\R^d$ as its center and $r>0$ as its radius, and the center $x$ will be omitted when it is the origin.\n\t\\item $|E|$: the volume of set $E\\subset\\R^d$.\n\t\\item $[k]$: the integer part of $k$.\n\n\\end{itemize}",
    "sketch": "To prove Theorem~\\ref{Main theorem}, the paper’s strategy is to “develop a refined noncommutative Calder\\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\\\"ormander condition (\\ref{L2 mean Hormander condition}).” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”\n\nThe proof of Theorem~\\ref{Main theorem} then proceeds via the following stated structure: (1) present the refined decomposition (Theorem~\\ref{nc CZ decomposition with convolution}); (2) in Section~\\ref{Section: Two reductions}, “start the proof of the main result (Theorem~\\ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\\ref{Second theorem} in Section~\\ref{Proof of the main theorem}.",
    "expanded_sketch": "To prove the main theorem, the paper’s strategy is to “develop a refined noncommutative Calder\\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\\\"ormander condition\n\\begin{align}\\label{L2 mean Hormander condition}\n\t\\sum_{m\\geq 1} H_2 (m) < \\infty,\n    \\end{align}\n.” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”\n\nIn establishing the main theorem, the proof then proceeds via the following stated structure: (1) present the refined decomposition. We first prove the following theorem.\n\n\\begin{theorem} \\label{nc CZ decomposition with convolution}\n\tFix $f\\in\\mN_{c,+}$, $\\lambda>0$, and $s\\in\\N$.\n\tLet $(q_n)_{n\\in\\Z}$ and $(p_n)_{n\\in\\Z}$ be the sequences of projections in Lemma \\ref{cuculescu's construction}, where $q_n = \\sum\\limits_{Q\\in\\mQ_n} q_Q \\chi_Q$, $p_n = \\sum\\limits_{Q\\in\\mQ_n} p_Q \\chi_Q$.\n\tSuppose $h\\in C_{c}^{\\infty} (\\R^d)$ satisfies\n    \\begin{align*}\n    \\supp h \\subset B_1,\\quad \\int_{\\R^d} h =1, \\quad h\\geq 0.\n    \\end{align*}\n    Define $h_r(x)= \\frac{1}{r^d} h(\\frac{x}{r})$ and $r_n= 2^{-n-1}\\sqrt{d}$.\n\tThen, $f$ admits the decomposition:\n\t\\begin{align*}\n\tf = g+ b\n\t  = g + \\widetilde{b}_{\\text{d}} + b_{\\text{d}}^0 + \\widetilde{b}_{\\text{off}} + b_{\\text{off}}^0,\n\t\\end{align*}\n\twhere $g$ is positive and $b$ is self-adjoint in $\\mN$.\n\tMore precisely, these components are defined as follows:\n\t    \\[g=qfq + \\sum\\limits_{n\\in\\Z}p_n f_n p_n,\\]\n        \\begin{gather*}\n        \\widetilde{b}_{\\text{d}} = \\sum\\limits_{n\\in\\Z} \\widetilde{b}_{\\text{d},n} = \\sum\\limits_{n\\in\\Z}\\sum\\limits_{Q\\in\\mQ_n}\\widetilde{b}_{\\text{d},n}^Q ,\\quad\n        b_{\\text{d}}^0 = \\sum\\limits_{n\\in\\Z} b_{\\text{d},n}^0=\\sum\\limits_{n\\in\\Z} \\sum\\limits_{Q\\in\\mQ_n}(b_{\\text{d},n}^Q-\\widetilde{b}_{\\text{d},n}^Q),\\\\\n        \\widetilde{b}_{\\text{off}} = \\sum\\limits_{n\\in\\Z} \\widetilde{b}_{\\text{off},n} = \\sum\\limits_{n\\in\\Z} \\sum\\limits_{Q\\in\\mQ_n} \\widetilde{b}_{\\text{off},n}^Q ,\\quad\n        b_{\\text{off}}^0 = \\sum\\limits_{n\\in\\Z} b_{\\text{off},n}^0 = \\sum\\limits_{n\\in\\Z}\\sum\\limits_{Q\\in\\mQ_n}(b_{\\text{off},n}^Q-\\widetilde{b}_{\\text{off},n}^Q),\n        \\end{gather*}\n     where for each $Q\\in\\mQ_n$,\n        \\begin{gather*}\n     \tb_{\\text{d},n}^Q = p_Q(f-f_n)p_Q\\chi_Q,\n     \t\\quad b_{\\text{off},n}^Q = p_Q(f-f_n)q_Q\\chi_Q + q_Q(f-f_n)p_Q\\chi_Q, \\\\\n     \t\\widetilde{b}_{\\text{d},n}^Q = b_{\\text{d},n}^Q * h_{r_n},\n     \t\\quad \\widetilde{b}_{\\text{off},n}^Q = b_{\\text{off},n}^Q * h_{r_n}.\n        \\end{gather*}\n    \\end{theorem}\n\n(2) next, “start the proof of the main result, reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\\ref{Second theorem} later. \\label{Proof of the main theorem}\nNote that every operator in a von Neumann algebra can be decomposed into a linear combination of four positive elements.\nBy this fact and the density of $\\mN_{c,+}$ i",
    "expanded_theorem": "\\label{Main theorem}\n\tSuppose $T$ is a Calder\\'on-Zygmund operator defined as in\n\\begin{align}\\label{def of Tf}\n\tT f(x) = \\int_{\\R^d} K(x,y) f(y) \\dif y, \\quad x\\notin \\overrightarrow{\\supp}\\ f,\n    \\end{align}\nassociated with a kernel satisfying\n\\begin{align} \\label{Size condition}\n       |K(x,y)| \\leq \\frac{C_1}{|x-y|^d} ;\n       \\end{align}\nand the $L_2$-mean H\\\"ormander condition\n\\begin{align}\\label{L2 mean Hormander condition}\n\t\\sum_{m\\geq 1} H_2 (m) < \\infty,\n    \\end{align}\n\tLet $T_{\\epsilon}$ be defined as in\n\\begin{align}\\label{def of T epsilon f}\n\tT_{\\epsilon}f(x) = \\int_{|x-y|>\\epsilon} K(x,y) f(y) \\dif y.\n    \\end{align}\n\tSuppose $(T_{\\epsilon})_{\\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\\infty$, that is, for any $f\\in L_{p_0}(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{L_{p_0}(\\mN;\\ell_{\\infty})} \\lesssim  \\left\\| f \\right\\| _{L_{p_0}(\\mN)}.\n\t\\]\n\tThen, $(T_{\\epsilon})_{\\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\\in L_1(\\mN)$,\n\t\\[\n\t\\norm{(T_{\\epsilon}f)_{\\epsilon>0}}_{\\Lambda_{1,\\infty}(\\mN;\\ell_{\\infty})} \\lesssim  \\left\\| f \\right\\| _{L_1(\\mN)}.\n\t\\]\nWe refer the reader to\n\\label{Appendix: Noncommutative Lp spaces and maximal norms}\nThis part consists of three subsections: noncommutative $L_p$ spaces, noncommutative maximal norms, and noncommutative square functions.\nAl\nfor the definitions of the noncommutative maximal norm $ \\left\\| \\cdot \\right\\| _{L_p(\\mN; \\ell_{\\infty})}$ and the noncommutative weak maximal norm $ \\left\\| \\cdot \\right\\| _{\\Lambda_{p,\\infty}(\\mN;\\ell_{\\infty})}$.,",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal M\\) be a von Neumann algebra with a normal semifinite faithful trace \\(\\tau\\), and set \\(\\mathcal N=L_\\infty(\\mathbb R^d)\\bar\\otimes \\mathcal M\\) with tensor trace \\(\\phi=\\int\\otimes \\tau\\). Suppose \\(T\\) is a Calder\\'on-Zygmund operator acting on \\(\\mathcal N\\)-valued functions by\n\\[\nTf(x)=\\int_{\\mathbb R^d} K(x,y)f(y)\\,dy\n\\]\nfor \\(x\\) outside the support of \\(f\\), where the kernel satisfies the size estimate\n\\[\n|K(x,y)|\\le \\frac{C_1}{|x-y|^d}.\n\\]\nAssume also that \\(K\\) satisfies the \\(L_2\\)-mean H\\\"ormander condition\n\\[\n\\sum_{m\\ge 1} H_2(m)<\\infty,\n\\]\nwhere\n\\[\nH_2(m)=\\sup_{y\\in\\mathbb R^d,\\,R>0}\\left(\\frac{(2^mR)^d}{|B_R|}\\int_{|v|\\le R}\\int_{2^mR\\le |x-y|\\le 2^{m+1}R}|K(x,y+v)-K(x,y)|^2\\,dx\\,dv\\right)^{1/2},\n\\]\nand \\(B_R\\) is the Euclidean ball of radius \\(R\\) centered at the origin. For \\(\\varepsilon>0\\), define the truncated operators\n\\[\nT_\\varepsilon f(x)=\\int_{|x-y|>\\varepsilon} K(x,y)f(y)\\,dy.\n\\]\nSuppose that for some \\(1<p_0<\\infty\\), the family \\((T_\\varepsilon)_{\\varepsilon>0}\\) is of strong type \\((p_0,p_0)\\), i.e.\n\\[\n\\|(T_\\varepsilon f)_{\\varepsilon>0}\\|_{L_{p_0}(\\mathcal N;\\ell_\\infty)}\\lesssim \\|f\\|_{L_{p_0}(\\mathcal N)}\n\\quad\\text{for all }f\\in L_{p_0}(\\mathcal N).\n\\]\nWhich quantitative estimate is valid under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "The family \\((T_\\varepsilon)_{\\varepsilon>0}\\) is of weak type \\((1,1)\\): for every \\(f\\in L_1(\\mathcal N)\\),\n\\[\n\\|(T_\\varepsilon f)_{\\varepsilon>0}\\|_{\\Lambda_{1,\\infty}(\\mathcal N;\\ell_\\infty)}\\lesssim \\|f\\|_{L_1(\\mathcal N)}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The family \\((T_\\varepsilon)_{\\varepsilon>0}\\) is of strong type \\((1,1)\\): for every \\(f\\in L_1(\\mathcal N)\\),\n\\[\n\\|(T_\\varepsilon f)_{\\varepsilon>0}\\|_{L_1(\\mathcal N;\\ell_\\infty)}\\lesssim \\|f\\|_{L_1(\\mathcal N)}.\n\\]"
        },
        {
          "label": "C",
          "text": "For every \\(f\\in L_1(\\mathcal N)\\) and every fixed \\(\\varepsilon>0\\), the truncated operator \\(T_\\varepsilon\\) is of weak type \\((1,1)\\):\n\\[\n\\|T_\\varepsilon f\\|_{\\Lambda_{1,\\infty}(\\mathcal N)}\\lesssim \\|f\\|_{L_1(\\mathcal N)}.\n\\]"
        },
        {
          "label": "D",
          "text": "There exists \\(p_0\\in(1,\\infty)\\) such that whenever \\((T_\\varepsilon)_{\\varepsilon>0}\\) is of strong type \\((p_0,p_0)\\), one has for every \\(f\\in L_1(\\mathcal N)\\),\n\\[\n\\|(T_\\varepsilon f)_{\\varepsilon>0}\\|_{\\Lambda_{1,\\infty}(\\mathcal N;\\ell_\\infty)}\\lesssim \\Big(\\sum_{m\\ge 1} H_2(m)^2\\Big)^{1/2}\\,\\|f\\|_{L_1(\\mathcal N)}.\n\\]"
        },
        {
          "label": "E",
          "text": "The family \\((T_\\varepsilon)_{\\varepsilon>0}\\) is of weak type \\((1,1)\\) provided the kernel satisfies only the \\(L_2\\)-mean H\\\"ormander condition and the size estimate; equivalently, for every \\(f\\in L_1(\\mathcal N)\\),\n\\[\n\\|(T_\\varepsilon f)_{\\varepsilon>0}\\|_{\\Lambda_{1,\\infty}(\\mathcal N;\\ell_\\infty)}\\lesssim \\|f\\|_{L_1(\\mathcal N)},\n\\]\nwithout assuming any strong type \\((p_0,p_0)\\) bound for \\((T_\\varepsilon)_{\\varepsilon>0}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "weak_vs_strong_endpoint_conclusion",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "maximal_family_conclusion_replaced_by_each_fixed_truncation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "summability_in_H2_replaced_by_square_summability_constant",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "dependence_on_assumed_strong_type_p0_p0",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the endpoint weak-(1,1) conclusion. It presents hypotheses and asks for the valid consequence, without giving away the answer directly."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct option is the theorem's endpoint conclusion almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options distinguish maximal-family weak type, fixed-ε weak type, nonuniform bounds, and an unjustified strong-(1,1) upgrade. Still, the item mainly tests recognition/recall of the stated theorem rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well-targeted: one is a weaker true statement, one adds an unnecessary stronger hypothesis, one weakens uniformity, and one overstates the endpoint to strong type. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall-oriented MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a theorem and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.06908v1",
    "paper_link": "http://arxiv.org/abs/2512.06908v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "maintheorem",
      "content": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\\ref{main-propo-first-order}, below. \nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.",
      "start_pos": 16280,
      "end_pos": 17547,
      "label": "main-theo"
    },
    "ref_dict": {
      "main-propo-first-order": "\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}",
      "definition": "\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}",
      "Eq1-01": "\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\"
    },
    "pre_theorem_intro_text_len": 8262,
    "pre_theorem_intro_text": "\\label{section---1}\n\n\\subsection{Purpose of this paper}\n\n\\paragraph{Theories of modified gravity}\n\nIn recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations. \nIndeed, only a few studies address the formulation and well-posedness of these theories \\cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}. \nMore recently, alternative, well-posed theories have been proposed; cf.~\\cite{Brady-Figueras,Figueras} (and the references therein).\n\nAmong the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \\cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \\emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \\cite{LeFlochMa23}.\n\n\\paragraph{Evolution in spherical symmetry.}\n\nIn this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \\cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \\cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \\cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \\emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.\n\n\\paragraph{Aim of this paper.}\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field. \n\n\\subsection{A first-order formulation of $f(R)$ gravity}\n\n\\paragraph{Action of $f(R)$ modified gravity.}\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g  \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+  \\big( g_{ab} \\, \\Box_g   - \\nabla_a\\nabla_b \\big)  \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free. \n\n\\paragraph{Main result.}\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below):  \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter  for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows. \n\n\\vskip.15cm",
    "context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm",
    "full_context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\n\\vskip.15cm\n\n\\vskip.15cm\n\nIn the modified field equations, there are terms that contain derivatives of the scalar\ncurvature, that is, third order derivatives of the metric functions $\\nu$ and $\\lambda$. This leads to an essential difficulty in dealing with the equations.\nFollowing LeFloch and Ma~\\cite{LeFlochMa17a}, we introduce an \\emph{augmented system}\n where the relation\n\\be\n\\rho = \\frac{1}{\\kappa}\\ln f'(R) \\quad \\text{(conformal formulation)} \n\\ee\nis no longer imposed but $\\rho$ \n is regarded as an independent variable.\nTo clarify the notation, we introduce a new independent variable denoted by \n\\be\n\\rhoh \\quad \\text{(augmented conformal formulation)} \n\\ee\nwhich plays the role of $\\rho$ and will coincides with $\\rho$ once the constraint below is enforced. \nIn view of Proposition~\\ref{prop essential f(R) Bondi}, this leads us to the following system \n\\bel{Bondi system diff augmented1.5}\n\\aligned\n\\del_r(\\nu+\\lambda)\n& = \\frac{3}{4}\\kappa^{2}r|\\del_r\\rhoh|^2 + 4\\pi r e^{- \\kappa\\rhoh} \\, |\\del_r\\phi|^2,\n\\\\\n\\del_r\\big(re^{\\nu- \\lambda}\\big)\n & = \\big(1 - r^{2}e^{-2\\kappa\\rhoh} \\big( \\Vstar(\\rhoh) + 8\\pi U(\\phi) \\big) \\big)e^{\\nu+\\lambda},\n \\\\\nD(\\del_r(r\\phi)) + {r \\over 2} e^{\\nu+\\lambda- \\kappa\\rhoh}U'(\\phi)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi \\\\\n& \\quad + \\frac{\\kappa r}{2}\\big(D\\rhoh \\del_r\\phi + D\\phi \\del_r\\rhoh\\big), \n\\\\\nD\\big(\\del_r(r\\rhoh)\\big) + {r \\over \\kappa} e^{\\nu+\\lambda} \\Wstar(\\rhoh)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rhoh\n\\\\\n& - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rhoh}\\del_r\\phi D\\phi\n + {r \\over \\kappa} e^{\\nu+\\lambda} \\frac{16\\pi}{3}e^{-2\\kappa\\rhoh}U(\\phi).\n\\endaligned\n\\ee\nObserve that, as before, the last equation of \\eqref{Bondi system diff augmented1.5} is equivalent to\n\\be\n\\label{box-tilde-rho}\n\\Box_{\\gt}\\rhoh - \\frac{2}{\\kappa} \\Wstar(\\rhoh)\n= - {8\\pi \\over 3\\kappa}e^{-2\\kappa\\rhoh}\\big(\\sigma+4 \\, U(\\phi)\\big).\n\\ee\nNow that we have defined the augmented system, the next question is whether a solution of \\eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \\eqref{equa-three-equa}, \\eqref{Bondi scalar}, and \\eqref{eq:599} \nmust additionally satisfy the condition\n\\bel{Bondi system diff augmented1.5 constraint}\nf'(R) = e^{\\kappa\\rhoh},\n\\ee\nwhich we regard as a nonlinear differential constraint on the solutions.\n Conversely, any classical solution of the original $f(R)$ system with $\\rho=\\frac{1}{\\kappa}\\ln f'(R)$ satisfies \\eqref{Bondi system diff augmented1.5} upon setting $\\rhoh=\\rho$; thus the two formulations are equivalent in the admissible class once \\eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)\n\nUnder the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\\ref{section---A2})) to the the following integro-differential system with the main unknowns $\\phi$ and $\\rho$: \n\\bel{notresysteme}\n\\aligned\nD(\\del_r(r\\phi))\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi\n\\\\\n & \\quad + \\frac{\\kappa r}{2}\\big(D\\rho \\del_r\\phi + D\\phi \\del_r\\rho\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\rho}U'(\\phi),\n\\\\\nD\\big(\\del_r(r\\rho)\\big)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rho\n\\\\\n& \\quad - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rho}\\del_r\\phi D\\phi- {r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\rho) - \\frac{16\\pi}{3}e^{-2\\kappa\\rho}U(\\phi)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficients $\\nu, \\lambda$ are given by \n\\bel{eq:403}\n\\aligned\n\\nu + \\lambda\n& = - \\frac{3}{4}\\kappa^{2} \\int_r^{+ \\infty}s|\\del_r\\rho|^2ds - 4\\pi\\int_r^{+ \\infty} e^{- \\kappa\\rho} \\, s|\\del_r\\phi|^2 \\, ds,\n\\\\\ne^{\\nu- \\lambda}\n& = {1 \\over r} \\int_0^r\\Big(1 - s^{2}e^{-2\\kappa\\rho} \\big( \\Vstar(\\rho) + 8\\pi U(\\phi) \\big)\\Big)e^{\\nu+\\lambda} \\, ds. \n\\endaligned\n\\ee\nWe refer to \\eqref{notresysteme} as the \\emph{augmented conformal system} of modified gravity.\n\nSuppose now that $h$ and $l$ are solutions to the system \\eqref{Bondi system int augmented3} defined on $[0,u_0]\\times [0,+ \\infty)$. If \n\\bel{center.condition.2}\nh,l\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big),\n\\ee\nthen the first and second regularity conditions at the center hold. In fact, it follows from the system \\eqref{Bondi system int augmented3} that\n\\be\n\\label{more-conditions-center}\n\\hb,\\,\\lb,\\,\\nu,\\,\\lambda\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big).\n\\ee\nIn view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, we get\n\\be\n\\aligned\n& r^2{e^{-\\kappa\\lb}}\\big(E-8\\pi T\\big)(D,D) \n\\\\\n& = r{e^{\\nu- \\lambda}}\\Big(\\frac{e^{\\nu- \\lambda}}{2}\\del_r(\\nu+\\lambda) - 2\\del_u\\lambda\\Big)\n- \\frac{3r^2\\kappa^{2}{ } }{2}(D\\lb)^2-8\\pi r^2{ }(D\\hb)^2.\n\\endaligned\n\\ee\nHence, from the regularity conditions, and taking into account \\eqref{more-conditions-center}, we deduce that\n\\be\n\\lim_{r\\to0}{\\Big(r^2e^{-\\kappa\\rho}\\big(E-8\\pi T\\big)(D,D)\\Big)}= 0, \n\\ee\nsince $\\nu,\\lambda=O(r)$ and $D\\hb,D\\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute \n\\be\nr\\del_r\\lb = l- \\lb, \\qquad r\\del_r\\hb = h- \\hb.\n\\ee \n This shows that $h=\\partial_r(r\\phi)$ and $l=\\partial_r(r\\rho)$ are $C^1$ on $[0,u_0]\\times[0,+\\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\\in C^0$, the right-hand sides of \\eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\\in C^1$, so the two formulations are equivalent in the admissible class.",
    "post_theorem_intro_text_len": 1131,
    "post_theorem_intro_text": "\\vskip.15cm\n\nWe point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\\phi \\, U'(\\phi) \\geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \\emph{defocusing}, so that the forward evolution will not be limited by the matter model. \nThe conditions ${U(\\phi) \\geq 0}$ and ${f(R) \\leq R \\, f'(R)}$ are required to prove that the Hawking-mass is non-negative. \n\n\\subsection{Outline of this paper.}\n\nIn Section~\\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\\ref{section---A2}, we analyze the regularity at the center. In Section~\\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.",
    "sketch": "In the discussion after Theorem~\\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”",
    "expanded_sketch": "In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”",
    "expanded_theorem": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.\n\n\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}\n\nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.\n\n\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}\n\nFurthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,",
    "theorem_type": [
      "Implication",
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
      },
      "choices": [
        {
          "label": "B",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "C",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "D",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "E",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "center regularity needed for equivalence",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "equivalence and Hawking-mass monotonicity clauses removed",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "direction/sign of Hawking-mass monotonicity along null directions",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "nonnegative Hawking mass incorrectly upgraded to constancy",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option directly. It states the hypotheses and asks for the equivalent formulation, but the correct answer must still be identified from several closely related alternatives."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct option is essentially the exact equivalence statement under the stated hypotheses. However, it is not a pure restatement because the distractors alter direction of implication, integration bounds, and regularity assumptions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish exact equivalence from weaker or tampered variants, especially regarding the admissible class and the metric integral formula. Still, the task is mainly precise recognition of a known theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: incorrect boundary/integration range, one-way implication instead of equivalence, dropped regularity hypotheses, and an unjustified extra conclusion. They are distinct and nontrivial."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-recognition-heavy MCQ. It avoids answer leakage and uses strong distractors, but it tests precision of recall/comparison more than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.06908v1",
    "paper_link": "http://arxiv.org/abs/2512.06908v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "maintheorem",
      "content": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\\ref{main-propo-first-order}, below. \nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.",
      "start_pos": 16280,
      "end_pos": 17547,
      "label": "main-theo"
    },
    "ref_dict": {
      "main-propo-first-order": "\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}",
      "definition": "\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}",
      "Eq1-01": "\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\"
    },
    "pre_theorem_intro_text_len": 8262,
    "pre_theorem_intro_text": "\\label{section---1}\n\n\\subsection{Purpose of this paper}\n\n\\paragraph{Theories of modified gravity}\n\nIn recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations. \nIndeed, only a few studies address the formulation and well-posedness of these theories \\cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}. \nMore recently, alternative, well-posed theories have been proposed; cf.~\\cite{Brady-Figueras,Figueras} (and the references therein).\n\nAmong the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \\cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \\emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \\cite{LeFlochMa23}.\n\n\\paragraph{Evolution in spherical symmetry.}\n\nIn this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \\cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \\cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \\cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \\emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.\n\n\\paragraph{Aim of this paper.}\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field. \n\n\\subsection{A first-order formulation of $f(R)$ gravity}\n\n\\paragraph{Action of $f(R)$ modified gravity.}\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g  \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+  \\big( g_{ab} \\, \\Box_g   - \\nabla_a\\nabla_b \\big)  \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free. \n\n\\paragraph{Main result.}\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below):  \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter  for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows. \n\n\\vskip.15cm",
    "context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm",
    "full_context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\n\\vskip.15cm\n\n\\vskip.15cm\n\nIn the modified field equations, there are terms that contain derivatives of the scalar\ncurvature, that is, third order derivatives of the metric functions $\\nu$ and $\\lambda$. This leads to an essential difficulty in dealing with the equations.\nFollowing LeFloch and Ma~\\cite{LeFlochMa17a}, we introduce an \\emph{augmented system}\n where the relation\n\\be\n\\rho = \\frac{1}{\\kappa}\\ln f'(R) \\quad \\text{(conformal formulation)} \n\\ee\nis no longer imposed but $\\rho$ \n is regarded as an independent variable.\nTo clarify the notation, we introduce a new independent variable denoted by \n\\be\n\\rhoh \\quad \\text{(augmented conformal formulation)} \n\\ee\nwhich plays the role of $\\rho$ and will coincides with $\\rho$ once the constraint below is enforced. \nIn view of Proposition~\\ref{prop essential f(R) Bondi}, this leads us to the following system \n\\bel{Bondi system diff augmented1.5}\n\\aligned\n\\del_r(\\nu+\\lambda)\n& = \\frac{3}{4}\\kappa^{2}r|\\del_r\\rhoh|^2 + 4\\pi r e^{- \\kappa\\rhoh} \\, |\\del_r\\phi|^2,\n\\\\\n\\del_r\\big(re^{\\nu- \\lambda}\\big)\n & = \\big(1 - r^{2}e^{-2\\kappa\\rhoh} \\big( \\Vstar(\\rhoh) + 8\\pi U(\\phi) \\big) \\big)e^{\\nu+\\lambda},\n \\\\\nD(\\del_r(r\\phi)) + {r \\over 2} e^{\\nu+\\lambda- \\kappa\\rhoh}U'(\\phi)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi \\\\\n& \\quad + \\frac{\\kappa r}{2}\\big(D\\rhoh \\del_r\\phi + D\\phi \\del_r\\rhoh\\big), \n\\\\\nD\\big(\\del_r(r\\rhoh)\\big) + {r \\over \\kappa} e^{\\nu+\\lambda} \\Wstar(\\rhoh)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rhoh\n\\\\\n& - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rhoh}\\del_r\\phi D\\phi\n + {r \\over \\kappa} e^{\\nu+\\lambda} \\frac{16\\pi}{3}e^{-2\\kappa\\rhoh}U(\\phi).\n\\endaligned\n\\ee\nObserve that, as before, the last equation of \\eqref{Bondi system diff augmented1.5} is equivalent to\n\\be\n\\label{box-tilde-rho}\n\\Box_{\\gt}\\rhoh - \\frac{2}{\\kappa} \\Wstar(\\rhoh)\n= - {8\\pi \\over 3\\kappa}e^{-2\\kappa\\rhoh}\\big(\\sigma+4 \\, U(\\phi)\\big).\n\\ee\nNow that we have defined the augmented system, the next question is whether a solution of \\eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \\eqref{equa-three-equa}, \\eqref{Bondi scalar}, and \\eqref{eq:599} \nmust additionally satisfy the condition\n\\bel{Bondi system diff augmented1.5 constraint}\nf'(R) = e^{\\kappa\\rhoh},\n\\ee\nwhich we regard as a nonlinear differential constraint on the solutions.\n Conversely, any classical solution of the original $f(R)$ system with $\\rho=\\frac{1}{\\kappa}\\ln f'(R)$ satisfies \\eqref{Bondi system diff augmented1.5} upon setting $\\rhoh=\\rho$; thus the two formulations are equivalent in the admissible class once \\eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)\n\nUnder the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\\ref{section---A2})) to the the following integro-differential system with the main unknowns $\\phi$ and $\\rho$: \n\\bel{notresysteme}\n\\aligned\nD(\\del_r(r\\phi))\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi\n\\\\\n & \\quad + \\frac{\\kappa r}{2}\\big(D\\rho \\del_r\\phi + D\\phi \\del_r\\rho\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\rho}U'(\\phi),\n\\\\\nD\\big(\\del_r(r\\rho)\\big)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rho\n\\\\\n& \\quad - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rho}\\del_r\\phi D\\phi- {r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\rho) - \\frac{16\\pi}{3}e^{-2\\kappa\\rho}U(\\phi)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficients $\\nu, \\lambda$ are given by \n\\bel{eq:403}\n\\aligned\n\\nu + \\lambda\n& = - \\frac{3}{4}\\kappa^{2} \\int_r^{+ \\infty}s|\\del_r\\rho|^2ds - 4\\pi\\int_r^{+ \\infty} e^{- \\kappa\\rho} \\, s|\\del_r\\phi|^2 \\, ds,\n\\\\\ne^{\\nu- \\lambda}\n& = {1 \\over r} \\int_0^r\\Big(1 - s^{2}e^{-2\\kappa\\rho} \\big( \\Vstar(\\rho) + 8\\pi U(\\phi) \\big)\\Big)e^{\\nu+\\lambda} \\, ds. \n\\endaligned\n\\ee\nWe refer to \\eqref{notresysteme} as the \\emph{augmented conformal system} of modified gravity.\n\nSuppose now that $h$ and $l$ are solutions to the system \\eqref{Bondi system int augmented3} defined on $[0,u_0]\\times [0,+ \\infty)$. If \n\\bel{center.condition.2}\nh,l\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big),\n\\ee\nthen the first and second regularity conditions at the center hold. In fact, it follows from the system \\eqref{Bondi system int augmented3} that\n\\be\n\\label{more-conditions-center}\n\\hb,\\,\\lb,\\,\\nu,\\,\\lambda\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big).\n\\ee\nIn view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, we get\n\\be\n\\aligned\n& r^2{e^{-\\kappa\\lb}}\\big(E-8\\pi T\\big)(D,D) \n\\\\\n& = r{e^{\\nu- \\lambda}}\\Big(\\frac{e^{\\nu- \\lambda}}{2}\\del_r(\\nu+\\lambda) - 2\\del_u\\lambda\\Big)\n- \\frac{3r^2\\kappa^{2}{ } }{2}(D\\lb)^2-8\\pi r^2{ }(D\\hb)^2.\n\\endaligned\n\\ee\nHence, from the regularity conditions, and taking into account \\eqref{more-conditions-center}, we deduce that\n\\be\n\\lim_{r\\to0}{\\Big(r^2e^{-\\kappa\\rho}\\big(E-8\\pi T\\big)(D,D)\\Big)}= 0, \n\\ee\nsince $\\nu,\\lambda=O(r)$ and $D\\hb,D\\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute \n\\be\nr\\del_r\\lb = l- \\lb, \\qquad r\\del_r\\hb = h- \\hb.\n\\ee \n This shows that $h=\\partial_r(r\\phi)$ and $l=\\partial_r(r\\rho)$ are $C^1$ on $[0,u_0]\\times[0,+\\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\\in C^0$, the right-hand sides of \\eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\\in C^1$, so the two formulations are equivalent in the admissible class.",
    "post_theorem_intro_text_len": 1131,
    "post_theorem_intro_text": "\\vskip.15cm\n\nWe point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\\phi \\, U'(\\phi) \\geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \\emph{defocusing}, so that the forward evolution will not be limited by the matter model. \nThe conditions ${U(\\phi) \\geq 0}$ and ${f(R) \\leq R \\, f'(R)}$ are required to prove that the Hawking-mass is non-negative. \n\n\\subsection{Outline of this paper.}\n\nIn Section~\\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\\ref{section---A2}, we analyze the regularity at the center. In Section~\\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.",
    "sketch": "In the discussion after Theorem~\\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”",
    "expanded_sketch": "In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”",
    "expanded_theorem": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.\n\n\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}\n\nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.\n\n\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}\n\nFurthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,",
    "theorem_type": [
      "Implication",
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
      },
      "choices": [
        {
          "label": "B",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "C",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "D",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        },
        {
          "label": "E",
          "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "center regularity needed for equivalence",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "equivalence and Hawking-mass monotonicity clauses removed",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "direction/sign of Hawking-mass monotonicity along null directions",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "nonnegative Hawking mass incorrectly upgraded to constancy",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It sets up the hypotheses and notation, but the decisive details in the answer choices—especially the center regularity assumptions and the precise Hawking-mass monotonicity directions—are not leaked."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-conclusion matching question: the stem lists hypotheses and asks for the resulting conclusion. The choices do introduce competing variants, so it is not a pure verbatim restatement, but it remains close to recall of a specific theorem rather than a genuinely new inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the solver must distinguish subtle differences involving regularity assumptions, equivalence claims, and monotonicity signs. However, the task is mostly precise theorem recognition, not derivation. Also, choice C is a weaker statement that is still true if A is true, which reduces the need for decisive generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and reflect realistic mathematical errors: omitting needed regularity, flipping a monotonicity direction, or overstating constancy. However, choice C is a weaker true statement rather than a genuinely false distractor, making the item technically ambiguous and lowering distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "Moderately good on leakage avoidance, but only middling as an MCQ. It mainly tests theorem recall, and the presence of a weaker true option makes the question ambiguous rather than cleanly discriminative."
    }
  },
  {
    "id": "2512.06970v1",
    "paper_link": "http://arxiv.org/abs/2512.06970v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{Main thm}\nLet $X/K$ be a smooth projective surface endowed with an elliptic fibration $\\pi \\colon X \\rightarrow \\mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\\mathcal{X}/ \\Spec(\\mathcal{O})$ be a smooth projective model of $X$ over some open subset of $\\Spec(\\Oo_{K})$. Then $\\Sigma(\\mathcal{X})$ is a finitely generated abelian group.",
      "start_pos": 11875,
      "end_pos": 12248,
      "label": "Main thm"
    },
    "ref_dict": {
      "Main thm": "\\begin{thm} \\label{Main thm}\nLet $X/K$ be a smooth projective surface endowed with an elliptic fibration $\\pi \\colon X \\rightarrow \\Pp^1_K$ with at least one singular fibre and a section. Let $\\mathcal{X}/ \\Spec(\\Oo)$ be a smooth projective model of $X$ over some open subset of $\\Spec(\\Oo_{K})$. Then $\\Sigma(\\mathcal{X})$ is a finitely generated abelian group. \n\\end{thm}",
      "final": "\\label{final}\nLet now $R$ be any localization of $\\Oo$ at to some prime $\\p$ satisfying (**). We denote by $\\mathcal{X}/R$ the model of $X$ over $R$ as in the previous section. We have a natural map $"
    },
    "pre_theorem_intro_text_len": 3329,
    "pre_theorem_intro_text": "Let $X$ be a smooth projective variety over a number field $K$. A conjecture originally attributed to Swinnerton-Dyer (see \\cite{zbMATH04061376}[Section 5] and also \\cite{MR1086888} and \\cite{MR1159204}) says that the Chow groups $\\mathrm{CH}^i(X)$ are finitely generated. At present, this conjecture is widely open, and we do not even know whether the $n$-torsion subgroup $\\mathrm{CH}^2(X)[n]$ is finitely generated when $X$ is a surface, although by Rojtman's theorem \\cite{MR577137} this is true over the algebraic closure of $K$. The strongest evidence concerning the finiteness of $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is due to \\cite{CT-R}[Theorem C], where the authors show that $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is finitely generated whenever $H^2(X, \\Oo_X) = 0$ (see also \\cite{MR1300892}). We refer the reader to \\cite{MR1744949} and \\cite{OTS}, which contain surveys regarding this problem and a more comprehensive account of the literature. \n\nIn general, to study $\\mathrm{CH}^2(X)$ one spreads out $X$ to a smooth projective model $\\mathcal{X} / \\mathcal{O}$, where $\\mathcal{O}$ is a localization of the ring of integers of $K$ at finitely many places, and considers the localization sequence \n$$\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) \\rightarrow \\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) \\rightarrow 0$$\n(note that this can be continued to the left using $K$-theory). Assume from now on that $H^1(X, \\Oo_X) = 0$, so, after further localizing $\\mathcal{O}$ if necessary, we have $\\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) = \\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ for every $\\mathfrak{p} \\subset \\mathcal{O}$. The difficulty when $H^2(X, \\Oo_X) \\neq 0$ arises from the fact that the rank of $\\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ can jump for infinitely many places (as for example happens for K3 surfaces), and one is not able to control the image of $\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ in $\\mathrm{CH}^2(\\mathcal{X})$. \n\nLet us call this image $\\Sigma(\\mathcal{X}) \\cong \\ker(\\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) )$. This group appeared in \\cite{MR1177312}, where the case $X = E \\times E$ is studied, with $E/ \\mathbb{Q}$ an elliptic curve. In particular, they show that $\\Sigma(\\mathcal{X})$ is finite when $K = \\mathbb{Q}$ and $E$ has complex multiplication (there are only finitely many such curves over $\\mathbb{Q}$). Moreover, the case where $E$ is an arbitrary elliptic curve over $\\mathbb{Q}$ is also investigated. The proof starts by interpreting the new divisors modulo $\\mathfrak{p}$ as graphs of the Frobenius, and then uses the theory of modular curves and their reduction modulo $\\mathfrak{p}$ developed in \\cite{MR337993} to control these cycles in $\\mathrm{CH}^2(\\mathcal{X})$. This is in fact one of the very few instances for which $H^2(X, \\Oo_X) \\neq 0$ and we nevertheless understand the role of vertical cycles in $\\mathrm{CH}^2(\\mathcal{X})$. See also \\cite{MR1417619}, where these ideas are pushed further to prove more finiteness results regarding self products of elliptic curves over $\\mathbb{Q}$. Our aim in this brief article is to generalize Mildenhall's finiteness to elliptic fibrations over $\\mathbb{P}^1$:",
    "context": "Let $X$ be a smooth projective variety over a number field $K$. A conjecture originally attributed to Swinnerton-Dyer (see \\cite{zbMATH04061376}[Section 5] and also \\cite{MR1086888} and \\cite{MR1159204}) says that the Chow groups $\\mathrm{CH}^i(X)$ are finitely generated. At present, this conjecture is widely open, and we do not even know whether the $n$-torsion subgroup $\\mathrm{CH}^2(X)[n]$ is finitely generated when $X$ is a surface, although by Rojtman's theorem \\cite{MR577137} this is true over the algebraic closure of $K$. The strongest evidence concerning the finiteness of $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is due to \\cite{CT-R}[Theorem C], where the authors show that $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is finitely generated whenever $H^2(X, \\Oo_X) = 0$ (see also \\cite{MR1300892}). We refer the reader to \\cite{MR1744949} and \\cite{OTS}, which contain surveys regarding this problem and a more comprehensive account of the literature.\n\nIn general, to study $\\mathrm{CH}^2(X)$ one spreads out $X$ to a smooth projective model $\\mathcal{X} / \\mathcal{O}$, where $\\mathcal{O}$ is a localization of the ring of integers of $K$ at finitely many places, and considers the localization sequence \n$$\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) \\rightarrow \\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) \\rightarrow 0$$\n(note that this can be continued to the left using $K$-theory). Assume from now on that $H^1(X, \\Oo_X) = 0$, so, after further localizing $\\mathcal{O}$ if necessary, we have $\\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) = \\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ for every $\\mathfrak{p} \\subset \\mathcal{O}$. The difficulty when $H^2(X, \\Oo_X) \\neq 0$ arises from the fact that the rank of $\\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ can jump for infinitely many places (as for example happens for K3 surfaces), and one is not able to control the image of $\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ in $\\mathrm{CH}^2(\\mathcal{X})$.\n\nLet us call this image $\\Sigma(\\mathcal{X}) \\cong \\ker(\\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) )$. This group appeared in \\cite{MR1177312}, where the case $X = E \\times E$ is studied, with $E/ \\mathbb{Q}$ an elliptic curve. In particular, they show that $\\Sigma(\\mathcal{X})$ is finite when $K = \\mathbb{Q}$ and $E$ has complex multiplication (there are only finitely many such curves over $\\mathbb{Q}$). Moreover, the case where $E$ is an arbitrary elliptic curve over $\\mathbb{Q}$ is also investigated. The proof starts by interpreting the new divisors modulo $\\mathfrak{p}$ as graphs of the Frobenius, and then uses the theory of modular curves and their reduction modulo $\\mathfrak{p}$ developed in \\cite{MR337993} to control these cycles in $\\mathrm{CH}^2(\\mathcal{X})$. This is in fact one of the very few instances for which $H^2(X, \\Oo_X) \\neq 0$ and we nevertheless understand the role of vertical cycles in $\\mathrm{CH}^2(\\mathcal{X})$. See also \\cite{MR1417619}, where these ideas are pushed further to prove more finiteness results regarding self products of elliptic curves over $\\mathbb{Q}$. Our aim in this brief article is to generalize Mildenhall's finiteness to elliptic fibrations over $\\mathbb{P}^1$:",
    "full_context": "Let $X$ be a smooth projective variety over a number field $K$. A conjecture originally attributed to Swinnerton-Dyer (see \\cite{zbMATH04061376}[Section 5] and also \\cite{MR1086888} and \\cite{MR1159204}) says that the Chow groups $\\mathrm{CH}^i(X)$ are finitely generated. At present, this conjecture is widely open, and we do not even know whether the $n$-torsion subgroup $\\mathrm{CH}^2(X)[n]$ is finitely generated when $X$ is a surface, although by Rojtman's theorem \\cite{MR577137} this is true over the algebraic closure of $K$. The strongest evidence concerning the finiteness of $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is due to \\cite{CT-R}[Theorem C], where the authors show that $\\mathrm{CH}^2(X)_{\\mathrm{tors}}$ is finitely generated whenever $H^2(X, \\Oo_X) = 0$ (see also \\cite{MR1300892}). We refer the reader to \\cite{MR1744949} and \\cite{OTS}, which contain surveys regarding this problem and a more comprehensive account of the literature.\n\nIn general, to study $\\mathrm{CH}^2(X)$ one spreads out $X$ to a smooth projective model $\\mathcal{X} / \\mathcal{O}$, where $\\mathcal{O}$ is a localization of the ring of integers of $K$ at finitely many places, and considers the localization sequence \n$$\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) \\rightarrow \\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) \\rightarrow 0$$\n(note that this can be continued to the left using $K$-theory). Assume from now on that $H^1(X, \\Oo_X) = 0$, so, after further localizing $\\mathcal{O}$ if necessary, we have $\\mathrm{Pic}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}}) = \\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ for every $\\mathfrak{p} \\subset \\mathcal{O}$. The difficulty when $H^2(X, \\Oo_X) \\neq 0$ arises from the fact that the rank of $\\operatorname{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ can jump for infinitely many places (as for example happens for K3 surfaces), and one is not able to control the image of $\\bigoplus_{\\mathfrak{p} \\subset \\mathcal{O}} \\mathrm{NS}(\\mathcal{X}_{\\Ff_{\\mathfrak{p}}})$ in $\\mathrm{CH}^2(\\mathcal{X})$.\n\nLet us call this image $\\Sigma(\\mathcal{X}) \\cong \\ker(\\mathrm{CH}^2(\\mathcal{X}) \\rightarrow \\mathrm{CH}^2(X) )$. This group appeared in \\cite{MR1177312}, where the case $X = E \\times E$ is studied, with $E/ \\mathbb{Q}$ an elliptic curve. In particular, they show that $\\Sigma(\\mathcal{X})$ is finite when $K = \\mathbb{Q}$ and $E$ has complex multiplication (there are only finitely many such curves over $\\mathbb{Q}$). Moreover, the case where $E$ is an arbitrary elliptic curve over $\\mathbb{Q}$ is also investigated. The proof starts by interpreting the new divisors modulo $\\mathfrak{p}$ as graphs of the Frobenius, and then uses the theory of modular curves and their reduction modulo $\\mathfrak{p}$ developed in \\cite{MR337993} to control these cycles in $\\mathrm{CH}^2(\\mathcal{X})$. This is in fact one of the very few instances for which $H^2(X, \\Oo_X) \\neq 0$ and we nevertheless understand the role of vertical cycles in $\\mathrm{CH}^2(\\mathcal{X})$. See also \\cite{MR1417619}, where these ideas are pushed further to prove more finiteness results regarding self products of elliptic curves over $\\mathbb{Q}$. Our aim in this brief article is to generalize Mildenhall's finiteness to elliptic fibrations over $\\mathbb{P}^1$:\n\nThen all but finitely many primes satisfy (*), and the Tate algorithm implies that we can resolve the singularities of the Weierstrass model $\\mathcal{X}'/R$ simultaneously over $R = \\Oo_{\\p}$. In particular, $X/K$ has good reduction at $\\p$, and all its reductions in positive characteristic maintain the same configuration of singular fibres. Thus we have in this case a smooth projective model $\\mathcal{X} / \\Spec(\\Oo)$, where $\\Oo$ is the localization of $\\Oo_{K}$ at the finitely many primes which do not satisfy (*), and an elliptic fibration $\\tilde{\\pi} \\colon \\mathcal{X} \\rightarrow \\mathbb{P}^1_{\\Oo}$. For any $\\p \\subset \\Oo$ we denote by $\\pi_{\\Ff_{\\p}} \\colon X_{\\Ff_{\\p}} \\rightarrow \\Pp^1_{\\Ff_{\\p}}$ the induced elliptic fibration over the residue field of $\\p$.\n\n\\begin{proof}\nPick any lift $f = f_0/f_1$ of $\\bar{f} = \\bar{f}_{0}/ \\bar{f}_{1}$, where\n$\\bar{f}_{0}, \\bar{f}_{1} \\in \\Ff_{\\p}[t]$ are coprime and\n$f_0,f_1 \\in R[t]$ have the same degrees as $\\bar{f}_{0}$ and $\\bar{f}_{1}$\nrespectively. Then $f_0$ and $f_1$ are also coprime and\n$v_{\\p}(f_0) = v_{\\p}(f_1) = 0$. For a local parameter $\\pi$ of $\\p$ we\nconsider lifts of the form\n\\[\nf_\\lambda = \\frac{f_0 + \\pi \\lambda}{f_1}, \\qquad \\lambda \\in R.\n\\]\nThen\n\\[\nf_1^3\\,P\\bigl((f_0 + \\pi \\lambda)/f_1\\bigr)\n= (f_0 + \\pi \\lambda)^3 + a_4 f_1^2 (f_0+\\pi \\lambda)+ a_6 f_1^3\n\\eqqcolon J_{\\lambda},\n\\]\nand the right-hand side is coprime to $f_1$ since $f_0$ and $f_1$ are coprime\nmodulo $\\p$. Thus it is enough to show that there are infinitely many\nchoices of $\\lambda$ such that $J_{\\lambda}$ is squarefree in $\\overline{K}[t]$.\nIf now, for a ring $S$, we denote by $S[t]_{N} \\cong S^{N+1}$ the\n$S$-module of polynomials of degree $\\leq N$, then the subset of\nsquarefree polynomials in $S[t]_{N}$ is a Zariski open subset defined\nover $\\Z$. Hence it is enough to show that $J_{\\lambda}$ is squarefree over $K[t]$\nfor infinitely many $\\lambda$. Now note that\n\\[\nQ(x,t) = x^3 + a_4 f_1^2 x+ a_6 f_1^3\n\\]\nsatisfies $Q(f_1 x , t) = f_1^3 P(x,t)$ and that $P(x,t)$ is irreducible\nin $K[x,t]$. If, for a contradiction, we had $Q(x,t) = Q_0(x,t) Q_1(x,t)$,\nthen\n\\[\nQ_0(f_1x,t)\\, Q_1(f_1x,t) = f_1^3 P(x,t),\n\\]\nwhich would imply, by degree considerations in $x$, that either $Q_0$ or\n$Q_1$ is constant in $x$, which is impossible unless the other factor\nis a unit. Thus $Q(x,t)$ is irreducible, and hence also\n$Q(f_0 + \\pi x,t)$ must be irreducible. Finally, by Hilbert\nirreducibility, we have infinitely many $x_0 \\in K$ such that\n$Q(f_0 + \\pi x_0,t)$ remains irreducible over $K$, and in particular, squarefree.\n\\end{proof}\nSo we can choose $L$ such that $\\Spec(L)$ is geoemtrically irreducible and the generic fiber of $\\mathcal{C}(D_{\\Ff_{\\p}})$ is a smooth projective hyperelliptic curve.\n\\begin{prop}\nThe induced map $\\phi \\colon \\mathcal{C}(D_{\\Ff_{\\p}}) \\rightarrow \\Pp^1_{R}$ is finite (of degree $2$) and flat. The special fibre of $\\mathcal{C}(D_{\\Ff_{\\p}})$ splits into two components $(\\Pp^1_{\\Ff_{\\p}})^{+} \\cup (\\Pp^1_{\\Ff_{\\p}})^{-}$ meeting at the zeroes and poles of $\\bar{g}$. The two components are swapped by the reduction modulo $\\p$ of the hyperelliptic involution $\\iota \\colon \\mathcal{C}(D_{\\Ff_{\\p}}) \\xrightarrow{\\sim} \\mathcal{C}(D_{\\Ff_{\\p}})$.\n\\end{prop}\n\n\\subsection{Chow groups and Proof of Theorem \\ref{Main thm}}\nChow groups of schemes over Dedekind domains and their functoriality are studied in Fulton's book \\cite{MR1644323}[Chapter 20]. We recall the main results that we are going to use (keeping the notation from the previous section). Let $\\mathcal{X}$ be a separated scheme of finite type over $\\mathcal{O}$ (or $R$) with generic fibre $X$ and special fibres $X_{\\Ff_{\\p}}$ (both may be empty in Fulton's setting). If $Z \\subset \\mathcal{X}$ is a closed integral subscheme we let\n\\[\n\\dim_{\\Oo}(Z) =\n\\begin{cases}\n \\dim(Z_{K}) & \\text{if } Z \\text{ is flat over } \\Oo, \\\\[4pt]\n \\dim(Z) - 1 & \\text{if } Z \\text{ is vertical}.\n\\end{cases}\n\\]\nThe group $\\mathrm{CH}_i(\\mathcal{X} / \\Oo)$ is then defined as the free abelian group generated by integral closed subschemes of relative dimension $i$ modulo rational equivalence (thus $\\mathrm{CH}^2(\\mathcal{X}) = \\mathrm{CH}_0(\\mathcal{X}/ \\Oo)$ in this notation). The specialization map\n\\[\n\\sigma_p \\colon \\mathrm{CH}_i(X) \\rightarrow \\mathrm{CH}_i(X_{\\Ff_{\\p}})\n\\]\nis defined in \\cite{MR1644323}[Section 20.3] in full generality. Moreover, if $Z \\subset X$ is a prime cycle and $\\mathcal{Z} \\subset \\mathcal{X}$ denotes its Zariski closure, which is flat over $\\Oo$, then\n\\[\n\\sigma_p(Z) = [\\mathcal{Z}_{\\Ff_{\\p}}] \\in \\mathrm{CH}_i(X_{\\Ff_{\\p}}),\n\\]\ni.e. $\\sigma_p(Z)$ is the element of $\\mathrm{CH}_i(X_{\\Ff_{\\p}})$ associated to the special fibre of $\\mathcal{Z}$, seen as a closed subscheme of $X_{\\Ff_{\\p}}$. The following lemma is well-known:\n\n\\begin{proof}\nLet $\\pi \\in \\Oo$ be a local parameter for $\\p$ and, if necessary, localize $\\Oo$ further to obtain $\\Oo'$ so that $\\pi$ has only $\\p$ as a simple zero on $\\Spec(\\Oo')$. Let now $\\pi_{| \\mathcal{Z}}$ be considered as a rational function on $\\mathcal{Z}$. Then by the definition of rational equivalence\n\\[\n\\mathrm{div}(\\pi_{| \\mathcal{Z}}) = [\\mathcal{Z}_{\\Ff_{\\p}}] = 0\n\\]\nin the Chow group $\\mathrm{CH}_{i-1}(\\mathcal{Z}/ \\Oo')$. Hence, by the fact that proper pushforward respects rational equivalence, we get that $[\\mathcal{Z}_{\\Ff_{\\p}}] = 0$ also in $\\mathrm{CH}_{i-1}(\\mathcal{X}/ \\Oo')$. Finally, this is also zero in $\\mathrm{CH}_{i-1}(\\mathcal{X}/ \\Oo)$ by functoriality.\n\\end{proof}\nLet now $X$ be an elliptic surface with a model $\\mathcal{X} / R$ as in the previous section. Let $\\psi \\colon \\mathcal{X}(D_{\\Ff_{\\p}}) \\rightarrow \\mathcal{X}$ be the $2\\!:\\!1$ covering corresponding to some new section $D_{\\Ff_{\\p}} \\in \\NS(X_{\\Ff_{\\p}})$. Let\n\\[\n\\psi_{\\Ff_{\\p}} \\colon \\mathcal{X}(D_{\\Ff_{\\p}})_{\\Ff_{\\p}} = (X_{\\Ff_{\\p}})^{+} \\cup (X_{\\Ff_{\\p}})^{-} \\longrightarrow X_{\\Ff_{\\p}}\n\\]\nbe the induced (finite, flat) morphism between special fibres. By construction\n\\[\n\\psi_{\\Ff_{\\p}}^{*}(D_{\\Ff_{\\p}}) = \\sigma_{\\p}(D^{+} + D^{-}),\n\\]\nwhere $D^{+} \\subset X(D_{\\Ff_{\\p}}) = \\mathcal{X}(D_{\\Ff_{\\p}})_{K}$ is the generic fibre of $\\mathcal{D}^{+} \\subset \\mathcal{X}(D_{\\Ff_{\\p}})$. Thus the previous lemma implies that $[D_{\\Ff_{\\p}}] \\in \\mathrm{CH}_0(\\mathcal{X} / R)$ satisfies\n\\[\n0 = \\psi^{*}([D_{\\Ff_{\\p}}]) \\in \\mathrm{CH}_0(\\mathcal{X}(D_{\\Ff_{\\p}}) / R).\n\\]\nHence\n\\[\n0 = \\psi_{*}(\\psi^{*}[D_{\\Ff_{\\p}}]) = 2 \\cdot [D_{\\Ff_{\\p}}] \\in \\mathrm{CH}_{0}(\\mathcal{X}/R).\n\\]\nThis shows that the element $(\\dots, 0, D_{\\Ff_{\\p}}, 0, \\dots) \\in \\bigoplus_{\\p \\subset \\Oo} \\NS(X_{\\Ff_{\\p}})$ has image in $\\mathrm{CH}_0(\\mathcal{X}/ \\Oo)[2]$. Finally, all the components of the singular fibres of $\\pi \\colon X \\rightarrow \\Pp^1_K$ are defined over some field extension $K'/K$ of degree $N$. Hence, if $(\\dots, 0, F, 0, \\dots) \\in \\bigoplus_{\\p \\subset \\Oo} \\NS(X_{\\Ff_{\\p}})$ describes a component of some singular fibre which is not defined over $K$, we must also have that\n\\[\nN \\cdot (\\dots, 0, F, 0, \\dots) = 0 \\quad \\text{in } \\mathrm{CH}_0(\\mathcal{X}/ \\Oo)\n\\]\nby a standard push–pull argument and so the image of $\\bigoplus_{\\p \\subset \\Oo} \\NS(X_{\\Ff_{\\p}})$ is contained in $\\mathrm{CH}_0(\\mathcal{X}/ \\Oo)[2N] = \\mathrm{CH}^2(\\mathcal{X})[2N]$, which is a finite group \\cite{CT-R}[Theorem 1.1.]. The result then follows since there are only finitely many primes which do not satisfy (**) and since $\\NS(X_{\\Ff_{\\p}})$ is always finitely generated.",
    "post_theorem_intro_text_len": 1110,
    "post_theorem_intro_text": "This applies to many surfaces of interest, e.g., (some) K3 surfaces. Also note that $H^2(X, \\Oo_X)$ can be arbitrarily large and that there are no restrictions on the number field $K$.\n\\begin{cor}\nLet $X/K$ be as in the theorem above. Then $\\mathrm{CH}^2(X)[n]$ is finite for every $n>0$ and the group $\\mathrm{CH}^2(X)$ is finitely generated if and only if $\\mathrm{CH}^2(\\mathcal{X})$ is finitely generated.\n\\end{cor}\nThis follows from \\cite{CT-R}[Theorem 1.1.], which says that $\\mathrm{CH}^2(\\mathcal{X})[n]$ is finite. To prove Theorem \\ref{Main thm} we use the relation between the Néron--Severi groups and the Mordell--Weil groups of elliptic surfaces. Our idea is to realize any new section of $\\pi_{\\Ff_{\\mathfrak{p}}} \\colon \\mathcal{X}_{\\Ff_{\\mathfrak{p}}} \\rightarrow \\mathbb{P}^1_{\\Ff_{\\mathfrak{p}}}$ as (a component of) the degeneration of a $2$-section $D \\subset X$ defined over $K$. This is made precise in Section \\ref{final}. Once we do this, the result will follow almost immediately from the formal properties of Chow groups, specialization maps and the aforementioned finiteness results.",
    "sketch": "To prove Theorem \\ref{Main thm} we use the relation between the N\\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \\(\\pi_{\\Ff_{\\mathfrak{p}}} \\colon \\mathcal{X}_{\\Ff_{\\mathfrak{p}}} \\to \\mathbb{P}^1_{\\Ff_{\\mathfrak{p}}}\\) as (a component of) the degeneration of a \\(2\\)-section \\(D \\subset X\\) defined over \\(K\\) (made precise in Section \\ref{final}). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.",
    "expanded_sketch": "To prove the main theorem, we use the relation between the N\\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \\(\\pi_{\\Ff_{\\mathfrak{p}}} \\colon \\mathcal{X}_{\\Ff_{\\mathfrak{p}}} \\to \\mathbb{P}^1_{\\Ff_{\\mathfrak{p}}}\\) as (a component of) the degeneration of a \\(2\\)-section \\(D \\subset X\\) defined over \\(K\\) (made precise later as follows. \\label{final}\nLet now $R$ be any localization of $\\Oo$ at to some prime $\\p$ satisfying (**). We denote by $\\mathcal{X}/R$ the model of $X$ over $R$ as in the previous section. We have a natural map $). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.",
    "expanded_theorem": "\\label{Main thm}\nLet $X/K$ be a smooth projective surface endowed with an elliptic fibration $\\pi \\colon X \\rightarrow \\mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\\mathcal{X}/ \\Spec(\\mathcal{O})$ be a smooth projective model of $X$ over some open subset of $\\Spec(\\Oo_{K})$. Then $\\Sigma(\\mathcal{X})$ is a finitely generated abelian group.",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Let $K$ be a number field, let $X/K$ be a smooth projective surface, and suppose $X$ is endowed with an elliptic fibration $\\pi\\colon X\\to \\mathbb{P}^1_K$ that has a section and at least one singular fibre. Let $\\mathcal{X}\\to \\operatorname{Spec}(\\mathcal{O})$ be a smooth projective model of $X$ over an open subset of $\\operatorname{Spec}(\\mathcal{O}_K)$. Define $\\Sigma(\\mathcal{X})$ to be the image of the localization map\n\\[\n\\bigoplus_{\\mathfrak p\\subset \\mathcal O} \\operatorname{Pic}(\\mathcal X_{\\mathbb F_{\\mathfrak p}})\\longrightarrow \\mathrm{CH}^2(\\mathcal X),\n\\]\nso equivalently $\\Sigma(\\mathcal{X})\\cong \\ker\\bigl(\\mathrm{CH}^2(\\mathcal X)\\to \\mathrm{CH}^2(X)\\bigr)$. Which statement is valid under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "$\\Sigma(\\mathcal{X})$ is a finitely generated abelian group."
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\Sigma(\\mathcal{X})$ is a finite abelian group."
        },
        {
          "label": "C",
          "text": "$\\Sigma(\\mathcal{X})$ is a countable abelian group."
        },
        {
          "label": "D",
          "text": "For every smooth projective model $\\mathcal{X}\\to \\operatorname{Spec}(\\mathcal O)$ of $X$ over an open subset of $\\operatorname{Spec}(\\mathcal O_K)$, the group $\\Sigma(\\mathcal{X})$ is contained in $\\mathrm{CH}^2(\\mathcal X)[m]$ for some integer $m\\ge 1$ depending only on $X/K$."
        },
        {
          "label": "E",
          "text": "If $\\mathfrak p\\subset \\mathcal O$ is any prime of good reduction, then every class in $\\operatorname{Pic}(\\mathcal X_{\\mathbb F_{\\mathfrak p}})$ maps to a torsion element of $\\mathrm{CH}^2(\\mathcal X)$; hence $\\Sigma(\\mathcal X)$ is a torsion abelian group."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "finite_vs_finitely_generated",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "finite_generation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniform_global_torsion_bound_for_all_vertical_classes",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "only_new_sections_and_fibre_components_are_shown_torsion_after_specialization",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal that \u0003\u0003Sigma(\\mathcal{X}) is finitely generated, nor does it strongly hint at that exact conclusion. It only sets up the geometric/arithmetic context and definition."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: under the stated hypotheses, the correct option is the main structural conclusion about \\Sigma(\\mathcal{X}). The question mostly asks the student to recognize the theorem rather than derive a new consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to distinguish 'finitely generated' from nearby alternatives such as 'finite,' 'countable,' or 'torsion,' but the item still mainly tests recall of the exact theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target natural failure modes: confusing finitely generated with finite, selecting a weaker true statement, or overclaiming torsion/uniform boundedness. They are distinct and well-designed."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall-based theorem-identification MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests genuine reasoning."
    }
  },
  {
    "id": "2512.07260v1",
    "paper_link": "http://arxiv.org/abs/2512.07260v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm1}\n\tLet $u\\in C_{loc}(\\rn)$ be a convex viscosity solution of \\eqref{eqma}, where $f:\\rn\\rightarrow{\\mathbb R}$ satisfies \n\t{\\textnormal{(H)}}. If $n\\geq3$, then there exist $b\\in\\rn$, $A\\in\\mathcal{A}$ and $u_p\\in C^{2,\\alpha}(\\rn)$ such that\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\\left|u(x)-\\left(\\frac{1}{2}x'Ax+b\\cdot x+u_p(x)\\right)\\right|\\leq C_0(1+|x|)^{2-\\min\\{n,\\beta\\}},\\ \\ \\ \\forall\\ x\\in\\rn,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\\cdots,n$,   \n    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\\beta$, ${\\alpha}$ and $a_1,\\cdots,a_n$.",
      "start_pos": 9111,
      "end_pos": 9738,
      "label": "thm1"
    },
    "ref_dict": {
      "lemeqv": "\\begin{lemma}\\label{lemeqv}\n\tLet $E$, $v$ and $f_T$ be as above. Then, for $s\\in(0,\\frac{\\az}{2})$, \n\t\\begin{equation*}\n\t\ta_{ij}(x)\\pat_{ij}(\\De_e^sv(x))\\geq F_s(x),\\ \\ \\ \\ \\forall x\\in\\rn,\\ e\\in E\\setminus\\{0\\},\n\t\\end{equation*}\n\twhere $a_{ij}(x)=cof_{ij}(D^2v(x))$ and \n\t\\begin{equation*}\n\t\tF_s(x):=n\\left(\\det(D^2v(x))^{\\frac{n-1}{n}}\\right)\\left(\\frac{f_T^{\\frac{1}{n}}(x+e)+f_T^{\\frac{1}{n}}(x-e)-2f_T^{\\frac{1}{n}}(x)}{|e|^{2s}}\\right).\n\t\\end{equation*}\n\\end{lemma}",
      "thm1": "\\begin{theorem}\\label{thm1}\n\tLet $u\\in C_{loc}(\\rn)$ be a convex viscosity solution of \\eqref{eqma}, where $f:\\rn\\rightarrow\\rr$ satisfies \n\t{\\textnormal{(H)}}. If $n\\geq3$, then there exist $b\\in\\rn$, $A\\in\\A$ and $u_p\\in C^{2,\\alpha}(\\rn)$ such that\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\\left|u(x)-\\left(\\frac{1}{2}x'Ax+b\\cdot x+u_p(x)\\right)\\right|\\leq C_0(1+|x|)^{2-\\min\\{n,\\beta\\}},\\ \\ \\ \\forall\\ x\\in\\rn,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\\cdots,n$,   \n    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\\be$, $\\az$ and $a_1,\\cdots,a_n$.\n\\end{theorem}",
      "eqma": "\\begin{equation}\\label{eqma}\n\t\\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}",
      "lemeF": "\\begin{lemma}\\label{lemeF}\n\tLet $F_s$ be the one obtained in Lemma \\ref{lemeqv}. Then we have \n\t\\begin{equation}\\label{eqlemeF1}\n\t\t|F_s(x)|\\leq \n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\tC_0(1+|x|)^{-\\min\\{\\be+2s,n\\}},&\\ \\ \\ \\be+2s\\neq n,\\ \\be\\neq n,\\\\\n\t\t\tC_0(1+|x|)^{-\\beta-2s}(\\ln(2+|x|)),&\\ \\ \\ \\be+2s=n,\\\\\n\t\t\tC_0(1+|x|)^{-\\beta-2s}(\\ln(2+|x|)),&\\ \\ \\ \\be=n,\n\t\t\\end{array}\n\t\t\\right.\\ \\ \\ \\ \\ \n\t\\end{equation}\n\tfor all $e\\in E\\setminus\\{0\\},\\ x\\in\\rn.$ The constant $C_0>0$ depends only $d_0,\\ d_1,\\ \\be,\\ n$ and $s$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 3959,
    "pre_theorem_intro_text": "Monge-Amp\\`{e}re equations are a class of fully nonlinear equations and can be found in several contexts of analysis and geometry. \nThe asymptotic behavior of solution to the Monge-Amp\\`{e}re equation \n\\begin{equation}\\label{eqma}\n\t\\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}\nas an extension of Liouville's theorem, has been investigated extensively in the last years, starting with the pioneering works by \nJ\\\"{o}rgens \\cite{j}, Calabi \\cite{ce} and Pogorelov \\cite{p}. They showed that any convex classical solution of \\eqref{eqma} with $f\\equiv1$ \nmust be a quadratic polynomial. \nCaffarelli \\cite{ca} extended the result from classical solution to viscosity solution.\nCaffarelli and Li \\cite{cl} \nconsidered the case $f$ is equal to $1$ outside a compact subset of $\\rn$ and Bao, Li and Zhang \\cite{blz} focused on a more general situation \nthat $f$ is a perturbation of $1$ near infinity. They all proved that any convex viscosity solution $u$ is close to a \nquadratic polynomial at infinity for $n\\geq3$ (there is an additional logarithmic term for $n=2$). In \\cite{lb}, Bao and Liu concentrated on \nthe regularity of $f$ and weakened it from $f\\in C^3$ in \\cite{blz} to $f\\in C^2$ by the spherical harmonic expansion of the solutions of \nnonhomogeneous linearized equations.  Bao and Qi \\cite{qb} developed a nonlocal method to obtain the asymptotic behavior of $u$ under \nthe hypothesis on regularity that $f$ is just H\\\"{o}lder continuous. In \\cite{jtx}, Jin, Tu and Xiong considered the case that the right-hand side of the equation is a measure.\n\nOn the other hand, $f\\equiv1$ is also a special periodic function. In the well known work \\cite{cl2}, Caffarelli and Li investigated the case \nthat $f\\in C^{\\alpha}$ is a positive periodic function and showed that the difference between $u$ and a quadratic polynomial is periodic. Li and Lu considered the \npositive periodic function $f\\in L^\\infty$ in a following work \\cite{ll}. As the counterpart of \\cite{blz} in the aperiodic case, Teixeira and Zhang \\cite{tz} studied the \nthe asymptotic behavior of solution to \\eqref{eqma}, in which the right hand side $f\\in C^{1,{\\alpha}}$ is asymptotically close to periodic function. \nThey showed that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.\n\nIn this paper we discuss the asymptotic behavior of solution $u$ to \\eqref{eqma} with a H\\\"{o}lder continuous term $f$, compared to the condition \n$f\\in C^{1,{\\alpha}}$ in \\cite{tz}. More precisely, our assumptions on $f$ are as follows:\n\nLet $f_p$ be a positive periodic function in $\\rn$ such that, for some constants $d_0>0$, $\\alpha\\in(0,1)$ and $a_1,\\cdots,a_n>0$ there hold\n\\begin{equation*}\n   \\begin{array}{rl}\n   \t   &d_0^{-1}\\leq f_p\\leq d_0,\\ \\ \\ [f_p]_{C^\\alpha(\\rn)}\\leq d_0,\\\\\n   \t   &f_p(x+a_ie_i)=f_p(x),\\ \\ \\ \\forall x\\in\\rn,\n   \\end{array}\n\\end{equation*}\nwhere $e_i=(0,\\cdots,0,1,0,\\cdots,0)$ denotes the $i$-th standard coordinate vector in $\\rn$, $i=1,\\cdots,n$.\n\nWe suppose that $f$ is asymptotically close to $f_p$ in the following sense: \n\\vspace{-0.2cm}\n\\begin{itemize}\n\t\\item[(H)] The function $f\\in C(\\rn)\\cap C_{loc}^{\\alpha}({\\rn\\setminus\\overline{\\mathcal{O}}})$ for \n\tsome bounded open subset $\\mathcal{O}\\subset\\rn$ and satisfies, for $x\\in\\rn\\setminus\\{0\\}$, \n\t\\begin{equation*}\\label{eqfcdinfi}\n\t\t\\begin{aligned}\n\t\t\t|x|^{\\beta}\\left|\\left(f-f_p\\right)(x)\\right|+\n\t\t\t|x|^{\\beta+\\alpha}\\left[f-f_p\\right]_{C^{\\alpha}(\\overline{B_{\\frac{|x|}{2}}(x)})}\\leq d_1\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tfor some constants $\\beta>2$ and $d_1>0$.\n\\end{itemize}\n\\vspace{-0.2cm}\nLet ${\\mathbb M}^{n\\times n}$ be the set of the real valued, $n\\times n$ matrices and \n\\begin{equation*}\n\t\\mathcal{A}:=\\left\\{A\\in{\\mathbb M}^{n\\times n}:\\ A\\text{ is symmetric, positive definite and }\\det(A)=\n\t\\int\\hspace{-1.05em}-_{\\raisebox{-5pt}{\\tiny$\\prod_{1\\leq i\\leq n}[0,a_i]$}}fdx\\right\\}.\n\\end{equation*}\n\nOur main result is",
    "context": "Monge-Amp\\`{e}re equations are a class of fully nonlinear equations and can be found in several contexts of analysis and geometry. \nThe asymptotic behavior of solution to the Monge-Amp\\`{e}re equation \n\\begin{equation}\\label{eqma}\n \\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}\nas an extension of Liouville's theorem, has been investigated extensively in the last years, starting with the pioneering works by \nJ\\\"{o}rgens \\cite{j}, Calabi \\cite{ce} and Pogorelov \\cite{p}. They showed that any convex classical solution of \\eqref{eqma} with $f\\equiv1$ \nmust be a quadratic polynomial. \nCaffarelli \\cite{ca} extended the result from classical solution to viscosity solution.\nCaffarelli and Li \\cite{cl} \nconsidered the case $f$ is equal to $1$ outside a compact subset of $\\rn$ and Bao, Li and Zhang \\cite{blz} focused on a more general situation \nthat $f$ is a perturbation of $1$ near infinity. They all proved that any convex viscosity solution $u$ is close to a \nquadratic polynomial at infinity for $n\\geq3$ (there is an additional logarithmic term for $n=2$). In \\cite{lb}, Bao and Liu concentrated on \nthe regularity of $f$ and weakened it from $f\\in C^3$ in \\cite{blz} to $f\\in C^2$ by the spherical harmonic expansion of the solutions of \nnonhomogeneous linearized equations. Bao and Qi \\cite{qb} developed a nonlocal method to obtain the asymptotic behavior of $u$ under \nthe hypothesis on regularity that $f$ is just H\\\"{o}lder continuous. In \\cite{jtx}, Jin, Tu and Xiong considered the case that the right-hand side of the equation is a measure.\n\nOn the other hand, $f\\equiv1$ is also a special periodic function. In the well known work \\cite{cl2}, Caffarelli and Li investigated the case \nthat $f\\in C^{\\alpha}$ is a positive periodic function and showed that the difference between $u$ and a quadratic polynomial is periodic. Li and Lu considered the \npositive periodic function $f\\in L^\\infty$ in a following work \\cite{ll}. As the counterpart of \\cite{blz} in the aperiodic case, Teixeira and Zhang \\cite{tz} studied the \nthe asymptotic behavior of solution to \\eqref{eqma}, in which the right hand side $f\\in C^{1,{\\alpha}}$ is asymptotically close to periodic function. \nThey showed that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.\n\nIn this paper we discuss the asymptotic behavior of solution $u$ to \\eqref{eqma} with a H\\\"{o}lder continuous term $f$, compared to the condition \n$f\\in C^{1,{\\alpha}}$ in \\cite{tz}. More precisely, our assumptions on $f$ are as follows:\n\nLet $f_p$ be a positive periodic function in $\\rn$ such that, for some constants $d_0>0$, $\\alpha\\in(0,1)$ and $a_1,\\cdots,a_n>0$ there hold\n\\begin{equation*}\n \\begin{array}{rl}\n &d_0^{-1}\\leq f_p\\leq d_0,\\ \\ \\ [f_p]_{C^\\alpha(\\rn)}\\leq d_0,\\\\\n &f_p(x+a_ie_i)=f_p(x),\\ \\ \\ \\forall x\\in\\rn,\n \\end{array}\n\\end{equation*}\nwhere $e_i=(0,\\cdots,0,1,0,\\cdots,0)$ denotes the $i$-th standard coordinate vector in $\\rn$, $i=1,\\cdots,n$.\n\nWe suppose that $f$ is asymptotically close to $f_p$ in the following sense: \n\\vspace{-0.2cm}\n\\begin{itemize}\n \\item[(H)] The function $f\\in C(\\rn)\\cap C_{loc}^{\\alpha}({\\rn\\setminus\\overline{\\mathcal{O}}})$ for \n some bounded open subset $\\mathcal{O}\\subset\\rn$ and satisfies, for $x\\in\\rn\\setminus\\{0\\}$, \n \\begin{equation*}\\label{eqfcdinfi}\n \\begin{aligned}\n |x|^{\\beta}\\left|\\left(f-f_p\\right)(x)\\right|+\n |x|^{\\beta+\\alpha}\\left[f-f_p\\right]_{C^{\\alpha}(\\overline{B_{\\frac{|x|}{2}}(x)})}\\leq d_1\n \\end{aligned}\n \\end{equation*}\n for some constants $\\beta>2$ and $d_1>0$.\n\\end{itemize}\n\\vspace{-0.2cm}\nLet ${\\mathbb M}^{n\\times n}$ be the set of the real valued, $n\\times n$ matrices and \n\\begin{equation*}\n \\mathcal{A}:=\\left\\{A\\in{\\mathbb M}^{n\\times n}:\\ A\\text{ is symmetric, positive definite and }\\det(A)=\n \\int\\hspace{-1.05em}-_{\\raisebox{-5pt}{\\tiny$\\prod_{1\\leq i\\leq n}[0,a_i]$}}fdx\\right\\}.\n\\end{equation*}\n\nOur main result is\n\n\\begin{equation}\\label{eqma}\n\t\\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}",
    "full_context": "Monge-Amp\\`{e}re equations are a class of fully nonlinear equations and can be found in several contexts of analysis and geometry. \nThe asymptotic behavior of solution to the Monge-Amp\\`{e}re equation \n\\begin{equation}\\label{eqma}\n \\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}\nas an extension of Liouville's theorem, has been investigated extensively in the last years, starting with the pioneering works by \nJ\\\"{o}rgens \\cite{j}, Calabi \\cite{ce} and Pogorelov \\cite{p}. They showed that any convex classical solution of \\eqref{eqma} with $f\\equiv1$ \nmust be a quadratic polynomial. \nCaffarelli \\cite{ca} extended the result from classical solution to viscosity solution.\nCaffarelli and Li \\cite{cl} \nconsidered the case $f$ is equal to $1$ outside a compact subset of $\\rn$ and Bao, Li and Zhang \\cite{blz} focused on a more general situation \nthat $f$ is a perturbation of $1$ near infinity. They all proved that any convex viscosity solution $u$ is close to a \nquadratic polynomial at infinity for $n\\geq3$ (there is an additional logarithmic term for $n=2$). In \\cite{lb}, Bao and Liu concentrated on \nthe regularity of $f$ and weakened it from $f\\in C^3$ in \\cite{blz} to $f\\in C^2$ by the spherical harmonic expansion of the solutions of \nnonhomogeneous linearized equations. Bao and Qi \\cite{qb} developed a nonlocal method to obtain the asymptotic behavior of $u$ under \nthe hypothesis on regularity that $f$ is just H\\\"{o}lder continuous. In \\cite{jtx}, Jin, Tu and Xiong considered the case that the right-hand side of the equation is a measure.\n\nOn the other hand, $f\\equiv1$ is also a special periodic function. In the well known work \\cite{cl2}, Caffarelli and Li investigated the case \nthat $f\\in C^{\\alpha}$ is a positive periodic function and showed that the difference between $u$ and a quadratic polynomial is periodic. Li and Lu considered the \npositive periodic function $f\\in L^\\infty$ in a following work \\cite{ll}. As the counterpart of \\cite{blz} in the aperiodic case, Teixeira and Zhang \\cite{tz} studied the \nthe asymptotic behavior of solution to \\eqref{eqma}, in which the right hand side $f\\in C^{1,{\\alpha}}$ is asymptotically close to periodic function. \nThey showed that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.\n\nIn this paper we discuss the asymptotic behavior of solution $u$ to \\eqref{eqma} with a H\\\"{o}lder continuous term $f$, compared to the condition \n$f\\in C^{1,{\\alpha}}$ in \\cite{tz}. More precisely, our assumptions on $f$ are as follows:\n\nLet $f_p$ be a positive periodic function in $\\rn$ such that, for some constants $d_0>0$, $\\alpha\\in(0,1)$ and $a_1,\\cdots,a_n>0$ there hold\n\\begin{equation*}\n \\begin{array}{rl}\n &d_0^{-1}\\leq f_p\\leq d_0,\\ \\ \\ [f_p]_{C^\\alpha(\\rn)}\\leq d_0,\\\\\n &f_p(x+a_ie_i)=f_p(x),\\ \\ \\ \\forall x\\in\\rn,\n \\end{array}\n\\end{equation*}\nwhere $e_i=(0,\\cdots,0,1,0,\\cdots,0)$ denotes the $i$-th standard coordinate vector in $\\rn$, $i=1,\\cdots,n$.\n\nWe suppose that $f$ is asymptotically close to $f_p$ in the following sense: \n\\vspace{-0.2cm}\n\\begin{itemize}\n \\item[(H)] The function $f\\in C(\\rn)\\cap C_{loc}^{\\alpha}({\\rn\\setminus\\overline{\\mathcal{O}}})$ for \n some bounded open subset $\\mathcal{O}\\subset\\rn$ and satisfies, for $x\\in\\rn\\setminus\\{0\\}$, \n \\begin{equation*}\\label{eqfcdinfi}\n \\begin{aligned}\n |x|^{\\beta}\\left|\\left(f-f_p\\right)(x)\\right|+\n |x|^{\\beta+\\alpha}\\left[f-f_p\\right]_{C^{\\alpha}(\\overline{B_{\\frac{|x|}{2}}(x)})}\\leq d_1\n \\end{aligned}\n \\end{equation*}\n for some constants $\\beta>2$ and $d_1>0$.\n\\end{itemize}\n\\vspace{-0.2cm}\nLet ${\\mathbb M}^{n\\times n}$ be the set of the real valued, $n\\times n$ matrices and \n\\begin{equation*}\n \\mathcal{A}:=\\left\\{A\\in{\\mathbb M}^{n\\times n}:\\ A\\text{ is symmetric, positive definite and }\\det(A)=\n \\int\\hspace{-1.05em}-_{\\raisebox{-5pt}{\\tiny$\\prod_{1\\leq i\\leq n}[0,a_i]$}}fdx\\right\\}.\n\\end{equation*}\n\nOur main result is\n\n\\begin{equation}\\label{eqma}\n\t\\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}\n\n\\begin{remark}\\label{remcf}\n It is clear that $f\\in C^\\az(\\rn\\setminus\\O)$. Moreover, the viscosity solution $u$ in Theorem \\ref{thm1} \n is really $C^{2,\\alpha}$ near infinity, see Section 3.\n\\end{remark}\n\nFor the case in $A_1$, we first write \n\\begin{gather*}\n \\Gamma_1:=\\int_{|y|\\leq\\frac{|x|}{2}}(-\\De)^sg(y)\\left(\\frac{1}{|x+e-y|^{n-2s}}-\\frac{1}{|x-y|^{n-2s}}\\right)dy,\\\\\n \\Gamma_2:=\\int_{|y|\\leq\\frac{|x|}{2}}(-\\De)^sg(y)\\left(\\frac{1}{|x-y|^{n-2s}}-\\frac{1}{|x-e-y|^{n-2s}}\\right)dy.\n\\end{gather*}\nNote that $|x+e|\\leq\\frac{|x|}{4}$ in $\\Omega_2$ and $(-\\De)^sg$ is bounded, we have, for $\\rho>0$ small, \n\\begin{equation*}\n \\begin{aligned}\n \\Gamma_1=&\\left(\\int_{B_{\\frac{|x|}{2}}\\setminus B_\\rho(x+e)}+\\int_{B_\\rho(x+e)}\\right)\n (-\\De)^sg(y)\\left(\\frac{1}{|x+e-y|^{n-2s}}-\\frac{1}{|x-y|^{n-2s}}\\right)dy\\\\\n \\leq&C_{n,s}\\int_{B_{\\frac{|x|}{2}}\\setminus B_\\rho(x+e)}|(-\\De)^sg(y)|\\frac{|e|}{|\\xi-y|^{n+1-2s}}dy+C_{n,s}(\\rho^{2s}+\\rho^n),\n \\end{aligned}\n\\end{equation*}\nwhere $\\xi$ is obtained by the Lagrange mean value theorem. In fact, $\\xi$ can be chosen such that $\\xi=x+\\theta e$ for $\\theta\\in[\\frac{1}{8},\\frac{3}{8}]$, \nwhich implies $|\\xi|\\geq|x|-\\frac{3}{8}|e|\\geq\\frac{17}{32}|x|$ since $|e|\\leq\\frac{5}{4}|x|$. \nMoreover, $|\\xi-y|\\geq\\frac{1}{32}|x|$ for $|y|\\leq\\frac{|x|}{2}$. Passing to the limit $\\rho\\rightarrow0$ one gets \n\\begin{equation*}\n |\\Gamma_1|\\leq\\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\\int_{|y|\\leq\\frac{|x|}{2}}|(-\\De)^sg(y)|dy.\n\\end{equation*}\nUsing the same argument we also get \n\\begin{equation*}\n |\\Gamma_2|\\leq\\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\\int_{|y|\\leq\\frac{|x|}{2}}|(-\\De)^sg(y)|dy.\n\\end{equation*}\nThus, noticing that $\\frac{|x|}{2}>1$, we have \n\\begin{equation}\\label{eqg}\n\\begin{aligned}\n \\int_{A_1}|(-\\Delta)^sg(y)||\\Psi|dy\n &\\leq\\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\\left(\\int_{B_{1}}+\\int_{B_{\\frac{|x|}{2}}\\setminus B_{1}}\\right)|(-\\Delta)^sg(y)|dy.\n\\end{aligned}\n\\end{equation}\nThe boundedness of $(-\\Delta)^sg$ implies\n\\begin{equation*}\n \\int_{B_{1}}|(-\\Delta)^sg(y)|dy\\leq C'\n\\end{equation*}\nfor some $C'>0$. On the other hand, equation \\eqref{eqlemFgd} yields that \n\\begin{align*}\n \\int_{B_{\\frac{|x|}{2}}\\setminus B_{1}}|(-\\Delta)^sg(y)|dy&\\leq C\\int_{B_{\\frac{|x|}{2}}\\setminus B_{1}}|y|^{-\\min\\{\\be,n\\}-2s}dy\\\\\n &\\leq\n \\left\\{\n \\begin{array}{ll}\n C|x|^{n-\\min\\{\\be,n\\}-2s},&\\ \\ \\ \\min\\{\\be,n\\}+2s\\neq n,\\\\\n C\\ln|x|,&\\ \\ \\ \\min\\{\\be,n\\}+2s=n.\n \\end{array}\n \\right.\n\\end{align*}\nIt follows that \n\\begin{align*}\n \\int_{B_{\\frac{|x|}{2}}}|(-\\Delta)^sg(y)|dy\n &\\leq\\left\\{\n \\begin{array}{ll}\n C(|x|^{n-\\min\\{\\be,n\\}-2s}+1),&\\ \\ \\ \\be+2s\\neq n,\\\\\n C(\\ln|x|+1),&\\ \\ \\ \\be+2s=n.\n \\end{array}\n \\right.\n\\end{align*}\nSince \n\\begin{align*}\n (|x|^{n-\\min\\{\\be,n\\}-2s}+1)\\cdot{|x|^{-n-1+2s}}&\\leq|x|^{-\\min\\{\\be,n-2s\\}-1},\n\\end{align*} \nwe use \\eqref{eqg} and \\eqref{eqex} to obtain \n\\begin{align*}\n \\frac{C}{|e|^{2s}}\\int_{A_1}|(-\\Delta)^sg(y)||\\Psi|dy\n &\\leq\\left\\{\n \\begin{array}{ll}\n C|e|^{1-2s}|x|^{-\\min\\{\\be,n-2s\\}-1},\\\\\n C|e|^{1-2s}|x|^{-n+2s-1}(\\ln|x|),\n \\end{array}\n \\right.\\\\\n &\\leq\\left\\{\n \\begin{array}{ll}\n C|x|^{-\\min\\{\\be+2s,n\\}},&\\ \\ \\ \\be+2s\\neq n,\\\\\n C|x|^{-\\beta-2s}(\\ln|x|),&\\ \\ \\ \\be+2s=n,\n \\end{array}\n \\right.\\ \\ \\ \\text{for}\\ |x|>2.\n\\end{align*}\n\nBy \\cite[Theorem 0.2]{cl2} (see also \\cite{l}), there exists $v_p\\in C^{2,\\az}(\\rn)$ such that \n\\begin{equation}\\label{eqvp}\n \\left\\{\n \\begin{array}{ll}\n \\det(I+D^2v_p)=(f_p)_T,\\ \\ \\ \\text{in}\\ \\rn,\\\\\n I+D^2v_p>0.\n \\end{array}\n \\right.\n\\end{equation}\nHere, the function $v_p$ is periodic and has the same period as that of $(f_p)_T$. Let \n\\begin{equation*}\n w(x):=v(x)-\\frac{1}{2}|x|^2-v_p(x).\n\\end{equation*}\nApplying Lemma \\ref{lemvh} and the facts that $\\De^s_e\\left(\\frac{1}{2}|x|^2\\right)=|e|^{2-2s}$ and $\\De^s_ev_p(x)=0$, one has\n\\begin{equation*}\n \\De^s_ew(x)-h(x)=\\De^s_ev(x)-|e|^{2-2s}-h(x)\\leq0,\\ \\ \\ \\ \\forall x\\in\\rn,\\ e\\in E.\n\\end{equation*}\nCombined with the equations \\eqref{eqv} and \\eqref{eqvp} we have\n\\begin{equation*}\n \\det(D^2v(x))-\\det(D^2(\\frac{1}{2}|x|^2+v_p(x)))=f_T(x)-(f_p)_T(x),\\ \\ \\ \\ \\forall x\\in\\rn,\n\\end{equation*}\nfrom which we get the equation for $w$:\n\\begin{equation*}\n \\tilde{a}_{ij}(x)\\pat_{ij}w(x)=f_T(x)-(f_p)_T(x),\\ \\ \\ \\ \\forall x\\in\\rn,\n\\end{equation*}\nwhere \n\\begin{equation*}\n \\tilde{a}_{ij}(x):=\\int_0^1cof_{ij}\\left[tD^2v(x)+(1-t)D^2\\left(\\frac{1}{2}|x|^2+v_p(x)\\right)\\right]dt.\n\\end{equation*}\nIn fact, we can use Proposition \\ref{pro2} and \\eqref{eqvp} to get that $\\tilde{a}_{ij}(x)$ is uniformly elliptic. On the other hand, we can \nuse the fundamental solution to construct (as that of $h$) a function $h_1$ which satisfies \n\\begin{equation*}\n \\tilde{a}_{ij}(x)\\pat_{ij}h_1(x)=(f_p)_T(x)-f_T(x),\\ \\ \\ \\ \\forall x\\in\\rn.\n\\end{equation*}\nTherefore, \n\\begin{equation}\\label{eqwh1}\n \\tilde{a}_{ij}(x)\\pat_{ij}\\left(w+h_1\\right)(x)=0,\\ \\ \\ \\ \\forall x\\in\\rn.\n\\end{equation}\nBy \\cite[Lemma 3.2]{lb} and assumption (H) we conclude\n\\begin{equation*}\n |h_1(x)|\\leq \n \\left\\{\n \\begin{array}{ll}\n C(1+|x|)^{2-\\min\\{\\be,n\\}},&\\ \\ \\ \\ \\be\\neq n,\\\\\n C(1+|x|)^{2-n}\\ln(2+|x|),&\\ \\ \\ \\ \\be=n.\n \\end{array}\n \\right. \n\\end{equation*}\n\nBy \\eqref{eqglim} we have $\\max_{B_1}g\\geq\\frac{3}{4}$, thus the linear function $l=a\\cdot x$ for $|a|\\geq\\frac{3}{4}$, where we have using the fact that \n$g(0)=l(0)=0$. It follows \n\\begin{equation}\\label{eqwro}\n \\frac{w(ry)}{M_r}-a\\cdot y\\rightarrow0\\ \\ \\ \\ \\text{as}\\ r\\rightarrow\\infty\\ \\ \\ \\ \\text{for }y\\in B_{\\frac{3}{4}},\n\\end{equation}\nfrom which we obtain for $r$ large that \n\\begin{equation}\\label{eqcon}\n w(2^Ne)>\\frac{M_r}{2r}2^N|e|,\n\\end{equation}\nwhere $e\\in E$ satisfies $a\\cdot e>\\frac{2}{3}$ and integer $N$ is chosen so that $2^N|e|<\\frac{3}{4}r\\leq2^{N+1}|e|$. Let $\\bar{l}$ be a linear function \nthat agrees with $w$ at $0$ and $e$. Then the function $\\overline{w}:=w-\\bar{l}$ satisfies $\\overline{w}(0)=\\overline{w}(e)=0$ and \n\\begin{equation*}\n \\overline{w}(2^Ne)>\\frac{M_r}{4r}2^N|e|\n\\end{equation*}\nfor $r$ large. Since, by Remark \\ref{remhss}, \n\\begin{equation*}\n \\De^s_e\\overline{w}(x)=\\De^s_ew(x)\\leq C|x|^{-2s-\\sigma},\\ \\ \\ \\ \\forall |x|\\geq1\n\\end{equation*} \nfor some $\\sigma>0$ small, we have \n\\begin{equation*}\n \\overline{w}(2e)\\leq2\\overline{w}(e)-\\overline{w}(0)+C|e|^{2s}|e|^{-2s-\\sigma}\\leq C|e|^{-\\sigma}.\n\\end{equation*}\nSimilarly, \n\\begin{align*}\n \\overline{w}(4e)&\\leq2\\overline{w}(2e)-\\overline{w}(0)+C|2e|^{2s}|2e|^{-2s-\\sigma}\\\\\n &\\leq 2C|e|^{-\\sigma}+C|2e|^{-\\sigma}.\n\\end{align*}\nBy induction, \n\\begin{align*}\n \\overline{w}(2^Ne)&\\leq C|e|^{-\\sigma}\\left(2^{N-1}+2^{N-2-\\sigma}+2^{N-3-2\\sigma}+\\cdots+2^{-\\sigma(N-1)}\\right)\\\\\n &\\leq 2^{N-1}C|e|^{-\\sigma}\\frac{1-2^{-N(1+\\sigma)}}{1-2^{-(1+\\sigma)}}\\leq C2^N|e|,\n\\end{align*}\na contradiction to \\eqref{eqcon}. Thus \\eqref{eqtmr} is proved.\n\nNow we rewrite \\eqref{eqwro} as \n\\begin{equation}\\label{eqwo}\n \\left|w(x)-\\frac{M_r}{r}a\\cdot x\\right|=o(1)M_r,\\ \\ \\ \\ \\ \\text{for }x\\in B_{\\frac{3}{4}r}.\n\\end{equation}\nLet \n\\begin{equation*}\n b:=\\lim_{r\\rightarrow\\infty}\\frac{M_r}{r}a\\ \\ \\ \\ \\text{and }\\ \\ \\ w_1(x):=w(x)-b\\cdot x. \n\\end{equation*}\nThen we get from \\eqref{eqwo} that $|w_1(x)|=o(|x|)$ for $|x|$ large, which gives \n\\begin{equation*}\n |w_1(x)|\\leq C+o(|x|),\\ \\ \\ \\ \\forall x\\in\\rn.\n\\end{equation*}\nIn other words, $w_1$ is bounded in $\\rn$ and so is $w_1+h_1$. We also have \n\\begin{equation*}\n \\tilde{a}_{ij}(x)\\pat_{ij}\\left(w+h_1\\right)(x)=0,\\ \\ \\ \\ \\forall x\\in\\rn.\n\\end{equation*}\nThus, by the Harnack inequality we see that $w_1+h_1\\equiv C$. Theorem \\ref{thm1} is proved. \n\\qed",
    "post_theorem_intro_text_len": 4141,
    "post_theorem_intro_text": "\\begin{remark}\\label{remcf}\n\tIt is clear that $f\\in C^{\\alpha}(\\rn\\setminus\\mathcal{O})$. Moreover, the viscosity solution $u$ in Theorem \\ref{thm1} \n\tis really $C^{2,\\alpha}$ near infinity, see Section 3.\n\\end{remark}\n\n\\begin{remark}\\label{rembeta}\n\tThe assumption $\\beta>2$ is essentially optimal, one can see the example in \\cite{blz,tz}.\n\\end{remark}\n\nThe major difficulties in this paper are caused by the lower regularity and the aperiodicity of $f$. To get the asymptotic behavior near infinity, \nthe authors both in \\cite{cl2} and \\cite{tz} paid much attention on the second-order increment of $u$. Their vital step is deducing a suitable \nequation of the increment. More specifically, in \\cite{cl2}, although $f$ is just H\\\"{o}lder continuous they obtained a homogeneous equation of the increment \nsince $f$ is periodic, while in \\cite{tz} the authors got a nonhomogeneous equation in which the nonhomogeneous term can be controlled since the higher \nregularity of $f$. \nIn the case $f\\in L^\\infty$ is periodic \\cite{ll}, the authors established the asymptotic results by following the main ideas in \n\\cite{cl2} with additional approximation arguments. \nIt is not the case in our paper and the second-order increment is not an appropriate object for us. We will propose a novel \nfractional order increment and deduce a nonhomogeneous equation of it. Then we employ the nonlocal ideas in \\cite{qb} and refine the \nnonlocal method to measure the decay of the nonhomogeneous term. The asymptotic behavior is finally established with the aid of the fractional order increment.\n\n\\subsection*{Notations and the structure of the article}\n\nThroughout the paper, we use the following notations:\n\nThe point $x\\in\\rn$ will also be written as $x=(x_1,x')=(x_1,\\cdots,x_n)$. The notation $|x|$ denotes the Euclidean norm of $x$.\n$B_r(x)\\subset\\rn$ denotes the ball centered at $x$ with radius $r$. \nWe drop the center if it coincides with the origin, i.e., $B_r=B_r(0)$. \n\nThe identity matrix is denoted by $I$. \nFor an $n\\times n$ matrix $B=(b_{ij})_{n\\times n}$, $1\\leq i,j\\leq n$, $cof_{ij}B$ stands for the algebraic cofactor of \nelement $b_{ij}$. We will always assume that $i,j\\in\\mathbb{N}_+$ and $i,j\\leq n$ in this paper \nexcept for additional explanation.\n\nGiven a function $u:\\rn\\rightarrow{\\mathbb R}$ and a point $x\\in\\rn$, we denote by $Du(x)$ and $\\Delta u(x)$ the gradient  vector and \nthe  Laplacian respectively.  \n\nLet $k\\in\\mathbb{N}$, $\\kappa\\in(0,1)$ and $\\Omega\\subset\\rn$ be open. For multi-indices \n$\\gamma=(\\gamma_1,\\cdots,\\gamma_n)\\in\\mathbb{N}^n$, we let $|\\gamma|$ denote the sum of its components. \nThe space $C^k(\\overline{\\Omega})$ consists of functions $u: \\overline{\\Omega}\\rightarrow{\\mathbb R}$, which admit derivatives \nup to order $k$, such that \\begin{equation*}\n\t\\|u\\|_{C^{k}(\\overline{\\Omega})}:=\\sup_{\\substack{ \\gamma\\in\\mathbb{N}^n\\\\|\\gamma|\\leq k}}\n\t\\sup_{x\\in{\\Omega}}\\left|D^\\gamma u(x)\\right|<+\\infty.\\end{equation*}\nThe H\\\"{o}lder space $C^{k,\\kappa}(\\overline{\\Omega})$ consists of function $u\\in C^k(\\overline{\\Omega})$ satisfying \n\\begin{gather*}\n\t\\|u\\|_{C^{k,\\kappa}(\\overline{\\Omega})}:=\\|u\\|_{C^{k}(\\overline{\\Omega})}+[u]_{C^{k,\\kappa}(\\overline{\\Omega})}<+\\infty, \n\\end{gather*} \nwhere the seminorm \n\\begin{equation*}\n\t[u]_{C^{k,\\kappa}(\\overline{\\Omega})}:=\\sup_{\\substack{ \\gamma\\in\\mathbb{N}^n\\\\|\\gamma|=k}}\n\t\\sup_{\\substack{x,y\\in{\\Omega} \\\\ x\\neq y}}\\frac{|D^\\gamma u(x)-D^\\gamma u(y)|}{|x-y|^\\kappa}.\n\\end{equation*}\nA function $u\\in C^{k,\\kappa}_{loc}({\\Omega})$ if $u\\in C^{k,\\kappa}(K)$ for all compact \n$K\\subset\\Omega$. For convenience, we may write $C^{0}(\\overline{\\Omega})$, $C^{k,\\kappa}(\\overline{\\Omega})$ as $C(\\overline{\\Omega})$, \n$C^{k+\\kappa}(\\overline{\\Omega})$,  respectively. \n\nThe paper is organized as follows: In Section 2, we introduce the fractional Laplacian and a series of relevant properties,  \nwhich will be applied in the sequel. In Section 3, we deduce the equation for the fractional order increment (Lemma \\ref{lemeqv}) and then \nshow the crucial estimate (Lemma \\ref{lemeF}) by a nonlocal method. Finally, Theorem \\ref{thm1} is proved in Section 4.",
    "sketch": "The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \\cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \\cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”\n\nThey also indicate the proof’s placement/structure: in Section 3 they “deduce the equation for the fractional order increment (Lemma \\ref{lemeqv}) and then show the crucial estimate (Lemma \\ref{lemeF}) by a nonlocal method,” and “finally, Theorem \\ref{thm1} is proved in Section 4.”",
    "expanded_sketch": "The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \\cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \\cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”\n\nThey also indicate the proof’s placement/structure: next they deduce the equation for the fractional order increment. \\begin{lemma}\\label{lemeqv}\n\tLet $E$, $v$ and $f_T$ be as above. Then, for $s\\in(0,\\frac{\\az}{2})$, \n\t\\begin{equation*}\n\t\ta_{ij}(x)\\pat_{ij}(\\De_e^sv(x))\\geq F_s(x),\\ \\ \\ \\ \\forall x\\in\\rn,\\ e\\in E\\setminus\\{0\\},\n\t\\end{equation*}\n\twhere $a_{ij}(x)=cof_{ij}(D^2v(x))$ and \n\t\\begin{equation*}\n\t\tF_s(x):=n\\left(\\det(D^2v(x))^{\\frac{n-1}{n}}\\right)\\left(\\frac{f_T^{\\frac{1}{n}}(x+e)+f_T^{\\frac{1}{n}}(x-e)-2f_T^{\\frac{1}{n}}(x)}{|e|^{2s}}\\right).\n\t\\end{equation*}\n\\end{lemma}\nThey then show the crucial estimate by a nonlocal method. \\begin{lemma}\\label{lemeF}\n\tLet $F_s$ be the one obtained in Lemma \\ref{lemeqv}. Then we have \n\t\\begin{equation}\\label{eqlemeF1}\n\t\t|F_s(x)|\\leq \n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\tC_0(1+|x|)^{-\\min\\{\\be+2s,n\\}},&\\ \\ \\ \\be+2s\\neq n,\\ \\be\\neq n,\\\\\n\t\t\tC_0(1+|x|)^{-\\beta-2s}(\\ln(2+|x|)),&\\ \\ \\ \\be+2s=n,\\\\\n\t\t\tC_0(1+|x|)^{-\\beta-2s}(\\ln(2+|x|)),&\\ \\ \\ \\be=n,\n\t\t\\end{array}\n\t\t\\right.\\ \\ \\ \\ \\ \n\t\\end{equation}\n\tfor all $e\\in E\\setminus\\{0\\},\\ x\\in\\rn.$ The constant $C_0>0$ depends only $d_0,\\ d_1,\\ \\be,\\ n$ and $s$.\n\\end{lemma}\nFinally, in establishing the main theorem, they complete the argument in the subsequent section.",
    "expanded_theorem": "\\label{thm1}\n\tLet $u\\in C_{loc}(\\rn)$ be a convex viscosity solution of \n\\begin{equation}\\label{eqma}\n\t\\det(D^2u)=f,\\ \\ \\ \\ \\ \\text{in}\\ \\mathbb{R}^n,\n\\end{equation}\n, where $f:\\rn\\rightarrow{\\mathbb R}$ satisfies \n\t{\\textnormal{(H)}}. If $n\\geq3$, then there exist $b\\in\\rn$, $A\\in\\mathcal{A}$ and $u_p\\in C^{2,\\alpha}(\\rn)$ such that\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\\left|u(x)-\\left(\\frac{1}{2}x'Ax+b\\cdot x+u_p(x)\\right)\\right|\\leq C_0(1+|x|)^{2-\\min\\{n,\\beta\\}},\\ \\ \\ \\forall\\ x\\in\\rn,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\\cdots,n$,   \n    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\\beta$, ${\\alpha}$ and $a_1,\\cdots,a_n$.,\n",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $n\\ge 3$, and let $u\\in C_{\\mathrm{loc}}(\\mathbb{R}^n)$ be a convex viscosity solution of\n\\[\n\\det(D^2u)=f\\quad \\text{in }\\mathbb{R}^n.\n\\]\nAssume there exist constants $d_0>0$, $d_1>0$, $\\alpha\\in(0,1)$, $\\beta>2$, and $a_1,\\dots,a_n>0$, together with a positive periodic function $f_p:\\mathbb{R}^n\\to\\mathbb{R}$ such that\n\\[\nd_0^{-1}\\le f_p\\le d_0,\\qquad [f_p]_{C^\\alpha(\\mathbb{R}^n)}\\le d_0,\n\\]\nand\n\\[\nf_p(x+a_i e_i)=f_p(x)\\quad \\text{for all }x\\in\\mathbb{R}^n,\\ i=1,\\dots,n.\n\\]\nAlso assume $f\\in C(\\mathbb{R}^n)\\cap C^{\\alpha}_{\\mathrm{loc}}(\\mathbb{R}^n\\setminus \\overline{\\mathcal O})$ for some bounded open set $\\mathcal O\\subset \\mathbb{R}^n$, and that for every $x\\in\\mathbb{R}^n\\setminus\\{0\n\\}$,\n\\[\n|x|^{\\beta}|(f-f_p)(x)|+|x|^{\\beta+\\alpha}[f-f_p]_{C^\\alpha(\\overline{B_{|x|/2}(x)})}\\le d_1.\n\\]\nDefine\n\\[\n\\mathcal A:=\\left\\{A\\in \\mathbb{R}^{n\\times n}: A\\text{ is symmetric, positive definite, and }\\det(A)=\\dashint_{\\prod_{i=1}^n[0,a_i]} f(x)\\,dx\\right\\},\n\\]\nwhere $\\dashint$ denotes the average over the rectangle $\\prod_{i=1}^n[0,a_i]$. Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exist $b\\in\\mathbb{R}^n$, $A\\in\\mathcal A$, and a function $u_p\\in C^{2,\\alpha}(\\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\\dots,n$, such that\n\\[\n\\left|u(x)-\\left(\\frac12 x^T A x+b\\cdot x+u_p(x)\\right)\\right|\\le C_0(1+|x|)^{2-\\min\\{n,\\beta\\}}\\qquad \\forall x\\in\\mathbb{R}^n,\n\\]\nwhere $C_0>0$ depends only on $d_0,d_1,n,\\beta,\\alpha,$ and $a_1,\\dots,a_n$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist $b\\in\\mathbb{R}^n$, $A\\in\\mathcal A$, and a function $u_p\\in C^{2,\\alpha}(\\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\\dots,n$, such that\n\\[\n\\left|u(x)-\\left(\\frac12 x^T A x+b\\cdot x+u_p(x)\\right)\\right|\\le C_0(1+|x|)^{2-\\min\\{n,\\beta+\\alpha\\}}\\qquad \\forall x\\in\\mathbb{R}^n,\n\\]\nwhere $C_0>0$ depends only on $d_0,d_1,n,\\beta,\\alpha,$ and $a_1,\\dots,a_n$."
        },
        {
          "label": "C",
          "text": "There exist $b\\in\\mathbb{R}^n$, $A\\in\\mathcal A$, and a function $u_p:\\mathbb{R}^n\\to\\mathbb{R}$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\\dots,n$, such that\n\\[\n\\left|u(x)-\\left(\\frac12 x^T A x+b\\cdot x+u_p(x)\\right)\\right|\\le C_0(1+|x|)^{2-\\min\\{n,\\beta\\}}\\qquad \\forall x\\in\\mathbb{R}^n,\n\\]\nfor some constant $C_0>0$."
        },
        {
          "label": "D",
          "text": "There exists a constant $C_0>0$, depending only on $d_0,d_1,n,\\beta,\\alpha,$ and $a_1,\\dots,a_n$, such that for every $A\\in\\mathcal A$ there exist $b\\in\\mathbb{R}^n$ and a function $u_p\\in C^{2,\\alpha}(\\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\\dots,n$, and\n\\[\n\\left|u(x)-\\left(\\frac12 x^T A x+b\\cdot x+u_p(x)\\right)\\right|\\le C_0(1+|x|)^{2-\\min\\{n,\\beta\\}}\\qquad \\forall x\\in\\mathbb{R}^n.\n\\]"
        },
        {
          "label": "E",
          "text": "There exist $b\\in\\mathbb{R}^n$, $A\\in\\mathcal A$, and a function $u_p\\in C^{2,\\alpha}(\\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\\dots,n$, such that\n\\[\n\\left|u(x)-\\left(\\frac12 x^T A x+b\\cdot x+u_p(x)\\right)\\right|\\le C_0(1+|x|)^{2-\\beta}\\qquad \\forall x\\in\\mathbb{R}^n,\n\\]\nwhere $C_0>0$ depends only on $d_0,d_1,n,\\beta,\\alpha,$ and $a_1,\\dots,a_n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "decay exponent tied to nonlocal estimate",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the stated $C^{2,\\alpha}$ regularity and parameter dependence details for $u_p$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "existential choice of the asymptotic matrix $A$",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "case split at $\\beta\\ge n$ producing the $\\min\\{n,\\beta\\}$ rate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It presents hypotheses and asks for the guaranteed asymptotic conclusion, without embedding the exact answer."
      },
      "TAS": {
        "score": 1,
        "justification": "This is very close to theorem-statement recall: the correct choice is essentially the sharp conclusion of the cited result under the stated hypotheses. The alternatives create some comparison, but the item is still largely a near-restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing the sharp rate, the role of min{n, β}, the uniform-for-all-x bound versus asymptotic little-o, and the determinant constraint on A. However, the task mostly tests recognition of the exact theorem statement rather than substantial derivation."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically meaningful and target common errors: B overstates the decay rate, D alters the quantifier/strength of the conclusion, and E corrupts the determinant condition on A. But C is a weaker statement that appears to be true if A is true, so it weakens the single-correct-answer structure."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated but theorem-recall-heavy MCQ. It avoids answer leakage and uses mostly plausible distractors, but it is close to a direct theorem restatement and is weakened by including a distractor that is also true in a weaker form."
    }
  },
  {
    "id": "2512.07362v1",
    "paper_link": "http://arxiv.org/abs/2512.07362v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.",
      "start_pos": 8781,
      "end_pos": 9185,
      "label": "th:main"
    },
    "ref_dict": {
      "th:main": "\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}",
      "TWS": "\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}",
      "pp": "\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}",
      "BC": "\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 4485,
    "pre_theorem_intro_text": "\\setcounter{equation}{0}\n\nWe consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\,  $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that  $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.",
    "context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}",
    "full_context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nWe now state our main theorem as follows.\n\nHereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.\n\n\\begin{proposition}\\label{le6}\nLet $s>0$ be given. Let $(\\overline{\\phi },\\overline{\\psi })$ and $(\\underline{\\phi},\\underline{\\psi})$ be a pair of bounded upper and lower solutions of \\eqref {TWS}. Then system \\eqref {TWS} admits a positive solution $(\\phi,\\psi)$ of class $C^1(\\R)$ such that\n\\begin{equation*}\n\\underline{\\phi }( z )\\leq \\phi(z)\\leq \\overline{\\phi}( z ),\\; \\upsi(z)\\leq \\psi(z)\\leq\\overline{\\psi}( z ),\\; z \\in \\mathbb{R}.\n\\end{equation*}\n\\end{proposition}\n\nFor a given $s>s^*$, we consider the quantity\n\\beaa\nA(\\lambda)=A(\\lambda;s):=d[I_2(\\lambda)-1]-s\\lambda+b,\\; \\lambda>0.\n\\eeaa\nIt follows from the definition of $s^*$ and the strict convexity of $A(\\lambda)$ that there are two positive constants $\\lambda_1<\\lambda_2<\\hat{\\ld}_2$ such that\n$$A(\\lambda_1)=A(\\ld_2)=0$$\nand $A(\\lambda)<0$ for all $\\lambda\\in(\\lambda_1,\\lambda_2)$. Also, set \n$$B(\\ld)=B(\\ld;s):=[I_1(\\ld)-1]-s\\ld.$$ \nSince $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\\ld_0\\in(0,\\min\\{\\ld_1,\\hat{\\ld}_1\\})$ small enough such that\n$$B(\\ld_0)<0.$$\nNow, for a fixed constant $\\mu\\in(1,\\min{\\{ \\lambda_2/ \\lambda_1,2\\}})$, we choose \n\\be\\label{q0}\nq>\\max\\Big\\{1,\\frac{2b}{-A(\\mu\\ld_1)}\\Big\\}.\n\\ee\nThen, the function $f(z) := e^{ -\\lambda_1z}-qe^{-\\mu \\lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\\in(z_0,\\infty)$, and there holds $f(z_M)>0$. Thus, using $a\\ge4>1$ and $d<b$, we can choose $\\delta$ such that\n$$0<\\delta<\\min\\Big\\{f(z_M),\\underbrace{1-\\frac{1}{a}}_{=a^*},\\frac12\\Big(1-\\frac{d}{b}\\Big)\\Big\\},$$\nand a point $z_1\\in(z_0,z_M)$ such that $f(z_1)=\\delta$. Note that\n\\be\\label{delta}\nd<b(1-2\\delta).\n\\ee\nFurthermore, the inequalities $z_1>z_0>0$, $\\ld_1>0$ and $\\mu>1$ imply\n\\be\\label{q}\n{e^{(\\mu-1)\\ld_1z_1}>e^{(\\mu-1)\\ld_1z_0}=q.}\n\\ee\nWith these choices of $\\mu,q,\\delta$, we finally choose $\\e$ such that\n\\be\\label{ep1} \n0<\\e<\\min{\\left\\{\\frac{\\delta}{1+s\\ld_1+a},\\frac{e^{(\\mu-1)\\lambda_1z_1}-q}{(1+s\\lambda_1+a)e^{(\\mu-1)\\lambda_1z_1}} \\right\\}}.\n\\ee\nNote that the constant $\\e$ is admissible, due to \\eqref{q}, and that $0<\\e<\\delta<1/2$.\n\n\\begin{lemma}\\label{la:upper-lower2}\nAssume that $a\\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\\in(0,\\infty)$. For $s=s^{*}$, the functions $(\\overline{\\phi},\\overline{\\psi})$ and~$(\\underline{\\phi},\\underline{\\psi})$ defined in \\eqref{uplosol2} are a pair of upper and lower solutions of \\eqref{TWS}.\n\\end{lemma}\n\nFirst, for any $s\\ge s^*$, the existence of a $C^1(\\R)$ solution $(\\phi,\\psi)$ to~\\eqref{TWS} such that $1/2\\le\\underline{\\phi}\\le\\phi\\le\\overline{\\phi}<1$ and $0<\\underline{\\psi}\\le\\psi\\le\\overline{\\psi}\\le1$ in $\\R$ follows from Lemmas~\\ref{la:upper-lower1}-\\ref{la:upper-lower2} and Proposition~\\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\\phi,\\psi)(\\infty)=(1,0)$.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\n\n\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1964,
    "post_theorem_intro_text": "Hereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nIt is worth to note that there are some difficulties in the study of system \\eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \\cite{CLS}. Thus system \\eqref{pp} is named as a {\\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \\eqref{pp} replaced by\n\\beaa\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-kUV](x,t), \\; x\\in\\bR,\\, t>0,\n\\eeaa\nin which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.",
    "sketch": "The post-theorem introduction outlines the proof of Theorem~\\ref{th:main} as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\\eqref{TWS}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** “Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.”",
    "expanded_sketch": "The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.",
    "expanded_theorem": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\nfor each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \\eqref{TWS} with boundary conditions \\eqref{BC} for $s < s^{*}$.",
    "theorem_type": [
      "Existence",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let \\(a,b,d>0\\) with \\(a\\ge 4\\) and \\(d<b\\), and set \\(a^*=1-\\frac1a\\). For \\(i=1,2\\), define the nonlocal dispersal operator\n\\[\n\\mathcal N_i[u](z)=\\int_{\\mathbb R}J_i(y)u(z-y)\\,dy-u(z),\n\\]\nwhere each kernel \\(J_i\\) is a nonnegative continuous symmetric probability density on \\(\\mathbb R\\) satisfying the exponential-moment condition: there exists \\(\\hat\\lambda_i\\in(0,\\infty]\\) such that \\(\\int_{\\mathbb R}J_i(y)e^{\\lambda y}\\,dy<\\infty\\) for \\(0<\\lambda<\\hat\\lambda_i\\), and this integral tends to \\(\\infty\\) as \\(\\lambda\\uparrow \\hat\\lambda_i\\). Let\n\\[\nI_2(\\lambda)=\\int_{\\mathbb R}J_2(y)e^{\\lambda y}\\,dy,\n\\qquad\ns^*=\\inf_{\\lambda\\in(0,\\hat\\lambda_2)}\\frac{d\\,[I_2(\\lambda)-1]+b}{\\lambda}.\n\\]\nConsider the traveling-wave profile system\n\\[\n\\begin{cases}\n\\mathcal N_1[\\phi](z)+s\\phi'(z)+a\\phi(z)(1-\\phi(z))-\\psi(z)=0, & z\\in\\mathbb R,\\\\\n d\\,\\mathcal N_2[\\psi](z)+s\\psi'(z)+b\\psi(z)\\left(1-\\dfrac{\\psi(z)}{\\phi(z)}\\right)=0, & z\\in\\mathbb R,\n\\end{cases}\n\\]\nwith boundary conditions\n\\[\n(\\phi,\\psi)(+\\infty)=(1,0),\\qquad (\\phi,\\psi)(-\\infty)=(a^*,a^*).\n\\]\nWhich statement holds for positive solutions \\((\\phi,\\psi)\\) of this boundary-value problem?",
      "correct_choice": {
        "label": "A",
        "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that \\(J_2\\) has compact support. For every \\(s<s^*\\), no positive solution exists."
      },
      "choices": [
        {
          "label": "B",
          "text": "A positive solution exists for every wave speed \\(s\\ge s^*\\). Moreover, for every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "C",
          "text": "For every \\(s>s^*\\), there exists a positive solution. In addition, for every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "D",
          "text": "A positive solution exists for every wave speed \\(s>s^*\\), and also for \\(s=s^*\\) without any additional assumption on \\(J_2\\). For every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "E",
          "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that both \\(J_1\\) and \\(J_2\\) have compact support. For every \\(s<s^*\\), no positive solution exists."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical-speed compact-support hypothesis at s=s^*",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the critical-speed existence clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of critical-speed existence on compact support of J_2",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "which kernel requires compact support at the critical speed",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the conclusion outright. It defines the setting and the threshold speed $s^*$, but the exact existence/nonexistence regime—especially the subtle critical-speed compact-support exception—is not leaked."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the theorem’s exact statement in this setting, so it is largely a recall/restatement question. Still, the options differ in meaningful ways at the critical speed, so it is not a pure tautology."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare the nuanced alternatives ($s>s^*$, $s=s^*$, compact support, and universal nonexistence for $s<s^*$). However, it mainly tests precise theorem recall rather than generating a conclusion from the equations."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and target natural failure modes: overclaiming existence at $s=s^*$, omitting the compact-support hypothesis, making the threshold too strict, or weakening the nonexistence quantifier. They are distinct and mathematically relevant."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it leans more toward exact-result recall than genuine generative reasoning."
    }
  },
  {
    "id": "2512.07362v1",
    "paper_link": "http://arxiv.org/abs/2512.07362v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.",
      "start_pos": 8781,
      "end_pos": 9185,
      "label": "th:main"
    },
    "ref_dict": {
      "th:main": "\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}",
      "TWS": "\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}",
      "pp": "\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}",
      "BC": "\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 4485,
    "pre_theorem_intro_text": "\\setcounter{equation}{0}\n\nWe consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\,  $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that  $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.",
    "context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}",
    "full_context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nWe now state our main theorem as follows.\n\nHereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.\n\n\\begin{proposition}\\label{le6}\nLet $s>0$ be given. Let $(\\overline{\\phi },\\overline{\\psi })$ and $(\\underline{\\phi},\\underline{\\psi})$ be a pair of bounded upper and lower solutions of \\eqref {TWS}. Then system \\eqref {TWS} admits a positive solution $(\\phi,\\psi)$ of class $C^1(\\R)$ such that\n\\begin{equation*}\n\\underline{\\phi }( z )\\leq \\phi(z)\\leq \\overline{\\phi}( z ),\\; \\upsi(z)\\leq \\psi(z)\\leq\\overline{\\psi}( z ),\\; z \\in \\mathbb{R}.\n\\end{equation*}\n\\end{proposition}\n\nFor a given $s>s^*$, we consider the quantity\n\\beaa\nA(\\lambda)=A(\\lambda;s):=d[I_2(\\lambda)-1]-s\\lambda+b,\\; \\lambda>0.\n\\eeaa\nIt follows from the definition of $s^*$ and the strict convexity of $A(\\lambda)$ that there are two positive constants $\\lambda_1<\\lambda_2<\\hat{\\ld}_2$ such that\n$$A(\\lambda_1)=A(\\ld_2)=0$$\nand $A(\\lambda)<0$ for all $\\lambda\\in(\\lambda_1,\\lambda_2)$. Also, set \n$$B(\\ld)=B(\\ld;s):=[I_1(\\ld)-1]-s\\ld.$$ \nSince $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\\ld_0\\in(0,\\min\\{\\ld_1,\\hat{\\ld}_1\\})$ small enough such that\n$$B(\\ld_0)<0.$$\nNow, for a fixed constant $\\mu\\in(1,\\min{\\{ \\lambda_2/ \\lambda_1,2\\}})$, we choose \n\\be\\label{q0}\nq>\\max\\Big\\{1,\\frac{2b}{-A(\\mu\\ld_1)}\\Big\\}.\n\\ee\nThen, the function $f(z) := e^{ -\\lambda_1z}-qe^{-\\mu \\lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\\in(z_0,\\infty)$, and there holds $f(z_M)>0$. Thus, using $a\\ge4>1$ and $d<b$, we can choose $\\delta$ such that\n$$0<\\delta<\\min\\Big\\{f(z_M),\\underbrace{1-\\frac{1}{a}}_{=a^*},\\frac12\\Big(1-\\frac{d}{b}\\Big)\\Big\\},$$\nand a point $z_1\\in(z_0,z_M)$ such that $f(z_1)=\\delta$. Note that\n\\be\\label{delta}\nd<b(1-2\\delta).\n\\ee\nFurthermore, the inequalities $z_1>z_0>0$, $\\ld_1>0$ and $\\mu>1$ imply\n\\be\\label{q}\n{e^{(\\mu-1)\\ld_1z_1}>e^{(\\mu-1)\\ld_1z_0}=q.}\n\\ee\nWith these choices of $\\mu,q,\\delta$, we finally choose $\\e$ such that\n\\be\\label{ep1} \n0<\\e<\\min{\\left\\{\\frac{\\delta}{1+s\\ld_1+a},\\frac{e^{(\\mu-1)\\lambda_1z_1}-q}{(1+s\\lambda_1+a)e^{(\\mu-1)\\lambda_1z_1}} \\right\\}}.\n\\ee\nNote that the constant $\\e$ is admissible, due to \\eqref{q}, and that $0<\\e<\\delta<1/2$.\n\n\\begin{lemma}\\label{la:upper-lower2}\nAssume that $a\\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\\in(0,\\infty)$. For $s=s^{*}$, the functions $(\\overline{\\phi},\\overline{\\psi})$ and~$(\\underline{\\phi},\\underline{\\psi})$ defined in \\eqref{uplosol2} are a pair of upper and lower solutions of \\eqref{TWS}.\n\\end{lemma}\n\nFirst, for any $s\\ge s^*$, the existence of a $C^1(\\R)$ solution $(\\phi,\\psi)$ to~\\eqref{TWS} such that $1/2\\le\\underline{\\phi}\\le\\phi\\le\\overline{\\phi}<1$ and $0<\\underline{\\psi}\\le\\psi\\le\\overline{\\psi}\\le1$ in $\\R$ follows from Lemmas~\\ref{la:upper-lower1}-\\ref{la:upper-lower2} and Proposition~\\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\\phi,\\psi)(\\infty)=(1,0)$.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\n\n\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1964,
    "post_theorem_intro_text": "Hereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nIt is worth to note that there are some difficulties in the study of system \\eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \\cite{CLS}. Thus system \\eqref{pp} is named as a {\\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \\eqref{pp} replaced by\n\\beaa\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-kUV](x,t), \\; x\\in\\bR,\\, t>0,\n\\eeaa\nin which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.",
    "sketch": "The post-theorem introduction outlines the proof of Theorem~\\ref{th:main} as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\\eqref{TWS}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** “Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.”",
    "expanded_sketch": "The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.",
    "expanded_theorem": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\nfor each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \\eqref{TWS} with boundary conditions \\eqref{BC} for $s < s^{*}$.",
    "theorem_type": [
      "Existence",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let \\(a,b,d>0\\) with \\(a\\ge 4\\) and \\(d<b\\), and set \\(a^*=1-\\frac1a\\). For \\(i=1,2\\), define the nonlocal dispersal operator\n\\[\n\\mathcal N_i[u](z)=\\int_{\\mathbb R}J_i(y)u(z-y)\\,dy-u(z),\n\\]\nwhere each kernel \\(J_i\\) is a nonnegative continuous symmetric probability density on \\(\\mathbb R\\) satisfying the exponential-moment condition: there exists \\(\\hat\\lambda_i\\in(0,\\infty]\\) such that \\(\\int_{\\mathbb R}J_i(y)e^{\\lambda y}\\,dy<\\infty\\) for \\(0<\\lambda<\\hat\\lambda_i\\), and this integral tends to \\(\\infty\\) as \\(\\lambda\\uparrow \\hat\\lambda_i\\). Let\n\\[\nI_2(\\lambda)=\\int_{\\mathbb R}J_2(y)e^{\\lambda y}\\,dy,\n\\qquad\ns^*=\\inf_{\\lambda\\in(0,\\hat\\lambda_2)}\\frac{d\\,[I_2(\\lambda)-1]+b}{\\lambda}.\n\\]\nConsider the traveling-wave profile system\n\\[\n\\begin{cases}\n\\mathcal N_1[\\phi](z)+s\\phi'(z)+a\\phi(z)(1-\\phi(z))-\\psi(z)=0, & z\\in\\mathbb R,\\\\\n d\\,\\mathcal N_2[\\psi](z)+s\\psi'(z)+b\\psi(z)\\left(1-\\dfrac{\\psi(z)}{\\phi(z)}\\right)=0, & z\\in\\mathbb R,\n\\end{cases}\n\\]\nwith boundary conditions\n\\[\n(\\phi,\\psi)(+\\infty)=(1,0),\\qquad (\\phi,\\psi)(-\\infty)=(a^*,a^*).\n\\]\nWhich statement holds for positive solutions \\((\\phi,\\psi)\\) of this boundary-value problem?",
      "correct_choice": {
        "label": "A",
        "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that \\(J_2\\) has compact support. For every \\(s<s^*\\), no positive solution exists."
      },
      "choices": [
        {
          "label": "B",
          "text": "A positive solution exists for every wave speed \\(s\\ge s^*\\). Moreover, for every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "C",
          "text": "For every \\(s>s^*\\), there exists a positive solution. In addition, for every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "D",
          "text": "A positive solution exists for every wave speed \\(s>s^*\\), and also for \\(s=s^*\\) without any additional assumption on \\(J_2\\). For every \\(s<s^*\\), no positive solution exists."
        },
        {
          "label": "E",
          "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that both \\(J_1\\) and \\(J_2\\) have compact support. For every \\(s<s^*\\), no positive solution exists."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical-speed compact-support hypothesis at s=s^*",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the critical-speed existence clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of critical-speed existence on compact support of J_2",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "which kernel requires compact support at the critical speed",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and hypotheses but does not directly reveal which existence/nonexistence statement is correct. The correct choice is not implied by wording cues alone."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: the stem presents the full hypotheses and asks for the exact conclusion. However, it is not a pure verbatim restatement because the options introduce competing variants about the critical speed and compact-support assumptions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some discrimination is required among subtle alternatives, especially the role of compact support of J2 at s=s*. But the task mainly tests recall of the theorem’s precise statement rather than substantial derivation or generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and well targeted: they vary in logically meaningful ways (dropping the critical-speed clause, overgeneralizing to all s>=s*, removing the compact-support condition, or assigning it to the wrong kernel). These reflect realistic confusion points."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than genuine generative reasoning."
    }
  },
  {
    "id": "2512.07560v2",
    "paper_link": "http://arxiv.org/abs/2512.07560v2",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thma",
      "content": "[\\cref{thm:main1,thm:mainsimplification}]\n\\label{thmfirst:intro}\nThe augmented vertically parametrized system $(C(\\kappa \\circ x^M), Lx - b)$ admits multiple positive zeros if and only if  there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ \nhas a solution.",
      "start_pos": 11415,
      "end_pos": 11826,
      "label": "thmfirst:intro"
    },
    "ref_dict": {
      "thm:mono": "\\begin{theorem}[Preclusion of multiple positive zeros]\\label{thm:mono}\n Let $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal and $L$ of full row rank.  Let  $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be  row and column partitions, and consider the  associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix}. \n\n If $\\mathcal{C}_{\\mathcal{S}}^\\sigma = \\varnothing$ for all orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$   compatible with $\\alpha$ and  all  $\\mathcal{S}\\in \\{-1,0,1\\}^{s\\times \\ell}$\n that are feasible with respect to $(P,\\sigma)$, then $F$ does not admit multiple positive zeros for any choice of parameter values.\n\\end{theorem}",
      "thm:full": "\\begin{theorem}[Characterization of multiple positive zeros when $P$ induces a forest]\\label{thm:full}\nLet $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal and $L$ of full row rank. \nLet  $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be  row and column partitions, and assume that the associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix} induces a forest. \n\n Then, $F$ admits multiple positive zeros for some  parameter values $\\k\\in \\R^{\\bar{m}}_{>0}$, $b\\in \\R^{n-\\bs}$  if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$ and a sign matrix  $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ that is feasible with respect to $(P,\\sigma)$  such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{theorem}",
      "def:simpif_reduced_matrix": "\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}"
    },
    "pre_theorem_intro_text_len": 4712,
    "pre_theorem_intro_text": "Polynomial systems that arise in applications  have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the  form\n\\begin{equation*}\n    C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system,  and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial  is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n    \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2  , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n    1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n    1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n}  \\, ,\n\\]\nwhere  $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters. \n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad  Lx = Ly = 0\\, .\\]\n\n Our approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a  family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that,  in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, ,  \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional  rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any   $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and    $A^\\sigma_{\\rho}$  defined by \n\\[ \n     (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad  (A^\\sigma_{\\rho})_{ik} =  P_{ik} \n     (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}),  \\qquad i \\in \\{1,\\dots,s\\}, \\  k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n  With these objects in place,  the \\bfemph{oriented characteristic system}  associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set   $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.",
    "context": "Polynomial systems that arise in applications have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the form\n\\begin{equation*}\n C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system, and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2 , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n 1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n 1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n} \\, ,\n\\]\nwhere $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters.\n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad Lx = Ly = 0\\, .\\]\n\nOur approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that, in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, , \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and $A^\\sigma_{\\rho}$ defined by \n\\[ \n (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad (A^\\sigma_{\\rho})_{ik} = P_{ik} \n (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}), \\qquad i \\in \\{1,\\dots,s\\}, \\ k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n With these objects in place, the \\bfemph{oriented characteristic system} associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.\n\n\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}",
    "full_context": "Polynomial systems that arise in applications have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the form\n\\begin{equation*}\n C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system, and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2 , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n 1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n 1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n} \\, ,\n\\]\nwhere $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters.\n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad Lx = Ly = 0\\, .\\]\n\nOur approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that, in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, , \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and $A^\\sigma_{\\rho}$ defined by \n\\[ \n (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad (A^\\sigma_{\\rho})_{ik} = P_{ik} \n (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}), \\qquad i \\in \\{1,\\dots,s\\}, \\ k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n With these objects in place, the \\bfemph{oriented characteristic system} associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.\n\n\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}\n\n\\begin{thmc}[\\cref{thm:full}]\nAssume that $P\\in \\R^{s\\times \\ell}$ induces a forest. \n Then the system $( C (\\k \\circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{thmc}\n\n\\begin{theorem}\n\\label{thm:mainsimplification}\n Let $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal with reduced matrix $\\oP\\in \\R^{\\bs\\times \\bl}$, and $L\\in \\R^{(n-\\bs)\\times n}$ has full rank. \n Let $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be row and column partitions, and consider the associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix}. \nThe following statements are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n \\item There exists $\\bar{\\rho} \\in M^\\top(\\mO_L)$ such that the characteristic system\n\\[ \nA_{\\bar{\\rho}}\\, \\bar{\\mu} = 0\\,, \\qquad \\oP\\, \\bar{\\mu} > 0\\,, \\qquad \\bar{\\mu}\\in \\R^{\\bl}_{>0}\n\\]\nhas a solution. \n \\item There exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$, and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system\n\\[ A^\\sigma_{\\rho} \\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\mathbb{R}^{\\ell}_{>0}\\]\nhas a solution. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{lemma}\\label[lemma]{mainlemma}\nLet $P\\in \\R^{s\\times \\ell}$, $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell},\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$ such that \n$\\mS$ is feasible with respect to $(P,\\sigma)$ and consider their associated $\\Lambda$-sets. \nAssume $P^\\sigma\\mu^*>0$ for some $\\mu^*\\in \\R^\\ell_{>0}$ and let $\\rho \\in \\R^m\\setminus \\mathcal{D}_{\\mS}^{\\sigma} $ such that $\\sign(A_{\\rho}^{\\sigma}) = \\mathcal{S}$. Then, there exists\n $Q \\in \\mathbb{R}^{s \\times \\ell}$ with $\\sign(Q) = \\mathcal{S}$ such that the system \n\\[ Q \\mu = 0,\\qquad P^\\sigma \\mu>0, \\qquad \\mu\\in \\R^\\ell_{>0} \\]\nhas a solution and, additionally, for all $i \\in [s]$, it holds\n \\begin{equation}\\label{eq2MainLemma} \n \\sum_{j \\in \\Lambda_i^{\\neq}}\\frac{Q_{ij} \\, \\mu_j}{e^{\\rho_i} - \\frac{\\sigma_{2j}}{\\sigma_{1j}}\\, e^{\\rho_{s + j}}} + \\sum_{j \\notin \\Lambda_i}P^{\\sigma}_{ij} \\mu_j > 0 \\, .\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nAs $\\rho \\notin \\mathcal{D}_{\\mS}^\\sigma$, if $I_{\\rho}^+ \\cup I_{\\rho}^{-}\\neq \\varnothing$, then there exists $\\mu \\in \\bigcap_{i \\in I_{\\rho}^+ \\cup I_{\\rho}^{-}}\\Gamma_{\\Lambda,i} \\neq \\varnothing$. If $I_{\\rho}^+ \\cup I_{\\rho}^{-}= \\varnothing$, then we let $\\mu=\\mu^*$. \nIn both cases $P^\\sigma \\mu>0$, and hence, by \\cref{prop:feasible}, there exists $\\widetilde{Q}\\in \\mathbb{R}^{s \\times \\ell} $ with $\\sign(\\wQ) = \\mathcal{S}$ such that $\\wQ \\mu = 0$. \nWe construct now a parametric matrix $Q(\\omega)$ for $\\omega = (\\omega_1,\\dots,\\omega_s) \\in \\mathbb{R}_{>0}^s$ such that, for every $i\\in [s]$, $\\omega_i$ only modifies the $i$-th row and it holds that \n\\begin{equation}\\label{eq:row_mod}\n\\sign(\\wQ_{ij})=\\sign(Q(\\omega)_{ij}) \\quad\\text{for all } j\\in [\\ell]\\, , \\qquad (Q(\\omega)\\mu)_i=0 \\, , \n\\end{equation} \nand \\eqref{eq2MainLemma} holds after choosing $\\omega_i$ either small or large enough.\n\nAssume that $P$ induces a forest. Let \n $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ be an orientation compatible with $\\alpha$, and $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ be feasible with respect to $(P,\\sigma)$. Then, for any $\\rho \\in\\mC_{\\mS}^{\\sigma}$, the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution.\n \\end{theorem}\n\\begin{proof} \nAs $\\rho \\in \\mathcal{C}_{\\mathcal{S}}^\\sigma$, we have in particular that $\\sign(A_{\\rho}^\\sigma) = \\mS$ and $\\rho \\notin \\mathcal{D}_{\\mS}^\\sigma$. Hence by \\cref{mainlemma}, there exist $\\bar{\\mu}\\in \\R^\\ell_{>0}$ and $Q \\in \\mathbb{R}^{s \\times \\ell}$ with $\\sign(Q) = \\mathcal{S}$ such that $Q \\bar{\\mu} = 0$, $P^\\sigma\\bar{\\mu} > 0$, and \n\\[\n \\sum_{j \\in \\Lambda_i^{\\neq}}\\frac{Q_{ij} \\, \\bar{\\mu}_j}{e^{\\rho_i} - \\frac{\\sigma_{2j}}{\\sigma_{1j}}\\,e^{\\rho_{s + j}}} + \\sum_{j \\notin \\Lambda_i}P^{\\sigma}_{ij} \\bar{\\mu}_j > 0 \n\\]\n for all $i \\in [s]$. We will now modify $\\bar{\\mu}$ to obtain $\\mu\\in \\R^\\ell_{>0}$ satisfying $A_\\rho^\\sigma \\mu=0$ and $P^\\sigma\\mu>0$.\n\nThen, $F$ admits multiple positive zeros for some parameter values $\\k\\in \\R^{\\bar{m}}_{>0}$, $b\\in \\R^{n-\\bs}$ if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$ and a sign matrix $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ that is feasible with respect to $(P,\\sigma)$ such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{theorem}\n\\begin{proof}\nIf $F$ admits multiple positive zeros, \\Cref{thm:mono} tells us that there exist $\\sigma$ and $\\mathcal{S}$ as in the statement such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$. Reciprocally, if $\\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$ for some $\\sigma$ and $\\mS$, then \\Cref{thm:multiSimplified} tells us that the oriented characteristic system at any $\\rho\\in \\mathcal{C}_{\\mS}^\\sigma$ has a solution. \nBy \\Cref{thm:mainsimplification}, this implies that \\Cref{thm:main1}(iii) holds and hence $F$ admits multiple positive zeros.\n\\end{proof}\n\n\\begin{example}\n\\label[example]{example:hybrid_histidine_kinase}\nWe illustrate \\Cref{thm:full} with an augmented vertical system that arises from the study of the steady states of a reaction network named \n\\emph{hybrid histidine kinase}. This network is known to admit multiple positive zeros \\cite{hhk, plos:identifying}. The stoichiometric matrix, the matrix of reactants, and a matrix defining the stoichiometric compatibility classes are:\n{\\small \\[ N = \\begin{pmatrix}\n -1 & 0 & 0 & 1 & 0 & 0 \\\\\n 1 & -1 & 0 & 0 & 1 & 0 \\\\\n 0 & 1 & -1 & -1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & -1 & 0 \\\\\n 0 & 0 & 0 & -1 & -1 & 1 \\\\\n 0 & 0 & 0 & 1 & 1 & -1 \\\\\n\\end{pmatrix}, \\quad M = \\begin{pmatrix} \n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1\\end{pmatrix}, \\quad L = \\begin{pmatrix} \n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1\\end{pmatrix}.\\] }After performing Gaussian elimination to $N$, the steady states of the network are the zeros of the augmented vertical system \n$F = (C(\\kappa \\circ x^M), Lx - b) \\in \\R[\\k,b,x]^6$\nwith\n \\[ C = \\begin{pmatrix} \n1 & 0 & 0 & 0 & 1 & -1 \\\\\n0 & 1 & 0 & 0 & 0 & -1 \\\\\n0 & 0 & 1 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & -1 \\end{pmatrix}\\, .\\]\nThe last two columns of $C$ give rise to the reduced matrix $\\oP \\in \\R^{4 \\times 2}$ from \\eqref{eqP}. In this example we have $\\bs=4, \\bar{m}=6$. We consider the row and column partitions to be $[\\bs] = \\{1,4\\} \\sqcup \\{2\\} \\sqcup \\{3\\}$ and $[\\bl] = \\{1\\} \\sqcup \\{2\\}$, so $s = 3,m=5$. We choose the orientation $\\sigma_+$. This gives rise to the following simplified reduced matrix $P$, oriented characteristic matrix $A^{\\sigma_+}_\\rho$ for $\\rho\\in \\R^5$, and associated graph $G_{P}$:",
    "post_theorem_intro_text_len": 7677,
    "post_theorem_intro_text": "It turns out that  the feasibility of the oriented characteristic system depends  on the signs of \\( A^{\\sigma}_{\\rho} \\). Then, an important observation is that  the strictly increasing nature of the exponential function, implies that the sign of each entry of   \\( A^{\\sigma}_{\\rho} \\) is either fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\), and can be expressed as a linear inequality in the entries of $\\rho$. \nFormally, \nwe first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions. Any sign matrix $\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$ not satisfying the conditions is called feasible. \nThen, for any $\\sigma\\in \\mathbb{R}^{2\\times \\ell}$ and feasible $\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$, we introduce a set  $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which we name the feasible ground set, and for which  deciding whether it is nonempty is equivalent to checking the feasibility of a linear system of equalities and inequalities. \n\n\\begin{thmb}[\\cref{thm:mono}]   \n If $\\mathcal{C}_{\\mathcal{S}}^\\sigma = \\varnothing$ for all orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s\\times \\ell}$, then $(C(\\kappa \\circ x^M), Lx - b)$ does not admit multiple positive zeros. \n\\end{thmb}\n\nThe converse of  Theorem B holds when   $P$ has a particular structure of zero entries. Namely, we consider the bipartite graph  whose nodes are the sets of rows and columns of $P$, and edges correspond to nonzero entries of $P$. If  the graph is a forest, we say that $P$ \\bfemph{induces a forest} (see \\cref{forest}). \n\n\\begin{thmc}[\\cref{thm:full}]\nAssume that  $P\\in \\mathbb{R}^{s\\times \\ell}$ induces a forest. \n Then the system $( C (\\kappa \\circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and  feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$  such that \n$ \\mathcal{C}_{\\mathcal{S}}^\\sigma \\neq \\varnothing$.\n\\end{thmc}\n\nWe conclude this introduction by briefly highlighting two applications where augmented vertically parametrized systems arise. As we have already hinted at, they arise when studying \n \\bfemph{reaction networks}, which are given by a collection of \n reactions \n\\[ \\sum_{i = 1}^n\\alpha_{ij}X_i \\ce{->[\\k_j]} \\sum_{i = 1}^n\\beta_{ij}X_i, \\qquad j=1,\\dots,m \\]\n where $\\alpha_{ij},\\beta_{ij}\\in \\Z_{\\geq 0}$ and $\\k_j\\in \\R_{>0}$ is called the reaction rate constant.  \nThe net production of the species along each reaction is encoded by the  stoichiometric matrix $N=(\\beta_{ij} - \\alpha_{ij})_ \\in\\mathbb{Z}^{n\\times m}$ and the reactants define the matrix $M  = (\\alpha_{ij}) \\in \\mathbb{Z}^{n \\times m}$.\nUnder the assumption  of mass-action kinetics, the evolution of the concentration of each species in time is  modelled by a system of ordinary differential equations of the form \n\\[ \\frac{d  x }{dt} =  N \\big( \\kappa \\circ x^M \\big)\\, .\\]\nMoreover, the trajectories are constrained in the stoichiometric compatibility classes, whose defining equations are of the form $Lx-b=0$ where $Lx=0$ are equations of $\\im(N)$ and $b$ depends on the initial condition of the system. \nBy letting $C$ be a full rank matrix with $\\ker(N)=\\ker(C)$, the positive steady states  within stoichiometric compatibility classes are the positive zeros of the\naugmented vertically parametrized  system $(C(\\kappa\\circ x^M),  Lx-b)$. \n\nAdmitting more than one positive steady state is  necessary  for the network to display multistability, which has been linked to several properties of biological systems \\cite{laurentkellershohn, ozbudak}. Thus, deciding upon the existence of multiple positive steady states for some parameter choice   has been an active topic of research in the field. To name a few of the many references, see for example \\cite{BanajiPantea2016injectivity,craciun-feinbergI,Conradi_2019, dickenstein_giaroli_perez,feinberg-def0,Feliu_2015, signconditions, wiuf2013power} and the references therein.  \n\nThe second application we would like to highlight arises when studying \\bfemph{singular points of hypersurfaces}. Namely, let $f \\in \\mathbb{R}[x]$ be an $n$-variate polynomial with a prescribed support  $\\mathcal{A} \\subset \\mathbb{Z}^n$   and signs $\\varepsilon\\colon \\mathcal{A} \\mapsto \\{-,+\\}$: \n\\[ f = \\sum_{a \\in \\mathcal{A}}\\varepsilon(a)\\kappa_{a}x^{a} \\, , \\qquad \\kappa \\in \\mathbb{R}_{>0}^{\\mathcal{A}} \\, .\\] \nThe singular positive zeros of $f$\nare the points of $\\mathbb{R}^n_{>0}$ satisfying $f = x_1\\frac{d}{dx_1} f = \\dots = x_n\\frac{d}{dx_n} f = 0$. By letting $A$ have as columns the elements of $\\mathcal{A}$ in some order, and $\\overline{A}$ be obtained by adding a top row of $1's$ to $A$ and multiplying the $i$-th column by $\\epsilon(a_i)$, \nthe singular positive zeros of $f$ are precisely the positive zeros of the  vertically parametrized system \n\\[ \\overline{A}( \\kappa \\circ x^A) \\, .  \\]\nThe existence of signed coefficients of $f$ giving rise to a polynomial with multiple singular positive  points depends on the combinatorial properties of the signed support and can be studied with the methods in this work. This question becomes relevant when studying $\\mathcal{A}$-discriminants \\cite{gkz1994}, the number of connected components of fewnomial hypersurfaces, e.g. \\cite{bihan2024bounds, feliutelek, telekdescartesrule}, or copositivity of polynomials and SONC decompositions \\cite{Ferrer:SONC}.\n\n\\subsection*{Structure of the Paper}\nIn \\Cref{section:characterization}, we present the characterization of multiple positive zeros for augmented vertically parametrized polynomial systems via the characteristic system. \\Cref{section:simplifications} introduces orientations, showing that the question of multiple zeros can be   reduced to the feasibility of the oriented characteristic system. In \\Cref{section:feasibility}, we analyze the feasibility of oriented characteristic systems. First, we establish sufficient conditions for the existence of multiple positive zeros by reducing the feasibility of the characteristic system to checking whether the relative interiors over a family of polyhedral cones are empty. Secondly, we assume that the simplified reduced matrix $P$ induces a forest and characterize the existence of multiple positive zeros using the same family of polyhedral cones. \n\n\\subsection*{Acknowledgments}\nThis project has been funded by the European Union under the Grant Agreement number 101044561, POSALG. Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or European Research Council (ERC). We thank Joan Ferrer, Oskar Henriksson and Nidhi Kaihnsa for useful discussions.\n\n\\medskip\n\\noindent\n\\textbf{Notation. } \nFor two vectors $\\alpha,\\beta \\in \\mathbb{R}^{m}$,  $\\alpha \\circ \\beta$  denotes its component-wise multiplication, i.e.\n$(\\alpha \\circ \\beta)_{i} = \\alpha_i\\beta_i$ for $i = 1,\\dots,m.$\nSimilarly, the operations $\\frac{\\alpha}{\\beta}, e^{\\alpha}, \\ln{\\alpha}$ are taken component-wise.\n\nFor a  vector $v\\in \\mathbb{R}^n$, we let $\\sign(v)\\in \\{-1,0,1\\}^n$ be obtained by taking the sign entry-wise. \nFor a set $V\\subseteq \\mathbb{R}^n$, we let $\\sign(V)=\\{\\sign(v) \\colon v\\in V\\}$. For a vector $v\\in \\mathbb{R}^n$,   $v>0$ is shorthand notation for   $v\\in \\mathbb{R}^n_{>0}$. \n\nFor an integer $n$ we let $[n]:=\\{1,\\dots,n\\}$. We  write $b \\in [a,c]^\\circ$ if $b$ is in the relative interior of the interval $[a,b]$, that is if either\n\\[ a < b < c\\quad  \\text{  or  } \\quad a = b = c\\, .\\]",
    "sketch": "To analyze Theorem~\\ref{thmfirst:intro}, the introduction explains that the feasibility of the oriented characteristic system “depends on the signs of \\( A^{\\sigma}_{\\rho} \\)”. Using that “the strictly increasing nature of the exponential function” fixes these signs, it states that each entry’s sign is either “fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\)”, and hence “can be expressed as a linear inequality in the entries of $\\rho$.”\n\nFormally, it sketches the following steps:\n\\begin{itemize}\n\\item “we first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions.”\n\\item Any sign matrix $\\mathcal{S}\\in\\{-1,0,1\\}^{s\\times\\ell}$ not excluded by these conditions is called “feasible”.\n\\item For any orientation $\\sigma\\in\\mathbb{R}^{2\\times\\ell}$ and feasible $\\mathcal{S}$, one “introduce[s] a set $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which [is] name[d] the feasible ground set,” such that deciding nonemptiness is “equivalent to checking the feasibility of a linear system of equalities and inequalities.”\n\\end{itemize}\n(Additionally, the text notes a sufficient nonexistence criterion via Theorem B and, under the extra assumption that $P$ “induces a forest,” an equivalence in terms of $\\mathcal{C}_{\\mathcal{S}}^\\sigma\\neq\\varnothing$ in Theorem C.)",
    "expanded_sketch": "To analyze Theorem~\\ref{thmfirst:intro}, the introduction explains that the feasibility of the oriented characteristic system “depends on the signs of \\( A^{\\sigma}_{\\rho} \\)”. Using that “the strictly increasing nature of the exponential function” fixes these signs, it states that each entry’s sign is either “fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\)”, and hence “can be expressed as a linear inequality in the entries of $\\rho$.”\n\nFormally, it sketches the following steps:\n\\begin{itemize}\n\\item “we first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions.”\n\\item Any sign matrix $\\mathcal{S}\\in\\{-1,0,1\\}^{s\\times\\ell}$ not excluded by these conditions is called “feasible”.\n\\item For any orientation $\\sigma\\in\\mathbb{R}^{2\\times\\ell}$ and feasible $\\mathcal{S}$, one “introduce[s] a set $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which [is] name[d] the feasible ground set,” such that deciding nonemptiness is “equivalent to checking the feasibility of a linear system of equalities and inequalities.”\n\\end{itemize}\n(Additionally, the text notes a sufficient nonexistence criterion via Theorem B and, under the extra assumption that $P$ “induces a forest,” an equivalence in terms of $\\mathcal{C}_{\\mathcal{S}}^\\sigma\\neq\\varnothing$ in Theorem C.)",
    "expanded_theorem": "[\\cref{thm:main1,thm:mainsimplification}]\n\\label{thmfirst:intro}\nThe augmented vertically parametrized system $(C(\\kappa \\circ x^M), Lx - b)$ admits multiple positive zeros if and only if  there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ \nhas a solution.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Consider an augmented vertically parametrized polynomial system\n\\[\nF(x;\\kappa,b)=\\big(C(\\kappa\\circ x^M),\\,Lx-b\\big),\n\\]\nwhere $C\\in\\mathbb{R}^{\\bar s\\times \\bar m}$, $M\\in\\mathbb{Z}^{n\\times \\bar m}$, $L$ has $n-\\bar s$ rows, $\\kappa\\in\\mathbb{R}_{>0}^{\\bar m}$, $b\\in\\mathbb{R}^{n-\\bar s}$, and $x\\in\\mathbb{R}_{>0}^n$. Say that $F$ admits multiple positive zeros if there exist parameter values $\\kappa\\in\\mathbb{R}_{>0}^{\\bar m}$ and $b\\in\\mathbb{R}^{n-\\bar s}$ and distinct points $x,y\\in\\mathbb{R}_{>0}^n$ such that $F(x;\\kappa,b)=F(y;\\kappa,b)=0$, equivalently,\n\\[\nC(\\kappa\\circ x^M)=C(\\kappa\\circ y^M)=0,\n\\qquad Lx=Ly=b.\n\\]\nLet $P\\in\\mathbb{R}^{s\\times \\ell}$ be the reduced matrix associated with the system, let $m=s+\\ell$, and for an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and $\\rho\\in\\mathbb{R}^m$ define\n\\[\n(P^{\\sigma})_{ik}=\\sigma_{1k}P_{ik},\n\\qquad\n(A^{\\sigma}_{\\rho})_{ik}=P_{ik}\\big(\\sigma_{1k}e^{\\rho_i}-\\sigma_{2k}e^{\\rho_{s+k}}\\big)\n\\quad (i\\in[s],\\ k\\in[\\ell]).\n\\]\nThe oriented characteristic system associated with $(P,\\sigma,\\rho)$ is\n\\[\nA^{\\sigma}_{\\rho}\\mu=0,\\qquad P^{\\sigma}\\mu>0,\\qquad \\mu\\in\\mathbb{R}_{>0}^{\\ell},\n\\]\nand $\\mathcal{G}^{\\sigma}\\subseteq\\mathbb{R}^m$ is the oriented ground set associated with $\\sigma$. Which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if for every orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ there exists a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        },
        {
          "label": "C",
          "text": "If there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$, then the augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros."
        },
        {
          "label": "D",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathbb{R}^{m}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        },
        {
          "label": "E",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the sign pattern of $A^{\\sigma}_{\\rho}$ is feasible and $P^{\\sigma}\\mu>0$ for some $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "existential choice of orientation",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the converse direction of the iff statement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "membership condition $\\rho\\in\\mathcal{G}^{\\sigma}$",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "replaced solvability of $A^{\\sigma}_{\\rho}\\mu=0$ by mere feasible sign-pattern data",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal choice A or state the exact equivalent condition outright. It sets up definitions and asks for the equivalent statement, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct option is a near-verbatim statement of the equivalence characterizing multiple positive zeros. It mainly asks the student to recognize the exact formal statement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to track subtle changes in quantifiers, positivity, and ground-set membership across the options, but the task is still primarily recognition of the canonical theorem statement rather than genuine derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they vary by plausible mathematical failure modes such as dropping a constraint, weakening strict inequalities, removing ground-set membership, or replacing existence by universality. They are distinct and nontrivial."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-designed recall question with high-quality distractors, but it is strongly theorem-statement recognition and therefore weak on tautology avoidance and only moderate on generative reasoning."
    }
  },
  {
    "id": "2512.07560v2",
    "paper_link": "http://arxiv.org/abs/2512.07560v2",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thma",
      "content": "[\\cref{thm:main1,thm:mainsimplification}]\n\\label{thmfirst:intro}\nThe augmented vertically parametrized system $(C(\\kappa \\circ x^M), Lx - b)$ admits multiple positive zeros if and only if  there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ \nhas a solution.",
      "start_pos": 11415,
      "end_pos": 11826,
      "label": "thmfirst:intro"
    },
    "ref_dict": {
      "thm:mono": "\\begin{theorem}[Preclusion of multiple positive zeros]\\label{thm:mono}\n Let $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal and $L$ of full row rank.  Let  $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be  row and column partitions, and consider the  associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix}. \n\n If $\\mathcal{C}_{\\mathcal{S}}^\\sigma = \\varnothing$ for all orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$   compatible with $\\alpha$ and  all  $\\mathcal{S}\\in \\{-1,0,1\\}^{s\\times \\ell}$\n that are feasible with respect to $(P,\\sigma)$, then $F$ does not admit multiple positive zeros for any choice of parameter values.\n\\end{theorem}",
      "thm:full": "\\begin{theorem}[Characterization of multiple positive zeros when $P$ induces a forest]\\label{thm:full}\nLet $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal and $L$ of full row rank. \nLet  $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be  row and column partitions, and assume that the associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix} induces a forest. \n\n Then, $F$ admits multiple positive zeros for some  parameter values $\\k\\in \\R^{\\bar{m}}_{>0}$, $b\\in \\R^{n-\\bs}$  if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$ and a sign matrix  $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ that is feasible with respect to $(P,\\sigma)$  such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{theorem}",
      "def:simpif_reduced_matrix": "\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}"
    },
    "pre_theorem_intro_text_len": 4712,
    "pre_theorem_intro_text": "Polynomial systems that arise in applications  have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the  form\n\\begin{equation*}\n    C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system,  and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial  is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n    \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2  , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n    1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n    1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n}  \\, ,\n\\]\nwhere  $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters. \n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad  Lx = Ly = 0\\, .\\]\n\n Our approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a  family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that,  in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, ,  \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional  rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any   $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and    $A^\\sigma_{\\rho}$  defined by \n\\[ \n     (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad  (A^\\sigma_{\\rho})_{ik} =  P_{ik} \n     (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}),  \\qquad i \\in \\{1,\\dots,s\\}, \\  k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n  With these objects in place,  the \\bfemph{oriented characteristic system}  associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set   $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.",
    "context": "Polynomial systems that arise in applications have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the form\n\\begin{equation*}\n C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system, and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2 , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n 1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n 1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n} \\, ,\n\\]\nwhere $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters.\n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad Lx = Ly = 0\\, .\\]\n\nOur approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that, in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, , \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and $A^\\sigma_{\\rho}$ defined by \n\\[ \n (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad (A^\\sigma_{\\rho})_{ik} = P_{ik} \n (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}), \\qquad i \\in \\{1,\\dots,s\\}, \\ k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n With these objects in place, the \\bfemph{oriented characteristic system} associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.\n\n\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}",
    "full_context": "Polynomial systems that arise in applications have often fixed support and their coefficients are functions of some parameters. \\bfemph{Vertically parametrized} polynomial systems \\cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \\cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \\cite{gkz1994} or geometric modelling \\cite{craciunsottile}. These are systems defined by a set of polynomials of the form\n\\begin{equation*}\n C \\big( \\mathbf{\\kappa} \\circ x^M \\big) \\ \\in \\mathbb{R}[\\kappa,x^{\\pm}]^{\\bar{s}}\n\\end{equation*}\nwith parameters $\\kappa=(\\kappa_1,\\dots,\\kappa_{\\bar{m}})$ and variables $x=(x_1,\\dots,x_n)$. Here $C \\in \\mathbb{R}^{\\bar{s} \\times \\bar{m}}$ is a full rank matrix, the columns of $M \\in \\mathbb{Z}^{n \\times \\bar{m}}$ give the exponents of the monomials of the system, and $\\kappa \\circ x^M$ indicates that \nthe $i$-th monomial is scaled by $\\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.\nFor instance, \n\\begin{align*}\n \\big( \\k_1x_1- 2 \\k_2x_1x_2^2 + \\k_3 x_2^2, \\k_2x_1x_2^2 - (\\k_3+\\k_5)x_2^2 , -\\k_4 x_1 + 3\\k_5 x_2^2\\big)\n\\end{align*}\nis a vertically parametrized system with \n\\[ C=\\begin{pmatrix}\n 1 & -2 & 1 & 0 & 0 \\\\ 0 & 1 & -1 & 0& -1 \\\\ 0 & 0 & 0& -1 & 3\n\\end{pmatrix}\\quad \\text{ and } \\quad \nM=\\begin{pmatrix}\n 1 & 1 & 0 & 1 & 0 \\\\ 0 & 2 & 2 & 0 & 2\n\\end{pmatrix}\\, . \\]\nSparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.\n\nMotivated by their origin in chemical reaction network theory, we include \nadditional $n - \\bar{s}$ linear entries and consider \\bfemph{augmented vertically parametrized} polynomial systems,\n\\[ \n\\big( C\\big( \\kappa \\circ x^M\\big) , Lx - b \\big) \\in \\mathbb{R}[\\kappa,b,x^{\\pm}]^{n} \\, ,\n\\]\nwhere $b=(b_1,\\dots,b_{n-\\bar{s}})$ are parameters.\n\nThe goal of this work is to decide whether an augmented vertically parametrized system \n\\bfemph{admits multiple positive zeros}, that is, whether there \nexists a choice of parameter values $\\kappa\\in \\mathbb{R}^{\\bar{m}}_{>0} $, $b\\in \\mathbb{R}^{n-\\bar{s}}$ and distinct $x,y \\in \\mathbb{R}_{>0}^n$ such that \n\\[ C(\\kappa \\circ x^M) = C(\\kappa \\circ y^M) =0 \\qquad Lx = Ly = 0\\, .\\]\n\nOur approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \\cite{Ji2011UniquenessOE}, but also \\cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \\cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.\nThe main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\\\"obner basis computations.\n\nTo understand the main ideas behind this system of linear equalities and inequalities, we assume that, in row reduced echelon form, $C$ is of the form \n\\[ \nC = \\begin{pmatrix} \\id_{\\bar{s}} & -\\bar{P} \n\\end{pmatrix}\\, , \n\\] \nand let \n$P \\in \\mathbb{R}^{s \\times \\ell}$ be obtained from $\\bar{P}$ by selecting one representative \nfor each set of pairwise proportional rows and columns (see \\eqref{def:simpif_reduced_matrix}).\nWe consider orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ after imposing certain conditions depending on $\\bar{P}$, see \\cref{section:simplifications}, and for any $\\rho \\in \\mathbb{R}^{m}$, where $m = s + \\ell$, we consider matrices $P^{\\sigma}$ and $A^\\sigma_{\\rho}$ defined by \n\\[ \n (P^{\\sigma})_{ik} = \\sigma_{1k}P_{ik}\\, , \\qquad (A^\\sigma_{\\rho})_{ik} = P_{ik} \n (\\sigma_{1k}\\,e^{\\rho_i} - \\sigma_{2k}\\,e^{\\rho_{s+ k}}), \\qquad i \\in \\{1,\\dots,s\\}, \\ k \\in \\{1,\\dots,\\ell\\}\\, .\n\\]\n With these objects in place, the \\bfemph{oriented characteristic system} associated with $(P,\\sigma,\\rho)$ is the linear system\n\\[\nA_{\\rho}^{\\sigma}\\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\R_{>0}^{\\ell}\\, .\n\\]\nFinally, we consider the oriented ground set $\\mathcal{G}^{\\sigma}\\subseteq \\mathbb{R}^m$ defined in \\cref{section:simplifications}.\n\n\\begin{equation}\n\\label{def:simpif_reduced_matrix}\nP_{ik} :=  \\oP_{r(i), c(k)} \\,, \\qquad i\\in [s], \\ k\\in [\\ell] \\, . \\end{equation}\n\n\\begin{thmc}[\\cref{thm:full}]\nAssume that $P\\in \\R^{s\\times \\ell}$ induces a forest. \n Then the system $( C (\\k \\circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{thmc}\n\n\\begin{theorem}\n\\label{thm:mainsimplification}\n Let $F=( C (\\k \\circ x^M), Lx-b) \\in \\R[\\k,b,x^\\pm]^n$ be an augmented vertical system with $C\\in \\R^{\\bs\\times \\bar{m}}$ principal with reduced matrix $\\oP\\in \\R^{\\bs\\times \\bl}$, and $L\\in \\R^{(n-\\bs)\\times n}$ has full rank. \n Let $(\\tau,r,\\gamma')$ and $(\\alpha,c,\\gamma)$ be row and column partitions, and consider the associated simplified reduced matrix $P\\in \\R^{s\\times \\ell}$ from \\eqref{def:simpif_reduced_matrix}. \nThe following statements are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n \\item There exists $\\bar{\\rho} \\in M^\\top(\\mO_L)$ such that the characteristic system\n\\[ \nA_{\\bar{\\rho}}\\, \\bar{\\mu} = 0\\,, \\qquad \\oP\\, \\bar{\\mu} > 0\\,, \\qquad \\bar{\\mu}\\in \\R^{\\bl}_{>0}\n\\]\nhas a solution. \n \\item There exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$, and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system\n\\[ A^\\sigma_{\\rho} \\mu = 0\\,, \\qquad P^{\\sigma}\\mu > 0\\,, \\qquad \\mu \\in \\mathbb{R}^{\\ell}_{>0}\\]\nhas a solution. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{lemma}\\label[lemma]{mainlemma}\nLet $P\\in \\R^{s\\times \\ell}$, $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell},\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$ such that \n$\\mS$ is feasible with respect to $(P,\\sigma)$ and consider their associated $\\Lambda$-sets. \nAssume $P^\\sigma\\mu^*>0$ for some $\\mu^*\\in \\R^\\ell_{>0}$ and let $\\rho \\in \\R^m\\setminus \\mathcal{D}_{\\mS}^{\\sigma} $ such that $\\sign(A_{\\rho}^{\\sigma}) = \\mathcal{S}$. Then, there exists\n $Q \\in \\mathbb{R}^{s \\times \\ell}$ with $\\sign(Q) = \\mathcal{S}$ such that the system \n\\[ Q \\mu = 0,\\qquad P^\\sigma \\mu>0, \\qquad \\mu\\in \\R^\\ell_{>0} \\]\nhas a solution and, additionally, for all $i \\in [s]$, it holds\n \\begin{equation}\\label{eq2MainLemma} \n \\sum_{j \\in \\Lambda_i^{\\neq}}\\frac{Q_{ij} \\, \\mu_j}{e^{\\rho_i} - \\frac{\\sigma_{2j}}{\\sigma_{1j}}\\, e^{\\rho_{s + j}}} + \\sum_{j \\notin \\Lambda_i}P^{\\sigma}_{ij} \\mu_j > 0 \\, .\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nAs $\\rho \\notin \\mathcal{D}_{\\mS}^\\sigma$, if $I_{\\rho}^+ \\cup I_{\\rho}^{-}\\neq \\varnothing$, then there exists $\\mu \\in \\bigcap_{i \\in I_{\\rho}^+ \\cup I_{\\rho}^{-}}\\Gamma_{\\Lambda,i} \\neq \\varnothing$. If $I_{\\rho}^+ \\cup I_{\\rho}^{-}= \\varnothing$, then we let $\\mu=\\mu^*$. \nIn both cases $P^\\sigma \\mu>0$, and hence, by \\cref{prop:feasible}, there exists $\\widetilde{Q}\\in \\mathbb{R}^{s \\times \\ell} $ with $\\sign(\\wQ) = \\mathcal{S}$ such that $\\wQ \\mu = 0$. \nWe construct now a parametric matrix $Q(\\omega)$ for $\\omega = (\\omega_1,\\dots,\\omega_s) \\in \\mathbb{R}_{>0}^s$ such that, for every $i\\in [s]$, $\\omega_i$ only modifies the $i$-th row and it holds that \n\\begin{equation}\\label{eq:row_mod}\n\\sign(\\wQ_{ij})=\\sign(Q(\\omega)_{ij}) \\quad\\text{for all } j\\in [\\ell]\\, , \\qquad (Q(\\omega)\\mu)_i=0 \\, , \n\\end{equation} \nand \\eqref{eq2MainLemma} holds after choosing $\\omega_i$ either small or large enough.\n\nAssume that $P$ induces a forest. Let \n $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ be an orientation compatible with $\\alpha$, and $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ be feasible with respect to $(P,\\sigma)$. Then, for any $\\rho \\in\\mC_{\\mS}^{\\sigma}$, the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution.\n \\end{theorem}\n\\begin{proof} \nAs $\\rho \\in \\mathcal{C}_{\\mathcal{S}}^\\sigma$, we have in particular that $\\sign(A_{\\rho}^\\sigma) = \\mS$ and $\\rho \\notin \\mathcal{D}_{\\mS}^\\sigma$. Hence by \\cref{mainlemma}, there exist $\\bar{\\mu}\\in \\R^\\ell_{>0}$ and $Q \\in \\mathbb{R}^{s \\times \\ell}$ with $\\sign(Q) = \\mathcal{S}$ such that $Q \\bar{\\mu} = 0$, $P^\\sigma\\bar{\\mu} > 0$, and \n\\[\n \\sum_{j \\in \\Lambda_i^{\\neq}}\\frac{Q_{ij} \\, \\bar{\\mu}_j}{e^{\\rho_i} - \\frac{\\sigma_{2j}}{\\sigma_{1j}}\\,e^{\\rho_{s + j}}} + \\sum_{j \\notin \\Lambda_i}P^{\\sigma}_{ij} \\bar{\\mu}_j > 0 \n\\]\n for all $i \\in [s]$. We will now modify $\\bar{\\mu}$ to obtain $\\mu\\in \\R^\\ell_{>0}$ satisfying $A_\\rho^\\sigma \\mu=0$ and $P^\\sigma\\mu>0$.\n\nThen, $F$ admits multiple positive zeros for some parameter values $\\k\\in \\R^{\\bar{m}}_{>0}$, $b\\in \\R^{n-\\bs}$ if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ compatible with $\\alpha$ and a sign matrix $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$ that is feasible with respect to $(P,\\sigma)$ such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$.\n\\end{theorem}\n\\begin{proof}\nIf $F$ admits multiple positive zeros, \\Cref{thm:mono} tells us that there exist $\\sigma$ and $\\mathcal{S}$ as in the statement such that \n$ \\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$. Reciprocally, if $\\mathcal{C}_{\\mS}^\\sigma \\neq \\varnothing$ for some $\\sigma$ and $\\mS$, then \\Cref{thm:multiSimplified} tells us that the oriented characteristic system at any $\\rho\\in \\mathcal{C}_{\\mS}^\\sigma$ has a solution. \nBy \\Cref{thm:mainsimplification}, this implies that \\Cref{thm:main1}(iii) holds and hence $F$ admits multiple positive zeros.\n\\end{proof}\n\n\\begin{example}\n\\label[example]{example:hybrid_histidine_kinase}\nWe illustrate \\Cref{thm:full} with an augmented vertical system that arises from the study of the steady states of a reaction network named \n\\emph{hybrid histidine kinase}. This network is known to admit multiple positive zeros \\cite{hhk, plos:identifying}. The stoichiometric matrix, the matrix of reactants, and a matrix defining the stoichiometric compatibility classes are:\n{\\small \\[ N = \\begin{pmatrix}\n -1 & 0 & 0 & 1 & 0 & 0 \\\\\n 1 & -1 & 0 & 0 & 1 & 0 \\\\\n 0 & 1 & -1 & -1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & -1 & 0 \\\\\n 0 & 0 & 0 & -1 & -1 & 1 \\\\\n 0 & 0 & 0 & 1 & 1 & -1 \\\\\n\\end{pmatrix}, \\quad M = \\begin{pmatrix} \n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1\\end{pmatrix}, \\quad L = \\begin{pmatrix} \n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1\\end{pmatrix}.\\] }After performing Gaussian elimination to $N$, the steady states of the network are the zeros of the augmented vertical system \n$F = (C(\\kappa \\circ x^M), Lx - b) \\in \\R[\\k,b,x]^6$\nwith\n \\[ C = \\begin{pmatrix} \n1 & 0 & 0 & 0 & 1 & -1 \\\\\n0 & 1 & 0 & 0 & 0 & -1 \\\\\n0 & 0 & 1 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & -1 \\end{pmatrix}\\, .\\]\nThe last two columns of $C$ give rise to the reduced matrix $\\oP \\in \\R^{4 \\times 2}$ from \\eqref{eqP}. In this example we have $\\bs=4, \\bar{m}=6$. We consider the row and column partitions to be $[\\bs] = \\{1,4\\} \\sqcup \\{2\\} \\sqcup \\{3\\}$ and $[\\bl] = \\{1\\} \\sqcup \\{2\\}$, so $s = 3,m=5$. We choose the orientation $\\sigma_+$. This gives rise to the following simplified reduced matrix $P$, oriented characteristic matrix $A^{\\sigma_+}_\\rho$ for $\\rho\\in \\R^5$, and associated graph $G_{P}$:",
    "post_theorem_intro_text_len": 7677,
    "post_theorem_intro_text": "It turns out that  the feasibility of the oriented characteristic system depends  on the signs of \\( A^{\\sigma}_{\\rho} \\). Then, an important observation is that  the strictly increasing nature of the exponential function, implies that the sign of each entry of   \\( A^{\\sigma}_{\\rho} \\) is either fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\), and can be expressed as a linear inequality in the entries of $\\rho$. \nFormally, \nwe first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions. Any sign matrix $\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$ not satisfying the conditions is called feasible. \nThen, for any $\\sigma\\in \\mathbb{R}^{2\\times \\ell}$ and feasible $\\mathcal{S} \\in \\{-1,0,1\\}^{s \\times \\ell}$, we introduce a set  $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which we name the feasible ground set, and for which  deciding whether it is nonempty is equivalent to checking the feasibility of a linear system of equalities and inequalities. \n\n\\begin{thmb}[\\cref{thm:mono}]   \n If $\\mathcal{C}_{\\mathcal{S}}^\\sigma = \\varnothing$ for all orientations $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s\\times \\ell}$, then $(C(\\kappa \\circ x^M), Lx - b)$ does not admit multiple positive zeros. \n\\end{thmb}\n\nThe converse of  Theorem B holds when   $P$ has a particular structure of zero entries. Namely, we consider the bipartite graph  whose nodes are the sets of rows and columns of $P$, and edges correspond to nonzero entries of $P$. If  the graph is a forest, we say that $P$ \\bfemph{induces a forest} (see \\cref{forest}). \n\n\\begin{thmc}[\\cref{thm:full}]\nAssume that  $P\\in \\mathbb{R}^{s\\times \\ell}$ induces a forest. \n Then the system $( C (\\kappa \\circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and  feasible $\\mathcal{S}\\in \\{-1,0,1\\}^{s \\times \\ell}$  such that \n$ \\mathcal{C}_{\\mathcal{S}}^\\sigma \\neq \\varnothing$.\n\\end{thmc}\n\nWe conclude this introduction by briefly highlighting two applications where augmented vertically parametrized systems arise. As we have already hinted at, they arise when studying \n \\bfemph{reaction networks}, which are given by a collection of \n reactions \n\\[ \\sum_{i = 1}^n\\alpha_{ij}X_i \\ce{->[\\k_j]} \\sum_{i = 1}^n\\beta_{ij}X_i, \\qquad j=1,\\dots,m \\]\n where $\\alpha_{ij},\\beta_{ij}\\in \\Z_{\\geq 0}$ and $\\k_j\\in \\R_{>0}$ is called the reaction rate constant.  \nThe net production of the species along each reaction is encoded by the  stoichiometric matrix $N=(\\beta_{ij} - \\alpha_{ij})_ \\in\\mathbb{Z}^{n\\times m}$ and the reactants define the matrix $M  = (\\alpha_{ij}) \\in \\mathbb{Z}^{n \\times m}$.\nUnder the assumption  of mass-action kinetics, the evolution of the concentration of each species in time is  modelled by a system of ordinary differential equations of the form \n\\[ \\frac{d  x }{dt} =  N \\big( \\kappa \\circ x^M \\big)\\, .\\]\nMoreover, the trajectories are constrained in the stoichiometric compatibility classes, whose defining equations are of the form $Lx-b=0$ where $Lx=0$ are equations of $\\im(N)$ and $b$ depends on the initial condition of the system. \nBy letting $C$ be a full rank matrix with $\\ker(N)=\\ker(C)$, the positive steady states  within stoichiometric compatibility classes are the positive zeros of the\naugmented vertically parametrized  system $(C(\\kappa\\circ x^M),  Lx-b)$. \n\nAdmitting more than one positive steady state is  necessary  for the network to display multistability, which has been linked to several properties of biological systems \\cite{laurentkellershohn, ozbudak}. Thus, deciding upon the existence of multiple positive steady states for some parameter choice   has been an active topic of research in the field. To name a few of the many references, see for example \\cite{BanajiPantea2016injectivity,craciun-feinbergI,Conradi_2019, dickenstein_giaroli_perez,feinberg-def0,Feliu_2015, signconditions, wiuf2013power} and the references therein.  \n\nThe second application we would like to highlight arises when studying \\bfemph{singular points of hypersurfaces}. Namely, let $f \\in \\mathbb{R}[x]$ be an $n$-variate polynomial with a prescribed support  $\\mathcal{A} \\subset \\mathbb{Z}^n$   and signs $\\varepsilon\\colon \\mathcal{A} \\mapsto \\{-,+\\}$: \n\\[ f = \\sum_{a \\in \\mathcal{A}}\\varepsilon(a)\\kappa_{a}x^{a} \\, , \\qquad \\kappa \\in \\mathbb{R}_{>0}^{\\mathcal{A}} \\, .\\] \nThe singular positive zeros of $f$\nare the points of $\\mathbb{R}^n_{>0}$ satisfying $f = x_1\\frac{d}{dx_1} f = \\dots = x_n\\frac{d}{dx_n} f = 0$. By letting $A$ have as columns the elements of $\\mathcal{A}$ in some order, and $\\overline{A}$ be obtained by adding a top row of $1's$ to $A$ and multiplying the $i$-th column by $\\epsilon(a_i)$, \nthe singular positive zeros of $f$ are precisely the positive zeros of the  vertically parametrized system \n\\[ \\overline{A}( \\kappa \\circ x^A) \\, .  \\]\nThe existence of signed coefficients of $f$ giving rise to a polynomial with multiple singular positive  points depends on the combinatorial properties of the signed support and can be studied with the methods in this work. This question becomes relevant when studying $\\mathcal{A}$-discriminants \\cite{gkz1994}, the number of connected components of fewnomial hypersurfaces, e.g. \\cite{bihan2024bounds, feliutelek, telekdescartesrule}, or copositivity of polynomials and SONC decompositions \\cite{Ferrer:SONC}.\n\n\\subsection*{Structure of the Paper}\nIn \\Cref{section:characterization}, we present the characterization of multiple positive zeros for augmented vertically parametrized polynomial systems via the characteristic system. \\Cref{section:simplifications} introduces orientations, showing that the question of multiple zeros can be   reduced to the feasibility of the oriented characteristic system. In \\Cref{section:feasibility}, we analyze the feasibility of oriented characteristic systems. First, we establish sufficient conditions for the existence of multiple positive zeros by reducing the feasibility of the characteristic system to checking whether the relative interiors over a family of polyhedral cones are empty. Secondly, we assume that the simplified reduced matrix $P$ induces a forest and characterize the existence of multiple positive zeros using the same family of polyhedral cones. \n\n\\subsection*{Acknowledgments}\nThis project has been funded by the European Union under the Grant Agreement number 101044561, POSALG. Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or European Research Council (ERC). We thank Joan Ferrer, Oskar Henriksson and Nidhi Kaihnsa for useful discussions.\n\n\\medskip\n\\noindent\n\\textbf{Notation. } \nFor two vectors $\\alpha,\\beta \\in \\mathbb{R}^{m}$,  $\\alpha \\circ \\beta$  denotes its component-wise multiplication, i.e.\n$(\\alpha \\circ \\beta)_{i} = \\alpha_i\\beta_i$ for $i = 1,\\dots,m.$\nSimilarly, the operations $\\frac{\\alpha}{\\beta}, e^{\\alpha}, \\ln{\\alpha}$ are taken component-wise.\n\nFor a  vector $v\\in \\mathbb{R}^n$, we let $\\sign(v)\\in \\{-1,0,1\\}^n$ be obtained by taking the sign entry-wise. \nFor a set $V\\subseteq \\mathbb{R}^n$, we let $\\sign(V)=\\{\\sign(v) \\colon v\\in V\\}$. For a vector $v\\in \\mathbb{R}^n$,   $v>0$ is shorthand notation for   $v\\in \\mathbb{R}^n_{>0}$. \n\nFor an integer $n$ we let $[n]:=\\{1,\\dots,n\\}$. We  write $b \\in [a,c]^\\circ$ if $b$ is in the relative interior of the interval $[a,b]$, that is if either\n\\[ a < b < c\\quad  \\text{  or  } \\quad a = b = c\\, .\\]",
    "sketch": "To analyze Theorem~\\ref{thmfirst:intro}, the introduction explains that the feasibility of the oriented characteristic system “depends on the signs of \\( A^{\\sigma}_{\\rho} \\)”. Using that “the strictly increasing nature of the exponential function” fixes these signs, it states that each entry’s sign is either “fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\)”, and hence “can be expressed as a linear inequality in the entries of $\\rho$.”\n\nFormally, it sketches the following steps:\n\\begin{itemize}\n\\item “we first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions.”\n\\item Any sign matrix $\\mathcal{S}\\in\\{-1,0,1\\}^{s\\times\\ell}$ not excluded by these conditions is called “feasible”.\n\\item For any orientation $\\sigma\\in\\mathbb{R}^{2\\times\\ell}$ and feasible $\\mathcal{S}$, one “introduce[s] a set $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which [is] name[d] the feasible ground set,” such that deciding nonemptiness is “equivalent to checking the feasibility of a linear system of equalities and inequalities.”\n\\end{itemize}\n(Additionally, the text notes a sufficient nonexistence criterion via Theorem B and, under the extra assumption that $P$ “induces a forest,” an equivalence in terms of $\\mathcal{C}_{\\mathcal{S}}^\\sigma\\neq\\varnothing$ in Theorem C.)",
    "expanded_sketch": "To analyze Theorem~\\ref{thmfirst:intro}, the introduction explains that the feasibility of the oriented characteristic system “depends on the signs of \\( A^{\\sigma}_{\\rho} \\)”. Using that “the strictly increasing nature of the exponential function” fixes these signs, it states that each entry’s sign is either “fixed (if $\\sigma_{1k}\\sigma_{2k} = -1$) or depends solely on the sign of \\( P_{ik}(\\sigma_{1k}\\rho_i - \\sigma_{2k}\\rho_{s+k}) \\)”, and hence “can be expressed as a linear inequality in the entries of $\\rho$.”\n\nFormally, it sketches the following steps:\n\\begin{itemize}\n\\item “we first derive simple conditions on the signs of the entries of \\( A^{\\sigma}_{\\rho} \\) that preclude the oriented characteristic system from having solutions.”\n\\item Any sign matrix $\\mathcal{S}\\in\\{-1,0,1\\}^{s\\times\\ell}$ not excluded by these conditions is called “feasible”.\n\\item For any orientation $\\sigma\\in\\mathbb{R}^{2\\times\\ell}$ and feasible $\\mathcal{S}$, one “introduce[s] a set $\\mathcal{C}_{\\mathcal{S}}^\\sigma$, which [is] name[d] the feasible ground set,” such that deciding nonemptiness is “equivalent to checking the feasibility of a linear system of equalities and inequalities.”\n\\end{itemize}\n(Additionally, the text notes a sufficient nonexistence criterion via Theorem B and, under the extra assumption that $P$ “induces a forest,” an equivalence in terms of $\\mathcal{C}_{\\mathcal{S}}^\\sigma\\neq\\varnothing$ in Theorem C.)",
    "expanded_theorem": "[\\cref{thm:main1,thm:mainsimplification}]\n\\label{thmfirst:intro}\nThe augmented vertically parametrized system $(C(\\kappa \\circ x^M), Lx - b)$ admits multiple positive zeros if and only if  there exist an orientation $\\sigma \\in \\{-1,0,1\\}^{2 \\times \\ell}$ and $\\rho \\in \\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ \nhas a solution.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Consider an augmented vertically parametrized polynomial system\n\\[\nF(x;\\kappa,b)=\\big(C(\\kappa\\circ x^M),\\,Lx-b\\big),\n\\]\nwhere $C\\in\\mathbb{R}^{\\bar s\\times \\bar m}$, $M\\in\\mathbb{Z}^{n\\times \\bar m}$, $L$ has $n-\\bar s$ rows, $\\kappa\\in\\mathbb{R}_{>0}^{\\bar m}$, $b\\in\\mathbb{R}^{n-\\bar s}$, and $x\\in\\mathbb{R}_{>0}^n$. Say that $F$ admits multiple positive zeros if there exist parameter values $\\kappa\\in\\mathbb{R}_{>0}^{\\bar m}$ and $b\\in\\mathbb{R}^{n-\\bar s}$ and distinct points $x,y\\in\\mathbb{R}_{>0}^n$ such that $F(x;\\kappa,b)=F(y;\\kappa,b)=0$, equivalently,\n\\[\nC(\\kappa\\circ x^M)=C(\\kappa\\circ y^M)=0,\n\\qquad Lx=Ly=b.\n\\]\nLet $P\\in\\mathbb{R}^{s\\times \\ell}$ be the reduced matrix associated with the system, let $m=s+\\ell$, and for an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and $\\rho\\in\\mathbb{R}^m$ define\n\\[\n(P^{\\sigma})_{ik}=\\sigma_{1k}P_{ik},\n\\qquad\n(A^{\\sigma}_{\\rho})_{ik}=P_{ik}\\big(\\sigma_{1k}e^{\\rho_i}-\\sigma_{2k}e^{\\rho_{s+k}}\\big)\n\\quad (i\\in[s],\\ k\\in[\\ell]).\n\\]\nThe oriented characteristic system associated with $(P,\\sigma,\\rho)$ is\n\\[\nA^{\\sigma}_{\\rho}\\mu=0,\\qquad P^{\\sigma}\\mu>0,\\qquad \\mu\\in\\mathbb{R}_{>0}^{\\ell},\n\\]\nand $\\mathcal{G}^{\\sigma}\\subseteq\\mathbb{R}^m$ is the oriented ground set associated with $\\sigma$. Which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if for every orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ there exists a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        },
        {
          "label": "C",
          "text": "If there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$, then the augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros."
        },
        {
          "label": "D",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathbb{R}^{m}$ such that the oriented characteristic system associated with $(P,\\sigma,\\rho)$ has a solution $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        },
        {
          "label": "E",
          "text": "The augmented vertically parametrized system $\\big(C(\\kappa\\circ x^M),Lx-b\\big)$ admits multiple positive zeros if and only if there exist an orientation $\\sigma\\in\\{-1,0,1\\}^{2\\times \\ell}$ and a vector $\\rho\\in\\mathcal{G}^{\\sigma}$ such that the sign pattern of $A^{\\sigma}_{\\rho}$ is feasible and $P^{\\sigma}\\mu>0$ for some $\\mu\\in\\mathbb{R}_{>0}^{\\ell}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "existential choice of orientation",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the converse direction of the iff statement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "membership condition $\\rho\\in\\mathcal{G}^{\\sigma}$",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "replaced solvability of $A^{\\sigma}_{\\rho}\\mu=0$ by mere feasible sign-pattern data",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It sets up notation and definitions, but does not state the theorem-level equivalence itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-identification item: the correct answer is the exact iff characterization, with distractors formed by small logical perturbations. It mainly tests recognition of the stated result rather than application."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare existential vs universal quantifiers, iff vs one-way implication, and the role of the ground set condition. However, the item still leans more toward recall/recognition of a theorem statement than genuine generative problem solving."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: quantifier strengthening (B), weaker true implication (C), dropping a necessary domain condition (D), and replacing exact solvability by weaker sign-feasibility data (E)."
      },
      "total_score": 5,
      "overall_assessment": "Technically well-constructed with strong distractors, but it is mostly a theorem-recall question rather than a non-tautological test of generative mathematical reasoning."
    }
  },
  {
    "id": "2512.07751v2",
    "paper_link": "http://arxiv.org/abs/2512.07751v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\n\tLet $1 \\le \\ell \\le k-1$ be such that $k \\ge 3$ and $(k,\\ell) \\neq (3,1)$, let $t = \\floor{\\frac{k}{k-\\ell}}(k - \\ell)$, and let $n$ be sufficiently large and divisible by $k-\\ell$.\n\tThen every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\\ell$-cycle.",
      "start_pos": 16866,
      "end_pos": 17264,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{thm} \\label{thm:main}\n\tLet $1 \\le \\ell \\le k-1$ be such that $k \\ge 3$ and $(k,\\ell) \\neq (3,1)$, let $t = \\floor{\\frac{k}{k-\\ell}}(k - \\ell)$, and let $n$ be sufficiently large and divisible by $k-\\ell$.\n\tThen every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\\ell$-cycle.\n\\end{thm}",
      "thm:Hamilton cycle extremal": "\\begin{restatable}{thm}{thmHamiltonCycleExtremal}\n\t\t\\label{thm:Hamilton cycle extremal}    \n\t\tLet $k \\ge 3$ and $\\ell \\in [k-1]$ be such that $(k,\\ell) \\neq (3,1)$, and set $t = \\floor{\\frac{k}{k-\\ell}}(k-\\ell)$.\n\t\tLet $1/n\\ll \\eps \\ll 1/k\\leq 1/3$ be such that $n$ is divisible by $k-\\ell$. \n\t\tSuppose $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\\delta^*(G)\\geq (1-1/t)n-(k-3)$. If there is a subset $A \\subseteq V(G)$ with $|A|=\\floor{n/t}$ that contains at most $\\eps n^2$ supported pairs, then $G$ has a Hamilton $\\ell$-cycle.\n\t\\end{restatable}",
      "thm:Hamilton cycle non-extremal": "\\begin{restatable}{thm}{thmHamiltonCycleNonExtremal}\n\t\t\\label{thm:Hamilton cycle non-extremal}\n\t\tLet $k \\ge 3$ and $\\ell \\in [k-1]$ be such that $(k,\\ell) \\neq (3,1)$, and set $t = \\floor{\\frac{k}{k-\\ell}}(k-\\ell)$. Let $1/n \\ll \\eps \\ll \\mu \\ll 1/k$ such that $n$ is divisible by $k-\\ell$. If $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\\delta^*(G)\\geq (1-1/t-\\eps)n$ such that every set of $\\floor{n/t}$ vertices of $G$ contains at least $\\mu n^2$ supported pairs, then $G$ has a Hamilton $\\ell$-cycle.\n\t\\end{restatable}",
      "rem:exact degree": "\\begin{rem}\\label{rem:exact degree}\n    A critical point to note is that this is the only part of the entire proof that utilises the exact bound $\\delta^*(G) \\ge n - \\floor{\\frac{n}{t}} - (k - 3)$ via \\cref{obs:min 1-degree,lem:disjoint edges cherries}. As mentioned in \\Cref{sec:intro}, we can prove \\cref{thm:Hamilton cycle extremal} and consequently \\cref{thm:main} with the weaker bound of $\\delta^*(G) \\ge n - \\floor{\\frac{n}{t}} - (k - 2)$ instead, for most values of $k$ and $\\ell$ (namely when $t \\ge \\ell+2$ or when $\\ell = k-1$) which is achieved by modifying \\cref{lem:disjoint edges cherries} to potentially allow for an additional cherry or some edges. We discuss how to suitably adapt the rest of the proof in \\Cref{sec:improving the bound}.\n\\end{rem}"
    },
    "pre_theorem_intro_text_len": 10124,
    "pre_theorem_intro_text": "\\label{sec:intro}\n    A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\\cite{dirac1952some}, states that any graph on $n\\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \\cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.\n\n\tA \\emph{$k$-uniform hypergraph} or simply a \\emph{$k$-graph} $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$, where each edge consists of exactly $k$ vertices. The most common generalisation of a minimum degree condition from graphs to hypergraphs is accomplished via the notion of a \\emph{minimum co-degree} -- which is the minimum $d$ such that every set of $k-1$ vertices in $G$ is contained in at least $d$ edges -- and we denote it here by $\\delta(G)$. The larger uniformity $k$ allows for multiple distinct ways to define a cycle in a $k$-graph. The cyclic structures we search for are \\emph{$\\ell$-cycles}, which are perhaps the most common extensions of graph cycles. Intuitively, we think of an $\\ell$-cycle as a spanning path formed by a cyclic set of edges such that each edge has exactly $\\ell$ vertices in common with the preceding edge (it could intersect other edges as well). More formally, given an $n$-vertex $k$-graph $G$ and any $\\ell \\in [k-1]$ such that $k - \\ell$ divides $n$, we define a \\emph{Hamilton $\\ell$-cycle} in $G$ to be an ordering of the vertices of $G$, say $v_1 \\dots v_n$, such that the vertices of the subsequence $v_{i(k - \\ell)+1} \\dots v_{i(k - \\ell) +k}$ form an edge for all $i \\ge 0$, where we view the indices cyclically modulo $n$. Since each edge of an $\\ell$-cycle contains $k-\\ell$ vertices that were not in the previous edge, we trivially require that $k - \\ell$ divides $n$ so that when we ``cycle around'' the vertices of $G$, the original sequence of edges gets repeated. We call $(k-1)$-cycles and $1$-cycles \\emph{tight} and \\emph{loose} cycles respectively.\n\nThe question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\\cite{katona1999hamiltonian} who conjectured that if $\\delta(G)\\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\\geq 3$ and $\\delta(G)\\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\\cite{rodl2011dirac}. Additionally, Liu and Liu \\cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\\ell \\in [k-1]$ such that $k - \\ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\\ell$-cycle will imply the existence of a perfect matching which necessitates $\\delta(G) \\ge n/2 - k$ as shown in \\cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\\delta(G) \\ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\\left\\lfloor n/k \\right\\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\\ell$-cycle, when $k-\\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\\ell$ does not divide $k$, a series of works by K\\\"uhn and Osthus~\\cite{kuhn2006loose}, Keevash, K\\\"uhn, Mycroft and Osthus~\\cite{keevash2011loose}, H\\`an and Schacht~\\cite{han2010dirac} and K\\\"uhn, Mycroft and Osthus~\\cite{kuhn2010hamilton} have established that \n\\[\n\\delta(G)\\geq \\frac{n}{\\ceil{\\frac{k}{k-\\ell}}(k-\\ell)}+o(n)\n\\]\nsuffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\\ell)=(3,2)$ by R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2011dirac}, $(k,\\ell)=(3,1)$ by Czygrinow and Molla~\\cite{czygrinow2014tight}, $(k,\\ell)=(4,2)$ by Garbe and Mycroft~\\cite{garbe2018hamilton}, and $k\\ge 3$ and $\\ell<k/2$ by Han and Zhao~\\cite{han2015minimum}. We refer the reader to the surveys~\\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.\n\nA drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \\emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.\n\nMultiple recent studies have investigated questions that arise from replacing a minimum co-degree with a minimum supported co-degree requirement in classical extremal hypergraph theoretic problems. The notion of supported co-degrees was introduced by Balogh, Lemons and Palmer~\\cite{balogh2021maximum} (they use the term ``positive co-degree'' instead) to formulate a generalisation of the Erd\\H{o}s-Ko-Rado theorem, providing bounds on the sizes of intersecting families satisfying a minimum supported co-degree condition. Subsequently, Frankl and Wang~\\cite{frankl2025intersecting} improved these bounds in almost all cases, and Spiro~\\cite{spiro2023t} has extended these studies to the more general case of $t$-intersecting families. Other papers have worked on generalisations of the Andr\\'asfai-Erd\\H{o}s-S\\'os theorem~\\cite{liu2024positive} and unique colourability of hypergraphs~\\cite{liu2024uniquely}.\n\nHalfpap, Lemons and Palmer~\\cite{halfpap2025positive} investigated variants of hypergraph Tur\\'an problems, and established the asymptotic supported co-degree threshold for a host $3$-graph to contain a copy of a fixed graph $F$ for several distinct $3$-graphs $F$, and Wu~\\cite{wu2025positive} has consequently addressed this question for other examples. Halfpap, Lemons and Palmer~\\cite{halfpap2025positive} also study ``jumps'' in this threshold, which has further been tackled by Balogh, Halfpap, Lidick\\'y, and Palmer~\\cite{balogh2024positive}. Pikhurko~\\cite{pikhurko2023limit} recently proved that these problems are well-defined for a generalisation to the ``minimum positive/supported $\\ell$-co-degree'' for any $\\ell \\in [k-1]$, providing analogous results to those Lo and Markstr\\\"om~\\cite{lo2014degree} regarding the usual notion of $\\ell$-co-degree.\n\nThe question of finding minimum supported co-degree conditions that ensure the existence of spanning structures in hypergraphs was first raised by Halfpap and Magnan~\\cite{halfpap2024positive}. They prove an exact optimal lower bound on the minimum supported co-degree required to guarantee a perfect matching in $3$-graphs and a slightly weaker bound for all higher uniformities, which was later improved to an exact tight bound by Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025matching}. Furthermore, Halfpap and Magnan also establish an exact best-possible minimum supported co-degree condition for Hamilton Berge cycles and an asymptotic one for $3$-uniform loose cycles. In a different direction, Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite{illingworth2025spanning} showed that an $n$-vertex $k$-graph with $\\delta^*(G)\\ge n/2 + o(n)$ contains a spanning $k$-sphere, asymptotically confirming a conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\\cite{georgakopoulos2022spanning}.\n\nThe problem of determining the optimal minimum supported co-degree for $k$-uniform $\\ell$-cycles was initially suggested by Halfpap and Magnan in~\\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\\ge 3$ and $\\ell\\in [k-1]$. \n\nOur main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\\ell$-cycles for all $k\\ge 3$ and $\\ell\\in [k-1]$ except $(k,\\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\\ell$, and is off by at most $1$ for all values of $k$ and $\\ell$. We discuss the optimality of our result in \\Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.",
    "context": "\\label{sec:intro}\n A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\\cite{dirac1952some}, states that any graph on $n\\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \\cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.\n\nThe question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\\cite{katona1999hamiltonian} who conjectured that if $\\delta(G)\\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\\geq 3$ and $\\delta(G)\\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\\cite{rodl2011dirac}. Additionally, Liu and Liu \\cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\\ell \\in [k-1]$ such that $k - \\ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\\ell$-cycle will imply the existence of a perfect matching which necessitates $\\delta(G) \\ge n/2 - k$ as shown in \\cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\\delta(G) \\ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\\left\\lfloor n/k \\right\\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\\ell$-cycle, when $k-\\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\\ell$ does not divide $k$, a series of works by K\\\"uhn and Osthus~\\cite{kuhn2006loose}, Keevash, K\\\"uhn, Mycroft and Osthus~\\cite{keevash2011loose}, H\\`an and Schacht~\\cite{han2010dirac} and K\\\"uhn, Mycroft and Osthus~\\cite{kuhn2010hamilton} have established that \n\\[\n\\delta(G)\\geq \\frac{n}{\\ceil{\\frac{k}{k-\\ell}}(k-\\ell)}+o(n)\n\\]\nsuffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\\ell)=(3,2)$ by R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2011dirac}, $(k,\\ell)=(3,1)$ by Czygrinow and Molla~\\cite{czygrinow2014tight}, $(k,\\ell)=(4,2)$ by Garbe and Mycroft~\\cite{garbe2018hamilton}, and $k\\ge 3$ and $\\ell<k/2$ by Han and Zhao~\\cite{han2015minimum}. We refer the reader to the surveys~\\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.\n\nA drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \\emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.\n\nThe problem of determining the optimal minimum supported co-degree for $k$-uniform $\\ell$-cycles was initially suggested by Halfpap and Magnan in~\\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\\ge 3$ and $\\ell\\in [k-1]$.\n\nOur main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\\ell$-cycles for all $k\\ge 3$ and $\\ell\\in [k-1]$ except $(k,\\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\\ell$, and is off by at most $1$ for all values of $k$ and $\\ell$. We discuss the optimality of our result in \\Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.",
    "full_context": "\\label{sec:intro}\n A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\\cite{dirac1952some}, states that any graph on $n\\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \\cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.\n\nThe question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\\cite{katona1999hamiltonian} who conjectured that if $\\delta(G)\\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\\geq 3$ and $\\delta(G)\\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\\cite{rodl2011dirac}. Additionally, Liu and Liu \\cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\\ell \\in [k-1]$ such that $k - \\ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\\ell$-cycle will imply the existence of a perfect matching which necessitates $\\delta(G) \\ge n/2 - k$ as shown in \\cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\\delta(G) \\ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\\left\\lfloor n/k \\right\\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\\ell$-cycle, when $k-\\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\\ell$ does not divide $k$, a series of works by K\\\"uhn and Osthus~\\cite{kuhn2006loose}, Keevash, K\\\"uhn, Mycroft and Osthus~\\cite{keevash2011loose}, H\\`an and Schacht~\\cite{han2010dirac} and K\\\"uhn, Mycroft and Osthus~\\cite{kuhn2010hamilton} have established that \n\\[\n\\delta(G)\\geq \\frac{n}{\\ceil{\\frac{k}{k-\\ell}}(k-\\ell)}+o(n)\n\\]\nsuffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\\ell)=(3,2)$ by R\\\"{o}dl, Ruci\\'nski and Szemer\\'edi~\\cite{rodl2011dirac}, $(k,\\ell)=(3,1)$ by Czygrinow and Molla~\\cite{czygrinow2014tight}, $(k,\\ell)=(4,2)$ by Garbe and Mycroft~\\cite{garbe2018hamilton}, and $k\\ge 3$ and $\\ell<k/2$ by Han and Zhao~\\cite{han2015minimum}. We refer the reader to the surveys~\\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.\n\nA drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \\emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.\n\nThe problem of determining the optimal minimum supported co-degree for $k$-uniform $\\ell$-cycles was initially suggested by Halfpap and Magnan in~\\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\\ge 3$ and $\\ell\\in [k-1]$.\n\nOur main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\\ell$-cycles for all $k\\ge 3$ and $\\ell\\in [k-1]$ except $(k,\\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\\ell$, and is off by at most $1$ for all values of $k$ and $\\ell$. We discuss the optimality of our result in \\Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.\n\n\\begin{ex}\\label{ex:lower bd weak}\n Let $k\\ge 3$, $\\ell \\in [k-1]$ and $t = \\floor{\\frac{k}{k-\\ell}}(k - \\ell)$. Let $n$ be chosen such that $k-\\ell$ divides $n$. Now, consider an $n$-vertex $k$-graph $G$ with a partition $A\\sqcup B$ of its vertex set such that $|A| = \\floor{n/t} + 1$ and $E(G)$ consists of all $k$-sets $e \\in \\binom{V(G)}{k}$ that satisfy $|e \\cap A| \\le 1$ (and so $A$ is a strong independent set). Suppose $S$ is any supported $(k-1)$-set. If $|S\\cap A| = 0$, then $S$ supports all vertices outside itself, and so $d_G^1(S) = n-k+1$. If $|S\\cap A| = 1$, then it supports all vertices in $B\\setminus S$, meaning that $d_G^1(S) = n - \\floor{n/t} - (k-2)$, and this is equal to $\\delta^*(G)$ since there is no supported set with $|S\\cap A| \\ge 2$. The following observation shows that $G$ cannot contains a Hamilton $\\ell$-cycle.\n \\end{ex}\n\n\\begin{ex}\\label{ex:lower bd strong}\n Next, we provide a specialised modification of the previous example. Let $k\\ge 3$ be odd, $\\ell = (k-1)/2$, and consequently $t = \\floor{\\frac{k}{k-\\ell}}(k - \\ell) = (k+1)/2 = \\ell+1 = k - \\ell$. Let $n$ be such that $t$ divides $n$ and $n/t+1$ is even. Yet again, we consider an $n$-vertex $k$-graph $G$ with a partition $A\\sqcup B$ of its vertex set with $|A| = n/t + 1$. Write $A = \\{a_1, \\dots, a_{|A|}\\}$ and let $E(G)$ consist of all $e\\in \\binom{V(G)}{k}$ such that $e\\cap A \\subseteq \\{a_{2i-1}, a_{2i}\\}$ for some $i\\in [|A|/2]$ (so the graph $\\partial^2[A]$ is a perfect matching). Clearly $|e\\cap A| \\le 2$ for all $e \\in E(G)$. If $S$ is any supported $(k-1)$-set, then, similar to the previous example, it is not too hard to see that\n \\[d_G^1(S) =\n \\left\\{\n \\begin{array}{ll}\n n - (k-1) & \\textnormal{if} \\ |e\\cap A|=0,\\\\\n |B| - (k - 2) +1 = |B| - (k-3) & \\textnormal{if} \\ |e\\cap A|=1,\\\\\n |B| - (k - 3) & \\textnormal{if} \\ |e\\cap A|=2,\\\\\n \\end{array}\n \\right.\n \\]\n and so $\\delta^*(G) = |B| - (k - 3) = n - n/t - (k - 2)$. We now argue that $G$ cannot contain a Hamilton $\\ell$-cycle, which shows that the bound in \\cref{thm:main} cannot be improved.\n\n\\begin{restatable}{thm}{thmHamiltonCycleNonExtremal}\n \\label{thm:Hamilton cycle non-extremal}\n Let $k \\ge 3$ and $\\ell \\in [k-1]$ be such that $(k,\\ell) \\neq (3,1)$, and set $t = \\floor{\\frac{k}{k-\\ell}}(k-\\ell)$. Let $1/n \\ll \\eps \\ll \\mu \\ll 1/k$ such that $n$ is divisible by $k-\\ell$. If $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\\delta^*(G)\\geq (1-1/t-\\eps)n$ such that every set of $\\floor{n/t}$ vertices of $G$ contains at least $\\mu n^2$ supported pairs, then $G$ has a Hamilton $\\ell$-cycle.\n \\end{restatable}\n\n\\begin{restatable}{thm}{thmHamiltonCycleExtremal}\n \\label{thm:Hamilton cycle extremal} \n Let $k \\ge 3$ and $\\ell \\in [k-1]$ be such that $(k,\\ell) \\neq (3,1)$, and set $t = \\floor{\\frac{k}{k-\\ell}}(k-\\ell)$.\n Let $1/n\\ll \\eps \\ll 1/k\\leq 1/3$ be such that $n$ is divisible by $k-\\ell$. \n Suppose $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\\delta^*(G)\\geq (1-1/t)n-(k-3)$. If there is a subset $A \\subseteq V(G)$ with $|A|=\\floor{n/t}$ that contains at most $\\eps n^2$ supported pairs, then $G$ has a Hamilton $\\ell$-cycle.\n \\end{restatable}\n\n\\begin{prop} \\label{prop:supported-l-paths}\n Let $G$ be a $k$-graph, where $k \\ge 3$, $\\ell \\in [k-1]$ and $t = \\floor{\\frac{k}{k-\\ell}}(k-\\ell)$.\n \\begin{enumerate}[label = \\rm(\\roman*)]\n \\item \\label{itm:extended-path-1}\n Let $P$ and $Q$ be two sequences of vertices in $G$ of length at least $t$, both of which support an $\\ell$-path, and whose vertices are disjoint. If the sequence consisting of the last $t$ vertices in $P$ followed by the first $t-1$ vertices in $Q$ is a tight path, then $PQ$ supports an $\\ell$-path. \n \\item \\label{itm:extended-path-2}\n Let $P_1, \\ldots, P_m$ be sequences of vertices in $G$ of length at least $t$, each of which supports an $\\ell$-path, and whose vertices are pairwise disjoint. If $P_iP_{i+1}$ supports an $\\ell$-path for every $i \\in [m]$ (addition of indices taken modulo $m$) then $P_1 \\ldots P_m$ is an $\\ell$-cycle.\n \\item \\label{itm:extended-path-3}\n Let $P = v_0 \\ldots v_r$ be a sequence of distinct vertices where $r$ is divisible by $t$. If every subsequence of at most $k$ consecutive vertices that contains at most one vertex $v_i$ with $i$ divisible by $t$ is supported by an edge, then $P$ supports an extended $\\ell$-path. \n \\item \\label{itm:extended-path-4}\n Let $P$ be a sequence of distinct vertices of length $r$ such that $r = -1 \\pmod{k-\\ell}$ and $r \\ge t-1$, and let $u,v$ be two vertices that do not appear in $P$, such that $uP$ and $Pv$ are tight paths (or supported sets if $r+1 \\le k$). Then $uPv$ supports an extended $\\ell$-path.\n \\end{enumerate}\n \\end{prop}\n \\begin{proof}\n For \\ref{itm:extended-path-1}, say that the length of $P$ is $p$ and that of $Q$ is $q$. Let $PQ = v_1 \\dots v_{p+q}$, where $P = v_1 \\dots v_p$ and $Q = v_{p+1} \\dots v_{p+q}$. Since $P$ and $Q$ both support $\\ell$-paths, we know that $p$ and $q$ are divisible by $k-\\ell$, implying the same is true of $p+q$. Furthermore, as $q\\ge t$, we also know that the last $t$ vertices of $PQ$ correspond to the last $t$ vertices of $Q$, and hence are supported. All that is left to show is that $v_{i(k-\\ell) + 1} \\ldots v_{i(k-\\ell)+k}$ is an edge for all $i\\le \\frac{p + q - t}{k-\\ell} - 1$. Since $P$ and $Q$ support $\\ell$-paths (and in particular their length is divisible by $k-\\ell$), we only need to restrict our attention to $i \\in [\\frac{p - t}{k - \\ell}, \\frac{p}{k - \\ell}-1]$. For this, it would suffice to argue that the subsequence $v_{i_1} \\dots v_{i_2}$ forms a tight path, where $i_1 = p - t + 1$ and $i_2 = p + \\ell$. By assumption, we know that $v_{p - t + 1} \\dots v_{p+t-1}$ is a tight path, and we know $t-1\\ge \\ell$ (see \\Cref{obs:useful t info}), which completes the proof.\n\nThese results contribute to the broader research theme of finding the best possible minimum supported co-degree to ensure a specified spanning substructure, an area which poses some tantalising questions. Our original motivation for working on this problem arose from Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite[Conjecture $1.4$]{illingworth2025spanning}, who conjectured that $\\delta^*(G) \\ge (1-1/k)n$ guarantees a tight Hamilton cycle. The main focus of their paper is the minimum supported co-degree required to ensure a spanning $(k-1)$-sphere -- a subset of edges in a $k$-graph that induces a homogeneous simplicial complex homeomorphic to the $(k-1)$-sphere $\\mathbb{S}^{k-1}$, and whose $0$-skeleton spans the entire vertex set. They tackle the following conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\\cite{georgakopoulos2022spanning} (paraphrased).\n \\begin{conj}\\label{conj:spheres}\n Every $k$-graph $G$ of order $n>k \\ge 2$ with $\\delta^*(G) \\ge n/2$ and with no isolated vertices contains a spanning copy of $\\mathbb{S}^{k-1}$.\\footnote{In the original phrasing of the conjecture $G$ is required to be tightly connected -- meaning that any two edges can be joined by a tight walk -- but this follows from the other assumptions.}\n \\end{conj}\n The chief result in~\\cite{georgakopoulos2022spanning} provides an asymptotically optimal minimum co-degree condition for a $3$-graph to contain a vertex-spanning copy of any \\emph{surface}, meaning an arbitrary connected, closed $2$-manifold. In particular, this addresses the minimum co-degree necessary to guarantee a spanning $2$-sphere in a $3$-graph.\n \\begin{thm}[{\\cite[Theorem 1.4]{georgakopoulos2022spanning}}]\\label{thm:spheres-codegree}\n Let $\\cS$ be an arbitrary surface and $\\eps>0$. Then any sufficiently large $n$-vertex $3$-graph with $\\delta(G) \\ge n/3 + \\eps n$ contains a spanning copy of $\\cS$. Moreover, for any $n \\in \\mathbb{N}$, there exists an $n$-vertex $3$-graph $H$ with $\\delta(H) = \\floor{\\frac{n}{3}} - 1$ such that there are at most $2\\ceil{\\frac{n}{3}}$ vertices in the $0$-skeleton of a copy of any surface in $H$.\n \\end{thm}",
    "post_theorem_intro_text_len": 4317,
    "post_theorem_intro_text": "This improves the aforementioned asymptotic results of Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025positive}, who showed that $\\delta^*(G) \\ge (1 - 1/t)n + o(n)$ suffices. Our methods are largely different from theirs. Our analysis splits into two regimes: the extremal and non-extremal (see \\Cref{sec:stability} for details). The extremal regime does not feature in \\cite{mycroft2025positive} at all, and for this we use ad-hoc structural analysis with lemmas about random matchings in bipartite graphs and about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs. For the non-extremal case, while Mycroft and Z\\'arate-Guer\\'en rely on a version of the regularity lemma for hypergraphs (called the weak regularity lemma) along with the absorption method, our approach stems from Lang's breakthrough work \\cite{lang2023tiling} on hypergraph tilings, which allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method. For the latter part, however, we use a novel idea from \\cite{mycroft2025positive} about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\\ell$-cycles in certain $k$-partite $k$-graphs, which is one of the steps in our proof. \n\nWe point out that Illingworth, Lang, M\\\"uyesser, Parczyk and Sgueglia~\\cite{illingworth2025spanning} conjectured that for all $k\\ge 3$, any sufficiently large $k$-graph with $\\delta^*(G) \\ge (1-1/k)n$ contains a tight Hamilton cycle, which is a special case of \\cref{thm:main} (up to the $k-3$ term). Observe that if $k$ divides $n$, then a tight Hamilton cycle in $G$ will contain a perfect matching. Thus, our result implies that $\\delta^*(G) \\ge (1 - 1/k)n - (k-3)$ ensures a perfect matching, which recovers a result of Mycroft and Z\\'arate-Guer\\'en~\\cite{mycroft2025matching} up to an additive constant of one, who show that a bound of $(1 - 1/k)n - (k - 2)$ suffices.\n\nAs alluded to above, the bound in \\cref{thm:main} is off by one in most cases. In fact, in many cases (namely whenever $t \\ge \\ell+2$ or when $\\ell = k-1$, that is, we are seeking a tight Hamilton cycle), our techniques can be adapted to prove \\cref{thm:main} with the improved bound of $\\delta^*(G) \\ge (1 - 1/t)n - (k - 2)$ instead (which, for instance, would immediately imply the previously discussed optimal perfect matching bound). \nHowever, since this will require some technical modifications, we choose to present a single unified proof for $\\delta^*(G) \\ge (1 - 1/t)n - (k - 3)$ that will work for all cases instead. We point out the part of our proof that requires the exact degree condition in \\Cref{rem:exact degree}, and briefly discuss how to suitably alter our proof and remove the extra one in the bound in all relevant cases in \\Cref{sec:improving the bound}.\n\nFinally, we remark that in a recent personal communication, Mycroft and Z\\'arate-Guer\\'en informed us that they are preparing a manuscript where they prove that any sufficiently large $3$-graph with no isolated vertices and $\\delta^*(G) \\ge n/2$ contains a loose Hamilton cycle. This improves the previous asymptotic bound of $n/2 + o(n)$ established by them~\\cite{mycroft2025positive} and Halfpap and Magnan~\\cite{halfpap2024positive} (which our methods can recover as well) and extends \\cref{thm:main} to include $(k,\\ell) = (3,1)$, the only case we do not handle.\n\n\\paragraph{Organisation of the paper.} In \\Cref{sec:overview}, we first provide lower bound constructions to prove the optimality of \\cref{thm:main}. We then split the proof of \\cref{thm:main} into two complementary cases and handle them separately in \\cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}. We also provide brief proof overviews for both these theorems in \\Cref{sec:proof-overviews}. Then in \\Cref{sec:prelims} we introduce some notation and definitions and provide some basic tools.\n\n\\Crefrange{sec:non-extremal}{sec:proof Hamilton paths} are dedicated to proving \\cref{thm:Hamilton cycle non-extremal}, and \\Crefrange{sec:extremal hypergraphs}{sec:Hamilton cycle extremal} prove \\cref{thm:Hamilton cycle extremal}. These two parts are treated largely independently, and can be read as such. We conclude with some open problems in \\Cref{sec:conclusion}.",
    "sketch": "The proof of \\cref{thm:main} is split into two regimes: the \\emph{extremal} and \\emph{non-extremal} cases (see \\Cref{sec:stability}). The two cases are then handled separately in \\cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.\n\nIn the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”\n\nIn the non-extremal regime, the approach “stems from Lang's breakthrough work \\cite{lang2023tiling} on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”\n\nThey also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in \\Cref{rem:exact degree},” and “briefly discuss how to suitably alter our proof and remove the extra one … in \\Cref{sec:improving the bound}.”",
    "expanded_sketch": "The proof of the main theorem is split into two regimes: the \\emph{extremal} and \\emph{non-extremal} cases (see \\Cref{sec:stability}). The two cases are then handled separately in \\cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.\n\nIn the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”\n\nIn the non-extremal regime, the approach “stems from Lang's breakthrough work Lang, \\emph{tiling} (2023) on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”\n\nThey also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in the following remark. \\begin{rem}\\label{rem:exact degree}\n    A critical point to note is that this is the only part of the entire proof that utilises the exact bound $\\delta^*(G) \\ge n - \\floor{\\frac{n}{t}} - (k - 3)$ via \\cref{obs:min 1-degree,lem:disjoint edges cherries}. As mentioned in \\Cref{sec:intro}, we can prove \\cref{thm:Hamilton cycle extremal} and consequently \\cref{thm:main} with the weaker bound of $\\delta^*(G) \\ge n - \\floor{\\frac{n}{t}} - (k - 2)$ instead, for most values of $k$ and $\\ell$ (namely when $t \\ge \\ell+2$ or when $\\ell = k-1$) which is achieved by modifying \\cref{lem:disjoint edges cherries} to potentially allow for an additional cherry or some edges. We discuss how to suitably adapt the rest of the proof in \\Cref{sec:improving the bound}.\n\\end{rem}” and “briefly discuss how to suitably alter our proof and remove the extra one … in \\Cref{sec:improving the bound}.”",
    "expanded_theorem": "\\label{thm:main}\n\tLet $1 \\le \\ell \\le k-1$ be such that $k \\ge 3$ and $(k,\\ell) \\neq (3,1)$, let $t = \\floor{\\frac{k}{k-\\ell}}(k - \\ell)$, and let $n$ be sufficiently large and divisible by $k-\\ell$.\n\tThen every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\\ell$-cycle.,",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $1\\le \\ell\\le k-1$ with $k\\ge 3$ and $(k,\\ell)\\neq (3,1)$, and define\n\\[\nt=\\left\\lfloor \\frac{k}{k-\\ell}\\right\\rfloor (k-\\ell).\n\\]\nAssume $n$ is sufficiently large and divisible by $k-\\ell$. Let $G$ be a $k$-uniform hypergraph on $n$ vertices with no isolated vertices. Define the minimum supported co-degree $\\delta^*(G)$ to be the maximum integer $d$ such that every $(k-1)$-subset of vertices that is contained in at least one edge of $G$ is contained in at least $d$ edges of $G$. Suppose that\n\\[\n\\delta^*(G)\\ge \\left(1-\\frac1t\\right)n-(k-3).\n\\]\nA Hamilton $\\ell$-cycle means a spanning $\\ell$-cycle, i.e. a cyclic ordering of all $n$ vertices such that each edge consists of $k$ consecutive vertices in the ordering and consecutive edges intersect in exactly $\\ell$ vertices. Under these hypotheses, which conclusion holds?",
      "correct_choice": {
        "label": "A",
        "text": "The hypergraph $G$ contains a Hamilton $\\ell$-cycle."
      },
      "choices": [
        {
          "label": "B",
          "text": "The hypergraph $G$ contains a Hamilton $\\ell$-cycle provided, in addition, that $k-\\ell$ divides $k$."
        },
        {
          "label": "C",
          "text": "The hypergraph $G$ contains an $\\ell$-cycle spanning all but at most $k-\\ell$ vertices."
        },
        {
          "label": "D",
          "text": "There exists a constant $n_0=n_0(k,\\ell)$ such that for every integer $n\\ge n_0$ divisible by $k-\\ell$, every $k$-uniform $n$-vertex hypergraph $G$ without isolated vertices and with minimum supported co-degree at least $\\left(1-\\frac1t\\right)n-(k-2)$ contains a Hamilton $\\ell$-cycle."
        },
        {
          "label": "E",
          "text": "The hypergraph $G$ contains a Hamilton $\\ell$-path."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "unnecessary divisibility restriction $k-\\ell\\mid k$",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped exact spanning conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "exact degree offset $(k-3)$ replaced by $(k-2)$ uniformly",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "cycle conclusion weakened to path despite spanning cyclic closure being the hard step",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the resulting conclusion, without directly naming the theorem's conclusion in advance."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem gives the full hypothesis set and the correct choice restates the theorem's exact conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some precision is needed to distinguish the exact conclusion from nearby variants such as a weaker spanning claim, an added unnecessary divisibility condition, or a shifted degree bound. However, the item mainly tests recall of the exact theorem rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one adds an unnecessary condition, one weakens the conclusion, one perturbs the threshold, and one replaces cycle by path. These reflect realistic failure modes and are clearly distinct."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically precise MCQ with strong distractors, but it is largely a theorem restatement and therefore only moderately effective at testing genuine generative reasoning."
    }
  },
  {
    "id": "2512.07773v1",
    "paper_link": "http://arxiv.org/abs/2512.07773v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{THM:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\omega$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1 (\\mathbb{T})} > z_0\\varepsilon^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noindent \nholds if and only if $(c_k) \\in \\ell^1$.",
      "start_pos": 16675,
      "end_pos": 17112,
      "label": "THM:main"
    },
    "ref_dict": {
      "minp": "\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}",
      "decay": "\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}",
      "weakNLS1": "\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}",
      "THM_LDP3": "\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\PP \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}",
      "COR:main": "\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\l^1$.\n\\end{corollary}",
      "random_data": "\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}",
      "PROP:linear2": "\\begin{proposition}\n\\label{PROP:linear2}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear2}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\PP \\big( \\|u(t)\\|_{L^\\infty (\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}, \n\\end{align}\n\n\\noi \nprovided $(c_k) \\in \\l^1$.\n\\end{proposition}",
      "obser": "\\begin{align}\\label{obser}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\sup_{x} |u(t,x) | > z_0\\eps^{-1/2} \\Big) = \\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1} > z_0\\eps^{-1/2} \\Big),\n\\end{align}",
      "THM_LDP4": "\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}",
      "THM:main": "\\begin{theorem}\n\\label{THM:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noi \nholds if and only if $(c_k) \\in \\l^1$.\n\\end{theorem}",
      "NLS": "\\begin{equation}\n\\label{NLS}\n\\begin{cases}\ni\\pa_t u + \\partial_x^2 u = \\pm \\,|u|^2\\, u,\\\\\nu(t,x)|_{t=0}=u_0.\n\\end{cases}\n\\end{equation}",
      "LDP2": "\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}",
      "LDP1": "\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}",
      "random1": "\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 7821,
    "pre_theorem_intro_text": "In this paper, we consider the following Cauchy problem of the one-dimensional cubic nonlinear Schr\\\"odinger equation (NLS) on the torus $\\mathbb{T}= \\mathbb{R} / (2 \\pi \\mathbb{Z})$: \n\\begin{equation}\n\\label{NLS}\n\\begin{cases}\ni\\pa_t u + \\partial_x^2 u = \\pm \\,|u|^2\\, u,\\\\\nu(t,x)|_{t=0}=u_0.\n\\end{cases}\n\\end{equation}\n\n\\noindent \nThe equation \\eqref{NLS} is fundamental in theoretical physics and applied mathematics, for describing the evolution of wave packets in various physical systems, such as nonlinear optics, fluids, and plasmas; see \\cite{SS99} for a review.\nBourgain \\cite{BO93} proved \\eqref{NLS} is deterministically global well-posed in $L^2 (\\mathbb{T})$.\nThe probabilistic study of \\eqref{NLS} has emerging importance since the seminal work of Bourgain on invariant measures \\cite{BO94,BO96}.\n\nMore recently, probabilistic analyses of \\eqref{NLS} have also found applications in oceanography, particularly in modelling rare extreme events such as the formation of {\\it rogue waves} in deep-sea dynamics~\\cite{DGV18,DGV19,GGKS21}. In particular, Garrido et al.~\\cite{GGKS21} established a large deviations principle (LDP) describing the probability of observing a wave of unusually large height in the weakly nonlinear regime. Their result relied on a strong exponential decay assumption on the Fourier coefficients of the random initial data, which plays a crucial role in their analysis.\nIn this paper, we improve the exponential decay condition on coefficients of initial data in \\cite{GGKS21} to a $\\ell^1$ decay condition. \n\nWe consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$.  \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables. \n\nThe study of the large deviation principle for solutions to \\eqref{NLS} was initiated by \\cite{DGV18,DGV19}, where the authors conjectured that the existence of an LDP can be used to predict the formation of rogue waves.\nIn particular, they considered \\eqref{NLS} with random initial data $u_0^N$ of the form\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\n\\noindent \nwhere $u$ is the solution to NLS \\eqref{NLS} with the truncated initial data \\eqref{random1}.\nDematteis et al, then proposed a theoretical framework of a large deviations principle (LDP) to quantify the likelihood of $\\mathcal D (t,z)$.\nIn particular, consider the minimization problem\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\noindent \nwhere\n\\[\n\\mathcal I (\\theta) = \\max_{y \\in \\mathcal{C}^{2N+1}} [\\langle y,\\theta \\rangle - S(y)] \\quad \\textup{ and } \\quad S(y) = \\log \\mathbb{E} e^{\\langle y, \\theta \\rangle}.\n\\]\n\n\\noindent\nThen Dematteis et al. claim that for $t >0$ \n\\begin{align}\n\\label{LDP1}\n\\log \\mathbb{P} (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\noindent \nas $z \\to \\infty$, provided \nthe minimization problem \\eqref{minp} has a unique solution.\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$. \n\nThe proof of~\\eqref{THM_LDP3} is based on the G\\\"artner--Ellis theorem combined with resonant approximation techniques. \nIn their analysis, the rapid decay condition~\\eqref{decay} plays a crucial role in both the linear and nonlinear estimates. \nThey further observed that alternative families of coefficients $(c_k)$ might also yield~\\eqref{LDP2}, and posed an \\textit{open question} concerning the identification of the optimal decay condition under which~\\eqref{LDP2} holds; see~\\cite[Remark~1.3]{GGKS21}. \nOne of the principal aims of the present work is to address this question.\n\nTo this purpose, we consider a LDP under Fourier-Lebesgue norm, which is stronger than the $L^\\infty (\\mathbb{T})$ norm used in \\eqref{THM_LDP3}.\nOur conventions for the Fourier transform are as follows:\n\\[\n\\widehat f (n) = \\frac1{{2\\pi}} \\int_0^{2\\pi} f(x) e^{ix n}dx \\quad \\textup{ and } \\quad f(x) = \\sum_{n\\in \\mathbb{Z}} \\widehat f(n) e^{ixn}\n\\]\n\n\\noindent \nfor functions on the circle $\\mathbb{T} = \\mathbb{R}/(2\\pi \\mathbb{Z})$.\nWith the above notations, we define the Fourier-Lebesgue norm by\n\\[\n\\begin{split} \n\\|f\\|_{\\mathcal{F} L^1 (\\mathbb{T})} = \\sum_{n \\in \\mathbb{Z}} |\\widehat f(n)|.\n\\end{split}\n\\]\n\n\\noindent \nIt is easy to see that\n\\[\n\\sup_{x \\in \\mathbb{T}} |f(x)| \\le \\|f\\|_{\\mathcal{F} L^1 (\\mathbb{T})}.\n\\]\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.",
    "context": "We consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$. \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables.\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.\n\n\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\n\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}",
    "full_context": "We consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$. \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables.\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.\n\n\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\n\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\T = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\PP \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noi \nWe note that $L^\\infty$ and $\\FL^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\FL^1$ share the same rate function.\nIn particular, for $t \\les \\eps^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\sup_{x} |u(t,x) | > z_0\\eps^{-1/2} \\Big) = \\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1} > z_0\\eps^{-1/2} \\Big),\n\\end{align}\n\n\\begin{remark}\n\\rm \nWe remark that the time scale $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ is critical;\nsee \\cite[Remark 1.7]{GGKS21} for more discussion on critical time.\nIn the limit $\\eps \\to 0^+$, the large deviation principle \\eqref{THM_LDP4} may hold for any fixed time $t > 0$.\n\\end{remark}\n\n\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\l^1$.\n\\end{corollary}\n\n\\begin{proposition}\n\\label{PROP:linear1}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear1}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\P \\big( \\|u(t)\\|_{\\FL^1(\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2},\n\\end{align}\n\nWe turn to the summation in \\eqref{main} over $k \\in B$. \nWe note that\n\\begin{align}\n\\label{main2}\n\\begin{split}\n\\eps & \\sum_{k \\in B} \\log \\left( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\exp\\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\right) \\\\\n& \\le \\eps \\sum_{k \\in B} \\log \\left( \\big( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\big) \\exp\\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\right) \\\\\n& \\le \\eps \\sum_{k \\in B} \\log \\big( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\big) + \\eps \\sum_{k \\in B} \\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\\\\n& \\le \\sqrt{\\pi}\\varepsilon^{1/2} \\sum_{k \\in B} \\, c_k\\, + \\frac14 \\sum_{k \\in B} c_k^2,\n\\end{split}\n\\end{align}\nwhich is again uniformly bounded as $(c_k) \\in \\l^1 \\subset \\l^2$, where we used $\\eps^{1/2} \\le c_k$ for $k \\in B$.\n\\end{proof}\n\n\\subsection{Linear LDP II - \\texorpdfstring{$L^\\infty$}{Lg} norm}\nIn this subsection,\nwe prove an LDP for the linear solution to \\eqref{linear} under $L^\\infty$ norm.\nIn particular, we shall prove\n\\begin{proposition}\n\\label{PROP:linear2}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear2}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\PP \\big( \\|u(t)\\|_{L^\\infty (\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}, \n\\end{align}\n\n\\noi \nwhere $\\mathcal B_\\eps$ is defined in \\eqref{Beps}.\nWe first note, see also \\cite[Proposition 4.1]{GGKS21},\\footnote{Please note that \\cite[Proposition 4.1]{GGKS21} only requires that $\\|c_k\\|_{\\l^1_k} < \\infty$.} that \n\\begin{align}\n\\label{pp400}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\big( \\| u_{\\rm app} (t) \\|_{L^\\infty (\\T)} > z_0 (\\eps^{-1/2} + \\eps^{-1/2 + \\dl}) \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}.\n\\end{align}\n\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}",
    "post_theorem_intro_text_len": 2107,
    "post_theorem_intro_text": "\\begin{remark}\n\\rm \nWe remark that the time scale $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ is critical;\nsee \\cite[Remark 1.7]{GGKS21} for more discussion on critical time.\nIn the limit $\\varepsilon \\to 0^+$, the large deviation principle \\eqref{THM_LDP4} may hold for any fixed time $t > 0$.\n\\end{remark}\n\n\\begin{remark}\n\\rm \nTo prove Theorem~\\ref{THM:main}, we first develop a novel approach to establish\nthe LDP for the linear Schr\\\"odinger equation under the sole assumption that\nthe initial data lies in $\\mathcal{F} L^1$. We then carry out the nonlinear analysis,\ntogether with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.\n\\end{remark}\n\nAs a consequence of Theorem \\ref{THM:main} and the observation \\eqref{obser}, we can improve \\eqref{THM_LDP3} in terms of the decay of $c_k$.\n\n\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\omega$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\ell^1$.\n\\end{corollary} \n\n\\begin{remark}\n\\rm \nWe note that Corollary \\ref{COR:main} improves \\cite[Theorem 1.1]{GGKS21} in the following sense.\nThe coefficients $(c_k)$ in \\cite[Theorem 1.1]{GGKS21} is required to decay condition \\eqref{decay}.\nIn Proposition \\ref{PROP:linear2}, it requires only $(c_k) \\in \\ell^1$.\nIt is expected that \\eqref{THM_LDP3} holds for initial data $u_0 \\in L^2 \\cap L^\\infty$, where $\\mathcal{F} L^1 \\subset L^2 \\cap L^\\infty$.\nHowever, our current argument can only handle $\\mathcal{F} L^1$ data.\n\\end{remark}\n\n\\begin{remark}\n\\rm \nRecently, in \\cite{BGMS25}, the authors studied rogue waves and large deviations\nfor two-dimensional pure-gravity deep-water waves, assuming initial data with\nexponentially decaying Fourier coefficients; see \\cite[(1.14)]{BGMS25}. It is\nexpected that our argument also applies in their setting and allows one to\nrelax this assumption to merely $\\ell^1$ decay of the Fourier coefficients.\nWe plan to pursue this in future work.\n\\end{remark}",
    "sketch": "To prove Theorem~\\ref{THM:main}, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\\\"odinger equation under the sole assumption that the initial data lies in $\\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.”",
    "expanded_sketch": "To prove the main theorem, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\\\"odinger equation under the sole assumption that the initial data lies in $\\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.”,",
    "expanded_theorem": "\\label{THM:main} \nLet $u(t,x)$ be solutions to \\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align} with initial data $u_0^\\omega$ given in \\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1 (\\mathbb{T})} > z_0\\varepsilon^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noindent \nholds if and only if $(c_k) \\in \\ell^1.\\,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $u(t,x)$ solve the weakly nonlinear Schr\\\"odinger equation on the torus $\\mathbb T$,\n\\[\ni\\partial_t u+\\Delta u=\\varepsilon^2|u|^2u,\n\\]\nwith random initial data\n\\[\nu_0(x)=\\sum_{k\\in\\mathbb Z} c_k g_k e^{ikx},\n\\]\nwhere $\\{g_k\\}_{k\\in\\mathbb Z}$ are independent identically distributed complex Gaussian random variables and $(c_k)$ is a sequence of coefficients. For any $t=\\mathcal O(\\varepsilon^{-1}|\\log\\varepsilon|)$ and any $z_0>0$, suppose one considers the large-deviation asymptotic\n\\[\n\\lim_{\\varepsilon\\to0^+}\\varepsilon\\log\\mathbb P\\Big(\\|u(t,\\cdot)\\|_{\\mathcal F L^1(\\mathbb T)}>z_0\\varepsilon^{-1/2}\\Big)\n= -\\frac{z_0^2}{\\sum_k c_k^2}.\n\\]\nWhich of the following statements is equivalent to this asymptotic formula?",
      "correct_choice": {
        "label": "A",
        "text": "The coefficient sequence satisfies $(c_k)\\in \\ell^1$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The coefficient sequence satisfies $(c_k)\\in \\ell^2$."
        },
        {
          "label": "C",
          "text": "The coefficient sequence satisfies $\\sum_k c_k^2<\\infty$."
        },
        {
          "label": "D",
          "text": "For each fixed $z_0>0$, there exists a time range $t=\\mathcal O(\\varepsilon^{-1}|\\log\\varepsilon|)$ on which the asymptotic formula holds whenever $(c_k)\\in \\ell^1\\cap \\ell^2$."
        },
        {
          "label": "E",
          "text": "The asymptotic formula holds if and only if the coefficient sequence satisfies $(c_k)\\in \\mathcal F L^1(\\mathbb T)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "sharp \\ell^1 threshold replaced by weaker square-summability",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the full equivalence and kept only the weaker consequence implied by $(c_k)\\in\\ell^1$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "replaced the iff criterion by a merely sufficient mixed assumption with altered quantifier structure",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "confused coefficient summability on $(c_k)$ with function-space membership notation",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the conclusion or single out the correct option. It gives the PDE, random data, norm, and time scale, but the actual asymptotic statement and the sharp \\(\\ell^1\\) condition appear only in the choices."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: it asks directly which limiting statement holds for the exact setup of a specific result. However, it is not a pure restatement because the options vary in logical strength (especially 'if' versus 'if and only if') and in the rate formula."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement (choice C) and from nearby false variants. Still, success depends heavily on recalling the theorem rather than deriving the answer from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing \\(\\ell^1\\) with \\(\\ell^2\\), dropping the 'only if' direction, mishandling the time-scale quantifier, and using an incorrect denominator. They are distinct and nontrivial."
      },
      "total_score": 6,
      "overall_assessment": "A solid, technically well-constructed MCQ with strong distractors and little answer leakage, but it remains fairly close to theorem recall rather than deeply generative reasoning."
    }
  },
  {
    "id": "2512.07839v1",
    "paper_link": "http://arxiv.org/abs/2512.07839v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: new theorem}\nFor any $m \\equiv 3 \\pmod{4}$ there exists $N$ such that for all $n \\geq N$ there exists an equilateral polygon of size n.",
      "start_pos": 4407,
      "end_pos": 4585,
      "label": "thm: new theorem"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 1699,
    "pre_theorem_intro_text": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.",
    "context": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.",
    "full_context": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.\n\n\\begin{abstract}\nWe study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ with $m \\equiv 3 \\pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \\geq N$, the lattice $\\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.\n\\end{abstract}\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.\n\nWe will give an existence proof in \\Cref{Section: Existence} followed by a explicit construction in \\Cref{Section: Construction}.\n\nFor any $m \\equiv 3 \\pmod{4}$ does there exist $N$ such that for all $n \\geq N$ there exists a \\textbf{convex} equilateral polygon of size n?\n\n\\begin{open} \nFor any $m \\equiv 3 \\pmod{4}$ does there exist an equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?\n\\end{open}\n\nFor any $m \\equiv 3 \\pmod{4}$ does there exist a \\textbf{convex} equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?",
    "post_theorem_intro_text_len": 129,
    "post_theorem_intro_text": "We will give an existence proof in \\Cref{Section: Existence} followed by a explicit construction in \\Cref{Section: Construction}.",
    "sketch": "The post-theorem introduction indicates the proof strategy: it will present \"an existence proof\" (in \\Cref{Section: Existence}) and then an \"explicit construction\" (in \\Cref{Section: Construction}).",
    "expanded_sketch": "The post-theorem introduction indicates the proof strategy: it will present \"an existence proof\" (in \\Cref{Section: Existence}) and then an \"explicit construction\" (in \\Cref{Section: Construction}).",
    "expanded_theorem": "\\label{thm: new theorem}\nFor any $m \\equiv 3 \\pmod{4}$ there exists $N$ such that for all $n \\geq N$ there exists an equilateral polygon of size n.,",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Let \\(\\Lambda(m)=L[(1,0),(0,\\sqrt m)]=\\{(a,b\\sqrt m):a,b\\in \\mathbb Z\\}\\) be the planar rectangular lattice associated to an integer \\(m\\equiv 3\\pmod 4\\). An equilateral polygon of size \\(n\\) means an \\(n\\)-gon whose vertices lie in \\(\\Lambda(m)\\) and whose side lengths are all equal. Which statement holds for such lattices?",
      "correct_choice": {
        "label": "A",
        "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains a convex equilateral polygon of size \\(n\\)."
        },
        {
          "label": "C",
          "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(n\\) such that the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."
        },
        {
          "label": "D",
          "text": "There exists an integer \\(N\\) such that for every integer \\(m\\equiv 3\\pmod 4\\) and every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."
        },
        {
          "label": "E",
          "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every odd integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\), and one may take \\(N\\) to be the largest prime factor of \\(m\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "equilateral_vs_convex_conclusion",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped_eventual_all_n_conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence_of_N_on_m",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "computational_check",
          "tampered_component": "threshold_equals_largest_prime_factor",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only defines the lattice and the notion of an equilateral polygon; it does not state or strongly hint at the eventual-for-all-large-n conclusion in choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to theorem recognition: choice A appears to be the target theorem statement, while the other options are nearby quantifier or strengthening/weakening variants. This is better than a pure restatement, but still only mildly non-tautological."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle ways (eventual existence vs mere existence, dependence of N on m, convexity, odd n only). However, the item mainly tests precise recall/discrimination of the theorem statement rather than substantial mathematical generation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: B is a plausible overstrengthening, C is a weaker true-looking statement, D tests quantifier dependence, and E adds a tempting but unjustified explicit threshold. They are distinct and reflect common theorem-misreading errors."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recognizing the exact statement than toward deep generative reasoning."
    }
  },
  {
    "id": "2512.07967v1",
    "paper_link": "http://arxiv.org/abs/2512.07967v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}",
      "start_pos": 4984,
      "end_pos": 5468,
      "label": "thm:mainthm"
    },
    "ref_dict": {
      "thm:mainthm": "\\begin{theorem}\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}\n\\end{theorem}",
      "Lagrapf": "\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}. \n\nThere is considerable interest in conditions implying that the characteristic cycle of\nthe intersection cohomology sheaf is irreducible. Lusztig (\\cite[13.7, p.~414]{MR1088333}) \nexpressed the `hope' that this may be the case for Schubert varieties in flag manifolds of \ntype A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in \n$F\\ell(8)$ (\\cite{MR1458969}, \\cite{MR1896039}), while it holds in $F\\ell(n)$ for $n\\le 7$.\nIrreducibility is also known for all Schubert \nvarieties in the standard Grassmannian (\\cite{MR1084458}), and more generally \nfor all Schubert varieties in cominuscule Grassmannians of \ntypes A, D, and E, see~\\cite{MR1451256, MihalceaSingh}.\n\nWe hope that Theorem~\\ref{thm:mainthm} may help in streamlining such verifications. \nFor instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve is trim, the irreducibility of the $\\IC$ characteristic cycle in this case (first established \nin \\cite{MR1642745}) follows directly from~Theorem~\\ref{thm:mainthm}.\n\nThe connection between the irreducibility of the $\\IC$ characteristic cycle and \nthe equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones\nin~\\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal \nindex formula of Dubson and Kashiwara. \nWe freely borrow ideas from~\\cite{MR2628830} \nin~\\S\\ref{thmpf}.\n\nA condition on fibers of small resolutions is considered in~\\cite{graham}, including a proof\nthat the condition implies the irreducibility of the IC characteristic cycles.\\medskip\n\n{\\em Acknowledgments.} This work was supported in part by an award from the Simons\nFoundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and\nLeonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this \nwork was carried out.\n\n\\section{Lagrangian push-forward}\\label{Lagrapf}\nLet $X$ be a nonsingular variety and denote by $T^* X$ the cotangent bundle of $X$.\nThe {\\em conormal variety\\/} $T^*_W X$ of a closed subvariety $W\\subseteq X$ is the \nclosure in $T^* X$ of the conormal variety to the nonsingular part $W^\\circ$ of $V$: \n$T^*_W X = \\overline{T^*_{W^\\circ} X}$. The conormal variety of $X$ itself is the\nzero-section $T^*_XX$ of the cotangent bundle. All conormal varieties have dimension\n$\\dim X$; they determine conormal {\\em cycles\\/} in $Z_{\\dim X} T^* X$.\nIt will also be convenient to take the projective completion of these constructions:\nwe will denote by $\\Tbb^* X$ the projective completion $\\Pbb(T^* X\\oplus \\one)$ \nof $T^* X$ (here, $\\Pbb$ denotes the projective bundle of lines) and by~$\\Tbb^*_WX$ \nthe closure of the conormal variety in $\\Tbb^* X$. \n\nFor a nonsingular variety $X$, we denote by $\\cL (X)$ the free abelian group of \n{\\em conical Lagrangian cycles\\/} in the cotangent bundle $T^* X$.\nConormal cycles are conical Lagrangian, and in fact (cf.~\\cite[Lemma~3]{MR1063344}) \n$\\cL (X)$ may be realized as the free abelian group on conormal cycles. \nFor a closed (and possibly singular) subvariety $V\\subseteq X$, we denote by\n$\\cL(V)$ the subgroup of $\\cL(X)$ generated by the conormal cycles $T^*_W X$\nwith $W\\subseteq V$. Thus, elements of~$\\cL (V)$ may be viewed as finite integer linear \ncombinations $\\sum_W m_W T^*_WX$ ranging over closed subvarieties $W$ of $V$.\nClearly $\\cL(V)$ is isomorphic to the group of algebraic cycles of~$V$, and in particular\nit is independent of the ambient nonsingular variety $X$.\n\nImportant invariants of $V$ may be expressed directly in terms of Lagrangian cycles \nby means of intersection-theoretic operations, after taking the projective completion. \nAs above, realize $V$ as a closed subvariety of a\nnonsingular variety $X$; the results will be independent of the choice of $X$. \nLet $\\pi: \\Tbb^*_V X\\to V$ be the natural projection; and let $\\cO(1)$ be the \ntautological line bundle on $\\Tbb^* X=\\Pbb(T^* X\\oplus \\one)$.\n\\begin{itemize}\n\\item[---]\nThe {\\em local Euler obstruction\\/} $\\Eu_V: V\\to \\Zbb$ is\n\\[\n\\Eu_V(p)= (-1)^{\\dim X-\\dim V} \\int c(\\pi^* TX|_V) c(\\cO(1))^{-1}\\cap s(\\pi^{-1}(p), \\Tbb^*_V X)\n\\]\nwhere $s(\\pi^{-1}(p), \\Tbb^*_V X)$ denotes the {\\em Segre class\\/} in the sense \nof~\\cite[Chapter~4]{85k:14004};\n\\item[---]\nThe {\\em Chern-Mather class\\/} $\\cma(V)\\in A_*V$ is\n\\[\n\\cma(V)=(-1)^{\\dim X-\\dim V} c(TX|_V)\\cap \\pi_* \\left(c(\\cO(1))^{-1}\\cap [\\Tbb^*_V X]\\right)\\saf.\n\\]\n\\end{itemize}\nEquivalent results were established by C.~Sabbah (\\cite[1.2.1, 1.2.2]{MR804052}); we also\naddress the reader to~\\cite{MR1063344} and~\\cite{MR2002g:14005} for clear treatments of these\nformulas. \n\nLet $\\cF(V)$ denote the abelian group of constructible functions $V\\to \\Zbb$.\nFollowing~\\cite[\\S1]{MR2002g:14005}, the relation between conormal cycles and local\nEuler obstructions is recorded by the homomorphism\n\\[\n\\CC: \\cF(V) \\to \\cL(V)\n\\]\ndefined by prescribing\n\\[\n\\Eu_W \\mapsto (-1)^{\\dim W} T^*_W X\n\\]\nfor all closed subvarieties $W$ of $V$. \nIn fact, $\\CC$ is an {\\em isomorphism:\\/} it simply matches a basis of $\\cF(V)$ with a \nbasis of $\\cL(V)$.\n\n\\begin{defin}\\label{def:CC}\nThe {\\em characteristic cycle\\/} $\\CC(\\alpha)$ of a constructible function $\\alpha$ on $V$ \nis the image of $\\alpha$ in $\\cL(V)$ under this isomorphism. \n\\qede\\end{defin}\n\nLikewise, the formula for Chern-Mather classes motivates the introduction of a homomorphism\n\\[\nc_*:\\cL(V) \\to A_*(V)\n\\]\ndefined on generators by\n\\[\nT^*_W X \\mapsto (-1)^{\\dim W} \\cma(W)\\saf.\n\\]\nThen the composition $c_*\\circ \\CC:\\cF(V) \\to A_*(V)$ agrees with the value at $V$ \nof MacPherson's natural transformation $\\cF \\leadsto A_*$, where $\\cF$ is taken as \na functor with push-forward defined by Euler characteristics of fibers; \nsee~\\cite{MR0361141} and~\\cite[Example 19.1.7]{85k:14004}. \nFor instance, the {\\em Chern-Schwartz-MacPherson class\\/} of a (possibly singular)\nalgebraic variety $V$ is the image $c_*(\\CC(\\one_V))\\in A_*V$ of the characteristic \ncycle in~$\\cL(V)$ of the constant function $\\one_V$.\nSabbah provides an alternative proof of the naturality of this assignment, which is\nthe main result of~\\cite{MR0361141}, by defining a covariant {\\em push-forward\\/}\n\\[\n\\varphi_*: \\cL(V') \\to \\cL(V'')\n\\]\nfor every proper map $\\varphi: V' \\to V''$, making $\\cL$ into a functor, in such a way \nthat the above homomorphisms define natural transformations\n\\[\n\\cF \\leadsto \\cL\\quad, \\quad \\cL \\leadsto A_*\n\\]\nwhose composition agrees with MacPherson's natural transformation.\n\nWe are interested in explicit formulas for this {\\em Lagrangian push-forward.\\/}\nAfter embedding~$V''$ in a nonsingular variety $X$ and replacing $V'$ by a resolution $Y$,\nwe can reduce to the case of a proper morphism $f:Y \\to X$ of nonsingular varieties. \nWe are specifically interested in the image $f_*(T^*_YY)$ of the zero-section in this\nsituation. \n\nTheorem~\\ref{thm:mainthm} will be a consequence of the following result. We recall the \ndefinition of `trim' given in the introduction.\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}\n\n\\begin{remark}\\label{rem:YtoS}\nLet $\\pi: Y\\to S$ be a proper surjective morphism, with $Y$ nonsingular, and \n$\\iota:S\\hookrightarrow X$ a closed embedding, with $X$ nonsingular; and let \n$f=\\iota\\circ \\pi: Y \\to X$ be the composition. \nAs explained above, $T^*_{f(Y)} X=T^*_S X$ may be viewed as an element of $\\cL(S)$;  \nProposition~\\ref{prop:Sabpf} states that if $\\pi$ is trim, then this is the image $\\pi_*(T^*_YY)$ \nunder $\\pi_*: \\cL(Y)\\to \\cL(S)$.\n\\qede\\end{remark}\n\nThe rest of this section is devoted to the proof of Proposition~\\ref{prop:Sabpf}.\\smallskip\n\n{\\em A priori,\\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles\n$\\sum_Z m_Z T^*_ZX$, with $Z\\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.\nThe task is to show that if $Y\\to f(Y)$ is trim and $Z\\subsetneq f(Y)$ is a {\\em proper\\/} \nsubvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.\nFollowing Sabbah, we write $f$ as the composition\n\\[\n\\xymatrix{\nY \\ar[r]^-\\gamma & Y\\times X \\ar[r]^-\\rho & X\n}\n\\]\nwhere $\\gamma$ is the graph of $f$ and $\\rho$ is the (smooth) projection.\nLet $p: \\rho^*(T^* X) \\to T^* X$ be the natural mophism. Then  (cf.~\\cite[\\S2]{MR804052})\nthe components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the\nirreducible components of the image\n\\begin{equation}\\label{eq:pint}\np \\left(\\rho^*(T^* X) \\cap T^*_{\\gamma(Y)}(Y\\times X)\\right)\n\\end{equation}",
      "eq:strMa": "\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}",
      "thmpf": "\\label{thmpf}\n\nLet $S$ be a variety admitting a trim resolution $\\pi: Y\\to S$; let $\\iota: S\\hookrightarrow X$\nbe an embedding in a nonsingular variety, and let $f=\\iota\\circ \\pi:Y \\to X$ be the compo",
      "Furrem": "\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co",
      "prop:Sabpf": "\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}",
      "prop:consma": "\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}",
      "def:contained": "\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}"
    },
    "pre_theorem_intro_text_len": 1160,
    "pre_theorem_intro_text": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.",
    "context": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.\n\n\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}",
    "full_context": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.\n\n\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}\n\n\\begin{abstract}\nWe introduce {\\em trim\\/} resolutions of complex algebraic varieties, a strengthening \nof the notion of small resolution. We prove that the characteristic cycle of the intersection \ncohomology sheaf of a variety admitting a trim resolution is irreducible and that for \nsuch varieties the stringy and Chern-Mather classes coincide.\n\\end{abstract}\n\nOur main result is the following.\n\nThe definition of characteristic cycle will be recalled below (\\S\\ref{Lagrapf}).\n`Stringy' Chern classes were defined in~\\cite{MR2183846, MR2304329} \n(see~\\cite{MR2280127} for a lean account). If $\\pi: Y\\to S$ is a crepant resolution, \n$\\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle\nof~$Y$; the equality stated in the theorem is\n\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}.\n\nImportant invariants of $V$ may be expressed directly in terms of Lagrangian cycles \nby means of intersection-theoretic operations, after taking the projective completion. \nAs above, realize $V$ as a closed subvariety of a\nnonsingular variety $X$; the results will be independent of the choice of $X$. \nLet $\\pi: \\Tbb^*_V X\\to V$ be the natural projection; and let $\\cO(1)$ be the \ntautological line bundle on $\\Tbb^* X=\\Pbb(T^* X\\oplus \\one)$.\n\\begin{itemize}\n\\item[---]\nThe {\\em local Euler obstruction\\/} $\\Eu_V: V\\to \\Zbb$ is\n\\[\n\\Eu_V(p)= (-1)^{\\dim X-\\dim V} \\int c(\\pi^* TX|_V) c(\\cO(1))^{-1}\\cap s(\\pi^{-1}(p), \\Tbb^*_V X)\n\\]\nwhere $s(\\pi^{-1}(p), \\Tbb^*_V X)$ denotes the {\\em Segre class\\/} in the sense \nof~\\cite[Chapter~4]{85k:14004};\n\\item[---]\nThe {\\em Chern-Mather class\\/} $\\cma(V)\\in A_*V$ is\n\\[\n\\cma(V)=(-1)^{\\dim X-\\dim V} c(TX|_V)\\cap \\pi_* \\left(c(\\cO(1))^{-1}\\cap [\\Tbb^*_V X]\\right)\\saf.\n\\]\n\\end{itemize}\nEquivalent results were established by C.~Sabbah (\\cite[1.2.1, 1.2.2]{MR804052}); we also\naddress the reader to~\\cite{MR1063344} and~\\cite{MR2002g:14005} for clear treatments of these\nformulas.\n\n{\\em A priori,\\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles\n$\\sum_Z m_Z T^*_ZX$, with $Z\\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.\nThe task is to show that if $Y\\to f(Y)$ is trim and $Z\\subsetneq f(Y)$ is a {\\em proper\\/} \nsubvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.\nFollowing Sabbah, we write $f$ as the composition\n\\[\n\\xymatrix{\nY \\ar[r]^-\\gamma & Y\\times X \\ar[r]^-\\rho & X\n}\n\\]\nwhere $\\gamma$ is the graph of $f$ and $\\rho$ is the (smooth) projection.\nLet $p: \\rho^*(T^* X) \\to T^* X$ be the natural mophism. Then (cf.~\\cite[\\S2]{MR804052})\nthe components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the\nirreducible components of the image\n\\begin{equation}\\label{eq:pint}\np \\left(\\rho^*(T^* X) \\cap T^*_{\\gamma(Y)}(Y\\times X)\\right)\n\\end{equation}\nwhere we view both $\\rho^*(T^* X)$ and $T^*_{\\gamma(Y)}(Y\\times X)$\nas subschemes of $T^* (Y\\times X)$. \nLet $T^*_ZX$ be one such component, and let $z$ be a general point of $Z$\nand $\\xi$ a general covector in the fiber $(T^*_ZX)_z$. For $(z,\\xi)$ to be in\nthe image~\\eqref{eq:pint}, there must be a point $(y,z)\\in \\gamma(Y)$ mapping to \n$z$, such that \n$\n(0,\\xi)\\in T^*_y Y\\oplus T^*_z X\\cong T^*_{(y,z)} (Y\\times X)\n$\nvanishes on $T_{(y,z)}\\gamma(Y)$.\n\nBy Proposition~\\ref{prop:consma}, $\\pi$ is small. It follows (cf.~\\cite[\\S6.2]{MR696691})\nthat the intersection cohomology sheaf $\\IC^\\bullet_S$ is the direct image of the intersection \ncohomology sheaf of $Y$. Since $Y$ is nonsingular, the latter is a shift of the constant\nsheaf. Therefore,\n\\[\n\\IC^\\bullet_S = R\\pi_* \\Qbb_Y[\\dim Y]\\saf.\n\\]\n(Concerning the shift, we follow the modern convention as in e.g., \n\\cite[Remark~4.2.4]{MR2525735}.)\nThe following formula for the stalk Euler characteristic of $\\IC^\\bullet_S$ at $z\\in S$\nis a consequence of standard properties of (derived) direct images:\n\\[\n\\chi_z(\\IC^\\bullet_S) = \n\\chi_z(R\\pi_* \\Qbb_Y[\\dim Y]) =\n(-1)^{\\dim S} \\sum_i (-1)^i \\dim H^i(\\pi^{-1}(z);\\Qbb) \n= (-1)^{\\dim S}\\chi(\\pi^{-1}(z))\\saf.\n\\]\nBy the first assertion in Theorem~\\ref{thm:mainthm}, this implies \n\\[\n\\chi_z(\\IC^\\bullet_S) = (-1)^{\\dim S}\\Eu_S(z)\\saf.\n\\]\nNow embed $S$ in a nonsingular variety $X$. \nThe characteristic cycle of $\\IC^\\bullet_S$ is\n\\[\nCC(\\chi_z(\\IC^\\bullet_S)) = CC((-1)^{\\dim S}\\Eu_S(z)) = T^*_SX\\saf,\n\\]\nand this verifies it is irreducible, completing the proof of \nTheorem~\\ref{thm:mainthm}.",
    "post_theorem_intro_text_len": 4013,
    "post_theorem_intro_text": "The definition of characteristic cycle will be recalled below (\\S\\ref{Lagrapf}).\n`Stringy' Chern classes were defined in~\\cite{MR2183846, MR2304329} \n(see~\\cite{MR2280127} for a lean account). If $\\pi: Y\\to S$ is a crepant resolution, \n$\\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle\nof~$Y$; the equality stated in the theorem is\n\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}. \n\nThere is considerable interest in conditions implying that the characteristic cycle of\nthe intersection cohomology sheaf is irreducible. Lusztig (\\cite[13.7, p.~414]{MR1088333}) \nexpressed the `hope' that this may be the case for Schubert varieties in flag manifolds of \ntype A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in \n$F\\ell(8)$ (\\cite{MR1458969}, \\cite{MR1896039}), while it holds in $F\\ell(n)$ for $n\\le 7$.\nIrreducibility is also known for all Schubert \nvarieties in the standard Grassmannian (\\cite{MR1084458}), and more generally \nfor all Schubert varieties in cominuscule Grassmannians of \ntypes A, D, and E, see~\\cite{MR1451256, MihalceaSingh}.\n\nWe hope that Theorem~\\ref{thm:mainthm} may help in streamlining such verifications. \nFor instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve is trim, the irreducibility of the $\\IC$ characteristic cycle in this case (first established \nin \\cite{MR1642745}) follows directly from~Theorem~\\ref{thm:mainthm}.\n\nThe connection between the irreducibility of the $\\IC$ characteristic cycle and \nthe equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones\nin~\\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal \nindex formula of Dubson and Kashiwara. \nWe freely borrow ideas from~\\cite{MR2628830} \nin~\\S\\ref{thmpf}.\n\nA condition on fibers of small resolutions is considered in~\\cite{graham}, including a proof\nthat the condition implies the irreducibility of the IC characteristic cycles.\\medskip\n\n{\\em Acknowledgments.} This work was supported in part by an award from the Simons\nFoundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and\nLeonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this \nwork was carried out.",
    "sketch": "Theorem~\\ref{thm:mainthm} is said to be “a consequence of considerations concerning Sabbah's formalism of {\\em conical Lagrangian cycles,\\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\\pi:Y\\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves (Proposition~\\ref{prop:Sabpf}) that for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown in~\\S\\ref{thmpf}.",
    "expanded_sketch": "To prove the main theorem, it is said to be “a consequence of considerations concerning Sabbah's formalism of {\\em conical Lagrangian cycles,\\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\\pi:Y\\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves the following proposition.\n\n\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}\n\nThat is, for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown later.\n\\label{thmpf}\n\nLet $S$ be a variety admitting a trim resolution $\\pi: Y\\to S$; let $\\iota: S\\hookrightarrow X$\nbe an embedding in a nonsingular variety, and let $f=\\iota\\circ \\pi:Y \\to X$ be the compo",
    "expanded_theorem": "\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}",
    "theorem_type": [
      "Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let $S$ be a complex algebraic variety that admits a trim resolution, i.e. there exists a proper birational morphism $\\pi:Y\\to S$ with $Y$ nonsingular such that, for each $0\\le d\\le \\dim Y$, if $Y_d\\subseteq Y$ denotes the locus where the differential $d\\pi$ has rank $d$, then $\\dim Y_d<d$ for every $d<\\dim Y$. In this setting, write $\\Eu_S(s)$ for the local Euler obstruction at $s\\in S$, write $\\cma(S)$ for the Chern--Mather class of $S$, and write $\\cstr(S)$ for the stringy Chern class (here, for a trim resolution $\\pi$, one may take $\\cstr(S)=\\pi_*(c(TY)\\cap [Y])$). Which statement holds for every such variety $S$?",
      "correct_choice": {
        "label": "A",
        "text": "For every trim resolution $\\pi:Y\\to S$ and every point $s\\in S$, one has $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every trim resolution $\\pi:Y\\to S$ and every point $s\\in S$, one has $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is the push-forward of the zero-section $T^*_Y Y$ for any trim resolution, hence may have several irreducible components."
        },
        {
          "label": "C",
          "text": "For every trim resolution $\\pi:Y\\to S$ and every point $s\\in S$, one has $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$."
        },
        {
          "label": "D",
          "text": "For every proper birational morphism $\\pi:Y\\to S$ with $Y$ nonsingular such that $\\dim Y_d\\le d$ for every $d<\\dim Y$, and for every point $s\\in S$, one has $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible."
        },
        {
          "label": "E",
          "text": "For every complex algebraic variety $S$ admitting a trim resolution, there exists a trim resolution $\\pi:Y\\to S$ such that for every $s\\in S$ one has $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$, and for this choice of $\\pi$ one has $\\cstr(S)=\\cma(S)$ in the Chow group of $S$; however the characteristic cycle of the intersection cohomology sheaf of $S$ need not be irreducible."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "irreducibility_from_lagrangian_pushforward",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "dropped_characteristic_cycle_irreducible_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "strict_trim_inequality",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "for_every_resolution_vs_there_exists_and_irreducibility_conclusion",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem sets up definitions and hypotheses but does not itself state the target conclusion. There is no direct textual leakage of the correct option from the stem alone."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: it asks which packaged conclusion holds under the stated trim-resolution hypothesis. The alternatives add quantifier and strength variations, so it is not purely tautological, but it remains a mild reformulation of a likely source theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish the strongest correct statement from a weaker true-looking option and from quantifier/condition tamperings. However, for someone who knows the theorem, the answer is mostly recognition rather than substantial generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: dropping a clause, weakening 'for every' to 'there exists', altering the trim inequality, and modifying the characteristic-cycle conclusion. They are distinct and nontrivial."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning."
    }
  },
  {
    "id": "2512.08275v1",
    "paper_link": "http://arxiv.org/abs/2512.08275v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.",
      "start_pos": 7135,
      "end_pos": 7553,
      "label": "main theorem"
    },
    "ref_dict": {
      "main theorem": "\\begin{theorem}\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.\n\\end{theorem}",
      "def1-9-24": "\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 2517,
    "pre_theorem_intro_text": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an  invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved  that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor)  which is also biholomorphically invariant.\nLater,  Mok and Yau \\cite{MY80}  removed the  boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important  invariant metrics  coincide. A classical conjecture  of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain.  Earlier,\nCheng \\cite{C79}  had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were  also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded  pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in  $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in  $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with  rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.",
    "context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}",
    "full_context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been used to work on many related \nproblems. In particular, in connection with our present work, we \nmention the papers by Wong \\cite{W77},\nKim \\cite{Kim}, Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\\\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nWe next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\\Omega\\subset{\\mathbb C}^n$.\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$ with $n>1$ and let $p\\in\\partial\\Omega$. \nWe say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$ such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$ and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$ are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\n\\begin{proposition}\\label{prop1-4-3}\nLet $\\Omega$ be a possibly unbounded pseudoconvex domain in $\\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\\in\\partial\\Omega$. Let $U$ be a neighborhood of $p$ such that $U\\cap\\Omega$ is connected, and on which the Bergman metric $g_{\\Omega}$ is well defined. Then its Bergman metric $g_\\Omega$ is K\\\"ahler-Einstein on $U\\cap\\Omega$ if and only if its Bergman canonical invariant $J_{\\Omega}\\equiv (n+1)^n\\frac{\\pi^n}{n!}$ on $U\\cap\\Omega$. \n\\end{proposition}\n\\begin{proof}\nBy Corollary \\ref{coro1-3-30}, for any smooth boundary point $q\\in U\\cap\\partial\\Omega$ near which $\\partial\\Omega$ is strongly pseudoconvex, one has \n \\begin{equation}\\label{e2-3-31}\n \\lim_{z\\rightarrow q}J_{\\Omega}(z)=\\frac{(n+1)^n\\pi^n}{n!}.\n \\end{equation}\n The Bergman metric $g_{\\Omega}$ {is} K\\\"ahler-Einstein if its Ricci curvature $R_{\\Omega}=c g_{\\Omega}$ for some constant $c$. By Corollary \\ref{coro1-3-30}, one has $c=-1$. Consequently, the K\\\"ahler-Einstein assumption implies that $\\log J_{\\Omega}$ is a pluriharmonic function on $U\\cap\\Omega$. Now, for any attached holomorphic disk $\\phi:\\Delta\\rightarrow\\Omega$ where $\\phi$ is holomorphic in $\\Delta:=\\{t\\in {\\mathbb C}: |t|<1\\}$, continuous up to $\\overline {\\Delta}$, and $\\phi(\\partial\\Delta)$ is contained in the smooth part of $ U\\cap \\partial\\Omega$, we have that $\\log\n J_{\\Omega}(\\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \\eqref{e2-3-31}, it assumes the value\n\\[\n\\log\\frac{(n+1)^n\\pi^n}{n!}\n\\]\neverywhere on $\\Delta$. Now, since $\\partial \\Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\\partial\\Omega$ near $q$. Since $\\log\n J_{\\Omega}$ is well defined in $U\\cap\\Omega$ on which it is real analytic, we conclude that $\\log J_{\\Omega}\\equiv\n \\log\\frac{(n+1)^n\\pi^n}{n!}$ over $U\\cap\\Omega$ as $U\\cap\\Omega$ is connected by definition. \nConversely, if $J_\\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\\\"ahler-Einstien. Thus, we have the conclusion of the proposition.\n\\end{proof}\n\\begin{remark}\\label{remark1-10-21}\n\nNote that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\\Omega$. \nThus, $J_\\Omega$ is a well-defined real-analytic function on $\\Omega\\setminus E$. Since $\\Omega\\setminus E$ is connected, $J_\\Omega$ is constant if and only if it is constant on some nonempty open subset of $\\Omega$. In particular, when $\\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\\Omega=c$ is a constant on a certain open subset of $\\Omega\\setminus E$ .\n In this case, $c=\\frac{(n+1)^n\\pi^n}{n!}$, and the Bergman space $A^2(\\Omega)$ separates holomorphic directions at any point in $\\Omega\\setminus E$ and thus the Bergman metric is well-defined in $\\Omega\\setminus E$.\n\\end{remark}\n\\section{Stability of the Bergman kernels}\\label{sec2}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nLet $\\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\\in\\partial\\Omega$ as defined in Definition~\\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that\n\\begin{enumerate}\n\\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\\{\\rho_{j}\\}_{j=1}^{m}$ with $m>1$ on $U$ such that\n\\begin{equation}\\label{9-24-a2}\n U\\cap\\Omega=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\};\n\\end{equation}\n\\item[(ii)] the vectors $\\partial\\rho_{1}(q),\\dots,\\partial\\rho_{m}(q)$ are linearly independent over $\\mathbb C$ for $q\\in U\\cap \\overline\\Omega$.\n\\end{enumerate}\nBy Lemma \\ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\\Subset U$ of $p$, we may assume that $p=0$ and \n\\begin{equation}\\label{9-24-a1}\nV\\cap\\Omega=\\{(z_1,\\cdots, z_n)\\in V: {\\rm Im~ }z_1>\\Phi_1(z, \\overline z), \\cdots, {\\rm Im}~ z_m>\\Phi_m(z, \\overline z)\\}\n\\end{equation}\nwhere $$\\Phi_1(z, \\overline z)=|z|^2+R_1(z), \\Phi_j(z, \\overline z)=\\sum a^j_{\\alpha\\overline\\beta} z_{\\alpha}\\overline z_{\\beta}+R_j(z), \\quad 2\\leq j\\leq m$$\nwith each remainder $R_j=\\mathcal O(|z|^3)$. Write $$U_0=\\{z\\in\\mathbb C^n: |z_j|<\\varepsilon_0, 1\\leq j\\leq n\\}$$ with $\\varepsilon_0\\ll 1$ such that $U_0\\Subset V$ and on $U_0$ one has\n$$|R_j(z)|\\leq \\frac{A_0}{2}|z|^2, \\forall z\\in U_0, 1\\leq j\\leq m,$$\nwhere $A_0$ is defined in (\\ref{10-27-a3}).\nWe first construct a bounded continuous plurisubharmonic function $\\psi$ in $\\Omega$ where $\\psi$ is strictly plurisubharmonic near $p=0$ as follows: \n\\begin{lemma}\\label{le3-28-1}\nAfter shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\\psi:\\Omega\\rightarrow (-\\infty, 0)$ such that $$\\psi (z)>-c_0, ~\\left(\\frac{\\partial^2\\psi(z)}{\\partial z_j\\partial\\overline z_k}\\right)\\geq c ~I_{n}, ~~z\\in U_0\\cap\\Omega$$for some constants $c_0>0,~c>0$.\n\\end{lemma}\n\\begin{proof}\nLet $(V, z)$ be the coordinates given as in (\\ref{9-24-a1}) and set $\\varphi=\\frac{A_0}4|z|^2-y_1$. Then $\\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\\varphi(0)=0, \\varphi(z)<0$ when $z\\in U_0\\cap\\overline\\Omega\\setminus \\{0\\}$. Take $r>0$ such that $\\overline {\\mathbb B^n(0, r)}\\Subset U_0$. Set $M=\\max\\{\\varphi(z): z\\in \\partial \\mathbb B^n(0, r)\\cap\\overline \\Omega\\}$. Then $M<0$. Now we define $\\psi$ as follows:\n\\begin{equation}\n \\psi=\\begin{cases}\n \\max\\{\\varphi(z), M\\}, &z\\in \\mathbb B^n(0, r)\\cap\\Omega\\\\\n M, &z\\in\\Omega\\setminus \\mathbb B^n(0, r).\n \\end{cases}\n\\end{equation}\nThen $\\psi$ is a bounded and continuous plurisubharmonic function on $\\Omega$ with $$\\psi(0)=0, ~~\\psi(z)<0, \\forall z\\in\\overline\\Omega\\setminus\\{0\\}.$$\nFurthermore, $\\psi$ is equal to $\\varphi$ near $0$ with $$\\left(\\frac{\\partial^2\\psi}{\\partial z_i\\partial\\overline z_j}\\right)= \\frac{A_0}4 I_n $$\nin some neighborhood $U$ of $0$ in $\\mathbb C^n$. Moreover, $\\psi>-c_0$ on $U$ for some positive constant $c_0$.\n\\end{proof}",
    "post_theorem_intro_text_len": 2280,
    "post_theorem_intro_text": "\\begin{corollary}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.\n\\end{corollary}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been  used  to work on many related \nproblems. In particular, in connection with our present work, we \nmention   the papers by Wong \\cite{W77},\nKim \\cite{Kim},  Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\\\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nThe  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.  \nTo obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\\Omega$.\nA direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \\cite{HX20}.",
    "sketch": "The ideas of the proof of Theorem~\\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”",
    "expanded_sketch": "The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”",
    "expanded_theorem": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.",
    "theorem_type": [
      "Implication",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb C^n\\) with \\(n>1\\) be a pseudoconvex domain, possibly unbounded. Assume that \\(\\Omega\\) has a strongly pseudoconvex polyhedral boundary point \\(p\\in \\partial\\Omega\\), meaning that there exist a neighborhood \\(U\\) of \\(p\\) in \\(\\mathbb C^n\\) and \\(C^2\\)-smooth strongly plurisubharmonic functions \\(\\rho_1,\\dots,\\rho_m:U\\to\\mathbb R\\) with \\(m>1\\) such that\n\\[\n\\Omega\\cap U=\\{z\\in U: \\rho_1(z)<0,\\dots,\\rho_m(z)<0\\},\n\\]\nand the complex differentials \\(\\{\\partial\\rho_1|_p,\\dots,\\partial\\rho_m|_p\\}\\) are linearly independent over \\(\\mathbb C\\). Under these assumptions, which statement about the Bergman metric of \\(\\Omega\\) holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \\(\\Omega^*\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and if the Bergman metric is Einstein on some open subset of \\(\\Omega^*\\), then \\(\\Omega\\) is biholomorphic to the unit ball."
        },
        {
          "label": "C",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined."
        },
        {
          "label": "D",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \\(\\Omega^*\\)."
        },
        {
          "label": "E",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman invariant function on \\(\\Omega^*\\) is constant and equal to that of the unit ball."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "conclusion-from-contradiction replaced by global classification",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the non-Einstein conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "quantifier on where Einstein fails",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "computational_check",
          "tampered_component": "constant-ball invariant asserted unconditionally instead of under Einstein hypothesis",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and relevant definitions but does not explicitly or implicitly reveal the correct conclusion. No option is singled out by wording in the prompt."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall/application question: the correct choice appears to match the intended theorem conclusion closely. However, it is not a pure tautology because the distractors include stronger, weaker, and altered statements."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact supported conclusion from a weaker true statement and stronger false variants, but the problem mainly tests recognition of the theorem rather than deeper generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one is weaker-but-true, one is an overstrong rigidity claim, one reverses the quantifier structure, and one asserts an unjustified ball-like invariant. These reflect realistic failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with good distractors and no answer leakage, but it remains mostly a theorem-recall item rather than a strongly generative reasoning question."
    }
  },
  {
    "id": "2512.08275v1",
    "paper_link": "http://arxiv.org/abs/2512.08275v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.",
      "start_pos": 7135,
      "end_pos": 7553,
      "label": "main theorem"
    },
    "ref_dict": {
      "main theorem": "\\begin{theorem}\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.\n\\end{theorem}",
      "def1-9-24": "\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 2517,
    "pre_theorem_intro_text": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an  invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved  that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor)  which is also biholomorphically invariant.\nLater,  Mok and Yau \\cite{MY80}  removed the  boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important  invariant metrics  coincide. A classical conjecture  of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain.  Earlier,\nCheng \\cite{C79}  had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were  also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded  pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in  $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in  $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with  rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.",
    "context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}",
    "full_context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been used to work on many related \nproblems. In particular, in connection with our present work, we \nmention the papers by Wong \\cite{W77},\nKim \\cite{Kim}, Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\\\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nWe next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\\Omega\\subset{\\mathbb C}^n$.\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$ with $n>1$ and let $p\\in\\partial\\Omega$. \nWe say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$ such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$ and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$ are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\n\\begin{proposition}\\label{prop1-4-3}\nLet $\\Omega$ be a possibly unbounded pseudoconvex domain in $\\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\\in\\partial\\Omega$. Let $U$ be a neighborhood of $p$ such that $U\\cap\\Omega$ is connected, and on which the Bergman metric $g_{\\Omega}$ is well defined. Then its Bergman metric $g_\\Omega$ is K\\\"ahler-Einstein on $U\\cap\\Omega$ if and only if its Bergman canonical invariant $J_{\\Omega}\\equiv (n+1)^n\\frac{\\pi^n}{n!}$ on $U\\cap\\Omega$. \n\\end{proposition}\n\\begin{proof}\nBy Corollary \\ref{coro1-3-30}, for any smooth boundary point $q\\in U\\cap\\partial\\Omega$ near which $\\partial\\Omega$ is strongly pseudoconvex, one has \n \\begin{equation}\\label{e2-3-31}\n \\lim_{z\\rightarrow q}J_{\\Omega}(z)=\\frac{(n+1)^n\\pi^n}{n!}.\n \\end{equation}\n The Bergman metric $g_{\\Omega}$ {is} K\\\"ahler-Einstein if its Ricci curvature $R_{\\Omega}=c g_{\\Omega}$ for some constant $c$. By Corollary \\ref{coro1-3-30}, one has $c=-1$. Consequently, the K\\\"ahler-Einstein assumption implies that $\\log J_{\\Omega}$ is a pluriharmonic function on $U\\cap\\Omega$. Now, for any attached holomorphic disk $\\phi:\\Delta\\rightarrow\\Omega$ where $\\phi$ is holomorphic in $\\Delta:=\\{t\\in {\\mathbb C}: |t|<1\\}$, continuous up to $\\overline {\\Delta}$, and $\\phi(\\partial\\Delta)$ is contained in the smooth part of $ U\\cap \\partial\\Omega$, we have that $\\log\n J_{\\Omega}(\\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \\eqref{e2-3-31}, it assumes the value\n\\[\n\\log\\frac{(n+1)^n\\pi^n}{n!}\n\\]\neverywhere on $\\Delta$. Now, since $\\partial \\Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\\partial\\Omega$ near $q$. Since $\\log\n J_{\\Omega}$ is well defined in $U\\cap\\Omega$ on which it is real analytic, we conclude that $\\log J_{\\Omega}\\equiv\n \\log\\frac{(n+1)^n\\pi^n}{n!}$ over $U\\cap\\Omega$ as $U\\cap\\Omega$ is connected by definition. \nConversely, if $J_\\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\\\"ahler-Einstien. Thus, we have the conclusion of the proposition.\n\\end{proof}\n\\begin{remark}\\label{remark1-10-21}\n\nNote that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\\Omega$. \nThus, $J_\\Omega$ is a well-defined real-analytic function on $\\Omega\\setminus E$. Since $\\Omega\\setminus E$ is connected, $J_\\Omega$ is constant if and only if it is constant on some nonempty open subset of $\\Omega$. In particular, when $\\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\\Omega=c$ is a constant on a certain open subset of $\\Omega\\setminus E$ .\n In this case, $c=\\frac{(n+1)^n\\pi^n}{n!}$, and the Bergman space $A^2(\\Omega)$ separates holomorphic directions at any point in $\\Omega\\setminus E$ and thus the Bergman metric is well-defined in $\\Omega\\setminus E$.\n\\end{remark}\n\\section{Stability of the Bergman kernels}\\label{sec2}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nLet $\\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\\in\\partial\\Omega$ as defined in Definition~\\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that\n\\begin{enumerate}\n\\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\\{\\rho_{j}\\}_{j=1}^{m}$ with $m>1$ on $U$ such that\n\\begin{equation}\\label{9-24-a2}\n U\\cap\\Omega=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\};\n\\end{equation}\n\\item[(ii)] the vectors $\\partial\\rho_{1}(q),\\dots,\\partial\\rho_{m}(q)$ are linearly independent over $\\mathbb C$ for $q\\in U\\cap \\overline\\Omega$.\n\\end{enumerate}\nBy Lemma \\ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\\Subset U$ of $p$, we may assume that $p=0$ and \n\\begin{equation}\\label{9-24-a1}\nV\\cap\\Omega=\\{(z_1,\\cdots, z_n)\\in V: {\\rm Im~ }z_1>\\Phi_1(z, \\overline z), \\cdots, {\\rm Im}~ z_m>\\Phi_m(z, \\overline z)\\}\n\\end{equation}\nwhere $$\\Phi_1(z, \\overline z)=|z|^2+R_1(z), \\Phi_j(z, \\overline z)=\\sum a^j_{\\alpha\\overline\\beta} z_{\\alpha}\\overline z_{\\beta}+R_j(z), \\quad 2\\leq j\\leq m$$\nwith each remainder $R_j=\\mathcal O(|z|^3)$. Write $$U_0=\\{z\\in\\mathbb C^n: |z_j|<\\varepsilon_0, 1\\leq j\\leq n\\}$$ with $\\varepsilon_0\\ll 1$ such that $U_0\\Subset V$ and on $U_0$ one has\n$$|R_j(z)|\\leq \\frac{A_0}{2}|z|^2, \\forall z\\in U_0, 1\\leq j\\leq m,$$\nwhere $A_0$ is defined in (\\ref{10-27-a3}).\nWe first construct a bounded continuous plurisubharmonic function $\\psi$ in $\\Omega$ where $\\psi$ is strictly plurisubharmonic near $p=0$ as follows: \n\\begin{lemma}\\label{le3-28-1}\nAfter shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\\psi:\\Omega\\rightarrow (-\\infty, 0)$ such that $$\\psi (z)>-c_0, ~\\left(\\frac{\\partial^2\\psi(z)}{\\partial z_j\\partial\\overline z_k}\\right)\\geq c ~I_{n}, ~~z\\in U_0\\cap\\Omega$$for some constants $c_0>0,~c>0$.\n\\end{lemma}\n\\begin{proof}\nLet $(V, z)$ be the coordinates given as in (\\ref{9-24-a1}) and set $\\varphi=\\frac{A_0}4|z|^2-y_1$. Then $\\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\\varphi(0)=0, \\varphi(z)<0$ when $z\\in U_0\\cap\\overline\\Omega\\setminus \\{0\\}$. Take $r>0$ such that $\\overline {\\mathbb B^n(0, r)}\\Subset U_0$. Set $M=\\max\\{\\varphi(z): z\\in \\partial \\mathbb B^n(0, r)\\cap\\overline \\Omega\\}$. Then $M<0$. Now we define $\\psi$ as follows:\n\\begin{equation}\n \\psi=\\begin{cases}\n \\max\\{\\varphi(z), M\\}, &z\\in \\mathbb B^n(0, r)\\cap\\Omega\\\\\n M, &z\\in\\Omega\\setminus \\mathbb B^n(0, r).\n \\end{cases}\n\\end{equation}\nThen $\\psi$ is a bounded and continuous plurisubharmonic function on $\\Omega$ with $$\\psi(0)=0, ~~\\psi(z)<0, \\forall z\\in\\overline\\Omega\\setminus\\{0\\}.$$\nFurthermore, $\\psi$ is equal to $\\varphi$ near $0$ with $$\\left(\\frac{\\partial^2\\psi}{\\partial z_i\\partial\\overline z_j}\\right)= \\frac{A_0}4 I_n $$\nin some neighborhood $U$ of $0$ in $\\mathbb C^n$. Moreover, $\\psi>-c_0$ on $U$ for some positive constant $c_0$.\n\\end{proof}",
    "post_theorem_intro_text_len": 2280,
    "post_theorem_intro_text": "\\begin{corollary}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.\n\\end{corollary}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been  used  to work on many related \nproblems. In particular, in connection with our present work, we \nmention   the papers by Wong \\cite{W77},\nKim \\cite{Kim},  Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\\\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nThe  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.  \nTo obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\\Omega$.\nA direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \\cite{HX20}.",
    "sketch": "The ideas of the proof of Theorem~\\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”",
    "expanded_sketch": "The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”",
    "expanded_theorem": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.",
    "theorem_type": [
      "Implication",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb C^n\\) with \\(n>1\\) be a pseudoconvex domain, possibly unbounded. Assume that \\(\\Omega\\) has a strongly pseudoconvex polyhedral boundary point \\(p\\in \\partial\\Omega\\), meaning that there exist a neighborhood \\(U\\) of \\(p\\) in \\(\\mathbb C^n\\) and \\(C^2\\)-smooth strongly plurisubharmonic functions \\(\\rho_1,\\dots,\\rho_m:U\\to\\mathbb R\\) with \\(m>1\\) such that\n\\[\n\\Omega\\cap U=\\{z\\in U: \\rho_1(z)<0,\\dots,\\rho_m(z)<0\\},\n\\]\nand the complex differentials \\(\\{\\partial\\rho_1|_p,\\dots,\\partial\\rho_m|_p\\}\\) are linearly independent over \\(\\mathbb C\\). Under these assumptions, which statement about the Bergman metric of \\(\\Omega\\) holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \\(\\Omega^*\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and if the Bergman metric is Einstein on some open subset of \\(\\Omega^*\\), then \\(\\Omega\\) is biholomorphic to the unit ball."
        },
        {
          "label": "C",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined."
        },
        {
          "label": "D",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \\(\\Omega^*\\)."
        },
        {
          "label": "E",
          "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman invariant function on \\(\\Omega^*\\) is constant and equal to that of the unit ball."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "conclusion-from-contradiction replaced by global classification",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the non-Einstein conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "quantifier on where Einstein fails",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "computational_check",
          "tampered_component": "constant-ball invariant asserted unconditionally instead of under Einstein hypothesis",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or otherwise reveal choice A. It gives only the hypotheses and asks for the valid consequence."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: the stem lists the hypotheses of a specific result and asks for its conclusion. However, it is not a pure verbatim restatement because the options vary in logical strength and quantification."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact conclusion from weaker true statements and stronger false variants, especially around quantifiers and rigidity claims. Still, the item mainly tests recognition of the theorem rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well targeted: one is weaker but true, one overstates the conclusion via ball-rigidity, one weakens the quantifier structure, and one asserts an unjustified invariant conclusion. These reflect realistic failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based."
    }
  },
  {
    "id": "2512.08369v1",
    "paper_link": "http://arxiv.org/abs/2512.08369v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm-main}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tIf $Q(A)$ is totally positive, then\n\t\\begin{enumerate}[\\rm(1)]\n\t\t\\item $A$ is totally positive;\n\t\t\\item $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive;\n\t\t\\item the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\\ge0$.\n\t\\end{enumerate}",
      "start_pos": 11004,
      "end_pos": 11414,
      "label": "thm-main"
    },
    "ref_dict": {
      "eq-A=Q": "\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n\t\\begin{array}{cc}\n\t\tI_1 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_2 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_3 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\dots.\n\\end{equation}",
      "thm-main": "\\begin{thm}\\label{thm-main}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tIf $Q(A)$ is totally positive, then\n\t\\begin{enumerate}[\\rm(1)]\n\t\t\\item $A$ is totally positive;\n\t\t\\item $\\rev{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive;\n\t\t\\item the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\\ge0$.\n\t\\end{enumerate}\t\n\\end{thm}",
      "prob-RZ": "\\begin{prob}\\label{prob-RZ}\n\tGiven a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n\tunder which conditions is the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\\ge0$,\n\tor equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?\n\\end{prob}",
      "prob-TP": "\\begin{prob}\\label{prob-TP}\n\tGiven a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n\tunder which conditions are both $A$ and its reversal $\\rev{A}:=[a_{n,n-k}]_{n,k\\ge0}$ totally positive?\n\\end{prob}",
      "thm-T": "\\begin{thm}\\label{thm-T}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tDefine\n\t\\setlength{\\arraycolsep}{3pt}\n\t$$\n\tM_{n,r}:=\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_0 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_r\n\t\t\\end{array}\n\t\\right]\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_1 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_{r-1}\n\t\t\\end{array}\n\t\\right]\n\t\\dots\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_r \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_0\n\t\t\\end{array}\n\t\\right],\n\t$$\n\twhere $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.\n\tLet $\\bal_n:=(a_{n,0},a_{n,1},\\dots,a_{n,n},0,0,\\dots)$ be the $n$th row of $A$.\n\tThen for each $r\\ge0$, we have\n\t$$\n\tT_r(\\bal_n)^T=M_{n,r}\\submat{n,n+1,n+2,\\dots,n+r}{0,1,2,\\dots,r},\n\t$$\n\twhere $T_r(\\bal_n)$ is the $r$th leading principal submatrix of $T_{\\infty}(\\bal_n)$,\n\tand $^T$ stands for transpose.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 5550,
    "pre_theorem_intro_text": "\\hspace*{\\parindent}\nFollowing Karlin \\cite{Kar68},\na finite or infinite matrix $A=[a_{n,k}]_{n,k\\ge0}$ is {\\it totally positive}\nif all its minors are nonnegative.\nSuch matrices have been widely studied in both pure and applied mathematics \\cite{Bre95,Bre96,Kar68,Pin10}.\nIn recent five years there has been great interest concerning total positivity of combinatorial matrices \\cite{CDD+21,CS24,LLY+22,LPW23,LWW23,MMW22,MW22,PSZ23,Sok23,Zhu21,Zhu24}.\nHowever, there are still many combinatorial matrices that are experimentally totally positive but without proof.\nFor instance, a longstanding conjecture of Brenti \\cite[Conjecture 6.10]{Bre96} asked the total positivity of the Eulerian triangle\n$$\n[A(n,k)]_{n,k\\ge0}=\n\\left[\n\t\\begin{array}{*{6}c}\n\t\t1 \\\\\n\t\t1 &  1 \\\\\n\t\t1 &  4 & 1 \\\\\n\t\t1 & 11 & 11 & 1 \\\\\n\t\t1 & 26 & 66 & 26 & 1 \\\\\n\t\t\\vdots &&&&& \\ddots\n\t\\end{array}\n\\right],\n$$\nwhere the {\\it Eulerian number} $A(n,k)$ counts the number of permutations of $[n+1]$ with $k$ descents,\nand satisfies the recurrence\n$$\nA(n,k)=(n-k+1)A(n-1,k-1)+(k+1)A(n-1,k)\n$$\nwith initial conditions $A(0,k)=\\del_{0k}$.\nAlthough the total positivity of the Eulerian triangle is hard to prove,\nthe first author {\\it et al.} \\cite{CDD+21},\nby using a planar network approach,\nshowed the total positivity of the reversal of the Stirling triangle of the second kind\n$$\n\\overleftarrow{S}:=[\\overleftarrow{S}(n,k)]_{n,k\\ge0}=\n\\left[\n\t\\begin{array}{*{6}c}\n\t\t1 \\\\\n\t\t1 &  1 \\\\\n\t\t1 &  3 &  1 \\\\\n\t\t1 &  6 &  7 & 1 \\\\\n\t\t1 & 10 & 25 & 15 & 1 \\\\\n\t\t\\vdots &&&&& \\ddots\n\t\\end{array}\n\\right],\n$$\nwhich enjoys a similar recurrence to the Eulerian triangle, i.e.,\n$$\n\\overleftarrow{S}(n,k)=(n-k+1)\\overleftarrow{S}(n-1,k-1)+\\overleftarrow{S}(n-1,k)\n$$\nwith initial conditions $\\overleftarrow{S}(0,k)=\\del_{0k}$.\nOn the other hand, it is known \\cite{Bre95,Zhu14,CLW15rec} that the Stirling triangle of the second kind is totally positive.\nSo it is natural to ask the following question.\n\n\\begin{prob}\\label{prob-TP}\n\tGiven a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n\tunder which conditions are both $A$ and its reversal $\\overleftarrow{A}:=[a_{n,n-k}]_{n,k\\ge0}$ totally positive?\n\\end{prob}\n\nOne particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\\dots$.\nThe reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$:\n$$\nT_{\\infty}(\\boldsymbol{\\al}):=[a_{i-j}]_{i,j\\ge0}=\n\\left[\n\t\\begin{array}{*{5}c}\n\t\ta_0 \\\\\n\t\ta_1 & a_0 \\\\\n\t\ta_2 & a_1 & a_0 \\\\\n\t\ta_3 & a_2 & a_1 & a_0 \\\\\n\t\t\\vdots &&&& \\ddots\n\t\\end{array}\n\\right].\n$$\nAn infinite nonnegative sequence $\\boldsymbol{\\al}$ is called a {\\it P\\'olya frequency sequence} (or shortly, {\\it PF sequence}),\nif its Toeplitz matrix $T_{\\infty}(\\boldsymbol{\\al})$ is totally positive.\nWe identify a finite sequence $a_0,a_1,\\dots,a_n$ with the infinite sequence $a_0,a_1,\\dots,a_n,0,0,\\dots$.\nA fundamental characterization for PF sequences is due to Schoenberg and Edrei,\nwhich states that the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$ is PF if and only if its generating function\n\\begin{equation}\\label{eq-PF}\n\t\\sum_{n\\ge0} a_n x^n=C x^k e^{\\gamma x} \\frac{\\prod_{j\\ge0} (1+\\al_j x)}{\\prod_{j\\ge0} (1-\\be_j x)},\n\\end{equation}\nwhere $C>0$, $k\\in\\mathbb{N}$, $\\al_j,\\be_j,\\gamma\\ge0$\nand $\\sum_{j\\ge0}(\\al_j+\\be_j)<\\infty$.\nIn particular, Aissen, Schoenberg and Whitney states that\na finite sequence of nonnegative numbers is PF if and only if its generating function is {\\it real-rooted}, that is,\nit is a constant polynomial or all of its zeros are real numbers (see \\cite[p. 399, 412]{Kar68} for instance).\nIn this sense, we refer the generating functions of PF sequences as {\\it PF formal power series}.\nHence, it gives rise to the another problem.\n\n\\begin{prob}\\label{prob-RZ}\n\tGiven a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n\tunder which conditions is the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\\ge0$,\n\tor equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?\n\\end{prob}\n\nThe objective of this paper is to give a common answer to Problems \\ref{prob-TP} and \\ref{prob-RZ}.\nAn efficient method of proving total positivity of matrices is to prove the total positivity of their (right) production matrices \\cite{DFR05},\nwhich has been frequently used in literature \\cite{CLW15rio,CLW15rec,PZ16,LLY+22,CS24,Zhu14,Zhu24}.\nHowever, this method does not work when proving total positivity of matrix reversals.\nSo we turn to consider the total positivity of left production matrices,\nand discover that it implies the total positivity of the matrix itself, its reversal, and the Toeplitz matrix of each row, simultaneously.\n\nLet $A$ be a lower triangular matix with nonzero diagonals,\nthen $A$ is invertible.\nFollowing Liang {\\it et al.} \\cite{LPW23}, call\n\\begin{equation}\\label{eq-leftprod}\nQ(A):=A\\cdot\n\\left[\n\t\\begin{array}{cc}\n\t\t1 & O \\\\\n\t\tO & A^{-1}\n\t\\end{array}\n\\right]\n\\end{equation}\nthe {\\it left production matrix} of $A$.\nThis implies that $Q(A)$ is also lower triangular with nonzero diagonals.\nMoreover, we have\n\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n\t\\begin{array}{cc}\n\t\tI_1 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_2 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_3 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\dots.\n\\end{equation}\nNote that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \\eqref{eq-A=Q}.\nIn such cases, we also call $Q(A)$ the left production matrix of $A$.",
    "context": "\\hspace*{\\parindent}\nFollowing Karlin \\cite{Kar68},\na finite or infinite matrix $A=[a_{n,k}]_{n,k\\ge0}$ is {\\it totally positive}\nif all its minors are nonnegative.\nSuch matrices have been widely studied in both pure and applied mathematics \\cite{Bre95,Bre96,Kar68,Pin10}.\nIn recent five years there has been great interest concerning total positivity of combinatorial matrices \\cite{CDD+21,CS24,LLY+22,LPW23,LWW23,MMW22,MW22,PSZ23,Sok23,Zhu21,Zhu24}.\nHowever, there are still many combinatorial matrices that are experimentally totally positive but without proof.\nFor instance, a longstanding conjecture of Brenti \\cite[Conjecture 6.10]{Bre96} asked the total positivity of the Eulerian triangle\n$$\n[A(n,k)]_{n,k\\ge0}=\n\\left[\n \\begin{array}{*{6}c}\n 1 \\\\\n 1 & 1 \\\\\n 1 & 4 & 1 \\\\\n 1 & 11 & 11 & 1 \\\\\n 1 & 26 & 66 & 26 & 1 \\\\\n \\vdots &&&&& \\ddots\n \\end{array}\n\\right],\n$$\nwhere the {\\it Eulerian number} $A(n,k)$ counts the number of permutations of $[n+1]$ with $k$ descents,\nand satisfies the recurrence\n$$\nA(n,k)=(n-k+1)A(n-1,k-1)+(k+1)A(n-1,k)\n$$\nwith initial conditions $A(0,k)=\\del_{0k}$.\nAlthough the total positivity of the Eulerian triangle is hard to prove,\nthe first author {\\it et al.} \\cite{CDD+21},\nby using a planar network approach,\nshowed the total positivity of the reversal of the Stirling triangle of the second kind\n$$\n\\overleftarrow{S}:=[\\overleftarrow{S}(n,k)]_{n,k\\ge0}=\n\\left[\n \\begin{array}{*{6}c}\n 1 \\\\\n 1 & 1 \\\\\n 1 & 3 & 1 \\\\\n 1 & 6 & 7 & 1 \\\\\n 1 & 10 & 25 & 15 & 1 \\\\\n \\vdots &&&&& \\ddots\n \\end{array}\n\\right],\n$$\nwhich enjoys a similar recurrence to the Eulerian triangle, i.e.,\n$$\n\\overleftarrow{S}(n,k)=(n-k+1)\\overleftarrow{S}(n-1,k-1)+\\overleftarrow{S}(n-1,k)\n$$\nwith initial conditions $\\overleftarrow{S}(0,k)=\\del_{0k}$.\nOn the other hand, it is known \\cite{Bre95,Zhu14,CLW15rec} that the Stirling triangle of the second kind is totally positive.\nSo it is natural to ask the following question.\n\n\\begin{prob}\\label{prob-TP}\n Given a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n under which conditions are both $A$ and its reversal $\\overleftarrow{A}:=[a_{n,n-k}]_{n,k\\ge0}$ totally positive?\n\\end{prob}\n\nOne particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\\dots$.\nThe reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$:\n$$\nT_{\\infty}(\\boldsymbol{\\al}):=[a_{i-j}]_{i,j\\ge0}=\n\\left[\n \\begin{array}{*{5}c}\n a_0 \\\\\n a_1 & a_0 \\\\\n a_2 & a_1 & a_0 \\\\\n a_3 & a_2 & a_1 & a_0 \\\\\n \\vdots &&&& \\ddots\n \\end{array}\n\\right].\n$$\nAn infinite nonnegative sequence $\\boldsymbol{\\al}$ is called a {\\it P\\'olya frequency sequence} (or shortly, {\\it PF sequence}),\nif its Toeplitz matrix $T_{\\infty}(\\boldsymbol{\\al})$ is totally positive.\nWe identify a finite sequence $a_0,a_1,\\dots,a_n$ with the infinite sequence $a_0,a_1,\\dots,a_n,0,0,\\dots$.\nA fundamental characterization for PF sequences is due to Schoenberg and Edrei,\nwhich states that the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$ is PF if and only if its generating function\n\\begin{equation}\\label{eq-PF}\n \\sum_{n\\ge0} a_n x^n=C x^k e^{\\gamma x} \\frac{\\prod_{j\\ge0} (1+\\al_j x)}{\\prod_{j\\ge0} (1-\\be_j x)},\n\\end{equation}\nwhere $C>0$, $k\\in\\mathbb{N}$, $\\al_j,\\be_j,\\gamma\\ge0$\nand $\\sum_{j\\ge0}(\\al_j+\\be_j)<\\infty$.\nIn particular, Aissen, Schoenberg and Whitney states that\na finite sequence of nonnegative numbers is PF if and only if its generating function is {\\it real-rooted}, that is,\nit is a constant polynomial or all of its zeros are real numbers (see \\cite[p. 399, 412]{Kar68} for instance).\nIn this sense, we refer the generating functions of PF sequences as {\\it PF formal power series}.\nHence, it gives rise to the another problem.\n\n\\begin{prob}\\label{prob-RZ}\n Given a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n under which conditions is the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\\ge0$,\n or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?\n\\end{prob}\n\nThe objective of this paper is to give a common answer to Problems \\ref{prob-TP} and \\ref{prob-RZ}.\nAn efficient method of proving total positivity of matrices is to prove the total positivity of their (right) production matrices \\cite{DFR05},\nwhich has been frequently used in literature \\cite{CLW15rio,CLW15rec,PZ16,LLY+22,CS24,Zhu14,Zhu24}.\nHowever, this method does not work when proving total positivity of matrix reversals.\nSo we turn to consider the total positivity of left production matrices,\nand discover that it implies the total positivity of the matrix itself, its reversal, and the Toeplitz matrix of each row, simultaneously.\n\nLet $A$ be a lower triangular matix with nonzero diagonals,\nthen $A$ is invertible.\nFollowing Liang {\\it et al.} \\cite{LPW23}, call\n\\begin{equation}\\label{eq-leftprod}\nQ(A):=A\\cdot\n\\left[\n \\begin{array}{cc}\n 1 & O \\\\\n O & A^{-1}\n \\end{array}\n\\right]\n\\end{equation}\nthe {\\it left production matrix} of $A$.\nThis implies that $Q(A)$ is also lower triangular with nonzero diagonals.\nMoreover, we have\n\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n \\begin{array}{cc}\n I_1 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_2 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_3 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\dots.\n\\end{equation}\nNote that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \\eqref{eq-A=Q}.\nIn such cases, we also call $Q(A)$ the left production matrix of $A$.",
    "full_context": "\\hspace*{\\parindent}\nFollowing Karlin \\cite{Kar68},\na finite or infinite matrix $A=[a_{n,k}]_{n,k\\ge0}$ is {\\it totally positive}\nif all its minors are nonnegative.\nSuch matrices have been widely studied in both pure and applied mathematics \\cite{Bre95,Bre96,Kar68,Pin10}.\nIn recent five years there has been great interest concerning total positivity of combinatorial matrices \\cite{CDD+21,CS24,LLY+22,LPW23,LWW23,MMW22,MW22,PSZ23,Sok23,Zhu21,Zhu24}.\nHowever, there are still many combinatorial matrices that are experimentally totally positive but without proof.\nFor instance, a longstanding conjecture of Brenti \\cite[Conjecture 6.10]{Bre96} asked the total positivity of the Eulerian triangle\n$$\n[A(n,k)]_{n,k\\ge0}=\n\\left[\n \\begin{array}{*{6}c}\n 1 \\\\\n 1 & 1 \\\\\n 1 & 4 & 1 \\\\\n 1 & 11 & 11 & 1 \\\\\n 1 & 26 & 66 & 26 & 1 \\\\\n \\vdots &&&&& \\ddots\n \\end{array}\n\\right],\n$$\nwhere the {\\it Eulerian number} $A(n,k)$ counts the number of permutations of $[n+1]$ with $k$ descents,\nand satisfies the recurrence\n$$\nA(n,k)=(n-k+1)A(n-1,k-1)+(k+1)A(n-1,k)\n$$\nwith initial conditions $A(0,k)=\\del_{0k}$.\nAlthough the total positivity of the Eulerian triangle is hard to prove,\nthe first author {\\it et al.} \\cite{CDD+21},\nby using a planar network approach,\nshowed the total positivity of the reversal of the Stirling triangle of the second kind\n$$\n\\overleftarrow{S}:=[\\overleftarrow{S}(n,k)]_{n,k\\ge0}=\n\\left[\n \\begin{array}{*{6}c}\n 1 \\\\\n 1 & 1 \\\\\n 1 & 3 & 1 \\\\\n 1 & 6 & 7 & 1 \\\\\n 1 & 10 & 25 & 15 & 1 \\\\\n \\vdots &&&&& \\ddots\n \\end{array}\n\\right],\n$$\nwhich enjoys a similar recurrence to the Eulerian triangle, i.e.,\n$$\n\\overleftarrow{S}(n,k)=(n-k+1)\\overleftarrow{S}(n-1,k-1)+\\overleftarrow{S}(n-1,k)\n$$\nwith initial conditions $\\overleftarrow{S}(0,k)=\\del_{0k}$.\nOn the other hand, it is known \\cite{Bre95,Zhu14,CLW15rec} that the Stirling triangle of the second kind is totally positive.\nSo it is natural to ask the following question.\n\n\\begin{prob}\\label{prob-TP}\n Given a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n under which conditions are both $A$ and its reversal $\\overleftarrow{A}:=[a_{n,n-k}]_{n,k\\ge0}$ totally positive?\n\\end{prob}\n\nOne particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\\dots$.\nThe reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$:\n$$\nT_{\\infty}(\\boldsymbol{\\al}):=[a_{i-j}]_{i,j\\ge0}=\n\\left[\n \\begin{array}{*{5}c}\n a_0 \\\\\n a_1 & a_0 \\\\\n a_2 & a_1 & a_0 \\\\\n a_3 & a_2 & a_1 & a_0 \\\\\n \\vdots &&&& \\ddots\n \\end{array}\n\\right].\n$$\nAn infinite nonnegative sequence $\\boldsymbol{\\al}$ is called a {\\it P\\'olya frequency sequence} (or shortly, {\\it PF sequence}),\nif its Toeplitz matrix $T_{\\infty}(\\boldsymbol{\\al})$ is totally positive.\nWe identify a finite sequence $a_0,a_1,\\dots,a_n$ with the infinite sequence $a_0,a_1,\\dots,a_n,0,0,\\dots$.\nA fundamental characterization for PF sequences is due to Schoenberg and Edrei,\nwhich states that the sequence $\\boldsymbol{\\al}=(a_n)_{n\\ge0}$ is PF if and only if its generating function\n\\begin{equation}\\label{eq-PF}\n \\sum_{n\\ge0} a_n x^n=C x^k e^{\\gamma x} \\frac{\\prod_{j\\ge0} (1+\\al_j x)}{\\prod_{j\\ge0} (1-\\be_j x)},\n\\end{equation}\nwhere $C>0$, $k\\in\\mathbb{N}$, $\\al_j,\\be_j,\\gamma\\ge0$\nand $\\sum_{j\\ge0}(\\al_j+\\be_j)<\\infty$.\nIn particular, Aissen, Schoenberg and Whitney states that\na finite sequence of nonnegative numbers is PF if and only if its generating function is {\\it real-rooted}, that is,\nit is a constant polynomial or all of its zeros are real numbers (see \\cite[p. 399, 412]{Kar68} for instance).\nIn this sense, we refer the generating functions of PF sequences as {\\it PF formal power series}.\nHence, it gives rise to the another problem.\n\n\\begin{prob}\\label{prob-RZ}\n Given a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n under which conditions is the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\\ge0$,\n or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?\n\\end{prob}\n\nThe objective of this paper is to give a common answer to Problems \\ref{prob-TP} and \\ref{prob-RZ}.\nAn efficient method of proving total positivity of matrices is to prove the total positivity of their (right) production matrices \\cite{DFR05},\nwhich has been frequently used in literature \\cite{CLW15rio,CLW15rec,PZ16,LLY+22,CS24,Zhu14,Zhu24}.\nHowever, this method does not work when proving total positivity of matrix reversals.\nSo we turn to consider the total positivity of left production matrices,\nand discover that it implies the total positivity of the matrix itself, its reversal, and the Toeplitz matrix of each row, simultaneously.\n\nLet $A$ be a lower triangular matix with nonzero diagonals,\nthen $A$ is invertible.\nFollowing Liang {\\it et al.} \\cite{LPW23}, call\n\\begin{equation}\\label{eq-leftprod}\nQ(A):=A\\cdot\n\\left[\n \\begin{array}{cc}\n 1 & O \\\\\n O & A^{-1}\n \\end{array}\n\\right]\n\\end{equation}\nthe {\\it left production matrix} of $A$.\nThis implies that $Q(A)$ is also lower triangular with nonzero diagonals.\nMoreover, we have\n\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n \\begin{array}{cc}\n I_1 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_2 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_3 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\dots.\n\\end{equation}\nNote that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \\eqref{eq-A=Q}.\nIn such cases, we also call $Q(A)$ the left production matrix of $A$.\n\nOne particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\\dots$.\nThe reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\\bal=(a_n)_{n\\ge0}$:\n$$\nT_{\\infty}(\\bal):=[a_{i-j}]_{i,j\\ge0}=\n\\left[\n \\begin{array}{*{5}c}\n a_0 \\\\\n a_1 & a_0 \\\\\n a_2 & a_1 & a_0 \\\\\n a_3 & a_2 & a_1 & a_0 \\\\\n \\vdots &&&& \\ddots\n \\end{array}\n\\right].\n$$\nAn infinite nonnegative sequence $\\bal$ is called a \\defn{P\\'olya frequency sequence} (or shortly, \\defn{PF sequence}),\nif its Toeplitz matrix $T_{\\infty}(\\bal)$ is totally positive.\nWe identify a finite sequence $a_0,a_1,\\dots,a_n$ with the infinite sequence $a_0,a_1,\\dots,a_n,0,0,\\dots$.\nA fundamental characterization for PF sequences is due to Schoenberg and Edrei,\nwhich states that the sequence $\\bal=(a_n)_{n\\ge0}$ is PF if and only if its generating function\n\\begin{equation}\\label{eq-PF}\n \\sum_{n\\ge0} a_n x^n=C x^k e^{\\ga x} \\frac{\\prod_{j\\ge0} (1+\\al_j x)}{\\prod_{j\\ge0} (1-\\be_j x)},\n\\end{equation}\nwhere $C>0$, $k\\in\\mathbb{N}$, $\\al_j,\\be_j,\\ga\\ge0$\nand $\\sum_{j\\ge0}(\\al_j+\\be_j)<\\infty$.\nIn particular, Aissen, Schoenberg and Whitney states that\na finite sequence of nonnegative numbers is PF if and only if its generating function is \\defn{real-rooted}, that is,\nit is a constant polynomial or all of its zeros are real numbers (see \\cite[p. 399, 412]{Kar68} for instance).\nIn this sense, we refer the generating functions of PF sequences as \\defn{PF formal power series}.\nHence, it gives rise to the another problem.\n\n\\begin{prob}\\label{prob-RZ}\n Given a lower triangular matrix $A=[a_{n,k}]_{n,k\\ge0}$,\n under which conditions is the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\\ge0$,\n or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?\n\\end{prob}\n\nLet $A$ be a lower triangular matix with nonzero diagonals,\nthen $A$ is invertible.\nFollowing Liang {\\it et al.} \\cite{LPW23}, call\n\\begin{equation}\\label{eq-leftprod}\nQ(A):=A\\cdot\n\\left[\n \\begin{array}{cc}\n 1 & O \\\\\n O & A^{-1}\n \\end{array}\n\\right]\n\\end{equation}\nthe \\defn{left production matrix} of $A$.\nThis implies that $Q(A)$ is also lower triangular with nonzero diagonals.\nMoreover, we have\n\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n \\begin{array}{cc}\n I_1 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_2 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\left[\n \\begin{array}{cc}\n I_3 & \\\\\n & Q(A)\n \\end{array}\n\\right]\n\\dots.\n\\end{equation}\nNote that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \\eqref{eq-A=Q}.\nIn such cases, we also call $Q(A)$ the left production matrix of $A$.\n\nRecently, Mao {\\it et al.} \\cite{MMW22} showed Theorem \\ref{thm-main} (1) whenever $A$ is a proper Riordan array,\nby using matrix decompositions and induction.\nHowever, sometimes Theorem \\ref{thm-main} (2) seems more useful.\nIn fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain\nwhenever neither its left nor right production matrix is totally positive (for instance, the matrix $\\rev{S}$),\nbut we can turn to consider its reversal $\\rev{A}$ which may have a totally positive left production matrix.\nIn this paper, we will prove the three results in Theorem \\ref{thm-main} simultaneously,\nby constructing a unified planar network that represent $A$, $\\rev{A}$, and Toeplitz matrices of rows of $A$, respectively,\nwhen selecting different sets of sources and sinks.\nFurthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.\n\n\\begin{thm}\\label{thm-T}\n Let $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n and $Q(A)$ be its left production matrix.\n Define\n \\setlength{\\arraycolsep}{3pt}\n $$\n M_{n,r}:=\n \\left[\n \\begin{array}{ccc}\n I_0 \\\\\n & Q_n(A) \\\\\n & & I_r\n \\end{array}\n \\right]\n \\left[\n \\begin{array}{ccc}\n I_1 \\\\\n & Q_n(A) \\\\\n & & I_{r-1}\n \\end{array}\n \\right]\n \\dots\n \\left[\n \\begin{array}{ccc}\n I_r \\\\\n & Q_n(A) \\\\\n & & I_0\n \\end{array}\n \\right],\n $$\n where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.\n Let $\\bal_n:=(a_{n,0},a_{n,1},\\dots,a_{n,n},0,0,\\dots)$ be the $n$th row of $A$.\n Then for each $r\\ge0$, we have\n $$\n T_r(\\bal_n)^T=M_{n,r}\\submat{n,n+1,n+2,\\dots,n+r}{0,1,2,\\dots,r},\n $$\n where $T_r(\\bal_n)$ is the $r$th leading principal submatrix of $T_{\\infty}(\\bal_n)$,\n and $^T$ stands for transpose.\n\\end{thm}\n\n\\begin{thm1}\n Let $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n and $Q(A)$ be its left production matrix.\n If $Q(A)$ is totally positive, then\n \\begin{enumerate}[\\rm(1)]\n \\item $A$ is totally positive;\n \\item $\\rev{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive.\n \\end{enumerate} \n\\end{thm1}\n\n\\begin{thm2}\n Let $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n and $Q(A)$ be its left production matrix.\n Define\n \\setlength{\\arraycolsep}{3pt}\n \\begin{equation}\\label{eq-M}\n M_{n,r}:=\n \\left[\n \\begin{array}{ccc}\n I_0 \\\\\n & Q_n(A) \\\\\n & & I_r\n \\end{array}\n \\right]\n \\left[\n \\begin{array}{ccc}\n I_1 \\\\\n & Q_n(A) \\\\\n & & I_{r-1}\n \\end{array}\n \\right]\n \\dots\n \\left[\n \\begin{array}{ccc}\n I_r \\\\\n & Q_n(A) \\\\\n & & I_0\n \\end{array}\n \\right],\n \\end{equation}\n where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.\n Let $\\bal_n:=(a_{n,0},a_{n,1},\\dots,a_{n,n},0,0,\\dots)$ be the $n$th row of $A$.\n Then for each $r\\ge0$, we have\n \\begin{equation}\\label{eq-Toep}\n T_r(\\bal_n)^T=M_{n,r}\\submat{n,n+1,n+2,\\dots,n+r}{0,1,2,\\dots,r},\n \\end{equation}\n where $T_r(\\bal_n)$ is the $r$th leading principal submatrix of $T_{\\infty}(\\bal_n)$,\n and $^T$ stands for transpose.\n\\end{thm2}\n\n\\begin{thm1}\n Let $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n and $Q(A)$ be its left production matrix.\n If $Q(A)$ is totally positive, then\n \\begin{enumerate}[\\rm(1)]\n \\item[\\rm(3)] the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\\ge0$.\n \\end{enumerate} \n\\end{thm1}\n\n\\begin{equation}\\label{eq-A=Q}\nA=Q(A)\n\\left[\n\t\\begin{array}{cc}\n\t\tI_1 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_2 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\left[\n\t\\begin{array}{cc}\n\t\tI_3 &  \\\\\n\t\t\t& Q(A)\n\t\\end{array}\n\\right]\n\\dots.\n\\end{equation}\n\n\\begin{thm}\\label{thm-main}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tIf $Q(A)$ is totally positive, then\n\t\\begin{enumerate}[\\rm(1)]\n\t\t\\item $A$ is totally positive;\n\t\t\\item $\\rev{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive;\n\t\t\\item the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\\ge0$.\n\t\\end{enumerate}\t\n\\end{thm}",
    "post_theorem_intro_text_len": 3101,
    "post_theorem_intro_text": "Recently, Mao {\\it et al.} \\cite{MMW22} showed Theorem \\ref{thm-main} (1) whenever $A$ is a proper Riordan array,\nby using matrix decompositions and induction.\nHowever, sometimes Theorem \\ref{thm-main} (2) seems more useful.\nIn fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain\nwhenever neither its left nor right production matrix is totally positive (for instance, the matrix $\\overleftarrow{S}$),\nbut we can turn to consider its reversal $\\overleftarrow{A}$ which may have a totally positive left production matrix.\nIn this paper, we will prove the three results in Theorem \\ref{thm-main} simultaneously,\nby constructing a unified planar network that represent $A$, $\\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively,\nwhen selecting different sets of sources and sinks.\nFurthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.\n\nFor a given a matrix $A$, denote by\n$$\nA\\submat{i_0, \\ldots, i_r}{j_0, \\ldots, j_r}\n\\quad \\text{and} \\quad\nA\\minor{i_0, \\ldots, i_r}{j_0, \\ldots, j_r}\n$$\nthe submatrix and the minor of $A$ determined by\nthe rows indexed $i_0 < \\cdots < i_r$ and\nthe columns indexed $j_0 < \\cdots < j_r$, respectively.\nLet\n$$\nA_r:=A\t\\left[\n\t\\begin{array}{c}\n\t\t0,1,\\dots,r \\\\\n\t\t0,1,\\dots,r\n\t\\end{array}\n\t\\right]\n\n$$\nbe the $r$th leading principal submatrix (of order $r+1$) of $A$.\n\n\\begin{thm}\\label{thm-T}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tDefine\n\t\\setlength{\\arraycolsep}{3pt}\n\t$$\n\tM_{n,r}:=\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_0 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_r\n\t\t\\end{array}\n\t\\right]\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_1 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_{r-1}\n\t\t\\end{array}\n\t\\right]\n\t\\dots\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_r \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_0\n\t\t\\end{array}\n\t\\right],\n\t$$\n\twhere $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.\n\tLet $\\bal_n:=(a_{n,0},a_{n,1},\\dots,a_{n,n},0,0,\\dots)$ be the $n$th row of $A$.\n\tThen for each $r\\ge0$, we have\n\t$$\n\tT_r(\\bal_n)^T=M_{n,r}\\submat{n,n+1,n+2,\\dots,n+r}{0,1,2,\\dots,r},\n\t$$\n\twhere $T_r(\\bal_n)$ is the $r$th leading principal submatrix of $T_{\\infty}(\\bal_n)$,\n\tand $^T$ stands for transpose.\n\\end{thm}\n\nThis paper is organized as follows.\nIn section 2, we introduce the Lindstr\\\"om-Gessel-Viennot lemma\nand planar network characterizations of totally positive matrices.\nIn section 3, we give proofs of Theorem \\ref{thm-main} and Theorem \\ref{thm-T}\nby constructing a unified planar network.\nAs applications, we present sufficient conditions for total positivity and real-rootedness properties in the exponential Riordan arrays and the iteration matrices (section 4),\nand the $n$-recursive matrices (section 5), repectively.\nExamples include matrices and polynomials formed by the Stirling numbers of both kinds (of type A and type B), the Lah numbers, the idempotent numbers, the Delannoy numbers, and the derangement numbers of type A and type B.",
    "sketch": "They propose to prove the three statements in Theorem~\\ref{thm-main} \"simultaneously, by constructing a unified planar network that represent $A$, $\\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks.\" They further state that they will \"identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.\" Concretely, Section 2 introduces the Lindstr\\\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and Section 3 gives the proofs of Theorem~\\ref{thm-main} (and Theorem~\\ref{thm-T}) \"by constructing a unified planar network.\"",
    "expanded_sketch": "They propose to prove the three statements in the main theorem simultaneously, by constructing a unified planar network that represent $A$, $\\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks. They further state that they will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$. Concretely, they first introduce the Lindstr\\\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and then they give the proofs of the main theorem and also establish the following theorem.\n\n\\begin{thm}\\label{thm-T}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tDefine\n\t\\setlength{\\arraycolsep}{3pt}\n\t$$\n\tM_{n,r}:=\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_0 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_r\n\t\t\\end{array}\n\t\\right]\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_1 \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_{r-1}\n\t\t\\end{array}\n\t\\right]\n\t\\dots\n\t\\left[\n\t\t\\begin{array}{ccc}\n\t\t\tI_r \\\\\n\t\t\t\t& Q_n(A) \\\\\n\t\t\t\t&     & I_0\n\t\t\\end{array}\n\t\\right],\n\t$$\n\twhere $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.\n\tLet $\\bal_n:=(a_{n,0},a_{n,1},\\dots,a_{n,n},0,0,\\dots)$ be the $n$th row of $A$.\n\tThen for each $r\\ge0$, we have\n\t$$\n\tT_r(\\bal_n)^T=M_{n,r}\\submat{n,n+1,n+2,\\dots,n+r}{0,1,2,\\dots,r},\n\t$$\n\twhere $T_r(\\bal_n)$ is the $r$th leading principal submatrix of $T_{\\infty}(\\bal_n)$,\n\tand $^T$ stands for transpose.\n\\end{thm}\n\nThey do this by constructing a unified planar network.",
    "expanded_theorem": "\\label{thm-main}\n\tLet $A=[a_{n,k}]_{n,k\\ge0}$ be a lower triangular matix,\n\tand $Q(A)$ be its left production matrix.\n\tIf $Q(A)$ is totally positive, then\n\t\\begin{enumerate}[\\rm(1)]\n\t\t\\item $A$ is totally positive;\n\t\t\\item $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive;\n\t\t\\item the row generating function $\\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\\ge0$.\n\t\\end{enumerate}",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let $A=[a_{n,k}]_{n,k\\ge 0}$ be a lower triangular matrix, and define its left production matrix by\n$$\nQ(A):=A\\begin{bmatrix}1&O\\\\ O&A^{-1}\\end{bmatrix},\n$$\nwhere $O$ denotes zero blocks. A matrix is called totally positive if all of its minors are nonnegative, and the reversal of $A$ is the lower triangular matrix $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge 0}$. Which statement holds for every such matrix $A$ for which $Q(A)$ is totally positive?",
      "correct_choice": {
        "label": "A",
        "text": "$A$ is totally positive, $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive, and for every $n\\ge0$ the row generating polynomial $\\sum_{k=0}^n a_{n,k}x^k$ is real-rooted."
      },
      "choices": [
        {
          "label": "B",
          "text": "$A$ is totally positive and for every $n\\ge0$ the row generating polynomial $\\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, but the reversal $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ need not be totally positive."
        },
        {
          "label": "C",
          "text": "$A$ is totally positive and $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive."
        },
        {
          "label": "D",
          "text": "$A$ is totally positive, $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive, and for every $n\\ge0$ the row generating polynomial $\\sum_{k=0}^n a_{n,k}x^k$ has only nonnegative coefficients."
        },
        {
          "label": "E",
          "text": "If $A$ is totally positive, $\\overleftarrow{A}=[a_{n,n-k}]_{n,k\\ge0}$ is totally positive, and for every $n\\ge0$ the row generating polynomial $\\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, then $Q(A)$ is totally positive."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "unified planar-network transfer to the reversal",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped the real-rootedness conclusion for row generating polynomials",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "property_confusion",
          "tampered_component": "real-rootedness of row generating polynomials replaced by coefficient nonnegativity",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "reversed the implication from consequences back to total positivity of the left production matrix",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the hypothesis on the left production matrix and asks for the valid conclusion. It does not explicitly state the three target conclusions, though mentioning the reversal signals that it may matter."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially a theorem-recall question: the correct option restates the full theorem-level conclusion from the stated hypothesis. It does not substantially transform the result into a new reasoning task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to compare a full conclusion against weaker, truncated, or partially false variants, but success mainly depends on recognizing the exact theorem rather than generating a nontrivial argument."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and distinct: one is a weaker true statement, others reflect common confusions about finite truncations, PF/real-rootedness implications, and whether reversal total positivity follows."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically polished MCQ with strong distractors and little answer leakage, but it functions mostly as direct theorem recognition rather than a genuinely generative reasoning problem."
    }
  },
  {
    "id": "2512.08391v1",
    "paper_link": "http://arxiv.org/abs/2512.08391v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{T2}\nAssume \\eqref{(2)}-\\eqref{(6)} hold. Then, \n\\begin{itemize}\n\\item[(i)] If $0<q<1-\\theta$, there exists a distributional solution $u\\in W_{0}^{1,(r_{j})}(\\Omega)$ to problem \\eqref{(1)}, such that \n\\begin{align}\n\\label{(17)}\nr_{j}=\\frac{N(\\overline{p}-\\theta)}{\\overline{p}(N-\\theta)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(ii)] If $1-\\theta<q\\leq1$, there exists a distributional solution $u\\in W_{0}^{1,(\\eta_{j})}(\\Omega)$ to problem \\eqref{(1)}, such that \n\\begin{align}\n\\label{(017)}\nr_{j}<\\frac{N(\\overline{p}-1+q)}{\\overline{p}(N+q-1)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(iii)] If $q>1$, there exists a distributional solution $u\\in W_{0}^{1,(p_{j})}(\\Omega)$ to problem \\eqref{(1)}.\n\\end{itemize}",
      "start_pos": 15877,
      "end_pos": 16644,
      "label": "T2"
    },
    "ref_dict": {
      "(1)": "\\begin{equation}\n\\label{(1)}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+b(x)\\sum\\limits_{j\\in J}\\frac{\\vert D_{j}u\\vert^{p_{j}}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}",
      "T2": "\\begin{theorem}\n\\label{T2}\nAssume \\eqref{(2)}-\\eqref{(6)} hold. Then, \n\\begin{itemize}\n\\item[(i)] If $0<q<1-\\theta$, there exists a distributional solution $u\\in W_{0}^{1,(r_{j})}(\\Omega)$ to problem \\eqref{(1)}, such that \n\\begin{align}\n\\label{(17)}\nr_{j}=\\frac{N(\\overline{p}-\\theta)}{\\overline{p}(N-\\theta)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(ii)] If $1-\\theta<q\\leq1$, there exists a distributional solution $u\\in W_{0}^{1,(\\eta_{j})}(\\Omega)$ to problem \\eqref{(1)}, such that \n\\begin{align}\n\\label{(017)}\nr_{j}<\\frac{N(\\overline{p}-1+q)}{\\overline{p}(N+q-1)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(iii)] If $q>1$, there exists a distributional solution $u\\in W_{0}^{1,(p_{j})}(\\Omega)$ to problem \\eqref{(1)}.\n\\end{itemize}\n\\end{theorem}",
      "(4)": "\\begin{align}\n&\\label{(2)}\nq>0,\\quad 0<\\theta<1,\\\\\n&\\label{(3)}\nf\\geq0,\\;\\; f\\not\\equiv0,\\quad f\\in L^{1}(\\Omega),\\\\\n&\\label{(4)}\n2\\leq p_{1}\\leq p_{2}\\leq ... \\leq p_{N-1}\\leq p_{N}\\quad \\mbox{and}\\quad 2\\leq \\overline{p}:=N\\left(\\sum\\limits_{j\\in J}\\frac{1}{p_{j}}\\right)^{-1}<N.\n\\end{align}",
      "(6)": "\\begin{align}\n&\\label{(5)}\n 0<\\alpha\\leq a(x)\\leq \\beta,\\\\\n&\\label{(6)}\n0<\\mu\\leq b(x)\\leq \\nu.\n\\end{align}",
      "P3": "\\begin{equation}\n\\label{P3}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\mbox{div}\\left(\\left[a(x)+u^{q}\\right]\\nabla u\\right)+b(x)\\frac{\\vert \\nabla u\\vert^{2}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}",
      "HH": "\\begin{align}\n\\label{HH}\n1\\leq s_{j}<\\frac{N(\\overline{p}-1)}{\\overline{p}(N-1)}p_{j},\\quad\\forall\\; j\\in J.\n\\end{align}",
      "HH1": "\\begin{equation}\\label{HH1}\n\\sum\\limits_{j\\in J}\\int_{\\Omega}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi\\,dx =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{equation}",
      "(3)": "\\begin{align}\n&\\label{(2)}\nq>0,\\quad 0<\\theta<1,\\\\\n&\\label{(3)}\nf\\geq0,\\;\\; f\\not\\equiv0,\\quad f\\in L^{1}(\\Omega),\\\\\n&\\label{(4)}\n2\\leq p_{1}\\leq p_{2}\\leq ... \\leq p_{N-1}\\leq p_{N}\\quad \\mbox{and}\\quad 2\\leq \\overline{p}:=N\\left(\\sum\\limits_{j\\in J}\\frac{1}{p_{j}}\\right)^{-1}<N.\n\\end{align}",
      "(2)": "\\begin{align}\n&\\label{(2)}\nq>0,\\quad 0<\\theta<1,\\\\\n&\\label{(3)}\nf\\geq0,\\;\\; f\\not\\equiv0,\\quad f\\in L^{1}(\\Omega),\\\\\n&\\label{(4)}\n2\\leq p_{1}\\leq p_{2}\\leq ... \\leq p_{N-1}\\leq p_{N}\\quad \\mbox{and}\\quad 2\\leq \\overline{p}:=N\\left(\\sum\\limits_{j\\in J}\\frac{1}{p_{j}}\\right)^{-1}<N.\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 11576,
    "pre_theorem_intro_text": "Anisotropic elliptic equations describe physical and biological phenomena where\ndiffusion or conductivity varies with direction, such as flow in heterogeneous\nporous media \\cite{R18}, epidemic propagation across irregular landscapes\n\\cite{R19}, and thermistor models exhibiting directional heat–electric\nresponses \\cite{HHH1,HHH2,HHH3}. Singular elliptic equations also arise in many\napplications, featuring blow-up behaviour, boundary singularities\n\\cite{CD,ZD}. These problems often require\ntechniques beyond the classical isotropic theory \\cite{V}, especially when\ncombined with nonstandard growth conditions \\cite{R1,R2}.  \n\nLet \\(\\Omega \\subset \\mathbb{R}^{N}\\) be a bounded open set, where \\(N \\geq 2\\), and let \n\\((p_{j}) = (p_{1}, \\ldots, p_{N}) \\in \\mathbb{R}^{N}\\) satisfy\n\\[\n1 \\leq p^{-} = \\min_{j \\in J}\\{p_{j}\\} \\leq p^{+} = \\max_{j \\in J}\\{p_{j}\\} < \\infty .\n\\]\n\nThe anisotropic Sobolev spaces \\(W^{1,(p_{j})}(\\Omega)\\) and \\(W_{0}^{1,(p_{j})}(\\Omega)\\) are defined respectively by\n\\[\n\\begin{aligned}\nW^{1,(p_{j})}(\\Omega)\n&= \\big\\{ v \\in W^{1,1}(\\Omega) :\\; D_{j}v \\in L^{p_{j}}(\\Omega)\\ \\text{for all } j \\in J \\big\\},\\\\[2mm]\nW_{0}^{1,(p_{j})}(\\Omega)\n&= \\big\\{ v \\in W_{0}^{1,1}(\\Omega) :\\; D_{j}v \\in L^{p_{j}}(\\Omega)\\ \\text{for all } j \\in J \\big\\}.\n\\end{aligned}\n\\]\n\nEquivalently, \\(W_{0}^{1,(p_{j})}(\\Omega)\\) is the closure of \\(C_{0}^{\\infty}(\\Omega)\\) with respect to the norm\n\\[\n\\|v\\|_{W_{0}^{1,(p_{j})}(\\Omega)}\n= \\sum_{j \\in J} \\|D_{j}v\\|_{L^{p_{j}}(\\Omega)}.\n\\]\n\nEndowed with this norm, \\(W_{0}^{1,(p_{j})}(\\Omega)\\) is a separable and reflexive Banach space. \nIts dual space is denoted by \\(W^{-1,(p'_{j})}(\\Omega)\\), where for each \\(j \\in J\\), \\(p_{j}'\\) is the conjugate exponent of \\(p_{j}\\), that is,\n\\[\n\\frac{1}{p_{j}} + \\frac{1}{p_{j}'} = 1.\n\\]\n\nThis article's main goal is to investigate the following problem.\n\\begin{equation}\n\\label{(1)}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+b(x)\\sum\\limits_{j\\in J}\\frac{\\vert D_{j}u\\vert^{p_{j}}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwhere $\\Omega$ is a bounded open subset of $\\mathbb{R}^{N}$ ($N\\geq 3$), $f$, $q$, $\\theta$, and $p_{j}$ are satisfies  the following conditions \n\\begin{align}\n&\\label{(2)}\nq>0,\\quad 0<\\theta<1,\\\\\n&\\label{(3)}\nf\\geq0,\\;\\; f\\not\\equiv0,\\quad f\\in L^{1}(\\Omega),\\\\\n&\\label{(4)}\n2\\leq p_{1}\\leq p_{2}\\leq ... \\leq p_{N-1}\\leq p_{N}\\quad \\mbox{and}\\quad 2\\leq \\overline{p}:=N\\left(\\sum\\limits_{j\\in J}\\frac{1}{p_{j}}\\right)^{-1}<N.\n\\end{align}\nFor some positive numbers $\\alpha,\\beta,\\mu$, and $\\nu$, we assume in this study that $a,b$ are measurable functions satisfying the following conditions.\n\\begin{align}\n&\\label{(5)}\n 0<\\alpha\\leq a(x)\\leq \\beta,\\\\\n&\\label{(6)}\n0<\\mu\\leq b(x)\\leq \\nu.\n\\end{align}\nThe isotropic case of problem \\eqref{(1)}, where $p_j$ is constant for all $j \\in J$, has been extensively studied by several authors, with notable contributions in \\cite{olivia, boca92, boca97, dalill}. Also, a problem like \\eqref{(1)} without the singular term was studied in more general context in \\cite{FR1,FA1}. \n\nA further motivation for this work comes from the isotropic analogue of problem \\eqref{(1)}, studied in \\cite{R0}. There, the authors considered\n\\begin{equation}\n\\label{P3}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\mbox{div}\\left(\\left[a(x)+u^{q}\\right]\\nabla u\\right)+b(x)\\frac{\\vert \\nabla u\\vert^{2}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwith $q>0$, $0<\\theta<1$, and $f\\in L^{1}(\\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the  absorption term $q$ and the singular term's exponent $\\theta$. \nThe present work investigates the same type of phenomenon in an \\textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\\overline{p}$ replaces the Laplacian structure. \n\nThe problem \\eqref{P3} in the case of $q = 0$ has been addressed by several authors, including \\cite{A1, A2, A3, A4, G1, G2}.\n\n In the context of anisotropic elliptic problems like \\eqref{(1)} with no absorption term (i.e., when $q=0$, $a\\equiv 1$, and $b\\equiv 0)$, Di Castro in \\cite{R2} addressed the following problem:\n\\begin{equation*}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)=f& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation*}\nwhere $p_j$ increases with respect to $j$ such that $2 - \\frac{1}{N} \\leq \\overline{p} < N$, and $f \\in L^1(\\Omega)$. The author demonstrated that this problem admits a  solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where\n\\begin{align}\n\\label{HH}\n1\\leq s_{j}<\\frac{N(\\overline{p}-1)}{\\overline{p}(N-1)}p_{j},\\quad\\forall\\; j\\in J.\n\\end{align}\nand for all  $\\Phi \\in C_0^1(\\Omega),$\n\\begin{equation}\\label{HH1}\n\\sum\\limits_{j\\in J}\\int_{\\Omega}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi\\,dx =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{equation}\n\nFurther, this problem has been studied under different assumptions on the data in \\cite{Stamp, BO1, BO2, OR1, BO3}, particularly when $p_j$ is constant for all $j \\in J$.\n\n In the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \\cite{R1} explored the following  general anisotropic problem\n\\begin{equation}\n\\label{P2}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+\\sum\\limits_{j\\in J}g_{j}(x,u,\\nabla u)=f& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwhere $f \\in L^1(\\Omega)$ and $g_j: \\Omega \\times \\mathbb{R} \\times \\mathbb{R}^N \\to \\mathbb{R}$ is a Carath\\'eodory function satisfying the following conditions\n\\begin{itemize}\n\\item[(H1)] $g_{j}(x,s,\\xi)\\mbox{sgn}(s)\\geq0$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$;\n\\item[(H2)] $\\vert g_{j}(x,s,\\xi)\\vert\\leq h(\\vert s\\vert)\\vert\\xi_{j}\\vert^{p_{j}}$, for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$, where $h:\\mathbb{R}\\rightarrow \\mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\\gamma>0$ for $\\vert t\\vert$ sufficiently large;\n\\item[(H3)] There exists $\\nu>0$ such that $\\vert g_{j}(x,s,\\xi)\\vert\\geq \\nu \\vert \\xi_{j}\\vert^{p_{j}}$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$.\n\\end{itemize}\nThe author investigated the existence and regularity of solutions to this problem, yielding the following results\n\\begin{itemize}\n\\item[$\\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality\n$$\\frac{\\overline{p}(N-1)}{N(\\overline{p}-1)}<p_{j}<\\frac{\\overline{p}(N-1)}{N-\\overline{p}},\\quad \\forall\\; j\\in J,$$\nthen there exists at least one  solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where $s_j$ is defined in \\eqref{HH} and \\eqref{HH1} holds.\n\\item[$\\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\\Omega)$.\n\\end{itemize} \n\nOn the other hand, when \\(b = 0\\) and \\(q \\neq 0\\), we have recently addressed, see \\cite{FH}, a corresponding problem involving a singular right-hand side of the type \\(\\displaystyle\\frac{f}{u^{l}}\\), with \\(l > 0\\).\n\nWe end this  brief history of the problem by recalling the paper \\cite{bacii}, in their work, the authors establish the existence of weak solutions for a broad class of anisotropic Dirichlet problems of the type\n\\[\n\\mathcal{A}u + \\Phi(x,u,\\nabla u) = \\Psi(u,\\nabla u) + \\mathcal{L}u + f \\quad \\text{in } \\Omega,\n\\]\nwhere \\(\\Omega \\subset \\mathbb{R}^N\\) \\((N \\ge 2)\\) is a bounded domain and \\(f \\in L^1(\\Omega)\\) is arbitrary.\nThe anisotropic principal operator is\n\\[\n\\mathcal{A}u = -\\sum_{j=1}^N \\partial_j\\left( \\vert\\partial_j u\\vert^{p_j - 2}\\partial_j u \\right),\n\\]\nand the lower-order term \\(\\Phi\\) is given by\n\\[\n\\Phi(x,u,\\nabla u) = \\left( 1 + \\sum_{j=1}^N \\mathfrak{a}_j \\vert\\partial_j u\\vert^{p_j} \\right)|u|^{m-2}u,\n\\]\nwith parameters \\(m,p_j>1\\), \\(\\mathfrak{a}_j \\ge 0\\), and \\(\\sum_{k=1}^N \\frac{1}{p_k} > 1\\).\nA key novelty of the study is the inclusion of a possibly singular gradient-dependent nonlinearity\n\\[\n\\Psi(u,\\nabla u) = \\sum_{j=1}^N |u|^{\\theta_j - 2}u , \\vert\\partial_j u\\vert^{q_j},\n\\]\nwhere \\(\\theta_j>0\\) and \\(0 \\le q_j < p_j\\).\nThe lower-order operator $\\mathcal{L}$ belongs to the general class $\\mathcal{BC}$ introduced in \\cite{baci}. \nThe class $\\mathcal{BC}$ consists of \\emph{bounded} operators\n\\[\n\\mathcal{L} : W_0^{1,(p_j)}(\\Omega) \\rightarrow W^{-1,(p_j')}(\\Omega)\n\\]\nthat satisfy the following structural conditions:\n\n\\begin{itemize}\n    \\item[(P$_1$)]  \n    The operator $\\mathcal{A} - \\mathcal{L}$ is coercive on $W_0^{1,(p_j)}(\\Omega)$, namely,\n    \\[\n    \\frac{\\langle \\mathcal{A}u - \\mathcal{L}u,\\, u \\rangle}{\\|u\\|_{W_0^{1,(p_j)}(\\Omega)}} \\to \\infty \n    \\qquad \\text{as } \\|u\\|_{W_0^{1,(p_j)}(\\Omega)} \\to \\infty .\n    \\]\n\n    \\item[(P$_2$)]  \n    If $u_l \\rightharpoonup u$ and $v_l \\rightharpoonup v$ weakly in $W_0^{1,(p_j)}(\\Omega)$, then\n    \\[\n    \\lim_{l \\to \\infty} \\langle \\mathcal{L}u_l,\\, v_l \\rangle\n    = \\langle \\mathcal{L}u,\\, v \\rangle .\n    \\]\n\\end{itemize}\n\nThese assumptions allow the authors to include a broad family of lower-order terms while preserving the coercivity and compactness properties needed for their existence theory.\n\nThe authors obtain existence results under two different regimes:\n\\begin{enumerate}\n\\item when \\(\\theta_j > 1\\) for every \\(j = 1,\\dots, N\\);\n\n\\item when \\(\\theta_j \\le 1\\) for at least one index \\(j\\).\n\\end{enumerate}\n   In the latter case, by assuming \\(f \\ge 0\\) almost everywhere in \\(\\Omega\\), they further prove the existence of nonnegative weak solutions.\n\n In our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\\frac{\\vert D_{j} u\\vert^{p_{j}}}{\\vert u\\vert^{\\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term \n $$\n\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)$$ \nwhose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.\n\\begin{definition}\n\\label{Definition3.1}\nLet the condition in \\eqref{(3)} hold. A function $u\\in W_{0}^{1,1}(\\Omega)$ is a distributional solution to problem \\eqref{(1)} if\n\\begin{align}\n\\label{(13)}\n& u>0\\quad\\quad \\mbox{ a.e. in}\\; \\Omega,\\\\\n\\label{(14)}\n& \\sum\\limits_{j\\in J}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\in (L^{\\sigma_{j}}(\\Omega))^N,\\quad \\mbox{for all}\\; \\sigma_{j}<p_{j}',\\\\\n\\label{(15)}\n& \\sum\\limits_{j\\in J} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\in L^{1}(\\Omega),\n\\end{align}\nand for all  $\\Phi \\in W_0^{1,d_j}(\\Omega),\\; d_j>p_j$\n\\begin{align}\n\\label{(16)}\n&\\displaystyle\\sum\\limits_{j\\in J}\\int_{\\Omega}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi +\\sum\\limits_{j\\in J}\\int_{\\Omega} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\Phi =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{align}\n\\end{definition}\n\\par Our main results are as follows",
    "context": "A further motivation for this work comes from the isotropic analogue of problem \\eqref{(1)}, studied in \\cite{R0}. There, the authors considered\n\\begin{equation}\n\\label{P3}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\mbox{div}\\left(\\left[a(x)+u^{q}\\right]\\nabla u\\right)+b(x)\\frac{\\vert \\nabla u\\vert^{2}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwith $q>0$, $0<\\theta<1$, and $f\\in L^{1}(\\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the absorption term $q$ and the singular term's exponent $\\theta$. \nThe present work investigates the same type of phenomenon in an \\textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\\overline{p}$ replaces the Laplacian structure.\n\nIn the context of anisotropic elliptic problems like \\eqref{(1)} with no absorption term (i.e., when $q=0$, $a\\equiv 1$, and $b\\equiv 0)$, Di Castro in \\cite{R2} addressed the following problem:\n\\begin{equation*}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)=f& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation*}\nwhere $p_j$ increases with respect to $j$ such that $2 - \\frac{1}{N} \\leq \\overline{p} < N$, and $f \\in L^1(\\Omega)$. The author demonstrated that this problem admits a solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where\n\\begin{align}\n\\label{HH}\n1\\leq s_{j}<\\frac{N(\\overline{p}-1)}{\\overline{p}(N-1)}p_{j},\\quad\\forall\\; j\\in J.\n\\end{align}\nand for all $\\Phi \\in C_0^1(\\Omega),$\n\\begin{equation}\\label{HH1}\n\\sum\\limits_{j\\in J}\\int_{\\Omega}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi\\,dx =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{equation}\n\nIn the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \\cite{R1} explored the following general anisotropic problem\n\\begin{equation}\n\\label{P2}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+\\sum\\limits_{j\\in J}g_{j}(x,u,\\nabla u)=f& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwhere $f \\in L^1(\\Omega)$ and $g_j: \\Omega \\times \\mathbb{R} \\times \\mathbb{R}^N \\to \\mathbb{R}$ is a Carath\\'eodory function satisfying the following conditions\n\\begin{itemize}\n\\item[(H1)] $g_{j}(x,s,\\xi)\\mbox{sgn}(s)\\geq0$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$;\n\\item[(H2)] $\\vert g_{j}(x,s,\\xi)\\vert\\leq h(\\vert s\\vert)\\vert\\xi_{j}\\vert^{p_{j}}$, for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$, where $h:\\mathbb{R}\\rightarrow \\mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\\gamma>0$ for $\\vert t\\vert$ sufficiently large;\n\\item[(H3)] There exists $\\nu>0$ such that $\\vert g_{j}(x,s,\\xi)\\vert\\geq \\nu \\vert \\xi_{j}\\vert^{p_{j}}$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$.\n\\end{itemize}\nThe author investigated the existence and regularity of solutions to this problem, yielding the following results\n\\begin{itemize}\n\\item[$\\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality\n$$\\frac{\\overline{p}(N-1)}{N(\\overline{p}-1)}<p_{j}<\\frac{\\overline{p}(N-1)}{N-\\overline{p}},\\quad \\forall\\; j\\in J,$$\nthen there exists at least one solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where $s_j$ is defined in \\eqref{HH} and \\eqref{HH1} holds.\n\\item[$\\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\\Omega)$.\n\\end{itemize}\n\n\\item when \\(\\theta_j \\le 1\\) for at least one index \\(j\\).\n\\end{enumerate}\n In the latter case, by assuming \\(f \\ge 0\\) almost everywhere in \\(\\Omega\\), they further prove the existence of nonnegative weak solutions.\n\nIn our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\\frac{\\vert D_{j} u\\vert^{p_{j}}}{\\vert u\\vert^{\\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term \n $$\n\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)$$ \nwhose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.\n\\begin{definition}\n\\label{Definition3.1}\nLet the condition in \\eqref{(3)} hold. A function $u\\in W_{0}^{1,1}(\\Omega)$ is a distributional solution to problem \\eqref{(1)} if\n\\begin{align}\n\\label{(13)}\n& u>0\\quad\\quad \\mbox{ a.e. in}\\; \\Omega,\\\\\n\\label{(14)}\n& \\sum\\limits_{j\\in J}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\in (L^{\\sigma_{j}}(\\Omega))^N,\\quad \\mbox{for all}\\; \\sigma_{j}<p_{j}',\\\\\n\\label{(15)}\n& \\sum\\limits_{j\\in J} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\in L^{1}(\\Omega),\n\\end{align}\nand for all $\\Phi \\in W_0^{1,d_j}(\\Omega),\\; d_j>p_j$\n\\begin{align}\n\\label{(16)}\n&\\displaystyle\\sum\\limits_{j\\in J}\\int_{\\Omega}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi +\\sum\\limits_{j\\in J}\\int_{\\Omega} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\Phi =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{align}\n\\end{definition}\n\\par Our main results are as follows",
    "full_context": "A further motivation for this work comes from the isotropic analogue of problem \\eqref{(1)}, studied in \\cite{R0}. There, the authors considered\n\\begin{equation}\n\\label{P3}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\mbox{div}\\left(\\left[a(x)+u^{q}\\right]\\nabla u\\right)+b(x)\\frac{\\vert \\nabla u\\vert^{2}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwith $q>0$, $0<\\theta<1$, and $f\\in L^{1}(\\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the absorption term $q$ and the singular term's exponent $\\theta$. \nThe present work investigates the same type of phenomenon in an \\textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\\overline{p}$ replaces the Laplacian structure.\n\nIn the context of anisotropic elliptic problems like \\eqref{(1)} with no absorption term (i.e., when $q=0$, $a\\equiv 1$, and $b\\equiv 0)$, Di Castro in \\cite{R2} addressed the following problem:\n\\begin{equation*}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)=f& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation*}\nwhere $p_j$ increases with respect to $j$ such that $2 - \\frac{1}{N} \\leq \\overline{p} < N$, and $f \\in L^1(\\Omega)$. The author demonstrated that this problem admits a solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where\n\\begin{align}\n\\label{HH}\n1\\leq s_{j}<\\frac{N(\\overline{p}-1)}{\\overline{p}(N-1)}p_{j},\\quad\\forall\\; j\\in J.\n\\end{align}\nand for all $\\Phi \\in C_0^1(\\Omega),$\n\\begin{equation}\\label{HH1}\n\\sum\\limits_{j\\in J}\\int_{\\Omega}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi\\,dx =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{equation}\n\nIn the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \\cite{R1} explored the following general anisotropic problem\n\\begin{equation}\n\\label{P2}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+\\sum\\limits_{j\\in J}g_{j}(x,u,\\nabla u)=f& \\hbox{in}\\;\\Omega, \\\\\n u =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nwhere $f \\in L^1(\\Omega)$ and $g_j: \\Omega \\times \\mathbb{R} \\times \\mathbb{R}^N \\to \\mathbb{R}$ is a Carath\\'eodory function satisfying the following conditions\n\\begin{itemize}\n\\item[(H1)] $g_{j}(x,s,\\xi)\\mbox{sgn}(s)\\geq0$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$;\n\\item[(H2)] $\\vert g_{j}(x,s,\\xi)\\vert\\leq h(\\vert s\\vert)\\vert\\xi_{j}\\vert^{p_{j}}$, for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$, where $h:\\mathbb{R}\\rightarrow \\mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\\gamma>0$ for $\\vert t\\vert$ sufficiently large;\n\\item[(H3)] There exists $\\nu>0$ such that $\\vert g_{j}(x,s,\\xi)\\vert\\geq \\nu \\vert \\xi_{j}\\vert^{p_{j}}$ for a.e. $x\\in \\Omega$, for all $s\\in \\mathbb{R}$ and $\\xi\\in \\mathbb{R}^{N}$.\n\\end{itemize}\nThe author investigated the existence and regularity of solutions to this problem, yielding the following results\n\\begin{itemize}\n\\item[$\\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality\n$$\\frac{\\overline{p}(N-1)}{N(\\overline{p}-1)}<p_{j}<\\frac{\\overline{p}(N-1)}{N-\\overline{p}},\\quad \\forall\\; j\\in J,$$\nthen there exists at least one solution $u \\in W_0^{1, (s_j)}(\\Omega)$, where $s_j$ is defined in \\eqref{HH} and \\eqref{HH1} holds.\n\\item[$\\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\\Omega)$.\n\\end{itemize}\n\n\\item when \\(\\theta_j \\le 1\\) for at least one index \\(j\\).\n\\end{enumerate}\n In the latter case, by assuming \\(f \\ge 0\\) almost everywhere in \\(\\Omega\\), they further prove the existence of nonnegative weak solutions.\n\nIn our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\\frac{\\vert D_{j} u\\vert^{p_{j}}}{\\vert u\\vert^{\\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term \n $$\n\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)$$ \nwhose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.\n\\begin{definition}\n\\label{Definition3.1}\nLet the condition in \\eqref{(3)} hold. A function $u\\in W_{0}^{1,1}(\\Omega)$ is a distributional solution to problem \\eqref{(1)} if\n\\begin{align}\n\\label{(13)}\n& u>0\\quad\\quad \\mbox{ a.e. in}\\; \\Omega,\\\\\n\\label{(14)}\n& \\sum\\limits_{j\\in J}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\in (L^{\\sigma_{j}}(\\Omega))^N,\\quad \\mbox{for all}\\; \\sigma_{j}<p_{j}',\\\\\n\\label{(15)}\n& \\sum\\limits_{j\\in J} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\in L^{1}(\\Omega),\n\\end{align}\nand for all $\\Phi \\in W_0^{1,d_j}(\\Omega),\\; d_j>p_j$\n\\begin{align}\n\\label{(16)}\n&\\displaystyle\\sum\\limits_{j\\in J}\\int_{\\Omega}\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u D_{j}\\Phi +\\sum\\limits_{j\\in J}\\int_{\\Omega} b(x)\\frac{\\vert D_{j}u\\vert^{p_{j}}}{u^{\\theta}}\\Phi =\\int_{\\Omega}f\\Phi\\,dx .\n\\end{align}\n\\end{definition}\n\\par Our main results are as follows\n\n\\item when \\(\\theta_j \\le 1\\) for at least one index \\(j\\).\n\\end{enumerate}\n In the latter case, by assuming \\(f \\ge 0\\) almost everywhere in \\(\\Omega\\), they further prove the existence of nonnegative weak solutions.\n\nWe first note that hypothesis \\eqref{(4)} already ensures that \n\\[\n1 < r_{j} \\le p_{j} \\qquad \\text{for every } j \\in J.\n\\] \nMoreover, by combining assumptions \\eqref{(2)} and \\eqref{(4)}, we obtain the stronger condition \n\\[\nr_{j} > p_{j} - 1, \\qquad \\forall\\, j \\in J.\n\\]\n\nThe following lemma will be of central importance in the remainder of the proof, and it will be especially needed in the passage to the limit.\n\\begin{lem}\\label{lmq}\nAssume that the assumptions \\eqref{(2)}-\\eqref{(6)} hold true.\nThen, we have\n\\begin{equation}\\label{ugest}\n\\int_{\\Omega} u_{n}^{q\\sigma_j}\\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_j(p_{j}-1)}\\;dx \\leq C\\quad\\quad \\forall \\sigma_j<p_j^\\prime.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\n Taking $ \\varphi(u_n)=1-\\frac{1}{(1-u_n)^\\lambda},$ with $\\lambda>1,$ as test function in the weak formulation \\eqref{(21)}, we get\n\\begin{equation*}\n(\\lambda-1)\\sum_{j\\in J}\\int_{\\Omega} \\frac{[a(x)+u_n^q]\\vert D_j u_n\\vert^{p_{j}}}{(1+u_n)^\\lambda}+\\sum_{j\\in J}\\int_{\\Omega}b(x)\\frac{u_{n}\\vert D_{j}u_{n}\\vert^{p_{j}}}{\\left( u_{n}+\\frac{1}{n}\\right)^{\\theta+1}}\\varphi =\\int_\\Omega f_n \\varphi(u_n).\n\\end{equation*}\nWe recall that there exists a positive constant $\\mathcal{C}$ such that\n$$(a(x)+u_n^q) \\geq \\mathcal{C}(1+u_n)^q,\\quad \\quad \\forall q>1.$$ \nDropping the positive term and using the fact that $\\vert \\varphi(u_n)\\vert\\leq 1,$ we obtain\n\\begin{equation}\\label{est1}\n\\sum_{j\\in J}\\int_{\\Omega} \\frac{\\vert D_j u_n\\vert^{p_j}}{(1+u_n)^{\\lambda-q}}\\leq\\frac{1}{\\mathcal{C}(\\lambda-1)}\\int_\\Omega \\vert f_n\\vert .\n\\end{equation}\nFor $1 < \\sigma_{j} < p'_{j} = \\frac{p_{j}}{p_{j} - 1}$, an application of H\\\"older's inequality together with estimate \\eqref{est1} yields, for every $j \\in J$,\n\\begin{align*}\n\\int_{\\Omega} u_{n}^{q \\sigma_{j}(p_{j}-1)}\\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \\leq C\\left( \\int_{\\Omega} (1+u_{n})^{\\frac{\\sigma_{j}(p_{j}-1)(\\lambda-q+p_{j} q)}{p_{j}-\\sigma_{j}(p_{j}-1)}}\\right)^{\\frac{p_{j}-\\sigma_{j}(p_{j}-1)}{p_{j}}}.\n\\end{align*}\nWe now introduce the parameter $\\sigma_{j} = \\theta\\frac{p_{j}}{p_{j}-1}$, with $\\theta \\in (0,1)$. Under this choice, one readily verifies that\n$$\n\\frac{p_{j}-\\sigma_{j}(p_{j}-1)}{p_{j}} = 1 - \\theta,\\quad \\forall\\; j\\in J,$$\nand\n$$ \\frac{\\sigma_{j}(p_{j}-1)(\\lambda-q+p_{j} q)}{p_{j}-\\sigma_{j}(p_{j}-1)} = \\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)],\\quad \\forall\\; j\\in J.\n$$\nSubstituting into the previous estimate leads to\n\\begin{equation}\n\\label{E11}\n\\int_{\\Omega} u_{n}^{q \\sigma_{j}(p_{j}-1)}\\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \\leq C\\left( \\int_{\\Omega}(1+ u_{n})^{\\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)]}\\right)^{1-\\theta}, \n\\end{equation}\nfor every $j\\in J$. Hence, we deduce the following inequality:\n\\begin{align*}\n&\\prod_{i\\in J}\\left(\\int_{\\Omega} u_{n}^{q \\sigma_{j}(p_{j}-1)}\\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)}\\;dx\\right)^{\\frac{1}{\\theta p_{j}}}\\\\\n&\\quad \\leq C\\prod_{i\\in J}\\left( \\int_{\\Omega} (1+u_{n})^{\\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)]}\\;dx\\right)^{\\frac{1-\\theta}{\\theta p_{j}}}.\n\\end{align*}\nBy invoking the anisotropic inequality \\eqref{(9)} with $\\delta_{j} = \\sigma_{j}(p_{j}-1) \\leq p_{j}$ (due to $\\theta < 1$), $t_{j} = q \\geq 0$, and $\\overline{\\delta} = \\theta \\overline{p}$, we obtain\n\\begin{equation}\\label{E12}\n\\left(\\int_{\\Omega}u_{n}^{r}\\right)^{\\frac{N}{\\theta\\overline{p}}-1} \\leq C\\prod_{i\\in J}\\left( \\int_{\\Omega} (1+u_{n})^{\\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)]}\\right)^{\\frac{1-\\theta}{\\theta p_{j}}}.\n\\end{equation}\nIn order for the preceding inequality to hold uniformly for all $j \\in J$, we must enforce that\n$$\nr = \\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)], \\quad \\forall\\, j\\in J.\n$$\nThis leads to the following system of equations:\n\\begin{align}\n& r=\\frac{1+q}{b_{j}(N-1)-1+\\frac{1}{\\theta p_{j}}}, \\quad \\forall j\\in J, \\label{S1} \\\\\n& r=\\frac{\\theta}{1-\\theta}[\\lambda+q(p_{j}-1)], \\quad \\forall j\\in J, \\label{S2} \\\\\n&\\sum_{i\\in J}b_{j}=1, \\quad b_{j} \\geq 0, \\quad \\forall j\\in J. \\label{S3}\n\\end{align}\nBy combining equations \\eqref{S1} and \\eqref{S3}, one deduces\n$$\nr = \\frac{N(1+q)\\theta \\overline{p}}{N-\\theta\\overline{p}}.\n$$\nIn parallel, equation \\eqref{S2} yields\n$$\n\\theta = \\frac{N(\\overline{p}-\\lambda+q)}{\\overline{p}[N(1+q)-q\\overline{p}-\\lambda+q]}.\n$$\nSince $q \\geq 0$ and $\\lambda > 1$, it follows that $0 < \\theta < 1$. Consequently, by combining these expressions, we arrive at\n$$\nr = \\frac{N(\\overline{p}-\\lambda+q)}{N-\\overline{p}}.\n$$\nUsing this value of $r$ in inequality \\eqref{E12}, we find\n$$\n\\left(\\int_{\\Omega}u_{n}^{r}\\right)^{\\frac{N}{\\theta\\overline{p}}-1} \\leq C\\left(\\int_{\\Omega}u_{n}^{r}\\right)^{\\frac{(1-\\theta)N}{\\theta \\overline{p}}}.\n$$\nSince the assumption $\\overline{p} < N$ ensures that the left-hand exponent exceeds the right-hand one, we conclude that $\\{u_{n}\\}$ is bounded in $L^{r}(\\Omega)$. This bound, together with estimate \\eqref{E11}, implies that\n$$\n\\int_{\\Omega} u_{n}^{q \\sigma_{j}(p_{j}-1)}\\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \\leq C, \\quad \\forall\\, j\\in J.\n$$\nGiven that $p_{j} > 2$ and $\\sigma_{j} > 1$ for all $j\\in\\mathcal{E}$, we apply Hölder’s inequality to obtain\n\\begin{align*}\n\\int_{\\Omega} u_{n}^{q \\sigma_{j}} \\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \n&= \\int_{\\left\\{u_{n}<1\\right\\}} u_{n}^{q \\sigma_{j}} \\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \\\\\n&\\qquad+ \\int_{\\left\\{u_{n} \\geq 1\\right\\}} u_{n}^{q \\sigma_{j}} \\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)}\\\\\n&\\leq \\int_{\\Omega} \\left\\vert D_{j} \\mathcal{T}_{1}(u_{n})\\right\\vert^{\\sigma_{j}(p_{j}-1)}\n+ \\int_{\\Omega} u_{n}^{q \\sigma_{j}(p_{j}-1)} \\left\\vert D_{j} u_{n}\\right\\vert^{\\sigma_{j}(p_{j}-1)} \\\\\n&\\leq C \\left(\\int_{\\Omega} \\left\\vert D_{j} \\mathcal{T}_{1}(u_{n})\\right\\vert^{p_{j}} \\right)^{\\frac{\\sigma_{j}(p_{j}-1)}{p_{j}}} + C \\\\\n&\\leq C.\n\\end{align*}\nThus, it follows that the sequence $\\left\\lbrace u_{n}^{q}\\left\\vert D_{j} u_{n}\\right\\vert^{p_{j}-1}\\right\\rbrace$ is uniformly bounded in $L^{\\sigma_{j}}(\\Omega)$ for all $1 < \\sigma_{j} < p'_{j}$ and for each $j \\in J$.\n\\end{proof}\nLet us specify some useful notation we will use from now on. For any \\(n\\in \\mathbb{N}^{*}\\) we define the following auxiliary functions:\n\\begin{equation*}\n\\mathcal{H}_{n}(l)=\\int_{0}^{l}\\frac{dt}{\\left(t+\\frac{1}{n}\\right)^{\\theta}},\n\\qquad\n\\mathcal{H}_{\\infty}(l)=\\int_{0}^{l}\\frac{dt}{t^{\\theta}},\n\\end{equation*}\nand, let $\\gamma=\\frac{\\nu}{\\alpha}>0,$ we denote\n\\begin{equation*}\n\\Psi_{n}(l)=e^{-\\gamma \\mathcal{H}_{n}(l)},\\qquad \\Psi_{\\infty}(l)=e^{-\\gamma \\mathcal{H}_{\\infty}(l)}.\n\\end{equation*}\nNote that, the function \\(\\mathcal{H}_{\\infty}\\) is well defined because \\(\\theta<1\\). Moreover, it's clear that\n$$\\lim_{n\\rightarrow \\infty}\\mathcal{H}_{n}(l)=\\mathcal{H}_{\\infty}(l)\\text{ and }\\lim_{n\\rightarrow \\infty}\\Psi_{n}(l)=\\Psi_{\\infty}(l).$$\nObserve that, for any \\(\\phi, u_{n}\\in W_{0}^{1,(p_{j})}(\\Omega)\\cap L^{\\infty}(\\Omega)\\) with \\(\\phi\\ge0\\) , we have \\(\\Psi_{n}(u_{n})\\phi\\in W_{0}^{1,(p_{j})}(\\Omega)\\cap L^{\\infty}(\\Omega)\\).",
    "post_theorem_intro_text_len": 2770,
    "post_theorem_intro_text": "We now give a brief description of the proof of our main theorem. The first step consists in removing the degeneracy of the principal operator and the singularity of the lower-order gradient term. For this purpose, we consider a sequence of approximate problems associated with \\eqref{(1)}, where both difficulties are regularized. In the second step, we establish suitable a priori estimates for the approximate solutions. The main challenge here is to control the term\n$$-\\sum_{j \\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\right)$$\nin an appropriate anisotropic Lebesgue space. Finally, by using Fatou’s lemma together with a standard compactness argument, we pass to the limit in the approximate problems and obtain a weak solution to the original equation.\n\\begin{remark}\nWe collect here several observations related to the assumptions of Theorem~\\ref{T2}.\n\nWe first note that hypothesis \\eqref{(4)} already ensures that  \n\\[\n1 < r_{j} \\le p_{j} \\qquad \\text{for every } j \\in J.\n\\]  \nMoreover, by combining assumptions \\eqref{(2)} and \\eqref{(4)}, we obtain the stronger condition  \n\\[\nr_{j} > p_{j} - 1, \\qquad \\forall\\, j \\in J.\n\\]\n\nWe also observe that the condition \\(1 - \\theta < q\\) appearing in Theorem~\\ref{T2}(ii) is equivalent to  \n\\[\n\\frac{N(\\overline{p}-\\theta)}{\\overline{p}(N-\\theta)} p_{j}  \n\\;<\\;\n\\frac{N(\\overline{p}-1+q)}{\\overline{p}(N+q-1)} p_{j},\n\\qquad \\forall\\, j \\in J.\n\\]  \nThis relation will be useful when comparing the integrability exponents arising in the proof.\n\nFurthermore, the hypothesis \\(q \\le 1\\) in Theorem~\\ref{T2}(ii) automatically implies that  \n\\[\n1 < \\eta_{j} < p_{j},\n\\]  \nwhich plays an important role in establishing the required estimates.\n\nIt is also worth mentioning that, in the isotropic case \\(p_{j} = 2\\) for all \\(j \\in J\\), Theorem~\\ref{T2} reduces to known regularity results for classical elliptic equations. In particular, our conclusions coincide with those of Theorem~1.1 in \\cite{R0}.\n\nFinally, by applying the Sobolev embedding theorem, we can describe the integrability of the solutions.  \nFor Theorem~\\ref{T2}(i), the solution \\(u\\) also satisfies  \n\\[\nu \\in L^{\\overline{r}^{*}}(\\Omega),\n\\qquad \n\\overline{r}^{*} = \\frac{N(\\overline{p}-\\theta)}{N-\\overline{p}}.\n\\]  \nFor Theorem~\\ref{T2}(ii), the solution enjoys the additional property  \n\\[\nu \\in L^{s}(\\Omega),\n\\qquad \\text{for every }\\; s < \\frac{N(\\overline{p}-1+q)}{\\,N-\\overline{p}\\,}.\n\\]\n\\end{remark}\nFor ease of reading, Section 2 begins with a brief recall of the main analytical tools of anisotropic Sobolev spaces and the definition of the approximate problem. We then state the essential a priori estimates Lemmas as well as some convergences results, which form the main step on the proof, and finally we pass to the limit.",
    "sketch": "To prove Theorem~\\ref{T2}, the authors give the following outline. First, they \"remove the degeneracy of the principal operator and the singularity of the lower-order gradient term\" by introducing \"a sequence of approximate problems associated with \\eqref{(1)}, where both difficulties are regularized.\" Second, they \"establish suitable a priori estimates for the approximate solutions\"; the main difficulty is \"to control the term\n$$-\\sum_{j \\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\right)$$\nin an appropriate anisotropic Lebesgue space.\" Finally, \"by using Fatou’s lemma together with a standard compactness argument,\" they \"pass to the limit in the approximate problems and obtain a weak solution to the original equation.\" They further indicate that Section~2 recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and \"finally we pass to the limit.\"",
    "expanded_sketch": "To prove the main theorem, the authors give the following outline. First, they “remove the degeneracy of the principal operator and the singularity of the lower-order gradient term” by introducing “a sequence of approximate problems associated with\n\\begin{equation}\n\\label{(1)}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+b(x)\\sum\\limits_{j\\in J}\\frac{\\vert D_{j}u\\vert^{p_{j}}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\n, where both difficulties are regularized.” Second, they “establish suitable a priori estimates for the approximate solutions”; the main difficulty is “to control the term\n$$-\\sum_{j \\in J} D_{j}\\left(u^{q}\\vert D_{j}u\\vert^{p_{j}-2}D_{j}u\\right)$$\nin an appropriate anisotropic Lebesgue space.” Finally, “by using Fatou’s lemma together with a standard compactness argument,” they “pass to the limit in the approximate problems and obtain a weak solution to the original equation.” They further indicate that the next part recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and “finally we pass to the limit.”",
    "expanded_theorem": "\\label{T2}\nAssume\n\\begin{align}\n&\\label{(2)}\nq>0,\\quad 0<\\theta<1,\\\\\n&\\label{(3)}\nf\\geq0,\\;\\; f\\not\\equiv0,\\quad f\\in L^{1}(\\Omega),\\\\\n&\\label{(4)}\n2\\leq p_{1}\\leq p_{2}\\leq ... \\leq p_{N-1}\\leq p_{N}\\quad \\mbox{and}\\quad 2\\leq \\overline{p}:=N\\left(\\sum\\limits_{j\\in J}\\frac{1}{p_{j}}\\right)^{-1}<N.\n\\end{align}\n--\n\\begin{align}\n&\\label{(5)}\n 0<\\alpha\\leq a(x)\\leq \\beta,\\\\\n&\\label{(6)}\n0<\\mu\\leq b(x)\\leq \\nu.\n\\end{align}\nhold. Then,\n\\begin{itemize}\n\\item[(i)] If $0<q<1-\\theta$, there exists a distributional solution $u\\in W_{0}^{1,(r_{j})}(\\Omega)$ to problem\n\\begin{equation}\n\\label{(1)}\n\\left\\{\\begin{array}{ll}\\displaystyle-\\sum\\limits_{j\\in J} D_{j}\\left(\\left[a(x)+u^{q}\\right]\\vert D_{j}u\\vert^{p_{j}-2} D_{j}u\\right)+b(x)\\sum\\limits_{j\\in J}\\frac{\\vert D_{j}u\\vert^{p_{j}}}{ u^{\\theta}}=f& \\hbox{in}\\;\\Omega, \\\\\nu>0& \\hbox{in}\\;\\Omega, \\\\\n u  =0 & \\hbox{on}\\; \\partial\\Omega, \\end{array}\n \\right.\n\\end{equation}\nsuch that\n\\begin{align}\n\\label{(17)}\nr_{j}=\\frac{N(\\overline{p}-\\theta)}{\\overline{p}(N-\\theta)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(ii)] If $1-\\theta<q\\leq1$, there exists a distributional solution $u\\in W_{0}^{1,(\\eta_{j})}(\\Omega)$ to the above problem such that\n\\begin{align}\n\\label{(017)}\nr_{j}<\\frac{N(\\overline{p}-1+q)}{\\overline{p}(N+q-1)}p_{j},\\quad \\forall \\; j\\in J.\n\\end{align}\n\\item[(iii)] If $q>1$, there exists a distributional solution $u\\in W_{0}^{1,(p_{j})}(\\Omega)$ to the above problem.\n\\end{itemize}",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(J=\\{1,\\dots,N\\}\\). Consider the anisotropic Dirichlet problem\n\\[\n\\begin{cases}\n-\\displaystyle\\sum_{j\\in J} D_j\\!\n\\left(\\left[a(x)+u^{q}\\right]\\,|D_j u|^{p_j-2}D_j u\\right)\n+ b(x)\\displaystyle\\sum_{j\\in J}\\frac{|D_j u|^{p_j}}{u^{\\theta}}=f & \\text{in } \\Omega,\\\\[2mm]\nu>0 & \\text{in } \\Omega,\\\\\nu=0 & \\text{on } \\partial\\Omega.\n\\end{cases}\n\\]\nAssume\n\\[\nq>0,\\qquad 0<\\theta<1,\n\\]\n\\[\nf\\ge 0,\\quad f\\not\\equiv 0,\\quad f\\in L^1(\\Omega),\n\\]\n\\[\n2\\le p_1\\le p_2\\le \\cdots \\le p_N,\n\\qquad\n2\\le \\overline p:=N\\left(\\sum_{j\\in J}\\frac1{p_j}\\right)^{-1}<N,\n\\]\nand the coefficients satisfy, a.e. in \\(\\Omega\\),\n\\[\n0<\\alpha\\le a(x)\\le \\beta,\n\\qquad\n0<\\mu\\le b(x)\\le \\nu.\n\\]\nHere \\(W_0^{1,(s_j)}(\\Omega)\\) denotes the anisotropic Sobolev space of zero-trace functions such that \\(D_j u\\in L^{s_j}(\\Omega)\\) for each \\(j\\), and a distributional solution means a function with zero trace, positive a.e. in \\(\\Omega\\), satisfying the equation in the sense of distributions. Which conclusion about existence and regularity is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "The problem admits a distributional solution with the following case-by-case regularity: (i) if \\(0<q<1-\\theta\\), then there exists \\(u\\in W_0^{1,(r_j)}(\\Omega)\\) with\n\\[\nr_j=\\frac{N(\\overline p-\\theta)}{\\overline p\\,(N-\\theta)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(ii) if \\(1-\\theta<q\\le 1\\), then there exists a distributional solution \\(u\\in W_0^{1,(\\eta_j)}(\\Omega)\\) with directional exponents satisfying\n\\[\n\\eta_j<\\frac{N(\\overline p-1+q)}{\\overline p\\,(N+q-1)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(iii) if \\(q>1\\), then there exists a distributional solution \\(u\\in W_0^{1,(p_j)}(\\Omega)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The problem admits a distributional solution with the following case-by-case regularity: (i) if \\(0<q\\le 1-\\theta\\), then there exists \\(u\\in W_0^{1,(r_j)}(\\Omega)\\) with\n\\[\nr_j=\\frac{N(\\overline p-\\theta)}{\\overline p\\,(N-\\theta)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(ii) if \\(1-\\theta<q\\le 1\\), then there exists a distributional solution \\(u\\in W_0^{1,(\\eta_j)}(\\Omega)\\) with directional exponents satisfying\n\\[\n\\eta_j\\le \\frac{N(\\overline p-1+q)}{\\overline p\\,(N+q-1)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(iii) if \\(q>1\\), then there exists a distributional solution \\(u\\in W_0^{1,(p_j)}(\\Omega)\\)."
        },
        {
          "label": "C",
          "text": "The problem admits a distributional solution with the following case-by-case regularity: (i) if \\(0<q<1-\\theta\\), then there exists \\(u\\in W_0^{1,(r_j)}(\\Omega)\\) with\n\\[\nr_j=\\frac{N(\\overline p-\\theta)}{\\overline p\\,(N-\\theta)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(ii) if \\(1-\\theta<q\\le 1\\), then there exists a distributional solution \\(u\\in W_0^{1,(\\eta_j)}(\\Omega)\\) for some directional exponents \\((\\eta_j)_{j\\in J}\\); \n(iii) if \\(q>1\\), then there exists a distributional solution \\(u\\in W_0^{1,(p_j)}(\\Omega)\\)."
        },
        {
          "label": "D",
          "text": "The problem admits a distributional solution with the following case-by-case regularity: (i) if \\(0<q<1-\\theta\\), then there exists \\(u\\in W_0^{1,(r_j)}(\\Omega)\\) with\n\\[\nr_j=\\frac{N(\\overline p-\\theta)}{\\overline p\\,(N-\\theta)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(ii) if \\(1-\\theta<q\\le 1\\), then there exists a distributional solution \\(u\\in W_0^{1,(\\eta_j)}(\\Omega)\\) with directional exponents satisfying\n\\[\n\\eta_j<\\frac{N(\\overline p-1+q)}{\\overline p\\,(N+q-1)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(iii) if \\(q>1\\), then there exists a distributional solution \\(u\\in W_0^{1,(p_j)}(\\Omega)\\), and moreover for every \\(q>0\\) one may choose the solution in the full energy space \\(W_0^{1,(p_j)}(\\Omega)\\)."
        },
        {
          "label": "E",
          "text": "The problem admits a distributional solution with the following case-by-case regularity: (i) if \\(0<q<1-\\theta\\), then there exists \\(u\\in W_0^{1,(r_j)}(\\Omega)\\) with\n\\[\nr_j=\\frac{N(\\overline p-\\theta)}{\\overline p\\,(N-\\theta)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(ii) if \\(1-\\theta\\le q\\le 1\\), then there exists a distributional solution \\(u\\in W_0^{1,(\\eta_j)}(\\Omega)\\) with directional exponents satisfying\n\\[\n\\eta_j<\\frac{N(\\overline p-1+q)}{\\overline p\\,(N+q-1)}\\,p_j,\\qquad \\forall j\\in J;\n\\]\n(iii) if \\(q>1\\), then there exists a distributional solution \\(u\\in W_0^{1,(p_j)}(\\Omega)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
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        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical threshold and strict upper bound in the intermediate regime",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "explicit quantitative upper bound on the exponents in case (ii)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "loss of full anisotropic energy regularity for small/intermediate q caused by the delicate estimate on the nonlinear term",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "excluded borderline case q=1-theta",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the full hypotheses of the PDE theorem, but no direct cue singles out choice A over the nearby variants."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-conclusion identification item: the stem states the assumptions and asks for the corresponding existence statement. The distractors introduce threshold and exponent variations, so it is not a pure verbatim restatement, but it remains close to theorem recall."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some discrimination among subtle alternatives (strict vs non-strict inequalities, exact regime splits, equality vs upper-bound regularity), but it mainly tests recognition/recall of a specific result rather than genuine derivation or generative mathematical reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "B, D, and E are plausible and mathematically targeted distractors. However, C is problematic because it reads like a weaker paraphrase that may also be taken as true unless the question explicitly asks for the most precise statement, creating potential ambiguity."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated but mostly theorem-recall MCQ. It avoids answer leakage and has several strong near-miss distractors, but it does not strongly test generative reasoning, and the weaker paraphrase option introduces ambiguity."
    }
  },
  {
    "id": "2512.08395v1",
    "paper_link": "http://arxiv.org/abs/2512.08395v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Hao_Kodaira_codim_1}\\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}\n    Let $X$ be a smooth projective variety of Kodaira dimension $\\kappa(X)=\\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\\'etale covering $X'\\rightarrow X^{min}$ such that any $\\mathbb{Q}$-factorialization $X''$ of $X'$ is a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.",
      "start_pos": 4374,
      "end_pos": 4933,
      "label": "Hao_Kodaira_codim_1"
    },
    "ref_dict": {
      "Hao_Kodaira_codim_1": "\\begin{theorem}\\label{Hao_Kodaira_codim_1}\\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}\n    Let $X$ be a smooth projective variety of Kodaira dimension $\\kappa(X)=\\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\\'etale covering $X'\\rightarrow X^{min}$ such that any $\\Q$-factorialization $X''$ of $X'$ is a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.\n\\end{theorem}",
      "main_result": "\\begin{theorem}\\label{main_result}\n    Let $X$ be a compact K\\\"ahler manifold of Kodaira dimension $\\kappa(X)=\\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then there exist a finite \\'etale cover $X'\\rightarrow X$ such that $X'$ is bimeromorphic to an elliptic fiber bundle $Z\\rightarrow B$ with trivial monodromy over a smooth projective base $B$ of general type.\n\\end{theorem}",
      "precise_main_result": "\\begin{theorem}\\label{precise_main_result}\n    Let $X$ be a compact K\\\"ahler manifold of Kodaira dimension $\\kappa(X)=\\dim X-1$ that admits a holomorphic 1-form without zeros. Then there is a smooth projective variety $Y$ of general type and of dimension $\\dim Y=\\dim X-1$, an elliptic curve $E=\\C/\\Lambda$, a cohomology class $\\eta\\in H^1(Y,\\OO_Y/\\Lambda)$ and a finite \\'etale cover $X'\\rightarrow X$ such that $X'$ is bimeromorphic to $(Y\\times E)^\\eta$.\n\\end{theorem}",
      "example_no_splitting": "\\begin{remark}\\label{example_no_splitting}\n    \\Cref{fundamental_properties_special_torus_bundles} allows us to construct a compact K\\\"ahler manifold $X$ of Kodaira dimension $\\kappa(X)=\\dim X-1$ that admits a holomorphic 1-form without zeros such that no finite \\'etale cover of $X$ splits into a product where one factor is an elliptic curve. This shows that \\Cref{Hao_Kodaira_codim_1} fails in the K\\\"ahler case. Indeed, let $C$ be a smooth projective curve of genus $g\\geq 2$, and let $E=\\C/\\Lambda$ be an elliptic curve. Consider the exact sequence\n    \\begin{equation*}\n        H^1(C,\\Lambda)\\longrightarrow H^1(C,\\OO_C)\\stackrel{\\mathrm{exp}}{\\longrightarrow}H^1(C,\\OO_C/\\Lambda)\\stackrel{c}{\\longrightarrow}H^2(C,\\Lambda),\n    \\end{equation*}\n    and choose a class $\\alpha\\in H^1(C,\\OO_C)$ that is not in the image of\n    \\begin{equation*}\n        H^1(C,\\Lambda)\\otimes\\Q\\longrightarrow H^1(C,\\OO_C).\n    \\end{equation*}\n    Since $H^1(C,\\OO_C)$ is a non-trivial complex vector space, and $H^1(C,\\Lambda)$ is a finitely generated abelian group, such an $\\alpha$ exists. Define $\\eta:=\\mathrm{exp}(\\alpha)$. Note that the choice of $\\alpha$ implies that $\\eta$ is non-torsion. The exactness of the above sequence implies that the Chern class $c(\\eta)$ vanishes. Thus, by \\Cref{fundamental_properties_special_torus_bundles} \\cref{torus_bundle_kaehler} and \\cref{1-forms_on_torus_bundles}, the space $X:=(C\\times E)^\\eta$ is a compact K\\\"ahler manifold that admits a holomorphic 1-form without zeros. Since $C$ is of genus $g\\geq 2$, $\\kappa(X)=\\dim C=\\dim X-1$. We claim that no finite \\'etale cover of $X$ splits into a product where one factor is an elliptic curve. To show this, suppose that a finite \\'etale cover $X'\\rightarrow X$ splits into a product $X'\\cong D\\times F$, where $F$ is an elliptic curve. Since $g(C)\\geq 2$, the composition\n    \\begin{equation*}\n        \\{x\\}\\times F\\longhookrightarrow D\\times F\\longrightarrow X\\longrightarrow C\n    \\end{equation*}\n    must be constant for every $x\\in D$, and hence there exists a morphism $D\\rightarrow C$ such that the composition $D\\times F\\rightarrow X\\rightarrow C$ factors through $D\\times F\\rightarrow D\\rightarrow C$. Since $D\\times F\\rightarrow X\\rightarrow C$ is surjective, the morphism $D\\rightarrow C$ cannot be constant. Therefore, the image $S\\subset X$ of $D\\times\\{0\\}\\rightarrow X$ defines a multisection of $X\\rightarrow C$. Thus, by \\Cref{fundamental_properties_special_torus_bundles} \\cref{torus_bundle_multisection_a}, the cohomology class $\\eta$ is torsion, which is a contradiction.\n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 463,
    "pre_theorem_intro_text": "Let $X$ be a smooth projective variety over $\\mathbb{C}$ with a nowhere vanishing holomorphic 1-form $\\omega\\in H^0(X,\\Omega^1_X)$. In \\cite{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, Popa and Schnell showed that the Kodaira dimension of $X$ is bounded by $\\kappa(X)\\leq\\dim X-1$. To prove this result, Popa-Schnell applied the theory of mixed Hodge modules. A fortiori, Hao gave the following geometric explanation of this behavior, for $\\kappa(X)=\\dim X-1$.",
    "context": "Let $X$ be a smooth projective variety over $\\mathbb{C}$ with a nowhere vanishing holomorphic 1-form $\\omega\\in H^0(X,\\Omega^1_X)$. In \\cite{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, Popa and Schnell showed that the Kodaira dimension of $X$ is bounded by $\\kappa(X)\\leq\\dim X-1$. To prove this result, Popa-Schnell applied the theory of mixed Hodge modules. A fortiori, Hao gave the following geometric explanation of this behavior, for $\\kappa(X)=\\dim X-1$.",
    "full_context": "Let $X$ be a smooth projective variety over $\\mathbb{C}$ with a nowhere vanishing holomorphic 1-form $\\omega\\in H^0(X,\\Omega^1_X)$. In \\cite{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, Popa and Schnell showed that the Kodaira dimension of $X$ is bounded by $\\kappa(X)\\leq\\dim X-1$. To prove this result, Popa-Schnell applied the theory of mixed Hodge modules. A fortiori, Hao gave the following geometric explanation of this behavior, for $\\kappa(X)=\\dim X-1$.\n\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\OO}{\\mathcal{O}}\n\\newcommand{\\pic}{\\mathrm{Pic}}\n\\newcommand{\\im}{\\mathrm{im}}\n\\newcommand{\\Hom}{\\mathrm{Hom}}\n\\newcommand{\\Ext}{\\mathrm{Ext}}\n\\newcommand{\\id}{\\mathrm{id}}\n\n\\title{Holomorphic one-forms without zeros on K\\\"ahler manifolds of Kodaira codimension one}\n\\author{Simon Pietig}\n\\subjclass{32Q15, 32J18, 32Q57, 32H04}\n\\address{Leibniz Universit\\\"at Hannover, Institut f\\\"ur Algebraische Geometrie, Welfengarten 1, 30167 Hannover}\n\\address{\\href{mailto:pietig@math.uni-hannover.de}{pietig@math.uni-hannover.de}}\n\\address{\\href{https://orcid.org/0009-0001-1800-6799}{Orcid iD}}\n\n@book{Eisenbud_Harris_3264,\n author = {Eisenbud, David and Harris, Joe},\n title = {3264 and all that: Intersection theory in algebraic geometry},\n year = {2016},\n publisher = {Cambridge Univ. Press}\n}\n\n@article{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One,\n author = {Hao, Feng},\n title = {Nowhere Vanishing Holomorphic One-Forms on Varieties of Kodaira Codimension One},\n journal = {Int. Math. Res. Not. IMRN},\n volume = {2024},\n number = {6},\n pages = {4501--4515},\n year = {2023}\n}\n\n@article{Popa_Schnell_Kodaira_dimension_zeros_1-forms,\n author = {Popa, Mihnea and Schnell, Christian},\n title = {Kodaira dimension and zeros of holomorphic one-forms},\n journal = {Ann. of Math.},\n volume = {179},\n number = {3},\n pages = {1109--1120},\n year = {2017}\n}\n\n@online{Hao_Schreieder_erratum,\n author = {Hao, Feng and Schreieder, Stefan},\n title = {Corrections to \"Holomorphic one-forms\" without zeros on threefolds},\n url = {https://www.iag.uni-hannover.de/fileadmin/iag/homepages/schreieder/publications/Correction_for_nowhere_vanishing_1-forms_on_3-folds_2.pdf},\n year = {2025}\n}\n\n@misc{church_gmm,\n title={Nowhere vanishing 1-forms on varieties admitting a good minimal model}, \n author={Church, Benjamin},\n year={2024},\n eprint={2410.22753},\n archivePrefix={arXiv},\n primaryClass={math.AG},\n url={https://arxiv.org/abs/2410.22753}, \n}\n\n@misc{Hao_Wang_Zhang_gmm,\n title={Good minimal models with nowhere vanishing holomorphic $1$-forms}, \n author={Hao, Feng and Wang, Zichang and Zhang, Lei},\n year={2024},\n eprint={2412.12582},\n archivePrefix={arXiv},\n primaryClass={math.AG},\n url={https://arxiv.org/abs/2412.12582}, \n}",
    "post_theorem_intro_text_len": 6254,
    "post_theorem_intro_text": "Recent work of Church \\cite{church_gmm} and of Hao, Wang and Zhang \\cite{Hao_Wang_Zhang_gmm} generalized Hao's result to a birational classification of smooth projective varieties with $g$ everywhere linearly independent holomorphic 1-forms assuming the abundance conjecture without any restrictions of the Kodaira dimension of $X$.\\par\nThe main result of our paper is the following generalization of \\Cref{Hao_Kodaira_codim_1} to the K\\\"ahler case. For a more precise version, see \\Cref{precise_main_result} down below.\n\\begin{theorem}\\label{main_result}\n    Let $X$ be a compact K\\\"ahler manifold of Kodaira dimension $\\kappa(X)=\\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then there exist a finite \\'etale cover $X'\\rightarrow X$ such that $X'$ is bimeromorphic to an elliptic fiber bundle $Z\\rightarrow B$ with trivial monodromy over a smooth projective base $B$ of general type.\n\\end{theorem}\n\\begin{remark}\n    In contrast to the projective situation, one cannot expect that the finite \\'etale covering space $X'$ of $X$ in \\Cref{main_result} is bimeromorphic to a product $Z\\cong B\\times E$, see \\Cref{example_no_splitting} below.\n\\end{remark}\n\\subsection{Outline of the argument}\nIn the projective case, Hao argues as follows: Let $X$ be a smooth projective variety over $\\mathbb{C}$ of Kodaira codimension 1 that admits a holomorphic 1-form $\\omega\\in H^0(X,\\Omega^1_X)$ without zeros. Let $X^{min}$ be a good minimal model of $X$, and consider the Iitaka fibration $f\\colon X^{min}\\rightarrow Y$. By \\cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, the holomorphic 1-form $\\omega$ cannot come from $Y$. In particular, the general fiber of $f$ is not contracted by the Albanese morphism. Therefore, it is mapped to a translate of a fixed elliptic curve $E\\subset Alb(X)$. As $Alb(X)$ is projective, we can dualize the inclusion $E\\subset Alb(X)$ to get a morphism\n\\begin{equation*}\n    \\varphi\\colon X^{min}\\longrightarrow Alb(X)\\longrightarrow E.\n\\end{equation*}\nThe variety $B$ in \\Cref{Hao_Kodaira_codim_1}, as well as the splitting $X''\\cong B\\times E$ of a $\\mathbb{Q}$-factorialization $X''\\rightarrow X'$ of a finite quasi-\\'etale cover $X'\\rightarrow X^{min}$ are then constructed from a general fiber of $\\varphi\\colon X^{min}\\rightarrow E$.\\par\nNow, let $X$ be a compact connected K\\\"ahler manifold of Kodaira codimension 1 that admits a holomorphic 1-form $\\omega\\in H^0(X,\\Omega^1_X)$ without zeros. Note that we cannot apply the result by Popa-Schnell \\cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}. Moreover, $Alb(X)$ is not projective, and therefore, the morphism $\\varphi$ does not exists in general.\\par\nTo overcome these obstacles, we invoke Lin's construction of an algebraic approximation via tautological models and families \\cite{Lin_algebraic_approximation_codim_1} to obtain the following diagram\n\\[\\begin{tikzcd}\n\t{\\mathcal{X}} && {\\mathcal{T}/G} \\\\\n\t& V && {Y/G.}\n\t\\arrow[\"\\sim\", dashed, from=1-1, to=1-3]\n\t\\arrow[\"\\pi\"', from=1-1, to=2-2]\n\t\\arrow[\"{\\Pi/G}\", from=1-3, to=2-2]\n\t\\arrow[\"p/G\"', from=1-3, to=2-4]\n\\end{tikzcd}\\]\nHere, $\\pi\\colon\\mathcal{X}\\rightarrow V$ is a deformation of $X\\cong\\mathcal{X}_0:=\\pi^{-1}(0)$ over a complex vector space $V$, and $\\mathcal{T}$ is a complex analytic variety on which a finite group $G$ acts. The morphism\n\\begin{equation*}\n    \\Pi\\colon\\mathcal{T}\\longrightarrow Y\\times V\\longrightarrow V\n\\end{equation*}\nis an explicitly constructed $G$-equivariant family of elliptic fibrations $p_v\\colon\\Pi^{-1}(v)\\rightarrow Y$ over a fixed smooth projective base $Y$, called the tautological family. The central elliptic fibration\n\\begin{equation*}\n    p_0/G\\colon\\Pi^{-1}(0)/G\\longrightarrow Y/G\n\\end{equation*}\nis bimeromorphic to the Iitaka fibration of $X$. Moreover, the deformation $\\pi\\colon\\mathcal{X}\\rightarrow V$ has the property that the set of points $v\\in V$ such that $\\mathcal{X}_v:=\\pi^{-1}(v)$ is a projective manifold, is dense in the euclidean topology. The two families $\\pi\\colon\\mathcal{X}\\rightarrow V$ and $\\Pi/G\\colon\\mathcal{T}/G\\rightarrow V$ are bimeromorphic over $V$. The elliptic fibrations $p_v\\colon\\Pi^{-1}(v)\\rightarrow Y$ are locally isomorphic over $Y$, i.e. there is an open cover $\\{U_i\\}_{i\\in I}$ such that for every $U_i$ the isomorphism class of the elliptic fibrations\n\\begin{equation*}\n    p_v\\colon\\Pi^{-1}(v)\\vert_{U_i}\\longrightarrow U_i\n\\end{equation*}\ndoes not depend on $v\\in V$.\\par\nWe then show that a small deformation of $X$ also admits a holomorphic 1-form without zeros. In particular, there is a point $v_0\\in V$ such that $\\mathcal{X}_{v_0}$ is a projective manifold that admits a holomorphic 1-form without zeros. As the Iitaka fibration of $\\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G\\colon\\Pi^{-1}(v_0)/G\\rightarrow Y/G$, and the elliptic fibrations $p_v\\colon\\Pi^{-1}(v)\\rightarrow Y$ are locally isomorphic over $Y$, we conclude that the general fiber of the Iitaka fibration of any $\\mathcal{X}_v$ is isomorphic to a fixed elliptic curve. This overcomes the first obstacle.\\par\nTo finish the argument, we show that the elliptic fibration $p_0\\colon\\Pi^{-1}(0)\\rightarrow Y$ is locally trivial, i.e. locally isomorphic to a product. It remains to carefully describe the $G$-action on this fibration. Taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\\rightarrow B$.\n\\subsection{Conventions and notation}\nBy a complex analytic variety, we mean a Hausdorff, second-countable, irreducible, and reduced complex space. A complex analytic variety is called K\\\"ahler if it admits a K\\\"ahler metric in the sense of \\cite[][II. 1]{Varouchas_Kaehler_spaces_proper_open_morphisms}. A compact complex analytic K\\\"ahler variety $X$ is called minimal if it is $\\mathbb{Q}$-factorial, has terminal singularities, and the canonical divisor $K_X$ is nef. It is furthermore called a good minimal model, if a positive multiple of $K_X$ is globally generated\n\\subsection*{Acknowledgments}\nI would like to thank Stefan Schreieder for suggesting this problem to me, for very helpful discussions, and for comments on this manuscript. The research was conducted in the framework of the DFG-funded research training group RTG 2965: From Geometry to Numbers.",
    "sketch": "In the projective case, Hao’s argument is summarized as follows: take a good minimal model $X^{\\min}$ and its Iitaka fibration $f\\colon X^{\\min}\\to Y$. By \\cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, the nowhere-vanishing $1$-form $\\omega$ “cannot come from $Y$”. Hence “the general fiber of $f$ is not contracted by the Albanese morphism”, so it maps to “a translate of a fixed elliptic curve $E\\subset \\operatorname{Alb}(X)$”. Since $\\operatorname{Alb}(X)$ is projective, one “dualize[s] the inclusion $E\\subset \\operatorname{Alb}(X)$ to get a morphism” $\\varphi\\colon X^{\\min}\\to \\operatorname{Alb}(X)\\to E$, and then “$B$ … as well as the splitting $X''\\cong B\\times E$ … are then constructed from a general fiber of $\\varphi\\colon X^{\\min}\\to E$.\n\nFor the K\\\"ahler generalization (Theorem~\\ref{main_result}), two obstacles are noted: one cannot apply Popa–Schnell, and $\\operatorname{Alb}(X)$ is not projective so “the morphism $\\varphi$ does not exists in general.” To overcome this, the authors “invoke Lin’s construction of an algebraic approximation via tautological models and families” to produce a deformation $\\pi\\colon \\mathcal{X}\\to V$ of $X$ and a $G$-equivariant tautological family of elliptic fibrations $\\Pi\\colon \\mathcal{T}\\to Y\\times V\\to V$, with $\\Pi^{-1}(0)/G\\to Y/G$ bimeromorphic to the Iitaka fibration of $X$, and with $\\mathcal{X}\\to V$ having a dense set of projective fibers; moreover $\\mathcal{X}$ and $\\mathcal{T}/G$ are “bimeromorphic over $V$” and the elliptic fibrations $p_v\\colon \\Pi^{-1}(v)\\to Y$ are “locally isomorphic over $Y$” (independent of $v$ on an open cover).\n\nThey then “show that a small deformation of $X$ also admits a holomorphic 1-form without zeros”, so there exists $v_0\\in V$ with $\\mathcal{X}_{v_0}$ projective and carrying such a form. Since the Iitaka fibration of $\\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G$ and the $p_v$ are locally isomorphic, they “conclude that the general fiber of the Iitaka fibration of any $\\mathcal{X}_v$ is isomorphic to a fixed elliptic curve,” addressing the first obstacle. “To finish the argument,” they “show that the elliptic fibration $p_0\\colon \\Pi^{-1}(0)\\to Y$ is locally trivial” (locally a product), then “carefully describe the $G$-action,” and “taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\\to B$” with trivial monodromy.",
    "expanded_sketch": "In the projective case, Hao’s argument is summarized as follows: take a good minimal model $X^{\\min}$ and its Iitaka fibration $f\\colon X^{\\min}\\to Y$. By Popa–Schnell, “Kodaira dimension and zeros of holomorphic 1-forms” (Thm. 2.1), the nowhere-vanishing $1$-form $\\omega$ “cannot come from $Y$”. Hence “the general fiber of $f$ is not contracted by the Albanese morphism”, so it maps to “a translate of a fixed elliptic curve $E\\subset \\operatorname{Alb}(X)$”. Since $\\operatorname{Alb}(X)$ is projective, one “dualize[s] the inclusion $E\\subset \\operatorname{Alb}(X)$ to get a morphism” $\\varphi\\colon X^{\\min}\\to \\operatorname{Alb}(X)\\to E$, and then “$B$ … as well as the splitting $X''\\cong B\\times E$ … are then constructed from a general fiber of $\\varphi\\colon X^{\\min}\\to E$.\n\nFor the K\\\"ahler generalization (Theorem~\\ref{main_result}), two obstacles are noted: one cannot apply Popa–Schnell, and $\\operatorname{Alb}(X)$ is not projective so “the morphism $\\varphi$ does not exists in general.” To overcome this, the authors “invoke Lin’s construction of an algebraic approximation via tautological models and families” to produce a deformation $\\pi\\colon \\mathcal{X}\\to V$ of $X$ and a $G$-equivariant tautological family of elliptic fibrations $\\Pi\\colon \\mathcal{T}\\to Y\\times V\\to V$, with $\\Pi^{-1}(0)/G\\to Y/G$ bimeromorphic to the Iitaka fibration of $X$, and with $\\mathcal{X}\\to V$ having a dense set of projective fibers; moreover $\\mathcal{X}$ and $\\mathcal{T}/G$ are “bimeromorphic over $V$” and the elliptic fibrations $p_v\\colon \\Pi^{-1}(v)\\to Y$ are “locally isomorphic over $Y$” (independent of $v$ on an open cover).\n\nThey then “show that a small deformation of $X$ also admits a holomorphic 1-form without zeros”, so there exists $v_0\\in V$ with $\\mathcal{X}_{v_0}$ projective and carrying such a form. Since the Iitaka fibration of $\\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G$ and the $p_v$ are locally isomorphic, they “conclude that the general fiber of the Iitaka fibration of any $\\mathcal{X}_v$ is isomorphic to a fixed elliptic curve,” addressing the first obstacle. “To finish the argument,” they “show that the elliptic fibration $p_0\\colon \\Pi^{-1}(0)\\to Y$ is locally trivial” (locally a product), then “carefully describe the $G$-action,” and “taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\\to B$” with trivial monodromy.",
    "expanded_theorem": "\\label{Hao_Kodaira_codim_1}\\\\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}\n    Let $X$ be a smooth projective variety of Kodaira dimension $\\kappa(X)=\\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\\'etale covering $X'\\rightarrow X^{min}$ such that any $\\mathbb{Q}$-factorialization $X''$ of $X'$ is a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.,",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $X$ be a smooth projective variety over $\\mathbb{C}$ with Kodaira dimension $\\kappa(X)=\\dim X-1$, and assume that $X$ admits a nowhere vanishing holomorphic $1$-form. A finite quasi-\\'etale covering means a finite surjective morphism that is \\`etale in codimension one, and a $\\mathbb{Q}$-factorialization of a normal variety $X'$ means a small birational morphism $X''\\to X'$ with $X''$ $\\mathbb{Q}$-factorial. Under these assumptions, which statement about minimal models of $X$ is valid?",
      "correct_choice": {
        "label": "A",
        "text": "For any minimal model $X^{\\min}$ of $X$, there exists a finite quasi-\\'etale covering $X'\\to X^{\\min}$ such that every $\\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve."
      },
      "choices": [
        {
          "label": "B",
          "text": "For any minimal model $X^{\\min}$ of $X$, there exists a finite quasi-\\'etale covering $X'\\to X^{\\min}$ such that $X'$ itself is isomorphic to a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve."
        },
        {
          "label": "C",
          "text": "For any minimal model $X^{\\min}$ of $X$, there exists a finite quasi-\\'etale covering $X'\\to X^{\\min}$ and a $\\mathbb{Q}$-factorialization $X''$ of $X'$ such that $X''$ is isomorphic to a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve."
        },
        {
          "label": "D",
          "text": "There exists a minimal model $X^{\\min}$ of $X$ such that for every finite quasi-\\'etale covering $X'\\to X^{\\min}$, every $\\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve."
        },
        {
          "label": "E",
          "text": "For any minimal model $X^{\\min}$ of $X$, there exists a finite \\`etale covering $X'\\to X^{\\min}$ such that every $\\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\\times E$, where $B$ is a minimal model with $\\kappa(B)=\\dim B-1$ and $E$ is an elliptic curve."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "need_for_Q-factorialization_before_splitting",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "universal_quantifier_on_every_Q-factorialization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "quantifier_order_on_minimal_model_and_cover",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "quasi-etale_vs_etale_and_general_type_of_base",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the target conclusion; it only gives assumptions and definitions needed to parse the options. There is no direct textual leakage of choice A."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a mere restatement of a theorem in the stem. The options differ by subtle quantifier structure and strength ('every' vs. 'some', quasi-étale vs. étale, cover vs. factorialization), so the item asks the student to distinguish competing conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some reasoning about logical strength and theorem precision, especially around quantifiers and the role of Q-factorialization. However, it is closer to exact theorem recognition than to substantial generative mathematical reasoning, and the ambiguity introduced by another true option weakens the pressure."
      },
      "DQS": {
        "score": 0,
        "justification": "The distractors are not fully well-formed because choice C is a weaker statement implied by A, so it is also valid. That makes the single-correct-answer format defective, even though B, D, and E are plausibly designed around common quantifier and regularity mistakes."
      },
      "total_score": 5,
      "overall_assessment": "Conceptually sophisticated and non-tautological, with no answer leakage, but flawed as an MCQ because at least one distractor is also true, undermining uniqueness and assessment quality."
    }
  },
  {
    "id": "2512.08409v1",
    "paper_link": "http://arxiv.org/abs/2512.08409v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .",
      "start_pos": 4422,
      "end_pos": 5550,
      "label": "thm:KPS"
    },
    "ref_dict": {
      "prop:Second-Isomorphism-Classification": "\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}",
      "rem:Link-remark": "\\begin{rem}\n\\label{rem:Link-remark}In the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nwe constructed for a threefold $X=X_{22}^{m}(v)$, $v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty\\}$,\nwhere $X_{22}^{m}(-4)=X_{22}^{MU}$, and a pair $(Z,Z')$ of $\\mathbb{G}_{m}$-stable\nlines of special type in $X$, a biregular involution $\\theta$ of\n$X$ semi-commuting with the $\\mathbb{G}_{m}$-action and exchanging\nthe lines $Z$ and $Z'$. Let us briefly indicate a complementary\ninterpretation of the construction of this involution $\\theta$. With\nthe notation of the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nthe composition of the double projection $\\psi_{Z}:X\\dashrightarrow V_{5}$\nfrom $Z$ with the projection $\\pi_{(\\psi_{Z})_{*}Z'}:V_{5}\\dashrightarrow Q$\nfrom the line $(\\psi_{Z})_{*}Z'$ is a $\\mathbb{G}_{m}$-equivariant\nbirational map which contracts the unique hyperplane section $S_{Z}$\nof $X$ of multiplicity $3$ along $Z$ onto a rational normal quartic\ncurve $\\Gamma^{Q}$, contracts the unique hyperplane section $R_{\\{Z,Z'\\}}$\nof $X$ singular along $Z\\cup Z'$ onto a rational normal cubic curve\n$C$ and maps the the unique hyperplane section $S_{Z'}$ of $X$\nsingular along $Z'$ onto a cubic section $S_{Z'}^{Q}$ of $Q$ singular\nalong $\\Gamma^{Q}$. Letting $\\alpha_{\\Gamma^{Q}}:\\hat{Q}\\to Q$ be\nthe blow-up of $\\Gamma^{Q}$, where $\\hat{Q}\\subset\\mathbb{P}^{16}$\nis a Fano threefold of Picard rank $2$ and degree $28$ of type 2.21\nin \\cite{MoMu81}, we obtain an induced $\\mathbb{G}_{m}$-equivariant\nbirational map \n\\[\n\\hat{\\gamma}=\\alpha_{(\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z})_{*}S_{Z}}^{-1}\\circ\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z}:X\\dashrightarrow\\hat{Q}\n\\]\nwhich fits into the following commutative diagram of $\\mathbb{G}_{m}$-equivariant\nbirational maps\n\\[\n\\xymatrix{ &  & R'_{\\{Z,Z'\\}}\\subseteq Y\\ar@{.>}[r]^{\\hat{\\chi}}\\ar@{->}[lld]_{\\hat{\\sigma}} & Y^{+}\\supseteq R^{+}{}_{\\{Z,Z'\\}}\\ar@{->}[rrd]^{\\hat{\\tau}}\\\\\nZ\\cup Z'\\subset X\\ar@{-->}[rrrrr]^{\\hat{\\gamma}} &  &  &  &  & \\hat{Q}\\supset\\hat{C}\\\\\n}\n\\]\nwhere $\\hat{\\sigma}:Y\\to X$ is the blow-up of $Z\\cup Z'$, $\\hat{\\chi}:Y\\dashrightarrow Y^{+}$\nis a small $\\mathbb{Q}$-factorial modification and $\\hat{\\tau}:Y^{+}\\to\\hat{Q}$\nis the contraction of the proper transform $R^{+}{}_{\\{Z,Z'\\}}$ of\n$R_{\\{Z,Z'\\}}$ onto a smooth curve $\\hat{C}\\subset\\hat{S}$ of bi-degree\n$(3,3)$. One can further check from the construction that $\\hat{\\gamma}$\nis given by the linear system of quadric sections of $X$ having multiplicity\n$3$ along $Z$ and $Z'$. The birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\nin the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nlifts to a biregular involution $\\iota_{\\hat{Q}}$ of $\\hat{Q}$ which\nnormalizes the lifted $\\mathbb{G}_{m}$-action and makes $\\hat{Q}$\ninto a Fano threefold of $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-Picard\nrank $1$, and by the construction of the associated biregular involution\n$\\theta$ of $X$, the above diagram is then a $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-equivariant\nSarkisov link. We refer the reader to the forthcoming article \\cite{DFK-next}\nfor a for a more in-depth and detailed study of these birational links\nbetween smooth prime Fano threefolds of degree 22 and blow-ups of\nsmooth quadric threefolds along rational normal quartic curves. \n\\end{rem}",
      "prop:The-crucial-involution": "\\begin{lem}\n\\label{prop:The-crucial-involution}Let $\\Gamma_{4}\\subset\\mathbb{P}^{4}$\nbe a rational normal quartic curve and let $T\\subset\\mathrm{Aut}(\\mathbb{P}^{4},\\Gamma_{4})\\cong\\mathrm{Aut}(\\Gamma_{4})\\cong\\mathrm{PGL}_{2}$\nbe a maximal torus. Then the following hold:\n\n1) For every smooth $T$-stable quadric $Q$ containing $\\Gamma_{4}$,\nthe linear system of quadric sections of $Q$ containing $\\Gamma_{4}$\ndetermines a birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\ncontracting the unique cubic section $S_{\\Gamma_{4}}$ of $Q$ singular\nalong $\\Gamma_{4}$ onto $\\Gamma_{4}$ and restricting to an automorphism\nof $Q\\setminus S_{\\Gamma_{4}}$ which semi-commutes with the induced\n$T$-action on it.\n\n2) Let $r,r'\\in\\Gamma_{4}$ be the two $T$-fixed points. Then the\nproper transform $(\\iota_{Q}^{-1})_{*}H_{r'}$ of the tangent hyperplane\nsection $H_{r'}$ to $Q$ at $r'$ is the unique integral quadric\nsection of $Q$ containing $\\Gamma_{4}$ and singular along the tangent\nline $L_{r}$ to $\\Gamma_{4}$ at $r$. The scheme $H_{r'}\\cap(\\iota_{Q}^{-1})_{*}H_{r'}$\nis the union of the line $L_{r',s}$ passing through $r'$ and the\nunique $T$-fixed point $s\\in L_{r}\\setminus\\{r\\}$ and of a rational\nnormal cubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$\nand passing through $s$. Moreover, $\\iota_{Q}$ maps $C_{2r',s}$\nisomorphically onto itself.\n\\end{lem}",
      "thm:KPS": "\\begin{thm}\n\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 540,
    "pre_theorem_intro_text": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):",
    "context": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):\n\n\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}",
    "full_context": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):\n\n\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}\n\n\\section*{Introduction}\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$.\n\nThe possibilities for $X$ were initially found by Prokhorov in \\cite{Pr90}\nand the articles \\cite{KPS18} and \\cite{KP18} completed later on\nthe structures of the automorphism groups ${\\rm Aut}(X_{22}^{a})$\nand ${\\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained\nalternative proof of Theorem \\ref{thm:KPS} and a complementary view\non certain of the constructions and results in these articles.\n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}\n\n$\\quad$ - If $\\Gamma=\\Gamma_{p}$ then $\\mathrm{Aut}(X,Z)=G_{p}$\nand $\\mathrm{Aut}(X,Z')\\subset\\mathrm{Aut}(X)$ is the conjugate of\n$\\mathrm{Aut}(X,Z)$ by the involution $\\theta$. Since the latter\nnormalizes $T$, $\\mathrm{Aut}(X,Z)$ and $\\theta$ generate a subgroup\nof $\\mathrm{Aut}(X)$ isomorphic to $\\mathrm{PGL}_{2}$, from which\nit follows, by Lemma \\ref{prop:line}, that $\\mathrm{Aut}^{0}(X)\\cong\\mathrm{PGL}_{2}$.\nFinally, since an element of $\\mathrm{Aut}(X)$ which stabilizes every\nline in $X$ is contained in the intersection of all Borel subgroups\nof $\\mathrm{Aut}^{0}(X)$, it must be trivial, which implies that\nthe canonically induced action of $\\mathrm{Aut}(X)$ on $(\\mathcal{H}_{X})_{\\mathrm{red}}\\cong\\mathbb{P}^{1}$\nis faithful. Thus, $\\mathrm{Aut}(X)$ identifies with a subgroup of\nthe automorphism group of $(\\mathcal{H}_{X})_{\\mathrm{red}}\\cong\\mathbb{P}^{1}$,\nhence is connected, equal to $\\mathrm{PGL}_{2}$. \n\\end{proof}\nIn the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nabove, we have used the following auxiliary result on the birational\ngeometry of smooth quadric threefolds with an effective action of\na $1$-dimensional torus. \n\\begin{lem}\n\\label{prop:The-crucial-involution}Let $\\Gamma_{4}\\subset\\mathbb{P}^{4}$\nbe a rational normal quartic curve and let $T\\subset\\mathrm{Aut}(\\mathbb{P}^{4},\\Gamma_{4})\\cong\\mathrm{Aut}(\\Gamma_{4})\\cong\\mathrm{PGL}_{2}$\nbe a maximal torus. Then the following hold:\n\nTo prove assertion 2), we can assume that $r=[0:0:0:0:1]$ and $r'=[1:0:0:0:0]$\nso that $H_{r'}=\\{w_{4}=0\\}\\cap Q$, $L_{r'}=\\{w_{2}=w_{3}=w_{4}=0\\}$\nand $L_{r}=\\{w_{0}=w_{1}=w_{2}=0\\}$. Then $(\\iota_{Q}^{-1})_{*}H_{r'}=\\{w_{0}w_{2}-w_{1}^{2}=0\\}\\cap Q$\nis an anti-canonically embedded del Pezzo surface of degree $4$ singular\nalong $L_{r}$ and $H_{r'}\\cap(\\iota_{Q}^{-1})_{*}H_{r'}=\\{w_{4}=w_{0}w_{2}-w_{1}^{2}=0\\}\\cap Q$\nis the union of the line $L_{r',s}=\\{w_{1}=w_{2}=w_{4}=0\\}$ where\n$s=[0:0:0:1:0]\\in L_{r}\\setminus\\{r\\}$ and of the rational normal\ncubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$ passing\nthrough $s$ given by the image of the morphism \n\\[\n\\alpha:\\mathbb{P}_{[u_{0}:u_{1}]}^{1}\\to\\mathbb{P}^{4},\\,[u_{0}:u_{1}]\\mapsto[u_{0}^{3}:u_{0}^{2}u_{1}:u_{0}u_{1}^{2}:(1-c^{2})u_{1}^{3}:0].\n\\]\nNow it is routine to verify from (\\ref{eq:The-involution}) that $\\iota_{Q}\\circ\\alpha=\\alpha\\circ\\iota_{c}$,\nwhere $\\iota_{c}$ is the involution $[u_{0}:u_{1}]\\mapsto[u_{1}:\\frac{c}{1-c^{2}}u_{0}]$\nof $\\mathbb{P}^{1}$. \n\\end{proof}\n\\begin{rem}\n\\label{rem:Link-remark}In the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nwe constructed for a threefold $X=X_{22}^{m}(v)$, $v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty\\}$,\nwhere $X_{22}^{m}(-4)=X_{22}^{MU}$, and a pair $(Z,Z')$ of $\\mathbb{G}_{m}$-stable\nlines of special type in $X$, a biregular involution $\\theta$ of\n$X$ semi-commuting with the $\\mathbb{G}_{m}$-action and exchanging\nthe lines $Z$ and $Z'$. Let us briefly indicate a complementary\ninterpretation of the construction of this involution $\\theta$. With\nthe notation of the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nthe composition of the double projection $\\psi_{Z}:X\\dashrightarrow V_{5}$\nfrom $Z$ with the projection $\\pi_{(\\psi_{Z})_{*}Z'}:V_{5}\\dashrightarrow Q$\nfrom the line $(\\psi_{Z})_{*}Z'$ is a $\\mathbb{G}_{m}$-equivariant\nbirational map which contracts the unique hyperplane section $S_{Z}$\nof $X$ of multiplicity $3$ along $Z$ onto a rational normal quartic\ncurve $\\Gamma^{Q}$, contracts the unique hyperplane section $R_{\\{Z,Z'\\}}$\nof $X$ singular along $Z\\cup Z'$ onto a rational normal cubic curve\n$C$ and maps the the unique hyperplane section $S_{Z'}$ of $X$\nsingular along $Z'$ onto a cubic section $S_{Z'}^{Q}$ of $Q$ singular\nalong $\\Gamma^{Q}$. Letting $\\alpha_{\\Gamma^{Q}}:\\hat{Q}\\to Q$ be\nthe blow-up of $\\Gamma^{Q}$, where $\\hat{Q}\\subset\\mathbb{P}^{16}$\nis a Fano threefold of Picard rank $2$ and degree $28$ of type 2.21\nin \\cite{MoMu81}, we obtain an induced $\\mathbb{G}_{m}$-equivariant\nbirational map \n\\[\n\\hat{\\gamma}=\\alpha_{(\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z})_{*}S_{Z}}^{-1}\\circ\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z}:X\\dashrightarrow\\hat{Q}\n\\]\nwhich fits into the following commutative diagram of $\\mathbb{G}_{m}$-equivariant\nbirational maps\n\\[\n\\xymatrix{ & & R'_{\\{Z,Z'\\}}\\subseteq Y\\ar@{.>}[r]^{\\hat{\\chi}}\\ar@{->}[lld]_{\\hat{\\sigma}} & Y^{+}\\supseteq R^{+}{}_{\\{Z,Z'\\}}\\ar@{->}[rrd]^{\\hat{\\tau}}\\\\\nZ\\cup Z'\\subset X\\ar@{-->}[rrrrr]^{\\hat{\\gamma}} & & & & & \\hat{Q}\\supset\\hat{C}\\\\\n}\n\\]\nwhere $\\hat{\\sigma}:Y\\to X$ is the blow-up of $Z\\cup Z'$, $\\hat{\\chi}:Y\\dashrightarrow Y^{+}$\nis a small $\\mathbb{Q}$-factorial modification and $\\hat{\\tau}:Y^{+}\\to\\hat{Q}$\nis the contraction of the proper transform $R^{+}{}_{\\{Z,Z'\\}}$ of\n$R_{\\{Z,Z'\\}}$ onto a smooth curve $\\hat{C}\\subset\\hat{S}$ of bi-degree\n$(3,3)$. One can further check from the construction that $\\hat{\\gamma}$\nis given by the linear system of quadric sections of $X$ having multiplicity\n$3$ along $Z$ and $Z'$. The birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\nin the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nlifts to a biregular involution $\\iota_{\\hat{Q}}$ of $\\hat{Q}$ which\nnormalizes the lifted $\\mathbb{G}_{m}$-action and makes $\\hat{Q}$\ninto a Fano threefold of $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-Picard\nrank $1$, and by the construction of the associated biregular involution\n$\\theta$ of $X$, the above diagram is then a $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-equivariant\nSarkisov link. We refer the reader to the forthcoming article \\cite{DFK-next}\nfor a for a more in-depth and detailed study of these birational links\nbetween smooth prime Fano threefolds of degree 22 and blow-ups of\nsmooth quadric threefolds along rational normal quartic curves. \n\\end{rem}",
    "post_theorem_intro_text_len": 4861,
    "post_theorem_intro_text": "The possibilities for $X$ were initially found by Prokhorov in \\cite{Pr90}\nand the articles \\cite{KPS18} and \\cite{KP18} completed later on\nthe structures of the automorphism groups ${\\rm Aut}(X_{22}^{a})$\nand ${\\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained\nalternative proof of Theorem \\ref{thm:KPS} and a complementary view\non certain of the constructions and results in these articles. \n\n\\medskip\n\nOur approach to the classification of isomorphism types of prime Fano\nthreefolds of degree $22$ with infinite automorphism follows the\nsame initial path as in \\cite{KP18,KPS18,Pr90}. It builds essentially\non Iskovskikh's \\emph{double projection from lines} \\cite{Isk89}\nwhich are birational maps relating the geometry of $X$ to that of\nthe smooth quintic del Pezzo threefold $V_{5}$ in $\\mathbb{P}^{6}$.\nThe principle stems from the observation that such a double projection\n$\\psi_{Z}:X\\dashrightarrow V_{5}$ from a line $Z$ in $X$ induces\na correspondence between subgroups of the stabilizer $\\mathrm{Aut}(X,Z)$\nof $Z$ which stabilize the base locus of $\\psi_{Z}$ and subgroups\nof the stabilizer $\\mathrm{Aut}(V_{5},\\Gamma)$ of a rational normal\nquintic curve $\\Gamma\\subset V_{5}$ contained in the base locus of\n$\\psi_{Z}^{-1}$ which stabilize the base locus of $\\psi_{Z}^{-1}$\n. This leads to determine equivalence classes of rational normal quintic\ncurves in $V_{5}$ with infinite stabilizers under the action of $\\mathrm{Aut}(V_{5})$.\nIn \\cite{KPS18}, this is done by exhibiting representatives of these\nclasses in the form of orbit closures of certain polynomials in Mukai-Umemura's\ndescription \\cite{MuUm83} of $V_{5}$ as an orbit closure of the\naction of $\\mathrm{PGL}_{2}$ on the projective space of homogeneous\npolynomials of degree $6$ in two variables. We propose a complementary\napproach building on the study of the equivariant geometry of inverse\nimages of rational normal quintic curves in $V_{5}$ in the universal\nfamily on the Hilbert scheme of lines on $V_{5}$ constructed by Furushima-Nakayama\n\\cite{FuNa89}. \n\nTo determine the exact automorphism group for each isomorphism type,\nthe main difficulty essentially reduces to establish the existence\nof a special involution of $X$ generating the displayed subgroup\n$\\mu_{2}$ in the case $X=X_{22}^{m}(v)$. In \\cite{KPS18}, the existence\nis taken for granted by an external result \\cite[Proposition 5.1]{DKK}\nwhich depends itself on Mukai's theory of varieties of sums of powers.\nLater, \\cite{KP18} provided a different argument for the existence\nwhich depends in particular on the fact established in \\cite{KPS18}\nthat the Hilbert schemes of conics in $X$ is isomorphic to $\\mathbb{P}^{2}$.\nIn contrast, we produce such a special involution in a geometric way,\nas a conjugate by an appropriate $\\mathbb{G}_{m}$-equivariant birational\nmap of a known involution on certain quadric threefolds with $\\mathbb{G}_{m}$-actions\nnormalizing the $\\mathbb{G}_{m}$-action, see the proof of Theorem\n\\ref{prop:Second-Isomorphism-Classification} and Lemma \\ref{prop:The-crucial-involution}\nas well as Remark \\ref{rem:Link-remark} for these constructions. \n\n\\medskip\n\nWe henceforth work over a fixed algebraically closed base field $k$\nof characteristic zero. The scheme of the article is the following.\nIn Section \\ref{sec:V5-Stuff}, we first review basic properties of\nthe construction of the Hilbert scheme of lines on $V_{5}$ and then\nproceed to the classification of rational normal quintic curves $\\Gamma\\subset V_{5}$\nwith infinite stabilizers up to the action of the automorphism group\nof $V_{5}$. In Section \\ref{sec:X_22-stuff}, we begin with a review\nof classical properties of the double projection from a line in a\nsmooth prime Fano threefold $X$ of degree $22$ and of the correspondence\nit induces between certain families of lines in $X$ and $V_{5}$\nand then re-derive Theorem \\ref{thm:KPS} from these results.\\\\\n\n\\textit{Acknowledgements.} We would like to express our sincere gratitude\nto Sasha Kuznetsov for his thorough readings of successive drafts\nof this article and his numerous constructive comments and suggestions,\nwhich contributed significantly to the clarity and mathematical accuracy\nof the results. \n\nThe present research was initiated during visits of the first and\nthe second authors at Saitama University and continued during a one-month\nvisit of the second and third authors at the University of Poitiers\nduring fall 2024 funded by the University of Poitiers . The authors\nare grateful to these institutions for their generous support and\nthe excellent working conditions offered. The second author was supported\nby JSPS KAKENHI Grant Number 22K03269, Royal Society International\nCollaboration Award ICA\\textbackslash 1\\textbackslash 23109 and\nAsian Young Scientist Fellowship. The third author was partially funded\nby JSPS KAKENHI Grant Number 23K03047.",
    "sketch": "The post-theorem introduction outlines an alternative proof of Theorem~\\ref{thm:KPS} based on Iskovskikh's \\emph{double projection from lines}. Concretely, one studies a double projection \\(\\psi_Z\\colon X\\dashrightarrow V_5\\) from a line \\(Z\\subset X\\), relating \\(X\\) to the smooth quintic del Pezzo threefold \\(V_5\\subset\\mathbb P^6\\). The key principle is that \\(\\psi_Z\\) induces a correspondence between subgroups of the stabilizer \\(\\mathrm{Aut}(X,Z)\\) that stabilize the base locus of \\(\\psi_Z\\) and subgroups of \\(\\mathrm{Aut}(V_5,\\Gamma)\\) (where \\(\\Gamma\\subset V_5\\) is a rational normal quintic curve contained in the base locus of \\(\\psi_Z^{-1}\\)) that stabilize the base locus of \\(\\psi_Z^{-1}\\). This reduces the classification to determining equivalence classes of rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers under \\(\\mathrm{Aut}(V_5)\\).\n\nInstead of the method of \\cite{KPS18} using Mukai--Umemura's description of \\(V_5\\) as an orbit closure for the \\(\\mathrm{PGL}_2\\)-action on degree-6 binary forms, the authors propose a \\\"complementary approach\\\" via \\\"the equivariant geometry of inverse images of rational normal quintic curves in \\(V_5\\) in the universal family on the Hilbert scheme of lines on \\(V_5\\)\\\" (Furushima--Nakayama).\n\nFor determining the \\emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \\(X=X_{22}^m(v)\\), one must establish the existence of a \\\"special involution\\\" generating the subgroup \\(\\mu_2\\). The authors' approach is to \\\"produce such a special involution in a geometric way\\\", namely as \\\"a conjugate by an appropriate \\(\\mathbb G_m\\)-equivariant birational map of a known involution on certain quadric threefolds with \\(\\mathbb G_m\\)-actions normalizing the \\(\\mathbb G_m\\)-action\\\".\n\nFinally, the proof strategy is organized by sections: first classify rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers (Section~\\ref{sec:V5-Stuff}); then review the double projection from a line in \\(X\\) and the induced correspondence between families of lines in \\(X\\) and \\(V_5\\), and \\\"re-derive Theorem~\\ref{thm:KPS} from these results\\\" (Section~\\ref{sec:X_22-stuff}).",
    "expanded_sketch": "The post-theorem introduction outlines an alternative proof of the main theorem based on Iskovskikh's \\emph{double projection from lines}. Concretely, one studies a double projection \\(\\psi_Z\\colon X\\dashrightarrow V_5\\) from a line \\(Z\\subset X\\), relating \\(X\\) to the smooth quintic del Pezzo threefold \\(V_5\\subset\\mathbb P^6\\). The key principle is that \\(\\psi_Z\\) induces a correspondence between subgroups of the stabilizer \\(\\mathrm{Aut}(X,Z)\\) that stabilize the base locus of \\(\\psi_Z\\) and subgroups of \\(\\mathrm{Aut}(V_5,\\Gamma)\\) (where \\(\\Gamma\\subset V_5\\) is a rational normal quintic curve contained in the base locus of \\(\\psi_Z^{-1}\\)) that stabilize the base locus of \\(\\psi_Z^{-1}\\). This reduces the classification to determining equivalence classes of rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers under \\(\\mathrm{Aut}(V_5)\\).\n\nInstead of the method of \\cite{KPS18} using Mukai--Umemura's description of \\(V_5\\) as an orbit closure for the \\(\\mathrm{PGL}_2\\)-action on degree-6 binary forms, the authors propose a \"complementary approach\" via \"the equivariant geometry of inverse images of rational normal quintic curves in \\(V_5\\) in the universal family on the Hilbert scheme of lines on \\(V_5\\)\" (Furushima--Nakayama).\n\nFor determining the \\emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \\(X=X_{22}^m(v)\\), one must establish the existence of a \"special involution\" generating the subgroup \\(\\mu_2\\). The authors' approach is to \"produce such a special involution in a geometric way\", namely as \"a conjugate by an appropriate \\(\\mathbb G_m\\)-equivariant birational map of a known involution on certain quadric threefolds with \\(\\mathbb G_m\\)-actions normalizing the \\(\\mathbb G_m\\)-action\".\n\nFinally, the proof strategy is organized by sections: first classify rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers; then review the double projection from a line in \\(X\\) and the induced correspondence between families of lines in \\(X\\) and \\(V_5\\), and re-derive the main theorem from these results.",
    "expanded_theorem": "\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .,",
    "theorem_type": [
      "Classification or Bijection",
      "Universal"
    ],
    "mcq": {
      "question": "Let $k$ be an algebraically closed field of characteristic $0$, and let ${\\mathscr M}_{22}^{\\circ}$ denote the class of smooth prime Fano threefolds $X$ of degree $(-K_X)^3=22$ whose automorphism group is infinite. Which statement holds for every such threefold $X$?",
      "correct_choice": {
        "label": "A",
        "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$, the normalizer of a maximal torus $\\mathbb G_m$ in $\\operatorname{PGL}_2$."
      },
      "choices": [
        {
          "label": "B",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)\\cong \\mathbb G_a\\rtimes \\mu_4$ as a Borel subgroup of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$, the normalizer of a maximal torus $\\mathbb G_m$ in $\\operatorname{PGL}_2$."
        },
        {
          "label": "C",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$."
        },
        {
          "label": "D",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m$, a maximal torus in $\\operatorname{PGL}_2$."
        },
        {
          "label": "E",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and there exists a choice of involution $\\theta_v$ such that after replacing $X_{22}^m(v)$ by an isomorphic model one has $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "excluded_parameter_-4",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "exact_descriptions_of_full_automorphism_groups",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "special_involution_extending_Gm_to_normalizer",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "for_every_v_vs_after_replacing_by_isomorphic_model",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states only the hypotheses and asks for the classification; it does not reveal the listed cases or the automorphism-group structure. There is no explicit or trivial answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct multiple-choice restatement of a classification theorem: under the stated hypotheses, select the exact classified objects. It primarily asks for recall of the theorem rather than application to a new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish the fully correct statement from subtle variants (e.g. the excluded parameter value -4, the discrete automorphism factor, and the weaker-true option), but the task is still mostly exact recall/discrimination rather than genuine generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one is weaker but true, while others alter specific structural details in ways that reflect realistic confusion about parameter ranges and automorphism groups."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall/discrimination MCQ with strong distractors and no answer leakage, but it is largely tautological as a direct restatement of a classification theorem and only modestly tests generative reasoning."
    }
  },
  {
    "id": "2512.08409v1",
    "paper_link": "http://arxiv.org/abs/2512.08409v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .",
      "start_pos": 4422,
      "end_pos": 5550,
      "label": "thm:KPS"
    },
    "ref_dict": {
      "prop:Second-Isomorphism-Classification": "\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}",
      "rem:Link-remark": "\\begin{rem}\n\\label{rem:Link-remark}In the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nwe constructed for a threefold $X=X_{22}^{m}(v)$, $v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty\\}$,\nwhere $X_{22}^{m}(-4)=X_{22}^{MU}$, and a pair $(Z,Z')$ of $\\mathbb{G}_{m}$-stable\nlines of special type in $X$, a biregular involution $\\theta$ of\n$X$ semi-commuting with the $\\mathbb{G}_{m}$-action and exchanging\nthe lines $Z$ and $Z'$. Let us briefly indicate a complementary\ninterpretation of the construction of this involution $\\theta$. With\nthe notation of the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nthe composition of the double projection $\\psi_{Z}:X\\dashrightarrow V_{5}$\nfrom $Z$ with the projection $\\pi_{(\\psi_{Z})_{*}Z'}:V_{5}\\dashrightarrow Q$\nfrom the line $(\\psi_{Z})_{*}Z'$ is a $\\mathbb{G}_{m}$-equivariant\nbirational map which contracts the unique hyperplane section $S_{Z}$\nof $X$ of multiplicity $3$ along $Z$ onto a rational normal quartic\ncurve $\\Gamma^{Q}$, contracts the unique hyperplane section $R_{\\{Z,Z'\\}}$\nof $X$ singular along $Z\\cup Z'$ onto a rational normal cubic curve\n$C$ and maps the the unique hyperplane section $S_{Z'}$ of $X$\nsingular along $Z'$ onto a cubic section $S_{Z'}^{Q}$ of $Q$ singular\nalong $\\Gamma^{Q}$. Letting $\\alpha_{\\Gamma^{Q}}:\\hat{Q}\\to Q$ be\nthe blow-up of $\\Gamma^{Q}$, where $\\hat{Q}\\subset\\mathbb{P}^{16}$\nis a Fano threefold of Picard rank $2$ and degree $28$ of type 2.21\nin \\cite{MoMu81}, we obtain an induced $\\mathbb{G}_{m}$-equivariant\nbirational map \n\\[\n\\hat{\\gamma}=\\alpha_{(\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z})_{*}S_{Z}}^{-1}\\circ\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z}:X\\dashrightarrow\\hat{Q}\n\\]\nwhich fits into the following commutative diagram of $\\mathbb{G}_{m}$-equivariant\nbirational maps\n\\[\n\\xymatrix{ &  & R'_{\\{Z,Z'\\}}\\subseteq Y\\ar@{.>}[r]^{\\hat{\\chi}}\\ar@{->}[lld]_{\\hat{\\sigma}} & Y^{+}\\supseteq R^{+}{}_{\\{Z,Z'\\}}\\ar@{->}[rrd]^{\\hat{\\tau}}\\\\\nZ\\cup Z'\\subset X\\ar@{-->}[rrrrr]^{\\hat{\\gamma}} &  &  &  &  & \\hat{Q}\\supset\\hat{C}\\\\\n}\n\\]\nwhere $\\hat{\\sigma}:Y\\to X$ is the blow-up of $Z\\cup Z'$, $\\hat{\\chi}:Y\\dashrightarrow Y^{+}$\nis a small $\\mathbb{Q}$-factorial modification and $\\hat{\\tau}:Y^{+}\\to\\hat{Q}$\nis the contraction of the proper transform $R^{+}{}_{\\{Z,Z'\\}}$ of\n$R_{\\{Z,Z'\\}}$ onto a smooth curve $\\hat{C}\\subset\\hat{S}$ of bi-degree\n$(3,3)$. One can further check from the construction that $\\hat{\\gamma}$\nis given by the linear system of quadric sections of $X$ having multiplicity\n$3$ along $Z$ and $Z'$. The birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\nin the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nlifts to a biregular involution $\\iota_{\\hat{Q}}$ of $\\hat{Q}$ which\nnormalizes the lifted $\\mathbb{G}_{m}$-action and makes $\\hat{Q}$\ninto a Fano threefold of $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-Picard\nrank $1$, and by the construction of the associated biregular involution\n$\\theta$ of $X$, the above diagram is then a $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-equivariant\nSarkisov link. We refer the reader to the forthcoming article \\cite{DFK-next}\nfor a for a more in-depth and detailed study of these birational links\nbetween smooth prime Fano threefolds of degree 22 and blow-ups of\nsmooth quadric threefolds along rational normal quartic curves. \n\\end{rem}",
      "prop:The-crucial-involution": "\\begin{lem}\n\\label{prop:The-crucial-involution}Let $\\Gamma_{4}\\subset\\mathbb{P}^{4}$\nbe a rational normal quartic curve and let $T\\subset\\mathrm{Aut}(\\mathbb{P}^{4},\\Gamma_{4})\\cong\\mathrm{Aut}(\\Gamma_{4})\\cong\\mathrm{PGL}_{2}$\nbe a maximal torus. Then the following hold:\n\n1) For every smooth $T$-stable quadric $Q$ containing $\\Gamma_{4}$,\nthe linear system of quadric sections of $Q$ containing $\\Gamma_{4}$\ndetermines a birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\ncontracting the unique cubic section $S_{\\Gamma_{4}}$ of $Q$ singular\nalong $\\Gamma_{4}$ onto $\\Gamma_{4}$ and restricting to an automorphism\nof $Q\\setminus S_{\\Gamma_{4}}$ which semi-commutes with the induced\n$T$-action on it.\n\n2) Let $r,r'\\in\\Gamma_{4}$ be the two $T$-fixed points. Then the\nproper transform $(\\iota_{Q}^{-1})_{*}H_{r'}$ of the tangent hyperplane\nsection $H_{r'}$ to $Q$ at $r'$ is the unique integral quadric\nsection of $Q$ containing $\\Gamma_{4}$ and singular along the tangent\nline $L_{r}$ to $\\Gamma_{4}$ at $r$. The scheme $H_{r'}\\cap(\\iota_{Q}^{-1})_{*}H_{r'}$\nis the union of the line $L_{r',s}$ passing through $r'$ and the\nunique $T$-fixed point $s\\in L_{r}\\setminus\\{r\\}$ and of a rational\nnormal cubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$\nand passing through $s$. Moreover, $\\iota_{Q}$ maps $C_{2r',s}$\nisomorphically onto itself.\n\\end{lem}",
      "thm:KPS": "\\begin{thm}\n\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 540,
    "pre_theorem_intro_text": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):",
    "context": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):\n\n\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}",
    "full_context": "Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over\nan algebraically closed field $k$ of characteristic zero form a family\n${\\mathscr{M}}_{22}$ of dimension 6 whose members are rational \\cite{Mu04}.\nOur main interest lies in the subfamily ${\\mathscr{M}}_{22}^{\\circ}$\nconsisting of threefolds with infinite automorphism groups, whose\nclassification was established by Kuznetsov, Prokhorov and Shramov\nin \\cite{KP18,KPS18} in a form which can be summarized as follows\n(see Theorem \\ref{prop:Second-Isomorphism-Classification}):\n\n\\begin{thm}\n\\label{prop:Second-Isomorphism-Classification} Up to isomorphism,\nthe following hold:\n\n1) There exists a unique threefold $X_{22}^{MU}$ such that $\\mathrm{Aut}^{0}(X_{22}^{MU})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$.\nMoreover, every line on $X_{22}^{MU}$ is of special type and $\\mathrm{Aut}(X_{22}^{MU})\\cong\\mathrm{PGL}_{2}$.\n\n2) There exists a unique threefold $X_{22}^{a}$ such that $\\mathrm{Aut}^{0}(X_{22}^{a})$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{a}$. Moreover,\n$X_{22}^{a}$ contains a unique line of special type and $\\mathrm{Aut}(X_{22}^{a})\\cong\\mathbb{G}_{a}\\rtimes\\mu_{4}$,\nwhere the action of $\\mu_{4}$ on $\\mathbb{G}_{a}$ is given by the\nnatural injective homomorphism $\\mu_{4}\\to\\mathrm{Aut}(\\mathbb{G}_{a})\\cong\\mathbb{G}_{m}$. \n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}\n\n\\section*{Introduction}\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$.\n\nThe possibilities for $X$ were initially found by Prokhorov in \\cite{Pr90}\nand the articles \\cite{KPS18} and \\cite{KP18} completed later on\nthe structures of the automorphism groups ${\\rm Aut}(X_{22}^{a})$\nand ${\\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained\nalternative proof of Theorem \\ref{thm:KPS} and a complementary view\non certain of the constructions and results in these articles.\n\n3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,\n$v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty,-4\\}$, such that $\\mathrm{Aut}^{0}(X_{22}^{m}(v))$\ncontains a Borel subgroup isomorphic to $\\mathbb{G}_{m}$. Each threefold\n$X_{22}^{m}(v)$ contains exactly two lines of special type and $\\mathrm{Aut}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}\\rtimes\\mu_{2}$,\nwhere the group $\\mu_{2}$ is generated by an involution $\\theta$\nwhich exchanges the two lines of special type in $X_{22}^{m}(v)$.\n\\end{thm}\n\n$\\quad$ - If $\\Gamma=\\Gamma_{p}$ then $\\mathrm{Aut}(X,Z)=G_{p}$\nand $\\mathrm{Aut}(X,Z')\\subset\\mathrm{Aut}(X)$ is the conjugate of\n$\\mathrm{Aut}(X,Z)$ by the involution $\\theta$. Since the latter\nnormalizes $T$, $\\mathrm{Aut}(X,Z)$ and $\\theta$ generate a subgroup\nof $\\mathrm{Aut}(X)$ isomorphic to $\\mathrm{PGL}_{2}$, from which\nit follows, by Lemma \\ref{prop:line}, that $\\mathrm{Aut}^{0}(X)\\cong\\mathrm{PGL}_{2}$.\nFinally, since an element of $\\mathrm{Aut}(X)$ which stabilizes every\nline in $X$ is contained in the intersection of all Borel subgroups\nof $\\mathrm{Aut}^{0}(X)$, it must be trivial, which implies that\nthe canonically induced action of $\\mathrm{Aut}(X)$ on $(\\mathcal{H}_{X})_{\\mathrm{red}}\\cong\\mathbb{P}^{1}$\nis faithful. Thus, $\\mathrm{Aut}(X)$ identifies with a subgroup of\nthe automorphism group of $(\\mathcal{H}_{X})_{\\mathrm{red}}\\cong\\mathbb{P}^{1}$,\nhence is connected, equal to $\\mathrm{PGL}_{2}$. \n\\end{proof}\nIn the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nabove, we have used the following auxiliary result on the birational\ngeometry of smooth quadric threefolds with an effective action of\na $1$-dimensional torus. \n\\begin{lem}\n\\label{prop:The-crucial-involution}Let $\\Gamma_{4}\\subset\\mathbb{P}^{4}$\nbe a rational normal quartic curve and let $T\\subset\\mathrm{Aut}(\\mathbb{P}^{4},\\Gamma_{4})\\cong\\mathrm{Aut}(\\Gamma_{4})\\cong\\mathrm{PGL}_{2}$\nbe a maximal torus. Then the following hold:\n\nTo prove assertion 2), we can assume that $r=[0:0:0:0:1]$ and $r'=[1:0:0:0:0]$\nso that $H_{r'}=\\{w_{4}=0\\}\\cap Q$, $L_{r'}=\\{w_{2}=w_{3}=w_{4}=0\\}$\nand $L_{r}=\\{w_{0}=w_{1}=w_{2}=0\\}$. Then $(\\iota_{Q}^{-1})_{*}H_{r'}=\\{w_{0}w_{2}-w_{1}^{2}=0\\}\\cap Q$\nis an anti-canonically embedded del Pezzo surface of degree $4$ singular\nalong $L_{r}$ and $H_{r'}\\cap(\\iota_{Q}^{-1})_{*}H_{r'}=\\{w_{4}=w_{0}w_{2}-w_{1}^{2}=0\\}\\cap Q$\nis the union of the line $L_{r',s}=\\{w_{1}=w_{2}=w_{4}=0\\}$ where\n$s=[0:0:0:1:0]\\in L_{r}\\setminus\\{r\\}$ and of the rational normal\ncubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$ passing\nthrough $s$ given by the image of the morphism \n\\[\n\\alpha:\\mathbb{P}_{[u_{0}:u_{1}]}^{1}\\to\\mathbb{P}^{4},\\,[u_{0}:u_{1}]\\mapsto[u_{0}^{3}:u_{0}^{2}u_{1}:u_{0}u_{1}^{2}:(1-c^{2})u_{1}^{3}:0].\n\\]\nNow it is routine to verify from (\\ref{eq:The-involution}) that $\\iota_{Q}\\circ\\alpha=\\alpha\\circ\\iota_{c}$,\nwhere $\\iota_{c}$ is the involution $[u_{0}:u_{1}]\\mapsto[u_{1}:\\frac{c}{1-c^{2}}u_{0}]$\nof $\\mathbb{P}^{1}$. \n\\end{proof}\n\\begin{rem}\n\\label{rem:Link-remark}In the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nwe constructed for a threefold $X=X_{22}^{m}(v)$, $v\\in\\mathbb{P}^{1}\\setminus\\{0,1,\\infty\\}$,\nwhere $X_{22}^{m}(-4)=X_{22}^{MU}$, and a pair $(Z,Z')$ of $\\mathbb{G}_{m}$-stable\nlines of special type in $X$, a biregular involution $\\theta$ of\n$X$ semi-commuting with the $\\mathbb{G}_{m}$-action and exchanging\nthe lines $Z$ and $Z'$. Let us briefly indicate a complementary\ninterpretation of the construction of this involution $\\theta$. With\nthe notation of the proof of Theorem \\ref{prop:Second-Isomorphism-Classification},\nthe composition of the double projection $\\psi_{Z}:X\\dashrightarrow V_{5}$\nfrom $Z$ with the projection $\\pi_{(\\psi_{Z})_{*}Z'}:V_{5}\\dashrightarrow Q$\nfrom the line $(\\psi_{Z})_{*}Z'$ is a $\\mathbb{G}_{m}$-equivariant\nbirational map which contracts the unique hyperplane section $S_{Z}$\nof $X$ of multiplicity $3$ along $Z$ onto a rational normal quartic\ncurve $\\Gamma^{Q}$, contracts the unique hyperplane section $R_{\\{Z,Z'\\}}$\nof $X$ singular along $Z\\cup Z'$ onto a rational normal cubic curve\n$C$ and maps the the unique hyperplane section $S_{Z'}$ of $X$\nsingular along $Z'$ onto a cubic section $S_{Z'}^{Q}$ of $Q$ singular\nalong $\\Gamma^{Q}$. Letting $\\alpha_{\\Gamma^{Q}}:\\hat{Q}\\to Q$ be\nthe blow-up of $\\Gamma^{Q}$, where $\\hat{Q}\\subset\\mathbb{P}^{16}$\nis a Fano threefold of Picard rank $2$ and degree $28$ of type 2.21\nin \\cite{MoMu81}, we obtain an induced $\\mathbb{G}_{m}$-equivariant\nbirational map \n\\[\n\\hat{\\gamma}=\\alpha_{(\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z})_{*}S_{Z}}^{-1}\\circ\\pi_{(\\psi_{Z})_{*}Z'}\\circ\\psi_{Z}:X\\dashrightarrow\\hat{Q}\n\\]\nwhich fits into the following commutative diagram of $\\mathbb{G}_{m}$-equivariant\nbirational maps\n\\[\n\\xymatrix{ & & R'_{\\{Z,Z'\\}}\\subseteq Y\\ar@{.>}[r]^{\\hat{\\chi}}\\ar@{->}[lld]_{\\hat{\\sigma}} & Y^{+}\\supseteq R^{+}{}_{\\{Z,Z'\\}}\\ar@{->}[rrd]^{\\hat{\\tau}}\\\\\nZ\\cup Z'\\subset X\\ar@{-->}[rrrrr]^{\\hat{\\gamma}} & & & & & \\hat{Q}\\supset\\hat{C}\\\\\n}\n\\]\nwhere $\\hat{\\sigma}:Y\\to X$ is the blow-up of $Z\\cup Z'$, $\\hat{\\chi}:Y\\dashrightarrow Y^{+}$\nis a small $\\mathbb{Q}$-factorial modification and $\\hat{\\tau}:Y^{+}\\to\\hat{Q}$\nis the contraction of the proper transform $R^{+}{}_{\\{Z,Z'\\}}$ of\n$R_{\\{Z,Z'\\}}$ onto a smooth curve $\\hat{C}\\subset\\hat{S}$ of bi-degree\n$(3,3)$. One can further check from the construction that $\\hat{\\gamma}$\nis given by the linear system of quadric sections of $X$ having multiplicity\n$3$ along $Z$ and $Z'$. The birational involution $\\iota_{Q}:Q\\dashrightarrow Q$\nin the proof of Theorem \\ref{prop:Second-Isomorphism-Classification}\nlifts to a biregular involution $\\iota_{\\hat{Q}}$ of $\\hat{Q}$ which\nnormalizes the lifted $\\mathbb{G}_{m}$-action and makes $\\hat{Q}$\ninto a Fano threefold of $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-Picard\nrank $1$, and by the construction of the associated biregular involution\n$\\theta$ of $X$, the above diagram is then a $(\\mathbb{G}_{m}\\rtimes\\mu_{2})$-equivariant\nSarkisov link. We refer the reader to the forthcoming article \\cite{DFK-next}\nfor a for a more in-depth and detailed study of these birational links\nbetween smooth prime Fano threefolds of degree 22 and blow-ups of\nsmooth quadric threefolds along rational normal quartic curves. \n\\end{rem}",
    "post_theorem_intro_text_len": 4861,
    "post_theorem_intro_text": "The possibilities for $X$ were initially found by Prokhorov in \\cite{Pr90}\nand the articles \\cite{KPS18} and \\cite{KP18} completed later on\nthe structures of the automorphism groups ${\\rm Aut}(X_{22}^{a})$\nand ${\\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained\nalternative proof of Theorem \\ref{thm:KPS} and a complementary view\non certain of the constructions and results in these articles. \n\n\\medskip\n\nOur approach to the classification of isomorphism types of prime Fano\nthreefolds of degree $22$ with infinite automorphism follows the\nsame initial path as in \\cite{KP18,KPS18,Pr90}. It builds essentially\non Iskovskikh's \\emph{double projection from lines} \\cite{Isk89}\nwhich are birational maps relating the geometry of $X$ to that of\nthe smooth quintic del Pezzo threefold $V_{5}$ in $\\mathbb{P}^{6}$.\nThe principle stems from the observation that such a double projection\n$\\psi_{Z}:X\\dashrightarrow V_{5}$ from a line $Z$ in $X$ induces\na correspondence between subgroups of the stabilizer $\\mathrm{Aut}(X,Z)$\nof $Z$ which stabilize the base locus of $\\psi_{Z}$ and subgroups\nof the stabilizer $\\mathrm{Aut}(V_{5},\\Gamma)$ of a rational normal\nquintic curve $\\Gamma\\subset V_{5}$ contained in the base locus of\n$\\psi_{Z}^{-1}$ which stabilize the base locus of $\\psi_{Z}^{-1}$\n. This leads to determine equivalence classes of rational normal quintic\ncurves in $V_{5}$ with infinite stabilizers under the action of $\\mathrm{Aut}(V_{5})$.\nIn \\cite{KPS18}, this is done by exhibiting representatives of these\nclasses in the form of orbit closures of certain polynomials in Mukai-Umemura's\ndescription \\cite{MuUm83} of $V_{5}$ as an orbit closure of the\naction of $\\mathrm{PGL}_{2}$ on the projective space of homogeneous\npolynomials of degree $6$ in two variables. We propose a complementary\napproach building on the study of the equivariant geometry of inverse\nimages of rational normal quintic curves in $V_{5}$ in the universal\nfamily on the Hilbert scheme of lines on $V_{5}$ constructed by Furushima-Nakayama\n\\cite{FuNa89}. \n\nTo determine the exact automorphism group for each isomorphism type,\nthe main difficulty essentially reduces to establish the existence\nof a special involution of $X$ generating the displayed subgroup\n$\\mu_{2}$ in the case $X=X_{22}^{m}(v)$. In \\cite{KPS18}, the existence\nis taken for granted by an external result \\cite[Proposition 5.1]{DKK}\nwhich depends itself on Mukai's theory of varieties of sums of powers.\nLater, \\cite{KP18} provided a different argument for the existence\nwhich depends in particular on the fact established in \\cite{KPS18}\nthat the Hilbert schemes of conics in $X$ is isomorphic to $\\mathbb{P}^{2}$.\nIn contrast, we produce such a special involution in a geometric way,\nas a conjugate by an appropriate $\\mathbb{G}_{m}$-equivariant birational\nmap of a known involution on certain quadric threefolds with $\\mathbb{G}_{m}$-actions\nnormalizing the $\\mathbb{G}_{m}$-action, see the proof of Theorem\n\\ref{prop:Second-Isomorphism-Classification} and Lemma \\ref{prop:The-crucial-involution}\nas well as Remark \\ref{rem:Link-remark} for these constructions. \n\n\\medskip\n\nWe henceforth work over a fixed algebraically closed base field $k$\nof characteristic zero. The scheme of the article is the following.\nIn Section \\ref{sec:V5-Stuff}, we first review basic properties of\nthe construction of the Hilbert scheme of lines on $V_{5}$ and then\nproceed to the classification of rational normal quintic curves $\\Gamma\\subset V_{5}$\nwith infinite stabilizers up to the action of the automorphism group\nof $V_{5}$. In Section \\ref{sec:X_22-stuff}, we begin with a review\nof classical properties of the double projection from a line in a\nsmooth prime Fano threefold $X$ of degree $22$ and of the correspondence\nit induces between certain families of lines in $X$ and $V_{5}$\nand then re-derive Theorem \\ref{thm:KPS} from these results.\\\\\n\n\\textit{Acknowledgements.} We would like to express our sincere gratitude\nto Sasha Kuznetsov for his thorough readings of successive drafts\nof this article and his numerous constructive comments and suggestions,\nwhich contributed significantly to the clarity and mathematical accuracy\nof the results. \n\nThe present research was initiated during visits of the first and\nthe second authors at Saitama University and continued during a one-month\nvisit of the second and third authors at the University of Poitiers\nduring fall 2024 funded by the University of Poitiers . The authors\nare grateful to these institutions for their generous support and\nthe excellent working conditions offered. The second author was supported\nby JSPS KAKENHI Grant Number 22K03269, Royal Society International\nCollaboration Award ICA\\textbackslash 1\\textbackslash 23109 and\nAsian Young Scientist Fellowship. The third author was partially funded\nby JSPS KAKENHI Grant Number 23K03047.",
    "sketch": "The post-theorem introduction outlines an alternative proof of Theorem~\\ref{thm:KPS} based on Iskovskikh's \\emph{double projection from lines}. Concretely, one studies a double projection \\(\\psi_Z\\colon X\\dashrightarrow V_5\\) from a line \\(Z\\subset X\\), relating \\(X\\) to the smooth quintic del Pezzo threefold \\(V_5\\subset\\mathbb P^6\\). The key principle is that \\(\\psi_Z\\) induces a correspondence between subgroups of the stabilizer \\(\\mathrm{Aut}(X,Z)\\) that stabilize the base locus of \\(\\psi_Z\\) and subgroups of \\(\\mathrm{Aut}(V_5,\\Gamma)\\) (where \\(\\Gamma\\subset V_5\\) is a rational normal quintic curve contained in the base locus of \\(\\psi_Z^{-1}\\)) that stabilize the base locus of \\(\\psi_Z^{-1}\\). This reduces the classification to determining equivalence classes of rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers under \\(\\mathrm{Aut}(V_5)\\).\n\nInstead of the method of \\cite{KPS18} using Mukai--Umemura's description of \\(V_5\\) as an orbit closure for the \\(\\mathrm{PGL}_2\\)-action on degree-6 binary forms, the authors propose a \\\"complementary approach\\\" via \\\"the equivariant geometry of inverse images of rational normal quintic curves in \\(V_5\\) in the universal family on the Hilbert scheme of lines on \\(V_5\\)\\\" (Furushima--Nakayama).\n\nFor determining the \\emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \\(X=X_{22}^m(v)\\), one must establish the existence of a \\\"special involution\\\" generating the subgroup \\(\\mu_2\\). The authors' approach is to \\\"produce such a special involution in a geometric way\\\", namely as \\\"a conjugate by an appropriate \\(\\mathbb G_m\\)-equivariant birational map of a known involution on certain quadric threefolds with \\(\\mathbb G_m\\)-actions normalizing the \\(\\mathbb G_m\\)-action\\\".\n\nFinally, the proof strategy is organized by sections: first classify rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers (Section~\\ref{sec:V5-Stuff}); then review the double projection from a line in \\(X\\) and the induced correspondence between families of lines in \\(X\\) and \\(V_5\\), and \\\"re-derive Theorem~\\ref{thm:KPS} from these results\\\" (Section~\\ref{sec:X_22-stuff}).",
    "expanded_sketch": "The post-theorem introduction outlines an alternative proof of the main theorem based on Iskovskikh's \\emph{double projection from lines}. Concretely, one studies a double projection \\(\\psi_Z\\colon X\\dashrightarrow V_5\\) from a line \\(Z\\subset X\\), relating \\(X\\) to the smooth quintic del Pezzo threefold \\(V_5\\subset\\mathbb P^6\\). The key principle is that \\(\\psi_Z\\) induces a correspondence between subgroups of the stabilizer \\(\\mathrm{Aut}(X,Z)\\) that stabilize the base locus of \\(\\psi_Z\\) and subgroups of \\(\\mathrm{Aut}(V_5,\\Gamma)\\) (where \\(\\Gamma\\subset V_5\\) is a rational normal quintic curve contained in the base locus of \\(\\psi_Z^{-1}\\)) that stabilize the base locus of \\(\\psi_Z^{-1}\\). This reduces the classification to determining equivalence classes of rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers under \\(\\mathrm{Aut}(V_5)\\).\n\nInstead of the method of \\cite{KPS18} using Mukai--Umemura's description of \\(V_5\\) as an orbit closure for the \\(\\mathrm{PGL}_2\\)-action on degree-6 binary forms, the authors propose a \"complementary approach\" via \"the equivariant geometry of inverse images of rational normal quintic curves in \\(V_5\\) in the universal family on the Hilbert scheme of lines on \\(V_5\\)\" (Furushima--Nakayama).\n\nFor determining the \\emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \\(X=X_{22}^m(v)\\), one must establish the existence of a \"special involution\" generating the subgroup \\(\\mu_2\\). The authors' approach is to \"produce such a special involution in a geometric way\", namely as \"a conjugate by an appropriate \\(\\mathbb G_m\\)-equivariant birational map of a known involution on certain quadric threefolds with \\(\\mathbb G_m\\)-actions normalizing the \\(\\mathbb G_m\\)-action\".\n\nFinally, the proof strategy is organized by sections: first classify rational normal quintic curves \\(\\Gamma\\subset V_5\\) with infinite stabilizers; then review the double projection from a line in \\(X\\) and the induced correspondence between families of lines in \\(X\\) and \\(V_5\\), and re-derive the main theorem from these results.",
    "expanded_theorem": "\\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging\nto ${\\mathscr{M}}_{22}^{\\circ}$ is isomorphic to one of the following\nthreefolds:\n\n$\\bullet$ A threefold $X_{22}^{{\\rm MU}}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{{\\rm MU}})={\\rm Aut}(X_{22}^{{\\rm MU}})\\cong{\\rm PGL}_{2}$\n(the Mukai-Umemura threefold),\n\n$\\bullet$ A threefold $X_{22}^{a}\\in{\\mathscr{M}}_{22}^{\\circ}$\nwith ${\\rm Aut}^{0}(X_{22}^{a})\\cong\\mathbb{G}_{a}$. Moreover, ${\\rm Aut}(X_{22}^{a})$\nis isomorphic to a subgroup $\\mathbb{G}_{a}\\rtimes\\mu_{4}$ of a Borel\nsubgroup $\\mathbb{G}_{a}\\rtimes\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$. \n\n$\\bullet$ A member of a one-parameter family of pairwise non isomorphic\nthreefold $X_{22}^{m}(v)\\in{\\mathscr{M}}_{22}^{\\circ}$ with ${\\rm Aut}^{0}(X_{22}^{m}(v))\\cong\\mathbb{G}_{m}$\nparametrized by $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$.\nMoreover, for every $v\\in{\\mathbb{P}}^{1}\\backslash\\{0,1,\\infty,-4\\}$,\n${\\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\\mathbb{G}_{m}\\rtimes\\mu_{2}$\nof a maximal torus $\\mathbb{G}_{m}$ of $\\mathrm{PGL}_{2}$ .,",
    "theorem_type": [
      "Classification or Bijection",
      "Universal"
    ],
    "mcq": {
      "question": "Let $k$ be an algebraically closed field of characteristic $0$, and let ${\\mathscr M}_{22}^{\\circ}$ denote the class of smooth prime Fano threefolds $X$ of degree $(-K_X)^3=22$ whose automorphism group is infinite. Which statement holds for every such threefold $X$?",
      "correct_choice": {
        "label": "A",
        "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$, the normalizer of a maximal torus $\\mathbb G_m$ in $\\operatorname{PGL}_2$."
      },
      "choices": [
        {
          "label": "B",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)\\cong \\mathbb G_a\\rtimes \\mu_4$ as a Borel subgroup of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$, the normalizer of a maximal torus $\\mathbb G_m$ in $\\operatorname{PGL}_2$."
        },
        {
          "label": "C",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$."
        },
        {
          "label": "D",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and for every such $v$, $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m$, a maximal torus in $\\operatorname{PGL}_2$."
        },
        {
          "label": "E",
          "text": "Every $X\\in {\\mathscr M}_{22}^{\\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\\mathrm{MU}}$, with $\\operatorname{Aut}^0(X_{22}^{\\mathrm{MU}})=\\operatorname{Aut}(X_{22}^{\\mathrm{MU}})\\cong \\operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\\operatorname{Aut}^0(X_{22}^a)\\cong \\mathbb G_a$, and $\\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\\mathbb G_a\\rtimes \\mu_4$ of a Borel subgroup $\\mathbb G_a\\rtimes \\mathbb G_m$ of $\\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\\in \\mathbb P^1\\setminus\\{0,1,\\infty,-4\\}$, with $\\operatorname{Aut}^0(X_{22}^m(v))\\cong \\mathbb G_m$, and there exists a choice of involution $\\theta_v$ such that after replacing $X_{22}^m(v)$ by an isomorphic model one has $\\operatorname{Aut}(X_{22}^m(v))\\cong \\mathbb G_m\\rtimes \\mu_2$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "excluded_parameter_-4",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "exact_descriptions_of_full_automorphism_groups",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "special_involution_extending_Gm_to_normalizer",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "for_every_v_vs_after_replacing_by_isomorphic_model",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly or implicitly reveal the keyed answer. It only specifies the class of Fano threefolds and asks which statement holds for all of them."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not a direct restatement of the stem, but it mainly asks for recall of a known classification theorem in nearly theorem-statement form. It tests recognition of the exact classification rather than a genuinely competing mathematical conclusion derived from the setup."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (excluded parameter values, full automorphism groups, normalizer vs torus). However, the task is mostly theorem recall/verification, and the presence of a weaker true option reduces the need for strong generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and target realistic errors, such as omitting the excluded value -4 or confusing the full automorphism group with its identity component. But choice C is a weaker true statement, so the distractor set is not cleanly single-correct, which significantly weakens item quality."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically sophisticated item with subtle near-miss options, but it is weakened by theorem-recall framing and, more seriously, by including a weaker true alternative, making the single-best-answer format ambiguous."
    }
  },
  {
    "id": "2512.08562v1",
    "paper_link": "http://arxiv.org/abs/2512.08562v1",
    "theorems_cnt": 6,
    "theorem": {
      "env_name": "theorem",
      "content": "[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of (\\ref{ILW}). Then for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of (\\ref{ILW}) verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}",
      "start_pos": 22335,
      "end_pos": 23327,
      "label": "thm1.1"
    },
    "ref_dict": {
      "1.9": "\\begin{align}\n\t\\langle f_{12},g_{12}\\rangle\n\t&:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}",
      "thm1.1": "\\begin{theorem}[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of (\\ref{ILW}). Then for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of (\\ref{ILW}) verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}\n\\end{theorem}",
      "th1.3": "\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}",
      "thm1.2": "\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}",
      "1.15": "\\begin{equation}\n\t\\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n\t\\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}",
      "ILW": "\\begin{align}\n\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}",
      "T1": "\\begin{align}\n\tv = \\frac{3}{\\delta} u\\left( \\frac{3}{\\delta}t,x\\right), \\label{T1}\n\\end{align}",
      "lagrange": "\\begin{align}\n\t\\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}",
      "1.16": "\\begin{align}\n\tu(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 16852,
    "pre_theorem_intro_text": "\\label{sec.1}\nIn this work, we consider the stability of $n$-soliton solutions for the intermediate long wave (ILW) equation\\cite{J,KAS,KKD}\n\\begin{align}\n\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}\nwhere $u=u(t,x)\\in\\mathbb{R}$ is a real-valued function, $T^\\delta $ is the inverse of Tilbert  transform $T$ with a large scale parameter $\\delta>0$. These two operators are defined as\n\\begin{align}\n\t(T^\\delta f)(x)&=\\frac{1}{2\\delta}\\text{P.V.}\\int_{\\mathbb{R}}\\coth \\frac{\\pi (y-x)}{2\\delta}f(y)dy,\\\\\\nonumber\n\t\t(Tf)(x)&=\\frac{1}{2\\delta}\\text{P.V.}\\int_{\\mathbb{R}}\\mathrm{cosech} \\frac{\\pi (y-x)}{2\\delta}f(y)dy,\n\\end{align}\nand P.V. indicates that the integral is to be computed in the principle value sense. Moreover, $T^\\delta$ is a zero Fourier multiplier, in the sense that $\\partial_x T^\\delta$ is the multiplier with symbol\n\\begin{align}\n\t\\sigma(\\partial_x T^\\delta)=\\widehat{\\partial_x T^\\delta}=-2\\pi\\xi\\coth(2\\pi\\delta \\xi).\n\\end{align}\n\nThe ILW equation (\\ref{ILW}) describes weakly nonlinear internal wave propagation in stratified fluids of finite depth $\\delta$. First derived by Kubota, Ko and Dobbs \\cite{KKD}, this model captures essential nonlinear and dispersive characteristics of wave dynamics. Joseph \\cite{J} subsequently formalized the derivation of the ILW equation by integrating the linear dispersion relation from \\cite{P} within Whitham's nonlocal framework \\cite{W}, and explicitly constructed its solitary wave solution. This represents a key advancement in establishing the integrability of the equation. Furthermore, the ILW equation itself arises in the context of the two-layer internal wave system \\cite{BLS, CGK}. A rigorous derivation of its system form in one or two spatial dimensions, including the case with a free upper surface, can be found in \\cite{Xu}.\n\nIn addition, the ILW equation (\\ref{ILW}) establishes a fundamental connection between two canonical models in water wave theory: the Korteweg-de Vries (KdV) equation for shallow water and the Benjamin-Ono (BO) equation for deep water \\cite{ABFS, AFS,HKO,SAF}, with shared initial data in appropriate Sobolev spaces. Specifically, the asymptotic behavior of the ILW solution is governed by the depth parameter $\\delta$. Let $u(t,x)$ be the solution of the ILW equation (\\ref{ILW}). Then, under the scaling transformation\n\\begin{align}\n\tv = \\frac{3}{\\delta} u\\left( \\frac{3}{\\delta}t,x\\right), \\label{T1}\n\\end{align}\nthe rescaled function $v$ converges (in a suitable functional sense, typically distributional or weak convergence) to the solution of the KdV equation\n\\begin{align}\n\tu_t + 2uu_x + \\frac{\\delta}{3}u_{xxx} = 0, \\label{KdV}\n\\end{align}\nin the shallow-water limit ($\\delta \\to 0$) and in the deep-water limit ($\\delta \\to \\infty$), (\\ref{T1}) will converge to the the solution of the BO equation\n\\begin{align}\n\tu_t + 2uu_x + Hu_{xx} = 0, \\label{BO}\n\\end{align}\nwhere $H$ denotes the Hilbert transform, defined by the principal value integral\n\\begin{align*}\n\t(Hf)(x)=\\frac{1}{\\pi}\\text{P.V.}\\int_{\\mathbb{R}}\\frac{f(y)}{y-x}\\mathrm{d}y.\n\\end{align*}\n\nAs a fundamental integrable system \\cite{AC,FA,KS}, the ILW equation (\\ref{ILW}) possesses an infinite-dimensional completely integrable Hamiltonian structure \\cite{KAS, Sa}. This integrability is manifested in the existence of an infinite hierarchy of conservation laws and a Lax-pair formulation.\nA key distinction between the ILW equation and the KdV equation lies in their dispersion operators \\cite{MPS,Sa,Xu}. Specifically, the ILW equation incorporates a singular nonlocal integro-differential operator $T^\\delta$, which fundamentally alters the asymptotic properties of its solutions. In particular, the soliton solutions of the ILW equation exhibit algebraic decay, in contrast to the exponential decay characteristic of KdV solitons. This difference in decay rates represents a key qualitative difference between the two models. Formally, the following quantities are conserved along the flow of the ILW equation \\cite{LR}\n\\begin{align}\n\tH_0(u)&=:\\int_{\\mathbb{R}}u\\text{d}x, \\\\\n    H_1(u)&=:\\frac{1}{2}\\int_{\\mathbb{R}}u^2\\text{d}x, \\\\\n\tH_2(u)&=:-\\int_{\\mathbb{R}}\\frac{1}{3}u^3+\\frac{1}{2}uT^\\delta u_x +\\frac{u^2}{2\\delta}\\text{d}x,\\\\\n\tH_3(u)&=:\\int_{\\mathbb{R}}\\frac{1}{4}u^4+\\frac{3}{4}u^2T^\\delta u_x+\\frac{3}{8}(T^\\delta u_x)^2+\\frac{1}{3\\delta}u^3+\\frac{1}{2\\delta}uT^\\delta u_x+\\frac{u^2}{8\\delta^2}\\text{d}x.\n\\end{align}\n\nIt is noteworthy that the energy space for the ILW equation, defined as the domain of the Hamiltonian $H_2(u)$, is $H^{\\frac{1}{2}}(\\mathbb{R})$.\nThis functional framework provides the natural setting for investigating solution properties.\nIn the context of weak solutions, Ginibre and Velo \\cite{GV} established the existence of global solutions satisfying\n$u \\in L^\\infty(\\mathbb{R}, H^1(\\mathbb{R})) \\cap L^2_{\\mathrm{loc}}(\\mathbb{R}, H^{\\frac{3}{2}}_{\\mathrm{loc}}(\\mathbb{R}))$.\nFor strong solutions, the first global well-posedness result in $H^s(\\mathbb{R})$ was obtained by \\cite{ABFS} for $s > \\frac{3}{2}$.\nSubsequent developments progressively lowered the regularity threshold: Molinet and Vento \\cite{MV} extended well-posedness to $s > \\frac{1}{2}$\n(where unconditional uniqueness holds), while Molinet, Pilod, and Vento \\cite{MPV} advanced it to $s > \\frac{1}{4}$.\nA significant breakthrough in the sharp low-regularity theory was achieved by Ifrim and Saut \\cite{IS}, who proved global well-posedness\nin $L^2(\\mathbb{R})$ for small initial data. Most recently, Gassot and Laurens \\cite{GAL} established sharp global well-posedness\nfor the ILW equation in $H^s(\\mathbb{T})$ for all $s > -\\frac{1}{2}$, thereby\ncompleting the well-posedness theory on the torus. Additionally, they proved the\ncontinuous convergence of solutions to those of the Benjamin-Ono equation in the\ndeep-water limit $\\delta \\to \\infty$.\n\nBased upon the facts above, the (\\ref{ILW}) equation can naturally be viewed as a Hamiltonian system of the form \\cite{KSA, LR}\n\\begin{align}\n\tu_t=\\mathcal{J}\\frac{\\delta H_2(u)}{\\delta u},\n\\end{align}\nwhere $\\mathcal{J}=\\partial_x$, and $\\frac{\\delta H_2(u)}{\\delta u}$ (subsequently simplified to $H_2'(u)$) refers to the variational derivative of $H_2(u)$ can be written as\n\\begin{align*}\n\t(\\frac{\\partial}{\\partial \\epsilon}H_2(u+\\epsilon f))\\Big|_{\\epsilon=0}=\\int_{\\mathbb{R}}\\frac{\\delta H_{2}}{\\delta u}(x)f(x)\\text{d}x.\n\\end{align*}\n\nMoreover, the ILW equation possesses a bi-Hamiltonian structure \\cite{Sa}. However, unlike the KdV equation, the bi-Hamiltonian structure of the ILW equation is highly nontrivial due to its formulation involving both the spatial derivative operator $\\partial_x$ and the singular integral operator $T^\\delta$. This operator composition aligns the ILW equation with structural features characteristic of completely integrable systems in two spatial dimensions.\n\nTo formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n\t\\langle f_{12},g_{12}\\rangle\n\t&:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n\t\\theta_{12}^{\\pm}\n\t&:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n\t\\mathcal{J}_{12}^{(1)}\n\t:= \\theta_{12}^{-}, \\quad\n\t\\mathcal{J}_{12}^{(2)}\n\t:= (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n\t(T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n\tf(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n\tu_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n\t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n\t\\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n\t\\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n\t\\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n\t&u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n\t&u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n\t&u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n\tu(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n\ta\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n\tT^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n\t\\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nIn recent years, the stability theory for solitary waves and multi-solitons has emerged as a prominent research direction. Within soliton theory, stability concepts are systematically categorized into four distinct types based on analytical methodology and robustness guarantees: (i) linear (spectral) stability, concerning the eigenvalue distribution of linearized operators; (ii) Lyapunov (dynamical) stability, established via positive definiteness of the second variation of Lyapunov functionals at soliton solutions; (iii) orbital (nonlinear) stability, which requires that solutions remain within a neighborhood of the soliton orbit under finite-amplitude disturbances; and (iv) asymptotic stability, demanding convergence to specific soliton profiles.\n\nAs fundamental models in integrable systems, the stability of solitary waves for both KdV and BO equations has been extensively investigated. For instance, Maddocks and Sachs \\cite{MS} established orbital stability of KdV multi-solitons in the energy space defined by conservation laws. Killip and Visan \\cite{KV} proved orbital stability of KdV multi-solitons in $H^{-1}(\\mathbb{R})$ using low-regularity conservation laws in \\cite{BP}. For the BO equation, Wang \\cite{LW} and Matsuno \\cite{Mat} studied dynamical stability of multi-soliton solutions. Additionally, Badreddine, Killip and Visan \\cite{BKP} proved that the multi-soliton solutions to the BO equation are uniformly orbitally stable in $H^s(\\mathbb{R})$ with $-\\frac{1}{2}<s<\\frac{1}{2}$.\n\nIn contrast to the extensive results for KdV and BO equations, stability analysis for solitary wave solutions of the ILW equation remains comparatively underdeveloped. Existing studies primarily focus on orbital stability. Albert and Bona \\cite{AB} employed a rigorous variational approach combined with spectral analysis to establish orbital stability of ILW solitary waves in its energy space $\\mathcal{X} = \\{g \\in L^2(\\mathbb{R})| \\int_\\mathbb{R}(1+\\alpha(k))|\\hat{g}|^2\\mathrm{d}k < \\infty\\}$. Subsequent work \\cite{A} extended these results to spaces $H^{\\frac{1}{2}}(\\mathbb{R})$, while \\cite{PCA} addressed linear and orbital stability of periodic wave solutions in periodic Sobolev spaces.\n\nConsidering the growing importance of understanding coherent structures in nonlocal dispersive equations, a systematic analysis of soliton and multi-soliton stability for the ILW equation represents a significant open research direction. Hence, in this work, we aim to establish dynamical stability results of $n$-soliton solutions to the ILW equation (\\ref{ILW}) in an appropriate Sobolev space.\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate  Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n\t\\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n\t\\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n\tH_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n\t\\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nThe demonstration of dynamical stability for the ILW equation relies on the recursion operators and the construction of a Lyapunov functional. This approach utilizes the trace formula derived from the inverse scattering transform (IST) framework to forge a link with conservation laws, in a manner analogous to techniques applied in the study of the Camassa-Holm equation \\cite{AYP, CGI, DKS}. The stability proof centers on a spectral analysis of the second variation $\\mathcal{L}_{n} = \\mathcal{S}_{n}''(U^{(n)})$ evaluated at a smooth $n$-soliton $U^{(n)}$, coupled with an eigenvalue computation for the Hessian matrix $D = \\left\\{\\frac{\\partial^2 \\mathcal{S}_{n}}{\\partial \\mu_{i} \\partial \\mu_{j}} \\right\\}$.\n\nThe spectral characterization of the linearized operator $\\mathcal{L}_{n}$ around the $n$-soliton $U^{(n)}$ is central to this endeavor. In the case of a single soliton, the operator $L_n = H''_{n+1}(Q_c) + c H''_{n}(Q_c)$ is analyzed. The inertia index $\\mathrm{in}(L_n)$ is determined by constructing the derivative operators $\\mathcal{J}L_n$ and $L_n\\mathcal{J}$ and leveraging IST theory, specifically employing Jost solutions and square eigenfunctions. For the general $n$-soliton, the recursive operator $\\mathcal{R}$ originating from the bi-Hamiltonian structure is applied to compute $\\mathrm{in}(\\mathcal{L}_n)$.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).",
    "context": "To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n \\langle f_{12},g_{12}\\rangle\n &:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n \\theta_{12}^{\\pm}\n &:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n \\mathcal{J}_{12}^{(1)}\n := \\theta_{12}^{-}, \\quad\n \\mathcal{J}_{12}^{(2)}\n := (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n (T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n f(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n u_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n &=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n \\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n \\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n \\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n &u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n &u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n &u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n u(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n a\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n T^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n \\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n \\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n \\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n H_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n \\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).",
    "full_context": "To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n \\langle f_{12},g_{12}\\rangle\n &:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n \\theta_{12}^{\\pm}\n &:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n \\mathcal{J}_{12}^{(1)}\n := \\theta_{12}^{-}, \\quad\n \\mathcal{J}_{12}^{(2)}\n := (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n (T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n f(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n u_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n &=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n \\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n \\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n \\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n &u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n &u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n &u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n u(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n a\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n T^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n \\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n \\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n \\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n H_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n \\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).\n\nThen we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.\n\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n \\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n \\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n \\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\n\\begin{align*}\n (\\mathcal{R}^\\star(u))^m u_x &= \\mathcal{J} (\\mathcal{R} (u))^m H_1'(u) = \\mathcal{J} (\\mathcal{R} (u))^m u, \\quad m \\in \\mathbb{N}.\n\\end{align*}\nFor subsequent applications, establishing the uniqueness and differentiability of $\\mathcal{R}(u)$ becomes essential. In contrast, the Korteweg-de Vries system ($\\delta = 3$) exhibits explicit structure. For Schwartz-class functions $u \\in \\mathcal{S}(\\mathbb{R})$, its recursion operator is $\\mathcal{R}_K(u) = -\\partial_x^2 - \\tfrac{2}{3}u - \\tfrac{2}{3}\\partial_x^{-1} u \\partial_x$, then $\\mathcal{R}_K'(u) = -\\tfrac{2}{3} - \\tfrac{2}{3}\\partial_x^{-1}(\\cdot\\partial_x)$.\n\\begin{prop}\\label{prop2.2}\n Given $u \\in H^{k+1}(\\mathbb{R})$ with $k \\geq 0$, there exists a unique linear operator\n \\[\n \\mathcal{R}(u): H^{k+1}(\\mathbb{R}) \\to H^{k}(\\mathbb{R}),\n \\]\n such that relations (\\ref{2.39}) and (\\ref{2.41}) hold true. Moreover, $\\mathcal{R}(u)$ is differentiable with respect to $u$.\n\\end{prop}\n\\begin{proof}\n The fundamental approach involves establishing a connection between the recursion operators $\\mathcal{R}(u)$ and $\\mathcal{R}_{12}$ (\\ref{1.15}). Assuming $u \\in \\mathcal{S}(\\mathbb{R})$, it follows from (\\ref{1.14}) and (\\ref{2.39}) that\n \\begin{align*}\n \\mathcal{S}(\\mathbb{R}) \\ni H_{m+1}^{\\prime}(u) & = \\frac{i}{2^{m+1}}\\mathcal{J}^{-1}\\int_{\\mathbb{R}} \\delta(x_{1} - x_{2}) \\theta_{12}^{-} \\mathcal{R}_{12}^{m+1} \\cdot 1 \\mathrm{d}x_{2} \\\\\n & = \\mathcal{R}(u)H_{m}^{\\prime}(u) \\\\\n & = \\mathcal{R}(u) \\frac{i}{2^m}\\mathcal{J}^{-1}\\int_{\\mathbb{R}} \\delta(x_{1} - x_{2}) \\theta_{12}^{-} \\mathcal{R}_{12}^{m} \\cdot 1 \\mathrm{d}x_{2}.\n \\end{align*}\n The uniqueness of $\\mathcal{R}(u)$ follows by an induction argument over $m$. Furthermore, the asymptotic behavior of the operator is characterized by\n \\[\n \\mathcal{R}(u) \\sim -T^\\delta\\partial_{x} + L(u),\n \\]\n where $L(u): \\mathcal{S}(\\mathbb{R}) \\mapsto \\mathcal{S}(\\mathbb{R})$ represents a differentiable higher-order remainder term. By standard density argument, $\\mathcal{R}(u)$ itself is differentiable and satisfies $\\mathcal{R}^{\\prime}(u) \\sim L^{\\prime}(u)$.\n\\end{proof}\n\\begin{re}\nThe propositions above will be shown in Section \\ref{sec.3} that understanding the spectral information of the (adjoint) recursion operators $\\mathcal{R} (u)$ and $\\mathcal{R}^\\star(u)$ is essential in proving the (spectral) stability of the ILW multi-solitons.\n\\end{re}\n\nNext, we will proceed to prove Theorem \\ref{th1.3}.\n\\subsection{The negative eigenvalues of $\\mathcal{L}_n$ :Proof of Theorem \\ref{th1.3}}\nIn this subsection, our goal is to prove the Theorem \\ref{th1.3}.\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{th1.3}}]\n The linearized operators around the $n$-solitons $\\mathcal{L}_{n} = S_{n}^{\\prime\\prime}(U^{(n)})$ possess $\\left[\\frac{n+1}{2}\\right]$ negative eigenvalues, as previously verified in (\\ref{3.47}). We now prove the expression (\\ref{negetive}) for these eigenvalues. As $t\\to \\infty$, the spectrum $\\sigma(\\mathcal{L}_{n}(t))$ converges to the union of the spectra $\\sigma(\\mathcal{L}_{n,j})$ where $\\mathcal{L}_{n,j} = S_{n}^{\\prime\\prime}(Q_{c_{j}})$, that is\n \\begin{align}\n \\sigma(\\mathcal{L}_{n}(t)) \\rightarrow \\bigcup_{j=1}^{n} \\sigma(\\mathcal{L}_{n,j}), \\quad \\text{as} \\quad t \\rightarrow +\\infty.\n \\end{align}\n Since the operators $\\mathcal{L}_{n}(t)$ are isoinertial for each $n$, the spectrum $\\sigma(\\mathcal{L}_{n}(t))$ is independent of $t$. Thus, the negative eigenvalues of $\\mathcal{L}_{n}$ match those of $\\mathcal{L}_{n,j}$ for all $j = 1, 2, \\dots, n$. According to Lemma \\ref{lem3.5}, $\\mathcal{L}_{n,j}$ has negative eigenvalues when $j = 2k-1$ and $1 \\leq k \\leq \\left[\\frac{n+1}{2}\\right]$. Specially, when $n=1$, one has $\\mathcal{L}_1=L_1$. Therefore, (\\ref{n=1}) indicates that $\\mathcal{L}_1$ has the negative $\\lambda_1<0$. With the methods in \\cite{BBSSB}, we could compute $\\lambda_1$, which is\n \\begin{align}\\label{1.30}\n \\lambda_1=-\\frac{c-\\frac{1}{\\delta}}{2a^2\\sin^2(a\\delta)}<0,\n \\end{align}\n Then for $n=2$, we have\n \\begin{align}\\label{lamb}\n \\mathcal{L}_{2,1}=(\\mathcal{R}(Q_{c_1})+c_2)\\mathcal{S}_1''(Q_{c_1}).\n \\end{align}\nIn terms of Lemma \\ref{lem3.2}, the operator $(\\mathcal{R}(Q_{c_1}) + c_{2})$ has an eigenvalue $c_{2} - c_{1} > 0$, with continuous spectrum $(c_2, +\\infty)$ whose generalized eigenfunctions are not in $L^{2}(\\mathbb{R})$. Therefore, the unique negative eigenvalues of $\\mathcal{L}_{2,1}$ is $(c_2-c_1)\\lambda_1<0$.\\\\\nAssuming the result holds for $n = p$, namely, the $\\left[\\frac{p+1}{2}\\right]$-th negative eigenvalue of $\\mathcal{L}_{p}$ is\n \\begin{align}\\label{4.48}\n \\lambda_{k}^{(p)} = -M^\\delta (c_{2k-1}-\\frac{1}{\\delta}) \\prod_{j \\neq 2k-1}^{p} (c_{j} - c_{2k-1}), \\quad k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right].\n \\end{align}\nThen for $n = p+1$ is even, we have $\\left[\\frac{p+1}{2}\\right] = \\left[\\frac{p+2}{2}\\right]$ and for $k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right]$, we have\n \\begin{align}\n \\mathcal{L}_{p+1,2k-1} =(\\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1}) \\mathcal{S}_{p}^{\\prime\\prime}(Q_{c_{2k-1}}).\n \\end{align}\n By Lemma \\ref{lem3.2}, the operator $(\\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1})$ has an eigenvalue $c_{p+1} - c_{2k-1} > 0$, with continuous spectrum $(c_{p+1}, +\\infty)$ whose generalized eigenfunctions are not in $L^{2}(\\mathbb{R})$. Therefore, the $\\left[\\frac{p+2}{2}\\right]$-th negative eigenvalues of $\\mathcal{L}_{p+1,2k-1}$ is\n \\begin{align}\\label{4.50}\n \\lambda_{k}^{(p+1)}= -M^\\delta (c_{2k-1}-\\frac{1}{\\delta}) \\prod_{j \\neq 2k-1}^{p+1} (c_{j} - c_{2k-1}), \\quad k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right].\n \\end{align}\nFor the case where $n = p+1$ is odd, we have $\\left[\\frac{p+1}{2}\\right] + 1 = \\left[\\frac{p+2}{2}\\right]$. The first $\\left[\\frac{p+1}{2}\\right]$ negative eigenvalues of $\\mathcal{L}_{p+1}$ are given by expression (\\ref{4.50}). We now compute the final negative eigenvalue. Consider\n\\begin{align}\n \\mathcal{L}_{p+1,p+1} =\\left( \\mathcal{R}(Q_{c_{p+1}}) + c_{j} \\right) \\tilde{S}_{p}^{\\prime\\prime}(Q_{c_{p+1}})\n\\end{align}\nwhere $\\tilde{S}_{p}$ denotes the action with wave speed $c_j$ in $S_p$ replaced by $c_{p+1}$ for some $1 \\leq j \\leq p$. By assumption (\\ref{4.48}), the discrete eigenvalue of $\\tilde{S}_{p}^{\\prime\\prime}(Q_{c_{p+1}})$ is\n\\begin{align}\n -M^\\delta (c_{p+1}-\\frac{1}{\\delta})\\prod_{l \\neq j}^{p} (c_l - c_{p+1}).\n\\end{align}\nThe operator $\\mathcal{R}(Q_{c_{p+1}}) + c_j$ has eigenvalue $c_j - c_{p+1} < 0$ by Lemma \\ref{lem3.2}, with continuous spectrum $(c_j, +\\infty)$ whose generalized eigenfunctions are not in $L^2(\\mathbb{R})$. Therefore, the last negative eigenvalue of $\\mathcal{L}_{p+1}$ is\n\\begin{align*}\n \\lambda_{\\left[\\frac{p+2}{2}\\right]}^{(p+1)} = -M^\\delta (c_{p+1}-\\frac{1}{\\delta}) \\prod_{l=1}^{p} (c_l - c_{p+1}).\n\\end{align*}\nThe proof of Theorem \\ref{th1.3} is completed by combining the eigenvalue expressions.\n\\end{proof}",
    "post_theorem_intro_text_len": 4644,
    "post_theorem_intro_text": "Since the stability result in \\cite{MS} coincides with orbital stability theorem in \\cite{GSS} for $n=1$, we obtain the following corollary regarding the orbital stability of ILW soliton solutions.\n\\begin{cor}\nThe single soliton solution of the ILW equation is orbital stable in $H^{\\frac{1}{2}}(\\mathbb{R})$.\n\\end{cor}\n\\begin{re}\nFor $n \\geq 2$, it should be noted that the stability result in \\cite{MS} aligns with the notion of orbital stability in \\cite{GSS} when the stability framework of \\cite{MS} is suitably extended. In Hamiltonian systems, invariant functionals (i.e., integrals of motion) generate Hamiltonian flows that commute with the original time evolution.\n\\end{re}\n\nThen we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.\n\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\nMoreover, something is interesting is that one can express the negative eigenvalues of the isoinertial operator $\\mathcal{L}_n$ in terms of the wave speeds $\\{c_j\\}^n_{j=1}$ as follows.\n\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}\n\\begin{re}\n\tSince the ILW equation serves as the bridge connecting KdV equation and BO equation. The stability results above are also valid for both equations. The specific results can be found in \\cite{WA} and \\cite{LW}, respectively.\n\\end{re}\n\nThe proof of our main stability result, Theorem \\ref{thm1.1}, proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data. Similarly, the proof of Theorem \\ref{thm1.2} on orbital stability is reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional. Finally, the proof of Theorem \\ref{th1.3} relies on the application of spectral analysis results. The proof strategy unfolds as follows.\n\nIn Section \\ref{sec.2}, we begin by establishing the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$. We then introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.\n\nSection \\ref{sec.3} is devoted to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$. We demonstrate that its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.\n\nIn Section \\ref{sec.4}, we apply the framework of constrained variational principles to establish the dynamical stability result in Theorem \\ref{thm1.1}. We then prove Theorem \\ref{thm1.2} on orbital stability of double solitons ($n=2$) through a refined spectral analysis of the linearized operator $\\mathcal{L}_2$, yielding a direct proof of orbital stability. Finally, we establish Theorem \\ref{th1.3} based on the spectral analysis results from Section \\ref{sec.3}.",
    "sketch": "The proof of Theorem~\\ref{thm1.1} “proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator.” “Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.”\n\nMore concretely, the stated strategy is organized by sections as follows:\n\\begin{itemize}\n\\item \\emph{Section~\\ref{sec.2}:} establish “the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”\n\\item \\emph{Section~\\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.”\n\\item \\emph{Section~\\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in Theorem~\\ref{thm1.1}.”\n\\end{itemize}\n\n(Additionally, the introduction notes that the orbital stability theorem (Theorem~\\ref{thm1.2}) is “reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that Theorem~\\ref{th1.3} “relies on the application of spectral analysis results.”)",
    "expanded_sketch": "The proof of the main theorem proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.\n\nMore concretely, the stated strategy is organized by sections as follows:\n\\begin{itemize}\n\\item \\emph{Section~\\ref{sec.2}:} establish “the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”\n\\item \\emph{Section~\\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.”\n\\item \\emph{Section~\\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in the main theorem.”\n\\end{itemize}\n\nAdditionally, the introduction notes that the orbital stability theorem\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\nis “reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that the following theorem “relies on the application of spectral analysis results”:\n\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}",
    "expanded_theorem": "[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of\n\\begin{align}\n\\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}\nThen for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of the equation above verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}",
    "theorem_type": [
      "Implication",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Let \\(n\\in\\mathbb N\\) with \\(n\\ge 1\\), let \\(\\mathbf c=(c_1,\\dots,c_n)\\) satisfy \\(0<c_1<\\cdots<c_n\\), and let \\(\\mathbf x=(x_1,\\dots,x_n)\\in\\mathbb R^n\\). Let \\(U^{(n)}(t,x;\\mathbf c,\\mathbf x)\\) be the corresponding \\(n\\)-soliton solution of the ILW equation\n\\[\nu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\qquad (t,x)\\in\\mathbb R\\times\\mathbb R,\n\\]\nwhere \\(T^\\delta\\) is the ILW nonlocal operator. Suppose \\(u(t)\\) is the solution with initial data \\(u(0)=u_0\\in H^{n/2}(\\mathbb R)\\). Which of the following conclusions holds about the orbit of the \\(n\\)-soliton family \\(\\{U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y):\\tau\\in\\mathbb R,\\ \\mathbf y=(y_1,\\dots,y_n)\\in\\mathbb R^n\\}\\) in the \\(H^{n/2}(\\mathbb R)\\) norm?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\tau\\in\\mathbb R,\\ \\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\) with the time parameter fixed."
        },
        {
          "label": "C",
          "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nIn particular, the orbit remains close to the \\(n\\)-soliton family in \\(H^{n/2}(\\mathbb R)\\)."
        },
        {
          "label": "D",
          "text": "There exists \\(\\eta>0\\) such that for every \\(\\epsilon>0\\), if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\tau\\in\\mathbb R,\\ \\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\) with a uniform stability radius independent of \\(\\epsilon\\)."
        },
        {
          "label": "E",
          "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the given \\(n\\)-soliton solution is asymptotically stable in \\(H^{n/2}(\\mathbb R)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "time-phase modulation parameter",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "free time-shift parameter \\(\\tau\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "quantifier order between \\(\\epsilon\\) and neighborhood size",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "orbital stability modulo symmetries replaced by fixed-profile/asymptotic stability",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the correct option. It asks for the valid uniform orbital stability formulation, and the answer is not trivially recoverable without knowing or reasoning through the role of the modulation parameters and quantifiers."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct choice is essentially the precise stability statement. However, it is not a pure restatement because the options vary in meaningful ways (time shift, quantifier order, fixed versus time-dependent orbit)."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish the strongest valid orbital stability claim from nearby false or weaker variants, especially regarding the extra time-shift parameter and quantifier structure. Still, it mainly tests recognition of the exact theorem form rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they reflect realistic mathematical confusions such as removing a modulation parameter, weakening to a pointwise-in-time existence statement, reversing quantifiers, or demanding an unrealistically fixed orbit for all time."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-centered MCQ: it avoids obvious leakage and has high-quality distractors, but it leans more toward precise statement recognition than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.08599v1",
    "paper_link": "http://arxiv.org/abs/2512.08599v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\np)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.",
      "start_pos": 9356,
      "end_pos": 11112,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:con": "\\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation}",
      "thm:Iw-main-int": "\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm}",
      "eq:Lp-eq": "\\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}",
      "eq:main": "\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}",
      "eq:contain": "\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation}",
      "thm:main": "\\begin{thm}\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\np)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 4409,
    "pre_theorem_intro_text": "\\subsection{Special value formula, Gross period and Hida canonical period}\n\nIn \\cite{Gro87} Gross provided a formula expressing the special\nvalues of $L$-functions via Heegner points. To describe his result,\nlet $f$ be a newform on $\\Gamma_0(N)$  of even weight $k$, and let $K$ be an imaginary\nquadratic field of discriminant $D$ such that $D$ is prime to $N$.\n\nAssume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nGross formula is generalized by Shouwu Zhang \\cite[Theorem 7.1]{Zha04} for $k=2$ and general $N$, and by Haiping Yuan \\cite{Yua05} for $k>2$.\n\nIn \\cite{BD07}  Bertolini and Darmon used Gross' special value\nformula to construct anticyclotomic $p$-adic $L$-functions, i.e.\n$p$-adic $L$-functions for anticyclotomic $\\BZ_p$-extension instead\nof the cyclotomic $\\BZ_p$-extension. In the same paper they proved\none divisibility for Iwasawa main conjecture for elliptic curves in\nthe setting of anticyclotomic $\\BZ_p$-extension (\\cite[Theorem\n1]{BD07}), and also gave new evidence for Birch and Swinnerton-Dyer\nconjecture (\\cite[Corollary 4]{BD07}).\n\nGross period serves as a bridge between the complex $L$-function and\nthe $p$-adic $L$-function. But it depends on the choice of an imaginary\nquadratic extension $K$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\n\\subsection{Comparison of periods under base change}\n\nThe goal of the present paper is to compare the canonical period of\na Hilbert modular form and that of its base change to a (totally)\nreal quadratic extension.\n\nTo make it precise, let $F$ be a totally real number field, and  $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.",
    "context": "Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\nTo make it precise, let $F$ be a totally real number field, and $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.",
    "full_context": "Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\nTo make it precise, let $F$ be a totally real number field, and $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.\n\nWe choose an imaginary quadratic extension $K$ of $F$ that is\ninertia at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd. Denote $KF'$\nby $K'$. Fix a place $\\mathfrak{p}$ of $F$ above $p$. Let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K$.\n\n\\begin{lem}\\label{lem:fitt} Let $S$ be the set of primes dividing $\\mathfrak{n}^-$. Then\n$$ \\mathrm{Fitt}(\\prod_{\\mathfrak{l}|\n\\mathfrak{n}^- }H^1((\\mathcal{K}_\\infty)_\\mathfrak{l},\nF^-_\\mathfrak{l}A_\\rho)^\\vee) =\\left(\\frac{L^S_p (\\mathcal{K}_\\infty,\nf)}{L_p (\\mathcal{K}_\\infty, f)}\\right).\n$$\n\\end{lem} This should be well known.\nHowever, the authors do not have a reference about it.\n\\begin{proof}\nBy definition (see Section \\ref{ss:selmer}) $H^{1}((\\mathcal{K}_{\\infty})_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho})$ is the inductive limit $\\lim\\limits_{\\overrightarrow{\\;\\;L\\;\\;}} H^{1}(L_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho}),$ where $L$ runs through all finite extensions of $K$ contained in $\\mathcal{K}_\\infty$. For each $L$, $H^{1}(L_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho})$ is isomorphic to\n$\\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})$ where $\\mathfrak{L}$ runs over all places of $L$ above $\\mathfrak{l}$,\nand sits in the following exact sequence\n{\\small $$ \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})\\rightarrowtail \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})\\twoheadrightarrow \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(I_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})^{G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}} . $$}\nAs $G_{L_{\\mathfrak{L}}}$ acts trivially on $F_\\mathfrak{l}^-A_\\rho$, we have\n$$ H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}}) \\cong F_{\\mathfrak{l}}^{-}A_{\\rho}. $$ If $L'$ is a finite extension of $L$ contained in $\\mathcal{K}_\\infty$, and $\\mathfrak{L}'$ is a place of $L'$ above $\\mathfrak{L}$, we have the following commutative diagram\n\\[ \\xymatrix{ H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})\\ar[rr]^{\\hskip45pt \\simeq}\\ar[d] && F_{\\mathfrak{l}}^{-}A_{\\rho} \\ar[d]^{\\times[G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}:G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}]} \\\\ H^1(G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L'_{\\mathfrak{L}'}}})\\ar[rr]^{\\hskip45pt\\simeq} && F_{\\mathfrak{l}}^{-}A_{\\rho}, } \\] where the left vertical arrow is the restriction map. Since both $L_{\\mathfrak{L}}$ and $L'_{\\mathfrak{L}'}$ are unramified over $K_\\mathfrak{l}$, $$[G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}:G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}]\n=|\\mathrm{Gal}(L'_{\\mathfrak{L}'}/L_{\\mathfrak{L}})|=[L'_{\\mathfrak{L}'}:L_{\\mathfrak{L}}]$$ is a power of $p$. Thus taking limit we have $$\\lim\\limits_{\\overrightarrow{\\;\\;L\\;\\;}}\\bigoplus_{\\mathfrak{L}: \\text{places of } L \\text{ above }\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})=0.$$\n\n\\begin{proof} The representation $\\bar{\\rho}_f$ factors through\n$\\mathrm{Gal}(F_1/F)$ for a finite Galois extension $F_1$ of $F$.\nTake a finite place $\\mathfrak{q}\\nmid p\\mathfrak{n}$. For each\n$\\mathfrak{l}|p\\mathfrak{n}$ take a unit $u_\\mathfrak{l}$ of\n$\\mathcal{O}_{F_{\\mathfrak{l}}}$ such that the image\n$u_\\mathfrak{l}$ in $k_{\\mathfrak{l}}$ by the natural homomorphism\n$$ \\mathcal{O}_{F_\\mathfrak{l}}\\rightarrow\n\\mathcal{O}_{F_\\mathfrak{l}}/\\mathfrak{l}\\cong k_\\mathfrak{l} $$\ngenerates the multiplicative group $k_\\mathfrak{l}^\\times$. Take a\nuniformizing element $\\omega_\\mathfrak{q}$ of\n$\\mathcal{O}_{F_{\\mathfrak{q}}}$. By the weak approximation theorem\nthere exists a totally positive element $a$ of $F$ such that\n$a_{\\mathfrak{l}}-u_{\\mathfrak{l}} \\in\n\\mathfrak{l}\\mathcal{O}_{F_\\mathfrak{l}}$ for each\n$\\mathfrak{l}|p\\mathfrak{n}$, and\n$a_\\mathfrak{q}-\\omega_\\mathfrak{q}\\in\n\\mathfrak{q}^2\\mathcal{O}_{F_\\mathfrak{q}}$. Put $F'=F(\\sqrt{a})$.\nThen each place dividing $p\\mathfrak{n}$ splits in $F'$. As $a$ is\ntotally positive, $F'$ is totally real. Let $f'$ be the base change\nof $f$ to $F'$.\n\n\\begin{thm}\\label{thm:Iw-main} Assume $f$ satisfies\n$(\\mathrm{CR}^+)$, $(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$,\n$(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\n\\begin{enumerate}\n\\item\\label{it:Iw-main-a}\nWe have \\begin{equation}\\label{eq:char} \\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee = (L_p(K_\\infty, f)). \\end{equation}\n\\item\\label{it:Iw-main-b} Let $\\mathcal{L}$ be $\\mathcal{K}_\\infty,\n\\mathcal{K}^-_\\infty$ or $K^-_J$. If $[F:\\BQ]$ is even, then\n$$ \\mathrm{char}\\ \\mathrm{Sel}(\\mathcal{L}, A_\\rho)^\\vee = (L_p(\\mathcal{L}, f))\n$$ in $\\mathcal{O}_\\mathfrak{P}[[\\mathrm{Gal}(\\mathcal{L}/K)]]_E$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} Let $F'$ be a totally real quadratic extension of $F$, as in Lemma \\ref{lem:quad-ext}. Then $[F':\\BQ]$ is even.\nLet $\\chi_{F'/F}$ be the quadratic character corresponding to the\nextension $F'/F$. Put $K'=F'K$. Let $\\mathcal{K}'^-_\\infty$ be the\nmaximal abelian anticyclotomic $\\BZ_p$-extension of $K'$ unramified\noutside $p$. Put $\\mathcal{K}'^+_\\infty=F_\\infty K'$,\n$\\mathcal{K}'_\\infty=\\mathcal{K}'^+_\\infty\\mathcal{K}'^-_\\infty$ and\n$\\Gamma_{K'}=\\mathrm{Gal}(\\mathcal{K}'_\\infty/K')$. Then\n$K'\\mathcal{K}_\\infty=F'\\mathcal{K}_\\infty$ is contained in\n$\\mathcal{K}'_\\infty$.\n\n\\begin{thm}\\label{thm:main-stronger} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nWe assume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary\nat each place above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume\nthat there exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, $\\mathfrak{q}$ is split in $F'$,\nand $\\bar{\\rho}_{f}$ is ramified at $\\mathfrak{q}$. Suppose that, if\n$\\mathfrak{l}'||\\mathfrak{n}_{f'}$ and\n$\\mathrm{N}(\\mathfrak{l}')\\equiv 1 \\ (\\mathrm{mod} \\ p)$, then\n$\\bar{\\rho}_{f'}$ is ramified at $\\mathfrak{l}'$.\n\n\\begin{conj}\\label{conj-mu} Let $F$ be a totally real number field, $K$ a quadratic imaginary\nextension of $F$. Let $F'$ be a finite totally real extension of\n$F$, and $f_{F'}$ be a Hilbert modular form over $F'$ that is\nordinary at each place of $F'$ above $p$. Then\n$$\\mu(L_p(F'K_\\infty, f_{F'}))=0.$$\n\\end{conj}\nCorollary \\ref{thm:mu} says that Conjecture \\ref{conj-mu} holds when\n$F'$ is a quadratic real extension of $F$ and $f_{F'}$ is the base\nchange of a Hilbert modular form over $F$ (that satisfies some\nconditions). It seems impossible to use the method in \\cite{CH18,\nHung} to prove Conjecture \\ref{conj-mu}, though it works when\n$F'=F$.",
    "post_theorem_intro_text_len": 5953,
    "post_theorem_intro_text": "We choose an imaginary quadratic extension $K$ of $F$ that is\ninertia at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd. Denote $KF'$\nby $K'$. Fix a place $\\mathfrak{p}$ of $F$  above $p$. Let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K$.\n\nUsing the canonical periods one attaches to $f$,\n$f\\otimes\\chi_{F'/F}$ and $f'$ the $p$-adic $L$-functions\n$$L_{p}(K_{\\infty} ,f), \\\nL_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F})  \\ \\text{ and }\nL_{p}(K'K_{\\infty} ,f').$$ All of them can be considered as elements\nof $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty} /K)]]$. The\nring $\\mathcal{O}_{\\mathfrak{P}}\\supset \\mathcal{O}_f$ is some\ncoefficient ring that is clearly defined in our context. To prove\nTheorem \\ref{thm:main} we only need to show\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation} in\n$\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty} /K)]].$\n\nOur strategy is to use Iwasawa main conjecture that provides an\nequality between a quantity measuring Selmer groups and $p$-adic\n$L$-functions. Actually what we need is Iwasawa main conjecture for\nHilbert modular forms in the anticyclotomic setting.\n\nIts proof is divided into two parts, one part proving one\ndivisibility by Ribet's method, and the other proving the converse\ndivisibility by Euler systems. The former divisibility was proved by\nSkinner and Urban \\cite{S-U} for elliptic modular forms, and was\nproved by Wan \\cite{Wan} for Hilbert modular forms. When\n$[F:\\mathbb{Q}]$ is odd, Wan needs the condition that Ihara Lemma\nfor Shimura curves holds. The latter divisibility was proved by\nBertolini and Darmon \\cite{BD07} for elliptic curves, and by Chida\nand Hsieh \\cite{CH15} for elliptic modular forms. It was proved by\nLongo \\cite{Longo} and Wang \\cite{Wang} for Hilbert modular forms\nassuming Ihara Lemma for Shimura curves. This condition was removed\nby the second author \\cite{Xie}.\n\nCombining results in \\cite{Wan} and \\cite{Xie} we obtain the\nfollowing theorem. Remark that we do not need Ihara Lemma for\nShimura curves even when $[F:\\mathbb{Q}]$ is odd, though such a\ncondition is needed in \\cite{Wan}.\n\n\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm} See our context for Conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and\n$(\\mathrm{Fuji}1$-$4)$.\n\nBy Theorem \\ref{thm:Iw-main-int} we have $$\n(L_{p}(K_{\\infty},f))=\\mathrm{char}\\\n\\mathrm{Sel}(K_{\\infty},A_{\\rho})^{\\vee} $$ and\n    $$  (L_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\ \\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}})^{\\vee}    .$$\nUsing the relation\n$$\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}})$$\nwe obtain \\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}\n\nLet $\\mathfrak{p}'$ be a place of $F'$ above $\\mathfrak{p}$. Let\n$K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K'$.    With $K'_\\infty$ instead of\n$K_\\infty$ we also have $$ (L_{p}(K_{\\infty}',f'))=\\mathrm{char}\\\n\\mathrm{Sel}(K_{\\infty}',A_{\\rho})^{\\vee},$$ from which we deduce\nthat \\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation} Combining (\\ref{eq:Lp-eq}) and (\\ref{eq:con}) we\nobtain\n\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation} As\n(\\ref{eq:main}) holds in\n$\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty}\n/K)]][\\frac{1}{p}]$, and as the $\\mu$-invariants of\n$L_{p}(K_{\\infty} ,f)$ and $L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F})$\nare zero, (\\ref{eq:contain}) implies (\\ref{eq:main}).\n\nThe paper is organized as follows. In Section \\ref{sec:a} we collect\nsome basic facts about automorphic forms on definite quaternion\nalgebras. In Section \\ref{sec:b} we recall the construction of\nanticyclotomic $p$-adic $L$-functions via Theta elements. In Section\n\\ref{sec:d}, we combine Wan's result in \\cite{Wan} with the result\nin \\cite{Xie} to prove Iwasawa main conjecture for Hilbert modular\nform in the anticyclotomic setting (under certain conditions).\nFinally we prove Theorem \\ref{thm:main} in Section \\ref{sec:e}.\n\nIn Sections \\ref{sec:a}-\\ref{sec:d}, the imaginary quadratic\nextension $K$ of $F$ is fixed. In Section \\ref{sec:e}, we will\nchoose a suitable $K$.\n\nThe authors thank B.H. Gross for his helpful advice.\n\n\\subsection*{Notations}\n\nWe fix a totally real number field \\(F\\) over \\(\\mathbb{Q}\\). Let\n\\(\\Sigma_{F}\\) be the set of all real embeddings of \\(F\\) and let\n\\(\\Sigma_{p}\\) be the set of all places of \\(F\\) above \\(p\\).\n\nFor a number field \\(L\\), and each place \\(v\\) of \\(L\\) above a\nprime number \\(l\\), let \\(|\\cdot|_{v}\\) or \\(|\\cdot|_{L_{v}}\\) be\nthe absolute value on \\(L_{v}\\) defined by\n\\(|x|_{v}=|N_{L_{v}/\\mathbb{Q}_{l}}(x)|_{l}\\), and  let\n\\(\\varpi_{v}\\) be a uniformizer of \\(\\mathcal{O}_{L_{v}}\\).\n\nLet $E$ be a finite extension of $\\mathbb{Q}$, $\\mathcal{O}$ be the ring of\nintegers in $E$ and $\\mathfrak{P}$ be a prime  of $\\mathcal{O}$\nabove $p$ such that $\\mathcal{O}_\\mathfrak{P}$ contains\n$\\mathcal{O}_f$. Let $\\omega$ be a uniformizer of\n$\\mathcal{O}_\\mathfrak{P}$. For each positive integer $n$ we put\n$\\mathcal{O}_{n}=\\mathcal{O}_\\mathfrak{P}/\\omega^n$. We consider\n$E_\\mathfrak{P}$, $\\mathcal{O}_\\mathfrak{P}$ and $\\mathcal{O}_n$ as\ncoefficient rings, and let $G_F=\\mathrm{Gal}(\\overline{F}/F)$ act trivially on them.\n\nLet $\\epsilon$ be the $p$-adic cyclotomic character of $G_F$.",
    "sketch": "Choose an imaginary quadratic extension $K/F$ (inert at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\\mathfrak{p}\\mid p$ of $F$ and let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic $\\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\\infty,f)$, $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ and $L_p(K'K_\\infty,f')$ in $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$. To prove Theorem~\\ref{thm:main} it suffices to show the ideal identity\n\\[\\tag{\\ref{eq:main}} (L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot (L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))\\]\ninside $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$.\n\nThe strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. By Theorem~\\ref{thm:Iw-main-int},\n\\[(L_p(K_\\infty,f))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho})^\\vee,\\qquad (L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho\\otimes\\chi_{F'/F}})^\\vee.\n\\]\nUsing the Selmer decomposition\n\\[\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}}),\\]\none gets\n\\[\\tag{\\ref{eq:Lp-eq}} (L_p(K_\\infty,f)\\cdot L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K'K_\\infty,A_{\\rho})^\\vee.\n\\]\nThen, letting $\\mathfrak{p}'\\mid\\mathfrak{p}$ in $F'$ and $K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic $\\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\\infty,f'))=\\mathrm{char}\\,\\mathrm{Sel}(K'_\\infty,A_{\\rho})^\\vee$, which implies the containment\n\\[\\tag{\\ref{eq:con}} (L_p(K'K_\\infty,f'))\\supseteq \\mathrm{char}\\,\\mathrm{Sel}(K'K_\\infty,A_{\\rho})^\\vee.\n\\]\nCombining \\eqref{eq:Lp-eq} and \\eqref{eq:con} yields\n\\[\\tag{\\ref{eq:contain}} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot L_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})).\n\\]\nFinally, since \\eqref{eq:main} holds after inverting $p$ and the $\\mu$-invariants of $L_p(K_\\infty,f)$ and $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ are zero, the containment \\eqref{eq:contain} implies the equality \\eqref{eq:main}, hence Theorem~\\ref{thm:main}.",
    "expanded_sketch": "Choose an imaginary quadratic extension $K/F$ (inert at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\\mathfrak{p}\\mid p$ of $F$ and let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic $\\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\\infty,f)$, $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ and $L_p(K'K_\\infty,f')$ in $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$. To prove the main theorem, it suffices to show the ideal identity\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}\ninside $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$.\n\nThe strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. We first use the following theorem.\n\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm}\nThus\n\\[(L_p(K_\\infty,f))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho})^\\vee,\\qquad (L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho\\otimes\\chi_{F'/F}})^\\vee.\n\\]\nUsing the Selmer decomposition\n\\[\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}}),\\]\none gets\n\\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}\nThen, letting $\\mathfrak{p}'\\mid\\mathfrak{p}$ in $F'$ and $K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic $\\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\\infty,f'))=\\mathrm{char}\\,\\mathrm{Sel}(K'_\\infty,A_{\\rho})^\\vee$, which implies the containment\n\\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation}\nCombining the equation above and the containment above yields\n\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation}\nFinally, since the displayed ideal identity above holds after inverting $p$ and the $\\mu$-invariants of $L_p(K_\\infty,f)$ and $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ are zero, the containment above implies the ideal identity\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}\nand this completes the proof of the main theorem.",
    "expanded_theorem": "\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\n p)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.",
    "theorem_type": [
      "Implication",
      "Equality or Bound"
    ],
    "mcq": {
      "question": "Let $F$ be a totally real field and let $F'/F$ be a real quadratic extension. Let $f$ be a Hilbert newform over $F$ of parallel even weight $k$, trivial central character, and level $\\mathfrak{n}_f$, and let $f'$ be the base change of $f$ to $F'$ with level $\\mathfrak{n}_{f'}$. Let $p$ be a prime such that $p\\ge \\max(k+2,7)$ and $p$ is unramified in both $F$ and $F'$. Assume that $\\mathfrak{n}_f$ is prime to $p$ and that $f$ is ordinary at every prime of $F$ above $p$. If $[F:\\mathbb{Q}]$ is odd, assume there exists a prime $\\mathfrak{q}$ such that $\\mathfrak{q}\\parallel \\mathfrak{n}_f$ (that is, $\\mathfrak{q}$ divides $\\mathfrak{n}_f$ exactly once) and $\\mathfrak{q}$ is split in $F'$. Let $\\rho_f$ be the $p$-adic Galois representation attached to $f$, and let $\\bar\\rho_f$ and $\\bar\\rho_{f'}$ denote the residual Galois representations attached to $f$ and $f'$, respectively. Suppose moreover that:\n\n1. the restrictions of $\\bar\\rho_f$ to $G_{F'(\\xi_p)}$ and to $G_{F'(\\sqrt{p^*})}$ are absolutely irreducible, where $p^*=(-1)^{\\frac{p-1}{2}}p$;\n2. if $k=2$, then for every place $v$ of $F$ above $p$, $a_v(f)^2\\not\\equiv 1\\pmod p$, and for every place $v'$ of $F'$ above $p$, $a_{v'}(f')^2\\not\\equiv 1\\pmod p$;\n3. for every prime $\\mathfrak{l}\\mid \\mathfrak{n}_f$, if $\\bar\\rho_f|_{I_{F_{\\mathfrak{l}}}}$ is absolutely irreducible, then $\\mathrm N(\\mathfrak{l})\\not\\equiv -1\\pmod p$, and for every prime $\\mathfrak{l}'\\mid \\mathfrak{n}_{f'}$, if $\\bar\\rho_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely irreducible, then $\\mathrm N(\\mathfrak{l}')\\not\\equiv -1\\pmod p$;\n4. $\\rho_f$ is a minimal modular lifting of $\\bar\\rho_f$.\n\nFor a modular form $g$, write its canonical period as $\\Omega_g^{\\mathrm{can}}=\\langle g,g\\rangle_{\\mathrm{Pet}}/\\eta_g$, where $\\langle g,g\\rangle_{\\mathrm{Pet}}$ is the Petersson inner product and $\\eta_g$ is the associated congruence number. Under these hypotheses, which statement about the canonical periods of $f$ and $f'$ is valid?",
      "correct_choice": {
        "label": "A",
        "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to a $p$-adic unit; equivalently, the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is a $p$-adic unit."
      },
      "choices": [
        {
          "label": "B",
          "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nexactly in the coefficient ring, i.e. the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is equal to $1$."
        },
        {
          "label": "C",
          "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=u\\, (\\Omega_f^{\\mathrm{can}})^2$$\nfor some nonzero element $u$ in the coefficient field; in particular, $\\Omega_{f'}^{\\mathrm{can}}$ and $(\\Omega_f^{\\mathrm{can}})^2$ agree up to multiplication by a scalar."
        },
        {
          "label": "D",
          "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to a $p$-adic unit whenever the restrictions of $\\bar\\rho_f$ to $G_{F(\\xi_p)}$ and to $G_{F(\\sqrt{p^*})}$ are absolutely irreducible; no hypothesis involving the corresponding restrictions over $F'$ is needed."
        },
        {
          "label": "E",
          "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to multiplication by an arbitrary $p$-power; equivalently, the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is merely $p$-adically integral."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "uniformity",
          "tampered_component": "unit-ambiguity from ideal identity and period normalization",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the requirement that the scalar be a p-adic unit",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "base-change irreducibility hypotheses over F'",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "mu=0 used to upgrade containment to equality up to unit",
          "template_used": "uniformity_effectivity"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses but does not reveal the conclusion. The correct choice is not explicitly stated, and the reader must distinguish it from closely related alternatives."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-recall question: under a long list of hypotheses, the task is to identify the exact conclusion. It is not a pure verbatim restatement because the options vary in strength and precision, but it remains close to theorem matching."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to separate 'up to a p-adic unit' from stronger or weaker formulations, but the main task is recognizing the precise theorem statement rather than generating a new argument or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: overstating to exact equality, weakening to mere scalar commensurability, allowing extra p-power ambiguity, or adding unnecessary hypotheses. They are distinct and well-designed."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it mainly tests precise recall of a known conclusion rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.08642v1",
    "paper_link": "http://arxiv.org/abs/2512.08642v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main}\n    Let $C\\subset\\mathbb CP^2$ be an algebraic plane curve of degree at most five. Then the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear and virtually polyfree. In particular, $\\pi_1(\\mathbb CP^2\\setminus C)$ is residually finite.",
      "start_pos": 27463,
      "end_pos": 27752,
      "label": "main"
    },
    "ref_dict": {
      "degree4": "\\label{degree4}\n\\noindent\\\\\nIf $C\\subset \\mathbb CP^2$ is an irreducible quartic curve that is not a 3-cuspidal quartic, then the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is abelian (cf. \\ci",
      "degree5": "\\label{degree5}\nThe presentation of fundamental groups of $\\mathbb{C}P^2 \\setminus\\ C$ for degree-5 algebraic curves $C$ has been classified by Degtyarev \\cite{Deg}. Based on this classification, we w",
      "q1": "\\begin{ques}\n\\label{q1}\n    Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}"
    },
    "pre_theorem_intro_text_len": 1860,
    "pre_theorem_intro_text": "Let $C$ be an algebraic curve in the projective plane $\\mathbb CP^2$.  We consider the following question.\n\n\\begin{ques}\n\\label{q1}\n    Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}\n\nThis question has been open for a long time. Zariski had already posed the question of the existence of non-residually finite groups, cf. § 1 and Appendix 1) of Chapter VIII of \\cite{Zar} (see also Libgober \\cite[Problem 3.1] {Lib}).\nEven in the low-degree cases (degree up to five), such a question was still unknown (see the survey article of Artal Bartolo-Cogolludo Agustın-Tokunaga \\cite{bartolo2008survey}, page 3). Toledo \\cite{Toledo} constructed a smooth projective variety with a non-residually finite fundamental group, answering in the negative a question of Serre. However, it is not clear whether one can choose the example as the complement of an algebraic curve with a non-residually finite fundamental group. Koberda and Suciu \\cite[Problem 1.9]{KS} asked Question \\ref{q1} for affine line arrangements. \n\nThe purpose of this article is to answer Question \\ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite\nsubnormal sequence\\begin{equation*}\n1<G_{1}\\trianglelefteq G_{2}\\trianglelefteq \\cdots \\trianglelefteq G_{n}=G\n\\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.",
    "context": "Let $C$ be an algebraic curve in the projective plane $\\mathbb CP^2$. We consider the following question.\n\n\\begin{ques}\n\\label{q1}\n Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}\n\nThis question has been open for a long time. Zariski had already posed the question of the existence of non-residually finite groups, cf. § 1 and Appendix 1) of Chapter VIII of \\cite{Zar} (see also Libgober \\cite[Problem 3.1] {Lib}).\nEven in the low-degree cases (degree up to five), such a question was still unknown (see the survey article of Artal Bartolo-Cogolludo Agustın-Tokunaga \\cite{bartolo2008survey}, page 3). Toledo \\cite{Toledo} constructed a smooth projective variety with a non-residually finite fundamental group, answering in the negative a question of Serre. However, it is not clear whether one can choose the example as the complement of an algebraic curve with a non-residually finite fundamental group. Koberda and Suciu \\cite[Problem 1.9]{KS} asked Question \\ref{q1} for affine line arrangements.\n\nThe purpose of this article is to answer Question \\ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite\nsubnormal sequence\\begin{equation*}\n1<G_{1}\\trianglelefteq G_{2}\\trianglelefteq \\cdots \\trianglelefteq G_{n}=G\n\\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.\n\n\\begin{ques}\n\\label{q1}\n    Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}",
    "full_context": "Let $C$ be an algebraic curve in the projective plane $\\mathbb CP^2$. We consider the following question.\n\n\\begin{ques}\n\\label{q1}\n Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}\n\nThis question has been open for a long time. Zariski had already posed the question of the existence of non-residually finite groups, cf. § 1 and Appendix 1) of Chapter VIII of \\cite{Zar} (see also Libgober \\cite[Problem 3.1] {Lib}).\nEven in the low-degree cases (degree up to five), such a question was still unknown (see the survey article of Artal Bartolo-Cogolludo Agustın-Tokunaga \\cite{bartolo2008survey}, page 3). Toledo \\cite{Toledo} constructed a smooth projective variety with a non-residually finite fundamental group, answering in the negative a question of Serre. However, it is not clear whether one can choose the example as the complement of an algebraic curve with a non-residually finite fundamental group. Koberda and Suciu \\cite[Problem 1.9]{KS} asked Question \\ref{q1} for affine line arrangements.\n\nThe purpose of this article is to answer Question \\ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite\nsubnormal sequence\\begin{equation*}\n1<G_{1}\\trianglelefteq G_{2}\\trianglelefteq \\cdots \\trianglelefteq G_{n}=G\n\\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.\n\n\\begin{ques}\n\\label{q1}\n    Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}\n\n\\begin{abstract}\nWe prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear and virtually polyfree. As a consequence, we answer positively the open question on the residual finiteness of these groups for all plane curves of degree at most $5$.\n\\end{abstract}\n\n\\begin{ques}\n\\label{q1}\n Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}\n\nThe purpose of this article is to answer Question \\ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite\nsubnormal sequence\\begin{equation*}\n1<G_{1}\\trianglelefteq G_{2}\\trianglelefteq \\cdots \\trianglelefteq G_{n}=G\n\\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.\n\nIt is well-known (cf.\\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\\subset\\mathbb CP^2$ of degree $\\le3$, the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is either finitely generated abelian or a free group $\\mathbb Z\\ast\\mathbb Z$. Thus $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \\cite{nori1983zariski} and Cogolludo-Agustin and Elduque \\cite{cogolludo2025quasi} to conclude that\nthat $\\pi_1(\\mathbb{C}P^2\\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\\mathbb{Z} \\ast \\mathbb{Z}/2\\mathbb{Z}$, or the semi-direct product $F_2 \\rtimes \\mathbb{Z}$, (see Subsection \\ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \\cite{degtyarev1987topology} obtained presentations of all the $\\pi_1(\\mathbb{C}P^2\\setminus C)$ when $\\deg C=5$. In Section \\ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.\n\n\\subsection{Curves of degree $\\le3$}\n\\noindent\\\\\nIt is well-known (cf.\\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\\subset\\mathbb CP^2$ of degree $\\le3$, $\\pi_1(\\mathbb CP^2\\setminus C)$ is either finitely generated abelian or a free group $\\mathbb Z\\ast\\mathbb Z$. Thus $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear and virtually polyfree.\n\n\\subsection{Quartic curves}\\label{degree4}\n\\noindent\\\\\nIf $C\\subset \\mathbb CP^2$ is an irreducible quartic curve that is not a 3-cuspidal quartic, then the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is abelian (cf. \\cite[p.130, 4.3]{dimca2012singularities}). When $C$ is a 3-cuspidal quartic $\\pi_1(\\mathbb CP^2\\setminus C) \\cong B_3(S^2)$, the sphere braid group (cf. \\cite{zariski1936poincare}), a finite group of order 12 and thus linear (see also\\cite[4.8, page 131]{dimca2012singularities}). Since any finitely generated abelian group is linear, we know that for any irreducible quartic curve $C$, $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear. Also, these groups are known to be virtually polyfree.\n\n\\begin{cor}\n If $C$ is a plane curve of degree $\\le 4$, then $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear and virtually polyfree.\n\\end{cor}\n\n\\section{Quintic}\\label{degree5}\nThe presentation of fundamental groups of $\\mathbb{C}P^2 \\setminus\\ C$ for degree-5 algebraic curves $C$ has been classified by Degtyarev \\cite{Deg}. Based on this classification, we will prove Theorem \\ref{main} for these curves by studying the presentations. \n\\subsection{Presentation of fundamental groups}\nIn this subsection, we will review the results proved by Degtyarev \\cite{Deg}. Before introducing the list of Degtyarev, we need some notations.\n\\begin{enumerate}\n\\item $F_n$ is the free group of rank $n$, and $B_n$ is the braid group with $n$-strands.\n\\item $T_{p,q}$ is the fundamental group of a toric link of type $(p, q)$. In particular, if $p = 2$, then\n$T_{2,2r}=\\langle a,b| (ab)^r=(ba)^r\\rangle$; if $\\text{gcd}(p,q)=1$ then $T_{p,q}=\\langle a,b| a^p=b^q\\rangle$.\n\\item $G(T)$ and $G_p(T)$, where $T\\in\\mathbb Z[t]$ is an integral polynomial, are the extensions\n$$1\\to \\mathbb Z[t]/T\\to G(T)\\to \\mathbb Z\\to 1$$ and\n$$1\\to (\\mathbb Z/{p\\mathbb Z})[t]/T\\to G(T)\\to \\mathbb Z\\to 1,$$ where the conjugation action of the generator of the quotient $\\mathbb Z$ on the kernel is the multiplication by $t$.\n\\item $\\text{Gr}\\langle p,q,r \\rangle :=\\langle a,b,c|a^p=b^q=c^r=abc\\rangle$.\n\\end{enumerate}\n\n\\begin{ques}\n\\label{q1}\n    Is the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ of an algebraic plane curve residually finite?\n\\end{ques}",
    "post_theorem_intro_text_len": 1542,
    "post_theorem_intro_text": "It is well-known (cf.\\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\\subset\\mathbb CP^2$ of degree $\\le3$, the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is either finitely generated abelian or a free group $\\mathbb Z\\ast\\mathbb Z$. Thus $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \\cite{nori1983zariski} and Cogolludo-Agustin and Elduque  \\cite{cogolludo2025quasi} to conclude that\nthat $\\pi_1(\\mathbb{C}P^2\\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\\mathbb{Z} \\ast \\mathbb{Z}/2\\mathbb{Z}$, or the semi-direct product $F_2 \\rtimes \\mathbb{Z}$, (see Subsection \\ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \\cite{degtyarev1987topology} obtained presentations of all the $\\pi_1(\\mathbb{C}P^2\\setminus C)$ when $\\deg C=5$. In Section \\ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.\n\n\\subsection*{Acknowledgments}\nWe would like to thank Anatoly Libgober, Jose Ignacio Cogolludo, Corey Bregman for useful discussions and Alex Degtyarev for pointing us the results from his thesis and book.",
    "sketch": "To prove Theorem~\\ref{main}, the argument is by degree.\n\n- For $\\deg C\\le 3$: it is \"well-known\" that $\\pi_1(\\mathbb CP^2\\setminus C)$ is \"either finitely generated abelian or a free group $\\mathbb Z\\ast\\mathbb Z$\"; hence it is linear.\n\n- For $\\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\\pi_1(\\mathbb CP^2\\setminus C)$ is one of an explicit list (\"a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\\mathbb{Z} \\ast \\mathbb{Z}/2\\mathbb{Z}$, or the semi-direct product $F_2 \\rtimes \\mathbb{Z}$\"), and these groups are \"all virtually poly-free and linear.\"\n\n- For $\\deg C=5$: Degtyarev \"obtained presentations\" of all groups $\\pi_1(\\mathbb{C}P^2\\setminus C)$ when $\\deg C=5$; the plan is to \"review the presentations\" and \"prove the virtual polyfreeness and linearity\" of these groups, with the \"key observation\" that \"most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.\"",
    "expanded_sketch": "To prove the main theorem, the argument is by degree.\n\n- For $\\deg C\\le 3$: it is \"well-known\" that $\\pi_1(\\mathbb CP^2\\setminus C)$ is \"either finitely generated abelian or a free group $\\mathbb Z\\ast\\mathbb Z$\"; hence it is linear.\n\n- For $\\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\\pi_1(\\mathbb CP^2\\setminus C)$ is one of an explicit list (\"a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\\mathbb{Z} \\ast \\mathbb{Z}/2\\mathbb{Z}$, or the semi-direct product $F_2 \\rtimes \\mathbb{Z}$\"), and these groups are \"all virtually poly-free and linear.\"\n\n- For $\\deg C=5$: Degtyarev \"obtained presentations\" of all groups $\\pi_1(\\mathbb{C}P^2\\setminus C)$ when $\\deg C=5$; the plan is to \"review the presentations\" and \"prove the virtual polyfreeness and linearity\" of these groups, with the \"key observation\" that \"most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.\"",
    "expanded_theorem": "\\label{main}\n    Let $C\\subset\\mathbb CP^2$ be an algebraic plane curve of degree at most five. Then the fundamental group $\\pi_1(\\mathbb CP^2\\setminus C)$ is linear and virtually polyfree. In particular, $\\pi_1(\\mathbb CP^2\\setminus C)$ is residually finite.",
    "theorem_type": [
      "Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(C\\subset \\mathbb{CP}^2\\) be an algebraic plane curve of degree at most \\(5\\). Recall that a group is called linear if it embeds into \\(\\mathrm{GL}(n,F)\\) for some field \\(F\\) and some positive integer \\(n\\), and it is called virtually polyfree if it has a finite-index subgroup admitting a finite subnormal series whose successive quotients are free groups. Which statement holds for every such curve \\(C\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The fundamental group \\(\\pi_1(\\mathbb{CP}^2\\setminus C)\\) is linear and virtually polyfree; in particular, it is residually finite."
      },
      "choices": [
        {
          "label": "B",
          "text": "The fundamental group \\(\\pi_1(\\mathbb{CP}^2\\setminus C)\\) is residually finite and virtually polyfree; in fact, for every such curve it admits a finite-index subgroup which is a free group."
        },
        {
          "label": "C",
          "text": "The fundamental group \\(\\pi_1(\\mathbb{CP}^2\\setminus C)\\) is residually finite."
        },
        {
          "label": "D",
          "text": "The fundamental group \\(\\pi_1(\\mathbb{CP}^2\\setminus C)\\) is linear; moreover, there exists a field \\(F\\) and a positive integer \\(n\\), depending only on the degree bound \\(5\\), such that every such group embeds into \\(\\mathrm{GL}(n,F)\\)."
        },
        {
          "label": "E",
          "text": "The fundamental group \\(\\pi_1(\\mathbb{CP}^2\\setminus C)\\) is either finitely generated abelian or a free group, and hence is linear and virtually polyfree."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "virtually polyfree vs virtually free",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "drops linear and virtually polyfree, keeping only the stated consequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "non-uniform linearity strengthened to uniform embedding dimension/field",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "extends the degree \\(\\le 3\\) classification to all degrees \\(\\le 5\\)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the theorem or uniquely point to the correct option. The recalled definitions of 'linear' and 'virtually polyfree' are needed to read several choices, not just the correct one."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct choice is essentially a direct statement of the target result for degree <= 5 curves, with only an immediate consequence appended. This makes the item close to theorem recall rather than a non-tautological inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to reject stronger or overextended variants (virtually free, uniform linearity, low-degree classification extended to degree 5) and to notice that one option is merely weaker. But if the theorem is known, the answer is mostly straightforward."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one is weaker but true, others are stronger false claims or overgeneralizations that reflect common failure modes in theorem recall and logical strength."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with good distractors and little answer leakage, but it scores low on tautology avoidance because the correct option essentially restates the theorem rather than forcing substantial generative reasoning."
    }
  },
  {
    "id": "2512.08669v1",
    "paper_link": "http://arxiv.org/abs/2512.08669v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "proposition",
      "content": "For any $d \\geqslant 1$ and $j \\in [0, d]$, we have $c_{d, 0}(j) = |s(d + 1, j + 1)|$.",
      "start_pos": 18292,
      "end_pos": 18415,
      "label": null
    },
    "ref_dict": {
      "eqn": "\\begin{equation} \\label{eqn}\n  d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}",
      "eq:polynomial def": "\\begin{equation} \\label{eq:polynomial def}\n  \\sum_{n \\geq 0} P_{d,k}(n)\\, x^n = \\frac{(1+x)^k}{(1-x)^{d+1}}\n\\end{equation}",
      "ehrbij": "\\begin{remark}\\label{ehrbij}\n    The fact that the objects described above are counted by $d!P_{d, k}(n)$ also follows by the interpretation of $P_{d, k}(n)$ as an Ehrhart polynomial. \n    More specifically, $P_{d, k}(n)$ counts points $\\bm x \\in \\ZZ^d$, such that\n    \\begin{itemize}\n        \\item $|x_1| + \\ldots +|x_k| + x_{k + 1} + \\ldots + x_d \\leq n$, and\n        \\item $x_{k + 1}, \\ldots, x_d \\geq 0$.\n    \\end{itemize}\n    One way to represent a sequence of length $d$ of integers with sum of absolute values at most $n$ is as a sequence consisting of $n$ balls and $d$ dots. \n    For example, with $n = 6$ and $d = 3$, the sequence $(2, 0, 3)$ (with only nonnegative terms) is represented as\n    \\begin{equation*}\n        \\emptycircle[0pt]\\, \\emptycircle[0pt]\\, {\\bullet}\\, {\\bullet}\\, \\emptycircle[0pt]\\, \\emptycircle[0pt]\\, \\emptycircle[0pt]\\, {\\bullet}\\, \\emptycircle[0pt].\n    \\end{equation*}\n    The total number of balls is $n$, the first entry is the number of balls before the first dot, and for any $i \\geq 2$, the $i$-th entry is the number of balls between the $(i - 1)$-th and $i$-th dots. \n    If we want the $i$-th entry to be negative, we move the $i$-th dot into the ball immediately before it. \n    For example, the sequence $(-1, 3, -2)$ is represented as\n    \\begin{equation*}\n        \\circled[0pt]{\\bullet}\\, \\emptycircle[0pt]\\, \\emptycircle[0pt]\\, \\emptycircle[0pt]\\, {\\bullet}\\, \\emptycircle[0pt]\\, \\circled[0pt]{\\bullet}.\n    \\end{equation*}\n    Hence, $P_{d, k}(n)$ counts such configurations with $n$ balls and $d$ dots where only the first $k$ dots can be in balls. \n    This interpretation can be used to show $d!P_{d, k}(n) = |W_{d, k}(n)|$.\n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 4721,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe Ehrhart polynomial \\(\\ehr_{\\mathcal P}(n)\\) of a lattice polytope \\(\\mathcal P\\) counts the number of integer lattice points in the dilate \\(n \\mathcal P\\). It is known that the Ehrhart polynomial of the cross-polytope\n\\begin{equation*}\n  \\mathcal P_d \\coloneqq \\left\\{\\bm x \\in \\mathbb{R}^d : |x_1| + |x_2| + \\ldots + |x_d| \\leqslant 1\\right\\}\n\\end{equation*}\nis determined by the generating function\n\\begin{equation*}\n  \\sum_{n \\geqslant 0} \\ehr_{\\mathcal P_d}(n)\\, x^n = \\frac{(1+x)^d}{(1-x)^{d+1}}.\n\\end{equation*}\nMore generally, if \\(k \\in [0,d]\\), the polynomial \\(P_{d,k}(n)\\) defined by\n\\begin{equation} \\label{eq:polynomial def}\n  \\sum_{n \\geqslant 0} P_{d,k}(n)\\, x^n = \\frac{(1+x)^k}{(1-x)^{d+1}}\n\\end{equation}\nis the Ehrhart polynomial of the \\((d-k)\\)-fold pyramid over \\(\\mathcal P_k\\). The relevant background can be found for example in \\cite{ccd}. The figure below illustrates \\(\\mathcal P_3\\) (left), the pyramid over \\(\\mathcal P_2\\) (middle) and \\(\\mathcal P_2\\) itself (right).\n\n\\bigskip\n\\begin{center}\n  \\begin{tikzpicture}[scale=1.7]\n    \\coordinate (1) at (1,0,0);\n    \\coordinate (-1) at (-1,0,0);\n    \\coordinate (2) at (0,1,0);\n    \\coordinate (-2) at (0,-1,0);\n    \\coordinate (3) at (0,0,1);\n    \\coordinate (-3) at (0,0,-1);\n\nll[gray!20] (1) -- (3) -- (-1) -- (2) -- cycle;\nll[gray!40] (1) -- (3) -- (-1) -- (-2) -- cycle;\n\n    \\draw[rpath, dashed]\n      (-3) -- (1)\n      (-3) -- (2)\n      (-3) -- (-1)\n      (-3) -- (-2);\n    \\draw[rpath]\n      (1) -- (3) -- (-1)\n      (2) -- (3) -- (-2);\n    \\draw[rpath] (1) -- (2) -- (-1) -- (-2) -- cycle;\n\n    \\begin{scope}[shift={(2.5,0,0)}]\n      \\coordinate (1) at (1,0,0);\n      \\coordinate (-1) at (-1,0,0);\n      \\coordinate (2) at (0,1,0);\n      \\coordinate (3) at (0,0,1);\n      \\coordinate (-3) at (0,0,-1);\n\nll[gray!20] (1) -- (3) -- (-1) -- (2) -- cycle;\n\n      \\draw[rpath, dashed]\n        (-3) -- (1)\n        (-3) -- (-1)\n        (-3) -- (2);\n      \\draw[rpath] (3) -- (2);\n      \\draw[rpath] (1) -- (3) -- (-1) -- (2) -- cycle;\n    \\end{scope}\n\n    \\begin{scope}[shift={(5,0,0)}]\n      \\coordinate (1) at (1,0,0);\n      \\coordinate (-1) at (-1,0,0);\n      \\coordinate (3) at (0,0,1);\n      \\coordinate (-3) at (0,0,-1);\n\nll[gray!20] (1) -- (3) -- (-1) -- (-3) -- cycle;\n      \\draw[rpath] (1) -- (3) -- (-1) -- (-3) -- cycle;\n    \\end{scope}\n  \\end{tikzpicture}\n\\end{center}\n\n\\noindent\nThe fact that \\(P_{d,k}(n)\\) is a polynomial is not immediately clear from \\eqref{eq:polynomial def}, but one can check that\n\\begin{equation*}\n  P_{d,k}(n) = \\sum_{i=0}^k \\binom{k}{i} \\binom{n + d - i}{d}.\n\\end{equation*}\n\nThe coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \\cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \\(c_{d,k}(j)\\) by\n\\begin{equation} \\label{eqn}\n  d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}\nwhere the factor \\(d!\\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \\(c_{d,k}(j)\\), and a bijective proof of equation \\eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \\cite[Chapter 4, Exercise 1(b)]{ecnewex}.\n\nThe outline of the paper is as follows. \nIn Section \\ref{sec:sri}, we construct a class of objects that are counted by the coefficients \\(c_{d, k}(j)\\), in particular showing that these coefficients are positive. \nIn Section \\ref{sec:bij}, we give a combinatorial argument that proves \\eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection. \nIn Section \\ref{sec:prop}, we use the combinatorial meaning we have for \\(c_{d, k}(j)\\) to obtain some properties of these coefficients. \nFinally, in Section \\ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.\n\n\\bigskip\\noindent\nBefore proceeding with the general results, we will briefly describe a proof strategy for the simplest case \\(k = 0\\). The same ideas will serve as our guiding light for the rest of the paper.\n\nThe numbers \\(c_{d, 0}(j)\\) are just unsigned Stirling numbers of the first kind. \nRecall that the (signed) Stirling number of the first kind \\(s(m, i)\\) is defined by\n\\begin{equation*}\n    s(m, i) \\coloneqq (-1)^{m - i} \\cdot |\\{\\sigma \\in S_m : \\sigma \\text{ has } i \\text{ cycles}\\}|.\n\\end{equation*}",
    "context": "The Ehrhart polynomial \\(\\ehr_{\\mathcal P}(n)\\) of a lattice polytope \\(\\mathcal P\\) counts the number of integer lattice points in the dilate \\(n \\mathcal P\\). It is known that the Ehrhart polynomial of the cross-polytope\n\\begin{equation*}\n \\mathcal P_d \\coloneqq \\left\\{\\bm x \\in \\mathbb{R}^d : |x_1| + |x_2| + \\ldots + |x_d| \\leqslant 1\\right\\}\n\\end{equation*}\nis determined by the generating function\n\\begin{equation*}\n \\sum_{n \\geqslant 0} \\ehr_{\\mathcal P_d}(n)\\, x^n = \\frac{(1+x)^d}{(1-x)^{d+1}}.\n\\end{equation*}\nMore generally, if \\(k \\in [0,d]\\), the polynomial \\(P_{d,k}(n)\\) defined by\n\\begin{equation} \\label{eq:polynomial def}\n \\sum_{n \\geqslant 0} P_{d,k}(n)\\, x^n = \\frac{(1+x)^k}{(1-x)^{d+1}}\n\\end{equation}\nis the Ehrhart polynomial of the \\((d-k)\\)-fold pyramid over \\(\\mathcal P_k\\). The relevant background can be found for example in \\cite{ccd}. The figure below illustrates \\(\\mathcal P_3\\) (left), the pyramid over \\(\\mathcal P_2\\) (middle) and \\(\\mathcal P_2\\) itself (right).\n\n\\bigskip\n\\begin{center}\n \\begin{tikzpicture}[scale=1.7]\n \\coordinate (1) at (1,0,0);\n \\coordinate (-1) at (-1,0,0);\n \\coordinate (2) at (0,1,0);\n \\coordinate (-2) at (0,-1,0);\n \\coordinate (3) at (0,0,1);\n \\coordinate (-3) at (0,0,-1);\n\nThe coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \\cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \\(c_{d,k}(j)\\) by\n\\begin{equation} \\label{eqn}\n d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}\nwhere the factor \\(d!\\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \\(c_{d,k}(j)\\), and a bijective proof of equation \\eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \\cite[Chapter 4, Exercise 1(b)]{ecnewex}.\n\nThe outline of the paper is as follows. \nIn Section \\ref{sec:sri}, we construct a class of objects that are counted by the coefficients \\(c_{d, k}(j)\\), in particular showing that these coefficients are positive. \nIn Section \\ref{sec:bij}, we give a combinatorial argument that proves \\eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection. \nIn Section \\ref{sec:prop}, we use the combinatorial meaning we have for \\(c_{d, k}(j)\\) to obtain some properties of these coefficients. \nFinally, in Section \\ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.\n\n\\bigskip\\noindent\nBefore proceeding with the general results, we will briefly describe a proof strategy for the simplest case \\(k = 0\\). The same ideas will serve as our guiding light for the rest of the paper.\n\nThe numbers \\(c_{d, 0}(j)\\) are just unsigned Stirling numbers of the first kind. \nRecall that the (signed) Stirling number of the first kind \\(s(m, i)\\) is defined by\n\\begin{equation*}\n s(m, i) \\coloneqq (-1)^{m - i} \\cdot |\\{\\sigma \\in S_m : \\sigma \\text{ has } i \\text{ cycles}\\}|.\n\\end{equation*}\n\n\\begin{equation} \\label{eqn}\n  d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}",
    "full_context": "The Ehrhart polynomial \\(\\ehr_{\\mathcal P}(n)\\) of a lattice polytope \\(\\mathcal P\\) counts the number of integer lattice points in the dilate \\(n \\mathcal P\\). It is known that the Ehrhart polynomial of the cross-polytope\n\\begin{equation*}\n \\mathcal P_d \\coloneqq \\left\\{\\bm x \\in \\mathbb{R}^d : |x_1| + |x_2| + \\ldots + |x_d| \\leqslant 1\\right\\}\n\\end{equation*}\nis determined by the generating function\n\\begin{equation*}\n \\sum_{n \\geqslant 0} \\ehr_{\\mathcal P_d}(n)\\, x^n = \\frac{(1+x)^d}{(1-x)^{d+1}}.\n\\end{equation*}\nMore generally, if \\(k \\in [0,d]\\), the polynomial \\(P_{d,k}(n)\\) defined by\n\\begin{equation} \\label{eq:polynomial def}\n \\sum_{n \\geqslant 0} P_{d,k}(n)\\, x^n = \\frac{(1+x)^k}{(1-x)^{d+1}}\n\\end{equation}\nis the Ehrhart polynomial of the \\((d-k)\\)-fold pyramid over \\(\\mathcal P_k\\). The relevant background can be found for example in \\cite{ccd}. The figure below illustrates \\(\\mathcal P_3\\) (left), the pyramid over \\(\\mathcal P_2\\) (middle) and \\(\\mathcal P_2\\) itself (right).\n\n\\bigskip\n\\begin{center}\n \\begin{tikzpicture}[scale=1.7]\n \\coordinate (1) at (1,0,0);\n \\coordinate (-1) at (-1,0,0);\n \\coordinate (2) at (0,1,0);\n \\coordinate (-2) at (0,-1,0);\n \\coordinate (3) at (0,0,1);\n \\coordinate (-3) at (0,0,-1);\n\nThe coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \\cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \\(c_{d,k}(j)\\) by\n\\begin{equation} \\label{eqn}\n d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}\nwhere the factor \\(d!\\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \\(c_{d,k}(j)\\), and a bijective proof of equation \\eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \\cite[Chapter 4, Exercise 1(b)]{ecnewex}.\n\nThe outline of the paper is as follows. \nIn Section \\ref{sec:sri}, we construct a class of objects that are counted by the coefficients \\(c_{d, k}(j)\\), in particular showing that these coefficients are positive. \nIn Section \\ref{sec:bij}, we give a combinatorial argument that proves \\eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection. \nIn Section \\ref{sec:prop}, we use the combinatorial meaning we have for \\(c_{d, k}(j)\\) to obtain some properties of these coefficients. \nFinally, in Section \\ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.\n\n\\bigskip\\noindent\nBefore proceeding with the general results, we will briefly describe a proof strategy for the simplest case \\(k = 0\\). The same ideas will serve as our guiding light for the rest of the paper.\n\nThe numbers \\(c_{d, 0}(j)\\) are just unsigned Stirling numbers of the first kind. \nRecall that the (signed) Stirling number of the first kind \\(s(m, i)\\) is defined by\n\\begin{equation*}\n s(m, i) \\coloneqq (-1)^{m - i} \\cdot |\\{\\sigma \\in S_m : \\sigma \\text{ has } i \\text{ cycles}\\}|.\n\\end{equation*}\n\n\\begin{equation} \\label{eqn}\n  d! P_{d,k}(n) = \\sum_{j=0}^d c_{d,k}(j) n^j,\n\\end{equation}\n\n\\renewcommand{\\epsilon}{\\varepsilon}\n \\renewcommand{\\phi}{\\varphi}\n \\renewcommand{\\emptyset}{\\varnothing}\n \\renewcommand{\\leq}{\\leqslant}\n \\renewcommand{\\geq}{\\geqslant}\n 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{bona_permutations_2000,\n AUTHOR = {B\\'ona, Mikl\\'os and McLennan, Andrew and White, Dennis},\n TITLE = {Permutations with roots},\n JOURNAL = {Random Struct. Algorithms},\n VOLUME = {17},\n YEAR = {2000},\n NUMBER = {2},\n PAGES = {157--167},\n ISSN = {1042-9832,1098-2418},\n DOI = {10.1002/1098-2418(200009)17:2<157::AID-RSA4>3.0.CO;2-2}\n}\n\nThe numbers \\(c_{d, 0}(j)\\) are just unsigned Stirling numbers of the first kind. \nRecall that the (signed) Stirling number of the first kind \\(s(m, i)\\) is defined by\n\\begin{equation*}\n s(m, i) \\coloneqq (-1)^{m - i} \\cdot |\\{\\sigma \\in S_m : \\sigma \\text{ has } i \\text{ cycles}\\}|.\n\\end{equation*}\n\n\\begin{proof}\n From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \\binom{n + d}{d}$. \n The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \\cdots (x - m + 1)$ (see, for example, \\cite[Proposition 1.3.7]{ec1}).\n\\end{proof}\n\nAlthough we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \\geq 0$,\n\\begin{equation*}\n d!P_{d, 0}(n) = \\sum_{j = 0}^d c_{d, 0}(j)n^j.\n\\end{equation*}\nEspecially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently. \nAlso note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \\geq 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.\n\n\\begin{lemma}\\label{lem:str}\n For any $d \\geq 1$ and $k, j \\in [0, d]$, we have\n \\begin{equation*}\n c_{d, k}(j) = \\sum_{i = 0}^k \\binom{k}{i} \\sum_{\\ell = 0}^{d - i} |s(d - i + 1, \\ell + 1)|s(i, j - \\ell).\n \\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n Note that for any permutation $\\sigma \\in S_m$ with $i$ cycles, the parity of $m - i$ is the same as the number of even cycles in $\\sigma$. \n We now interpret the expression from Lemma \\ref{lem:str} for $c_{d, k}(j)$ as the signed count of the following objects: \n Permutations of size $d + 1$ with $j + 1$ cycles where\n \\begin{itemize}\n \\item any cycle with all elements in $[k]$ is colored either red or blue,\n \\item any cycle containing an element not in $[k]$ is uncolored, and\n \\item the sign associated to the permutation is the number of even cycles that are colored red.\n \\end{itemize}\n\n\\begin{lemma}\n For any $d \\geq 1$ and $k \\in [0, d]$, we have $c_{d, k}(0) = d!$ and $c_{d, k}(d) = 2^k$.\n\\end{lemma}\n\n\\begin{lemma}\n For any $d \\geq 1$ and $k, j \\in [0, d]$, we have $c_{d, k}(j) \\leq c_{d + 1, k}(j)$.\n\\end{lemma}",
    "post_theorem_intro_text_len": 3204,
    "post_theorem_intro_text": "\\begin{proof}\n    From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \\binom{n + d}{d}$. \n    The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \\cdots (x - m + 1)$ (see, for example, \\cite[Proposition 1.3.7]{ec1}).\n\\end{proof}\n\nAlthough we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \\geqslant 0$,\n\\begin{equation*}\n    d!P_{d, 0}(n) = \\sum_{j = 0}^d c_{d, 0}(j)n^j.\n\\end{equation*}\nEspecially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently. \nAlso note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \\geqslant 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.\n\nTo do this, we construct objects counted by $d!P_{d, 0}(n) = d!\\binom{n + d}{d}$. \nOne choice is objects that consist of a permutation of size $d$ with $n$ balls distributed among the $d + 1$ slots created by the terms of the permutation. \nFor example, for $d = 9$ and $n = 6$, such an object is\n\\begin{equation*}\n    \\emptycircle\\, 281\\, \\emptycircle\\, 9\\, \\emptycircle\\, \\emptycircle\\, 354\\, \\emptycircle\\, 67\\, \\emptycircle\\,.\n\\end{equation*}\nIt can also be shown that these objects are counted by $d!P_{d, 0}(n)$ using the interpretation of $P_{d, 0}(n)$ as an Ehrhart polynomial (see Remark \\ref{ehrbij}).\n\nWe interpret $|s(d + 1, j + 1)|n^j$ as choosing a permutation of size $d + 1$ with $j + 1$ cycles as well as assigning the labels $1, 2, \\ldots, j$ to $n$ balls. \nSuch a pair for $d = 8$, $j = 3$, and $n = 4$ is\n\\begin{equation*}\n    (2) (7\\ 3\\ 5\\ 4) (8\\ 1) (9\\ 6)\\quad \\text{and}\\quad \\emptycircle\\, \\overset{1,3}{\\emptycircle}\\, \\emptycircle\\, \\overset{2}{\\emptycircle}\\,.\n\\end{equation*}\n\nWe write cycles in a permutation in \\emph{canonical cycle form}: each cycle has its largest element first, and the cycles are ordered in increasing order of their largest elements. \nThe labels on balls indicate which cycles must be placed before it. \nThe label $1, 3$ on the second ball in the example above indicates that the first and third cycle should be placed before the second ball. \nWe do this by placing the cycles as they are into that slot and then removing the brackets. \nThis is just an instance of the fundamental bijection on permutations (see \\cite[Proposition 1.3.1]{ec1}). \nThe cycle containing $d + 1$ is placed after the last ball and then the brackets and $d + 1$ are deleted. \nHence, the object corresponding to the example above is\n\\begin{equation*}\n    \\emptycircle\\, 281\\, \\emptycircle\\, \\emptycircle\\, 7354\\, \\emptycircle\\, 6.\n\\end{equation*}\n\nIt can be checked that this is a bijection and gives us a proof that for all $n \\geqslant 0$,\n\\begin{equation*}\n    d!P_{d, 0}(n) = \\sum_{j = 0}^d |s(d + 1, j + 1)| n^j\n\\end{equation*}\nand in particular $c_{d, 0}(j) = |s(d + 1, j + 1)|$. \nWe will generalize the interpretation we have for $c_{d, 0}(j)$ in Section \\ref{sec:sri} and then in Section \\ref{sec:bij}, we will present a bijection generalizing the one we have just seen.",
    "sketch": "From the definition of $P_{d,k}$, $P_{d,0}(n)=\\binom{n+d}{d}$. The theorem then follows since $s(m,i)$ is the coefficient of $x^i$ in $x(x-1)\\cdots(x-m+1)$.\n\nEquivalently, to show $c_{d,0}(j)=|s(d+1,j+1)|$, it suffices to prove that for all $n\\ge 0$,\n\\[\n d!P_{d,0}(n)=\\sum_{j=0}^d |s(d+1,j+1)|\\,n^j.\n\\]\nA combinatorial proof is outlined by:\n(1) constructing objects counted by $d!P_{d,0}(n)=d!\\binom{n+d}{d}$, e.g. “a permutation of size $d$ with $n$ balls distributed among the $d+1$ slots created by the terms of the permutation”;\n(2) interpreting $|s(d+1,j+1)|n^j$ as “choosing a permutation of size $d+1$ with $j+1$ cycles as well as assigning the labels $1,2,\\ldots,j$ to $n$ balls”, with cycles written in canonical cycle form and ball labels indicating “which cycles must be placed before it”;\n(3) producing the permutation-with-slots object by placing the cycles into the indicated slots and “removing the brackets” (an instance of “the fundamental bijection on permutations”), then placing the cycle containing $d+1$ after the last ball and deleting the brackets and $d+1$.\nIt is stated that “It can be checked that this is a bijection,” yielding the identity above and hence $c_{d,0}(j)=|s(d+1,j+1)|$.",
    "expanded_sketch": "From the definition of $P_{d,k}$, $P_{d,0}(n)=\\binom{n+d}{d}$. The theorem then follows since $s(m,i)$ is the coefficient of $x^i$ in $x(x-1)\\cdots(x-m+1)$.\n\nEquivalently, to show $c_{d,0}(j)=|s(d+1,j+1)|$, it suffices to prove that for all $n\\ge 0$,\n\\[\n d!P_{d,0}(n)=\\sum_{j=0}^d |s(d+1,j+1)|\\,n^j.\n\\]\nA combinatorial proof is outlined by:\n(1) constructing objects counted by $d!P_{d,0}(n)=d!\\binom{n+d}{d}$, e.g. “a permutation of size $d$ with $n$ balls distributed among the $d+1$ slots created by the terms of the permutation”;\n(2) interpreting $|s(d+1,j+1)|n^j$ as “choosing a permutation of size $d+1$ with $j+1$ cycles as well as assigning the labels $1,2,\\ldots,j$ to $n$ balls”, with cycles written in canonical cycle form and ball labels indicating “which cycles must be placed before it”;\n(3) producing the permutation-with-slots object by placing the cycles into the indicated slots and “removing the brackets” (an instance of “the fundamental bijection on permutations”), then placing the cycle containing $d+1$ after the last ball and deleting the brackets and $d+1$.\nIt is stated that “It can be checked that this is a bijection,” yielding the identity above and hence $c_{d,0}(j)=|s(d+1,j+1)|$.",
    "expanded_theorem": "For any $d \\geqslant 1$ and $j \\in [0, d]$, we have $c_{d, 0}(j) = |s(d + 1, j + 1)|$.",
    "theorem_type": [
      "Universal"
    ],
    "mcq": {
      "question": "For integers \\(d \\geq 1\\) and \\(0 \\leq j \\leq d\\), let the polynomial \\(P_{d,0}(n)\\) be defined by the generating function\n\\[\n\\sum_{n \\geq 0} P_{d,0}(n)x^n=\\frac{1}{(1-x)^{d+1}},\n\\]\nand define integers \\(c_{d,0}(j)\\) by\n\\[\nd!\\,P_{d,0}(n)=\\sum_{r=0}^d c_{d,0}(r)n^r.\n\\]\nAlso, for \\(m\\ge 0\\) and \\(0\\le i\\le m\\), let the signed Stirling number of the first kind be\n\\[\ns(m,i)=(-1)^{m-i}\\,\\bigl|\\{\\sigma\\in S_m:\\sigma\\text{ has }i\\text{ cycles}\\}\\bigr|.\n\\]\nWhich statement holds for every such pair \\((d,j)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\(c_{d,0}(j)=|s(d+1,j+1)|\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\(c_{d,0}(j)=|s(d,j)|\\)."
        },
        {
          "label": "C",
          "text": "\\(c_{d,0}(j)\\ge 0\\)."
        },
        {
          "label": "D",
          "text": "\\(c_{d,0}(j)=|s(d+1,j)|\\)."
        },
        {
          "label": "E",
          "text": "\\(c_{d,0}(j)=(-1)^{d-j}|s(d+1,j+1)|\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "shift_in_both_stirling_indices",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "exact_identification_with_unsigned_stirling_number",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "cycle_count_shift_j_to_j_plus_1",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "unsigned_vs_signed_stirling_conversion",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions but does not explicitly state the Stirling-number identity or otherwise reveal choice A. There is no direct answer leakage."
      },
      "TAS": {
        "score": 2,
        "justification": "The item is not a mere restatement of a theorem in the stem; it asks the solver to connect a generating-function/coefficient definition to a precise Stirling-number formula and distinguish among close variants."
      },
      "GPS": {
        "score": 1,
        "justification": "Identifying A requires meaningful reasoning through binomial/rising-factorial expansions and the correct Stirling index shift. However, the presence of choice C, which is also true, lowers generative pressure because a weaker conclusion can be selected without deriving the exact identity."
      },
      "DQS": {
        "score": 0,
        "justification": "The distractors are mathematically themed and some are plausible index/sign confusions, but choice C is a genuine weaker true statement. That makes the item ambiguous rather than cleanly discriminative, which is a serious distractor-quality flaw."
      },
      "total_score": 5,
      "overall_assessment": "The question has no answer leakage and is substantively non-tautological, but it is flawed as an MCQ because it includes at least one additional true option (C). This ambiguity substantially weakens its ability to test precise generative reasoning."
    }
  },
  {
    "id": "2512.08676v1",
    "paper_link": "http://arxiv.org/abs/2512.08676v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}",
      "start_pos": 6333,
      "end_pos": 6799,
      "label": "thm:KKT-circle"
    },
    "ref_dict": {
      "thm:KKT-circle": "\\begin{theorem}[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}\n\\end{theorem}",
      "definition-rotation": "\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 2055,
    "pre_theorem_intro_text": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.",
    "context": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.",
    "full_context": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nBefore stating the main results, we need to introduce some notation.\n\nFor every finite measurable cover\n\\[\n\\mathbb{S}^{n-1} \\subset F_1\\cup\\cdots\\cup F_t,\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{n-1}\\big(R_{\\theta_0}(F_{m_0})\n \\cap E_i^{\\mathbb{S}^{n-1}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathbb{S}^{n-1}$.\n\\end{theorem}\n\n\\begin{theorem}[Rotational power surfaces]\\label{thm:power-surface-Raimi}\nLet $n\\ge 3$ and $k>0$. Define\n\\[\n\\mathcal{S}_{k,R}\n := \\big\\{(x',x_n)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\n : x_n = |x'|^{\\,k},\\ 0<|x'|\\le R\\big\\}.\n\\]\nLet\n$\\sigma_{k,R}$ be the normalized surface measure on $\\mathcal{S}_{k,R}$.\nThen there exists a measurable partition\n$\\{E_i^{\\mathcal{S}_{k,R}}\\}_{i=1}^r$ of $\\mathcal{S}_{k,R}$ with the following property:\nFor every finite measurable cover\n\\[\n\\mathcal{S}_{k,R} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{k,R}\\big(R_{\\theta_0}(F_{m_0})\n \\cap E_i^{\\mathcal{S}_{k,R}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nIn particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.\n\\end{theorem}\n\nFor every finite measurable cover\n\\[\n\\mathcal{C}_{R,\\Omega} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\mu_{\\text{\\tiny $R,\\Omega$}}\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^{\\mathcal{C}_{R,\\Omega}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r,\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathcal{C}_{R,\\Omega}$.\n\\end{theorem}\n\nWe now present the following general theorem, of which the spherical, cylindrical, and rotational power surface cases are specific instances.\n\\begin{theorem}\\label{thm:general-circle-bundle}\nLet $(X,\\mu)$ be a probability space equipped with a measurable, measure-preserving action\n\\[\n\\{R_\\theta\\}_{\\theta\\in C},\\qquad C:=\\mathbb{R}/\\mathbb{Z},\n\\]\nof the circle group $C$. Assume there exist\n\\begin{itemize}\n \\item a probability space $(Y,\\nu)$,\n \\item a measurable set $N\\subset X$ with $\\mu(N)=0$,\n \\item a measurable bijection\n \\[\n \\Phi : C\\times Y \\longrightarrow X\\setminus N,\n \\]\n\\end{itemize}\nsatisfying:\n\\begin{enumerate}\n \\item[(i)] (\\emph{Equivariance}) For all $\\theta,\\alpha\\in C$ and $y\\in Y$,\n \\[\n \\Phi(\\theta+\\alpha,y) = R_\\alpha(\\Phi(\\theta,y)).\n \\]\n \\item[(ii)] (\\emph{Product disintegration}) For every bounded measurable function $f:X\\to\\mathbb{R}$,\n \\[\n \\int_X f(x)\\,d\\mu(x)\n = \\int_C \\int_Y f(\\Phi(\\theta,y))\\,d\\nu(y)\\,d\\mu_1(\\theta),\n \\]\n where $\\mu_1$ is the normalized Lebesgue measure on $C$.\n\\end{enumerate}\nLet $\\{E_i^C\\}_{i=1}^r$ be a measurable partition of $C$ with the Raimi property from Theorem~\\ref{thm:KKT-circle}. Define \n\\begin{equation*}\n \\begin{cases}\n E_1^X := \\Phi(E_1^C\\times Y) \\cup N,\\\\\n E_i^X:=\\Phi(E_i^C\\times Y), ~~~~~~~~2 \\leq i\\leq r.\n \\end{cases}\n\\end{equation*}\nThen $\\{E_i^X\\}_{i=1}^r$ is a measurable partition of $X$ with the following property:\n\nNow define a measurable partition of $C$ by\n\\[\n C_m := \\{\\theta\\in C : m(\\theta)=m\\},\\qquad 1\\le m\\le t.\n\\]\nBy construction, for every $\\theta\\in C_m$ we have\n\\begin{equation}\\label{eq:Am-lower-bound}\n \\nu(A_m(\\theta))\\ge \\frac1t.\n\\end{equation}\nBy applying Theorem~ \\ref{thm:KKT-circle} to the partition $\\{C_m\\}_{m=1}^t$ of $C$, there exist an index $m_0\\in\\{1,\\dots,t\\}$ and a rotation\n\\[\n R_{\\theta_0}: C\\to C,\\qquad R_{\\theta_0}(\\theta)=\\theta+\\theta_0,\n\\]\nsuch that\n\\begin{equation}\\label{eq:KKT-circle-hit}\n \\mu_1\\big(R_{\\theta_0}(C_{m_0})\\cap E_i^C\\big)>0,\n \\qquad\\, \\forall\\, 1\\le i\\le r.\n\\end{equation}\n\nTo complete the proof of the theorem, it remains to use \\eqref{eq:KKT-circle-hit} to show that\n\\begin{align}\\label{ineq:intersection-rotation-E_i^X}\n \\mu\\bigl(R_{\\theta_0}(F_{m_0}) \\cap E_i^X\\bigr) > 0\n \\qquad \\text{for all } 1 \\le i \\le r.\n\\end{align}\nLet \n\\[\n R_{\\theta_0}(\\theta,y):=(\\theta+\\theta_0,y)\n\\]\nbe the induced rotation on $C\\times Y$. By equivariance, we have \n\\[\n R_{\\theta_0}(F_{m_0})\\cap (X\\setminus N)\n = R_{\\theta_0}\\big(\\Phi(F_{m_0}^{\\mathrm e})\\big)\n = \\Phi\\big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\big),\n\\]\nwhich, together with the definition of $E_i^X$, gives\n\\[\n R_{\\theta_0}(F_{m_0})\\cap E_i^X\n = \\Phi\\Big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\cap (E_i^C\\times Y)\\Big)\n \\cup \\Big(R_{\\theta_0}(F_{m_0})\\cap N\\cap E_i^X\\Big).\n\\]\nBy the assumption on $N$, the definition of definition of $F_{m_0}^{\\mathrm e}$, and the rotation on $C\\times Y$, we have\n\\begin{align*}\n \\mu\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^X\\big)\n =& (\\mu_1\\times\\nu)\\big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\cap (E_i^C\\times Y)\\big) \\\\\n = &\\int_{C}\\int_Y \n \\mathbf{1}_{R_{\\theta_0}(F_{m_0}^{\\mathrm e})}(\\theta,y)\n \\mathbf{1}_{E_i^C}(\\theta)\\,d\\nu(y)\\,d\\mu_1(\\theta) \\\\\n =& \\int_{C}\\int_Y \n \\mathbf{1}_{A_{m_0}(\\theta-\\theta_0)}(y)\\mathbf{1}_{E_i^C}(\\theta)\n \\,d\\nu(y)\\,d\\mu_1(\\theta) \\\\\n =& \\int_{C} \\nu\\big(A_{m_0}(\\theta-\\theta_0)\\big)\\,\\mathbf{1}_{E_i^C}(\\theta)\\,d\\mu_1(\\theta).\n\\end{align*}\nBy restricting the outer integral to those $\\theta$ satisfying \n$\\theta-\\theta_0\\in C_{m_0}$, i.e.\\ $\\theta\\in R_{\\theta_0}(C_{m_0})$, and using \\eqref{eq:Am-lower-bound} together with \\eqref{eq:KKT-circle-hit}, we obtain\n\\begin{align*}\n \\mu\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^X\\big)\n \\geq & \\int_{R_{\\theta_0}(C_{m_0})\\cap E_i^C} \\nu\\big(A_{m_0}(\\theta-\\theta_0)\\big)\\,d\\mu_1(\\theta) \\\\\n &\\ge \\frac1t\\,\\mu_1\\big(R_{\\theta_0}(C_{m_0})\\cap E_i^C\\big)>0\n\\end{align*}\nThis is precisely \\eqref{ineq:intersection-rotation-E_i^X}. This completes the proof.\n\\end{proof}\n\n\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}\n\n\\begin{theorem}[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}\n\\end{theorem}",
    "post_theorem_intro_text_len": 5949,
    "post_theorem_intro_text": "The theorem provides a partition of the circle with the property that every finite measurable cover admits a translate meeting each partition element in positive measure.  This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \\textit{a hidden one-dimensional symmetry}. This answers a question raised in \\cite{KKT25} concerning the extension\nfrom the circle $C$ to the unit sphere $\\mathbb{S}^{n-1}\\subset \\mathbb{R}^n$.\n\nBefore stating the main results, we need to introduce some notation.\n\nFor notational convenience, given a point \n$x=(x_1,\\dots,x_n)\\in\\mathbb{R}^n$, we write\n\\[\nx' := (x_1,\\dots,x_{n-1}) \\in \\mathbb{R}^{\\,n-1},\n\\qquad\nx'' := (x_3,\\dots,x_{n-1}) \\in \\mathbb{R}^{\\,n-3}.\n\\]\nThroughout the paper, we denote by $C=\\mathbb{R}/\\mathbb{Z}$ the circle group. Let $\\{E_i^C\\}_{i=1}^r$ be the measurable partition of $C$ provided by Theorem~\\ref{thm:KKT-circle}.  \n\nWe also introduce the rotation\n\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}\nthat is, rotation by angle $2\\pi\\theta$ in the $(x_1,x_2)$–plane.\n\n\\begin{theorem}[Spheres]\\label{thm:sphere-Raimi}\nLet $n\\ge 3$,\n\\[\n\\mathbb{S}^{n-1} := \\{x\\in\\mathbb{R}^n : |x|=1\\}\n\\]\nbe the unit sphere equipped with the normalized surface measure $\\sigma_{n-1}$.\nThen there exists a measurable partition $\\{E_i^{\\mathbb{S}^{n-1}}\\}_{i=1}^r$ of $\\mathbb{S}^{n-1}$ such that:\n\nFor every finite measurable cover\n\\[\n\\mathbb{S}^{n-1} \\subset F_1\\cup\\cdots\\cup F_t,\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{n-1}\\big(R_{\\theta_0}(F_{m_0})\n    \\cap E_i^{\\mathbb{S}^{n-1}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathbb{S}^{n-1}$.\n\\end{theorem}\n\n\\begin{theorem}[Rotational power surfaces]\\label{thm:power-surface-Raimi}\nLet $n\\ge 3$ and $k>0$. Define\n\\[\n\\mathcal{S}_{k,R}\n := \\big\\{(x',x_n)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\n        : x_n = |x'|^{\\,k},\\ 0<|x'|\\le R\\big\\}.\n\\]\nLet\n$\\sigma_{k,R}$ be the normalized surface measure on $\\mathcal{S}_{k,R}$.\nThen there exists a measurable partition\n$\\{E_i^{\\mathcal{S}_{k,R}}\\}_{i=1}^r$ of $\\mathcal{S}_{k,R}$ with the following property:\nFor every finite measurable cover\n\\[\n\\mathcal{S}_{k,R} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{k,R}\\big(R_{\\theta_0}(F_{m_0})\n       \\cap E_i^{\\mathcal{S}_{k,R}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nIn particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.\n\\end{theorem}\n\n\\begin{theorem}[Cylindrical surface]\\label{thm:cylinder-Raimi}\nLet $n\\ge 3$, $R>0$, and $\\Omega\\subset\\mathbb{R}^{n-2}$ be a\nbounded Borel set. Define the cylindrical surface\n\\[\n\\mathcal{C}_{R,\\Omega}\n:= \\big\\{(x_1,x_2, x'', x_n)\\in\\mathbb{R}^2\\times\\mathbb{R}^{n-2}\n : x_1^2+x_2^2=R^2,\\ (x'', x_n)\\in\\Omega\\big\\}.\n\\]\nLet $\\mu_{\\text{\\tiny $R,\\Omega$}}$ be the normalized surface measure on $\\mathcal{C}_{R,\\Omega}$.\nThen there exists a measurable partition $\\{E_i^{\\mathcal{C}_{R,\\Omega}}\\}_{i=1}^r$ of $\\mathcal{C}_{R,\\Omega}$ with the following property:\n\nFor every finite measurable cover\n\\[\n\\mathcal{C}_{R,\\Omega} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\mu_{\\text{\\tiny $R,\\Omega$}}\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^{\\mathcal{C}_{R,\\Omega}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r,\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathcal{C}_{R,\\Omega}$.\n\\end{theorem}\n\n\\medskip\n\\paragraph{Sketch of proof.}\nThe essential observation is that spheres, rotational power surfaces, and cylinders all carry a natural measure-preserving action of the circle $C$ given by rotation in the $(x_1,x_2)$-plane.  Moreover, each of these surfaces admits a measurable, measure-preserving trivialization\n\\[\n\\Phi : C\\times Y \\longrightarrow X\\setminus N,\n\\]\nwhere $Y$ is a suitable parameter space and $N$ is a lower-dimensional set of measure zero.  This allows the surface measure on $X$ to disintegrate as\n\\[\nd\\mu_X(x)\n = d\\mu_1(\\theta)\\, d\\nu_Y(y),\n\\]\nmirroring the product structure of $C\\times Y$.\nThe equivariance identity\n\\[\n\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\n\\]\nthen enables the measurable Raimi partition on the base circle to be lifted directly to a partition of $X$.\n\nThe proofs of our three geometric theorems reduce to verifying the structural hypotheses of the general circle-bundle theorem introduced in Section~\\ref{section-main-result}.  Once this framework is in place, the partition constructed in the previous work of Kang, Koh, and the author on the circle $C$ automatically induces the desired Raimi-type partitions on $\\mathbb{S}^{n-1}$, on rotational power surfaces, and on circular cylindrical surfaces.\n\nOur approach naturally leads to the following open question: \nLet $M$ be a compact hyperbolic surface with its normalized area measure. \nDoes $M$ admit a measurable Raimi--type partition (in the sense of this paper), even though it has no circle action and therefore lies outside our circle--bundle framework?\n\n{\\bf The paper is organized as follows.}\nIn Section~\\ref{section-main-result}, we establish the general circle-bundle theorem, which serves as the main structural tool of the paper.  Sections~\\ref{section-sphere}, \\ref{section-power-surface}, and \\ref{section-cylinder} are devoted to the proofs of the three principal applications: the sphere, the rotational power surfaces, and the circular cylindrical surfaces, respectively.  Each result follows by verifying the hypotheses of the general theorem and applying the measurable Raimi partition on the base circle.",
    "sketch": "The post-theorem text does not sketch a proof of Theorem~\\ref{thm:KKT-circle}; it only states that Theorem~\\ref{thm:KKT-circle} provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \\paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \\(\\Phi: C\\times Y\\to X\\setminus N\\) leading to disintegration \\(d\\mu_X=d\\mu_1(\\theta)\\,d\\nu_Y(y)\\), uses equivariance \\(\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from Kang--Koh--Tran on \\(C\\).",
    "expanded_sketch": "The post-theorem text does not sketch a proof of the main theorem; it only states that the main theorem provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \\paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \\(\\Phi: C\\times Y\\to X\\setminus N\\) leading to disintegration \\(d\\mu_X=d\\mu_1(\\theta)\\,d\\nu_Y(y)\\), uses equivariance \\(\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from the main theorem on \\(C\\).",
    "expanded_theorem": "[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}",
    "theorem_type": [
      "Existential–Universal",
      "Universal–Existential"
    ],
    "mcq": {
      "question": "Let $C:=\\mathbb{R}/\\mathbb{Z}$ be the circle group with normalized Haar measure $\\mu_1$, and for $\\theta\\in C$ let the rotation $R_\\theta:C\\to C$ be given by $R_\\theta(x)=x+\\theta \\pmod 1$. Fix integers $r,t\\in\\mathbb{N}$ with $r,t\\ge 2$. Which statement holds about measurable partitions of $C$ and measurable covers of $C$?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and a rotation $R_\\theta$ with \\[\\mu_1\\bigl(R_\\theta(F_m)\\cap E_i\\bigr)>0\\qquad\\text{for all }1\\le i\\le r.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and a rotation $R_\\theta$ with \\[\\mu_1\\bigl(R_\\theta(F_m)\\cap E_i\\bigr)=1/r\\qquad\\text{for all }1\\le i\\le r.\\]"
        },
        {
          "label": "C",
          "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$, a rotation $R_\\theta$, and an index $i_0\\in\\{1,\\dots,r\\}$ with \\[\\mu_1\\bigl(R_\\theta(F_m)\\cap E_{i_0}\\bigr)>0.\\]"
        },
        {
          "label": "D",
          "text": "For every measurable partition $C=\\bigcup_{i=1}^r E_i$ and every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and a rotation $R_\\theta$ with \\[\\mu_1\\bigl(R_\\theta(F_m)\\cap E_i\\bigr)>0\\qquad\\text{for all }1\\le i\\le r.\\]"
        },
        {
          "label": "E",
          "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ and a rotation $R_\\theta$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exists an index $m\\in\\{1,\\dots,t\\}$ with \\[\\mu_1\\bigl(R_\\theta(F_m)\\cap E_i\\bigr)>0\\qquad\\text{for all }1\\le i\\le r.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "positivity_only_not_prescribed_density",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "all_parts_simultaneously",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "existential_choice_of_partition",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "rotation_depends_on_cover",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct statement or uniquely signal choice A. It asks which statement holds, and the alternatives differ by subtle quantifier and strength changes rather than by obvious cues."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not tautological from the stem alone, but it does function largely as a theorem-identification question: the correct option appears to be the exact intended theorem statement, while the others are nearby strengthenings or weakenings."
      },
      "GPS": {
        "score": 2,
        "justification": "Answering correctly requires genuine reasoning about quantifier order, existence versus universality, and the difference between positivity, exact measure constraints, and cover-dependent rotations. The strongest valid conclusion is not obvious."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and plausible. They reflect common failure modes: overstrengthening to exact densities, weakening to a single partition cell, replacing existential with universal quantification, and mishandling dependence on the cover."
      },
      "total_score": 7,
      "overall_assessment": "A strong MCQ with little answer leakage and high-quality distractors that test careful reasoning about quantifiers and statement strength, though it is somewhat theorem-recognition based rather than fully non-tautological."
    }
  },
  {
    "id": "2512.08726v1",
    "paper_link": "http://arxiv.org/abs/2512.08726v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{teoremaexistenciaB}\nAssume that $a>0$, $\\sigma>1$, $\\alpha>\\frac{1}{2}$, $\\beta>\\frac{1}{2}$ and $0\\leq s<\\frac{3}{2}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Then,\n  there exist an instant $T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0$ and a unique mild solution $(u,\\theta)\\in C_T(\\dot{H}_{a,\\sigma}^s)$ for the Boussinesq equations \\emph{(\\ref{micropolar})} that satisfies\n    \\begin{align*}\n  &\\|(u,\\theta)\\|_{L^\\infty_T(\\dot{H}_{a,\\sigma}^{s})}\n\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n  \\end{align*}",
      "start_pos": 11207,
      "end_pos": 11820,
      "label": "teoremaexistenciaB"
    },
    "ref_dict": {
      "lemalorenz": "\\begin{lemma}[see \\cite{lorenz}]\\label{lemalorenz}\n\tLet $a>0$, $\\sigma> 1$ and $s\\in[0,\\frac{3}{2})$. Then, there exists a positive constant $C_{a,\\sigma,s}$ such that, for all $f,g\\in \\dot{H}_{a,\\sigma}^{s}(\\mathbb{R}^3)$, we have\n\t$$\\|fg\\|_{\\dot{H}_{a,\\sigma}^{s}}\\leq C_{a,\\sigma,s}\t\\|f\\|_{\\dot{H}_{a,\\sigma}^{s}}\\|g\\|_{\\dot{H}_{a,\\sigma}^{s}}.$$\n\\end{lemma}",
      "wilber15": "\\begin{align}\\label{wilber15}\nu(t)= e^{- t(-\\Delta)^{\\alpha}}u_0 - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(u\\cdot\\nabla u)\\,d\\tau + \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(\\theta e_3)\\,d\\tau,\\quad\\forall t>0.\n\\end{align}",
      "wilber33": "\\begin{align}\\label{wilber33}\n\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2&\\geq\n\\frac{a^{2\\sigma_0+1}C_{s,\\alpha,\\sigma,\\sigma_0,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{(2\\alpha-1)[2(s\\sigma+\\sigma_0)+1]}{3\\alpha\\sigma}}}\\exp\\left\\{\\frac{aC'_{\\sigma,s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2\\alpha-1}{3\\alpha\\sigma}}}\\right\\},\\quad\\forall t\\in [0,T^*).\n\\end{align}",
      "teoremaexistenciaB": "\\begin{theorem}\\label{teoremaexistenciaB}\nAssume that $a>0$, $\\sigma>1$, $\\alpha>\\frac{1}{2}$, $\\beta>\\frac{1}{2}$ and $0\\leq s<\\frac{3}{2}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Then,\n  there exist an instant $T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0$ and a unique mild solution $(u,\\theta)\\in C_T(\\dot{H}_{a,\\sigma}^s)$ for the Boussinesq equations \\emph{(\\ref{micropolar})} that satisfies\n    \\begin{align*}\n  &\\|(u,\\theta)\\|_{L^\\infty_T(\\dot{H}_{a,\\sigma}^{s})}\n\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n  \\end{align*}\n \\end{theorem}",
      "wilber23": "\\begin{align}\\label{wilber23}\n\\nonumber\\|I_{31}^{(2)}\\|_{\\dot{H}_{a,\\sigma}^s}&\\leq\\int_{\\kappa_m}^{\\kappa_n}\\|e^{-(\\kappa_n-\\tau)(- \\Delta)^\\alpha}\\mathbb{P}[\\theta e_3]\\|_{\\dot{H}_{a,\\sigma}^s}\\,d\\tau\\\\\n&\\leq \\int_{\\kappa_m}^{T^*}\\|\\theta\\|_{\\dot{H}_{a,\\sigma}^s}\\,d\\tau\\leq C_{a,\\sigma,s}(T^*-\\kappa_m).\n\\end{align}",
      "linearL1eL2": "\\begin{align}\\label{linearL1eL2}\nL_1(w,v)(t)= \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}[ve_3]\\,d\\tau \\quad \\hbox{    and    }\\quad  L_2(w,v)(t)= 0,\n\\end{align}",
      "teoremaB": "\\begin{theorem}\\label{teoremaB}\nAssume that $a>0$, $\\sigma>1$, $\\alpha\\geq 1, \\beta \\geq 1, 0\\leq s<\\frac{3}{2}$ and $n\\in \\mathbb{N}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Consider that $(u,\\theta)\\in C([0,T^*);\\dot{H}_{a,\\sigma}^s)$ is the  solution\nfor the Boussinesq equations \\emph{(\\ref{micropolar})} in the maximal time interval $0\\leq t < T^*$  given in Theorem \\emph{\\ref{teoremaexistenciaB}}. If $T^*<\\infty$, then the following statements hold:\n      \\begin{enumerate}\n  \\item[\\textbf{\\emph{i)}}]  $\\displaystyle \\limsup_{t\\nearrow T^*} \\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^{(n-1)}},\\sigma}^s}=+\\infty$;\n  \\item[\\textbf{\\emph{ii)}}] $\\displaystyle\\int_{t}^{T^*} [\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(\\tau)\\|_{L^1}\n^{\\frac{2\\alpha}{2\\alpha-1}}+\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(\\tau)\\|_{L^1}^{\\frac{2\\beta}{2\\beta-1}}]\\,d\\tau=\\infty$;\n \\item[\\textbf{\\emph{iii)}}]  $\\displaystyle \\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\alpha}{2\\alpha-1}}\n    +\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\beta}{2\\beta-1}}\\geq  (e^{C(T^*-t)}-1)^{-1};$\n  \\item [\\textbf{\\emph{iv)}}] $\\displaystyle \\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^n},\\sigma}^s}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^n},\\sigma}^s}^{\\frac{2\\beta}{2\\beta-1}}\\gtrsim(e^{C(T^{*}-t)}-1)^{-1}.$\n  \\end{enumerate}\nfor all $t\\in [0,T^{*})$ and $C>0$ is a positive constant.\n\\end{theorem}",
      "wilber26": "\\begin{align}\\label{wilber26}\n \\nonumber&\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(T)\\|_{L^1}+ \\int_{t}^{T}\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}\\mathcal{F}[(-\\Delta)^{\\alpha} u](\\tau)\\|_{L ^1}\\,d\\tau+ \\int_{t}^{T}\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}\\mathcal{F}[(-\\Delta)^{\\beta} \\theta](\\tau)\\|_{L ^1}\\,d\\tau\\\\\n\\nonumber&\\leq \\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}\n+\\int_{t}^{T}\\int_{\\mathbb{R}^3}e^{\\frac{a}{\\sigma}|\\xi|^{\\frac{1}{\\sigma}}}[|\\widehat{(u\\cdot\\nabla \\displaystyle u)}(\\tau)|+|\\widehat{\\theta}(\\tau)|+|\\widehat{(u\\cdot\\nabla \\displaystyle \\theta)}(\\tau)|]\\,d\\xi d\\tau\\\\\n&\\leq  \\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}\n+\\int_{t}^{T}\\int_{\\mathbb{R}^3}e^{\\frac{a}{\\sigma}|\\xi|^{\\frac{1}{\\sigma}}}[|\\widehat{(u\\cdot\\nabla \\displaystyle u)}(\\tau)|+|\\widehat{(u\\cdot\\nabla \\displaystyle \\theta)}(\\tau)|]\\,d\\xi d\\tau+\\int_{t}^{T}\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}\\widehat{\\theta}(\\tau)\\|_{L^1} d\\tau.\n\\end{align}",
      "wilber9": "\\begin{align}\\label{wilber9}\n\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\beta}{2\\beta-1}}\\geq  [e^{C_{\\alpha,\\beta}(T^*-t)}-1]^{-1},\n\\end{align}",
      "wilber4": "\\begin{align}\\label{wilber4}\n\\nonumber\\frac{d}{dt}\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2+\\|u(t)\\|_{\\dot{H}_{a,\\sigma}^{s+\\alpha}}^2+\\|\\theta(t)\\|_{\\dot{H}_{a,\\sigma}^{s+\\beta}}^2 &\\leq C_{s,\\alpha,\\beta} [\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\beta}{2\\beta-1}}+1]\\\\\n&\\quad\\times\\|(u,\\theta)\\|_{\\dot{H}_{a,\\sigma}^{s}}^{2}.\n\\end{align}",
      "linearL": "\\begin{align}\\label{linearL}\nL(w,v)(t)=(L_1(w,v)(t),L_2(w,v)(t)),\\quad\\forall t>0,\n\\end{align}",
      "wilber16": "\\begin{align}\\label{wilber16}\n\\theta(t)= e^{- t(-\\Delta)^{\\beta}}\\theta_0 - \\int_{0}^te^{-  (t-\\tau)(-\\Delta)^{\\beta}} [u\\cdot \\nabla\\theta]\\,d\\tau ,\\quad\\forall t>0.\n\\end{align}",
      "esqueci": "\\begin{align}\\label{esqueci}\n\\frac{d}{dt}\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2+\\|u(t)\\|_{\\dot{H}_{a,\\sigma}^{s+\\alpha}}^2+\\|\\theta(t)\\|_{\\dot{H}_{a,\\sigma}^{s+\\beta}}^2 &\\leq C_{a,\\sigma,s,\\alpha,\\beta}\\left( \\|(u,\\theta)\\|_{\\dot{H}_{a,\\sigma}^s}^{\\frac{2\\alpha}{2\\alpha-1}} +\n \\|(u,\\theta)\\|_{\\dot{H}_{a,\\sigma}^s}^{\\frac{2\\beta}{2\\beta-1}}+1\\right)\\|(u,\\theta)\\|_{\\dot{H}_{a,\\sigma}^s}^2.\n\\end{align}",
      "wilber24": "\\begin{align}\\label{wilber24}\n\\nonumber\\|(u,b)(T)\\|_{\\dot{H}_{a,\\sigma}^s}^2&\\leq \\|(u,b)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2\\exp \\{C_{s,\\alpha,\\beta}\\int_t^T [\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\beta}{2\\beta-1}}+1]\\,d\\tau \\}\\\\\n&\\leq \\|(u,b)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2e^{C_{s,\\alpha,\\beta}(T-t)}\\exp \\{C_{s,\\alpha,\\beta}\\int_t^T [\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|e^{\\frac{a}{\\sigma}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})\\|_{L^{1}}^{\\frac{2\\beta}{2\\beta-1}}]\\,d\\tau \\},\n\\end{align}",
      "pontofixo": "\\begin{lemma}[see \\cite{pontofixo}]\\label{pontofixo}\n\t\tLet $(X,\\|\\cdot\\|)$ be a Banach space, $L:X\\rightarrow X$ a continuous linear operator and\n $B:X\\times X\\rightarrow X$ a continuous bilinear operator, i.e., there exist positive constants $C_1$ and $C_2$ such that\n $$\\|L(x)\\|\\leq C_1 \\|x\\|,\\quad \\|B(x,y)\\|\\leq C_2 \\|x\\|\\|y\\|,\\quad \\forall x,y\\in X.$$\n Then, for each $C_1\\in (0,1)$ and $x_0\\in X$ that satisfy $4C_2\\|x_0\\| <(1-C_1)^2$, one has that the equation \n $$a=x_0+B(a,a)+L(a), \\quad a\\in X,$$\n admits solution $x\\in X$. Moreover, $x$ satisfies the inequality $\\|x\\|\\leq \\frac{2\\|x_0\\|}{1-C_1}$ and this is the only one such that $\\|x\\|\\leq  \\frac{1-C_1}{2C_2}.$\n\t\\end{lemma}",
      "wilber32": "\\begin{align}\\label{wilber32}\n\\frac{1}{2}\\frac{d}{dt}\\|(u,\\theta)(t)\\|_{L^2}^2+\\|(-\\Delta)^{\\frac{\\alpha}{2}}u(t)\\|_{L^2}^2+\\|(-\\Delta)^{\\frac{\\beta}{2}}\\theta(t)\\|_{L^2}^2 &\\leq \\|(u,\\theta)(t)\\|_{L^2}^{2},\\quad\\forall t\\in [0,T^*).\n\\end{align}",
      "corolario": "\\begin{corollary}\\label{corolario}\nAssume that $a>0$, $\\sigma>1$, $\\alpha=\\beta\\geq 1 \\hbox{  and  } 0\\leq s<\\frac{3}{2}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s\\cap L^2$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Consider that $(u,\\theta)\\in C([0,T^*);\\dot{H}_{a,\\sigma}^s)$ is the  solution\nfor the Boussinesq equations \\emph{(\\ref{micropolar})} in the maximal time interval $0\\leq t < T^*$  given in Theorem \\emph{\\ref{teoremaexistenciaB}}. If $T^*<\\infty$, then the following statement holds:\n   $$\n   \\displaystyle\n\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s} \\gtrsim  {(e^{C_1(T^*-t)}-1)^{\\varrho_1}}\n\\exp\\{C_2(e^{C_1(T^*-t)}-1)^{\\varrho_2}\\},\n$$\nfor all \\(t \\in [0, T^{*})\\), where $\\varrho_1 := \\frac{(1 - 2\\alpha)\\,[\\,2(s\\sigma + \\sigma_{0}) + 1\\,]}{6\\alpha\\sigma}<0$, $\\varrho_2:=\\frac{1-2\\alpha}{3\\alpha\\sigma}<0$ and \\(2\\sigma_{0}\\) denotes the integer part of \\(2\\sigma\\mu\\) \\emph{(}with \\(\\mu > \\tfrac{3}{2}\\)\\emph{)}.  \nHere, $C_1=C_1(\\alpha)$ and $C_2=C_2(a,\\alpha,\\sigma,s,u_0,\\theta_0,T^*)$ are constants. In particular\n$$\n\\lim_{t \\nearrow T^*}\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s} = +\\infty,\n$$\nexponentially.\n\\end{corollary}",
      "lwfinal": "\\begin{align}\\label{lwfinal}\n\\displaystyle\\|L(w,v)\\|_{X}\\leq  T \\|(w,v)\\|_{X},\\quad\\forall (w,v)\\in X.\n\\end{align}",
      "lemanovow1": "\\begin{lemma}[see \\cite{robert}]\\label{lemanovow1}\nLet $a\\geq0$, $\\sigma\\geq1$, $s\\in\\mathbb{R}$ and $\\theta\\geq1$. The following inequality holds:\n$$\\|f\\|_{\\dot{H}_{a,\\sigma}^{s+1}}\\leq \\|f\\|_{\\dot{H}_{a,\\sigma}^{s}}^{1-\\frac{1}{\\theta}}\\|f\\|_{\\dot{H}_{a,\\sigma}^{s+\\theta}}^{\\frac{1}{\\theta}}.$$\n  \\end{lemma}",
      "wilber17": "\\begin{align}\\label{wilber17}\n\\nonumber\\displaystyle\\|B_1((w,v),(\\gamma,\\phi))(t)\\|_{\\dot{H}_{a,\\sigma}^s}&\\leq  \\int_0^t\\Big(\\int_{\\mathbb{R}^3} |\\xi|^{2}e^{-2(t-\\tau)|\\xi|^{2\\alpha}}|\\xi|^{2s} e^{2a|\\xi|^{\\frac{1}{\\sigma}}} |\\mathcal{F}(\\gamma\\otimes w)(\\xi)|^2\\,d\\xi\\Big)^{\\frac{1}{2}}\\,d\\tau\\\\\n\\nonumber&\\leq C_{\\alpha} \\int_0^t (t-\\tau)^{-\\frac{1}{2\\alpha}}\\Big(\\int_{\\mathbb{R}^3} |\\xi|^{2s} e^{2a|\\xi|^{\\frac{1}{\\sigma}}} |\\mathcal{F}(\\gamma\\otimes w)(\\xi)|^2\\,d\\xi\\Big)^{\\frac{1}{2}}\\,d\\tau\\\\\n&\\leq C_{\\alpha} \\int_0^t (t-\\tau)^{-\\frac{1}{2\\alpha}} \\|(\\gamma\\otimes w)(\\tau)\\|_{\\dot{H}_{a,\\sigma}^{s}}\\,d\\tau.\n\\end{align}",
      "wilber25": "\\begin{align}\\label{wilber25}\n \\frac{1}{2}\\partial_t|\\widehat{u}(t)|^2+|\\xi|^{2\\alpha}|\\widehat{ u}|^2 &\\leq\n|\\widehat{u}\\cdot \\widehat{u\\cdot\\nabla \\displaystyle u}|+|\\widehat{u}_3 \\widehat{\\theta}|.\n\\end{align}",
      "w1": "\\begin{align}\n\\nonumber\\displaystyle\\|B_1((w,v),(\\gamma,\\phi))(t)\\|_{\\dot{H}_{a,\\sigma}^s}\n&\\leq C_{a,\\sigma,s,\\alpha}\\|\\gamma\\|_{L_T^{\\infty}(\\dot{H}_{a,\\sigma}^{s})} \\|w\\|_{L_T^{\\infty}(\\dot{H}_{a,\\sigma}^{s})}\\int_0^t (t-\\tau)^{-\\frac{1}{2\\alpha}} d\\tau\\\\\n\\label{w1}&\\leq C_{a,\\sigma,s,\\alpha}T^{1-\\frac{1}{2\\alpha}}\\|(w,v)\\|_{X} \\|(\\gamma,\\phi)\\|_{X},\n\\end{align}",
      "i)n=1": "\\begin{align}\\label{i)n=1}\n \\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{\\sqrt{\\sigma}},\\sigma}^s}^{\\frac{2\\alpha}{2\\alpha-1}}\n+\\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{\\sqrt{\\sigma}},\\sigma}^s}^{\\frac{2\\beta}{2\\beta-1}}\\geq C_{a,\\sigma,s,\\alpha,\\beta}[e^{C_{\\alpha,\\beta}(T^{*}_a-t)}-1]^{-1},\n\\end{align}",
      "micropolar": "\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\,  \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}",
      "wilber19": "\\begin{align}\\label{wilber19}\n\\nonumber &I_2=(I_{21}^{(1)}+I_{21}^{(2)},I_{22}):= \\Big(\\int_{0}^{\\kappa_m}[e^{- (\\kappa_m-\\tau)(-\\Delta)^\\alpha}-e^{- (\\kappa_n-\\tau)(-\\Delta)^\\alpha}] \\mathbb{P}[u\\cdot \\nabla u]\\,d\\tau+\\\\ &\n\\int_{0}^{\\kappa_m}[e^{-  (\\kappa_n-\\tau)(-\\Delta)^\\alpha}-e^{- (\\kappa_m-\\tau)(-\\Delta)^\\alpha}] \\mathbb{P}[\\theta e_3]\\,d\\tau,\\int_{0}^{\\kappa_m}[e^{- (\\kappa_m-\\tau)(-\\Delta)^\\beta}-e^{- (\\kappa_n-\\tau)(-\\Delta)^\\beta}] (u\\cdot \\nabla \\theta)\\,d\\tau\\Big),\n\\end{align}",
      "3": "\\begin{align}\\label{3}\n\\frac{1}{2}\\frac{d}{dt}\\|u(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2+\\|u(t)\\|_{\\dot{H}_{a,\\sigma}^{s+\\alpha}}^2 &\\leq\n|\\langle u, u\\cdot\\nabla u\\rangle_{\\dot{H}_{a,\\sigma}^s}|+  |\\langle u_3,\\theta\\rangle_{\\dot{H}_{a,\\sigma}^s} |.\n\\end{align}",
      "normal2": "\\begin{align}\\label{normal2}\n \\|(u,\\theta)(t)\\|_{L^2}\\leq e^{T^*}\\|(u_0,\\theta_0)\\|_{L^2},\\quad\\forall t\\in [0,T^*).\n\\end{align}",
      "NS": "\\begin{equation}\\label{NS}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p  \n\\;=\\;\n\\Delta u, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0.\n\n\\end{array}\n\\right.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5824,
    "pre_theorem_intro_text": "\\hspace{0.5cm}\nIn this paper, we consider the following three-dimensional Boussinesq equations:\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\,  \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $u(x,t) = \\left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\\right) \\in \\mathbb{R}^3$ denotes the incompressible velocity field, and \n$\\theta(x,t) \\in \\mathbb{R}$ represents the temperature of the fluid. The parameters $\\alpha$ and $\\beta$ belong to \n$(\\tfrac{1}{2},\\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \\eqref{micropolar} is assumed to be divergence-free, that is,\n$$\n\\operatorname{div} u_{0} = 0,\n$$\nand we denote by $(-\\Delta)^{\\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by\n$$\n\\mathcal{F}[(-\\Delta)^{\\gamma} f](\\xi) = |\\xi|^{2\\gamma} \\widehat{f}(\\xi) .\n$$\n\nObserve that, in the case $\\theta \\equiv 0$ and $\\alpha = 1$, the system \\eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations\n\\begin{equation}\\label{NS}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p  \n\\;=\\;\n\\Delta u, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0.\n\n\\end{array}\n\\right.\n\\end{equation}\n\nThe Boussinesq equations (\\ref{micropolar}) are simplified yet powerful models widely used in the study of oceanic and atmospheric dynamics. By incorporating buoyancy effects while filtering out sound waves, they provide a mathematically tractable framework for describing stratified fluid flows under the influence of gravity. These equations also arise in several other areas of Physics, including thermal convection, geophysical fluid dynamics, and plasma modeling; see, for instance, \\cite{MR2245751,MR2540168} for more details. \n\nIt is worth noting, however, that although \\eqref{micropolar} with fractional dissipation may initially appear to be a purely mathematical generalization, there are geophysical scenarios in which the Boussinesq equations  with a fractional Laplacian (\\ref{micropolar}) naturally arise. A typical example occurs in the middle atmosphere, where upward-travelling flows experience changes due to variations in atmospheric properties, even though the incompressibility and Boussinesq approximations remain valid. In this regime, the effects of kinematic and thermal diffusion are attenuated by the thinning of the atmosphere, an anomalous behavior that can be effectively modeled using a spatial fractional Laplacian. See \\cite{MR2379269, Gill1982} for further discussions.\n\nThere is an extensive literature, in both two and three dimensions, concerning the well-posedness and qualitative properties of the system \\eqref{micropolar} in a variety of functional settings (see, for instance, \\cite{MR4884564,MR3503190,MR4795499,MR3808344,MR3008326,MR3759571,cilon1} and the references therein).\n\nIn this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \\eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \\ge 0$, $\\sigma \\ge 1$, and $s \\in \\mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by\n$$\n\\dot{H}_{a,\\sigma}^s=\\dot{H}_{a,\\sigma}^s(\\mathbb{R}^3)\n    = \\left\\{ f \\in \\mathcal{S}' : \\widehat{f} \\in L^1_{\\operatorname{loc}}\\,\\, \\text{and} \\,\\, \\ \n       \\|f\\|_{\\dot{H}_{a,\\sigma}^s}:=\\Big[\\int_{\\mathbb{R}^3} |\\xi|^{2s} e^{2a |\\xi|^{1/\\sigma}} |\\widehat{f}(\\xi)|^2\\, d\\xi\\Big]^{\\frac{1}{2}} < \\infty\n      \\right\\},\n$$\nwhere $\\widehat{f}$ denotes the Fourier transform\n$$\n\\mathcal{F}(f)(\\xi)=\\widehat{f}(\\xi):=\\int_{\\mathbb{R}^3}e^{-i\\xi\\cdot x}f(x)\\,dx.\n$$\nEssentially, $\\dot{H}_{a,\\sigma}^s \\equiv L^{2}\\!(|\\xi|^{s} e^{a|\\xi|^{1/\\sigma}}\\, d\\xi)$ and, in particular, $\\dot{H}_{0,\\sigma}^s \\equiv \\dot{H}^s$, \\textit{i.e.}, the classical (homogeneous) Sobolev space.  This class of functions plays a crucial role because, according to  Paley--Wiener Theorem (see \\cite{HormanderALPDO1}, Chapter~7, for further details), for $\\sigma=1$ a function $ f $ belongs to the space $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip\n\n$$\nS_a = \\{ x + i y  \\in \\mathbb{C}^d: x, y \\in \\mathbb{R}^d,\\ |y| < a \\},\n$$\nsuch that\n$$\n    \\sup_{|y| < a} \\| F(x + i y) \\|_{\\dot{H}^s} < \\infty.\n$$\nIn other words, the parameter $ a \\geq 0 $ determines the width of the complex strip to which \nfunctions in $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ can be analytically extended. For $\\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies. \n\nConsequently, the study of \ndifferential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \\cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.\n\nThe first main result of this work establishes the local well-posedness of the Cauchy problem \\eqref{micropolar} in  Sobolev-Gevrey class. Recall that, we say that $(u,\\theta)=(u,\\theta)(x,t)$ is a mild solution to the Boussinesq equations (\\ref{micropolar}) if this application satisfies the associated integral formulations (\\ref{wilber15}) and (\\ref{wilber16}), which are established via the fractional heat semigroup.",
    "context": "\\hspace{0.5cm}\nIn this paper, we consider the following three-dimensional Boussinesq equations:\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\, \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $u(x,t) = \\left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\\right) \\in \\mathbb{R}^3$ denotes the incompressible velocity field, and \n$\\theta(x,t) \\in \\mathbb{R}$ represents the temperature of the fluid. The parameters $\\alpha$ and $\\beta$ belong to \n$(\\tfrac{1}{2},\\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \\eqref{micropolar} is assumed to be divergence-free, that is,\n$$\n\\operatorname{div} u_{0} = 0,\n$$\nand we denote by $(-\\Delta)^{\\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by\n$$\n\\mathcal{F}[(-\\Delta)^{\\gamma} f](\\xi) = |\\xi|^{2\\gamma} \\widehat{f}(\\xi) .\n$$\n\nObserve that, in the case $\\theta \\equiv 0$ and $\\alpha = 1$, the system \\eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations\n\\begin{equation}\\label{NS}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \n\\;=\\;\n\\Delta u, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\nIn this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \\eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \\ge 0$, $\\sigma \\ge 1$, and $s \\in \\mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by\n$$\n\\dot{H}_{a,\\sigma}^s=\\dot{H}_{a,\\sigma}^s(\\mathbb{R}^3)\n = \\left\\{ f \\in \\mathcal{S}' : \\widehat{f} \\in L^1_{\\operatorname{loc}}\\,\\, \\text{and} \\,\\, \\ \n \\|f\\|_{\\dot{H}_{a,\\sigma}^s}:=\\Big[\\int_{\\mathbb{R}^3} |\\xi|^{2s} e^{2a |\\xi|^{1/\\sigma}} |\\widehat{f}(\\xi)|^2\\, d\\xi\\Big]^{\\frac{1}{2}} < \\infty\n \\right\\},\n$$\nwhere $\\widehat{f}$ denotes the Fourier transform\n$$\n\\mathcal{F}(f)(\\xi)=\\widehat{f}(\\xi):=\\int_{\\mathbb{R}^3}e^{-i\\xi\\cdot x}f(x)\\,dx.\n$$\nEssentially, $\\dot{H}_{a,\\sigma}^s \\equiv L^{2}\\!(|\\xi|^{s} e^{a|\\xi|^{1/\\sigma}}\\, d\\xi)$ and, in particular, $\\dot{H}_{0,\\sigma}^s \\equiv \\dot{H}^s$, \\textit{i.e.}, the classical (homogeneous) Sobolev space. This class of functions plays a crucial role because, according to Paley--Wiener Theorem (see \\cite{HormanderALPDO1}, Chapter~7, for further details), for $\\sigma=1$ a function $ f $ belongs to the space $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip\n\n$$\nS_a = \\{ x + i y \\in \\mathbb{C}^d: x, y \\in \\mathbb{R}^d,\\ |y| < a \\},\n$$\nsuch that\n$$\n \\sup_{|y| < a} \\| F(x + i y) \\|_{\\dot{H}^s} < \\infty.\n$$\nIn other words, the parameter $ a \\geq 0 $ determines the width of the complex strip to which \nfunctions in $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ can be analytically extended. For $\\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.\n\nConsequently, the study of \ndifferential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \\cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.\n\nThe first main result of this work establishes the local well-posedness of the Cauchy problem \\eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\\theta)=(u,\\theta)(x,t)$ is a mild solution to the Boussinesq equations (\\ref{micropolar}) if this application satisfies the associated integral formulations (\\ref{wilber15}) and (\\ref{wilber16}), which are established via the fractional heat semigroup.\n\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\,  \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\n\n\\begin{align}\\label{wilber15}\nu(t)= e^{- t(-\\Delta)^{\\alpha}}u_0 - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(u\\cdot\\nabla u)\\,d\\tau + \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(\\theta e_3)\\,d\\tau,\\quad\\forall t>0.\n\\end{align}\n\n\\begin{align}\\label{wilber16}\n\\theta(t)= e^{- t(-\\Delta)^{\\beta}}\\theta_0 - \\int_{0}^te^{-  (t-\\tau)(-\\Delta)^{\\beta}} [u\\cdot \\nabla\\theta]\\,d\\tau ,\\quad\\forall t>0.\n\\end{align}",
    "full_context": "\\hspace{0.5cm}\nIn this paper, we consider the following three-dimensional Boussinesq equations:\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\, \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $u(x,t) = \\left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\\right) \\in \\mathbb{R}^3$ denotes the incompressible velocity field, and \n$\\theta(x,t) \\in \\mathbb{R}$ represents the temperature of the fluid. The parameters $\\alpha$ and $\\beta$ belong to \n$(\\tfrac{1}{2},\\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \\eqref{micropolar} is assumed to be divergence-free, that is,\n$$\n\\operatorname{div} u_{0} = 0,\n$$\nand we denote by $(-\\Delta)^{\\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by\n$$\n\\mathcal{F}[(-\\Delta)^{\\gamma} f](\\xi) = |\\xi|^{2\\gamma} \\widehat{f}(\\xi) .\n$$\n\nObserve that, in the case $\\theta \\equiv 0$ and $\\alpha = 1$, the system \\eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations\n\\begin{equation}\\label{NS}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \n\\;=\\;\n\\Delta u, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\nIn this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \\eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \\ge 0$, $\\sigma \\ge 1$, and $s \\in \\mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by\n$$\n\\dot{H}_{a,\\sigma}^s=\\dot{H}_{a,\\sigma}^s(\\mathbb{R}^3)\n = \\left\\{ f \\in \\mathcal{S}' : \\widehat{f} \\in L^1_{\\operatorname{loc}}\\,\\, \\text{and} \\,\\, \\ \n \\|f\\|_{\\dot{H}_{a,\\sigma}^s}:=\\Big[\\int_{\\mathbb{R}^3} |\\xi|^{2s} e^{2a |\\xi|^{1/\\sigma}} |\\widehat{f}(\\xi)|^2\\, d\\xi\\Big]^{\\frac{1}{2}} < \\infty\n \\right\\},\n$$\nwhere $\\widehat{f}$ denotes the Fourier transform\n$$\n\\mathcal{F}(f)(\\xi)=\\widehat{f}(\\xi):=\\int_{\\mathbb{R}^3}e^{-i\\xi\\cdot x}f(x)\\,dx.\n$$\nEssentially, $\\dot{H}_{a,\\sigma}^s \\equiv L^{2}\\!(|\\xi|^{s} e^{a|\\xi|^{1/\\sigma}}\\, d\\xi)$ and, in particular, $\\dot{H}_{0,\\sigma}^s \\equiv \\dot{H}^s$, \\textit{i.e.}, the classical (homogeneous) Sobolev space. This class of functions plays a crucial role because, according to Paley--Wiener Theorem (see \\cite{HormanderALPDO1}, Chapter~7, for further details), for $\\sigma=1$ a function $ f $ belongs to the space $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip\n\n$$\nS_a = \\{ x + i y \\in \\mathbb{C}^d: x, y \\in \\mathbb{R}^d,\\ |y| < a \\},\n$$\nsuch that\n$$\n \\sup_{|y| < a} \\| F(x + i y) \\|_{\\dot{H}^s} < \\infty.\n$$\nIn other words, the parameter $ a \\geq 0 $ determines the width of the complex strip to which \nfunctions in $ \\dot{H}_{a,1}^s(\\mathbb{R}^d) $ can be analytically extended. For $\\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.\n\nConsequently, the study of \ndifferential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \\cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.\n\nThe first main result of this work establishes the local well-posedness of the Cauchy problem \\eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\\theta)=(u,\\theta)(x,t)$ is a mild solution to the Boussinesq equations (\\ref{micropolar}) if this application satisfies the associated integral formulations (\\ref{wilber15}) and (\\ref{wilber16}), which are established via the fractional heat semigroup.\n\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\nu_t\n\\;\\!+\\,\nu \\cdot \\nabla u\n\\,+\\,\n\\nabla \\;\\!p \\:\\!\\,+\\,  \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\n\\theta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\n\\theta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\n\n\\begin{align}\\label{wilber15}\nu(t)= e^{- t(-\\Delta)^{\\alpha}}u_0 - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(u\\cdot\\nabla u)\\,d\\tau + \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(\\theta e_3)\\,d\\tau,\\quad\\forall t>0.\n\\end{align}\n\n\\begin{align}\\label{wilber16}\n\\theta(t)= e^{- t(-\\Delta)^{\\beta}}\\theta_0 - \\int_{0}^te^{-  (t-\\tau)(-\\Delta)^{\\beta}} [u\\cdot \\nabla\\theta]\\,d\\tau ,\\quad\\forall t>0.\n\\end{align}\n\nThe first main result of this work establishes the local well-posedness of the Cauchy problem \\eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\\theta)=(u,\\theta)(x,t)$ is a mild solution to the Boussinesq equations (\\ref{micropolar}) if this application satisfies the associated integral formulations (\\ref{wilber15}) and (\\ref{wilber16}), which are established via the fractional heat semigroup.\n\nNext, we investigate the qualitative behavior of the local solution provided by Theorem \\ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.\n\nFirst of all, apply Helmholtz's projector $\\mathbb{P}$, defined via Fourier transform (see \\cite{PN} for more details)\n$$\n\\mathcal{F}[\\mathbb{P}f](\\xi) := \\widehat{f}(\\xi) - \\frac{\\widehat{f}(\\xi)\\cdot \\xi}{|\\xi|^2}\\, \\xi,\n$$\nthe heat semigroup $e^{-(t-\\tau)(-\\Delta)^{\\alpha}}$ to the first equation in (\\ref{micropolar}), and integrate the result obtained over $[0,t]$ to obtain\n\\begin{align}\\label{wilber15}\nu(t)= e^{- t(-\\Delta)^{\\alpha}}u_0 - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(u\\cdot\\nabla u)\\,d\\tau + \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(\\theta e_3)\\,d\\tau,\\quad\\forall t>0.\n\\end{align}\nSimilarly, by using the heat semigroup $e^{- (t-\\tau)(-\\Delta)^{\\beta}}$ in the second equation of (\\ref{micropolar}), and integrating over $[0,t]$, we deduce\n\\begin{align}\\label{wilber16}\n\\theta(t)= e^{- t(-\\Delta)^{\\beta}}\\theta_0 - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\beta}} [u\\cdot \\nabla\\theta]\\,d\\tau ,\\quad\\forall t>0.\n\\end{align}\nHence, we are able to write the following equality:\n\\begin{align}\\label{estimativaprojecao3}\n(u,\\theta)(t)&= (e^{- t(-\\Delta)^{\\alpha}}u_0,e^{- t (-\\Delta)^{\\beta}}\\theta_0) + B((u,\\theta),(u,\\theta))(t)+L(u,\\theta)(t),\\quad\\forall t>0.\n\\end{align}\nwhere\n\\begin{align}\\label{bilinearB}\nB((w,v),(\\gamma,\\phi))(t)=(B_1((w,v),(\\gamma,\\phi)),B_2((w,v),(\\gamma,\\phi)))(t),\\quad\\forall t>0,\n\\end{align}\nand also\n\\begin{align}\\label{linearL}\nL(w,v)(t)=(L_1(w,v)(t),L_2(w,v)(t)),\\quad\\forall t>0,\n\\end{align}\nwith\n\\begin{align}\\label{bilinearB1}\nB_1((w,v),(\\gamma,\\phi))(t)= - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}(w\\cdot\\nabla \\gamma)\\,d\\tau,\n\\end{align}\n\\begin{align}\\label{bilinearB2}\nB_2((w,v),(\\gamma,\\phi))(t)= - \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\beta}} [w\\cdot \\nabla \\phi]\\,d\\tau,\n\\end{align}\n\\begin{align}\\label{linearL1eL2}\nL_1(w,v)(t)= \\int_{0}^te^{- (t-\\tau)(-\\Delta)^{\\alpha}} \\mathbb{P}[ve_3]\\,d\\tau \\quad \\hbox{ and }\\quad L_2(w,v)(t)= 0,\n\\end{align}\nfor all $w,v,\\gamma,\\phi\\in C_T(\\dot{H}_{a,\\sigma}^s)$ ($T>0$ will be revealed below). Denote $X= [C_T (\\dot{H}_{a,\\sigma}^{s})]^3$ $\\times C_T (\\dot{H}_{a,\\sigma}^{s}) (\\equiv C_T (\\dot{H}_{a,\\sigma}^{s})$ throughout this work)$\\footnote{Here, $\\|(f,g)\\|_{X}:=[\\|f\\|^2_{L^\\infty_T(\\dot{H}_{a,\\sigma}^s)}+\\|g\\|^2_{L^\\infty_T(\\dot{H}_{a,\\sigma}^s)}]^{\\frac{1}{2}}$}$ and notice that $B$ and $L$ are bilinear and linear operators on $X\\times X$ and $X$, respectively.\n\nAt last, notice that \n\\begin{align}\\label{mudanca1}\n\\displaystyle\\|e^{- t(-\\Delta)^{\\alpha}} u_0\\|^2_{\\dot{H}_{a,\\sigma}^s}&=\\int_{\\mathbb{R}^3} e^{-2 t|\\xi|^{2\\alpha}}|\\xi|^{2s} e^{2a|\\xi|^{\\frac{1}{\\sigma}}} |\\widehat{u}_0(\\xi)|^2\\,d\\xi\\leq \\|u_0\\|_{\\dot{H}_{a,\\sigma}^{s}}^2,\\quad\\forall t\\in [0,T],\n\\end{align}\nand also that\n\\begin{align}\\label{mudanca2}\n\\displaystyle\\|e^{- t(-\\Delta)^{\\beta}} \\theta_0\\|_{\\dot{H}_{a,\\sigma}^s}\\leq \\|\\theta_0\\|_{\\dot{H}_{a,\\sigma}^{s}},\\quad\\forall t\\in [0,T].\n\\end{align}\nFrom (\\ref{mudanca1}) and (\\ref{mudanca2}), it follows that\n\\begin{align}\\label{dadoincial}\n\\|(e^{-t(-\\Delta)^{\\alpha}}u_0,e^{-t (-\\Delta)^{\\beta}}\\theta_0)\\|_{X}\\leq\\|(u_0,\\theta_0)\\|_{\\dot{H}_{a,\\sigma}^{s}}.\n\\end{align}\nThus, we are able to determine $T>0$ as follows: \n$$T< \\min\\Big\\{\\Big[\\sqrt{8C_{a,\\sigma,s,\\alpha,\\beta}\\|(u_0,\\theta_0)\\|_{\\dot{H}_{a,\\sigma}^{s}}}+1\\Big]^{-\\frac{4\\alpha}{2\\alpha-1}},\\Big[\\sqrt{8C_{a,\\sigma,s,\\alpha,\\beta}\\|(u_0,\\theta_0)\\|_{\\dot{H}_{a,\\sigma}^{s}}}+1\\Big]^{-\\frac{4\\beta}{2\\beta-1}}\\Big\\},$$\nwhere $C_{a,\\sigma,s,\\alpha,\\beta}$ is established in (\\ref{w5}). Therefore, by Lemma \\ref{pontofixo}, there exists a unique solution $(u,\\theta)\\in X$ for the equation (\\ref{estimativaprojecao3}) (it is enough to take a look at (\\ref{lwfinal}), (\\ref{w5}) and (\\ref{dadoincial})) that satisfies the following inequality:\n$$\\|(u,\\theta)\\|_{X}\\leq \\frac{1-T}{2C_{a,\\sigma,s,\\alpha,\\beta}[T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}]}.$$\n\nNow, let us assume that $\\displaystyle \\limsup_{t\\nearrow T^*} \\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^{s}} < \\infty$. Thus, apply Theorem \\ref{teoremaexistenciaB} to obtain a positive constant $C_{a,\\sigma,s}$ such that\n\\begin{align}\\label{limitacao}\n\\displaystyle \\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^{s}} \\leq C_{a,\\sigma,s},\\quad\\forall\\,t\\in[0,T^*).\n\\end{align}\nThereby, by integrating over $[0,t]$ the inequality (\\ref{esqueci}) and using (\\ref{limitacao}), we can write the following inequality:\n\\begin{align*}\n&\\nonumber\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2+\\int_0^t\\|u(\\tau)\\|_{\\dot{H}_{a,\\sigma}^{s+\\alpha}}^2+\\int_0^t\\|\\theta(\\tau)\\|_{\\dot{H}_{a,\\sigma}^{s+\\beta}}^2\\,d\\tau \\leq \\|(u_0,\\theta_0)\\|_{\\dot{H}_{a,\\sigma}^s}^2+C_{a,\\sigma,s,\\alpha,\\beta}T^*,\n\\end{align*}\nfor all $t\\in[0,T^*).$ As a result, particularly, one has\n\\begin{align}\\label{e4}\n\\int_0^t\\|u(\\tau)\\|_{\\dot{H}_{a,\\sigma}^{s+\\alpha}}^2d\\tau+\\int_0^t\\|\\theta(\\tau)\\|_{\\dot{H}_{a,\\sigma}^{s+\\beta}}^2\\,d\\tau \\leq C_{a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0,T^*},\\quad\\forall\\,t\\in[0,T^*).\n\\end{align}\n\nOn the other hand, take $L^2$-inner product of the Boussinesq equations (\\ref{micropolar}), with $u$ and $\\theta$, respectively, and integrate the results obtained over $[0,t]$ in order to be capable of writing the following inequality:\n\\begin{align}\\label{wilber32}\n\\frac{1}{2}\\frac{d}{dt}\\|(u,\\theta)(t)\\|_{L^2}^2+\\|(-\\Delta)^{\\frac{\\alpha}{2}}u(t)\\|_{L^2}^2+\\|(-\\Delta)^{\\frac{\\beta}{2}}\\theta(t)\\|_{L^2}^2 &\\leq \\|(u,\\theta)(t)\\|_{L^2}^{2},\\quad\\forall t\\in [0,T^*).\n\\end{align}\nBy Gronwall's Lemma, we have \n\\begin{align}\\label{normal2}\n \\|(u,\\theta)(t)\\|_{L^2}\\leq e^{T^*}\\|(u_0,\\theta_0)\\|_{L^2},\\quad\\forall t\\in [0,T^*).\n\\end{align}\nThereby, by using (\\ref{wilber12}) and (\\ref{normal2}), one infers\n\\begin{align*}\n\\frac{C_{s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2s(2\\alpha-1)}{3\\alpha}}}\\left(\\frac{C'_{\\sigma,s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2\\alpha-1}{3\\alpha\\sigma}}}\\right)^k \\leq \\|(u,\\theta)(t)\\|_{\\dot{H}^{s+\\frac{k}{2\\sigma}}}^2,\\quad\\forall t\\in [0,T^*).\n\\end{align*}\nAs a consequence, we have the inequality below:\n\\begin{align}\\label{wnovo1}\n\\frac{C_{s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2s(2\\alpha-1)}{3\\alpha}}}\\frac{\\left(\\frac{2aC'_{\\sigma,s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2\\alpha-1}{3\\alpha\\sigma}}}\\right)^k }{k!} &\\leq \\int_{\\mathbb{R}^3} \\frac{(2a|\\xi|^{\\frac{1}{\\sigma}})^k}{k!}|\\xi|^{2s}|(\\widehat{u},\\widehat{\\theta})(t)|^{2}\\,d\\xi,\\quad\\forall t\\in [0,T^*).\n\\end{align}\nHence, by using Monotone Convergence Theorem in the inequality (\\ref{wnovo1}), we deduce\n\\begin{align}\\label{wilber14}\n\\nonumber\\frac{C_{s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2s(2\\alpha-1)}{3\\alpha}}}\\sum_{k=2\\sigma_0+1}^{\\infty}\\frac{\\left(\\frac{2aC'_{\\sigma,s,\\alpha,u_0,\\theta_0,T^*}}{[e^{C_\\alpha(T^*-t)}-1]^{\\frac{2\\alpha-1}{3\\alpha\\sigma}}}\\right)^k }{k!} &\\leq \\int_{\\mathbb{R}^3} \\sum_{k=2\\sigma_0+1}^{\\infty}\\frac{(2a|\\xi|^{\\frac{1}{\\sigma}})^k}{k!}|\\xi|^{2s}|(\\widehat{u},\\widehat{\\theta})(t)|^{2}\\,d\\xi\\\\\n&\\leq \\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s}^2,\n\\end{align}\nfor all $t\\in [0,T^*)$, where $2\\sigma_0$ is the integer part of $2\\sigma\\mu$.",
    "post_theorem_intro_text_len": 7769,
    "post_theorem_intro_text": "The proof of this result relies on a fixed-point argument. However, for both the classical and the fractional Boussinesq equations, a major difficulty arises in handling the coupling terms $\\theta e_{3}$ and $(u \\cdot \\nabla)\\theta$, which demand delicate estimates within the chosen functional framework.\n\nNext, we investigate the qualitative behavior of the local solution provided by Theorem \\ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.\n\n\\begin{theorem}\\label{teoremaB}\nAssume that $a>0$, $\\sigma>1$, $\\alpha\\geq 1, \\beta \\geq 1, 0\\leq s<\\frac{3}{2}$ and $n\\in \\mathbb{N}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Consider that $(u,\\theta)\\in C([0,T^*);\\dot{H}_{a,\\sigma}^s)$ is the  solution\nfor the Boussinesq equations \\emph{(\\ref{micropolar})} in the maximal time interval $0\\leq t < T^*$  given in Theorem \\emph{\\ref{teoremaexistenciaB}}. If $T^*<\\infty$, then the following statements hold:\n      \\begin{enumerate}\n  \\item[\\textbf{\\emph{i)}}]  $\\displaystyle \\limsup_{t\\nearrow T^*} \\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^{(n-1)}},\\sigma}^s}=+\\infty$;\n  \\item[\\textbf{\\emph{ii)}}] $\\displaystyle\\int_{t}^{T^*} [\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(\\tau)\\|_{L^1}\n^{\\frac{2\\alpha}{2\\alpha-1}}+\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(\\tau)\\|_{L^1}^{\\frac{2\\beta}{2\\beta-1}}]\\,d\\tau=\\infty$;\n \\item[\\textbf{\\emph{iii)}}]  $\\displaystyle \\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\alpha}{2\\alpha-1}}\n    +\\|e^{\\frac{a}{\\sigma(\\sqrt{\\sigma})^{(n-1)}}|\\cdot|^{\\frac{1}{\\sigma}}}(\\widehat{u},\\widehat{\\theta})(t)\\|_{L^1}^{\\frac{2\\beta}{2\\beta-1}}\\geq  (e^{C(T^*-t)}-1)^{-1};$\n  \\item [\\textbf{\\emph{iv)}}] $\\displaystyle \\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^n},\\sigma}^s}^{\\frac{2\\alpha}{2\\alpha-1}}+\n\\|(u,\\theta)(t)\\|_{\\dot{H}_{\\frac{a}{(\\sqrt{\\sigma})^n},\\sigma}^s}^{\\frac{2\\beta}{2\\beta-1}}\\gtrsim(e^{C(T^{*}-t)}-1)^{-1}.$\n  \\end{enumerate}\nfor all $t\\in [0,T^{*})$ and $C>0$ is a positive constant.\n\\end{theorem}\n\n As a corollary, we obtain that the local solutions provided by Theorem \\ref{teoremaexistenciaB} exhibit exponential growth, which in turn ensures finite-time blow-up with an explicit exponential rate.\n\n\\begin{corollary}\\label{corolario}\nAssume that $a>0$, $\\sigma>1$, $\\alpha=\\beta\\geq 1 \\hbox{  and  } 0\\leq s<\\frac{3}{2}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s\\cap L^2$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Consider that $(u,\\theta)\\in C([0,T^*);\\dot{H}_{a,\\sigma}^s)$ is the  solution\nfor the Boussinesq equations \\emph{(\\ref{micropolar})} in the maximal time interval $0\\leq t < T^*$  given in Theorem \\emph{\\ref{teoremaexistenciaB}}. If $T^*<\\infty$, then the following statement holds:\n   $$\n   \\displaystyle\n\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s} \\gtrsim  {(e^{C_1(T^*-t)}-1)^{\\varrho_1}}\n\\exp\\{C_2(e^{C_1(T^*-t)}-1)^{\\varrho_2}\\},\n$$\nfor all \\(t \\in [0, T^{*})\\), where $\\varrho_1 := \\frac{(1 - 2\\alpha)\\,[\\,2(s\\sigma + \\sigma_{0}) + 1\\,]}{6\\alpha\\sigma}<0$, $\\varrho_2:=\\frac{1-2\\alpha}{3\\alpha\\sigma}<0$ and \\(2\\sigma_{0}\\) denotes the integer part of \\(2\\sigma\\mu\\) \\emph{(}with \\(\\mu > \\tfrac{3}{2}\\)\\emph{)}.  \nHere, $C_1=C_1(\\alpha)$ and $C_2=C_2(a,\\alpha,\\sigma,s,u_0,\\theta_0,T^*)$ are constants. In particular\n$$\n\\lim_{t \\nearrow T^*}\\|(u,\\theta)(t)\\|_{\\dot{H}_{a,\\sigma}^s} = +\\infty,\n$$\nexponentially.\n\\end{corollary}\n\n\\bigskip\n\n\\begin{remark}\nLet us point out  relevant  improvements presented by our main  results.\n\\begin{enumerate}\n  \\item \\underline{Solution spaces}: In this text, we have approached Gevrey classes as solution spaces. Thus, we have decided to compare our contributions to some results established by R.~Guterres, W.~G. Melo, N.~F. Rocha and T.~S.~R. Santos \\cite{robert} (see also \\cite{wilberr2,wilberr3,wilberr4,wilberr5,wilberr6,wilberr7,MR3504420,MR4632081,MR2169876,MR4369830} and references therein), which ones are similar to ours. First of all, Theorem \\ref{teoremaexistenciaB} presents weaker regularities for the mild solutions than Theorem $1.1$ obtained in \\cite{robert} (check also Theorem $1.4$ in \\cite{robert}, which is related to Theorem \\ref{teoremaB} and Corollary \\ref{corolario}). This fact occurs because we have applied Lemma \\ref{lemalorenz} instead of Lemma $2.3$  presented in \\cite{robert}, and have worked in a slightly different way as well  (see (\\ref{wilber17}) and (\\ref{w1}) below for more details). However, this choice led us to not consider the usual homogenous Sobolev spaces (see the proof of Lemma $2.9$ in \\cite{lorenz}). In addition, at last, it is worth to notice that $s<\\frac{3}{2}$ is our assumption due to the completeness of Sobolev-Gevrey space (see also Lemma \\ref{lemalorenz}) and, furthermore, $\\alpha,\\beta\\geq 1$ in Theorem \\ref{teoremaB} and Corollary \\ref{corolario} because of the use of Lemma \\ref{lemanovow1} (it is a technical issue).   \n  \\item \\underline{Boussinesq equations}: The term $\\theta e_3$ given in (\\ref{micropolar}) is the main mathematical difference between the Boussinesq equations and the generalized MHD equations studied by \\cite{robert} (and, consequently, the Navier-Stokes equations (\\ref{NS})) in this paper. Because of this term, we have made the use of a different fixed point theorem (see Lemma \\ref{pontofixo} and Lemma $2.1$ in \\cite{lorenz}). More specifically, we were supposed to add new equalities and estimates  in order to obtain our conclusions (see, for example, (\\ref{linearL}), (\\ref{linearL1eL2})--(\\ref{lwfinal}), (\\ref{3}), (\\ref{esqueci}), (\\ref{wilber19})--(\\ref{wilber23}), (\\ref{wilber4}), (\\ref{wilber24}), (\\ref{wilber25}), (\\ref{wilber26})--(\\ref{wilber9}), (\\ref{i)n=1}), (\\ref{wilber32}), (\\ref{normal2}) and (\\ref{wilber33})).\n  \\item \\underline{New lower bounds}: As a consequence of the results discussed in the last item above, we have proved new blow-up criteria for local mild solutions of the Boussinesq equations (\\ref{micropolar}) as, for instance, Theorem \\ref{teoremaB} \\textbf{iii)} and \\textbf{iv)}, and Corollary \\ref{corolario} (compare these results to Theorem $1.4$ iii), iv) and v) in \\cite{robert}). Lastly, it is important to point out that other information has been showed in this work, see Theorem \\ref{teoremaexistenciaB} and Theorem \\ref{teoremaB} \\textbf{i)} and \\textbf{ii)}.\n\\end{enumerate}\n\n\\end{remark}\n\n\\bigskip\n\n\\noindent\\textbf{Notations. }For  $(X,\\|\\cdot\\|_{X})$ be a normed space and $I \\subset \\mathbb{R}$ an interval. We define  \n$$\nC(I;X)=\\{f:I\\to X \\text{ continuous}\\}, \\qquad \n\\|f\\|_{L^{\\infty}(I;X)}=\\sup_{t\\in I}\\|f(t)\\|_{X},\n$$\nand for $T>0$, we  write $C_{T}(X)=C([0,T];X)$. For $1 \\le p < \\infty$, the Bochner space $L^{p}(I;X)$ is equipped with the norm  \n$$\n\\|f\\|_{L^{p}(I;X)}=\\left(\\int_{I}\\|f(t)\\|_{X}^{p}\\,dt\\right)^{1/p},\n$$\nand we denote $L_{T}^{p}(X)=L^{p}([0,T];X)$ with$\\footnote{Here, $\\|(f,g)\\|_{L^1}:=\\|f\\|_{L^1}+\\|g\\|_{L^1}$  and $\\|(f,g)\\|_{\\dot{H}_{a,\\sigma}^s}:=[\\|f\\|^{2}_{\\dot{H}_{a,\\sigma}^s}+\\|g\\|^{2}_{\\dot{H}_{a,\\sigma}^s}]^{\\frac{1}{2}}.$}$ $1\\leq p \\leq \\infty$. Furthermore, for the functions $f$ and $g=(g_1,g_2,g_3)$, the tensor product is  defined by $f \\otimes g := (g_{1}f, g_{2}f, g_{3}f)$.\n\n\\bigskip\n\n\\noindent\\textbf{Organization of the paper.}\nThe structure of the paper is as follows. \nIn Section~\\ref{secaonotacoes}, we present the auxiliary lemmas and technical tools required for the proofs of our main results. \nSection~$3$ is devoted to the proofs of Theorems~\\ref{teoremaexistenciaB} and~\\ref{teoremaB}, as well as Corollary~\\ref{corolario}, each of which is addressed in a separate subsection.",
    "sketch": "The post-theorem text states that “The proof of this result relies on a fixed-point argument.” It also highlights the main technical obstacle: “a major difficulty arises in handling the coupling terms $\\theta e_{3}$ and $(u \\cdot \\nabla)\\theta$, which demand delicate estimates within the chosen functional framework.”",
    "expanded_sketch": "The post-theorem text states that “The proof of this result relies on a fixed-point argument.” It also highlights the main technical obstacle: “a major difficulty arises in handling the coupling terms $\\theta e_{3}$ and $(u \\cdot \\nabla)\\theta$, which demand delicate estimates within the chosen functional framework.”",
    "expanded_theorem": "\\label{teoremaexistenciaB}\nAssume that $a>0$, $\\sigma>1$, $\\alpha>\\frac{1}{2}$, $\\beta>\\frac{1}{2}$ and $0\\leq s<\\frac{3}{2}$. Let $(u_0,\\theta_0)\\in \\dot{H}_{a,\\sigma}^s$ such that $\\hbox{\\emph{div}}\\,u_0=0$. Then,\n  there exist an instant $T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0$ and a unique mild solution $(u,\\theta)\\in C_T(\\dot{H}_{a,\\sigma}^s)$ for the Boussinesq equations\n\\begin{equation}\\label{micropolar}\n\\left\\{\n\\begin{array}{l}\\nu_t\n\\;\\!+\\,\\nu \\cdot \\nabla u\n\\,+\\,\\nabla \\;\\!p \\:\\!\\,+\\,  \\:\\!(-\\Delta)^\\alpha u\n\\;=\\;\\ntheta e_3, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\theta_t\n\\;\\!+\\,\\nu \\cdot \\nabla \\theta\\,+\\, \\,(-\\Delta)^\\beta \\theta\n\\;=\\;\\n0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\n\\mbox{div}\\:u  \\;=\\; 0, \\quad x\\in \\mathbb{R}^3, \\quad t > 0,\\\\\\nu(x,0) \\,=\\, u_0(x),\n\\;\\,\\ntheta(x,0) \\,=\\, \\theta_0(x), \\quad x\\in \\mathbb{R}^3,\n\\end{array}\n\\right.\n\\end{equation}\nthat satisfies\n    \\begin{align*}\n  &\\|(u,\\theta)\\|_{L^\\infty_T(\\dot{H}_{a,\\sigma}^{s})}\n\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n  \\end{align*}",
    "theorem_type": [
      "Uniqueness",
      "Existence"
    ],
    "mcq": {
      "question": "Let\n\\[\n\\dot H_{a,\\sigma}^s(\\mathbb R^3)=\\Big\\{f\\in\\mathcal S' : \\widehat f\\in L^1_{\\mathrm{loc}},\\ \\|f\\|_{\\dot H_{a,\\sigma}^s}:=\\Big(\\int_{\\mathbb R^3}|\\xi|^{2s}e^{2a|\\xi|^{1/\\sigma}}|\\widehat f(\\xi)|^2\\,d\\xi\\Big)^{1/2}<\\infty\\Big\\},\n\\]\nand write \\(C_T(\\dot H_{a,\\sigma}^s)=C([0,T];\\dot H_{a,\\sigma}^s)\\). Consider the three-dimensional Boussinesq system on \\(\\mathbb R^3\\times(0,\\infty)\\):\n\\[\n\\begin{cases}\nu_t+u\\cdot\\nabla u+\\nabla p+(-\\Delta)^\\alpha u=\\theta e_3,\\\\\n\\theta_t+u\\cdot\\nabla\\theta+(-\\Delta)^\\beta\\theta=0,\\\\\n\\operatorname{div}u=0,\\\\\nu(x,0)=u_0(x),\\quad \\theta(x,0)=\\theta_0(x),\n\\end{cases}\n\\]\nwhere \\(e_3=(0,0,1)\\). A mild solution means a pair \\((u,\\theta)\\) satisfying the corresponding Duhamel formulas\n\\[\nu(t)=e^{-t(-\\Delta)^\\alpha}u_0-\\int_0^t e^{-(t-\\tau)(-\\Delta)^\\alpha}\\mathbb P(u\\cdot\\nabla u)(\\tau)\\,d\\tau+\\int_0^t e^{-(t-\\tau)(-\\Delta)^\\alpha}\\mathbb P(\\theta e_3)(\\tau)\\,d\\tau,\n\\]\n\\[\n\\theta(t)=e^{-t(-\\Delta)^\\beta}\\theta_0-\\int_0^t e^{-(t-\\tau)(-\\Delta)^\\beta}(u\\cdot\\nabla\\theta)(\\tau)\\,d\\tau,\n\\]\nfor \\(t>0\\), where \\(\\mathbb P\\) is the Leray projector. If \\(a>0\\), \\(\\sigma>1\\), \\(\\alpha>\\tfrac12\\), \\(\\beta>\\tfrac12\\), and \\(0\\le s<\\tfrac32\\), and if \\((u_0,\\theta_0)\\in \\dot H_{a,\\sigma}^s\\) with \\(\\operatorname{div}u_0=0\\), which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a time \\(T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0\\) and a unique mild solution \\((u,\\theta)\\in C([0,T];\\dot H_{a,\\sigma}^s)\\) of the above Boussinesq system with initial data \\((u_0,\\theta_0)\\). Moreover,\n\\[\n\\|(u,\\theta)\\|_{L_T^\\infty(\\dot H_{a,\\sigma}^s)}\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a time \\(T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0\\) and a unique mild solution \\((u,\\theta)\\in C([0,T];\\dot H_{a,\\sigma}^s)\\) of the above Boussinesq system with initial data \\((u_0,\\theta_0)\\) for every \\(0\\le s\\le \\tfrac32\\). Moreover,\n\\[\n\\|(u,\\theta)\\|_{L_T^\\infty(\\dot H_{a,\\sigma}^s)}\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a time \\(T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0\\) and at least one mild solution \\((u,\\theta)\\in C([0,T];\\dot H_{a,\\sigma}^s)\\) of the above Boussinesq system with initial data \\((u_0,\\theta_0)\\)."
        },
        {
          "label": "D",
          "text": "There exists a time \\(T=T(a,\\sigma,s,\\alpha,\\beta)>0\\), depending only on \\(a,\\sigma,s,\\alpha,\\beta\\), such that for every \\((u_0,\\theta_0)\\in \\dot H_{a,\\sigma}^s\\) with \\(\\operatorname{div}u_0=0\\), there is a unique mild solution \\((u,\\theta)\\in C([0,T];\\dot H_{a,\\sigma}^s)\\) of the above Boussinesq system. Moreover,\n\\[\n\\|(u,\\theta)\\|_{L_T^\\infty(\\dot H_{a,\\sigma}^s)}\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}+T^{1-\\frac{1}{2\\beta}}}.\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a time \\(T=T(a,\\sigma,s,\\alpha,\\beta,u_0,\\theta_0)>0\\) and a unique mild solution \\((u,\\theta)\\in C([0,T];\\dot H_{a,\\sigma}^s)\\) of the above Boussinesq system with initial data \\((u_0,\\theta_0)\\). Moreover,\n\\[\n\\|(u,\\theta)\\|_{L_T^\\infty(\\dot H_{a,\\sigma}^s)}\\lesssim \\frac{1-T}{T^{1-\\frac{1}{2\\alpha}}}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "endpoint regularity range for s",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "uniqueness and a priori bound",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of lifespan on the initial data",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "two-parameter dissipation structure in the norm estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses, definitions, and the mild formulation, but it does not explicitly reveal the correct conclusion. The correct option is not stated or strongly implied beyond the general expectation of a local well-posedness result."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: under the listed assumptions, the student is asked to pick the exact theorem statement. The correct choice is a near-direct restatement of the result rather than an application to a new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants involving endpoint regularity, uniqueness versus existence, dependence of the lifespan on initial data, and the precise a priori bound. However, the task is still mainly recognition of the exact theorem rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: extending to the endpoint case, weakening uniqueness, making the lifespan uniform in the data, or dropping one dissipation parameter from the estimate. They are distinct and well aligned with realistic misconceptions."
      },
      "total_score": 5,
      "overall_assessment": "A solid multiple-choice theorem-identification item with strong distractors and little answer leakage, but it is largely a direct theorem restatement rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.08817v1",
    "paper_link": "http://arxiv.org/abs/2512.08817v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex (see \\Cref{def:nonconvex}),\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$.",
      "start_pos": 480016,
      "end_pos": 480703,
      "label": "thm:main"
    },
    "ref_dict": {
      "def:latticeword": "\\begin{definition}\\label{def:latticeword}\n    Given a sequence of sorted clasps $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, a lattice word $L$ has no \\emph{$\\underline C$-descents} if $L = L_1\\cdots L_m$ with $\\type(L_i) = \\underline c_i, \\forall i$ and each $L_i$ has no descents.\n    The set of all lattice words with no $\\underline C$-descents is denoted $\\mathsf L(\\underline C)$, and the set of all balanced lattice words with no $\\underline C$-descents is denoted $\\BL(\\underline C)$.\n\\end{definition}",
      "thm:intro-swap": "\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\swap$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm}",
      "thm:main": "\\begin{thm}\n    \\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\inv(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex (see \\Cref{def:nonconvex}),\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\inv\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$. \n\\end{thm}",
      "thm:intro-sorted": "\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\L(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm}",
      "def:nonconvex": "\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 5574,
    "pre_theorem_intro_text": "\\subsection{Webs and web bases}\n\nConsider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nThere are in general many relations (see \\cite{Cautis-Kamnitzer-Morrison}) between the invariants $[W]_q$ of webs with boundary conditions $\\underline{a}$. A \\emph{web basis} is a subset of the web invariants forming a basis. Kuperberg gave a $U_q(\\mathfrak{sl}_3)$ web basis consisting of the \\emph{non-elliptic} webs. Beyond early applications for quantum link invariants \\cite{Khovanov}, the non-elliptic basis has also found application to skein modules \\cite{Le.Sikora}, dimer models \\cite{Douglas-Kenyon-Shi}, and dynamical algebraic combinatorics \\cite{Petersen-Pylyavskyy-Rhoades}, among other areas. \n\nSince Kuperberg's work, much effort has gone into constructing higher-rank web bases, with additional special properties. In \\cite{GPPSS-sl4}, the first \\emph{rotation-invariant} $U_q(\\mathfrak{sl}_4)$ web basis was constructed using \\emph{hourglass plabic graphs}. This is the basis of \\emph{(top) fully reduced} web invariants $\\W_{\\underline a}$. Fully reduced hourglass plabic graphs also recover Kuperberg's basis for $U_q(\\mathfrak{sl}_3)$.\n\nAn obvious limitation of the fully reduced web bases is that they are heretofore only known for tensor products of irreducibles indexed by fundamental weights. To extend this basis to arbitrary tensor products of irreducibles, we need to understanding the \\emph{clasping} of these webs. Kuperberg gave clasping rules for $U_q(\\mathfrak{sl}_3)$, and these clasped webs have appeared, for example, in well-known conjectures of Fomin--Pylyavskyy \\cite{Fomin-Pylyavskyy-advances} relating basis webs to cluster algebras.\n\n\\subsection{Clasped webs}\n\\emph{Clasped webs} provide for an extension of the diagrammatic calculus to spaces of morphisms between tensor products of general finite-dimensional (type-$1$) irreducible representations $V_q(\\lambda)$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.",
    "context": "Consider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}",
    "full_context": "Consider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{remark}\n The fact that the web invariants not killed by the projection $\\pi_{\\underline C}$ form a basis in $\\inv\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \\cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \\cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties. \n\\end{remark}\n\n\\begin{remark}\n \\label{rem:schur}\n For any sequence of weights $\\underline \\lambda = (\\lambda_1,\\dots, \\lambda_m)$, the Littlewood--Richardson rule implies that the coefficient of $s_\\mu$ when $s_{\\lambda_1}\\cdots s_{\\lambda_m}$ when expressed in the Schur polynomial basis is equal to the number of $\\underline \\lambda$-Littlewood--Richardson tableaux of shape $\\mu$. In particular, if $\\lambda_i=\\weight(\\underline c_i)$, then\n \\[|\\RT(\\underline C)| = \\dim \\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right).\\]\n\\end{remark}\n\nWe are now ready to prove the theorem for sorted clasps.\n\\begin{thm}\\label{thm:sorted}\n Fix boundary conditions $\\underline a \\in [3]^n$ and a \\emph{sorted} clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, and let $\\lambda_i = \\weight(\\underline c_i)$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C} \\coloneqq \\{\\pi_{\\underline C}([W]) \\mid [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker\\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent for $W\\in \\mathcal W_{\\underline a}$:\n \\begin{enumerate}\n \\item $[W] \\notin \\ker\\pi_{\\underline C}$.\n \\item $W$ is non-convex.\n \\item There are no local boundary configurations from \\Cref{fig:bad_configs} occurring within any clasp of $W$.\n \\item The lattice word $\\L(W) = \\partial\\sep_W$ has no $\\underline C$-descents.\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\n\\begin{thm}\n Let $\\underline C = (\\underline c_1, \\dots, \\underline c_m)$ be a clasp sequence on $\\underline a\\in [3]^n$, with $\\weight(\\underline c_i) = \\lambda_i$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C}\\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker \\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent:\n \\begin{enumerate}\n \\item $[W]\\notin \\ker\\pi_{\\underline C}$,\n \\item $W$ is non-convex,\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n \\label{thm:main_general}\n\\end{thm}\n\n\\begin{thm}\\label{thm:sorted3}\n Fix boundary conditions $\\underline a \\in [2]^n$ and a {sorted} clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, and let $\\lambda_i = \\weight(\\underline c_i)$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C} \\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker\\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent for $W\\in \\mathcal W_{\\underline a}$:\n \\begin{enumerate}\n \\item $[W] \\notin \\ker\\pi_{\\underline C}$.\n \\item $W$ is non-convex.\n \\item There are no local boundary configurations from \\Cref{fig:badconf_3} occurring within any clasp of $W$.\n \\item The lattice word $\\L(W) = \\partial\\sep_W$ has no $\\underline C$-descents.\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\n\\begin{proof}\n By \\Cref{prop:nonconvex}, $\\W_{\\underline C}$ spans the invariant space, and we show that $|\\W_{\\underline C}|\\le \\dim\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ while simultaneously proving the equivalence of (1), (2), (3) and (4).(1)$\\implies$ (2) is \\Cref{prop:nonconvex}.(2)$\\implies$(3) is seen directly from \\Cref{fig:badconf_3}. (3)$\\implies$(4) is proven exactly like \\Cref{prop:descent}, using the KK-growth rules instead. There are far fewer cases here, since each descent is of one of the following types:\n \\begin{itemize}\n \\item $ab$ with $a<b$,\n \\item $\\overline{ab}$ with $a>b$, or\n \\item $1\\overline 1$.\n \\end{itemize}\n\n\\begin{thm}\n Let $\\underline C = (\\underline c_1, \\dots, \\underline c_m)$ be a clasp sequence on $\\underline a\\in [2]^n$, with $\\weight(\\underline c_i) = \\lambda_i$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C}\\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker \\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent:\n \\begin{enumerate}\n \\item $[W]\\notin \\ker\\pi_{\\underline C}$,\n \\item $W$ is non-convex,\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\\end{thm}\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}",
    "post_theorem_intro_text_len": 3257,
    "post_theorem_intro_text": "\\begin{remark}\n    The fact that the web invariants not killed by the projection $\\pi_{\\underline C}$ form a basis in $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \\cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \\cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties. \n\\end{remark}\n\nIn the case when $\\underline C$ is \\emph{sorted}, i.e., each $\\underline c_i$ is a weakly increasing tuple, the following theorem gives us more criteria to easily check for non-convexity.\n\n\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\mathcal{L}(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm}\n\nThe map $\\partial\\mathrm{sep}$ appearing in \\Cref{thm:intro-sorted}(5) is the bijection from \\cite{GPPSS-sl4} between elements of $\\W_{\\underline a}$ and (lattice words of) the associated \\emph{fluctuating tableaux} \\cite{fluctuating-paper}, certain generalizations of standard Young tableaux (which correspond to the case $\\underline{a}=(1,1,\\ldots,1)$).\n\nA key tool in the proof of \\Cref{thm:main} is the \\emph{swap} map of \\Cref{sec:sorting}, which is of interest even outside the context of clasping. It gives bijections between the fully reduced web bases of \\cite{GPPSS-sl4} for any ordering of the same tensor factors.\n\n\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\mathsf{swap}$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm}\n\nFor ease of exposition, in most of the paper we deal only with the classical groups $\\SL_r$. All of our results apply also to invariants of the quantum group $U_q(\\mathfrak{sl}_r)$, but this extension will be straightforward. \n\n\\subsection{Outline}\n\nIn \\Cref{sec:prelim} we recall background on web invariants and on the combinatorial constructions introduced in \\cite{GPPSS-sl4}. In \\Cref{sec:clasped-webs} we set up some machinery for clasping webs. In \\Cref{sec:descents} we establish a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web. In \\Cref{Sec:Fund} we show that lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space. This is applied in \\Cref{sec:main-sorted-proof} to prove \\Cref{thm:intro-sorted}. \\Cref{sec:sorting} introduces the swap map which is used to prove \\Cref{thm:intro-swap} and thereby \\Cref{thm:main}, after some verifications for $r=2$ and $3$, which take place in \\Cref{sec:r-2-3}.",
    "sketch": "A proof outline for \\Cref{thm:main} is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \\cite{GPPSS-sl4} (\\Cref{sec:prelim}). Next it develops “some machinery for clasping webs” (\\Cref{sec:clasped-webs}). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (\\Cref{sec:descents}). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (\\Cref{Sec:Fund}); this is used to prove the sorted case \\Cref{thm:intro-sorted} (\\Cref{sec:main-sorted-proof}). Finally, a “key tool in the proof of \\Cref{thm:main} is the \\emph{swap} map” (\\Cref{sec:sorting}), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving \\Cref{thm:intro-swap}) and “thereby \\Cref{thm:main}, after some verifications for $r=2$ and $3$” carried out in \\Cref{sec:r-2-3}.",
    "expanded_sketch": "A proof outline for the main theorem is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \\cite{GPPSS-sl4} (Next it proves these preliminary results.). Next it develops “some machinery for clasping webs” (Next it proves these results.). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (Next it proves these results.). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (Next it proves these results.); this is used to prove the following theorem.\n\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\L(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm} (Next it proves these results.). Finally, a “key tool in the proof of the main theorem is the \\emph{swap} map” (Next it proves these results.), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving the following theorem.\n\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\swap$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm} ) and “thereby the main theorem, after some verifications for $r=2$ and $3$” carried out in Next it proves these results..",
    "expanded_theorem": "\\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex, where:\n        \\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $r\\in\\{2,3,4\\}$. Let $\\mathcal W_{\\underline a}$ be the fully reduced web basis of $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline C=(\\underline c_1,\\dots,\\underline c_m)$ be a clasp sequence for the boundary conditions $\\underline a$, with $\\lambda_i=\\mathrm{wt}(\\underline c_i)$. For a fully reduced hourglass plabic graph $W$, say that $W$ is non-convex with respect to $\\underline C$ if for every $i\\in[m]$ and every $\\underline c_i$-cut path $\\gamma$, one has $\\mathrm{wt}(\\gamma)\\ge \\mathrm{wt}(\\underline c_i)$ in the dominance order. Which statement exactly describes the web basis elements whose clasped images under $\\pi_{\\underline C}$ give the invariant space $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$?",
      "correct_choice": {
        "label": "A",
        "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; for each clasp $\\underline c_i$ there exists a $\\underline c_i$-cut path $\\gamma$ with $\\mathrm{wt}(\\gamma)\\ge \\mathrm{wt}(\\underline c_i)$ in the dominance order. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        },
        {
          "label": "C",
          "text": "The clasped web invariants $\\pi_{\\underline C}([W]_q)$ coming from basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ with $[W]_q\\notin \\ker \\pi_{\\underline C}$ span $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$; in particular, every web satisfying the equivalent conditions $[W]_q\\notin \\ker \\pi_{\\underline C}$, non-convexity, and having no trip that starts and ends in the same clasp contributes such a spanning vector."
        },
        {
          "label": "D",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; $W$ has no trip whose two endpoints lie in adjacent clasps. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        },
        {
          "label": "E",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is partially convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "forall-cut-path quantifier in non-convexity",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "basis conclusion weakened to spanning conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "trip obstruction changed from same clasp to adjacent clasps",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "non-convex replaced by partially convex",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for an equivalent condition plus the resulting basis statement, without naming the key conclusion about trips in the same clasp."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem presents the setup and asks for the statement equivalent to those conditions and the accompanying basis result. It is very close to restating the theorem rather than posing an independent problem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact theorem statement from nearby variants such as 'spans' versus 'forms a basis' or altered endpoint conditions. However, the task is mainly precise recall/recognition rather than substantive generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they vary quantifier dependence, weaken 'basis' to 'spans,' or subtly alter the forbidden-trip condition. These reflect realistic failure modes in recalling or parsing the theorem."
      },
      "total_score": 5,
      "overall_assessment": "A technically strong but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses good distractors, but it is largely tautological and only moderately tests genuine reasoning."
    }
  },
  {
    "id": "2512.08817v1",
    "paper_link": "http://arxiv.org/abs/2512.08817v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex (see \\Cref{def:nonconvex}),\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$.",
      "start_pos": 480016,
      "end_pos": 480703,
      "label": "thm:main"
    },
    "ref_dict": {
      "def:latticeword": "\\begin{definition}\\label{def:latticeword}\n    Given a sequence of sorted clasps $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, a lattice word $L$ has no \\emph{$\\underline C$-descents} if $L = L_1\\cdots L_m$ with $\\type(L_i) = \\underline c_i, \\forall i$ and each $L_i$ has no descents.\n    The set of all lattice words with no $\\underline C$-descents is denoted $\\mathsf L(\\underline C)$, and the set of all balanced lattice words with no $\\underline C$-descents is denoted $\\BL(\\underline C)$.\n\\end{definition}",
      "thm:intro-swap": "\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\swap$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm}",
      "thm:main": "\\begin{thm}\n    \\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\inv(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex (see \\Cref{def:nonconvex}),\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\inv\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$. \n\\end{thm}",
      "thm:intro-sorted": "\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\L(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm}",
      "def:nonconvex": "\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 5574,
    "pre_theorem_intro_text": "\\subsection{Webs and web bases}\n\nConsider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nThere are in general many relations (see \\cite{Cautis-Kamnitzer-Morrison}) between the invariants $[W]_q$ of webs with boundary conditions $\\underline{a}$. A \\emph{web basis} is a subset of the web invariants forming a basis. Kuperberg gave a $U_q(\\mathfrak{sl}_3)$ web basis consisting of the \\emph{non-elliptic} webs. Beyond early applications for quantum link invariants \\cite{Khovanov}, the non-elliptic basis has also found application to skein modules \\cite{Le.Sikora}, dimer models \\cite{Douglas-Kenyon-Shi}, and dynamical algebraic combinatorics \\cite{Petersen-Pylyavskyy-Rhoades}, among other areas. \n\nSince Kuperberg's work, much effort has gone into constructing higher-rank web bases, with additional special properties. In \\cite{GPPSS-sl4}, the first \\emph{rotation-invariant} $U_q(\\mathfrak{sl}_4)$ web basis was constructed using \\emph{hourglass plabic graphs}. This is the basis of \\emph{(top) fully reduced} web invariants $\\W_{\\underline a}$. Fully reduced hourglass plabic graphs also recover Kuperberg's basis for $U_q(\\mathfrak{sl}_3)$.\n\nAn obvious limitation of the fully reduced web bases is that they are heretofore only known for tensor products of irreducibles indexed by fundamental weights. To extend this basis to arbitrary tensor products of irreducibles, we need to understanding the \\emph{clasping} of these webs. Kuperberg gave clasping rules for $U_q(\\mathfrak{sl}_3)$, and these clasped webs have appeared, for example, in well-known conjectures of Fomin--Pylyavskyy \\cite{Fomin-Pylyavskyy-advances} relating basis webs to cluster algebras.\n\n\\subsection{Clasped webs}\n\\emph{Clasped webs} provide for an extension of the diagrammatic calculus to spaces of morphisms between tensor products of general finite-dimensional (type-$1$) irreducible representations $V_q(\\lambda)$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.",
    "context": "Consider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}",
    "full_context": "Consider a tensor product $V_q^{\\underline{a}} = \\bigotimes_{i=1}^n V_q(\\omega_{a_i})$ of irreducible complex representations of $U_q(\\mathfrak{sl}_r)$ indexed by fundamental weights $\\omega_1,\\ldots, \\omega_{r-1}$. \\emph{Webs}, introduced by Kuperberg \\cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}'},V_q^{\\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:\n\\[\n\\mathsf{Inv}(V_q^{\\underline{a}}) \\coloneqq \\hom_{U_q(\\mathfrak{sl}_r)}(V_q^{\\underline{a}},\\mathbb{C}(q)).\n\\]\n\nWebs $W$ representing elements $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\\ldots,b_n$ are each incident to a single edge, colored $\\omega_{a_i}$. For this reason, we call $\\underline{a}$ the \\emph{boundary conditions} of the web. We follow the conventions (of e.g. \\cite{Sikora} and \\cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.\n\nFix a partition of $[n]= I_1\\sqcup \\cdots \\sqcup I_m$, with each $I_i$ an interval. This defines the \\emph{clasp sequence} $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ where $\\underline c_i = (a_j)_{j\\in I_i} \\in [r-1]^{|I_i|}$ is the $i$-th \\emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\\{b_j\\ |\\ j\\in I_i\\}$ also as the $i$-th clasp, with $\\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\\underline c_i$ is defined to be $\\mathsf{wt}(\\underline c_i) = \\sum_{j\\in I_i}\\omega_{a_j}\\in \\Lambda^+,$ where $\\Lambda^+$ is the set of dominant integral weights for $G=\\SL_r(\\mathbb{C})$.\n\nFix a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ for boundary conditions $\\underline a$, and let $\\lambda_i = \\mathsf{wt}(\\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\\mathfrak{sl}_r)$-representations $V_q(\\lambda_i)\\xhookrightarrow{} V_q^{\\underline c_i}.$ This induces an inclusion of $U_q(\\mathfrak{sl}_r)$-representations\n\\[\n\\bigotimes_{i=1}^m V_q(\\lambda_i) \\xhookrightarrow{} \\bigotimes_{i=1}^m V_q^{\\underline c_i} = V_q^{\\underline a}.\n\\]\nTherefore, we obtain a surjective map of invariant spaces\n\\[\\pi_{\\underline C}:\\mathsf{Inv}(V_q^{\\underline a}) \\to \\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right).\\]\nThe image $\\pi_{\\underline{C}}([W]_q)$ of a web invariant $[W]_q \\in \\mathsf{Inv}(V_q^{\\underline{a}})$ is a \\emph{clasped web invariant}.\n\n\\subsection{Webs as hourglass plabic graphs}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}\n\n\\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \\cite{two-column, GPPSS-sl4}; see \\Cref{sec:hpg-background}. In these, edges colored by $\\omega_{a}$ are drawn as twisted ``hourglass\" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \\emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \\cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \\cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\\pi_{\\underline C}$.\n\n\\begin{remark}\n The fact that the web invariants not killed by the projection $\\pi_{\\underline C}$ form a basis in $\\inv\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \\cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \\cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties. \n\\end{remark}\n\n\\begin{remark}\n \\label{rem:schur}\n For any sequence of weights $\\underline \\lambda = (\\lambda_1,\\dots, \\lambda_m)$, the Littlewood--Richardson rule implies that the coefficient of $s_\\mu$ when $s_{\\lambda_1}\\cdots s_{\\lambda_m}$ when expressed in the Schur polynomial basis is equal to the number of $\\underline \\lambda$-Littlewood--Richardson tableaux of shape $\\mu$. In particular, if $\\lambda_i=\\weight(\\underline c_i)$, then\n \\[|\\RT(\\underline C)| = \\dim \\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right).\\]\n\\end{remark}\n\nWe are now ready to prove the theorem for sorted clasps.\n\\begin{thm}\\label{thm:sorted}\n Fix boundary conditions $\\underline a \\in [3]^n$ and a \\emph{sorted} clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, and let $\\lambda_i = \\weight(\\underline c_i)$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C} \\coloneqq \\{\\pi_{\\underline C}([W]) \\mid [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker\\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent for $W\\in \\mathcal W_{\\underline a}$:\n \\begin{enumerate}\n \\item $[W] \\notin \\ker\\pi_{\\underline C}$.\n \\item $W$ is non-convex.\n \\item There are no local boundary configurations from \\Cref{fig:bad_configs} occurring within any clasp of $W$.\n \\item The lattice word $\\L(W) = \\partial\\sep_W$ has no $\\underline C$-descents.\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\n\\begin{thm}\n Let $\\underline C = (\\underline c_1, \\dots, \\underline c_m)$ be a clasp sequence on $\\underline a\\in [3]^n$, with $\\weight(\\underline c_i) = \\lambda_i$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C}\\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker \\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent:\n \\begin{enumerate}\n \\item $[W]\\notin \\ker\\pi_{\\underline C}$,\n \\item $W$ is non-convex,\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n \\label{thm:main_general}\n\\end{thm}\n\n\\begin{thm}\\label{thm:sorted3}\n Fix boundary conditions $\\underline a \\in [2]^n$ and a {sorted} clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$, and let $\\lambda_i = \\weight(\\underline c_i)$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C} \\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker\\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent for $W\\in \\mathcal W_{\\underline a}$:\n \\begin{enumerate}\n \\item $[W] \\notin \\ker\\pi_{\\underline C}$.\n \\item $W$ is non-convex.\n \\item There are no local boundary configurations from \\Cref{fig:badconf_3} occurring within any clasp of $W$.\n \\item The lattice word $\\L(W) = \\partial\\sep_W$ has no $\\underline C$-descents.\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\n\\begin{proof}\n By \\Cref{prop:nonconvex}, $\\W_{\\underline C}$ spans the invariant space, and we show that $|\\W_{\\underline C}|\\le \\dim\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ while simultaneously proving the equivalence of (1), (2), (3) and (4).(1)$\\implies$ (2) is \\Cref{prop:nonconvex}.(2)$\\implies$(3) is seen directly from \\Cref{fig:badconf_3}. (3)$\\implies$(4) is proven exactly like \\Cref{prop:descent}, using the KK-growth rules instead. There are far fewer cases here, since each descent is of one of the following types:\n \\begin{itemize}\n \\item $ab$ with $a<b$,\n \\item $\\overline{ab}$ with $a>b$, or\n \\item $1\\overline 1$.\n \\end{itemize}\n\n\\begin{thm}\n Let $\\underline C = (\\underline c_1, \\dots, \\underline c_m)$ be a clasp sequence on $\\underline a\\in [2]^n$, with $\\weight(\\underline c_i) = \\lambda_i$. Then a basis for $\\inv_G\\left(\\bigotimes_{i=1}^m V(\\lambda_i)\\right)$ is given by\n \\[\\mathcal W_{\\underline C}\\coloneqq \\{\\pi_{\\underline C}([W]): [W]\\in \\mathcal W_{\\underline a}\\setminus \\ker \\pi_{\\underline C}\\}.\\]\n Moreover, the following are equivalent:\n \\begin{enumerate}\n \\item $[W]\\notin \\ker\\pi_{\\underline C}$,\n \\item $W$ is non-convex,\n \\item $W$ has no trips that start and end in the same clasp.\n \\end{enumerate}\n\\end{thm}\n\n\\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}",
    "post_theorem_intro_text_len": 3257,
    "post_theorem_intro_text": "\\begin{remark}\n    The fact that the web invariants not killed by the projection $\\pi_{\\underline C}$ form a basis in $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \\cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \\cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties. \n\\end{remark}\n\nIn the case when $\\underline C$ is \\emph{sorted}, i.e., each $\\underline c_i$ is a weakly increasing tuple, the following theorem gives us more criteria to easily check for non-convexity.\n\n\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\mathcal{L}(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm}\n\nThe map $\\partial\\mathrm{sep}$ appearing in \\Cref{thm:intro-sorted}(5) is the bijection from \\cite{GPPSS-sl4} between elements of $\\W_{\\underline a}$ and (lattice words of) the associated \\emph{fluctuating tableaux} \\cite{fluctuating-paper}, certain generalizations of standard Young tableaux (which correspond to the case $\\underline{a}=(1,1,\\ldots,1)$).\n\nA key tool in the proof of \\Cref{thm:main} is the \\emph{swap} map of \\Cref{sec:sorting}, which is of interest even outside the context of clasping. It gives bijections between the fully reduced web bases of \\cite{GPPSS-sl4} for any ordering of the same tensor factors.\n\n\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\mathsf{swap}$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm}\n\nFor ease of exposition, in most of the paper we deal only with the classical groups $\\SL_r$. All of our results apply also to invariants of the quantum group $U_q(\\mathfrak{sl}_r)$, but this extension will be straightforward. \n\n\\subsection{Outline}\n\nIn \\Cref{sec:prelim} we recall background on web invariants and on the combinatorial constructions introduced in \\cite{GPPSS-sl4}. In \\Cref{sec:clasped-webs} we set up some machinery for clasping webs. In \\Cref{sec:descents} we establish a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web. In \\Cref{Sec:Fund} we show that lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space. This is applied in \\Cref{sec:main-sorted-proof} to prove \\Cref{thm:intro-sorted}. \\Cref{sec:sorting} introduces the swap map which is used to prove \\Cref{thm:intro-swap} and thereby \\Cref{thm:main}, after some verifications for $r=2$ and $3$, which take place in \\Cref{sec:r-2-3}.",
    "sketch": "A proof outline for \\Cref{thm:main} is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \\cite{GPPSS-sl4} (\\Cref{sec:prelim}). Next it develops “some machinery for clasping webs” (\\Cref{sec:clasped-webs}). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (\\Cref{sec:descents}). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (\\Cref{Sec:Fund}); this is used to prove the sorted case \\Cref{thm:intro-sorted} (\\Cref{sec:main-sorted-proof}). Finally, a “key tool in the proof of \\Cref{thm:main} is the \\emph{swap} map” (\\Cref{sec:sorting}), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving \\Cref{thm:intro-swap}) and “thereby \\Cref{thm:main}, after some verifications for $r=2$ and $3$” carried out in \\Cref{sec:r-2-3}.",
    "expanded_sketch": "A proof outline for the main theorem is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \\cite{GPPSS-sl4} (Next it proves these preliminary results.). Next it develops “some machinery for clasping webs” (Next it proves these results.). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (Next it proves these results.). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (Next it proves these results.); this is used to prove the following theorem.\n\\begin{thm}\n    \\label{thm:intro-sorted}\n    In the setting of \\Cref{thm:main}, assume further that $\\underline C$ is sorted. Then the following additional conditions are equivalent to \\Cref{thm:main}(1)-(3):\n    \\begin{enumerate}\n        \\item[(4)] No clasp contains a ``bad\" local configuration (see \\Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).\n        \\item[(5)] The lattice word $\\L(W) = \\partial\\mathrm{sep}(W)$ has no $\\underline C$-descents (see \\Cref{def:latticeword}).\n    \\end{enumerate}\n\\end{thm} (Next it proves these results.). Finally, a “key tool in the proof of the main theorem is the \\emph{swap} map” (Next it proves these results.), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving the following theorem.\n\\begin{thm}\n\\label{thm:intro-swap}\n    The map $\\swap$ gives bijections between the fully reduced web bases $\\W_{\\underline a}$ for all permutations of the boundary conditions $\\underline{a}$.\n\\end{thm} ) and “thereby the main theorem, after some verifications for $r=2$ and $3$” carried out in Next it proves these results..",
    "expanded_theorem": "\\label{thm:main}\n    Let $r \\in \\{2,3,4\\}$, let $\\W_{\\underline a}$ be the fully reduced web basis for $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline{C}$ be a clasp sequence for $\\underline{a}$ with weights $\\lambda_1,\\ldots,\\lambda_m$. Then the following are equivalent for $[W]_q \\in \\W_{\\underline a}$:\n    \\begin{enumerate}\n        \\item $[W]_q\\notin \\ker\\pi_{\\underline C}$,\n        \\item $W$ is non-convex, where:\n        \\begin{definition}\\label{def:nonconvex}\n    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\\underline C = (\\underline c_1,\\dots, \\underline c_m)$ is \\emph{non-convex} (with respect to $\\underline C$) if for each $i\\in [m]$ and every $\\underline c_i$-cut path $\\gamma$, $\\weight(\\gamma)\\ge \\weight(\\underline c_i)$ in the dominance ordering on $\\Lambda^+$. We say that $W$ is \\emph{partially convex} if it is not non-convex.\n\\end{definition}\n        \\item $W$ has no trips that start and end in the same clasp.\n    \\end{enumerate}\n    Moreover, the clasped web invariants for these webs $W$ form a basis for $\\mathsf{Inv}\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $r\\in\\{2,3,4\\}$. Let $\\mathcal W_{\\underline a}$ be the fully reduced web basis of $\\mathsf{Inv}(V_q^{\\underline a})$, and let $\\underline C=(\\underline c_1,\\dots,\\underline c_m)$ be a clasp sequence for the boundary conditions $\\underline a$, with $\\lambda_i=\\mathrm{wt}(\\underline c_i)$. For a fully reduced hourglass plabic graph $W$, say that $W$ is non-convex with respect to $\\underline C$ if for every $i\\in[m]$ and every $\\underline c_i$-cut path $\\gamma$, one has $\\mathrm{wt}(\\gamma)\\ge \\mathrm{wt}(\\underline c_i)$ in the dominance order. Which statement exactly describes the web basis elements whose clasped images under $\\pi_{\\underline C}$ give the invariant space $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$?",
      "correct_choice": {
        "label": "A",
        "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; for each clasp $\\underline c_i$ there exists a $\\underline c_i$-cut path $\\gamma$ with $\\mathrm{wt}(\\gamma)\\ge \\mathrm{wt}(\\underline c_i)$ in the dominance order. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        },
        {
          "label": "C",
          "text": "The clasped web invariants $\\pi_{\\underline C}([W]_q)$ coming from basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ with $[W]_q\\notin \\ker \\pi_{\\underline C}$ span $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$; in particular, every web satisfying the equivalent conditions $[W]_q\\notin \\ker \\pi_{\\underline C}$, non-convexity, and having no trip that starts and ends in the same clasp contributes such a spanning vector."
        },
        {
          "label": "D",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is non-convex; $W$ has no trip whose two endpoints lie in adjacent clasps. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        },
        {
          "label": "E",
          "text": "They are exactly the basis elements $[W]_q\\in \\mathcal W_{\\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\\notin \\ker \\pi_{\\underline C}$; $W$ is partially convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\\pi_{\\underline C}([W]_q)$ for precisely these webs form a basis of $\\mathsf{Inv}\\!\\left(\\bigotimes_{i=1}^m V_q(\\lambda_i)\\right)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "forall-cut-path quantifier in non-convexity",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "basis conclusion weakened to spanning conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "trip obstruction changed from same clasp to adjacent clasps",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "non-convex replaced by partially convex",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal option A. It introduces the definition of non-convexity, but several options reuse or modify that notion, so the correct answer is not leaked in a trivial way."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially asking for the exact statement of a theorem characterizing which webs survive under clasping and form a basis. This is very close to direct theorem restatement rather than a genuinely new inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the distractors alter quantifiers, weaken 'basis' to 'span,' or change the trip obstruction. However, the task mainly tests precise recall/recognition of the theorem rather than generating the conclusion from underlying principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are mathematically close, distinct, and target common errors such as quantifier weakening, confusing basis with spanning, and replacing the correct obstruction by a nearby but false one."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with high-quality distractors and little answer leakage, but it is mostly a near-verbatim recall item rather than a non-tautological, generative reasoning question."
    }
  },
  {
    "id": "2512.08863v1",
    "paper_link": "http://arxiv.org/abs/2512.08863v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "headthm",
      "content": "[\\autoref{thm_main}]\n\t\\label{thmA}\n\tLet $X$ be an equidimensional projective scheme over a field $\\mathbb{k}$.\n\tLet $\\mathscr{L}$ be an ample line bundle on $X$ and $W \\subseteq Z$ be two closed subschemes of $X$.\n\tThen the following three conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$.\n\t\t\\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.\n\t\t\\item $\\deg_\\mathscr{L}\\left(s^i(Z, X)\\right) = \\deg_\\mathscr{L}\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate}",
      "start_pos": 83817,
      "end_pos": 84330,
      "label": "thmA"
    },
    "ref_dict": {
      "corB": "\\begin{headcor}[\\autoref{thm_main_zeta}]\n\t\\label{corB}\n\tLet $I \\subseteq J \\subset R$ be two homogeneous ideals in a polynomial ring $R = \\kk[x_0, \\ldots,x_n]$.\n\tThen the following two conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $J$ is integral over $I$.\n\t\t\\item $\\zeta_I(t) = \\zeta_J(t)$.\n\t\\end{enumerate} \n\\end{headcor}",
      "thmA": "\\begin{headthm}[\\autoref{thm_main}]\n\t\\label{thmA}\n\tLet $X$ be an equidimensional projective scheme over a field $\\kk$.\n\tLet $\\LL$ be an ample line bundle on $X$ and $W \\subseteq Z$ be two closed subschemes of $X$.\n\tThen the following three conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$.\n\t\t\\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.\n\t\t\\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate} \n\\end{headthm}",
      "thm_main": "\\begin{theorem}\n\t\\label{thm_main}\n\tAssume \\autoref{setup_segre_int_dep}.\n\tThen the following five conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$. \n\t\t\\item ${j_Z}_*\\left(s(Z, X)\\right) = {j_W}_*\\left(s(W, X)\\right)$ in $A_*(X)$.\n\t\t\\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\beta^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate} \n\\end{theorem}",
      "thm_main_zeta": "\\begin{corollary}\n\t\\label{thm_main_zeta}\n\tLet $J \\subset R = \\kk[x_0,\\ldots,x_n]$ be a homogeneous ideal such that $J \\supseteq I$.\n\tThen the following two conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $J$ is integral over $I$.\n\t\t\\item $\\zeta_I(t) = \\zeta_J(t)$.\n\t\\end{enumerate} \n\\end{corollary}",
      "prop_hard": "\\begin{proposition}\n\t\\label{prop_hard}\n\tAssume that $\\Rees(\\II_W)$ is not of finite type as a module over $\\Rees(\\II_Z)$.\n\t Then we have the following strict inequality \n\t $$\n\t \\lim_{n \\to \\infty} \\frac{h^0\\left(X, \\II_W^{n}/\\II_Z^{n} \\otimes \\LL^{\\otimes 2pn} \\right)}{n^d} \\;>\\; 0.\n\t $$\n\\end{proposition}",
      "examp": "\\begin{example}[Line bundles that are big and nef do not suffice in \\autoref{thm_main}]\n\t\\label{examp}\n\tAssume $\\kk$ is algebraically closed.\n\tConsider the following two successive blow-ups: \n\t\\begin{enumerate}[\\rm (i)]\n\t\t\\item Let $\\pi_1 : Y = \\Bl_{p}\\left(\\PP_\\kk^3\\right) \\rightarrow \\PP_\\kk^3$ be the blow-up of $\\PP_{\\kk}^3$ along a closed point $p \\in \\PP_\\kk^3$.\n\t\tNotice that the exceptional divisor $E_p\\PP_\\kk^3$ of $Y=\\Bl_{p}\\left(\\PP_\\kk^3\\right)$ is isomorphic to $\\PP_{\\kk}^2$.\n\t\t\\item Choose a line $C$ in $\\PP_\\kk^2 \\cong E_p\\PP_\\kk^3$.\n\t\tLet $\\pi_2 : X = \\Bl_C(Y) \\rightarrow Y$ be the blow-up of $Y$ along $C$.\n\t\tNotice that $X$ is a smooth threefold. \n\t\tLet $E = E_CY$ be the exceptional divisor of $X = \\Bl_C(Y)$.\n\t\\end{enumerate}\n\tLet $\\pi = \\pi_1 \\circ \\pi_2 : X \\rightarrow \\PP_\\kk^3$ be the natural projection.\n\tLet $\\mathcal{L} = \\pi^*\\big(\\OO_{\\PP_\\kk^3}(1)\\big)$ be the line bundle on $X$  given as the pullback of $\\OO_{\\PP_\\kk^3}(1)$.\n\tNotice that $\\mathcal{L}$ is big and nef.\n\tLet $k \\ge 2$ and consider the effective Cartier divisor $kE$.\n\tAs closed subschemes,  we have $E \\subsetneq kE \\subset X$ (although they have the same support).\n\tWe also obtain that their Segre classes are different\n\t$$\n\ts(E, X) \\;=\\; E - E^2 + E^3  \\;\\neq\\; kE -k^2E^2 + k^3E^3  \\;=\\; s(kE, X);\n\t$$\t\n\there $E^i$ denotes the self-intersection of $E$\n\t(see \\cite[Proposition 4.1(a)]{FULTON_INTER}).\n\tHowever, we can compute the vanishings\n\t$$\n\t\\deg_{\\mathcal{L}}\\left(E\\right) \\;=\\; \\deg_{\\mathcal{L}}\\left(E^2\\right) \\;=\\; \\deg_{\\mathcal{L}}\\left(E^3\\right) \\;=\\; 0.\n\t$$\n\tThis shows that the big and nef line bundle $\\mathcal{L}$ cannot be used in \\autoref{thm_main}.\n\\end{example}"
    },
    "pre_theorem_intro_text_len": 1724,
    "pre_theorem_intro_text": "The \\emph{Segre class} associated with an embedding of schemes plays a central role in Fulton--MacPherson's intersection theory and underlies numerous applications.\nA detailed account of Segre classes and their applications can be found in \\cite{FULTON_INTER,ALUFFI_SURVEY}.\n\nA central and defining property of Segre classes is their \\emph{birational invariance}.\nIndeed, if $Z$ is a proper closed subvariety of a variety $X$ over a field $\\mathbb{k}$, this invariance yields \n$$\ns(Z, X) \\;=\\; \\eta_*\\left(s(E, \\mathcal{B})\\right) \\;=\\; \\eta_*\\left(c\\left(N_E\\mathcal{B}\\right)^{-1} \\smallfrown [E]\\right) \\;=\\; \\eta_*\\left(\\frac{[E]}{1+E}\\right) \\;\\in\\; A_*(Z),\n$$\nwhere $\\pi : \\mathcal{B} = \\Bl_Z(X) \\rightarrow X$ is the blow-up of $X$ along $Z$, $\\eta : E = E_ZX \\rightarrow Z$ is the exceptional divisor, and $N_E\\mathcal{B}$ is the normal bundle to $E$ in $\\mathcal{B}$ (see \\cite[\\S 4.2]{FULTON_INTER}).\nThus, by birational invariance, computing Segre classes reduces to the case of effective Cartier divisors, whose Segre classes admit an explicit formula since they are regularly embedded.\n\nThe birational invariance of Segre classes also implies that they depend only on the \\emph{integral closure} of the underlying ideal sheaf.\nMore precisely, if we let $Z'= V\\left(\\overline{\\II_Z}\\right) \\subset X$ where $\\overline{\\II_Z} \\subset \\OO_X$ is the integral closure of the ideal sheaf $\\II_Z$ of $Z$, then the Segre class $s(Z, X) \\in A_*(Z)$ of $Z$ equals the Segre class $s(Z', X) \\in A_*(Z')$ of $Z'$ (since $Z'_{\\rm red} = Z_{\\rm red}$, we can identify their Chow groups).\n\n\\smallskip\n\nOur main result demonstrates that Segre classes, in fact, determine integral dependence, leading to the following criterion:",
    "context": "The \\emph{Segre class} associated with an embedding of schemes plays a central role in Fulton--MacPherson's intersection theory and underlies numerous applications.\nA detailed account of Segre classes and their applications can be found in \\cite{FULTON_INTER,ALUFFI_SURVEY}.\n\nA central and defining property of Segre classes is their \\emph{birational invariance}.\nIndeed, if $Z$ is a proper closed subvariety of a variety $X$ over a field $\\mathbb{k}$, this invariance yields \n$$\ns(Z, X) \\;=\\; \\eta_*\\left(s(E, \\mathcal{B})\\right) \\;=\\; \\eta_*\\left(c\\left(N_E\\mathcal{B}\\right)^{-1} \\smallfrown [E]\\right) \\;=\\; \\eta_*\\left(\\frac{[E]}{1+E}\\right) \\;\\in\\; A_*(Z),\n$$\nwhere $\\pi : \\mathcal{B} = \\Bl_Z(X) \\rightarrow X$ is the blow-up of $X$ along $Z$, $\\eta : E = E_ZX \\rightarrow Z$ is the exceptional divisor, and $N_E\\mathcal{B}$ is the normal bundle to $E$ in $\\mathcal{B}$ (see \\cite[\\S 4.2]{FULTON_INTER}).\nThus, by birational invariance, computing Segre classes reduces to the case of effective Cartier divisors, whose Segre classes admit an explicit formula since they are regularly embedded.\n\nThe birational invariance of Segre classes also implies that they depend only on the \\emph{integral closure} of the underlying ideal sheaf.\nMore precisely, if we let $Z'= V\\left(\\overline{\\II_Z}\\right) \\subset X$ where $\\overline{\\II_Z} \\subset \\OO_X$ is the integral closure of the ideal sheaf $\\II_Z$ of $Z$, then the Segre class $s(Z, X) \\in A_*(Z)$ of $Z$ equals the Segre class $s(Z', X) \\in A_*(Z')$ of $Z'$ (since $Z'_{\\rm red} = Z_{\\rm red}$, we can identify their Chow groups).\n\n\\smallskip\n\nOur main result demonstrates that Segre classes, in fact, determine integral dependence, leading to the following criterion:\n\n\\begin{theorem}\n\t\\label{thm_main}\n\tAssume \\autoref{setup_segre_int_dep}.\n\tThen the following five conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$. \n\t\t\\item ${j_Z}_*\\left(s(Z, X)\\right) = {j_W}_*\\left(s(W, X)\\right)$ in $A_*(X)$.\n\t\t\\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\beta^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate} \n\\end{theorem}",
    "full_context": "The \\emph{Segre class} associated with an embedding of schemes plays a central role in Fulton--MacPherson's intersection theory and underlies numerous applications.\nA detailed account of Segre classes and their applications can be found in \\cite{FULTON_INTER,ALUFFI_SURVEY}.\n\nA central and defining property of Segre classes is their \\emph{birational invariance}.\nIndeed, if $Z$ is a proper closed subvariety of a variety $X$ over a field $\\mathbb{k}$, this invariance yields \n$$\ns(Z, X) \\;=\\; \\eta_*\\left(s(E, \\mathcal{B})\\right) \\;=\\; \\eta_*\\left(c\\left(N_E\\mathcal{B}\\right)^{-1} \\smallfrown [E]\\right) \\;=\\; \\eta_*\\left(\\frac{[E]}{1+E}\\right) \\;\\in\\; A_*(Z),\n$$\nwhere $\\pi : \\mathcal{B} = \\Bl_Z(X) \\rightarrow X$ is the blow-up of $X$ along $Z$, $\\eta : E = E_ZX \\rightarrow Z$ is the exceptional divisor, and $N_E\\mathcal{B}$ is the normal bundle to $E$ in $\\mathcal{B}$ (see \\cite[\\S 4.2]{FULTON_INTER}).\nThus, by birational invariance, computing Segre classes reduces to the case of effective Cartier divisors, whose Segre classes admit an explicit formula since they are regularly embedded.\n\nThe birational invariance of Segre classes also implies that they depend only on the \\emph{integral closure} of the underlying ideal sheaf.\nMore precisely, if we let $Z'= V\\left(\\overline{\\II_Z}\\right) \\subset X$ where $\\overline{\\II_Z} \\subset \\OO_X$ is the integral closure of the ideal sheaf $\\II_Z$ of $Z$, then the Segre class $s(Z, X) \\in A_*(Z)$ of $Z$ equals the Segre class $s(Z', X) \\in A_*(Z')$ of $Z'$ (since $Z'_{\\rm red} = Z_{\\rm red}$, we can identify their Chow groups).\n\n\\smallskip\n\nOur main result demonstrates that Segre classes, in fact, determine integral dependence, leading to the following criterion:\n\n\\begin{theorem}\n\t\\label{thm_main}\n\tAssume \\autoref{setup_segre_int_dep}.\n\tThen the following five conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$. \n\t\t\\item ${j_Z}_*\\left(s(Z, X)\\right) = {j_W}_*\\left(s(W, X)\\right)$ in $A_*(X)$.\n\t\t\\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\t\\item $\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\beta^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate} \n\\end{theorem}\n\n\\begin{corollary}\n \\label{cor_numerical_snapper}\n The following statements hold:\n \\begin{enumerate}[\\rm (i)]\n \\item The terms of $P_{\\LL, Z}(m, n)$ of degree $d$ are given as follows\n $$\n P_{\\LL, Z}(m, n) \\;=\\; \\sum_{i=0}^d \\frac{\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right)}{i!(d-i)!} \\,m^{d-i} n^{i} \\;+\\; \\text{{\\rm(}lower degree terms{\\rm)}}.\n $$\n \\item For all $m \\in \\ZZ$ and $n \\gg 0$, we have the equality\n $$\n P_{\\LL, Z}(m, n) \\;=\\; \\chi\\big(\\II_Z^n \\otimes \\LL^{\\otimes (m+n)}\\big). \n $$\n \\item If $\\LL$ is very ample on $X$, we get the equality\n $$\n P_{\\LL, Z}(m, n) \\;=\\; h^0\\big(X, \\II_Z^n \\otimes \\LL^{\\otimes (m+n)}\\big)\n $$\n for all $m \\gg 0$ and $n \\gg 0$.\n \\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n (i) By \\cite[Theorem B.7, Definition B.8]{KLEIMAN_PICARD}, we have \n $$\n \\chi\\left(\\pi^*(\\LL)^{\\otimes m} \\otimes \\OO_{\\mathcal{B}^\\LL}(n)\\right) \\;=\\; \\sum_{i=0}^d \\frac{a_{i}}{i!(d-i)!} \\,m^{d-i} n^{i} \\;+\\; \\text{(lower degree terms)}\n $$\n where $a_{i}= \\int c_1\\left(\\pi^*(\\LL)\\right)^{d-i} c_1\\left(\\OO_{\\mathcal{B}^\\LL}(1)\\right)^i \\smallfrown \\left[\\mathcal{B}\\right]$.\n Then the equalities \n $$\n a_i \\;=\\; \\int c_1\\left(\\LL\\right)^{d-i} \\smallfrown \\pi_*\\left( c_1\\left(\\OO_{\\mathcal{B}^\\LL}(1)\\right)^{i} \\smallfrown \\left[\\mathcal{B}\\right]\\right) \\;=\\; \\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right)\n $$\n follow from the projection formula and \\autoref{prop_blow_up_formula}.\n\n\\begin{lemma}\n \\label{lem_num_equiv}\n The following conditions are equivalent: \n \\begin{enumerate}[\\;\\;\\rm (a)]\n \\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n \\item $\\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n \\item $\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\beta^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n \\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n (a) $\\Rightarrow$ (b): \n By utilizing \\autoref{eq_vogel_Z}, we obtain\n \\begin{align*}\n \\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) &\\;=\\; \\int c_1(\\LL)^{d-i} \\smallfrown \\nu^i(\\underline{\\sigma}, X) \\\\\n &\\;=\\; \\sum_{j=0}^i\\binom{i-1}{j-1} p^{i-j} \\int c_1(\\LL)^{d-i} c_1(\\LL)^{i-j} \\smallfrown s^j(Z, X) \\\\ \n &\\;=\\; \\sum_{j=0}^i\\binom{i-1}{j-1} p^{i-j} \\deg_\\LL\\left(s^j(Z, X)\\right).\n \\end{align*}\n Similarly, \\autoref{eq_vogel_W} yields \n $$\n \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right) \\;=\\; \\sum_{j=0}^i\\binom{i-1}{j-1} p^{i-j} \\deg_\\LL\\left(s^j(W, X)\\right).\n $$ \n So the implication (a) $\\Rightarrow$ (b) follows.\n\n\\begin{theorem}\n \\label{thm_main}\n Assume \\autoref{setup_segre_int_dep}.\n Then the following five conditions are equivalent:\n \\begin{enumerate}[\\;\\;\\rm (a)]\n \\item $\\II_W$ is integral over $\\II_Z$. \n \\item ${j_Z}_*\\left(s(Z, X)\\right) = {j_W}_*\\left(s(W, X)\\right)$ in $A_*(X)$.\n \\item $\\deg_\\LL\\left(s^i(Z, X)\\right) = \\deg_\\LL\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n \\item $\\deg_\\LL\\left(\\nu^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\nu^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n \\item $\\deg_\\LL\\left(\\beta^i(\\underline{\\sigma}, X)\\right) = \\deg_\\LL\\left(\\beta^i(\\underline{\\eta}, X)\\right)$ for all $i \\ge 0$.\n \\end{enumerate} \n\\end{theorem}\n\\begin{proof}\n (a) $\\Rightarrow$ (b): This implication is a consequence of the birational invariance of Segre classes. \n For completeness, we include a short argument.\n Assume that $\\overline{\\II_Z} = \\overline{\\II_W}$.\n Thus $Z$ and $W$ have the same support, and so we may identify their Chow groups.\n Let $X_1,\\ldots,X_k$ be the irreducible components of $X$ and $m_i$ be the geometric multiplicity of $X_i$ in $X$.\n Let $Z_i = Z \\cap X_i$ and $W_i = W \\cap X_i$.\n By \\cite[Lemma 4.2]{FULTON_INTER}, we have $s(Z, X) = \\sum_{i=1}^k m_i s(Z_i, X_i) \\in A_*(Z)$ and $s(W, X) = \\sum_{i=1}^k m_i s(W_i, X_i) \\in A_*(W)$.\n Thus to show the equality $s(Z, X) = s(W, X) \\in A_*(Z) = A_*(W)$, we may assume that $X$ is irreducible.\n\n\\begin{proposition}\n \\label{prop_hard}\n Assume that $\\Rees(\\II_W)$ is not of finite type as a module over $\\Rees(\\II_Z)$.\n Then we have the following strict inequality \n $$\n \\lim_{n \\to \\infty} \\frac{h^0\\left(X, \\II_W^{n}/\\II_Z^{n} \\otimes \\LL^{\\otimes 2pn} \\right)}{n^d} \\;>\\; 0.\n $$\n\\end{proposition}\n\\begin{proof}\n To simplify notation, let \n $$\n \\sC \\;:=\\; \\Rees^{\\LL^{\\otimes p}}(\\II_Z) \\;=\\; \\bigoplus_{n\\ge 0} \\II_Z^n \\otimes \\LL^{\\otimes pn} \\quad \\text{ and } \\quad \\sD \\;:=\\; \\Rees^{\\LL^{\\otimes p}}(\\II_W) \\;=\\; \\bigoplus_{n\\ge 0} \\II_W^n \\otimes \\LL^{\\otimes pn}.\n $$\n Let $\\pi : \\sY = \\fProj_X\\left(\\sD\\right) \\rightarrow X$ be the natural projection.\n We cannot have that $W_{\\rm red} = X_{\\rm red}$, because otherwise the ideal sheaves $\\II_Z \\subseteq \\II_W$ would be nilpotent, and so they would have the same integral closure (equal to the nilradical ideal sheaf of $X$). \n It then follows that $\\sY$ is equidimensional of dimension $d$ (see \\autoref{rem_blowup_irr_comp}). \n By assumption, we have that $\\fProj_X\\left(\\sD/\\sC_+\\sD\\right) \\neq \\varnothing$, where $\\sC_+\\sD \\subset \\sD$ denotes the graded ideal in $\\sD$ generated by $\\sC_+ = \\bigoplus_{n\\ge 1} \\sC_n$.\n Let $y \\in \\sY$ be a closed point lying in $\\fProj_X\\left(\\sD/\\sC_+\\sD\\right)$.\n Let $\\II_y \\subset \\OO_\\sY$ be the ideal sheaf of $y$. Let $\\aaa \\subset \\sD$ be a graded ideal in $\\sD$ whose sheafification is $\\II_y = \\widetilde{\\aaa}$ and such that $\\aaa \\supset \\sC_+\\sD$.\n Since $\\pi$ is a closed morphism, the closed point $y \\in \\sY$ is mapped to a closed point $x = \\pi(y)$ in $X$.\n\n\\begin{example}[Line bundles that are big and nef do not suffice in \\autoref{thm_main}]\n \\label{examp}\n Assume $\\kk$ is algebraically closed.\n Consider the following two successive blow-ups: \n \\begin{enumerate}[\\rm (i)]\n \\item Let $\\pi_1 : Y = \\Bl_{p}\\left(\\PP_\\kk^3\\right) \\rightarrow \\PP_\\kk^3$ be the blow-up of $\\PP_{\\kk}^3$ along a closed point $p \\in \\PP_\\kk^3$.\n Notice that the exceptional divisor $E_p\\PP_\\kk^3$ of $Y=\\Bl_{p}\\left(\\PP_\\kk^3\\right)$ is isomorphic to $\\PP_{\\kk}^2$.\n \\item Choose a line $C$ in $\\PP_\\kk^2 \\cong E_p\\PP_\\kk^3$.\n Let $\\pi_2 : X = \\Bl_C(Y) \\rightarrow Y$ be the blow-up of $Y$ along $C$.\n Notice that $X$ is a smooth threefold. \n Let $E = E_CY$ be the exceptional divisor of $X = \\Bl_C(Y)$.\n \\end{enumerate}\n Let $\\pi = \\pi_1 \\circ \\pi_2 : X \\rightarrow \\PP_\\kk^3$ be the natural projection.\n Let $\\mathcal{L} = \\pi^*\\big(\\OO_{\\PP_\\kk^3}(1)\\big)$ be the line bundle on $X$ given as the pullback of $\\OO_{\\PP_\\kk^3}(1)$.\n Notice that $\\mathcal{L}$ is big and nef.\n Let $k \\ge 2$ and consider the effective Cartier divisor $kE$.\n As closed subschemes, we have $E \\subsetneq kE \\subset X$ (although they have the same support).\n We also obtain that their Segre classes are different\n $$\n s(E, X) \\;=\\; E - E^2 + E^3 \\;\\neq\\; kE -k^2E^2 + k^3E^3 \\;=\\; s(kE, X);\n $$ \n here $E^i$ denotes the self-intersection of $E$\n (see \\cite[Proposition 4.1(a)]{FULTON_INTER}).\n However, we can compute the vanishings\n $$\n \\deg_{\\mathcal{L}}\\left(E\\right) \\;=\\; \\deg_{\\mathcal{L}}\\left(E^2\\right) \\;=\\; \\deg_{\\mathcal{L}}\\left(E^3\\right) \\;=\\; 0.\n $$\n This shows that the big and nef line bundle $\\mathcal{L}$ cannot be used in \\autoref{thm_main}.\n\\end{example}\n\\begin{proof}\n We only need to show the claimed vanishings. \n The projection formula yields \n $$\n \\deg_\\mathcal{L}(E) \\;=\\; \\int c_1\\big(\\OO_{\\PP_\\kk^3}(1)\\big)^2 \\smallfrown \\pi_*\\left([E]\\right) \\quad \\text{ and } \\quad \\deg_\\mathcal{L}(E^2) \\;=\\; \\int c_1\\big(\\OO_{\\PP_\\kk^3}(1)\\big) \\smallfrown \\pi_*\\left(E \\cdot [E]\\right).\n $$\n By construction, $E$ is mapped onto the point $p$ via $\\pi$.\n Since $[E] \\in A_2(X)$ and $E\\cdot [E] \\in A_1(X)$, by dimension reasons and the definition of proper pushforward, we get $\\pi_*([E]) = 0$ and $\\pi_*(E \\cdot [E]) = 0$ (see \\cite[\\S 1.4]{FULTON_INTER}).\n It then follows that $\\deg_{\\mathcal{L}}\\left(E\\right) = \\deg_{\\mathcal{L}}\\left(E^2\\right) = 0$.",
    "post_theorem_intro_text_len": 4154,
    "post_theorem_intro_text": "In \\autoref{thmA}, even if one is not interested in integral dependence, the equivalence (b) $\\Leftrightarrow$ (c) seems interesting in its own right.  \nWe note that \\autoref{thmA} is sharp in the sense that the ampleness hypothesis appears to be indispensable. \n\\autoref{examp} illustrates this by exhibiting a big and nef line bundle for which the conclusion of \\autoref{thmA} does not hold.\n\nThe proof of \\autoref{thmA} relies on van Gastel’s foundational result \\cite{VanGastel}, which establishes the connection between Fulton–MacPherson’s intersection theory \\cite{FULTON_INTER} and St\\\"uckrad–Vogel’s intersection theory \\cite{VOGEL,FOV}. \nIndeed, our starting point is van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes.\nAn advantage of considering the Vogel cycle is that it opens the door to applying technical positivity results, including the one established in \\autoref{prop_hard}.\n\n\\smallskip\n\nAs an application of \\autoref{thmA}, we show that Aluffi’s Segre zeta function \\cite{aluffi2017segre} furnishes a criterion for the integral dependence of homogeneous ideals in a polynomial ring. \nThis power series records information about the Segre classes that arise when the ideal is extended to projective spaces of arbitrarily large dimension (see \\autoref{sect_zeta} for further details). \nWe obtain the following corollary:\n\n\\begin{headcor}[\\autoref{thm_main_zeta}]\n\t\\label{corB}\n\tLet $I \\subseteq J \\subset R$ be two homogeneous ideals in a polynomial ring $R = \\mathbb{k}[x_0, \\ldots,x_n]$.\n\tThen the following two conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $J$ is integral over $I$.\n\t\t\\item $\\zeta_I(t) = \\zeta_J(t)$.\n\t\\end{enumerate} \n\\end{headcor}\n\nThe idea of detecting integral dependence through numerical invariants originates in Rees’ seminal work \\cite{REES}.\nRees proved that in an equidimensional and universally catenary Noetherian local ring $(R, \\mm)$, two $\\mm$-primary ideals $I \\subseteq J$ have the same integral closure if and only if they share the same Hilbert–Samuel multiplicity.\nSince then, the search for numerical criteria to characterize integral dependence has been an important research topic in algebraic geometry, commutative algebra and singularity\ntheory.\nMuch effort has gone into extending Rees’ theorem to arbitrary ideals, modules, and more broadly, to algebras (see \\cite{BOEGER,TEISSIER_CYC,TEISSIER_RES2, KLEIMAN_THORUP_GEOM,KLEIMAN_THORUP_MIXED, GG, FLENNER_MANARESI,SUV_MULT, GAFFNEY, CIU, UV_CRIT_MOD,     UV_NUM_CRIT,   PTUV, cid2023relative, cidruiz2024polar, CRPU2, BLQ, das2024numerical}). \nIn this direction, two notable results are: \n\\begin{itemize}[\\;--]\n\t\\item Teissier's Principle of Specialization of Integral Dependence (PSID) \\cite{TEISSIER_CYC}, \\cite[Appendice I]{TEISSIER_RES2}.\n\tGiven a map of germs of analytic spaces,  the PSID asserts that\n\tfor a family of zero-dimensional ideals with constant Hilbert-Samuel multiplicity, a section is integrally\n\tdependent on the total family if and only if it is integrally dependent along the fibers corresponding to a dense open  subset of the base.\n\t\\item An extension of Rees’ theorem to arbitrary ideals (in an equidimensional and universally catenary Noetherian local ring $(R, \\mm)$) was established by Polini, Trung, Ulrich, and Validashti \\cite{PTUV}, using the multiplicity sequence introduced by Achilles and Manaresi \\cite{ACH_MANA}. (In the analytic setting, an analogous result had been obtained earlier by Gaffney and Gassler \\cite{GG}.)\n\\end{itemize}\nNote that these criteria are local in nature, whereas \\autoref{thmA} furnishes a global criterion.\n\n\\medskip\n\n  \\noindent\n\\textbf{Outline.} \nThe structure of the paper is as follows. \nIn \\autoref{sect_prelim}, we introduce the notation used throughout and recall the necessary preliminary results. \n\\autoref{sect_van_Gastel} develops the relation between Segre classes and Vogel cycles and shows how this connection can be applied. \nThe proof of \\autoref{thmA} is given in \\autoref{sect_main}. \nFinally, \\autoref{sect_zeta} contains the proof of \\autoref{corB}.",
    "sketch": "The post-theorem introduction indicates that the proof of \\autoref{thmA} \\emph{relies on van Gastel’s foundational result} \\cite{VanGastel}, which \"establishes the connection between Fulton–MacPherson’s intersection theory\" \\cite{FULTON_INTER} and \"St\\\"uckrad–Vogel’s intersection theory\" \\cite{VOGEL,FOV}. The stated \\emph{starting point} is \"van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes.\" The introduction further explains the strategy/advantage: \"considering the Vogel cycle\" allows one to apply \"technical positivity results, including the one established in \\autoref{prop_hard}.\" (The full proof is deferred: \"The proof of \\autoref{thmA} is given in \\autoref{sect_main}.\")",
    "expanded_sketch": "The post-theorem introduction indicates that, in establishing the main theorem, the proof \\emph{relies on van Gastel’s foundational result} \\cite{VanGastel}, which \"establishes the connection between Fulton–MacPherson’s intersection theory\" \\cite{FULTON_INTER} and \"St\\\"uckrad–Vogel’s intersection theory\" \\cite{VOGEL,FOV}. The stated \\emph{starting point} is \"van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes.\" The introduction further explains the strategy/advantage: \"considering the Vogel cycle\" allows one to apply \"technical positivity results, including the one established in the following proposition.\" \\begin{proposition}\n\t\\label{prop_hard}\n\tAssume that $\\Rees(\\II_W)$ is not of finite type as a module over $\\Rees(\\II_Z)$.\n\t Then we have the following strict inequality \n\t $$\n\t \\lim_{n \\to \\infty} \\frac{h^0\\left(X, \\II_W^{n}/\\II_Z^{n} \\otimes \\LL^{\\otimes 2pn} \\right)}{n^d} \\;>\\; 0.\n\t $$\n\\end{proposition} (The full proof is deferred: the proof of the main theorem is given later.)",
    "expanded_theorem": "To prove the main theorem, let $X$ be an equidimensional projective scheme over a field $\\mathbb{k}$.\n\t\\label{thmA}\n\tLet $\\mathscr{L}$ be an ample line bundle on $X$ and $W \\subseteq Z$ be two closed subschemes of $X$.\n\tThen the following three conditions are equivalent:\n\t\\begin{enumerate}[\\;\\;\\rm (a)]\n\t\t\\item $\\II_W$ is integral over $\\II_Z$.\n\t\t\\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.\n\t\t\\item $\\deg_\\mathscr{L}\\left(s^i(Z, X)\\right) = \\deg_\\mathscr{L}\\left(s^i(W, X)\\right)$ for all $i \\ge 0$.\n\t\\end{enumerate}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $X$ be an equidimensional projective scheme over a field $\\mathbb{k}$, let $\\mathscr L$ be an ample line bundle on $X$, and let $W\\subseteq Z\\subseteq X$ be closed subschemes. Write $s(Z,X)$ and $s(W,X)$ for their Segre classes, viewed in the Chow group $A_*(X)$, and let $s^i(Z,X)$ and $s^i(W,X)$ denote the codimension-$i$ components. For such a component $\\alpha$, define its $\\mathscr L$-degree by\n$$\n\\deg_{\\mathscr L}(\\alpha):=\\int c_1(\\mathscr L)^{\\dim X-i}\\smallfrown \\alpha.\n$$\nWhich statement is equivalent to the ideal sheaf $\\mathcal I_W$ being integral over $\\mathcal I_Z$?",
      "correct_choice": {
        "label": "A",
        "text": "It is equivalent to either of the following, and hence to both: the Segre classes agree in $A_*(X)$, $$s(Z,X)=s(W,X),$$ and, equivalently, their $\\mathscr L$-degrees agree in every codimension, $$\\deg_{\\mathscr L}(s^i(Z,X))=\\deg_{\\mathscr L}(s^i(W,X))\\quad\\text{for all } i\\ge 0.$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "It is equivalent to the equality of Segre classes after pushforward to numerical degrees: $$\\deg_{\\mathscr L}(s^i(Z,X))=\\deg_{\\mathscr L}(s^i(W,X))\\quad\\text{for all }0\\le i\\le \\dim X-1,$$ and hence also to $$s(Z,X)=s(W,X)\\text{ in }A_*(X).$$"
        },
        {
          "label": "C",
          "text": "It implies, and is in particular sufficient for, the equality of all $\\mathscr L$-degrees of the Segre components: $$\\deg_{\\mathscr L}(s^i(Z,X))=\\deg_{\\mathscr L}(s^i(W,X))\\quad\\text{for all } i\\ge 0.$$"
        },
        {
          "label": "D",
          "text": "It is equivalent to requiring that for every ample line bundle $\\mathscr M$ on $X$, the numerical equalities $$\\deg_{\\mathscr M}(s^i(Z,X))=\\deg_{\\mathscr M}(s^i(W,X))\\quad\\text{for all } i\\ge 0$$ hold; equivalently, $$s(Z,X)=s(W,X)\\text{ in }A_*(X).$$"
        },
        {
          "label": "E",
          "text": "It is equivalent to the equality of the reduced Segre classes, namely that $$s(Z_{\\mathrm{red}},X)=s(W_{\\mathrm{red}},X)\\text{ in }A_*(X),$$ and equivalently that $$\\deg_{\\mathscr L}(s^i(Z_{\\mathrm{red}},X))=\\deg_{\\mathscr L}(s^i(W_{\\mathrm{red}},X))\\quad\\text{for all } i\\ge 0.$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "all-codimensions requirement",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "equivalence dropped to one-way implication from integrality to degree equalities",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "fixed ample line bundle replaced by uniformity over all ample line bundles",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "scheme-structure sensitivity of Segre classes replaced by reduced-subscheme equality",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the equivalence outright; it only introduces the relevant objects and notation. While the setup strongly signals that Segre classes and their degrees are central, it does not explicitly leak which precise equivalence is correct."
      },
      "TAS": {
        "score": 0,
        "justification": "This is very close to a direct restatement of the underlying theorem: integrality of ideal sheaves is equivalent to equality of Segre classes, equivalently equality of the corresponding degrees. The correct choice essentially reproduces that theorem verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to distinguish the exact theorem from nearby variants such as one-way implication, strengthened quantification, or reduced-subscheme replacement. However, for a student who knows the theorem, the correct answer is largely recall rather than substantial generation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: truncating the codimension range, weakening equivalence to implication, over-strengthening to all ample line bundles, and incorrectly passing to reduced subschemes. They are distinct and well aligned with conceptual misunderstandings."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically well-constructed recall-style MCQ with strong distractors, but it is highly theorem-restatement-based and thus only moderately effective at testing genuine generative reasoning."
    }
  },
  {
    "id": "2512.08899v1",
    "paper_link": "http://arxiv.org/abs/2512.08899v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main-prague}\n\tLet\\footnote{In all our statements, $\\varepsilon$ is a fixed constant and $n$ tends to infinity.} $\\varepsilon \\in (0,1)$ and $(\\log n)^{2+\\varepsilon} n^{-1} \\le p < n^{-\\varepsilon}$. \n\tThen, with high probability we have\n\t\\begin{align*}\n\t\t\\mathrm{pdim}(G_{n,p}) = O_{\\varepsilon} \\left ( \\dfrac{pn \\log n}{\\log pn} \\right ).\n\t\\end{align*}",
      "start_pos": 8466,
      "end_pos": 8826,
      "label": "thm:main-prague"
    },
    "ref_dict": {
      "thm:main-dem": "\\begin{theorem}\\label{thm:main-dem}\n\tLet $\\eps \\in (0,2^{-10})$ and $(\\log n)^{2+\\eps} n^{-1} \\le p < n^{-\\eps}$. \n\tLet $G$ be an $n$-vertex $(\\eps,p)$-typical graph and $k = \\eps 2^{-10}p^{-1} \\log pn$.\n\tThen, the following holds for all sufficiently large $n$. With probability at least $1- n^{-2^{-11}\\log pn}$, in the random greedy independent set process, we have\n\t\\[ d_i(v) = (1 \\pm f_i)\\td_i\\] \n\tfor all $v \\in V_i$ and all $i \\in [k-1]$.\n\\end{theorem}",
      "thm:main-prague": "\\begin{theorem}\\label{thm:main-prague}\n\tLet\\footnote{In all our statements, $\\eps$ is a fixed constant and $n$ tends to infinity.} $\\eps \\in (0,1)$ and $(\\log n)^{2+\\eps} n^{-1} \\le p < n^{-\\eps}$. \n\tThen, with high probability we have\n\t\\begin{align*}\n\t\t\\mathrm{pdim}(G_{n,p}) = O_{\\eps} \\left ( \\dfrac{pn \\log n}{\\log pn} \\right ).\n\t\\end{align*}\n\\end{theorem}",
      "eq:pdim-idim": "\\begin{align}\\label{eq:pdim-idim}\n\t\\idim(G) \\le \\pdim(G) \\le \\idim(G) + 1. \n\\end{align}",
      "eq:lower-bound-MRSS": "\\begin{align}\\label{eq:lower-bound-MRSS}\n\t\\text{pdim}(G_{n,p}) \\ge \\dfrac{pn \\log \\frac{1}{p} }{5\\log n }\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 3205,
    "pre_theorem_intro_text": "Motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem~\\cite{Deuber,ErdosHajnalPosa,Rodl},\nNešetřil, Pultr and Rödl~\\cite{Nesetril1977-pj,Nesetril1978-km} introduced the notion of the Prague dimension of a graph in the 1970s.\nFor a graph $G$, the Prague dimension $\\text{pdim}(G)$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph.\nThis is closely related to $\\text{idim}(G)$, defined as the minimum number of partial clique factors whose union equals $\\overline{G}$, the complement of $G$.\nIn fact, \nNešetřil and Rödl~\\cite{Nesetril1978-km} showed that for every graph $G$, we have\n\\begin{align}\\label{eq:pdim-idim}\n\t\\mathrm{idim}(G) \\le \\mathrm{pdim}(G) \\le \\mathrm{idim}(G) + 1. \n\\end{align}\n\nIn 1977, Lovász, Nešetřil, and Pultr~\\cite{Lovasz1980-kw} showed that a graph on $n\\ge2$ vertices has Prague dimension at most $n-1$ and classified all graphs which achieve this bound.\nGeneral upper and lower bounds were obtained by Alon~\\cite{Alon1986-oc}, who proved that if $G$ is a graph on $n$ vertices with minimum degree at least 1 and maximum degree $\\Delta$, then $\\log (n/ \\Delta)/ \\log 2 \\le \\text{pdim}(G) = O(\\Delta^2\\log n)$; all our asymptotic expressions are meant as $n \\to \\infty$, and all logarithms are natural.\nThe upper bound was later improved by Eaton and Rödl~\\cite{Eaton1996-ap}, who showed that \n\\begin{align*}\n\t\\text{pdim}(G) = O(\\Delta \\log n).\n\\end{align*}\nMoreover, they proved that this is best possible up to a factor of $1/(\\log \\Delta + \\log \\log (n/2\\Delta) )$.\nWhen $\\Delta$ is a constant, these results provide the order of magnitude of $\\text{pdim}(G)$.\nPrecise asymptotics are known for matchings, trees and cycles~\\cite{Alon1986-oc,Lovasz1980-kw,Poljak1981-lt}, hypercubes~\\cite{Krivka} and Kneser graphs $K(n,k)$ for constant values of $k$~\\cite{Poljak1978-um}.\nFor more details, we refer the reader to~\\cite{Kantor}.\n\nNešetřil and Rödl~\\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\\Omega_p(n/\\log n)$ with high probability for constant values of $p$.\nMany years later, Guo, Patton and Warnke~\\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\\cite{Pippenger1989-ak} type\nedge-colouring result for random hypergraphs with large uniformities.\nMore recently, Molnar, Rödl, Sales and Schacht~\\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.\nVia a double counting argument, they showed that for $p \\gg n^{-2}$, we have\\footnote{For any two functions $f$ and $g$, we write $f \\ll g$ to mean $f = o(g)$.}\n\\begin{align}\\label{eq:lower-bound-MRSS}\n\t\\text{pdim}(G_{n,p}) \\ge \\dfrac{pn \\log \\frac{1}{p} }{5\\log n }\n\\end{align}\nwith high probability.\nFurthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\\varepsilon > 0$ and $p$ such that $n^{-1/3} \\log^{4/3} n \\ll p \\ll n^{-\\varepsilon}$, we have $\\text{pdim}(G_{n,p}) = \\Theta_{\\varepsilon}(pn)$ with high probability.\nIn our first main result, we prove an upper bound that is valid for almost the entire range of $p$.",
    "context": "Motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem~\\cite{Deuber,ErdosHajnalPosa,Rodl},\nNešetřil, Pultr and Rödl~\\cite{Nesetril1977-pj,Nesetril1978-km} introduced the notion of the Prague dimension of a graph in the 1970s.\nFor a graph $G$, the Prague dimension $\\text{pdim}(G)$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph.\nThis is closely related to $\\text{idim}(G)$, defined as the minimum number of partial clique factors whose union equals $\\overline{G}$, the complement of $G$.\nIn fact, \nNešetřil and Rödl~\\cite{Nesetril1978-km} showed that for every graph $G$, we have\n\\begin{align}\\label{eq:pdim-idim}\n \\mathrm{idim}(G) \\le \\mathrm{pdim}(G) \\le \\mathrm{idim}(G) + 1. \n\\end{align}\n\nIn 1977, Lovász, Nešetřil, and Pultr~\\cite{Lovasz1980-kw} showed that a graph on $n\\ge2$ vertices has Prague dimension at most $n-1$ and classified all graphs which achieve this bound.\nGeneral upper and lower bounds were obtained by Alon~\\cite{Alon1986-oc}, who proved that if $G$ is a graph on $n$ vertices with minimum degree at least 1 and maximum degree $\\Delta$, then $\\log (n/ \\Delta)/ \\log 2 \\le \\text{pdim}(G) = O(\\Delta^2\\log n)$; all our asymptotic expressions are meant as $n \\to \\infty$, and all logarithms are natural.\nThe upper bound was later improved by Eaton and Rödl~\\cite{Eaton1996-ap}, who showed that \n\\begin{align*}\n \\text{pdim}(G) = O(\\Delta \\log n).\n\\end{align*}\nMoreover, they proved that this is best possible up to a factor of $1/(\\log \\Delta + \\log \\log (n/2\\Delta) )$.\nWhen $\\Delta$ is a constant, these results provide the order of magnitude of $\\text{pdim}(G)$.\nPrecise asymptotics are known for matchings, trees and cycles~\\cite{Alon1986-oc,Lovasz1980-kw,Poljak1981-lt}, hypercubes~\\cite{Krivka} and Kneser graphs $K(n,k)$ for constant values of $k$~\\cite{Poljak1978-um}.\nFor more details, we refer the reader to~\\cite{Kantor}.\n\nNešetřil and Rödl~\\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\\Omega_p(n/\\log n)$ with high probability for constant values of $p$.\nMany years later, Guo, Patton and Warnke~\\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\\cite{Pippenger1989-ak} type\nedge-colouring result for random hypergraphs with large uniformities.\nMore recently, Molnar, Rödl, Sales and Schacht~\\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.\nVia a double counting argument, they showed that for $p \\gg n^{-2}$, we have\\footnote{For any two functions $f$ and $g$, we write $f \\ll g$ to mean $f = o(g)$.}\n\\begin{align}\\label{eq:lower-bound-MRSS}\n \\text{pdim}(G_{n,p}) \\ge \\dfrac{pn \\log \\frac{1}{p} }{5\\log n }\n\\end{align}\nwith high probability.\nFurthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\\varepsilon > 0$ and $p$ such that $n^{-1/3} \\log^{4/3} n \\ll p \\ll n^{-\\varepsilon}$, we have $\\text{pdim}(G_{n,p}) = \\Theta_{\\varepsilon}(pn)$ with high probability.\nIn our first main result, we prove an upper bound that is valid for almost the entire range of $p$.",
    "full_context": "Motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem~\\cite{Deuber,ErdosHajnalPosa,Rodl},\nNešetřil, Pultr and Rödl~\\cite{Nesetril1977-pj,Nesetril1978-km} introduced the notion of the Prague dimension of a graph in the 1970s.\nFor a graph $G$, the Prague dimension $\\text{pdim}(G)$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph.\nThis is closely related to $\\text{idim}(G)$, defined as the minimum number of partial clique factors whose union equals $\\overline{G}$, the complement of $G$.\nIn fact, \nNešetřil and Rödl~\\cite{Nesetril1978-km} showed that for every graph $G$, we have\n\\begin{align}\\label{eq:pdim-idim}\n \\mathrm{idim}(G) \\le \\mathrm{pdim}(G) \\le \\mathrm{idim}(G) + 1. \n\\end{align}\n\nIn 1977, Lovász, Nešetřil, and Pultr~\\cite{Lovasz1980-kw} showed that a graph on $n\\ge2$ vertices has Prague dimension at most $n-1$ and classified all graphs which achieve this bound.\nGeneral upper and lower bounds were obtained by Alon~\\cite{Alon1986-oc}, who proved that if $G$ is a graph on $n$ vertices with minimum degree at least 1 and maximum degree $\\Delta$, then $\\log (n/ \\Delta)/ \\log 2 \\le \\text{pdim}(G) = O(\\Delta^2\\log n)$; all our asymptotic expressions are meant as $n \\to \\infty$, and all logarithms are natural.\nThe upper bound was later improved by Eaton and Rödl~\\cite{Eaton1996-ap}, who showed that \n\\begin{align*}\n \\text{pdim}(G) = O(\\Delta \\log n).\n\\end{align*}\nMoreover, they proved that this is best possible up to a factor of $1/(\\log \\Delta + \\log \\log (n/2\\Delta) )$.\nWhen $\\Delta$ is a constant, these results provide the order of magnitude of $\\text{pdim}(G)$.\nPrecise asymptotics are known for matchings, trees and cycles~\\cite{Alon1986-oc,Lovasz1980-kw,Poljak1981-lt}, hypercubes~\\cite{Krivka} and Kneser graphs $K(n,k)$ for constant values of $k$~\\cite{Poljak1978-um}.\nFor more details, we refer the reader to~\\cite{Kantor}.\n\nNešetřil and Rödl~\\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\\Omega_p(n/\\log n)$ with high probability for constant values of $p$.\nMany years later, Guo, Patton and Warnke~\\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\\cite{Pippenger1989-ak} type\nedge-colouring result for random hypergraphs with large uniformities.\nMore recently, Molnar, Rödl, Sales and Schacht~\\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.\nVia a double counting argument, they showed that for $p \\gg n^{-2}$, we have\\footnote{For any two functions $f$ and $g$, we write $f \\ll g$ to mean $f = o(g)$.}\n\\begin{align}\\label{eq:lower-bound-MRSS}\n \\text{pdim}(G_{n,p}) \\ge \\dfrac{pn \\log \\frac{1}{p} }{5\\log n }\n\\end{align}\nwith high probability.\nFurthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\\varepsilon > 0$ and $p$ such that $n^{-1/3} \\log^{4/3} n \\ll p \\ll n^{-\\varepsilon}$, we have $\\text{pdim}(G_{n,p}) = \\Theta_{\\varepsilon}(pn)$ with high probability.\nIn our first main result, we prove an upper bound that is valid for almost the entire range of $p$.\n\nNešetřil and Rödl~\\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\\Omega_p(n/\\log n)$ with high probability for constant values of $p$.\nMany years later, Guo, Patton and Warnke~\\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\\cite{Pippenger1989-ak} type\nedge-colouring result for random hypergraphs with large uniformities.\nMore recently, Molnar, Rödl, Sales and Schacht~\\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.\nVia a double counting argument, they showed that for $p \\gg n^{-2}$, we have\\footnote{For any two functions $f$ and $g$, we write $f \\ll g$ to mean $f = o(g)$.}\n\\begin{align}\\label{eq:lower-bound-MRSS}\n \\text{pdim}(G_{n,p}) \\ge \\dfrac{pn \\log \\frac{1}{p} }{5\\log n }\n\\end{align}\nwith high probability.\nFurthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\\eps > 0$ and $p$ such that $n^{-1/3} \\log^{4/3} n \\ll p \\ll n^{-\\eps}$, we have $\\text{pdim}(G_{n,p}) = \\Theta_{\\eps}(pn)$ with high probability.\nIn our first main result, we prove an upper bound that is valid for almost the entire range of $p$.\n\nIn particular, by combining our result with the lower bound in~\\eqref{eq:lower-bound-MRSS}, we obtain that for all $\\eps > 0$ and $p$ such that $ n^{-1+\\eps} \\le p \\le n^{-\\eps}$, with high probability the Prague dimension of $G_{n,p}$ is $\\Theta_{\\eps}(pn)$.\nAs it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\\log n$ is a natural barrier for the range of $p$ our method can handle.\n\nOur methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\\overline{\\theta}_1(G)$.\nInitially, one may conjecture that with high probability $\\overline{\\theta}_1(G)$ should be of order $n^2/\\alpha(G)^2$ for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\\overline{G}$.\nFrieze and Reed~\\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\\H{o}s, Spencer, and West~\\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.\nSurprisingly, this is not true for the sparse range.\nIn~\\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\\eps >0$ and $n^{-1} \\ll p \\le 1-\\eps$, with high probability we have\n\\begin{align*}\n \\overline{\\theta}_1(G_{n,p}) = \\Omega \\left ( \\dfrac{n^2 \\log \\frac{1}{p} }{\\alpha^2(G_{n,p})} \\right ).\n\\end{align*}\nIt is worth mentioning that due to the work of Matula~\\cite{Matula1970-ka}, Bollobás and Erd\\H{o}s~\\cite{Bollobas1976-al}, Grimmett and McDiarmid~\\cite{Grimmett1975-vs}, and Frieze~\\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\\Theta(p^{-1}\\log pn)$ for $n^{-1} \\ll p \\le 1- \\eps$, for any fixed $\\eps > 0$.\nAlso note that\n$\\log \\frac{1}{p} = \\Theta_{\\eps}(\\log n)$ for $ n^{-1}\\ll p \\ll n^{-\\eps}$.\n\n\\begin{theorem}\\label{thm:main}\n Let $\\eps\\in (0,1)$ and $(\\log n)^{2+\\eps} n^{-1} \\le p < n^{-\\eps}$.\n Then, with high probability we have\n \\begin{align*}\n \\overline{\\theta}_1(G_{n,p}) = \\Theta_{\\eps} \\left ( \\dfrac{n^2\\log n}{\\alpha^2(G_{n,p})} \\right ).\n \\end{align*}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:prob-non-edge}\n Let $\\eps \\in (0,2^{-10})$ and $(\\log n)^{2+\\eps} n^{-1} \\le p < n^{-\\eps}$. \n Let $G$ be an $n$-vertex $(\\eps,p)$-typical graph and $k = \\eps 2^{-10} p^{-1} \\log pn $.\n Then, for all distinct $u, v \\in V(G)$ such that $uv \\notin E(G)$, we have\n \\begin{align*}\n \\Pr{v \\in I_k} = \\big(1+o_{\\eps}(1) \\big ) \\dfrac{k}{n} \\qquad \n \\text{and} \\qquad \\Pr{u,v \\in I_k} = \\big (1+o_{\\eps}(1) \\big ) \\left ( \\dfrac{k}{n} \\right )^2.\n \\end{align*}\n\\end{theorem}\n\nLet $\\big(J_{i,j}: i \\in [t], j \\in [s] \\big )$ be a collection of $ts$ independent copies of $I_k = I_k(G)$.\n For each $i \\in [t]$, define $\\mathcal{I}_i$ to be the collection of vertex-disjoint subsets of $V(G)$ given by \n \\begin{align*}\n I_{i,j} \\coloneqq J_{i,j} \\setminus \\big ( J_{i,1} \\cup \\cdots \\cup J_{i,j-1} \\big )\n \\end{align*}\n for all $j \\in [s]$. \n By~\\eqref{eq:pdim-idim}, it suffices to show that with high probability every non-edge of $G$ is covered by some $I_{i,j}$.\n For $i \\in [t]$, the probability that a given non-edge $uv$ is covered by an independent set in the partition $\\mathcal{I}_i$ is equal to\n \\begin{align}\\label{eq:prob-non-edge}\n \\sum_{j=1}^{s}\\Pr{u,v \\in I_{i,j}} & = \\sum_{j=1}^{s} \\Pr{u,v \\in J_{i,j}} \\prod_{\\ell\\le j-1} \\Pr{ \\{ u,v \\} \\cap J_{i,\\ell} = \\emptyset}\\nonumber\\\\\n & = \\sum_{j=1}^{s} \\Pr{u,v \\in I_k} \\Pr{ \\{ u,v \\} \\cap I_k = \\emptyset}^{j-1}.\n \\end{align}\n By the inclusion-exclusion principle, the probability that neither $u$ nor $v$ is covered by $I_k$ is \n \\begin{align}\\label{eq:prob-non-edge-2}\n \\Pr{ \\{ u,v \\} \\cap I_k = \\emptyset} &= 1 - \\Pr{u \\in I_k} - \\Pr{v \\in I_k} + \\Pr{u,v \\in I_k} \\nonumber \\\\\n & = 1 - \\Theta_{\\eps} \\left ( \\dfrac{k}{n} \\right ) = 1 - \\Theta_{\\eps}(s^{-1}).\n \\end{align}\n Here we use Theorem~\\ref{thm:prob-non-edge} and the bound $k = o(n)$, which follows from the fact that $p \\gg n^{-1}\\log n$.\n\nLet us also recall the definition of the error terms that we use to control the deviation of the actual degrees from the expected trajectory. \nFor $i \\ge 0$, we set\n\\begin{align}\\label{eq:error-function}\n f_0 \\coloneqq 4\\log n \\sqrt{ \\dfrac{\\log pn}{pn} + p } \\qquad \\text{and} \\qquad f_i := \\left ( \\dfrac{1+16p}{1-p} \\right )^i f_0.\n\\end{align}\nNote that the error is increasing, yet we have \n\\begin{align}\\label{eq:bound-f}\n f_{i-1} < f_{i} \\le f_k \\le (pn)^{2^{-5}\\eps} f_0 = o(1) \\qquad \\text{and} \\qquad\n \\dfrac{1}{1\\pm f_i} = 1 \\pm \\dfrac{3}{2}f_i\n\\end{align}\nfor all $i \\in [k]$ and $(\\log n)^{2+\\eps} n^{-1} \\le p < n^{-\\eps}$.\nTo derive the inequality $f_k \\le (pn)^{2^{-5}\\eps}f_0$, we simply use that $(1+16p)(1-p)^{-1} \\le 1+2^5p \\le e^{2^5p}$.\nIt follows from~\\ref{item:P1} that\n\\begin{align}\\label{eq:initial-parameters-2}\n |V_i| = (1\\pm f_i)(1-p)^i n .\n\\end{align}\nfor all $i \\in [k]$. \nOther useful estimates are\n\\begin{align}\\label{eq:asymptotics}\n (1-3p)^k \\ge e^{-8pk} \\ge (np)^{-2^{-7}\\eps} \\ge n^{-2^{-7}(\\eps - \\eps^2)} \\ge n^{-2^{-7}\\eps } \\quad\n \\text{and} \\quad f_i > f_0 \\ge n^{-1/2}\n \\end{align}\nfor all $i \\in [k]$. \nAbove we use that $1-x \\ge e^{-2x}$ for all $x \\in (0,1/2)$.\n\nBy Freedman's inequality, and using that $f_0 = o(1)$ and $\\td_0 = pn$, we have \n \\begin{align}\\label{eq:freedman-appl-1}\n \\Pr{ X_{v,i}^{-}-X_{v,0}^{-} \\ge \\dfrac{f_0\\td_0}{2}} & \\le \n \\exp\\left(-\\frac{(f_0\\td_0)^2}{2^{12}(p^2n+\\log n)(pn + f_0\\td_0)}\\right) \\nonumber \\\\\n & \\le \\exp\\left(-\\frac{f_0^2pn}{2^{13}(p^2n+\\log n)}\\right) \\nonumber \\\\\n & \\le \\exp\\left(-\\frac{\\log^2 n \\log pn + p^2n \\log^2 n }{2^{9}(p^2n+\\log n)}\\right).\n \\end{align}\n In the last inequality, we use the definition of $f_0$ (see~\\eqref{eq:error-function}).\n Let $E$ denote the absolute value of the exponent in the last term. \n If $p^2n \\ge \\log n$, then $E \\ge 2^{-9}p^2n\\log^2n(p^2n+\\log n)^{-1} \\ge 2^{-10}\\log^2n$, and hence\n the right-hand side of~\\eqref{eq:freedman-appl-1} is at most $\\exp(-2^{-10}\\log^2 n)$; if $p^2n < \\log n$, then $E \\ge 2^{-9}\\log^2 n \\log pn(p^2n+\\log n)^{-1} \\ge 2^{-10}\\log n \\log pn$, and hence\n the right-hand side of~\\eqref{eq:freedman-appl-1} is at most $\\exp(-2^{-10}\\log n \\log pn)$.\n Therefore, in both cases we have\n \\begin{align*} \\Pr{X_{v,i}^{-}-X_{v,0}^{-} \\ge \\dfrac{f_0\\td_0}{2}} \\le n^{-2^{-10}\\log pn} \\quad \\text{and} \\quad \n \\Pr{-X_{v,i}^{+}+X_{v,0}^{+} \\ge \\dfrac{f_0\\td_0}{2}} \\le n^{-2^{-10}\\log pn}.\n \\end{align*}\n Summing over all $v \\in V_0$ and $i \\in [k-1]$, we obtain that $\\Pr{\\tau < k} \\le n^{-2^{-11}\\log pn}.$\n\\end{proof}",
    "post_theorem_intro_text_len": 6752,
    "post_theorem_intro_text": "In particular, by combining our result with the lower bound in~\\eqref{eq:lower-bound-MRSS}, we obtain that for all $\\varepsilon > 0$ and $p$ such that $ n^{-1+\\varepsilon} \\le p \\le n^{-\\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $\\Theta_{\\varepsilon}(pn)$.\nAs it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\\log n$ is a natural barrier for the range of $p$ our method can handle.\n\nTo prove Theorem~\\ref{thm:main-prague}, instead of working directly with the Prague dimension, we use~\\eqref{eq:pdim-idim} and upper bound $\\mathrm{idim}(G_{n,p})$.\nOur main idea is to select independent sets with size proportional to $\\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen.\nWe then control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.\n\nOur strategy is inspired by the work of Bennett and Bohman~\\cite{Bennett2016-di}, who use a random greedy independent set process to lower bound the independence number of regular hypergraphs under certain degree and codegree conditions.\nFor graphs, we also improve on their work by handling pseudorandom graphs with logarithmic degrees rather than regular graphs with degrees growing polynomially with the number of vertices.\nOur analysis is done via the differential equation method, which was popularised in the combinatorial community by Wormald~\\cite{Wormald-1,Wormald-2}.\nAs can be seen in the references given in~\\cite{Wormald-2}, the roots of the method may be traced back further, see for example the works of Kurtz~\\cite{Kurtz1970-aa}\nand Karp and Sipser~\\cite{Karp1981-aa}.\n\nOur methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\\overline{\\theta}_1(G)$.\nInitially, one may conjecture that with high probability $\\overline{\\theta}_1(G)$ should be of order $n^2/\\alpha(G)^2$  for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\\overline{G}$.\nFrieze and Reed~\\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\\H{o}s, Spencer, and West~\\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.\nSurprisingly, this is not true for the sparse range.\nIn~\\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\\varepsilon >0$ and $n^{-1} \\ll p \\le 1-\\varepsilon$, with high probability we have\n\\begin{align*}\n\t\\overline{\\theta}_1(G_{n,p}) = \\Omega \\left ( \\dfrac{n^2 \\log \\frac{1}{p} }{\\alpha^2(G_{n,p})} \\right ).\n\\end{align*}\nIt is worth mentioning that due to the work of Matula~\\cite{Matula1970-ka}, Bollobás and Erd\\H{o}s~\\cite{Bollobas1976-al}, Grimmett and McDiarmid~\\cite{Grimmett1975-vs}, and Frieze~\\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\\Theta(p^{-1}\\log pn)$ for $n^{-1} \\ll p \\le 1- \\varepsilon$, for any fixed $\\varepsilon > 0$.\nAlso note that\n$\\log \\frac{1}{p} = \\Theta_{\\varepsilon}(\\log n)$ for $ n^{-1}\\ll p \\ll n^{-\\varepsilon}$. \n\nRecently, Molnar, Rödl, Sales and Schacht~\\cite{MolnarRodlSalesSchacht} obtained a matching upper bound for every $\\varepsilon > 0$ and $p$ such that $n^{-1/2} \\log^{5/4} n \\ll p \\ll n^{-\\varepsilon}$. \nTheir argument heavily relies on the fact that the number of independent sets in $G_{n,p}$ is concentrated around its expectation with high probability.\nAs this no longer holds for $p \\ll n^{-1/2}$ (see Corollary~19 in~\\cite{Coja-Efthymiou}), there is a barrier at $n^{-1/2}$ for the range of $p$ their method can handle.\nOur second main theorem overcomes this natural barrier and extends their result to a much wide range of~$p$.\n\n\\begin{theorem}\\label{thm:main}\n\tLet $\\varepsilon\\in (0,1)$ and $(\\log n)^{2+\\varepsilon} n^{-1} \\le p < n^{-\\varepsilon}$.\n\tThen, with high probability we have\n\t\\begin{align*}\n\t\t\\overline{\\theta}_1(G_{n,p}) = \\Theta_{\\varepsilon} \\left ( \\dfrac{n^2\\log n}{\\alpha^2(G_{n,p})} \\right ).\n\t\\end{align*}\n\\end{theorem}\n\nIt is worth noting we were only able to overcome the previous obstacles because for many host graphs the independent set generated by our random greedy process is not uniformly distributed.\nAs an example, let $1 \\ll k \\ll n$ and take a complete bipartite graph $K[A,B]$ with $|A| = n/3$ and $|B| = 2n/3$.\nNow let $I$ be a uniformly-chosen random independent $k$-set and let $J$ be the independent set generated by our random greedy process.\nOne can check that if $u, v \\in A$, then $\\mathbb{P}\\left[u,v \\in I\\right] \\ll \\mathbb{P}\\left[u,v \\in J\\right]$.\n\nNow let us mention some of the historic background related to $\\overline{\\theta}_1$.\nIn the 1960s, Erd\\H{o}s, Goodman and Pósa~\\cite{Erdos1966-so} introduced the parameter $\\theta_1(G)$, defined as the minimum number of cliques required to cover all the edges of $G$.\nNotice that $\\overline{\\theta}_1(G) = \\theta_1(\\overline{G})$, and hence one may think of $\\theta_1(G)$ and $\\overline{\\theta}_1(G)$ as the dual versions of each other.\nIn~\\cite{Erdos1966-so}, the authors showed that $\\theta_1(G) \\le \\lfloor n^2/4 \\rfloor$ for every $n$-vertex graph $G$, which is attained by balanced complete bipartite graphs.\nIn 1994, Alon~\\cite{Alon1986-oc} showed that if $G$ is a $n$-vertex graph with maximum degree $\\Delta$, then $\\theta_1(\\overline{G}) = O(\\Delta^2 \\log n)$.\nEaton and Rödl~\\cite{Eaton1996-ap} showed that this bound is close to best possible, as there are $n$-vertex graphs $G$ with $\\theta_1(\\overline{G}) = \\Omega(\\Delta^2 \\log n / \\log \\Delta)$.\n\nFrieze and Reed~\\cite{Frieze1995-xa} showed that with high probability $\\theta_1(G_{n,p}) = \\Theta_p(n^2/\\log n)$ for constant values of $p$.\nGuo, Patton and Warnke~\\cite{Guo2023-wq} extended these results much further by showing that for every fixed $\\gamma \\in (0,1)$ and $p$ such that $n^{-2} \\le p \\le 1-\\gamma$, with high probability, we have $\\theta_1(G_{n,p}) = \\Theta (n^2p /\\log_{1/p}^{2} n )$.\nAs results in~\\cite{Guo2023-wq} left the range where $p$ tends to 1 open, it is more convenient to just consider $\\overline{\\theta}_1(G_{n,p})$ for $p$ tending to 0. \n\nThe rest of the paper is divided as follows.\nIn Section \\ref{sec:proofs}, we present the proofs of our main theorems; in Section~\\ref{sec:random-greedy}, we analyse our random greedy independent set process using the differential equation method; in Section~\\ref{sec:prob-cover}, we prove the main technical result of the paper, namely Theorem~\\ref{thm:main-dem}; and in Section \\ref{sec:preliminaries}, we present some tools and standard lemmas that will be used in the analysis of our random process.",
    "sketch": "To prove Theorem~\\ref{thm:main-prague}, the authors say they do not work directly with the Prague dimension; instead they “use~\\eqref{eq:pdim-idim} and upper bound $\\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”",
    "expanded_sketch": "To prove the main theorem, the authors say they do not work directly with the Prague dimension; instead they “use\n\\begin{align}\\label{eq:pdim-idim}\n\t\\idim(G) \\le \\pdim(G) \\le \\idim(G) + 1. \n\\end{align}\nand upper bound $\\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”",
    "expanded_theorem": "\\label{thm:main-prague}\n\tLet\\footnote{In all our statements, $\\varepsilon$ is a fixed constant and $n$ tends to infinity.} $\\varepsilon \\in (0,1)$ and $(\\log n)^{2+\\varepsilon} n^{-1} \\le p < n^{-\\varepsilon}$. \n\tThen, with high probability we have\n\t\\begin{align*}\n\t\t\\mathrm{pdim}(G_{n,p}) = O_{\\varepsilon} \\left ( \\dfrac{pn \\log n}{\\log pn} \\right ).\n\t\\end{align*},",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $G_{n,p}$ be the Erd\\H{o}s--R\\'enyi random graph on $n$ vertices, where each edge is present independently with probability $p$. For a graph $G$, its Prague dimension $\\mathrm{pdim}(G)$ is the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Fix $\\varepsilon \\in (0,1)$, assume $n \\to \\infty$, and suppose the edge probability satisfies $(\\log n)^{2+\\varepsilon} n^{-1} \\le p < n^{-\\varepsilon}$. Which quantitative estimate holds with high probability (that is, with probability tending to $1$ as $n\\to\\infty$)?",
      "correct_choice": {
        "label": "A",
        "text": "With high probability, \\[\\mathrm{pdim}(G_{n,p}) = O_{\\varepsilon}\\!\\left(\\frac{pn\\log n}{\\log(pn)}\\right).\\] Equivalently, there exists a constant $C_{\\varepsilon}>0$ depending only on $\\varepsilon$ such that, with high probability, \\[\\mathrm{pdim}(G_{n,p}) \\le C_{\\varepsilon}\\frac{pn\\log n}{\\log(pn)}.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "With high probability, \\[\\mathrm{pdim}(G_{n,p}) = O_{\\varepsilon}\\!\\left(\\frac{pn\\log n}{\\log(1/p)}\\right).\\] Equivalently, there exists a constant $C_{\\varepsilon}>0$ depending only on $\\varepsilon$ such that, with high probability, \\[\\mathrm{pdim}(G_{n,p}) \\le C_{\\varepsilon}\\frac{pn\\log n}{\\log(1/p)}.\\]"
        },
        {
          "label": "C",
          "text": "With high probability, \\[\\mathrm{pdim}(G_{n,p}) = O_{\\varepsilon}(pn\\log n).\\] Equivalently, there exists a constant $C_{\\varepsilon}>0$ depending only on $\\varepsilon$ such that, with high probability, \\[\\mathrm{pdim}(G_{n,p}) \\le C_{\\varepsilon}\\,pn\\log n.\\]"
        },
        {
          "label": "D",
          "text": "With high probability, \\[\\mathrm{pdim}(G_{n,p}) = O\\!\\left(\\frac{pn\\log n}{\\log(pn)}\\right).\\] Equivalently, there exists an absolute constant $C>0$ independent of $\\varepsilon$ such that, with high probability, \\[\\mathrm{pdim}(G_{n,p}) \\le C\\frac{pn\\log n}{\\log(pn)}.\\]"
        },
        {
          "label": "E",
          "text": "With high probability, \\[\\mathrm{pdim}(G_{n,p}) = \\Theta_{\\varepsilon}\\!\\left(\\frac{pn\\log n}{\\log(pn)}\\right).\\] Equivalently, there exist constants $c_{\\varepsilon},C_{\\varepsilon}>0$ depending only on $\\varepsilon$ such that, with high probability, \\[c_{\\varepsilon}\\frac{pn\\log n}{\\log(pn)}\\le \\mathrm{pdim}(G_{n,p}) \\le C_{\\varepsilon}\\frac{pn\\log n}{\\log(pn)}.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "logarithmic denominator from process length/coverage",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the improving factor \\(\\log(pn)\\) in the denominator",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the implied constant on \\(\\varepsilon\\) and the admissible \\(p\\)-range",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "upper bound upgraded to matching lower bound without the separate lower-bound argument",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option and gives only the hypotheses and terminology needed for the result. There is no direct phrasing that singles out choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem states the assumptions and asks which asymptotic conclusion holds with high probability. The correct choice is basically the theorem statement itself rather than a conclusion derived from intermediate reasoning."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways: the denominator, loss of a logarithmic factor, dependence on epsilon, and promotion of an upper bound to a Theta statement. Still, the item mainly tests recognition of the exact result rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible. They reflect realistic failure modes: confusing log(pn) with log(1/p), accepting a weaker true bound, overlooking epsilon-dependence of constants, or overstating an upper bound as a matching asymptotic."
      },
      "total_score": 5,
      "overall_assessment": "A moderately good MCQ with excellent distractors and no answer leakage, but it is largely a direct restatement of a theorem and therefore only weakly tests generative reasoning."
    }
  },
  {
    "id": "2512.08994v1",
    "paper_link": "http://arxiv.org/abs/2512.08994v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main theorem}\n  Let $\\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\\max(\\deg R_1, \\dots, \\deg R_m)$, and let $S$ denote the set of indices $1 \\le i \\le m$ for which $\\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \\[ \\left|I_n(R_1, \\dots, R_m)-\\frac{I(n)}{q^m}\\right| \\le \\frac{q^m-1}{q^m} \\cdot \\min_{0 \\le u+v < n} \\left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\\sqrt{S_2(u, v)}\\right) \\] where \\[ S_1(u, v) = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] and \\[ S_2(u, v) = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\]",
      "start_pos": 7078,
      "end_pos": 7947,
      "label": "main theorem"
    },
    "ref_dict": {
      "lemmas": "\\label{lemmas}\n\\subsection{Vaughan's identity in $\\mathbb{F}_q[x]$}\nIn \\cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\\mathbb{N}$ is utilized. This bound en",
      "main proof": "\\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2))  \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m}  \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}",
      "future directions": "\\label{future directions}\nIn this section, we provide several research questions that allow for potential extensions of this work. \n\nFirst, we note that our error bounds rely on the assumption that $n",
      "prelim": "\\label{prelim}\nThroughout this paper, we will let $q$ be an odd prime power with prime divisor $p$. We will use $\\mathcal{M}(n)$ to denote the set of polynomials of degree $n$ in $\\mathbb{F}_q[x]$, an"
    },
    "pre_theorem_intro_text_len": 3557,
    "pre_theorem_intro_text": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases. \n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}. \n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.",
    "context": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.\n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}.\n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.",
    "full_context": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.\n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}.\n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.\n\n\\begin{lemma}\\label{Vaughan's identity}\n Let $\\Psi: \\mathbb{F}_q[x] \\to \\mathbb{C}$ be a function with $|\\Psi(f)| \\le 1$. If $u$ and $v$ are two integers in $[1, n]$ with $u+v>n$, then \\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\Psi(f)\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2}, \\] with \\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\Psi(gh)\\right| \\] and \\[ \\Sigma_2 = \\max_{v \\le i \\le n-u} \\max_{\\deg g_1 = n-i} \\sum_{\\deg g_2 = n-i} \\left|\\sum_{\\deg h = i} \\Psi(hg_1)\\overline{\\Psi(hg_2)}\\right|, \\] where both sums above are taken over monic polynomials.\n\\end{lemma}\nIn particular, it suffices to obtain upper bounds on $\\Sigma_1$ and $\\Sigma_2$. Using polarization, we will reduce the computation to computing character sums on multilinear polynomials.\n\n\\begin{lemma}\\label{Polarization result}\n Let $P_1(f), \\dots, P_m(f)$ be polynomials in the coefficients of a $k$-degree polynomial $f$. Let $j=\\max(\\deg P_1, \\dots, \\deg P_m)$ and $S$ denote the set of indices $i \\in [1, m]$ for which $\\deg P_i = j$. Then we have \\[ \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right| \\le \\left(q^{k}\\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right|\\right)^{1/2^j},\\] where the sum above is over monic $f$. \n\\end{lemma}\n\\begin{proof}\n Note that \\begin{align*}\n \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right|^2 & = \\sum_{\\deg f = k} \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(P_i(f+h))\\overline{\\psi_i(P_i(f))} \\\\ &= \\left|\\sum_{\\deg f = k} \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(\\Delta_h P_i(f))\\right| \\\\ &\\le \\sum_{\\deg f = k}\\left| \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(\\Delta_h P_i(f))\\right|. \n \\end{align*}\n Repeatedly squaring in this fashion, we obtain that at the $j$th squaring, \\[ \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right|^{2^j} \\le q^{k} \\left|\\sum_{h_1, \\dots, h_j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(h_1, \\dots, h_j)) \\right|, \\] as desired. \n\\end{proof}\nRecall that the purpose of this application of polarization is to obtain upper bounds on the sums $\\Sigma_1$ and $\\Sigma_2$ from Lemma \\ref{Vaughan's identity} upon selecting $\\Psi(f) = \\prod_{i=1}^m \\psi_i(R_i(f))$. Indeed, we may consider the following choices of $P_i$, which are both polynomials in the coefficients of their arguments. \n\\begin{itemize}\n \\item For $\\Sigma_1$, let $(P_i)_g(f)=R_i(fg)$ for all $f \\in \\mathcal{M}(n-\\deg g)$ and $1 \\le i \\le m$. \n \\item For $\\Sigma_2$, let $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$ for all $f \\in \\mathcal{M}(n - \\max(\\deg g_1, \\deg g_2))$ and $1 \\le i \\le m$. \n\\end{itemize}\n\n\\subsection{Exponential sums of multilinear polynomials}\nNote that Lemma \\ref{Polarization result} requires us to only work with exponential sums of multilinear polynomials going forward. In order to obtain useful bounds on the exponential sums of multilinear polynomials, we require a corollary of a recent result on the comparison between the analytic rank and partition rank of tensors.\n\\begin{defn}\n The \\emph{analytic rank} $\\operatorname{AR}(T)$ of $T$ is defined as \\[ \\operatorname{AR}(T) := -\\log_q\\left(\\frac{1}{q^n} \\sum_{(x_1, \\dots, x_n) \\in \\mathbb{F}_q^n} \\psi_0(T(x_1, \\dots, x_n))\\right). \\] The \\emph{partition rank} $\\operatorname{PR}(T)$ of $T$ is defined as the smallest number $r$ such that $T$ can be written as the sum of $r$ reducible homogeneous polynomials. \n\\end{defn}\n\n\\begin{lemma}\\label{char sums and ranks}\nFor all polynomials $f$ of degree $n<p$, $\\operatorname{PR}(f^{\\circ}) \\ge \\operatorname{rank}(f)$. Consequently, for any tuple $(P_1, \\dots, P_m)$ of polynomials and nontrivial tuple $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$, there exists a constant $c>0$ dependent only on $\\max(\\deg P_1, \\dots, \\deg P_m)$ such that \\[ \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| \\le q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\] where $j = \\max(\\deg P_1, \\dots, \\deg P_m)<p$ and $S$ is the set of indices $1 \\le i \\le m$ for which $\\deg P_i = j$. \n\\end{lemma}\n\\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m} \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}",
    "post_theorem_intro_text_len": 687,
    "post_theorem_intro_text": "The structure of this paper is as follows. We begin by stating the preliminaries assumed throughout this paper in Section \\ref{prelim}. Next, in Section \\ref{lemmas}, we state and prove the lemmas that play a key role in the proof of our main result. Our approach, based on character sums, uses a function field analogue of Vaughan's identity, polarization to reduce the problem to working with multilinear exponential sums, and partition rank to bound the multilinear exponential sums. In Section \\ref{main proof}, we prove the main result by applying the lemmas from Section \\ref{lemmas}, and we end with Section \\ref{future directions}, in which we discuss future research directions.",
    "sketch": "To prove Theorem~\\ref{main theorem}, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from Section~\\ref{lemmas}” in Section~\\ref{main proof}.",
    "expanded_sketch": "To prove the main theorem, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from \\label{lemmas}\n\\subsection{Vaughan's identity in $\\mathbb{F}_q[x]$}\nIn \\cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\\mathbb{N}$ is utilized. This bound en” in \\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2))  \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m}  \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}\n”.",
    "expanded_theorem": "\\label{main theorem}\n  Let $\\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\\max(\\deg R_1, \\dots, \\deg R_m)$, and let $S$ denote the set of indices $1 \\le i \\le m$ for which $\\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \\[ \\left|I_n(R_1, \\dots, R_m)-\\frac{I(n)}{q^m}\\right| \\le \\frac{q^m-1}{q^m} \\cdot \\min_{0 \\le u+v < n} \\left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\\sqrt{S_2(u, v)}\\right) \\] where \\[ S_1(u, v) = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] and \\[ S_2(u, v) = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\],",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb{F}_q\\) be a finite field of odd order \\(q\\) and characteristic \\(p\\), and let \\(n<p\\). Let \\(R_1,\\dots,R_m\\) be polynomial functions of the coefficients of monic degree-\\(n\\) polynomials over \\(\\mathbb{F}_q\\). Define \\(I_n(R_1,\\dots,R_m)\\) to be the number of monic irreducible polynomials \\(f\\in \\mathbb{F}_q[x]\\) of degree \\(n\\) such that \\(R_1(f)=\\cdots=R_m(f)=0\\), and let \\(I(n)\\) denote the number of monic irreducible polynomials of degree \\(n\\) over \\(\\mathbb{F}_q\\). Set \\(j:=\\max(\\deg R_1,\\dots,\\deg R_m)\\), let \\(S=\\{1\\le i\\le m: \\deg R_i=j\\}\\), and define \\((P_i)_g(f):=R_i(fg)\\) and \\((P_i)_{g_1,g_2}(f):=R_i(g_1f)-R_i(g_2f)\\). For nonnegative integers \\(u,v\\) with \\(u+v<n\\), define\n\\[\nS_1(u,v)=\\sum_{0\\le d\\le u+v}\\sum_{\\deg g=d} q^{(n-d)+\\frac{n-d}{2^j}-c\\,\\operatorname{rank}(((P_i)_g)_{i\\in S})}\n\\]\nand\n\\[\nS_2(u,v)=\\max_{v\\le k\\le n-u}\\max_{\\deg g_1=n-k}\\sum_{\\deg g_2=n-k} q^{k-c\\,\\operatorname{rank}(((P_i)_{g_1,g_2})_{i\\in S})},\n\\]\nwhere the degree-constrained sums are over monic polynomials. Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v\\le n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that for every pair of nonnegative integers \\(u,v\\) with \\(u+v<n\\),\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}S_2(u,v)\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j,m\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "admissible_range_u_plus_v",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_minimization_over_u_v",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "square_root_on_S2",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence_of_constant_and_prefactor",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct estimate or single out option A. It only sets up notation and asks which uniform bound holds."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially an exact theorem-recall item: the task is to recognize the correct formal statement of a quoted estimate among near-verbatim variants, rather than derive a new conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some careful comparison is needed because the options differ in subtle quantifiers, index sets, and error terms. However, the item mainly tests memory/statement matching, not genuinely generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are mathematically plausible, close to the target statement, and reflect realistic failure modes such as altered minimization range, replacing a sharp statement by a weaker true one, using the wrong rank family, or losing the square root."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed in terms of distractor plausibility and lack of answer leakage, but weak as a reasoning assessment because it is largely a theorem-statement recognition problem."
    }
  },
  {
    "id": "2512.08994v1",
    "paper_link": "http://arxiv.org/abs/2512.08994v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main theorem}\n  Let $\\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\\max(\\deg R_1, \\dots, \\deg R_m)$, and let $S$ denote the set of indices $1 \\le i \\le m$ for which $\\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \\[ \\left|I_n(R_1, \\dots, R_m)-\\frac{I(n)}{q^m}\\right| \\le \\frac{q^m-1}{q^m} \\cdot \\min_{0 \\le u+v < n} \\left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\\sqrt{S_2(u, v)}\\right) \\] where \\[ S_1(u, v) = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] and \\[ S_2(u, v) = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\]",
      "start_pos": 7078,
      "end_pos": 7947,
      "label": "main theorem"
    },
    "ref_dict": {
      "lemmas": "\\label{lemmas}\n\\subsection{Vaughan's identity in $\\mathbb{F}_q[x]$}\nIn \\cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\\mathbb{N}$ is utilized. This bound en",
      "main proof": "\\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2))  \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m}  \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}",
      "future directions": "\\label{future directions}\nIn this section, we provide several research questions that allow for potential extensions of this work. \n\nFirst, we note that our error bounds rely on the assumption that $n",
      "prelim": "\\label{prelim}\nThroughout this paper, we will let $q$ be an odd prime power with prime divisor $p$. We will use $\\mathcal{M}(n)$ to denote the set of polynomials of degree $n$ in $\\mathbb{F}_q[x]$, an"
    },
    "pre_theorem_intro_text_len": 3557,
    "pre_theorem_intro_text": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases. \n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}. \n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.",
    "context": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.\n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}.\n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.",
    "full_context": "The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \\cite{Hansen92}, who conjectured that for all prime powers $q \\ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \\cite{Wan97} proved the Hansen-Mullen conjecture for $n \\ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.\n\nIn general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \\textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \\cite{moshkovitz2024quasilinearrelationpartitionanalytic}.\n\nSince it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\\dots+a_{n-1}x^{n-1}+x^n \\in \\mathbb{F}_q[x]$ whose coefficient vector $(a_0, \\dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \\cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \\cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.\n\nThe goal of this paper is as follows. \n\\begin{quote}\n\\textsl{Fix polynomials $R_1, \\dots, R_m$ on the coefficients of monic polynomials in $\\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \\dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \\dots, R_m(f)) = (0, \\dots, 0)$.}\n\\end{quote}\nSince on average, the polynomials $R_1$, \\dots, $R_m$ take on equidistributed values, we should expect to have \\[ I_n(R_1, \\dots, R_m) \\approx \\frac{I(n)}{q^m}, \\] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right|$.\n\n\\begin{lemma}\\label{Vaughan's identity}\n Let $\\Psi: \\mathbb{F}_q[x] \\to \\mathbb{C}$ be a function with $|\\Psi(f)| \\le 1$. If $u$ and $v$ are two integers in $[1, n]$ with $u+v>n$, then \\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\Psi(f)\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2}, \\] with \\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\Psi(gh)\\right| \\] and \\[ \\Sigma_2 = \\max_{v \\le i \\le n-u} \\max_{\\deg g_1 = n-i} \\sum_{\\deg g_2 = n-i} \\left|\\sum_{\\deg h = i} \\Psi(hg_1)\\overline{\\Psi(hg_2)}\\right|, \\] where both sums above are taken over monic polynomials.\n\\end{lemma}\nIn particular, it suffices to obtain upper bounds on $\\Sigma_1$ and $\\Sigma_2$. Using polarization, we will reduce the computation to computing character sums on multilinear polynomials.\n\n\\begin{lemma}\\label{Polarization result}\n Let $P_1(f), \\dots, P_m(f)$ be polynomials in the coefficients of a $k$-degree polynomial $f$. Let $j=\\max(\\deg P_1, \\dots, \\deg P_m)$ and $S$ denote the set of indices $i \\in [1, m]$ for which $\\deg P_i = j$. Then we have \\[ \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right| \\le \\left(q^{k}\\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right|\\right)^{1/2^j},\\] where the sum above is over monic $f$. \n\\end{lemma}\n\\begin{proof}\n Note that \\begin{align*}\n \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right|^2 & = \\sum_{\\deg f = k} \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(P_i(f+h))\\overline{\\psi_i(P_i(f))} \\\\ &= \\left|\\sum_{\\deg f = k} \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(\\Delta_h P_i(f))\\right| \\\\ &\\le \\sum_{\\deg f = k}\\left| \\sum_{\\deg h < k} \\prod_{i=1}^m \\psi_i(\\Delta_h P_i(f))\\right|. \n \\end{align*}\n Repeatedly squaring in this fashion, we obtain that at the $j$th squaring, \\[ \\left|\\sum_{\\deg f = k} \\prod_{i=1}^m \\psi_i(P_i(f))\\right|^{2^j} \\le q^{k} \\left|\\sum_{h_1, \\dots, h_j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(h_1, \\dots, h_j)) \\right|, \\] as desired. \n\\end{proof}\nRecall that the purpose of this application of polarization is to obtain upper bounds on the sums $\\Sigma_1$ and $\\Sigma_2$ from Lemma \\ref{Vaughan's identity} upon selecting $\\Psi(f) = \\prod_{i=1}^m \\psi_i(R_i(f))$. Indeed, we may consider the following choices of $P_i$, which are both polynomials in the coefficients of their arguments. \n\\begin{itemize}\n \\item For $\\Sigma_1$, let $(P_i)_g(f)=R_i(fg)$ for all $f \\in \\mathcal{M}(n-\\deg g)$ and $1 \\le i \\le m$. \n \\item For $\\Sigma_2$, let $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$ for all $f \\in \\mathcal{M}(n - \\max(\\deg g_1, \\deg g_2))$ and $1 \\le i \\le m$. \n\\end{itemize}\n\n\\subsection{Exponential sums of multilinear polynomials}\nNote that Lemma \\ref{Polarization result} requires us to only work with exponential sums of multilinear polynomials going forward. In order to obtain useful bounds on the exponential sums of multilinear polynomials, we require a corollary of a recent result on the comparison between the analytic rank and partition rank of tensors.\n\\begin{defn}\n The \\emph{analytic rank} $\\operatorname{AR}(T)$ of $T$ is defined as \\[ \\operatorname{AR}(T) := -\\log_q\\left(\\frac{1}{q^n} \\sum_{(x_1, \\dots, x_n) \\in \\mathbb{F}_q^n} \\psi_0(T(x_1, \\dots, x_n))\\right). \\] The \\emph{partition rank} $\\operatorname{PR}(T)$ of $T$ is defined as the smallest number $r$ such that $T$ can be written as the sum of $r$ reducible homogeneous polynomials. \n\\end{defn}\n\n\\begin{lemma}\\label{char sums and ranks}\nFor all polynomials $f$ of degree $n<p$, $\\operatorname{PR}(f^{\\circ}) \\ge \\operatorname{rank}(f)$. Consequently, for any tuple $(P_1, \\dots, P_m)$ of polynomials and nontrivial tuple $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$, there exists a constant $c>0$ dependent only on $\\max(\\deg P_1, \\dots, \\deg P_m)$ such that \\[ \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| \\le q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\] where $j = \\max(\\deg P_1, \\dots, \\deg P_m)<p$ and $S$ is the set of indices $1 \\le i \\le m$ for which $\\deg P_i = j$. \n\\end{lemma}\n\\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m} \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}",
    "post_theorem_intro_text_len": 687,
    "post_theorem_intro_text": "The structure of this paper is as follows. We begin by stating the preliminaries assumed throughout this paper in Section \\ref{prelim}. Next, in Section \\ref{lemmas}, we state and prove the lemmas that play a key role in the proof of our main result. Our approach, based on character sums, uses a function field analogue of Vaughan's identity, polarization to reduce the problem to working with multilinear exponential sums, and partition rank to bound the multilinear exponential sums. In Section \\ref{main proof}, we prove the main result by applying the lemmas from Section \\ref{lemmas}, and we end with Section \\ref{future directions}, in which we discuss future research directions.",
    "sketch": "To prove Theorem~\\ref{main theorem}, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from Section~\\ref{lemmas}” in Section~\\ref{main proof}.",
    "expanded_sketch": "To prove the main theorem, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from \\label{lemmas}\n\\subsection{Vaughan's identity in $\\mathbb{F}_q[x]$}\nIn \\cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\\mathbb{N}$ is utilized. This bound en” in \\begin{proof}\nNote that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\\frac{1}{n!}f^{\\circ}(x, \\dots x)$. Thus, \\[ \\operatorname{rank}(f) = \\operatorname{rank}(f^{\\circ}(x, \\dots x)) \\le \\operatorname{PR}(f^{\\circ}), \\] proving the first part. Let $L$ denote the set of nontrivial $\\mathbb{F}_q$-combinations of $\\{P_i: i \\in S\\}$. Observe that \\begin{align*} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\prod_{i \\in S} \\psi_i(P_i^{\\circ}(x)) \\right| = \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(\\sum_{i \\in S} \\varphi(\\psi_i)P_i^{\\circ}(x)\\right) \\right| \\le \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| \\end{align*} and \\begin{align*} \\max_{P \\in L} \\left|\\sum_{x \\in (\\mathbb{F}_q^k)^j} \\psi_0\\left(P^{\\circ}(x)\\right) \\right| &\\le \\max_{P \\in L} q^{n-\\operatorname{AR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{PR}(P^{\\circ})} \\\\ &\\le \\max_{P \\in L} q^{n-c \\cdot \\operatorname{rank}(P)} \\\\ &= q^{n-c \\cdot \\operatorname{rank}((P_i)_{i \\in S})}, \\end{align*} which proves the second part.\n\\end{proof}\n\n\\section{Proof of main result}\\label{main proof}\nWe are ready to prove our main result.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nBy the orthogonality of characters, we have \\[ I_n(R_1, \\dots, R_m) = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m)} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] Thus, \\[ I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m} = \\frac{1}{q^m} \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)). \\] We may rewrite the above as \\begin{align*} \\sum_{f \\in \\mathcal{P}(n)} \\prod_{i=1}^m \\psi_i(R_i(f)) &= \\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{f \\in \\mathcal{P}(n)}\\prod_{i=1}^m \\psi_i(R_i(f)) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O\\left(\\sum_{f \\in \\mathcal{M}(n)} \\mathbf{1}_{\\Lambda(f) > 1}\\right) \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(I(n)-I(n/2))  \\\\ &= \\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f) \\prod_{i=1}^m \\psi_i(R_i(f)) + O(q^{n/2}/n) \\end{align*} Therefore, \\[ \\left|I_n(R_1, \\dots, R_m) - \\frac{I(n)}{q^m}\\right| \\le \\sum_{(\\psi_1, \\dots, \\psi_m) \\neq \\mathbf{0}} \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^{m}  \\psi_i(R_i(f))\\right| \\] by the triangle inequality. By Lemma \\ref{Vaughan's identity}, we have for all nontrivial tuples $(\\psi_1, \\dots, \\psi_m)$ of additive characters on $\\mathbb{F}_q$ that\n\\[ \\left|\\sum_{f \\in \\mathcal{M}(n)} \\Lambda(f)\\prod_{i=1}^m \\psi_i(R_i(f))\\right| \\ll n\\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\\Sigma_2^{1/2} \\] \nfor all choices of $u$ and $v$ with $u+v<n$, where \n\\[ \\Sigma_1 = \\sum_{\\deg g \\le u+v} \\left|\\sum_{\\deg h = n - \\deg g} \\prod_{i=1}^m \\psi_i((P_i)_g(h)) \\right| \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} \\left|\\sum_{\\deg h = k} \\prod_{i=1}^m \\psi_i((P_i)_{g_1, g_2}(h)) \\right|. \\] \nBy Lemma \\ref{Polarization result}, we have \n\\[ \\Sigma_1 \\le \\sum_{0 \\le d \\le u+v} q^{\\frac{n-d}{2^j}} \\sum_{\\deg g = d} \\left|\\sum_{x \\in (\\mathbb{F}_q^d)^{n-d}} \\prod_{i \\in S} \\psi_i((P_i)_g^{\\circ}(x))\\right|^{1/2^j} \\] \nand\n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{\\frac{k}{2^j}}\\left|\\sum_{x \\in (\\mathbb{F}_q^{n-k})^k} \\prod_{i \\in S} \\psi_i((P_i)_{g_1, g_2}^{\\circ}(x))\\right|^{1/2^j}. \\] \nThus, by Lemma \\ref{char sums and ranks}, we have \n\\[ \\Sigma_1 = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] \nand \n\\[ \\Sigma_2 = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\] The desired result follows. \n\\end{proof}\n”.",
    "expanded_theorem": "\\label{main theorem}\n  Let $\\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\\max(\\deg R_1, \\dots, \\deg R_m)$, and let $S$ denote the set of indices $1 \\le i \\le m$ for which $\\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \\[ \\left|I_n(R_1, \\dots, R_m)-\\frac{I(n)}{q^m}\\right| \\le \\frac{q^m-1}{q^m} \\cdot \\min_{0 \\le u+v < n} \\left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\\sqrt{S_2(u, v)}\\right) \\] where \\[ S_1(u, v) = \\sum_{0 \\le d \\le u+v} \\sum_{\\deg g = d} q^{(n-d)+\\frac{n-d}{2^j}-c \\cdot \\operatorname{rank}(((P_i)_g)_{i \\in S})} \\] and \\[ S_2(u, v) = \\max_{v \\le k \\le n-u} \\max_{\\deg g_1 = n-k} \\sum_{\\deg g_2 = n-k} q^{k-c \\cdot \\operatorname{rank}(((P_i)_{g_1, g_2})_{i \\in S})}. \\],",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb{F}_q\\) be a finite field of odd order \\(q\\) and characteristic \\(p\\), and let \\(n<p\\). Let \\(R_1,\\dots,R_m\\) be polynomial functions of the coefficients of monic degree-\\(n\\) polynomials over \\(\\mathbb{F}_q\\). Define \\(I_n(R_1,\\dots,R_m)\\) to be the number of monic irreducible polynomials \\(f\\in \\mathbb{F}_q[x]\\) of degree \\(n\\) such that \\(R_1(f)=\\cdots=R_m(f)=0\\), and let \\(I(n)\\) denote the number of monic irreducible polynomials of degree \\(n\\) over \\(\\mathbb{F}_q\\). Set \\(j:=\\max(\\deg R_1,\\dots,\\deg R_m)\\), let \\(S=\\{1\\le i\\le m: \\deg R_i=j\\}\\), and define \\((P_i)_g(f):=R_i(fg)\\) and \\((P_i)_{g_1,g_2}(f):=R_i(g_1f)-R_i(g_2f)\\). For nonnegative integers \\(u,v\\) with \\(u+v<n\\), define\n\\[\nS_1(u,v)=\\sum_{0\\le d\\le u+v}\\sum_{\\deg g=d} q^{(n-d)+\\frac{n-d}{2^j}-c\\,\\operatorname{rank}(((P_i)_g)_{i\\in S})}\n\\]\nand\n\\[\nS_2(u,v)=\\max_{v\\le k\\le n-u}\\max_{\\deg g_1=n-k}\\sum_{\\deg g_2=n-k} q^{k-c\\,\\operatorname{rank}(((P_i)_{g_1,g_2})_{i\\in S})},\n\\]\nwhere the degree-constrained sums are over monic polynomials. Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v\\le n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that for every pair of nonnegative integers \\(u,v\\) with \\(u+v<n\\),\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\frac{q^m-1}{q^m}\\cdot \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}S_2(u,v)\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "There exists a constant \\(c>0\\), depending only on \\(j,m\\), such that\n\\[\n\\left|I_n(R_1,\\dots,R_m)-\\frac{I(n)}{q^m}\\right|\\le \\min_{0\\le u+v<n}\\left(nS_1(u,v)+n^{5/2}q^{\\,n-(u+v)/2}\\sqrt{S_2(u,v)}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "admissible_range_u_plus_v",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_minimization_over_u_v",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "square_root_on_S2",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence_of_constant_and_prefactor",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option directly; it only sets up notation and hypotheses. The answer must be identified from the choices."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-statement recall item: after presenting all assumptions and definitions, it asks which quantitative estimate holds. The correct choice is basically the theorem’s conclusion verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "Some care is needed to distinguish subtle variants involving the minimization, the range of u+v, the square root on S_2, and constant dependence. However, the task is still mostly recognition/recall of the exact statement rather than genuine derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: changing the admissible range, dropping the minimization, removing the square root, or altering the constant/prefactor. They are distinct and nontrivial."
      },
      "total_score": 5,
      "overall_assessment": "A strong theorem-precision MCQ with high-quality distractors and no answer leakage, but it is largely a direct restatement/recognition task rather than a genuinely generative reasoning problem."
    }
  },
  {
    "id": "2512.09183v1",
    "paper_link": "http://arxiv.org/abs/2512.09183v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "mainthm",
      "content": "\\label{thm:ANN}\nIf $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\\ge q_i$, and $p_1q_2-p_2q_1=\\pm2$, then the disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in $\\CPb$.",
      "start_pos": 297606,
      "end_pos": 297904,
      "label": "thm:ANN"
    },
    "ref_dict": {
      "s:data": "\\label{s:data}\n\n\\Cref{table:triples} contains \\emph{all} triples of pairs of integers, $(p_1,q_1)$, $(p_2,q_2)$, $(p_3,q_3)$ such that:\n\\begin{itemize}\n\\item there exist rational homology balls $B_1$,",
      "fig:Apq": "\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=14cm]{./figures/Apq.pdf}\n\\caption{{\\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\\gcd(p,q)=2$.} The blue attaching circle for the 2-handle in the second diagram is the $(2,-1)$ cable, relative to the torus framing, of the torus knot $T_{-p/2,q/2}$.}\n\\label{fig:Apq}\n\\end{figure}",
      "s:compare": "\\begin{proof}[Proof of Corollaries \\ref{cor:Bpq} and \\ref{cor:Lpq}]\nFor this we note that by \\Cref{thm:2Farey}, either $\\dfrac{p}{q}$ or $\\dfrac{p}{p-q}$ appears in a 2-Farey triple.  It follows that $B_{p,q}$ embeds smoothly in $\\CPb$ and hence so does its boundary $L(p^2,pq-1)$.  Finally note that an embedding of a 3-manifold into a 4-manifold does not depend on the orientation of either.\n\\end{proof}\n\n\\section{A common framework for known results}\\label{s:compare}\n\nIn this section we make an observation which attempts to put \\Cref{thm:ANN} in a common framework with previous results of Hacking--Prokhorov \\cite{hp}, Evans--Smith \\cite{es}, and Lisca--Parma \\cite{HD2}.  Each of these results gives a binary tree, together with an embedding of a triple of rational balls in $\\CP$ for each node of the tree.  The first example of this is the famous Markov tree which contains all triples of solutions of \n\\begin{equation*}\np_1^2+p_2^2+p_3^2=3p_1p_2p_3.\n\\end{equation*}\nThis features in each of \\cite{es,hp,HD2}.  Two further trees of triples arise from \\cite{HD2}, and the 2-Farey tree underlies \\Cref{thm:ANN}.  The following definition may be interpreted in terms of the proofs of \\Cref{prop:2Femb} and  \\cite[Proposition 4.6]{HD1}.  In each case we have in mind three curves lying on or near nested tori, and the recursive rule is given by handlesliding one curve over another.\n\n\\begin{defn}\nA \\emph{signed slide triple tree} is an infinite rooted binary tree with a triple $\\left(\\left(\\dfrac{p_1}{q_1},\\delta_1\\right),\\left(\\dfrac{p_2}{q_2},\\delta_2\\right),\\left(\\dfrac{p_3}{q_3},\\delta_3\\right)\\right)$ at each node and the recursive rule  \n\n{\\centering\n\\includegraphics[width=\\textwidth]{./figures/sliderule.pdf}\n\\par}\n\\noindent\nwhere $\\delta_i\\in\\{\\pm1\\}$, $x_1=p_2q_3-p_3q_2$ and $x_3=p_1q_2-p_2q_1$.\n\\end{defn}\n\nWe note that the following gives new recursive descriptions of the triples appearing in \\cite[Theorem 1.2(2),(3)]{HD2}.\n\\begin{thm}\\label{thm:trees}\nLet $\\left(\\left(\\dfrac{p_1}{q_1},\\delta_1\\right),\\left(\\dfrac{p_2}{q_2},\\delta_2\\right),\\left(\\dfrac{p_3}{q_3},\\delta_3\\right)\\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of\n\\begin{enumerate}\n\\item \\triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},\n\\item \\triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.\n\\end{enumerate}\nThen the disjoint union $\\bigsqcup\\delta_i B_{p_i,q_i}$ embeds smoothly in $\\CP$.  The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \\cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \\Cref{thm:ANN}.\n\\end{thm}\n\n\\begin{proof}[Proof sketch]\nFor the case of the first root, which yields the Markov tree, we appeal to \\cite{perling}, which refers to $\\{q_1,q_2,q_3\\}$ as the \\emph{T-weights} of the Markov triple $\\{p_1,p_2,p_3\\}$.  The identities \\cite[(1)]{perling} show that $x_1=3p_1$ and $x_3=3p_3$, and then the recursive rule above agrees with the mutation rules for Markov triples and their T-weights \\cite[\\S2 and Lemma 3.1]{perling}.  The first three rows of the tree are shown below.  We omit the $\\delta_i$ which are all equal to $+1$.\n\n{\\centering\n\\includegraphics[width=10cm]{./figures/MarkovTree.pdf}\n\\par}\n\\noindent\n\nThe second root above yields the triples of rational balls embedding in $\\CP$ given by Lisca--Parma in \\cite[Theorem 1.2(2)]{HD2}.  \nOne can show that $x_1=\\delta_1p_1$ and $x_3=\\delta_3p_3$ and that, as for the standard Markov equation, the recursive rule given above agrees with the mutation rule for the triples of solutions and corresponding weights of the Markov-type equation\n\\begin{equation*}\n\\delta_1 p_1^2+\\delta_2 p_2^2 +\\delta_3 p_3^2=p_1 p_2 p_3,\\end{equation*}\nwith $\\{\\delta_1, \\delta_2, \\delta_3\\}=\\{1,-1,-1\\}$ and weights\n\\begin{align*}\nq_1&\\equiv\\delta_2p_2/p_3\\equiv-\\delta_3p_3/p_2\\pmod{p_1},\\\\\nq_2&\\equiv\\delta_3p_3/p_1\\equiv-\\delta_1p_1/p_3\\pmod{p_2},\\\\\nq_3&\\equiv\\delta_1p_1/p_2\\equiv-\\delta_2p_2/p_1\\pmod{p_3}.\n\\end{align*}\nEach weight $q_i$ is strictly between $0$ and $p_i$ unless $p_i=1$, in which case we take $q_i=1$.\n\nThe first entries of the resulting tree of triples are shown below.  In each triple the underlined entry has $\\delta_i=1$.  Note that $B_{-p,-q}=B_{p,q}$, but the negative signs in the tree below are needed for the recursion.\n\n{\\centering\n\\includegraphics[width=10cm]{./figures/HD2secondtree.pdf}\n\\par}\n\\noindent\n\nThe first few entries of the tree arising from the third root in the statement are shown below, as well as two extra nodes before the root.  In each triple the underlined entry has $\\delta_i=-1$.\n\n{\\centering\n\\includegraphics[width=10cm]{./figures/HD2thirdtree.pdf}\n\\par}\n\\noindent\nWe claim that this contains precisely the triples of rational balls embedding in $\\CP$ given by Lisca--Parma in \\cite[Theorem 1.2(3)]{HD2}.  To see this, for each node, let $\\gamma_i=(p_i,q_i)$,  and take\n$$(x_1,x_2,x_3)=(\\gamma_2\\cdot\\gamma_3,\\gamma_1\\cdot\\gamma_3,\\gamma_1\\cdot\\gamma_2),$$\nwhere $(p,q)\\cdot(r,s)=ps-qr$.  For example, if we consider the first triple \n$\\left(\\frac{1}{-1},\\frac{1}{\\underline{0}},\\frac{-1}{-1}\\right)$, we get\n$$\\gamma_1=(1,-1), \\gamma_2=(1,0), \\gamma_3=(-1,-1) \\text{ and } (x_1,x_2,x_3)=(-1,-2,1),$$\nand for the second triple $\\left(\\frac{1}{-1},\\frac{2}{1},\\frac{1}{\\underline{0}}\\right)$ we obtain\n$$\\gamma_1=(1,-1), \\gamma_2=(2,1), \\gamma_3=(1,0) \\text{ and } (x_1,x_2,x_3)=(-1,-1,-3).$$\n\nWe claim that every triple in this tree satisfies the following Markov-type equation\n\\begin{equation}\n\\delta_1 x_1^2+\\delta_2 x_2^2 +\\delta_3 x_3^2=x_1 x_2 x_3-4,\\label{eq:3-2}\n\\end{equation}\nand also\n\\begin{equation}\n\\delta_1 p_1^2+\\delta_2 p_2^2 +\\delta_3 p_3^2-\\delta_1\\delta_2p_1 p_2 x_3-\\delta_1\\delta_3p_1p_3x_2-\\delta_2\\delta_3p_2p_3x_1-p_1p_3x_1x_3=0.\\label{eq:3-3}\n\\end{equation}\nThese are equations (3-2) and (3-3) in \\cite{HD2}.\n\nIt is straightforward to check these equations are satisfied for the triple $\\left(\\frac{1}{-1},\\frac{1}{\\underline{0}},\\frac{-1}{-1}\\right)$, which forms the base case for a structural induction.  We then check that triples given by the recursive rule continue to satisfy \\eqref{eq:3-2} and \\eqref{eq:3-3}.\n\nSuppose that we have $\\delta_2=-1$ and that\n$$\\gamma_1=(p_1,q_1), \\gamma_2=(p_2,q_2), \\gamma_3=(p_3,q_3),$$\nand thus\n$$(x_1,x_2,x_3)=(p_2q_3-p_3q_2,p_1q_3-p_3q_1,p_1q_2-p_2q_1).$$\nBy induction, we assume that \\eqref{eq:3-2} and \\eqref{eq:3-3} hold for these variables.\nMoving left down the tree, the recursive rule leads to $\\delta_3=-1$,\n$$\\widehat{\\gamma}_1=(p_1,q_1), \\widehat{\\gamma}_2=(-p_3-x_1p_2,-q_3-x_1q_2), \\widehat{\\gamma}_3=(p_2,q_2),$$\nand \n\\begin{equation}\n(\\widehat{x}_1,\\widehat{x}_2,\\widehat{x}_3)=(x_1,x_3,-x_2-x_1x_3).\\label{eq:mut}\n\\end{equation}\nWe verify that \\eqref{eq:3-2} holds:\n\\begin{align*}\n&\\widehat{x}_1^2+\\widehat{x}_2^2-\\widehat{x}_3^2-\\widehat{x}_1\\widehat{x}_2\\widehat{x}_3+4\\\\\n&=x_1^2+x_3^2-x_2^2-2x_1x_2x_3-x_1^2x_3^2+x_1x_2x_3+x_1^2x_3^2+4\\\\\n&=x_1^2-x_2^2+x_3^2-x_1x_2x_3+4\\\\\n&=0,\n\\end{align*}\nand also \\eqref{eq:3-3}:\n\\begin{align*}\n&\\widehat{p}_1^2+\\widehat{p}_2^2-\\widehat{p}_3^2-\\widehat{p}_1 \\widehat{p}_2 \\widehat{x}_3+\\widehat{p}_1\\widehat{p}_3\\widehat{x}_2+\\widehat{p}_2\\widehat{p}_3\\widehat{x}_1-\\widehat{p}_1\\widehat{p}_3\\widehat{x}_1\\widehat{x}_3\\\\\n&=p_1^2+p_3^2+2p_2p_3x_1+p_2^2x_1^2-p_2^2-p_1p_3x_2-p_1p_2x_1x_2-p_1p_3x_1x_3-p_1p_2x_1^2x_3\\\\\n&\\quad+p_1p_2x_3-p_2p_3x_1-p_2^2x_1^2+p_1p_2x_1x_2+p_1p_2x_1^2x_3\\\\\n&=p_1^2-p_2^2+p_3^2+p_1p_2x_3-p_1p_3x_2+p_2p_3x_1-p_1p_3x_1x_3\\\\\n&=0,\n\\end{align*}\nin both cases using induction for the final equality.\n\nThe other cases (moving right down the tree from $\\delta_2=-1$, or starting with $\\delta_2=1$) are minor variants of this.\n\nThe proof of \\cite[Theorem 1.2(3)]{HD2} shows that the triples therein come from solutions to \\eqref{eq:3-2} and \\eqref{eq:3-3} such that the triples $(x_1,x_2,x_3)$ are obtained from the minimal solution $(1,2,1)$ by mutations, sign changes, and permutations as in \\eqref{eq:mut}.  By comparison, we conclude that our  signed slide triple tree contains all of the triples of rational balls in the family given in \\cite[Theorem 1.2(3)]{HD2}.\n\nFor the last case, one can show by induction that $x_1=x_3=2$ for all nodes, and then the recursion rule is easily seen to agree with that of the 2-Farey tree.\n\\end{proof}",
      "s:balls": "\\label{s:balls}\nWe use Hirzebruch--Jung continued fractions, with the following notation:\n$$[a_1,a_2,\\ldots,a_k]\n:=a_1-\\frac{1}{a_2-\\raisebox{-3mm}{$\\ddots$\n\\raisebox{-2mm}{${-\\frac{1}{\\displaystyle{a",
      "s:ANN": "\\label{s:ANN}\n\nGiven an ordered $n$-tuple $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dots,\\dfrac{p_n}{q_n}\\right)$, with $\\gcd(p_i,q_i)\\in\\{1,2\\}$, we define a Kirby diagram of a manifold $X=X_{\\frac{p",
      "thm:ANN": "\\begin{mainthm}\\label{thm:ANN}\nIf $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\\ge q_i$, and $p_1q_2-p_2q_1=\\pm2$, then the disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in $\\CPb$.\n\\end{mainthm}",
      "s:homotopyCP2s": "\\label{s:homotopyCP2s}\n\nIn this section we describe some methods for finding rational balls bounded by lens spaces which embed in a homotopy $\\CP$ or $\\CPb$.\nThese methods are suitable for use in a co",
      "q:exoticCP2s": "\\begin{quest}\\label{q:exoticCP2s}\nAre all homotopy $\\CP$'s constructed in this section diffeomorphic to the standard $\\CP$?\n\\end{quest}",
      "thm:trees": "\\begin{thm}\\label{thm:trees}\nLet $\\left(\\left(\\dfrac{p_1}{q_1},\\delta_1\\right),\\left(\\dfrac{p_2}{q_2},\\delta_2\\right),\\left(\\dfrac{p_3}{q_3},\\delta_3\\right)\\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of\n\\begin{enumerate}\n\\item \\triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},\n\\item \\triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.\n\\end{enumerate}\nThen the disjoint union $\\bigsqcup\\delta_i B_{p_i,q_i}$ embeds smoothly in $\\CP$.  The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \\cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \\Cref{thm:ANN}.\n\\end{thm}",
      "s:2Fareytree": "\\begin{proof}\nFirst note that by reversing the first diagram in either \\Cref{fig:Bpq} or \\ref{fig:Apq}, we see that $B_{p,q}\\cong B_{p,p-q}$.\n\nNow take $k\\in\\Z$.  Start with $B_{p,kp+q}$ as in the second diagram of either \\Cref{fig:Bpq} or \\ref{fig:Apq}, with the dotted circle replaced by a $\\langle0\\rangle$-framed circle as in the proof of Lemma $\\ref{lem:Bpq}$.  Then the  diffeomorphism $B_{p,q}\\cong B_{p,kp+ q}$ follows by performing Rolfsen twists \\cite[Section 5.3]{GS} along this $\\langle0\\rangle$-framed circle.\n\\end{proof}\n\n\\section{The 2-Farey tree}\\label{s:2Fareytree}\n\nWe define a variant of the classical Farey tree \\cite{aigner}, which we refer to as the 2-Farey tree.  This is a binary tree with ordered triples of fractions, called 2-Farey triples, at each node.  The triple at the root is $\\left(\\frac10,\\frac32,\\frac22\\right)$ and the recursive rule is given by \n\n{\\centering\n\\includegraphics[width=7cm]{./figures/2Fareyrule.pdf}\n\\par}\n\\noindent\n\nThe first three rows of the tree, and two further nodes, are shown below.\n\n{\\centering\n\\includegraphics[width=10cm]{./figures/2Fareytree.pdf}\n\\par}\n\\noindent\nObserve that each triple contains two reduced fractions and one fraction $p/q$ with $\\gcd(p,q)=2$.\n\nFor comparison, part of the standard Farey tree is shown below.  It uses the same recursive rule as above but the triple at the root is $\\left(\\frac10,\\frac11,\\frac01\\right)$.\n\n{\\centering\n\\includegraphics[width=10cm]{./figures/Fareytree.pdf}\n\\par}\n\\noindent\nNote that the 2-Farey tree is obtained from the subtree of the Farey tree on the left, beginning at $\\left(\\frac10,\\frac31,\\frac21\\right)$, by multiplying all denominators by 2.\n\nThe following theorem  follows immediately from the corresponding facts for the classical Farey tree found in the wonderful book by Aigner \\cite[Section 3.2]{aigner}.\n\n\\begin{thm}\\label{thm:2Farey}\nEvery fraction $p/q$ with $q$ even, $p\\ge q$, and $\\gcd(p,q)\\le2$ appears in the $2$-Farey tree.  Every such fraction with $p>2$ is generated as a mediant exactly once.  For each $2$-Farey triple $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dfrac{p_3}{q_3}\\right)$, we have  \n\\begin{equation}\\label{eqn:2Farey}\np_2=p_1+p_3,\\quad q_2=q_1+q_3,\\quad \\mathrm{and}\\quad p_iq_{i+1}-p_{i+1}q_i=2\\ \\mathrm{for}\\  i\\in\\{1,2\\}.  \n\\end{equation}\nFurthermore:\n\n\\begin{enumerate}[(i)]\n\\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.\n\\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying\n$0\\le q_i\\le p_i$ and $p_1q_2-p_2q_1=\\pm2$.  After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_1+p_2}{q_1+q_2},\\dfrac{p_2}{q_2}\\right)$ is a $2$-Farey triple.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rmk}\nOne could alternatively choose the triple $\\left(\\frac20,\\frac31,\\frac11\\right)$ at the root of the 2-Farey tree.  The resulting tree would be obtained from that above by replacing each fraction $p/q$ by $p/(p-q)$, and reflecting the tree and each triple from left to right.  It contains all triples $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dfrac{p_3}{q_3}\\right)$ with $p_i\\equiv q_i \\pmod2$ and  satisfying \\eqref{eqn:2Farey}.\nSince by \\Cref{lem:addp} we have $B_{p,q}=B_{p,p-q}$, this would result in the same embeddings of rational balls in $\\CPb$ in the next section.\n\nWe could also consider an expanded 2-Farey tree obtained by multiplying denominators by 2 in all nodes of the full Farey tree shown above.  It again follows from \\Cref{lem:addp} that this does not yield any more embedding results than those coming from the 2-Farey tree we have selected, though we will see that for the base of our induction, it is helpful to include two more nodes at the base of the tree.\n\\end{rmk}\n\n\\section{Embeddings of 2-Farey triples}\\label{s:ANN}\n\nGiven an ordered $n$-tuple $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dots,\\dfrac{p_n}{q_n}\\right)$, with $\\gcd(p_i,q_i)\\in\\{1,2\\}$, we define a Kirby diagram of a manifold $X=X_{\\frac{p_1}{q_1},\\frac{p_2}{q_2},\\dots,\\frac{p_n}{q_n}}$ as follows.  Let $r_i=1-i/(n+1)$ for $i=1,\\dots,n$, so that\n$$1>r_1>r_2>\\dots>r_n>0,$$\nand $\\varepsilon=1/(4n)$, so that the intervals $[r_i-\\varepsilon,r_i+\\varepsilon]$ are pairwise disjoint in $(0,1)$.\n\nThe manifold $X$ is a handlebody with one $0$-handle, one $1$-handle, and $n$ $2$-handles.  To draw the diagram, start with a dotted circle drawn as a circle of diameter 3.  For each $1\\le i\\le n$, draw the framed knot $K_{p_i,q_i}$ from \\Cref{s:balls} in an $\\varepsilon$-neighbourhood of the torus which is distance $r_i$ from the dotted circle.  An example is shown in Figure \\ref{fig:Base}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=14cm]{./figures/Base.pdf}\n\\caption{{\\bf The manifold $X_{\\frac10,\\frac12,\\frac02}$.} The black arrow indicates a handle slide that may be used to simplify the diagram.}\n\\label{fig:Base}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:LPembed}\nFor any $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dots,\\dfrac{p_n}{q_n}\\right)$, with $\\gcd(p_i,q_i)\\in\\{1,2\\}$, the disjoint union\n$$\\bigsqcup_{i=1}^n B_{p_i,q_i}$$\nembeds smoothly in $X_{\\frac{p_1}{q_1},\\frac{p_2}{q_2},\\dots,\\frac{p_n}{q_n}}$.\n\\end{lemma}\n\\begin{proof}\nThis follows exactly as in the proof of \\cite[Lemma 4.1]{HD2}.  We give a sketch here for convenience.  Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\\frac{p_1}{q_1},\\frac{p_2}{q_2},\\dots,\\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$.  This is a copy of $S^1\\times B^3$.  For convenience, view $B^3$ as the unit ball in $\\R^3$.  Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\\times S_\\zeta$, where $S_\\zeta$ is the intersection of the unit sphere with the plane $z=\\zeta$.  Here we  take $\\zeta(r)$ to be a monotone decreasing bijection from $(0,\\infty)$ to $(-1,1)$.\n\nFor each $1\\le i\\le n$, the set of points $(x,y,z)\\in B^3$ with $z\\in [\\zeta(r_i-\\varepsilon),\\zeta(r_i+\\varepsilon)]$ is again a 3-ball, after smoothing corners, which we may denote by $B^3_i$.  The disjoint union $\\bigsqcup_{i=1}^n S^1\\times B^3_i$ is embedded in $S^1\\times B^3$, and since the 2-handle for each $B_{p_i,q_i}$ is attached to the boundary of $S^1\\times B^3_i$, the statement follows.\n\\end{proof}",
      "fig:Bpq": "\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=14cm]{./figures/Bpq.pdf}\n\\caption{{\\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\\gcd(p,q)=1$.} The second diagram should be closed like a braid closure, so that there is one dotted circle in black and one 2-handle attaching circle, which is a copy of the torus knot $T_{-p,q}$, in blue.  The framing on the 2-handle is $-1$ relative to the torus framing.}\n\\label{fig:Bpq}\n\\end{figure}"
    },
    "pre_theorem_intro_text_len": 1782,
    "pre_theorem_intro_text": "\\label{s:intro}\nA basic, yet fundamental, question in 4-manifold topology is the following.\nGiven a smooth 4-manifold $X$ and a class $\\mathcal{Y}$ of 3-manifolds, which $Y \\in \\mathcal{Y}$ smoothly embed in $X$?\nIn this paper, we focus on the case when $\\mathcal{Y}$ is the class of lens spaces, which are the simplest 3-manifolds after the sphere, and on the simplest possible 4-manifold in which they can embed\\footnote{It is known since Hantzsche's work~\\cite{Hantzsche} that lens spaces cannot embed in $S^4$.}: the complex projective plane $\\CP$.\n\nWhenever a lens space $L$ embeds in $\\CP$, it splits it into two components, one of which is a \\emph{rational homology ball}, a 4-manifold with the rational homology of a point.\nTherefore, asking for the embedding of $L$ into $\\CP$ is equivalent to asking for the embedding into $\\CP$ of a rational ball whose boundary is $L$.\nNote that lens spaces that bound rational homology balls have been classified by Lisca~\\cite{Lisca-ribbon}.\n\nEarly results of the existence of this type of embedding come from algebraic geometry~\\cite{Manetti, hp} and symplectic topology~\\cite{es}, as well as from~\\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.\nThese results indicate that it is interesting to look at embeddings of \\emph{triples} of rational homology balls in $\\CP$, a question which is related  (but not equivalent) to asking about disjoint embeddings of lens spaces.\n\nHere we push the ideas of~\\cite{HD2} and of~\\cite{EMPR} further, and we obtain many more embeddings of \\emph{triples} of rational homology balls with lens space boundary in $\\CPb$;\nreversing all orientations gives embeddings in $\\CP$.\nOur triples give previously unknown examples, except possibly when $\\min\\{p_i\\} \\le 2$.",
    "context": "\\label{s:intro}\nA basic, yet fundamental, question in 4-manifold topology is the following.\nGiven a smooth 4-manifold $X$ and a class $\\mathcal{Y}$ of 3-manifolds, which $Y \\in \\mathcal{Y}$ smoothly embed in $X$?\nIn this paper, we focus on the case when $\\mathcal{Y}$ is the class of lens spaces, which are the simplest 3-manifolds after the sphere, and on the simplest possible 4-manifold in which they can embed\\footnote{It is known since Hantzsche's work~\\cite{Hantzsche} that lens spaces cannot embed in $S^4$.}: the complex projective plane $\\CP$.\n\nWhenever a lens space $L$ embeds in $\\CP$, it splits it into two components, one of which is a \\emph{rational homology ball}, a 4-manifold with the rational homology of a point.\nTherefore, asking for the embedding of $L$ into $\\CP$ is equivalent to asking for the embedding into $\\CP$ of a rational ball whose boundary is $L$.\nNote that lens spaces that bound rational homology balls have been classified by Lisca~\\cite{Lisca-ribbon}.\n\nEarly results of the existence of this type of embedding come from algebraic geometry~\\cite{Manetti, hp} and symplectic topology~\\cite{es}, as well as from~\\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.\nThese results indicate that it is interesting to look at embeddings of \\emph{triples} of rational homology balls in $\\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.\n\nHere we push the ideas of~\\cite{HD2} and of~\\cite{EMPR} further, and we obtain many more embeddings of \\emph{triples} of rational homology balls with lens space boundary in $\\CPb$;\nreversing all orientations gives embeddings in $\\CP$.\nOur triples give previously unknown examples, except possibly when $\\min\\{p_i\\} \\le 2$.",
    "full_context": "\\label{s:intro}\nA basic, yet fundamental, question in 4-manifold topology is the following.\nGiven a smooth 4-manifold $X$ and a class $\\mathcal{Y}$ of 3-manifolds, which $Y \\in \\mathcal{Y}$ smoothly embed in $X$?\nIn this paper, we focus on the case when $\\mathcal{Y}$ is the class of lens spaces, which are the simplest 3-manifolds after the sphere, and on the simplest possible 4-manifold in which they can embed\\footnote{It is known since Hantzsche's work~\\cite{Hantzsche} that lens spaces cannot embed in $S^4$.}: the complex projective plane $\\CP$.\n\nWhenever a lens space $L$ embeds in $\\CP$, it splits it into two components, one of which is a \\emph{rational homology ball}, a 4-manifold with the rational homology of a point.\nTherefore, asking for the embedding of $L$ into $\\CP$ is equivalent to asking for the embedding into $\\CP$ of a rational ball whose boundary is $L$.\nNote that lens spaces that bound rational homology balls have been classified by Lisca~\\cite{Lisca-ribbon}.\n\nEarly results of the existence of this type of embedding come from algebraic geometry~\\cite{Manetti, hp} and symplectic topology~\\cite{es}, as well as from~\\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.\nThese results indicate that it is interesting to look at embeddings of \\emph{triples} of rational homology balls in $\\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.\n\nHere we push the ideas of~\\cite{HD2} and of~\\cite{EMPR} further, and we obtain many more embeddings of \\emph{triples} of rational homology balls with lens space boundary in $\\CPb$;\nreversing all orientations gives embeddings in $\\CP$.\nOur triples give previously unknown examples, except possibly when $\\min\\{p_i\\} \\le 2$.\n\nEarly results of the existence of this type of embedding come from algebraic geometry~\\cite{Manetti, hp} and symplectic topology~\\cite{es}, as well as from~\\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.\nThese results indicate that it is interesting to look at embeddings of \\emph{triples} of rational homology balls in $\\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.\n\nWhen $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.\nKirby diagrams for these rational balls are given in Figures~\\ref{fig:Bpq} and~\\ref{fig:Apq}.\n\n\\begin{maincor}\\label{cor:Bpq}\nLet $p\\ge q$ be nonnegative integers. If $\\gcd(p,q)=2$ or if $p$ is odd and $\\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\\CPb$.\n\\end{maincor}\n\n\\begin{maincor}\\label{cor:Lpq}\nLet $p\\ge q$ be nonnegative integers. If $\\gcd(p,q)=2$ or if $p$ is odd and $\\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\\CP$.\n\\end{maincor}\n\n\\begin{enumerate}[(i)]\n\\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.\n\\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying\n$0\\le q_i\\le p_i$ and $p_1q_2-p_2q_1=\\pm2$. After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_1+p_2}{q_1+q_2},\\dfrac{p_2}{q_2}\\right)$ is a $2$-Farey triple.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{lemma}\\label{lem:LPembed}\nFor any $\\left(\\dfrac{p_1}{q_1},\\dfrac{p_2}{q_2},\\dots,\\dfrac{p_n}{q_n}\\right)$, with $\\gcd(p_i,q_i)\\in\\{1,2\\}$, the disjoint union\n$$\\bigsqcup_{i=1}^n B_{p_i,q_i}$$\nembeds smoothly in $X_{\\frac{p_1}{q_1},\\frac{p_2}{q_2},\\dots,\\frac{p_n}{q_n}}$.\n\\end{lemma}\n\\begin{proof}\nThis follows exactly as in the proof of \\cite[Lemma 4.1]{HD2}. We give a sketch here for convenience. Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\\frac{p_1}{q_1},\\frac{p_2}{q_2},\\dots,\\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$. This is a copy of $S^1\\times B^3$. For convenience, view $B^3$ as the unit ball in $\\R^3$. Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\\times S_\\zeta$, where $S_\\zeta$ is the intersection of the unit sphere with the plane $z=\\zeta$. Here we take $\\zeta(r)$ to be a monotone decreasing bijection from $(0,\\infty)$ to $(-1,1)$.\n\nWe note that the following gives new recursive descriptions of the triples appearing in \\cite[Theorem 1.2(2),(3)]{HD2}.\n\\begin{thm}\\label{thm:trees}\nLet $\\left(\\left(\\dfrac{p_1}{q_1},\\delta_1\\right),\\left(\\dfrac{p_2}{q_2},\\delta_2\\right),\\left(\\dfrac{p_3}{q_3},\\delta_3\\right)\\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of\n\\begin{enumerate}\n\\item \\triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},\n\\item \\triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},\n\\item \\triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.\n\\end{enumerate}\nThen the disjoint union $\\bigsqcup\\delta_i B_{p_i,q_i}$ embeds smoothly in $\\CP$. The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \\cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \\Cref{thm:ANN}.\n\\end{thm}\n\n\\begin{prop}\\label{prop:Addc}\nSuppose that we have two rational numbers, $p/q = [a_1,\\dots,a_m]$ and $r/s = [b_1,\\dots,b_n]$, and an integer $c$ such that:\n\\begin{itemize}\n\\item $L(p,q)$ and $L(r,s)$ bound rational homology balls,\n\\item letting $t/u = [a_m,\\dots,a_1,c,b_1,\\dots,b_n]$, $L(t,u)$ also bounds a rational homology ball, and\n\\item $p$, $r$, and $t$ are pairwise coprime.\n\\end{itemize}\nThen there exists a homotopy $\\CP$ or $\\CPb$, $X$, and an embedding $B \\sqcup B' \\sqcup (-B'') \\hookrightarrow X$, where $B$, $B'$, and $B''$ are three rational homology balls bounding $L(p,q)$, $L(r,s)$, and $L(t,u)$, respectively, constructed using handles of index at most $2$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=14cm]{./figures/Apq.pdf}\n\\caption{{\\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\\gcd(p,q)=2$.} The blue attaching circle for the 2-handle in the second diagram is the $(2,-1)$ cable, relative to the torus framing, of the torus knot $T_{-p/2,q/2}$.}\n\\label{fig:Apq}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=14cm]{./figures/Bpq.pdf}\n\\caption{{\\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\\gcd(p,q)=1$.} The second diagram should be closed like a braid closure, so that there is one dotted circle in black and one 2-handle attaching circle, which is a copy of the torus knot $T_{-p,q}$, in blue.  The framing on the 2-handle is $-1$ relative to the torus framing.}\n\\label{fig:Bpq}\n\\end{figure}\n\n\\begin{mainthm}\\label{thm:ANN}\nIf $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\\ge q_i$, and $p_1q_2-p_2q_1=\\pm2$, then the disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in $\\CPb$.\n\\end{mainthm}",
    "post_theorem_intro_text_len": 7797,
    "post_theorem_intro_text": "When $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.\nKirby diagrams for these rational balls are given in Figures~\\ref{fig:Bpq} and~\\ref{fig:Apq}.\n\nAn immediate corollary of Theorem~\\ref{thm:ANN} above is the following.\n\n\\begin{maincor}\\label{cor:Bpq}\nLet $p\\ge q$ be nonnegative integers.  If $\\gcd(p,q)=2$ or if $p$ is odd and $\\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\\CPb$.\n\\end{maincor}\n\nIn terms of embeddings of lens spaces, the following corollary shows that, among lens spaces of the form $L(p^2,pq-1)$ that bound a rational homology ball,\nroughly three-quarters  embed in $\\CP$.\n\n\\begin{maincor}\\label{cor:Lpq}\nLet $p\\ge q$ be nonnegative integers.  If $\\gcd(p,q)=2$ or if $p$ is odd and $\\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\\CP$.\n\\end{maincor}\n\nThe inspiration for the main theorem comes from the data extracted from the output of two more general constructions, that we describe in detail in Section~\\ref{s:homotopyCP2s}, which give a large collection of embeddings of triples of rational homology balls in  homotopy $\\CP$'s.\nWe expect that all these homotopy $\\CP$'s are in fact standard (see \\Cref{q:exoticCP2s} below).\n\nIn a companion paper~\\cite{GOobst},  we provide obstructions to the existence of embeddings of such triples, and work in the direction of  a conjecture of Koll\\'ar \\cite{Kollar}; see also~\\cite{JPP} for related recent work. It is worth noting that the embeddings of triples of Theorem~\\ref{thm:ANN} account for a very large fraction of the unobstructed triples.\n\nThe proof of Theorem~\\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.\nIt uses structural induction on a simple variant of the Farey tree, which we will define in Section~\\ref{s:2Fareytree}. \nThe Kirby calculus argument is related to Lisca and Parma's horizontal decompositions, and is informed by the Kirby diagrams of~\\cite{EMPR}.\nThe main idea is to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\\times D^3$.\n\nThe handlesliding used to prove Theorem~\\ref{thm:ANN} suggests another point of view on the results of Lisca and Parma in \\cite{HD2}.  In \\Cref{thm:trees} we use this point of view to derive new recursive descriptions of the triples of rational balls embedded in $\\CP$ obtained in that paper.\n\n\\subsection*{History and motivation}\nAs mentioned above, the motivation for studying embeddings of lens spaces in $\\CP$ comes from algebraic geometry.\nInspired by consequences of the orbifold Bogomolov--Miyaoka--Yau inequality, Koll\\'ar asks in~\\cite{Kollar} whether every compact, orientable, simply-connected 4-manifold $M$ with $H_1(M) = 0$, $H_2(M)\\cong\\mathbb{Z}$, and with boundary a union of spherical 3-manifolds $Y_1, \\dots, Y_n$, satisfies:\n\\[\n\\sum_{i=1}^n \\left(1-\\frac1{|\\pi_1(Y_i)|}\\right) \\le 3.\n\\]\nLens spaces are finite cyclic quotients of $S^3$, so they are spherical 3-manifolds, and indeed the complements of the rational homology balls that we embed in $\\CP$ are manifolds satisfying the assumption in Koll\\'ar's question.\n\nA prominent family of examples of such manifolds originates from weighted projective planes $\\mathbb{P}(a \\sd b \\sd c)$ by removing small open neighbourhoods of their (potentially) singular points $(0\\sd 0\\sd 1)$, $(0\\sd 1\\sd 0)$, and $(1\\sd 0\\sd 0)$.\nWhen $(a,b,c)$ satisfies $a^2 + b^2 + c^2 = 3abc$, the weighted projective plane $\\mathbb{P}(a^2 \\sd b^2 \\sd c^2)$ admits a deformation to $\\CP$.\nSuch triples are called \\emph{Markov triples}, and the deformation gives rise to an embedding of three rational homology balls, each with boundary a lens space, into $\\CP$.\nWhenever an embedding of this type exists, the orders of the fundamental groups are coprime perfect squares, and the inequality conjectured by Koll\\'ar implies that $M$ has at most three boundary components.\n\nIn the realm of complex algebraic geometry, this is in fact the only instance of such embeddings, as was shown by Hacking and Prokhorov~\\cite{hp}.\nThe corresponding result in the symplectic category is also known to hold, by work of Evans and Smith~\\cite{es}.\nExplicit Kirby diagrams for the associated 4-manifolds with boundary were exhibited by Etnyre, Min, Piccirillo, and Roy~\\cite{EMPR}.\nTopologically, however, the situation is quite different: earlier work of the second author~\\cite{nonsymp},\n and of Lisca and Parma~\\cite{LPStein,HD1, HD2}, shows that there are infinitely many embeddings of rational balls in $\\CP$ that do not arise symplectically.\n\nWe would like to stress that, among the rational homology balls $\\pm B_{p,q}$, the only ones that carry some `good' complex or symplectic structure are the those for which $\\gcd(p,q) = 1$, with the positive sign.\nBy `good' here we mean that either they are diffeomorphic to a Milnor fibre in the complex setup \\cite{wahl}, or that they have a symplectic structure with respect to which the boundary is convex in the symplectic setup \\cite{LiscaSympFill}.\nIn particular, since in Theorem~\\ref{thm:ANN} we have embeddings of $B_{p,q}$ in $\\CPb$, none of the nontrival rational homology balls in any of the triples, with the exception of $B_{2,0}$, bear any complex or symplectic significance.\n\n\\subsection*{Notation and conventions}\nUnless otherwise stated, homology is taken with integer coefficients. The lens space $L(p,q)$ is the quotient of $S^3 \\subset \\mathbb{C}^2$ by the action of the group generated by  $\\begin{pmatrix}\\zeta & 0 \\\\ 0 & \\zeta^q\\end{pmatrix}$, where $\\zeta$ is a primitive $p^{\\rm th}$ root of $1$, which is also the link of the cyclic quotient singularity of type $\\frac1p(1,q)$. With this description, $L(p,q)$ is obtained by doing $-p/q$-surgery on the unknot in $S^3$.\n\n\\subsection*{Organisation of the paper}\nIn Section~\\ref{s:balls} we give different descriptions of the balls $B_{p,q}$.\nIn Section~\\ref{s:2Fareytree} we introduce the 2-Farey tree and state some of its properties, which we use in Section~\\ref{s:ANN} to prove Theorem~\\ref{thm:ANN}.\nIn Section~\\ref{s:compare} we put the results from Theorem~\\ref{thm:ANN} in a common framework with those of Lisca and Parma~\\cite{HD2}, and also give new recursive descriptions of the examples obtained in \\cite{HD2}.\nIn Section~\\ref{s:homotopyCP2s} we describe two constructions of embeddings of triples of rational balls, coming from plumbings, and one construction of embeddings of rational balls and pairs of rational balls, coming from knots with lens space surgeries.\nFinally, in Section~\\ref{s:data} we share and discuss some data.\n\n\\subsection*{Acknowledgements}\nWe would like to warmly thank Yank{\\i} Lekili for a question which initiated this collaboration.\nWe also thank Nikolas Adaloglou, Giulia Carfora, Johannes Hauber, Woohyeok Jo, Paolo Lisca, Marco Marengon, Jongil Park, and Kyungbae Park for stimulating conversations.\nWe would like to thank Andrea Parma for providing helpful data and clarifications about his papers.\nThe first author thanks the University of Glasgow for their hospitality.\nThe second thanks Oxford University and Nantes University for their hospitality.\nPart of this work has been carried out at several conferences and workshops, including \\emph{Surfaces in 4-manifolds} in le Croisic, \\emph{4-vari\\'et\\'es: \\`a travers les dimensions}\nat CIRM in Marseilles, \\emph{The low-dimensional workshop} during the \\emph{Singularities and low-dimensional topology} research semester at the Erd\\H{o}s Center in Budapest and \\emph{Combinatorial and gauge-theoretical methods in low-dimensional topology and geometry} at the Centro De Giorgi in Pisa.",
    "sketch": "The post-theorem introduction says: \"The proof of Theorem~\\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.\" More precisely, it \"uses structural induction on a simple variant of the Farey tree\" (to be defined in Section~\\ref{s:2Fareytree}). The Kirby-calculus part is \"related to Lisca and Parma's horizontal decompositions\" and \"informed by the Kirby diagrams of~\\cite{EMPR}.\" The \"main idea\" is \"to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\\times D^3$.\"",
    "expanded_sketch": "The post-theorem introduction says: “The proof of the main theorem is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.” More precisely, it “uses structural induction on a simple variant of the Farey tree” (to be defined next). The Kirby-calculus part is “related to Lisca and Parma's horizontal decompositions” and “informed by the Kirby diagrams of~\\cite{EMPR}.” The “main idea” is “to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\\times D^3$.”",
    "expanded_theorem": "\\label{thm:ANN}\nIf $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\\ge q_i$, and $p_1q_2-p_2q_1=\\pm2$, then the disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in $\\CPb$.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(B_{p,q}\\) denote the paper’s rational homology ball associated to a pair of nonnegative integers \\((p,q)\\); in the cases relevant here it has boundary the lens space \\(L(p^2,pq-1)\\). Suppose \\(p_1,p_2,q_1,q_2\\) are nonnegative integers such that \\(q_1\\) and \\(q_2\\) are even, \\(p_i\\ge q_i\\) for \\(i=1,2\\), and\n\\[\np_1q_2-p_2q_1=\\pm 2.\n\\]\nWhich of the following conclusions about smooth embeddings into \\(\\overline{\\mathbb{CP}}^{\\,2}\\) (the complex projective plane with reversed orientation) holds?",
      "correct_choice": {
        "label": "A",
        "text": "The disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,\\,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in \\(\\overline{\\mathbb{CP}}^{\\,2}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,\\,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in \\(\\overline{\\mathbb{CP}}^{\\,2}\\) provided one also assumes that \\(p_1\\) and \\(p_2\\) are coprime."
        },
        {
          "label": "C",
          "text": "There exists a smooth embedding in \\(\\overline{\\mathbb{CP}}^{\\,2}\\) of the disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}.\n\\]"
        },
        {
          "label": "D",
          "text": "The disjoint union\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{p_1+p_2,\\,q_1+q_2}\n\\]\nadmits a smooth orientation-preserving embedding in \\(\\overline{\\mathbb{CP}}^{\\,2}\\) whenever \\(q_1\\) and \\(q_2\\) are even, \\(p_i\\ge q_i\\), and \\(p_1q_2-p_2q_1=\\pm 1\\)."
        },
        {
          "label": "E",
          "text": "For every such choice of \\(p_1,p_2,q_1,q_2\\), there is a smooth orientation-preserving embedding in \\(\\overline{\\mathbb{CP}}^{\\,2}\\) of\n\\[\nB_{p_1,q_1}\\sqcup B_{p_2,q_2}\\sqcup B_{r,s}\n\\]\nfor some pair of nonnegative integers \\((r,s)\\) depending on \\((p_1,p_2,q_1,q_2)\\), not necessarily equal to \\((p_1+p_2,q_1+q_2)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "added unnecessary coprimality hypothesis on p_1,p_2",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped the third summand B_{p_1+p_2,q_1+q_2}",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "Farey-determinant condition ±2 replaced by ±1",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "specific third ball determined by Farey sum replaced by unspecified existential third summand",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and asks which conclusion holds, but it does not explicitly reveal the correct conclusion. The correct option is not signaled by wording in the stem."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct choice is essentially the theorem statement itself under the stated hypotheses. The item mainly tests recognition of the exact result rather than deriving a conclusion from broader principles."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some need to compare nearby variants of the theorem (extra hypothesis, weakened conclusion, altered determinant condition, existential variant), but the task is still mostly theorem recall rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one adds an unnecessary coprimality assumption, one gives a weaker true-looking statement, one alters the determinant condition, and one replaces the specific third summand by an existential claim. These reflect realistic confusion points."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall-based MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of the theorem and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.09203v1",
    "paper_link": "http://arxiv.org/abs/2512.09203v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}",
      "start_pos": 10873,
      "end_pos": 11856,
      "label": "themain"
    },
    "ref_dict": {
      "eqgmoment": "\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}",
      "eq+1": "\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\ppmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}",
      "eqtbB": "\\begin{align}\n&E_{M,N}\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}",
      "rem1": "\\begin{remark}\\label{rem1}\nObserve that $\\ve(f,\\chi)$ is independent of $\\chi$ when $f$ is a Maa\\ss~form.\nWhen $\\ve(f,\\chi)=\\ve(f)=-1$, summing both sides of equation \\eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\\ve(f,\\chi)$ depends on $\\chi$ at most though its parity $\\chi(-1)$. The symmetry also implies that the total contribution of such characters $\\chi$ satisfying $\\ve(f,\\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:\n\\begin{itemize}\n  \\item for a Maa\\ss~form $f$, we take $\\ve(f)=1$,\n  \\item for a holomorphic $f$, we consider only characters with $\\chi(-1)=\\ve(f)$.\n\\end{itemize}\nIn both cases, it holds that\n\\[\n\\ve(f,\\chi)=1,\n\\]\nwhich we assume henceforth.\n\\end{remark}",
      "themain": "\\begin{theorem}\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 4881,
    "pre_theorem_intro_text": "\\subsection{Moments of twisted $L$-functions}\nThe study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \\cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.\n\nAsymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and  Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by  Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by  Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.",
    "context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}",
    "full_context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\n\n\\subsection{Twisted $L$-functions}\nWe review some basic facts on twisted $L$-functions in this section; for further details, see \\cite[Section 2.3]{BFK+17a}.\nLet $\\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\\lambda(n)$. The twisted form $f\\otimes\\chi$ remains cuspidal for the congruence subgroup $\\Gamma_0(q^2)$ with nebentypus $\\chi^2$ (see e.g \\cite[Propositions 14.19 \\& 14.20]{IK04}). Its $L$-function is given by\n\\[\nL(s,f\\otimes\\chi)=\\sum_{n\\ge1}\\frac{\\lambda(n)\\chi(n)}{n^{s}}=\\prod_p\\left(1-\\frac{\\lambda(p)\\chi(p)}{p^s}+\\frac{\\chi^2(p)}{p^{2s}}\\r)^{-1},\\ \\ \\text{for}\\ \\ \\re(s)>1,\n\\]\nwhere the product is over primes $p$.\nThe completed $L$-function is defined by\n\\begin{align*}\n\\Lambda(s,f\\otimes\\chi)=q^s L_\\infty(s,f\\otimes\\chi)L(s,f\\otimes\\chi),\n\\end{align*}\nwith\n\\begin{align*}\nL_\\infty(s,f\\otimes\\chi)=\\begin{cases}\n(2\\pi)^{-\\frac{k-1}2-s}\\Gamma\\left(\\frac{k-1}2+s\\r), &\\text{if} \\ f\\ \\text{is holomorphic of weight}\\ k,\\\\\n\\pi^{-s-\\ma}\\Gamma\\left(\\frac{s+i\\kappa+\\ma}2\\r)\\Gamma\\left(\\frac{s-i\\kappa+\\ma}2\\r), &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form with eigenvalue}\\ \\tfrac14+\\kappa^2.\n\\end{cases}\n\\end{align*}\nThis completed $L$-function admits analytic continuation to $\\mathbb{C}$ and satisfies the functional equation (see e.g \\cite[Theorem 14.17, Proposition 14.20]{IK04}):\n\\[\n\\Lambda(s,f\\otimes\\chi)=\\ve(f\\otimes\\chi)\\Lambda(1-s,f\\otimes\\ol{\\chi}),\n\\]\nwhere the root number is given by\n\\begin{align*}\n\\ve(f\\otimes\\chi)=\\begin{cases}\n\\ve(f)\\ve_\\chi^2, &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n\\chi(-1)\\ve(f)\\ve_\\chi^2, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\end{align*}\nand $\\ve(f)=\\pm 1$ denotes the root number of $L(s,f)$.\n\nUsing the identity\n\\[\nS(hq\\ol{b}, m;a_1k)=e\\left(-m\\frac{\\ol{a}_1}{b}\\right)S_{\\infty,1/a_1}(hq,m;\\gammaup),\n\\]\nwhere $\\gammaup=a_1\\ssqrt{b}k$ and $Q=a_1b$, we derive from \\eqref{eqdefmS} that\n\\begin{align*}\n\\mS(a_1,r)=&\\frac{\\sqrt{b}}{aN_1}\\sum_{h\\sim H}\\sum_{m\\sim M^*} \\lambda(m)e\\left(-m\\frac{\\ol{a}_1}{b}\\right)\\\\\n&\\times\\sum_{\\gammaup}\\frac 1{\\gammaup}S_{\\infty,1/a_1}(hq, m;\\gammaup)W\\left(\\frac {r\\gammaup}{a_1\\ssqrt{b}N_1}\\r)\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right).\n\\end{align*}\nPerforming the variable substitution $(bx-hq)/N\\rightarrow x$ in \\eqref{eqV0} yields\n\\[\n\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right)=\\frac{N}{b}\\Omega\\Bigg(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup},\\frac{4\\pi\\ssqrt{mN}}{\\gammaup},h,m\\Bigg),\n\\]\nwhere\n\\[\n\\Omega(y,z,h,m)=\\int_{0}^\\infty W(x)W\\left(\\frac {hq+xN}M\\r) \\mJ_{+}\\left(\\ssqrt{y^2+xz^2}\\right)\\d x\n\\]\nwith\n\\begin{equation}\\label{eqyz}\ny\\asymp Y:=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}\\gg q^\\ve,\\quad z\\asymp \\frac{r\\ssqrt{M^*N}}{a_1\\ssqrt{b}N_1},\\quad h\\asymp H,\\quad m\\asymp M^*.\n\\end{equation}\nFor $M\\gg q^\\ve N$, it is easy to see from \\eqref{eqlargeM} and \\eqref{eqyz} that\n\\[\ny\\gg z\\quad \\text{and}\\quad y\\gg z^2q^{-\\ve}.\n\\]\nThen Lemma \\ref{lemdecomJ} yields the decomposition\n\\[\n\\Omega(y,z,h,m)=\\Omega_+(y,z,h,m)e^{iy}+\\Omega_-(y,z,h,m)e^{-iy}+O_A\\left(q^{-A}\\r),\n\\]\nwhere the functions $\\Omega_{\\pm}$ satisfy\n\\begin{align}\\label{eqWpmd}\ny^{j_1}z^{j_2}h^{j_3}m^{j_4}\\frac{\\partial^{j_1+j_2+j_3+j_4}}{\\partial y^{j_1}\\partial z^{j_2}\\partial h^{j_3}\\partial m^{j_4}}\\Omega_{\\pm}(y,z,h,m)\\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\\ve} Y^{-\\frac12}.\n\\end{align}\nAfter separating variables in $\\Omega_\\pm$ via the Mellin transform, we bound $\\mS(a_1,r)$ by sums of the form\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\sum_{\\gammaup}\\frac1{\\gammaup}S_{\\infty,1/a_1}(hq,m;\\gammaup)\\Theta \\B(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup}\\B)\\B|\n\\end{equation*}\nfor some coefficients satisfying $\\alpha_h\\ll h^\\ve$, $\\beta_m\\ll m^\\ve$ and a test function\n\\[\n\\Theta (y)=W\\B(\\frac yY\\B)\\exp(\\pm iy)y^{-u}\n\\]\nwith $\\re(u)=\\ve$.\n\nAn application of the Kuznetsov formula (Lemma \\ref{lemKf}) followed by spectral truncation at parameter $Y^{\\frac12+\\ve}$ ( Lemma \\ref{lemBe}) produces\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\left(\\mH+\\mM+\\mE\\r)\\B|,\n\\end{equation*}\nwhere the spectral contributions decompose as\n\\[\n\\mH=\\sum_{\\substack{2\\le k\\le Y^{\\frac12+\\ve}\\\\ k ~\\text{even}}}\\sum_{f\\in \\mathcal{B}_k(Q)}\\tilde{\\Theta} (k)\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m),\n\\]\n\\[\n\\mM=\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\hat{\\Theta} (\\kappa_f)\\frac{\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)},\n\\]\n\\[\n\\mE=\\frac1{4\\pi}\\sum_{\\mc}\\int_{|\\kappa|\\ll Y^{\\frac12+\\ve}}\\hat{\\Theta} (\\kappa)\\ssqrt{hqm} ~\\frac{\\bar\\rho_{\\mc}(hq,\\kappa)~\\rho_{1/{a_1},\\mc}(m,\\kappa)}{\\cosh(\\pi\\kappa)}\\d \\kappa,\n\\]\nrepresenting the holomorphic cusp forms, Maa\\ss\\ cusp forms, and the Eisenstein series respectively.\nWe foucus our detailed analysis on the Maa\\ss\\ case (the holomorphic and Eisenstein cases are identical). Since $\\hat{\\Theta} (\\kappa_f)\\ll Y^{-\\frac12+\\ve}$ (see \\eqref{eqphi1}), the contribution of the Maa\\ss\\ case is bounded by\n\\[\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\frac{\\ssqrt{hqm} ~\\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)}\\B|,\n\\]\nwhich, after applying the Cauchy-Schwarz inequality, can be bounded by\n\\begin{align*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}&\\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{H<h\\le 2H}\\alpha_h \\ssqrt{hq} ~\\rho_f(hq)\\B|^2\\B)^{\\frac12}\\\\\n&\\times \\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{M^*<m\\le 2M^*}\\beta_m\\lambda(m)\\ssqrt{m} ~\\rho_f(a_1^{-1},m)\\B|^2\\B)^{\\frac12}.\n\\end{align*}\nWe may handle the second spectral sum via Lemma \\ref{lemsls}, but for the first spectral sum, an application of Lemma \\ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \\cite[Theorems 11.3-11.6]{GWZ25}.\n\\begin{lemma}\\label{thmsls}\nLet $Q,s\\in\\mathbb{N}$, $K, N\\ge 1$, and let $(a_n)_{n\\in[N,2N]}$ be complex coefficients. Then we have\n\\[\n\\sum_{\\substack{2\\le k\\le K\\\\ k~ \\text{even}}}\\Gamma(k)\\sum_{f\\in\\mathcal{B}_k(Q)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2, \\ \\ \\ \\sum_{\\mc}\\int_{-K}^K\\frac1{\\cosh(\\pi\\kappa)}\\B|\\sum_n a_n\\ssqrt{sn}~\\rho_{\\mc}(\\pm sn,\\kappa)\\B|^2\\d\\kappa,\n\\]\n\\[\n\\sum_{\\substack{f\\in\\mathcal{B}(Q)\\\\ |\\kappa_f|\\le K}}\\frac1{\\cosh(\\pi\\kappa_f)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2\n\\]\nare bounded by\n\\[\n\\ll (sQKN)^\\ve\\B(K+\\frac{s^{\\frac12}}{Q^{\\frac12}}\\B)\\B(K+\\frac{(s,Q)^{\\frac12}N}{Q^{\\frac12}}\\B)N\\|a_n\\|_\\infty.\n\\]\n\\end{lemma}\nApplying Lemma \\ref{thmsls} to the first spectral sum and Lemma \\ref{lemsls} to the second one reduces the Maa\\ss\\ contribution to\n\\begin{equation*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\B(H^{\\frac12}{M^*}^{\\frac12}\\left(Y+\\frac{q}{a_1b}\\r)^{\\frac14} \\left(Y+\\frac{(a_1b,q)H^2}{a_1b}\\r)^{\\frac14}\\left(Y+\\frac{M^*}{a_1b}\\r)^{\\frac12}\\B).\n\\end{equation*}\nTaking\n\\[\n H=\\frac{M}q,\\ \\ \\ M^*= q^\\ve\\frac{a_1^2N_1^2}{r^2}\\frac{bM}{N^2}, \\ \\ \\ Y=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}=\\frac{M}{N},\n\\]\nand observing that\n$\\frac{M^*}{a_1b}\\ll Y$ ( since $N_1^2\\ll N/a$), we obtain\nfor the Maa\\ss\\ contribution\n\\begin{align*}\n&\\ll q^\\ve\\frac{N}{r\\ssqrt{q}}\\left(Y+\\frac{q}{ab}\\r)^{\\frac14} \\left(Y+\\frac{(ab,q)H^2}{ab}\\r)^{\\frac14}Y^{\\frac12}\\\\\n&\\ll q^\\ve r^{-1}\\B(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac 54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac 34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}}+\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac 34}}\\B).\n\\end{align*}",
    "post_theorem_intro_text_len": 5321,
    "post_theorem_intro_text": "\\begin{remark}\nThe dependence on  $\\theta_f$ in the error term of Theorem \\ref{themain} arises from the trivial estimate \\eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\\ve(f)=1$ for a Maa\\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \\ref{rem1}). In the Maa\\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \\cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \\cite[Thoerem 1.1]{KZ23}.\n\\end{remark}\n\nSetting $a=b=1$ in Theorem \\ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.\n\n\\begin{corollary}\\label{cor}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$, we have\n\\begin{align*}\nM_{f,E}(q)=c_f\\prod_{p\\mid q}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(q^{-\\frac{1-2\\theta_f }{22+16\\theta_f }+\\ve}\\r),\n\\end{align*}\nwhere $c_f$ is the same constant as previously defined.\n\\end{corollary}\nFor the holomorphic case, Deligne's well-known work \\cite{Del74} gives $\\theta_f=0$, whereas for the Maa\\ss~form we may take $\\theta_f= 7/{64}$ due to Kim--Sarnak \\cite{Kim03}. This yields an error term as sharp as the best bound \\cite{KZ23} known for prime moduli.\n\nThrough the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum\n\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}\nwhere $a, b,M,N\\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\\delta$-method or Jutila's circle method to eliminate the congruence condition (see \\cite{BM15}). For \\eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.\n\nThe diagonal terms (where $bm=an$) in \\eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.\n\nIn the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum\n\\begin{align}\\label{eqAdef}\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m) \\tau(n) W\\left(\\frac {bm}M\\r) W\\left(\\frac {an}N\\r),\n\\end{align}\nwhere $W$ are some smooth functions.\nFor $M\\ge N$, we establish the bound\n\\[\nA_q(a,b,M,N)\\ll  q^\\ve\\Bigg(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}} +\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac34}}\\Bigg)\n\\]\nwith an analogous result holding for $N\\ge M$. Compared to the bound in \\cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\\frac12}$ and recovering a saving $(ab)^{-\\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \\cite{GWZ25}.\n\nFor the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \\cite{KSWX23} concerning arbitrary moduli.\nThe approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.\n\n\\subsection{Notation}\nWe adopt the standard convention that $\\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\\ve$ as well as on the fixed cusp form $f$.\nThroughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying\n\\begin{align}\\label{eqW}\nW^{(j)}(x)\\ll_{j}1,\n\\end{align}\nfor all integers $j\\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\\nequiv 2\\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.",
    "sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\\ref{themain}) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \\eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”",
    "expanded_sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \n\\begin{align}\n&E_{M,N}\\\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}\n,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”",
    "expanded_theorem": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment\nM_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),\nwhere φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define\nc_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,\nand\nc_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),\nwith c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?",
      "correct_choice": {
        "label": "A",
        "text": "One has\nM_{f,E}(q;a,b) = c_f · ∏_{p∣qab} (1 − λ(p)/p + 1/p^2)(1 − 1/p^2)^(−2) · ((c_{a,b}+c_{b,a})/(ab)^(1/2)) · (L(1,f)^2/ζ(2)) + O(q^(−1/20+ε)a^(3/10) + q^(−(1−2θ_f)/(22+16θ_f)+ε) b^((3+2θ_f)/(11+8θ_f)))."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}(ab)^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon}(ab)^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "C",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{\\varepsilon} b^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "D",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon} a^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "E",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon} b^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "separate balanced/unbalanced dependence on a and b in the error term",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the power-saving q-decay in the second error term",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "assignment of the second-regime parameter dependence to b rather than a",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "two-sided diagonal contribution c_{a,b}+c_{b,a}",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct formula; it only states the hypotheses and definitions. The correct choice must be identified from the options rather than being directly implied by wording in the question."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific theorem's full asymptotic statement. The question asks which exact asymptotic formula holds, so it is very close to restating the result itself."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some discrimination required among subtle variants of the main term and error terms, but the task is mostly theorem recognition/recall rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening explicit exponents, dropping the symmetric term, or conflating the separate dependence on a and b. They are distinct and nontrivial."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall question with strong distractors and no answer leakage, but it is largely a theorem-restatement item rather than a genuine reasoning-based MCQ."
    }
  },
  {
    "id": "2512.09203v1",
    "paper_link": "http://arxiv.org/abs/2512.09203v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}",
      "start_pos": 10873,
      "end_pos": 11856,
      "label": "themain"
    },
    "ref_dict": {
      "eqgmoment": "\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}",
      "eq+1": "\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\ppmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}",
      "eqtbB": "\\begin{align}\n&E_{M,N}\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}",
      "rem1": "\\begin{remark}\\label{rem1}\nObserve that $\\ve(f,\\chi)$ is independent of $\\chi$ when $f$ is a Maa\\ss~form.\nWhen $\\ve(f,\\chi)=\\ve(f)=-1$, summing both sides of equation \\eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\\ve(f,\\chi)$ depends on $\\chi$ at most though its parity $\\chi(-1)$. The symmetry also implies that the total contribution of such characters $\\chi$ satisfying $\\ve(f,\\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:\n\\begin{itemize}\n  \\item for a Maa\\ss~form $f$, we take $\\ve(f)=1$,\n  \\item for a holomorphic $f$, we consider only characters with $\\chi(-1)=\\ve(f)$.\n\\end{itemize}\nIn both cases, it holds that\n\\[\n\\ve(f,\\chi)=1,\n\\]\nwhich we assume henceforth.\n\\end{remark}",
      "themain": "\\begin{theorem}\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 4881,
    "pre_theorem_intro_text": "\\subsection{Moments of twisted $L$-functions}\nThe study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \\cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.\n\nAsymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and  Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by  Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by  Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.",
    "context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}",
    "full_context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\n\n\\subsection{Twisted $L$-functions}\nWe review some basic facts on twisted $L$-functions in this section; for further details, see \\cite[Section 2.3]{BFK+17a}.\nLet $\\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\\lambda(n)$. The twisted form $f\\otimes\\chi$ remains cuspidal for the congruence subgroup $\\Gamma_0(q^2)$ with nebentypus $\\chi^2$ (see e.g \\cite[Propositions 14.19 \\& 14.20]{IK04}). Its $L$-function is given by\n\\[\nL(s,f\\otimes\\chi)=\\sum_{n\\ge1}\\frac{\\lambda(n)\\chi(n)}{n^{s}}=\\prod_p\\left(1-\\frac{\\lambda(p)\\chi(p)}{p^s}+\\frac{\\chi^2(p)}{p^{2s}}\\r)^{-1},\\ \\ \\text{for}\\ \\ \\re(s)>1,\n\\]\nwhere the product is over primes $p$.\nThe completed $L$-function is defined by\n\\begin{align*}\n\\Lambda(s,f\\otimes\\chi)=q^s L_\\infty(s,f\\otimes\\chi)L(s,f\\otimes\\chi),\n\\end{align*}\nwith\n\\begin{align*}\nL_\\infty(s,f\\otimes\\chi)=\\begin{cases}\n(2\\pi)^{-\\frac{k-1}2-s}\\Gamma\\left(\\frac{k-1}2+s\\r), &\\text{if} \\ f\\ \\text{is holomorphic of weight}\\ k,\\\\\n\\pi^{-s-\\ma}\\Gamma\\left(\\frac{s+i\\kappa+\\ma}2\\r)\\Gamma\\left(\\frac{s-i\\kappa+\\ma}2\\r), &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form with eigenvalue}\\ \\tfrac14+\\kappa^2.\n\\end{cases}\n\\end{align*}\nThis completed $L$-function admits analytic continuation to $\\mathbb{C}$ and satisfies the functional equation (see e.g \\cite[Theorem 14.17, Proposition 14.20]{IK04}):\n\\[\n\\Lambda(s,f\\otimes\\chi)=\\ve(f\\otimes\\chi)\\Lambda(1-s,f\\otimes\\ol{\\chi}),\n\\]\nwhere the root number is given by\n\\begin{align*}\n\\ve(f\\otimes\\chi)=\\begin{cases}\n\\ve(f)\\ve_\\chi^2, &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n\\chi(-1)\\ve(f)\\ve_\\chi^2, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\end{align*}\nand $\\ve(f)=\\pm 1$ denotes the root number of $L(s,f)$.\n\nUsing the identity\n\\[\nS(hq\\ol{b}, m;a_1k)=e\\left(-m\\frac{\\ol{a}_1}{b}\\right)S_{\\infty,1/a_1}(hq,m;\\gammaup),\n\\]\nwhere $\\gammaup=a_1\\ssqrt{b}k$ and $Q=a_1b$, we derive from \\eqref{eqdefmS} that\n\\begin{align*}\n\\mS(a_1,r)=&\\frac{\\sqrt{b}}{aN_1}\\sum_{h\\sim H}\\sum_{m\\sim M^*} \\lambda(m)e\\left(-m\\frac{\\ol{a}_1}{b}\\right)\\\\\n&\\times\\sum_{\\gammaup}\\frac 1{\\gammaup}S_{\\infty,1/a_1}(hq, m;\\gammaup)W\\left(\\frac {r\\gammaup}{a_1\\ssqrt{b}N_1}\\r)\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right).\n\\end{align*}\nPerforming the variable substitution $(bx-hq)/N\\rightarrow x$ in \\eqref{eqV0} yields\n\\[\n\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right)=\\frac{N}{b}\\Omega\\Bigg(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup},\\frac{4\\pi\\ssqrt{mN}}{\\gammaup},h,m\\Bigg),\n\\]\nwhere\n\\[\n\\Omega(y,z,h,m)=\\int_{0}^\\infty W(x)W\\left(\\frac {hq+xN}M\\r) \\mJ_{+}\\left(\\ssqrt{y^2+xz^2}\\right)\\d x\n\\]\nwith\n\\begin{equation}\\label{eqyz}\ny\\asymp Y:=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}\\gg q^\\ve,\\quad z\\asymp \\frac{r\\ssqrt{M^*N}}{a_1\\ssqrt{b}N_1},\\quad h\\asymp H,\\quad m\\asymp M^*.\n\\end{equation}\nFor $M\\gg q^\\ve N$, it is easy to see from \\eqref{eqlargeM} and \\eqref{eqyz} that\n\\[\ny\\gg z\\quad \\text{and}\\quad y\\gg z^2q^{-\\ve}.\n\\]\nThen Lemma \\ref{lemdecomJ} yields the decomposition\n\\[\n\\Omega(y,z,h,m)=\\Omega_+(y,z,h,m)e^{iy}+\\Omega_-(y,z,h,m)e^{-iy}+O_A\\left(q^{-A}\\r),\n\\]\nwhere the functions $\\Omega_{\\pm}$ satisfy\n\\begin{align}\\label{eqWpmd}\ny^{j_1}z^{j_2}h^{j_3}m^{j_4}\\frac{\\partial^{j_1+j_2+j_3+j_4}}{\\partial y^{j_1}\\partial z^{j_2}\\partial h^{j_3}\\partial m^{j_4}}\\Omega_{\\pm}(y,z,h,m)\\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\\ve} Y^{-\\frac12}.\n\\end{align}\nAfter separating variables in $\\Omega_\\pm$ via the Mellin transform, we bound $\\mS(a_1,r)$ by sums of the form\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\sum_{\\gammaup}\\frac1{\\gammaup}S_{\\infty,1/a_1}(hq,m;\\gammaup)\\Theta \\B(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup}\\B)\\B|\n\\end{equation*}\nfor some coefficients satisfying $\\alpha_h\\ll h^\\ve$, $\\beta_m\\ll m^\\ve$ and a test function\n\\[\n\\Theta (y)=W\\B(\\frac yY\\B)\\exp(\\pm iy)y^{-u}\n\\]\nwith $\\re(u)=\\ve$.\n\nAn application of the Kuznetsov formula (Lemma \\ref{lemKf}) followed by spectral truncation at parameter $Y^{\\frac12+\\ve}$ ( Lemma \\ref{lemBe}) produces\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\left(\\mH+\\mM+\\mE\\r)\\B|,\n\\end{equation*}\nwhere the spectral contributions decompose as\n\\[\n\\mH=\\sum_{\\substack{2\\le k\\le Y^{\\frac12+\\ve}\\\\ k ~\\text{even}}}\\sum_{f\\in \\mathcal{B}_k(Q)}\\tilde{\\Theta} (k)\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m),\n\\]\n\\[\n\\mM=\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\hat{\\Theta} (\\kappa_f)\\frac{\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)},\n\\]\n\\[\n\\mE=\\frac1{4\\pi}\\sum_{\\mc}\\int_{|\\kappa|\\ll Y^{\\frac12+\\ve}}\\hat{\\Theta} (\\kappa)\\ssqrt{hqm} ~\\frac{\\bar\\rho_{\\mc}(hq,\\kappa)~\\rho_{1/{a_1},\\mc}(m,\\kappa)}{\\cosh(\\pi\\kappa)}\\d \\kappa,\n\\]\nrepresenting the holomorphic cusp forms, Maa\\ss\\ cusp forms, and the Eisenstein series respectively.\nWe foucus our detailed analysis on the Maa\\ss\\ case (the holomorphic and Eisenstein cases are identical). Since $\\hat{\\Theta} (\\kappa_f)\\ll Y^{-\\frac12+\\ve}$ (see \\eqref{eqphi1}), the contribution of the Maa\\ss\\ case is bounded by\n\\[\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\frac{\\ssqrt{hqm} ~\\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)}\\B|,\n\\]\nwhich, after applying the Cauchy-Schwarz inequality, can be bounded by\n\\begin{align*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}&\\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{H<h\\le 2H}\\alpha_h \\ssqrt{hq} ~\\rho_f(hq)\\B|^2\\B)^{\\frac12}\\\\\n&\\times \\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{M^*<m\\le 2M^*}\\beta_m\\lambda(m)\\ssqrt{m} ~\\rho_f(a_1^{-1},m)\\B|^2\\B)^{\\frac12}.\n\\end{align*}\nWe may handle the second spectral sum via Lemma \\ref{lemsls}, but for the first spectral sum, an application of Lemma \\ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \\cite[Theorems 11.3-11.6]{GWZ25}.\n\\begin{lemma}\\label{thmsls}\nLet $Q,s\\in\\mathbb{N}$, $K, N\\ge 1$, and let $(a_n)_{n\\in[N,2N]}$ be complex coefficients. Then we have\n\\[\n\\sum_{\\substack{2\\le k\\le K\\\\ k~ \\text{even}}}\\Gamma(k)\\sum_{f\\in\\mathcal{B}_k(Q)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2, \\ \\ \\ \\sum_{\\mc}\\int_{-K}^K\\frac1{\\cosh(\\pi\\kappa)}\\B|\\sum_n a_n\\ssqrt{sn}~\\rho_{\\mc}(\\pm sn,\\kappa)\\B|^2\\d\\kappa,\n\\]\n\\[\n\\sum_{\\substack{f\\in\\mathcal{B}(Q)\\\\ |\\kappa_f|\\le K}}\\frac1{\\cosh(\\pi\\kappa_f)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2\n\\]\nare bounded by\n\\[\n\\ll (sQKN)^\\ve\\B(K+\\frac{s^{\\frac12}}{Q^{\\frac12}}\\B)\\B(K+\\frac{(s,Q)^{\\frac12}N}{Q^{\\frac12}}\\B)N\\|a_n\\|_\\infty.\n\\]\n\\end{lemma}\nApplying Lemma \\ref{thmsls} to the first spectral sum and Lemma \\ref{lemsls} to the second one reduces the Maa\\ss\\ contribution to\n\\begin{equation*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\B(H^{\\frac12}{M^*}^{\\frac12}\\left(Y+\\frac{q}{a_1b}\\r)^{\\frac14} \\left(Y+\\frac{(a_1b,q)H^2}{a_1b}\\r)^{\\frac14}\\left(Y+\\frac{M^*}{a_1b}\\r)^{\\frac12}\\B).\n\\end{equation*}\nTaking\n\\[\n H=\\frac{M}q,\\ \\ \\ M^*= q^\\ve\\frac{a_1^2N_1^2}{r^2}\\frac{bM}{N^2}, \\ \\ \\ Y=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}=\\frac{M}{N},\n\\]\nand observing that\n$\\frac{M^*}{a_1b}\\ll Y$ ( since $N_1^2\\ll N/a$), we obtain\nfor the Maa\\ss\\ contribution\n\\begin{align*}\n&\\ll q^\\ve\\frac{N}{r\\ssqrt{q}}\\left(Y+\\frac{q}{ab}\\r)^{\\frac14} \\left(Y+\\frac{(ab,q)H^2}{ab}\\r)^{\\frac14}Y^{\\frac12}\\\\\n&\\ll q^\\ve r^{-1}\\B(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac 54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac 34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}}+\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac 34}}\\B).\n\\end{align*}",
    "post_theorem_intro_text_len": 5321,
    "post_theorem_intro_text": "\\begin{remark}\nThe dependence on  $\\theta_f$ in the error term of Theorem \\ref{themain} arises from the trivial estimate \\eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\\ve(f)=1$ for a Maa\\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \\ref{rem1}). In the Maa\\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \\cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \\cite[Thoerem 1.1]{KZ23}.\n\\end{remark}\n\nSetting $a=b=1$ in Theorem \\ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.\n\n\\begin{corollary}\\label{cor}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$, we have\n\\begin{align*}\nM_{f,E}(q)=c_f\\prod_{p\\mid q}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(q^{-\\frac{1-2\\theta_f }{22+16\\theta_f }+\\ve}\\r),\n\\end{align*}\nwhere $c_f$ is the same constant as previously defined.\n\\end{corollary}\nFor the holomorphic case, Deligne's well-known work \\cite{Del74} gives $\\theta_f=0$, whereas for the Maa\\ss~form we may take $\\theta_f= 7/{64}$ due to Kim--Sarnak \\cite{Kim03}. This yields an error term as sharp as the best bound \\cite{KZ23} known for prime moduli.\n\nThrough the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum\n\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}\nwhere $a, b,M,N\\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\\delta$-method or Jutila's circle method to eliminate the congruence condition (see \\cite{BM15}). For \\eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.\n\nThe diagonal terms (where $bm=an$) in \\eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.\n\nIn the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum\n\\begin{align}\\label{eqAdef}\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m) \\tau(n) W\\left(\\frac {bm}M\\r) W\\left(\\frac {an}N\\r),\n\\end{align}\nwhere $W$ are some smooth functions.\nFor $M\\ge N$, we establish the bound\n\\[\nA_q(a,b,M,N)\\ll  q^\\ve\\Bigg(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}} +\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac34}}\\Bigg)\n\\]\nwith an analogous result holding for $N\\ge M$. Compared to the bound in \\cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\\frac12}$ and recovering a saving $(ab)^{-\\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \\cite{GWZ25}.\n\nFor the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \\cite{KSWX23} concerning arbitrary moduli.\nThe approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.\n\n\\subsection{Notation}\nWe adopt the standard convention that $\\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\\ve$ as well as on the fixed cusp form $f$.\nThroughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying\n\\begin{align}\\label{eqW}\nW^{(j)}(x)\\ll_{j}1,\n\\end{align}\nfor all integers $j\\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\\nequiv 2\\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.",
    "sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\\ref{themain}) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \\eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”",
    "expanded_sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \n\\begin{align}\n&E_{M,N}\\\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}\n,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”",
    "expanded_theorem": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment\nM_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),\nwhere φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define\nc_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,\nand\nc_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),\nwith c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?",
      "correct_choice": {
        "label": "A",
        "text": "One has\nM_{f,E}(q;a,b) = c_f · ∏_{p∣qab} (1 − λ(p)/p + 1/p^2)(1 − 1/p^2)^(−2) · ((c_{a,b}+c_{b,a})/(ab)^(1/2)) · (L(1,f)^2/ζ(2)) + O(q^(−1/20+ε)a^(3/10) + q^(−(1−2θ_f)/(22+16θ_f)+ε) b^((3+2θ_f)/(11+8θ_f)))."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}(ab)^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon}(ab)^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "C",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{\\varepsilon} b^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "D",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon} a^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        },
        {
          "label": "E",
          "text": "One has\nM_{f,E}(q;a,b) = c_f \\cdot \\prod_{p\\mid qab} (1 - \\lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \\cdot \\frac{c_{a,b}}{(ab)^{1/2}} \\cdot \\frac{L(1,f)^2}{\\zeta(2)} + O\\!\\left(q^{-1/20+\\varepsilon}a^{3/10} + q^{-(1-2\\theta_f)/(22+16\\theta_f)+\\varepsilon} b^{(3+2\\theta_f)/(11+8\\theta_f)}\\right)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "separate balanced/unbalanced dependence on a and b in the error term",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the power-saving q-decay in the second error term",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "assignment of the second-regime parameter dependence to b rather than a",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "two-sided diagonal contribution c_{a,b}+c_{b,a}",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and definitions but does not reveal the conclusion. The correct asymptotic formula is not stated or strongly hinted at in the prompt itself."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-identification question: the setup is followed by 'Which statement holds?'. Although the options introduce nearby alternatives, the task is still largely selecting the exact theorem conclusion rather than resolving a broader mathematical issue."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning or precise recall is needed because the choices differ in subtle but meaningful ways (symmetry in c_{a,b}+c_{b,a}, dependence on a versus b, and strength of q-decay). However, it does not strongly induce generative reasoning from the hypotheses alone."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrong combined error terms, weakened q-saving, swapped variable dependence, and omission of the symmetric c_{b,a} term."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/discrimination than genuine generative reasoning."
    }
  },
  {
    "id": "2512.09490v1",
    "paper_link": "http://arxiv.org/abs/2512.09490v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "lemma",
      "content": "\\label{lem:anest}\nLet $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\\times g_2$. Then the following hold for all measurable $S\\subset M_1\\times M_2$ and $(m_1+m_2-1)$–rectifiable $\\Sigma$:\n\\begin{enumerate}\n\\item[(i)] For every set $S \\subset M_1 \\times M_2$,\n\\begin{equation}\\label{eq:anest1}\n  \\vv_{g_+^t}(S) = t^{m_2} \\,\\vv_g(S).\n\\end{equation}\n\\item[(ii)] If $\\Sigma \\subset M_1 \\times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \\leq 1$,\n\\begin{equation}\\label{eq:anest2}\n  \\hh_{g_+^t}^{m_1+m_2-1}(\\Sigma) \n  \\leq t^{m_2-1}\\,\\hh_g^{m_1+m_2-1}(\\Sigma).\n\\end{equation}\n\\item[(iii)] Equality holds in \\eqref{eq:anest2} if and only if the $g$-normal of $\\Sigma$ is tangent to $M_2$, up to a $\\hh_g^{m_1+m_2-1}$-null set.\n\\end{enumerate}",
      "start_pos": 16915,
      "end_pos": 17703,
      "label": "lem:anest"
    },
    "ref_dict": {
      "prp:prpc": "\\begin{proposition}\n\\label{prp:prpc}\n\\mbox{}\nUnder the hypotheses of Proposition~\\ref{prp:pr} there holds\n\\begin{equation}\n\\label{eq:prpc}\n\\ptl E_i\\to \\ptl E\\quad\\text{in the Hausdorff topology}.\n\\end{equation}\n\\end{proposition}",
      "prp:pr": "\\begin{proposition}\n\\label{prp:pr}\n\\mbox{}\nAssume $0<\\be<1$, $t_i\\uparrow\\infty\\,$ and  $\\{E_i\\}_{i\\in\\nn}$ be a sequence of compact symmetrized isoperimetric sets in $ \\esf\\times t_i N$ of volume $\\vv_{g_+^{t_i}}(E_i)=\\be \\vv_{g_+^{t_i}}(\\esf\\times N)$. Let $t_0>0$ and $s_i=t_0\\,t_i^{-1}$. \n\\begin{enum}\n\\item\nThen, possibly passing to a non-relabeling subsequence, $\\{E_i\\}_{i\\in\\nn}$ converges both in $L^1$ and Hausdorff topology of $\\esf\\times t_0N$,  to a finite perimeter set $E$ which is a cylinderoid, i.e $E=\\esf\\times \\pi_2(E)$. \n\\item Moreover $\\pi_2(E)$ is an isoperimetric set in $t_0N$, of volume fraction $\\be$.\n\\item Furthermore\n\\begin{equation}\n\\label{eq:ane00}\n\\vv_{g_+^{t_0}}(E_i)=\\vv_{g_+^{t_0}}(\\esf\\times \\pi_2(E)) \\,\\,\\text{and}\\,\\,\\pp_{g_+^{t_0}}(E_i)\\le\\pp_{g_+^{t_0}}(\\esf\\times \\pi_2(E))\n\\end{equation}\n\\end{enum}\n\\end{proposition}",
      "Thm:sts": "\\begin{proposition}\n\\label{Thm:sts}\nLet $\\Sigma\\subset N$ be a smooth constant–mean–curvature hypersurface.\nIf $\\Sigma$ is stable, then $tM\\times\\Sigma$ is stable for all sufficiently small $t>0$.\n\\end{proposition}",
      "prp:reduce": "\\begin{proposition}\n\\label{prp:reduce}\nIf Theorem \\ref{thm:main} is true when the first factor is a round sphere, then it holds in general, i.e, when the first factor is any compact Riemannian manifold.\n\\end{proposition}",
      "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $M,N$ be compact Riemannian manifolds and $\\be\\in (0,1)$. Assume that the boundaries of the isoperimetric regions in $N$ of volume fraction $\\be$ are of class $C^{2,\\alpha}$.\n\\begin{enum}\n\\item Then there exists $t_0=t_0(\\be)>0$ so that  for every $0<t\\le t_0$\nevery isoperimetric region of volume $\\be\\vv(t M \\times N)$ in $t M \\times N$, is of the form  $ M \\times S$, where $S$ is an isoperimetric region in $N$.\n\nBy scaling isotropically, we get equivalently.\n\\item\nFor every $s\\ge s_0$\nevery isoperimetric region of volume $\\be \\vv(M \\times s N)$ in $M \\times s N$, is of the form  $M \\times S$, where $S$ is an isoperimetric region in $s N$ and $s_0=t_0^{-1}$.\n\\end{enum}\n\\end{theorem}",
      "eq:anest2": "\\begin{equation}\\label{eq:anest2}\n  \\hh_{g_+^t}^{m_1+m_2-1}(\\Sg) \n  \\leq t^{m_2-1}\\,\\hh_g^{m_1+m_2-1}(\\Sg).\n\\end{equation}",
      "eq:stab": "\\begin{equation}\\label{eq:stab}\n- \\int_\\Sg u \\,\\big(\\Delta u + \\ric(\\nu,\\nu)u + |\\sg|^2 u\\big) \\,\\geq 0,\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5562,
    "pre_theorem_intro_text": "The classical isoperimetric problem seeks to determine, within a given ambient space, the subsets of prescribed volume that minimize boundary measure.\nIt is one of the most fundamental variational problems in geometry.\n\nIn products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.\nIn products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \\cite{peri}.\nMorgan~\\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.\nIn the spherical case, Pedrosa~\\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\\mathbb{S}^n\\times\\mathbb{R}$.\nIn Riemannian cylinders $M\\times\\mathbb{R}$, Duzaar and Steffen~\\cite{du-st} proved that large-volume minimizers are slabs.\nThis was later extended to $M\\times\\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\\cite{manstr}.\n\nA natural question was originally proposed by M. Hutchings — as reported by F. Morgan in~\\cite{morpol} — who conjectured that under anisotropic scaling, the isoperimetric regions should align with a cylindrical product structure. More concretely, if one rescales one factor of a product, do minimizers eventually acquire a cylindrical structure of the form $M \\times S$, where $S$ is an isoperimetric region in the other factor, once the scaling parameter is sufficiently small (or large)?\n\nIn this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \\(N\\) are of class \\(C^{2,\\alpha}\\), a condition that holds, for example, when \\(\\dim N\\leqslant 8\\) or for small volumes (and their complements).\n\nWe begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.\nIn the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\\mathbb{S}^m\\times N$ (Proposition~\\ref{prp:reduce})\n\\,(see also the related symmetrization results of Morgan--Howe--Harman~\\cite{mosym}).\nWe then analyze \\emph{cylinderoids} $M_1\\times S$ and symmetrized competitors in $\\mathbb{S}^m\\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\\ref{prp:pr} and~\\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,\nall relevant competitors have $C^{2,\\alpha}$ boundaries.  \nThis allows us to restrict the analysis to $C^{2,\\alpha}$ graphical \nperturbations of $\\Sigma_t$.\nNext, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\\Sigma\\subset N$ is a stable constant mean curvature hypersurface, then $tM\\times\\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\\ref{Thm:sts}).\nFinally, assuming the $C^{2,\\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\\cite{rqi} to the present product setting around $\\Sg_t:=tM\\times\\Sigma$. \nThis analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\\Sg_t$ to be cylinderoids (Theorem~\\ref{thm:main}).\n\nWe begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.\n\nLet $(M,g)$ be a Riemannian manifold. For a measurable set $S \\subset M$, we denote by \n\\[\n  \\vv_g(S) \\quad \\text{and} \\quad \\pp_g(S)\n\\]\nthe Riemannian volume and perimeter, respectively. The $\\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\\hh^\\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:\n\\begin{equation}\\label{eq:est}\n  \\hh_{t^2 g}^{\\alpha}(S) = t^{\\alpha}\\,\\hh_g^{\\alpha}(S), \n  \\qquad \\vv_{t^2 g}(S) = t^m\\,\\vv_g(S), \n  \\qquad \\pp_{t^2 g}(S) = t^{m-1}\\,\\pp_g(S),\n\\end{equation}\nwhere $m = \\dim M$.  \nThroughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \\emph{$t$-homothety} of $(M,g)$.\n\nLet $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \\times g_2$.  \nWe define two anisotropic product scalings:\n\\begin{itemize}\n  \\item The \\emph{right anisotropic $t$-homothety} is\n  \\[\n    (M_1 \\times M_2, g_1 \\times t^2 g_2),\n  \\]\n  denoted by $(M_1 \\times M_2, g_+^t)$ or simply $M_1 \\times tM_2$.\n  \\item The \\emph{left anisotropic $t$-homothety} is\n  \\[\n    (M_1 \\times M_2, t^2 g_1 \\times g_2),\n  \\]\n  denoted by $(M_1 \\times M_2, g_-^t)$ or simply $tM_1 \\times M_2$.\n\\end{itemize}\nThese anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.",
    "context": "In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.\nIn products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \\cite{peri}.\nMorgan~\\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.\nIn the spherical case, Pedrosa~\\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\\mathbb{S}^n\\times\\mathbb{R}$.\nIn Riemannian cylinders $M\\times\\mathbb{R}$, Duzaar and Steffen~\\cite{du-st} proved that large-volume minimizers are slabs.\nThis was later extended to $M\\times\\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\\cite{manstr}.\n\nIn this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \\(N\\) are of class \\(C^{2,\\alpha}\\), a condition that holds, for example, when \\(\\dim N\\leqslant 8\\) or for small volumes (and their complements).\n\nWe begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.\nIn the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\\mathbb{S}^m\\times N$ (Proposition~\\ref{prp:reduce})\n\\,(see also the related symmetrization results of Morgan--Howe--Harman~\\cite{mosym}).\nWe then analyze \\emph{cylinderoids} $M_1\\times S$ and symmetrized competitors in $\\mathbb{S}^m\\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\\ref{prp:pr} and~\\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,\nall relevant competitors have $C^{2,\\alpha}$ boundaries. \nThis allows us to restrict the analysis to $C^{2,\\alpha}$ graphical \nperturbations of $\\Sigma_t$.\nNext, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\\Sigma\\subset N$ is a stable constant mean curvature hypersurface, then $tM\\times\\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\\ref{Thm:sts}).\nFinally, assuming the $C^{2,\\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\\cite{rqi} to the present product setting around $\\Sg_t:=tM\\times\\Sigma$. \nThis analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\\Sg_t$ to be cylinderoids (Theorem~\\ref{thm:main}).\n\nWe begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.\n\nLet $(M,g)$ be a Riemannian manifold. For a measurable set $S \\subset M$, we denote by \n\\[\n \\vv_g(S) \\quad \\text{and} \\quad \\pp_g(S)\n\\]\nthe Riemannian volume and perimeter, respectively. The $\\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\\hh^\\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:\n\\begin{equation}\\label{eq:est}\n \\hh_{t^2 g}^{\\alpha}(S) = t^{\\alpha}\\,\\hh_g^{\\alpha}(S), \n \\qquad \\vv_{t^2 g}(S) = t^m\\,\\vv_g(S), \n \\qquad \\pp_{t^2 g}(S) = t^{m-1}\\,\\pp_g(S),\n\\end{equation}\nwhere $m = \\dim M$. \nThroughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \\emph{$t$-homothety} of $(M,g)$.\n\nLet $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \\times g_2$. \nWe define two anisotropic product scalings:\n\\begin{itemize}\n \\item The \\emph{right anisotropic $t$-homothety} is\n \\[\n (M_1 \\times M_2, g_1 \\times t^2 g_2),\n \\]\n denoted by $(M_1 \\times M_2, g_+^t)$ or simply $M_1 \\times tM_2$.\n \\item The \\emph{left anisotropic $t$-homothety} is\n \\[\n (M_1 \\times M_2, t^2 g_1 \\times g_2),\n \\]\n denoted by $(M_1 \\times M_2, g_-^t)$ or simply $tM_1 \\times M_2$.\n\\end{itemize}\nThese anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.",
    "full_context": "In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.\nIn products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \\cite{peri}.\nMorgan~\\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.\nIn the spherical case, Pedrosa~\\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\\mathbb{S}^n\\times\\mathbb{R}$.\nIn Riemannian cylinders $M\\times\\mathbb{R}$, Duzaar and Steffen~\\cite{du-st} proved that large-volume minimizers are slabs.\nThis was later extended to $M\\times\\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\\cite{manstr}.\n\nIn this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \\(N\\) are of class \\(C^{2,\\alpha}\\), a condition that holds, for example, when \\(\\dim N\\leqslant 8\\) or for small volumes (and their complements).\n\nWe begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.\nIn the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\\mathbb{S}^m\\times N$ (Proposition~\\ref{prp:reduce})\n\\,(see also the related symmetrization results of Morgan--Howe--Harman~\\cite{mosym}).\nWe then analyze \\emph{cylinderoids} $M_1\\times S$ and symmetrized competitors in $\\mathbb{S}^m\\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\\ref{prp:pr} and~\\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,\nall relevant competitors have $C^{2,\\alpha}$ boundaries. \nThis allows us to restrict the analysis to $C^{2,\\alpha}$ graphical \nperturbations of $\\Sigma_t$.\nNext, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\\Sigma\\subset N$ is a stable constant mean curvature hypersurface, then $tM\\times\\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\\ref{Thm:sts}).\nFinally, assuming the $C^{2,\\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\\cite{rqi} to the present product setting around $\\Sg_t:=tM\\times\\Sigma$. \nThis analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\\Sg_t$ to be cylinderoids (Theorem~\\ref{thm:main}).\n\nWe begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.\n\nLet $(M,g)$ be a Riemannian manifold. For a measurable set $S \\subset M$, we denote by \n\\[\n \\vv_g(S) \\quad \\text{and} \\quad \\pp_g(S)\n\\]\nthe Riemannian volume and perimeter, respectively. The $\\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\\hh^\\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:\n\\begin{equation}\\label{eq:est}\n \\hh_{t^2 g}^{\\alpha}(S) = t^{\\alpha}\\,\\hh_g^{\\alpha}(S), \n \\qquad \\vv_{t^2 g}(S) = t^m\\,\\vv_g(S), \n \\qquad \\pp_{t^2 g}(S) = t^{m-1}\\,\\pp_g(S),\n\\end{equation}\nwhere $m = \\dim M$. \nThroughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \\emph{$t$-homothety} of $(M,g)$.\n\nLet $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \\times g_2$. \nWe define two anisotropic product scalings:\n\\begin{itemize}\n \\item The \\emph{right anisotropic $t$-homothety} is\n \\[\n (M_1 \\times M_2, g_1 \\times t^2 g_2),\n \\]\n denoted by $(M_1 \\times M_2, g_+^t)$ or simply $M_1 \\times tM_2$.\n \\item The \\emph{left anisotropic $t$-homothety} is\n \\[\n (M_1 \\times M_2, t^2 g_1 \\times g_2),\n \\]\n denoted by $(M_1 \\times M_2, g_-^t)$ or simply $tM_1 \\times M_2$.\n\\end{itemize}\nThese anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.\n\n\\begin{proof}\nPart (i) follows directly from computing the Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$. \nFor (ii), at a regular point of $\\Sg$ the $g$-unit exterior normal vector $\\nu$ can be decomposed as $\\nu = a\\nu_1 + b\\nu_2$ so that $a^2+b^2=1$ and $\\nu_1$, $\\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$ restricted to $\\Sg$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \\eqref{eq:anest2}. \nAssertion (iii) follows from the fact that $\\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.\n\n\\end{proof}\nThe following local isoperimetric inequalities are classical, see Duzaar and Steffen \\cite{du-st}.\n\\begin{lemma}\n\\label{lem:ine}\nLet $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\\hh^m(M)$, there exist positive constants $\\ga_1(M,v_0),\\ga_2(M,v_0)$ such that for any set $S\\subset M$ with $(m-1)$-rectifiable boundary and $0<\\hh^m(S)<v_0$ the following isoperimetric inequalities hold\n\\begin{equation}\n\\label{eq:Ine1}\n\\hh^{m-1}(\\ptl S)\\ge \\ga_1(M,v_0)\\,\\hh^m(S)\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:Ine2}\n\\hh^{m-1}(\\ptl S)\\ge \\ga_2 (M,v_0)\\,\\hh^m(S)^{(m-1)/m},\n\\end{equation}\n\\end{lemma}\n\nWe are now ready to prove the main results of the section.\n\\begin{proposition}\n\\label{prp:pr}\n\\mbox{}\nAssume $0<\\be<1$, $t_i\\uparrow\\infty\\,$ and $\\{E_i\\}_{i\\in\\nn}$ be a sequence of compact symmetrized isoperimetric sets in $ \\esf\\times t_i N$ of volume $\\vv_{g_+^{t_i}}(E_i)=\\be \\vv_{g_+^{t_i}}(\\esf\\times N)$. Let $t_0>0$ and $s_i=t_0\\,t_i^{-1}$. \n\\begin{enum}\n\\item\nThen, possibly passing to a non-relabeling subsequence, $\\{E_i\\}_{i\\in\\nn}$ converges both in $L^1$ and Hausdorff topology of $\\esf\\times t_0N$, to a finite perimeter set $E$ which is a cylinderoid, i.e $E=\\esf\\times \\pi_2(E)$. \n\\item Moreover $\\pi_2(E)$ is an isoperimetric set in $t_0N$, of volume fraction $\\be$.\n\\item Furthermore\n\\begin{equation}\n\\label{eq:ane00}\n\\vv_{g_+^{t_0}}(E_i)=\\vv_{g_+^{t_0}}(\\esf\\times \\pi_2(E)) \\,\\,\\text{and}\\,\\,\\pp_{g_+^{t_0}}(E_i)\\le\\pp_{g_+^{t_0}}(\\esf\\times \\pi_2(E))\n\\end{equation}\n\\end{enum}\n\\end{proposition}\n\\begin{proof}\nOwing to \\eqref{eq:anest1} we get\n\\begin{equation}\n\\label{eq:thm0}\n\\vv_{g_+^{t_0}}(E_i)=\\be\\vv_{g_+^{t_0}}(\\esf\\times N) :=v_0\\quad\\text{for every}\\,\\,i\\in\\nn. \n\\end{equation}\nSince $N$ is compact, there exists an isoperimetric region $S\\subset N$ of volume $v_0\\,\\vv(\\esf)^{-1}$. As $E_i$ are isoperimetric by assumption, we have\n\\begin{equation}\n\\label{eq:cona}\n\\vv_{g_+^{t_i}}(\\esf\\times S)= \\vv_{g_+^{t_i}}(E_i) \\,\\,\\text{and}\\,\\,\\pp_{g_+^{t_i}}(E_i)\\le\\pp_{g_+^{t_i}}(\\esf\\times S)\n\\end{equation}\nThanks to (ii) and (iii) of Lemma \\ref{lem:anest}, we deduce \n\\begin{equation}\n\\label{eq:conb}\n\\frac{\\pp_{g_+^{t_0}}(E_i)}{\\pp_{g_+^{t_0}}(\\esf\\times S)}=\\frac{\\pp_{g_+^{s_it_i}}(E_i)}{\\pp_{g_+^{s_it_i}}(\\esf\\times S)}\\le\\frac{s_i^{n-1}\\pp_{g_+^{t_i}}(E_i)}{s_i^{n-1}\\pp_{g_+^{t_i}}(\\esf\\times S)}\\le 1\n\\end{equation} \nConsequently, $\\{\\pp_{g_+^{t_0}}(E_i)\\}_{i\\in\\nn}$ is bounded and so, possibly passing to a subsequence, $E_i\\to E$ in the $L^1(\\esf\\times t_0 N)$ topology. Hence, by the semi-continuity of the perimeter, we get\n\\begin{equation}\n\\label{eq:con1}\n\\pp_{g_+^{t_0}}(E)\\le\\liminf\\pp_{g_+^{t_0}}(E_i)\\quad\\text{and}\\,\\,\\vv_{g_+^{t_0}}(E)=\\be\\vv_{g_+^{t_0}}(\\esf\\times N) :=v_0\n\\end{equation}\nOwing to Lemma \\ref{lem:h=l}, possibly passing to a non-relabeling subsequence, we get $E_i\\to E$ in the Hausdorff topology of $\\esf\\times t_0 N$. We argue by contradiction, assume that $E\\not=\\esf\\times \\pi_2(E)$. Then by Lemma \\ref{lem:ww} there are $\\de>0$ and $w_0\\in (0,\\vv(\\esf))$ so that \n\\begin{equation}\n\\label{eq:con2}\n\\vv_{g_+^{t_0}}(E_i{\\{w_0\\}})>\\de\\quad\\text{for sufficiently large}\\,\\,i\\in\\nn.\n\\end{equation}\nHence, by \\eqref{eq:anest1}, we get\n\\begin{equation}\n\\label{eq:con3}\n\\frac{\\vv_{g_+^{t_i}}(E_i{\\{w_0\\}})}{\\vv_{g_+^{t_i}}(E_i)}=\\frac{\\vv_{g_+^{t_0}}(E_i{\\{w_0\\}})}{\\vv_{{g_+^{t_0}}}(E_i)}= \\frac{\\vv_{{g_+^{t_0}}}(E_i{\\{w_0\\}})}{\\be\\vv_{{g_+^{t_0}}}(\\esf\\times N)}\\ge \\frac{\\de}{\\be\\vv_{{g_+^{t_0}}}(\\esf\\times N)} :=c_1>0,\n\\end{equation}\nfor sufficiently large $i\\in\\nn$. Observe that since $ E_i\\{w_0\\}$ is symmetrized there holds\n\\begin{equation}\n\\label{eq:3a}\n\\pi_2(\\ptl E_i\\cap E_i\\{w_0\\})=\\pi_2( E_i\\{w_0\\}) := \\Xi_i.\n\\end{equation}\nNow owing to \\eqref{eq:con3} and Fubini's Theorem we obtain\n\\begin{equation}\n\\label{eq:con4}\n\\vv_{g_+^{t_i}}(E_i)\\le c_1^{-1}\\int_{\\Xi_i}\\hh_{g_1}^m( E_i\\cap \\pi_2^{-1}(y))\\,d\\hh_{{t_i}^2g_2}^n\n\\end{equation}\n\n\\begin{proof}\nLet $\\bar{g}=g_1\\times\\bar{g}_2$, where $\\bar{g}_2=g_2|_{T\\Sg\\times T\\Sg}$. \nNote that the outer unit normal $\\nu$ on $tM\\times \\Sg$ is tangent to the second factor. Consequently,\n\\begin{equation}\n\\label{eq:curvs}\n\\ric^{g_-^{t}}(\\nu,\\nu)=\\ric^{\\,g_2}(\\nu,\\nu), \n\\qquad \n\\sg^{\\bar{g}_-^{t}}=\\sg^{\\,\\bar{g}_2},\n\\end{equation}\nwhere $\\ric$ denotes Ricci tensor and $\\sg$ second fundamental form. Hence\n\\begin{equation}\n\\label{eq:sts00}\nq(x,\\vsg):=\\ric^{g_-^{t}}(\\nu,\\nu)+|\\sg^{\\bar{g}_-^{t}}|^2\n= \\ric^{\\,g_2}(\\nu,\\nu)+|\\sg^{\\,\\bar{g}_2}|^2 :=q(\\vsg).\n\\end{equation}\nSet\n\\begin{equation}\n\\label{eq:sts0}\nt_0^2=\\frac{\\la_1(g_1)}{\\|q\\|_\\infty}, \\qquad t<t_0.\n\\end{equation}\nAs a consequence of the product spectral decomposition, it suffices to show that\n\\begin{equation}\n\\label{eq:sts1}\n-\\int_{tM\\times\\Sg} f\\big(\\Delta f+qf\\big)\\geq0 \\qquad \\text{for all smooth } f \\text{ with } \\int_{tM\\times\\Sg} f=0,\n\\end{equation}\nwhere $f(x,\\vsg)=u(x)v(\\vsg)$, $x\\in M$, $\\vsg\\in \\Sg$ and $\\Delta=\\Delta^{\\bar{g}_-^{t}}$. Owing to \\eqref{eq:delprt}\n\\begin{equation}\n\\label{eq:sts2}\n\\begin{split}\n-\\int_{tM\\times\\Sg}f(\\Delta f+qf)\n&=-\\int_{tM\\times\\Sg} v^2 u\\Delta^{t^2g_1} u \n -\\int_{tM\\times\\Sg} u^2 v \\Delta^{\\bar{g}_2} v\n -\\int_{tM\\times\\Sg} q u^2 v^2.\n\\end{split}\n\\end{equation}\nApplying Stokes’ theorem:\n\\begin{equation}\n\\label{eq:sts3}\n-\\int_{tM\\times\\Sg} v^2 u \\Delta^{t^2g_1}u \n= \\int_{tM} |\\nabla^{t^2g_1}u|^2_{t^2g_1}\\,\\int_{\\Sg} v^2,\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:sts4}\n-\\int_{tM\\times\\Sg} u^2 v \\Delta^{\\bar{g}_2}v \n= \\int_{tM} u^2 \\,\\int_{\\Sg} |\\nabla^{\\bar{g}_2}v|^2_{\\bar{g}_2}.\n\\end{equation}\n\n\\section{Appendix}\nIn this appendix we recall the fibrewise rearrangement used throughout the paper.\nWe consider here a symmetrization first introduced by Ros~\\cite{ros} and later generalized by Morgan~\\cite{morpol} for warped product manifolds with density.\nLet $M,N$ be compact Riemannian manifolds. Let $\\esf$ be a round sphere with the same volume and dimension as $M$ and with isoperimetric profile not exceeding that of $M$. Fix a point $p_0\\in \\esf$ and consider the foliation of the geodesic balls centered at $p_0$.\nLet $E\\subset M\\times N$ be a set of finite perimeter. Replace each non-empty horizontal slice with the unique geodesic ball in the foliation centred at $p_0$ having the same volume. We denote by $\\sym E \\subset \\esf\\times N$ this set, which we call the symmetrized set of~$E$. Then, there holds\n\\begin{equation}\n\\label{eq:sym1}\n\\vv_{\\esf\\times N}(\\sym E)=\\vv_{M\\times N} (E)\\quad \\text{and}\\ \\quad \\pp_{\\esf\\times N}(\\sym E)\\le \\pp_{M\\times N}(E).\n\\end{equation}\nConsequently,\n\\begin{equation}\n\\label{eq:sym2}\nI_{M\\times N}\\ge I_{\\esf\\times N}\n\\end{equation}",
    "post_theorem_intro_text_len": 3674,
    "post_theorem_intro_text": "\\begin{proof}\nPart (i) follows directly from computing the Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$.  \nFor (ii), at a regular point of $\\Sigma$ the $g$-unit exterior normal vector $\\nu$ can be decomposed as $\\nu = a\\nu_1 + b\\nu_2$ so that $a^2+b^2=1$ and $\\nu_1$, $\\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$ restricted to $\\Sigma$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \\eqref{eq:anest2}.  \nAssertion (iii) follows from the fact that $\\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.  \n\n\\end{proof}\nThe following local isoperimetric inequalities are classical, see Duzaar and Steffen \\cite{du-st}.\n\\begin{lemma}\n\\label{lem:ine}\nLet $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\\hh^m(M)$, there exist positive constants  $\\ga_1(M,v_0),\\ga_2(M,v_0)$ such that for any set $S\\subset M$ with $(m-1)$-rectifiable boundary and $0<\\hh^m(S)<v_0$ the following isoperimetric inequalities hold\n\\begin{equation}\n\\label{eq:Ine1}\n\\hh^{m-1}(\\partial S)\\geqslant \\ga_1(M,v_0)\\,\\hh^m(S)\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:Ine2}\n\\hh^{m-1}(\\partial S)\\geqslant \\ga_2 (M,v_0)\\,\\hh^m(S)^{(m-1)/m},\n\\end{equation}\n\\end{lemma}\n\nThe \\emph{isoperimetric profile} of a Riemannian manifold $(M,g)$ is defined by\n\\[\n  I_g(v) = \\inf \\big\\{ \\pp_g(E) : \\vv_g(E) = v \\big\\}.\n\\]\nA set $E \\subset M$ is called an \\emph{isoperimetric region} if $\\pp_g(E) = I_g(\\vv_g(E))$. Existence of isoperimetric regions in compact manifolds is guaranteed by standard compactness arguments in geometric measure theory, see \\cite{Maggi}. Moreover, classical regularity theory implies that the boundary of an isoperimetric\nregion is a smooth hypersurface except for a singular set of Hausdorff codimension at least $8$,  as proved by Morgan \\cite{moreg} and by Gonzales–Massari–Tamanini \\cite{gomato}. In Chapter 13.2 of \\cite{Morgan2016}, Morgan, following methods of Smale \\cite{Smale1999},  describes constructions of irregular isoperimetric hypersurfaces in manifolds of dimension greater than seven. Furthermore, Morgan–Johnson \\cite{morjohn} and Nardulli \\cite{nard} show that, for small volumes (and their complements), isoperimetric hypersurfaces are smooth.\n\nIsoperimetric regions are considered up to sets of measure zero. \nIn particular, every isoperimetric set is equivalent to its closure, and its perimeter can be computed on the topological boundary:\n\\begin{equation}\\label{eq:ww4}\n\\mathcal{P}(E) = \\mathcal{P}(\\overline{E}) = \\hh^{m-1}(\\partial E).\n\\end{equation}\n\nLet $\\Sigma\\subset M$ be an isoperimetric hypersurface. Standard first and second variation computations\n(see for instance Barbosa--do Carmo--Eschenburg~\\cite{bd})\nimply that the regular part $\\reg(\\Sigma)$ has constant mean curvature and satisfies a stability inequality.\nSpecifically, for every smooth compactly supported function $u:\\reg(\\Sigma)\\to\\mathbb{R}$ with zero mean one has\n\\begin{equation}\\label{eq:stab}\n- \\int_\\Sigma u \\,\\big(\\Delta u + \\ric(\\nu,\\nu)u + |\\sigma|^2 u\\big) \\,\\geq 0,\n\\end{equation}\n\nwhere $\\nu$ is the exterior unit normal to $\\Sigma$, $\\Delta$ is the Laplace operator on $\\Sigma$, $|\\sigma|^2$ is the squared norm of the second fundamental form, and $\\ric$ denotes the Ricci tensor of the ambient manifold $M$. \nWe denote by $J_\\Sigma$ the Jacobi operator, which acts on smooth functions $u$ as follows:\n\\[\n  J_\\Sigma u = \\Delta u + \\ric(\\nu,\\nu)\\,u + |\\sigma|^2 u.\n\\]\nIf the inequality in \\eqref{eq:stab} is strict for all nontrivial $u$, we say that $\\sigma$ is \\emph{strictly stable}.",
    "sketch": "Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$,” giving \\eqref{eq:anest1}.\n\nFor (ii), “at a regular point of $\\Sigma$ the $g$-unit exterior normal vector $\\nu$ can be decomposed as $\\nu=a\\nu_1+b\\nu_2$ so that $a^2+b^2=1$ and $\\nu_1$, $\\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which “yields inequality \\eqref{eq:anest2}” for $t\\le 1$.\n\nFor (iii), “$\\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in \\eqref{eq:anest2}.",
    "expanded_sketch": "Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \\times M_2,g)$ to $(M_1 \\times M_2,g_+^t)$,” giving \\eqref{eq:anest1}.\n\nFor (ii), “at a regular point of $\\Sigma$ the $g$-unit exterior normal vector $\\nu$ can be decomposed as $\\nu=a\\nu_1+b\\nu_2$ so that $a^2+b^2=1$ and $\\nu_1$, $\\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which for $t\\le 1$ yields\n\\begin{equation}\\label{eq:anest2}\n  \\hh_{g_+^t}^{m_1+m_2-1}(\\Sg) \n  \\leq t^{m_2-1}\\,\\hh_g^{m_1+m_2-1}(\\Sg).\n\\end{equation}\n\nFor (iii), “$\\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in the equation above.",
    "expanded_theorem": "\\label{lem:anest}\nLet $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\\times g_2$. Then the following hold for all measurable $S\\subset M_1\\times M_2$ and $(m_1+m_2-1)$–rectifiable $\\Sigma$:\n\\begin{enumerate}\n\\item[(i)] For every set $S \\subset M_1 \\times M_2$,\n\\begin{equation}\\label{eq:anest1}\n  \\vv_{g_+^t}(S) = t^{m_2} \\,\\vv_g(S).\n\\end{equation}\n\\item[(ii)] If $\\Sigma \\subset M_1 \\times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \\leq 1$,\n\\begin{equation}\\label{eq:anest2}\n  \\hh_{g_+^t}^{m_1+m_2-1}(\\Sg) \n  \\leq t^{m_2-1}\\,\\hh_g^{m_1+m_2-1}(\\Sg).\n\\end{equation}\n\\item[(iii)] Equality holds in the equation above if and only if the $g$-normal of $\\Sigma$ is tangent to $M_2$, up to a $\\hh_g^{m_1+m_2-1}$-null set.\n\\end{enumerate}",
    "theorem_type": [
      "Inequality or Bound",
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Let \\((M_i,g_i)\\) for \\(i=1,2\\) be Riemannian manifolds of dimensions \\(m_i\\), let \\(g=g_1\\times g_2\\) on \\(M_1\\times M_2\\), and for \\(t>0\\) define the right anisotropic \\(t\\)-homothety metric by \\(g_+^t=g_1\\times t^2 g_2\\). For a measurable set \\(S\\subset M_1\\times M_2\\), write \\(\\mathrm{vol}_{g}(S)\\) for its Riemannian volume, and for an \\((m_1+m_2-1)\\)-rectifiable set \\(\\Sigma\\subset M_1\\times M_2\\), write \\(\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma)\\) for its \\((m_1+m_2-1)\\)-dimensional Hausdorff measure with respect to \\(g\\). Which quantitative statement holds for volume and hypersurface measure under \\(g_+^t\\)?",
      "correct_choice": {
        "label": "A",
        "text": "For every measurable set \\(S\\subset M_1\\times M_2\\), one has\n\\[\n\\mathrm{vol}_{g_+^t}(S)=t^{m_2}\\,\\mathrm{vol}_{g}(S).\n\\]\nMoreover, if \\(\\Sigma\\subset M_1\\times M_2\\) is \\((m_1+m_2-1)\\)-rectifiable and \\(t\\le 1\\), then\n\\[\n\\mathcal H^{m_1+m_2-1}_{g_+^t}(\\Sigma)\n\\le t^{m_2-1}\\,\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma).\n\\]\nEquality in this last inequality holds if and only if the \\(g\\)-normal to \\(\\Sigma\\) is tangent to the \\(M_2\\)-factor, up to a \\(\\mathcal H^{m_1+m_2-1}_{g}\\)-null set."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every measurable set \\(S\\subset M_1\\times M_2\\), one has\n\\[\n\\mathrm{vol}_{g_+^t}(S)=t^{m_2}\\,\\mathrm{vol}_{g}(S).\n\\]\nMoreover, if \\(\\Sigma\\subset M_1\\times M_2\\) is \\((m_1+m_2-1)\\)-rectifiable, then for every \\(t>0\\),\n\\[\n\\mathcal H^{m_1+m_2-1}_{g_+^t}(\\Sigma)\n\\le t^{m_2-1}\\,\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma).\n\\]\nEquality in this last inequality holds if and only if the \\(g\\)-normal to \\(\\Sigma\\) is tangent to the \\(M_2\\)-factor, up to a \\(\\mathcal H^{m_1+m_2-1}_{g}\\)-null set."
        },
        {
          "label": "C",
          "text": "For every measurable set \\(S\\subset M_1\\times M_2\\), one has\n\\[\n\\mathrm{vol}_{g_+^t}(S)=t^{m_2}\\,\\mathrm{vol}_{g}(S).\n\\]\nMoreover, if \\(\\Sigma\\subset M_1\\times M_2\\) is \\((m_1+m_2-1)\\)-rectifiable and \\(t\\le 1\\), then\n\\[\n\\mathcal H^{m_1+m_2-1}_{g_+^t}(\\Sigma)\n\\le t^{m_2-1}\\,\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma).\n\\]"
        },
        {
          "label": "D",
          "text": "For every measurable set \\(S\\subset M_1\\times M_2\\), one has\n\\[\n\\mathrm{vol}_{g_+^t}(S)=t^{m_1}\\,\\mathrm{vol}_{g}(S).\n\\]\nMoreover, if \\(\\Sigma\\subset M_1\\times M_2\\) is \\((m_1+m_2-1)\\)-rectifiable and \\(t\\le 1\\), then\n\\[\n\\mathcal H^{m_1+m_2-1}_{g_+^t}(\\Sigma)\n\\le t^{m_1-1}\\,\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma).\n\\]\nEquality in this last inequality holds if and only if the \\(g\\)-normal to \\(\\Sigma\\) is tangent to the \\(M_1\\)-factor, up to a \\(\\mathcal H^{m_1+m_2-1}_{g}\\)-null set."
        },
        {
          "label": "E",
          "text": "For every measurable set \\(S\\subset M_1\\times M_2\\), one has\n\\[\n\\mathrm{vol}_{g_+^t}(S)=t^{m_2}\\,\\mathrm{vol}_{g}(S).\n\\]\nMoreover, if \\(\\Sigma\\subset M_1\\times M_2\\) is \\((m_1+m_2-1)\\)-rectifiable and \\(t\\le 1\\), then\n\\[\n\\mathcal H^{m_1+m_2-1}_{g_+^t}(\\Sigma)\n= t^{m_2-1}\\,\\mathcal H^{m_1+m_2-1}_{g}(\\Sigma).\n\\]\nIn particular, the hypersurface measure always scales exactly by \\(t^{m_2-1}\\), independently of the orientation of the \\(g\\)-normal."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "range_restriction_t_le_1",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_equality_characterization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "which_factor_is_scaled_and_normal_component",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "jacobian_factor_(t^2a^2+b^2)^{1/2}",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state or strongly hint at the correct option; it asks for an equivalent characterization of the equality case without revealing which factor or which quantifier structure is right."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall/restatement question: it asks which statement is equivalent to the equality case, and the correct option is the theorem-style characterization itself."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish 'if and only if' from one-way implication, an a.e. null-set condition from pointwise regularity, and the correct factor direction. But it mostly tests precise recall of the equality characterization rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: confusing the factor, weakening to a one-way implication, overstrengthening to every regular point, and replacing the exact condition by a misleading constant-angle condition."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed distractors and no answer leakage, but the item is largely theorem-recall and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.09564v2",
    "paper_link": "http://arxiv.org/abs/2512.09564v2",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thmintro",
      "content": "[Theorem~\\ref{thm:mainthm} \\& Corollary~\\ref{cor:spec}]\n\\label{thm:1}\nLet $G$ be simply connected. The coordinate ring of the Vinberg monoid $\\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible. \n\nThe map $\\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.",
      "start_pos": 11929,
      "end_pos": 12384,
      "label": "thm:1"
    },
    "ref_dict": {
      "thm:mainthm": "\\begin{theorem}\n\\label{thm:mainthm}\nLet $\\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\\Sigma=\\wt{J}_\\fro\\backslash I'$. The isomorphism $\\Psi_\\bfi$ restricts to isomorphisms\n\\[\n\\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[G\\x ^Z T],\\qquad \\overline{\\CU}(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[\\Env G].\n\\]\nIn other words, we have the following commutative diagram\n\\begin{equation}\\label{eq:cr}\n    \\begin{tikzcd}\n        & \\CU(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] & \\BC[G^{w_0,w_0}\\x ^Z T] \\\\\n        & \\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] \\arrow[u,hook] & \\BC[G\\x^Z T] \\arrow[u,hook] \\\\\n        & \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[u,hook] \\arrow[r,\"\\sim\"] & \\BC[\\Env G] \\arrow[u,hook]\n    \\end{tikzcd}\n\\end{equation}\n\\end{theorem}",
      "thm:2": "\\begin{thmintro}[Theorem~\\ref{thm:reductiveMonoidCluster}]\n\\label{thm:2}\nLet $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.\n\\end{thmintro}",
      "thm:3": "\\begin{thmintro}\n[Theorem~\\ref{thm:affineAbelianCluster}]\n\\label{thm:3}\nLet $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\\mathbb{C}^k$ (See Section \\ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.\n\\end{thmintro}",
      "thm:reductiveMonoidCluster": "\\begin{theorem}\n\\label{thm:reductiveMonoidCluster}\nLet $H$ be a connected reductive group with simply connected derived group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra $\\overline{\\mathcal{U}}^\\Sigma$, whose further partially compactification $\\overline{\\mathcal{U}}$ along all frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.\n\\end{theorem}",
      "cor:spec": "\\begin{cor}\\label{cor:spec}\nThe diagram\n\\begin{equation}\n    \\begin{tikzcd}\n    \\Env G \\arrow[dr,\"\\pi\"'] \\arrow[rr,\"\\sim\"] && \\Spec \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[dl,\"\\pi^{I'}\"] \\\\ & \\BC^I\n    \\end{tikzcd}\n\\end{equation}\ncommutes.\n\\end{cor}",
      "thm:1": "\\begin{thmintro}[Theorem~\\ref{thm:mainthm} \\& Corollary~\\ref{cor:spec}]\n\\label{thm:1}\nLet $G$ be simply connected. The coordinate ring of the Vinberg monoid $\\Env G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible. \n\nThe map $\\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.\n\\end{thmintro}",
      "thm:affineAbelianCluster": "\\begin{theorem}\\label{thm:affineAbelianCluster}\nLet $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ isomorphic to an affine space $\\mathbb{C}^n$. Then the coordinate ring $\\BC[M]$ is a partially compactified upper cluster algebra with all frozen variables not invertible.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2813,
    "pre_theorem_intro_text": "\\label{sec:intro}\nIn this paper, all varieties are defined over $\\BC$.\n\n\\subsection{The Vinberg monoid}\n\\label{sec:intro:monoid}\nAn \\emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \\emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \\cite{Put84} and Renner \\cite{Ren85}, and it has attracted many attentions since then. \n\nIn the work of \\cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \\emph{Vinberg monoid}. The Vinberg monoid $\\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\\textnormal{Env } G$ by a base change. \n\nMoreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map \n\\[\n\\pi: \\textnormal{Env } G\\longrightarrow \\mathbb{C}^r\n\\]\nonto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\\pi^{-1}(\\mathbf{0})$ is the \\emph{asymptotic semigroup} considered by Vinberg \\cite{Vin95a}.\n\nThe Vinberg monoid together with the map $\\pi$ have found significant applications in recent developments of the geometric representation theory, see \\citelist{\\cite{BNS16}\\cite{FKM20}} for example.\n\n\\subsection{Cluster algebras}\n\\label{sec:intro:cluster}\n\nThe theory of cluster algebras was developed by Fomin and Zelevinsky \\cite{FZ02} to study total positivity and dual canonical bases in semisimple groups. The coordinate rings of many varieties related to algebraic groups have shown to be (upper) cluster algebras (See \\citelist{\\cite{BFZ05}\\cite{Wil13}\\cite{Sco06}\\cite{CGG+25}\\cite{GLSS}} for example.). \n\nIn contrast, cluster algebra structures on algebraic monoids have received little attention. It seems that the only nontrivial example is the cluster algebra structure on the space of $n\\times n$ matrices from \\cite{FWZ20}*{Theorem~6.6.1}.  \n\nThe existence of a cluster structure on a variety has a wide range of consequences. For example, it provides a total positivity structure, Poisson structure \\cite{GSV10}, and a cluster theoretic canonical basis on its coordinate ring \\cite{GHKK18}.\n\nIn this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \\emph{partially compactified (upper) cluster algebras} from \\cite{GHKK18} to highlight this choice of convention. \n\n\\subsection{Main results}\n\\label{sec:intro:main}\n\nThe main result of this paper is the follows.",
    "context": "\\subsection{The Vinberg monoid}\n\\label{sec:intro:monoid}\nAn \\emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \\emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \\cite{Put84} and Renner \\cite{Ren85}, and it has attracted many attentions since then.\n\nIn the work of \\cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \\emph{Vinberg monoid}. The Vinberg monoid $\\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\\textnormal{Env } G$ by a base change.\n\nMoreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map \n\\[\n\\pi: \\textnormal{Env } G\\longrightarrow \\mathbb{C}^r\n\\]\nonto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\\pi^{-1}(\\mathbf{0})$ is the \\emph{asymptotic semigroup} considered by Vinberg \\cite{Vin95a}.\n\nIn this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \\emph{partially compactified (upper) cluster algebras} from \\cite{GHKK18} to highlight this choice of convention.\n\n\\subsection{Main results}\n\\label{sec:intro:main}\n\nThe main result of this paper is the follows.\n\n\\begin{cor}\\label{cor:spec}\nThe diagram\n\\begin{equation}\n    \\begin{tikzcd}\n    \\Env G \\arrow[dr,\"\\pi\"'] \\arrow[rr,\"\\sim\"] && \\Spec \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[dl,\"\\pi^{I'}\"] \\\\ & \\BC^I\n    \\end{tikzcd}\n\\end{equation}\ncommutes.\n\\end{cor}\n\n\\begin{theorem}\n\\label{thm:mainthm}\nLet $\\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\\Sigma=\\wt{J}_\\fro\\backslash I'$. The isomorphism $\\Psi_\\bfi$ restricts to isomorphisms\n\\[\n\\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[G\\x ^Z T],\\qquad \\overline{\\CU}(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[\\Env G].\n\\]\nIn other words, we have the following commutative diagram\n\\begin{equation}\\label{eq:cr}\n    \\begin{tikzcd}\n        & \\CU(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] & \\BC[G^{w_0,w_0}\\x ^Z T] \\\\\n        & \\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] \\arrow[u,hook] & \\BC[G\\x^Z T] \\arrow[u,hook] \\\\\n        & \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[u,hook] \\arrow[r,\"\\sim\"] & \\BC[\\Env G] \\arrow[u,hook]\n    \\end{tikzcd}\n\\end{equation}\n\\end{theorem}",
    "full_context": "\\subsection{The Vinberg monoid}\n\\label{sec:intro:monoid}\nAn \\emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \\emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \\cite{Put84} and Renner \\cite{Ren85}, and it has attracted many attentions since then.\n\nIn the work of \\cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \\emph{Vinberg monoid}. The Vinberg monoid $\\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\\textnormal{Env } G$ by a base change.\n\nMoreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map \n\\[\n\\pi: \\textnormal{Env } G\\longrightarrow \\mathbb{C}^r\n\\]\nonto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\\pi^{-1}(\\mathbf{0})$ is the \\emph{asymptotic semigroup} considered by Vinberg \\cite{Vin95a}.\n\nIn this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \\emph{partially compactified (upper) cluster algebras} from \\cite{GHKK18} to highlight this choice of convention.\n\n\\subsection{Main results}\n\\label{sec:intro:main}\n\nThe main result of this paper is the follows.\n\n\\begin{cor}\\label{cor:spec}\nThe diagram\n\\begin{equation}\n    \\begin{tikzcd}\n    \\Env G \\arrow[dr,\"\\pi\"'] \\arrow[rr,\"\\sim\"] && \\Spec \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[dl,\"\\pi^{I'}\"] \\\\ & \\BC^I\n    \\end{tikzcd}\n\\end{equation}\ncommutes.\n\\end{cor}\n\n\\begin{theorem}\n\\label{thm:mainthm}\nLet $\\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\\Sigma=\\wt{J}_\\fro\\backslash I'$. The isomorphism $\\Psi_\\bfi$ restricts to isomorphisms\n\\[\n\\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[G\\x ^Z T],\\qquad \\overline{\\CU}(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[\\Env G].\n\\]\nIn other words, we have the following commutative diagram\n\\begin{equation}\\label{eq:cr}\n    \\begin{tikzcd}\n        & \\CU(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] & \\BC[G^{w_0,w_0}\\x ^Z T] \\\\\n        & \\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] \\arrow[u,hook] & \\BC[G\\x^Z T] \\arrow[u,hook] \\\\\n        & \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[u,hook] \\arrow[r,\"\\sim\"] & \\BC[\\Env G] \\arrow[u,hook]\n    \\end{tikzcd}\n\\end{equation}\n\\end{theorem}\n\nThe main result of this paper is the follows.\n\nCombining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \\ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids.\n\n\\begin{thmintro}\n[Theorem~\\ref{thm:affineAbelianCluster}]\n\\label{thm:3}\nLet $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\\mathbb{C}^k$ (See Section \\ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.\n\\end{thmintro}\n\nLet us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\\to \\BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\\times n$ matrices. In this case, Theorem \\ref{thm:3} recovers the well-known cluster structures, presented in \\cite{FWZ20}*{Theorem~6.6.1}.\n\n\\begin{thmintro}[Theorem~\\ref{thm:reductiveMonoidCluster}]\n\\label{thm:2}\nLet $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.\n\\end{thmintro}\n\n\\begin{theorem}\n\\label{thm:reductiveMonoidCluster}\nLet $H$ be a connected reductive group with simply connected derived group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra $\\overline{\\mathcal{U}}^\\Sigma$, whose further partially compactification $\\overline{\\mathcal{U}}$ along all frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:affineAbelianCluster}\nLet $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ isomorphic to an affine space $\\mathbb{C}^n$. Then the coordinate ring $\\BC[M]$ is a partially compactified upper cluster algebra with all frozen variables not invertible.\n\\end{theorem}\n\n\\begin{example}\nLet $M=M_n$ be the monoid of $n\\times n$ matrices, so $G(M)=GL_n$ and $G_M=SL_n$. In this case, the subalgebra $\\BC[M]^{G_M\\times G_M}$ of invariant functions is isomorphic to the polynomial algebra in one variable, generated by the determinant. Hence the abelianization $A(M)$ is isomorphic to the affine line. Theorem~\\ref{thm:affineAbelianCluster} gives a partially compactified upper cluster algebra structure on $\\BC[M_n]$, which coincides with the cluster structure in \\cite{FWZ20}*{Theorem 6.6.1}.\n\\end{example}\n\n\\begin{cor}\\label{cor:spec}\nThe diagram\n\\begin{equation}\n    \\begin{tikzcd}\n    \\Env G \\arrow[dr,\"\\pi\"'] \\arrow[rr,\"\\sim\"] && \\Spec \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[dl,\"\\pi^{I'}\"] \\\\ & \\BC^I\n    \\end{tikzcd}\n\\end{equation}\ncommutes.\n\\end{cor}\n\n\\begin{thmintro}[Theorem~\\ref{thm:mainthm} \\& Corollary~\\ref{cor:spec}]\n\\label{thm:1}\nLet $G$ be simply connected. The coordinate ring of the Vinberg monoid $\\Env G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible. \n\nThe map $\\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.\n\\end{thmintro}\n\n\\begin{theorem}\n\\label{thm:mainthm}\nLet $\\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\\Sigma=\\wt{J}_\\fro\\backslash I'$. The isomorphism $\\Psi_\\bfi$ restricts to isomorphisms\n\\[\n\\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[G\\x ^Z T],\\qquad \\overline{\\CU}(\\wt{\\bfs}(\\bfi))\\overset{\\sim}{\\rightarrow}\\BC[\\Env G].\n\\]\nIn other words, we have the following commutative diagram\n\\begin{equation}\\label{eq:cr}\n    \\begin{tikzcd}\n        & \\CU(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] & \\BC[G^{w_0,w_0}\\x ^Z T] \\\\\n        & \\overline{\\CU}^\\Sigma(\\wt{\\bfs}(\\bfi)) \\arrow[r,\"\\sim\"] \\arrow[u,hook] & \\BC[G\\x^Z T] \\arrow[u,hook] \\\\\n        & \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[u,hook] \\arrow[r,\"\\sim\"] & \\BC[\\Env G] \\arrow[u,hook]\n    \\end{tikzcd}\n\\end{equation}\n\\end{theorem}",
    "post_theorem_intro_text_len": 4958,
    "post_theorem_intro_text": "Combining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \\ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids. \n\n\\begin{thmintro}\n[Theorem~\\ref{thm:affineAbelianCluster}]\n\\label{thm:3}\nLet $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\\mathbb{C}^k$ (See Section \\ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.\n\\end{thmintro}\n\nLet us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\\to \\BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\\times n$ matrices. In this case, Theorem \\ref{thm:3} recovers the well-known cluster structures, presented in \\cite{FWZ20}*{Theorem~6.6.1}.\n\nBy inverting the appropriate frozen variables, we obtain cluster algebra structures on any connected reductive group.\n\n\\begin{thmintro}[Theorem~\\ref{thm:reductiveMonoidCluster}]\n\\label{thm:2}\nLet $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.\n\\end{thmintro}\n\nIn the case when $H$ is semisimple, Theorem \\ref{thm:2} was obtained by Qin and Yakimov \\cite{QY25}. Independently, Oya \\cite{Oya25} proved the same result under the further condition that $H$ is not of type $F_4$. In the semisimple case, there are no additional invertible frozen variables. This is also reflected by the fact that a reductive monoid with semisimple unit group is necessarily a group.\n\n\\subsection{Strategy of the proof}\n\\label{sec:intro:strategy}\n\nThe first key ingredient of the proof of Theorem~\\ref{thm:1} is the construction of a framed group $\\widetilde{G}$ of $G$. Let $I$ be the set of simple roots of $G$ with $|I|=r$. We define $\\widetilde{I}$ by adding $r$ vertices labeled $i'$ and a simple edge joining $i$ and $i'$ for each $i\\in I$. The framed group $\\widetilde{G}$ is defined to be the corresponding Kac--Moody group. We then construct an isomorphism between the unit group of $\\textnormal{Env } G$ to the standard Levi subgroup of $\\widetilde{G}$ associated with $I$. Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \\cite{Wil13} (See also \\cite{SW21}), we obtain a cluster structure on an open part $(\\textnormal{Env } G)^\\circ$ of $\\textnormal{Env } G$.\n\nThe second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\\textnormal{Env } G-(\\textnormal{Env } G)^\\circ$. This is achieved by applying the string parametrization of the dual canonical basis.\n\nWe note that the technique of extending root data was inspired by the thickening map introduced by Bao and He \\cite{BH24}. Similar ideas appeared in the recent works \\citelist{\\cite{Qin25}\\cite{FH25}} as well.\n\n\\subsection{Future problems}\n\\label{sec:intro:future}\n\nLet us close the introduction by mentioning some future directions.\n\nFirstly, reductive monoids admit standard Poisson structures and quantization \\cite{BS25}. We expect that all the results in the current paper can be extended to the quantum setting.\n\nSecondly, we focus only on upper cluster algebras in this paper. For simply-connected simple group $G$ not of type $F_4$, Oya \\cite{Oya25} proved that the upper cluster algebra structure on $\\mathbb{C}[G]$ coincides with the ordinary cluster algebra in the sense of Fomin and Zelevinsky. A quantum analogue of this coincidence was recently established by Oya, Qin, and Yakimov \\cite{OQY25}. It is therefore a natural question whether the upper cluster algebras studied in this paper also coincide with ordinary cluster algebras.\n\nThirdly, the Vinberg monoid $\\textnormal{Env } G$ is isomorphic to the total coordinate ring of the wonderful compactification $\\mathbf{X}$ of $G_{ad}=G/Z$. It would be interesting to explore the connections between the cluster algebra structure on $\\textnormal{Env } G$ developed in the current paper with the total positivity on $\\mathbf{X}$ studied by He \\cite{He04}. \n\n\\vspace{.2cm}\n\n\\noindent {\\bf Acknowledgment: } We are grateful to Huanchen Bao for valuable discussions and suggestions during the writing of this paper. JS is grateful to Fan Qin for helpful discussions at the early stage of the paper. JS is supported by the Glorious Sun Charity Fund.",
    "sketch": "The post-theorem introduction contains a proof strategy for Theorem~\\ref{thm:1}:\n\n1. **Construct a framed group and relate it to $\\textnormal{Env }G$.** “The first key ingredient of the proof of Theorem~\\ref{thm:1} is the construction of a framed group $\\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\\in I$, and let $\\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\\textnormal{Env } G$ to the standard Levi subgroup of $\\widetilde{G}$ associated with $I$.”\n\n2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \\cite{Wil13} (See also \\cite{SW21}), we obtain a cluster structure on an open part $(\\textnormal{Env } G)^\\circ$ of $\\textnormal{Env } G$.”\n\n3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\\textnormal{Env } G-(\\textnormal{Env } G)^\\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”\n\n(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \\cite{BH24}.”)",
    "expanded_sketch": "The post-theorem introduction contains a proof strategy for the main theorem:\n\n1. **Construct a framed group and relate it to $\\textnormal{Env }G$.** “The first key ingredient of the proof of the main theorem is the construction of a framed group $\\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\\in I$, and let $\\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\\textnormal{Env } G$ to the standard Levi subgroup of $\\widetilde{G}$ associated with $I$.”\n\n2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \\cite{Wil13} (See also \\cite{SW21}), we obtain a cluster structure on an open part $(\\textnormal{Env } G)^\\circ$ of $\\textnormal{Env } G$.”\n\n3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\\textnormal{Env } G-(\\textnormal{Env } G)^\\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”\n\n(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \\cite{BH24}.”)",
    "expanded_theorem": "[To prove the main theorem \\& we first prove the following corollary. \\begin{cor}\\label{cor:spec}\nThe diagram\n\\begin{equation}\n    \\begin{tikzcd}\n    \\Env G \\arrow[dr,\"\\pi\"'] \\arrow[rr,\"\\sim\"] && \\Spec \\overline{\\CU}(\\wt{\\bfs}(\\bfi)) \\arrow[dl,\"\\pi^{I'}\"] \\\\ & \\BC^I\n    \\end{tikzcd}\n\\end{equation}\ncommutes.\n\\end{cor}]\n\\label{thm:1}\nLet $G$ be simply connected. The coordinate ring of the Vinberg monoid $\\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible. \n\nThe map $\\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.,",
    "theorem_type": [
      "Classification or Bijection",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $G$ be a simply connected semisimple group, and let $\\Env G$ denote its Vinberg monoid with canonical flat morphism $\\pi:\\Env G\\to \\mathbb{C}^I$. Fix the seed $\\widetilde{\\mathbf{s}}(\\mathbf{i})$ attached to a double reduced word $\\mathbf{i}$ for $(w_0,w_0)$, and let $\\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$ be the partially compactified upper cluster algebra in which frozen variables are not inverted. Which affine scheme, together with its map to $\\mathbb{C}^I$, is identified with $\\Env G$?",
      "correct_choice": {
        "label": "A",
        "text": "$\\Spec \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$; equivalently, $\\mathbb{C}[\\Env G]\\cong \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$ as a partially compactified upper cluster algebra with all frozen variables noninvertible, and under the induced isomorphism the morphism $\\pi$ agrees with the specialization map $\\pi^{I'}$ of the relevant frozen variables to $\\mathbb{C}^I$, so the asymptotic semigroup is obtained by setting those frozen variables equal to $0$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\Spec \\overline{\\mathcal U}^{\\Sigma}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$; equivalently, $\\mathbb{C}[G\\times^{Z}T]\\cong \\overline{\\mathcal U}^{\\Sigma}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$, and under this identification the morphism $\\pi$ is the restriction of the specialization map $\\pi^{I'}:\\Spec \\overline{\\mathcal U}^{\\Sigma}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))\\to \\mathbb{C}^I$."
        },
        {
          "label": "C",
          "text": "$\\Spec \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$; equivalently, $\\mathbb{C}[\\Env G]\\cong \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$ as a partially compactified upper cluster algebra with frozen variables not inverted."
        },
        {
          "label": "D",
          "text": "$\\Spec \\mathcal U(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$; equivalently, the open part $(\\Env G)^\\circ$ is all of $\\Env G$, so $\\mathbb{C}[\\Env G]\\cong \\mathcal U(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$, and the morphism $\\pi$ is already given on this upper cluster algebra by specializing the relevant frozen variables to $\\mathbb{C}^I$."
        },
        {
          "label": "E",
          "text": "$\\Spec \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$; equivalently, $\\mathbb{C}[\\Env G]\\cong \\overline{\\mathcal U}(\\widetilde{\\mathbf{s}}(\\mathbf{i}))$ as a partially compactified upper cluster algebra with all frozen variables noninvertible, and under the induced isomorphism the morphism $\\pi$ agrees with the specialization map obtained by setting all frozen variables equal to $0$, so the asymptotic semigroup is the whole fibrewise image of this map."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "open-part-vs-full-monoid target",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "compatibility of the isomorphism with the map $\\pi$ and the asymptotic-semigroup specialization statement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "boundary divisor controlled by frozen-variable valuations",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "which frozen variables are specialized and how $\\pi$ maps to $\\mathbb{C}^I$",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It names the relevant objects and asks for the precise identification, but the correct choice must still be distinguished from nearby variants involving the open part, the wrong compactification, or incorrect specialization behavior."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall/restatement of a theorem: the question asks which scheme and map are identified with the Vinberg monoid, and the correct option reproduces that theorem-level statement almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required because several options are close and differ in subtle but mathematically meaningful ways (e.g. full monoid vs open part, compatibility with the map to \\(\\mathbb{C}^I\\), and which frozen variables are specialized). However, it primarily tests precise recall rather than deeper generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and target realistic failure modes: confusing \\(\\Env G\\) with its open part, replacing the relevant algebra by a \\(\\Sigma\\)-variant, omitting the map-compatibility clause, or misstating the specialization to \\(\\mathbb{C}^I\\). They are distinct and mathematically aligned with common misunderstandings."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically precise MCQ with strong distractors and little answer leakage, but it is mostly a theorem-recall item rather than a non-tautological or strongly generative reasoning question."
    }
  },
  {
    "id": "2512.09598v1",
    "paper_link": "http://arxiv.org/abs/2512.09598v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic.",
      "start_pos": 105649,
      "end_pos": 105840,
      "label": "t1"
    },
    "ref_dict": {
      "t1": "\\begin{theorem}\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic. \n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1869,
    "pre_theorem_intro_text": "In this paper we are interested in the following well-known conjecture of Hindman.\n\n\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n    Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.  \n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}.  The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors  have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations.  Our main result is the following, which was previously only known in the case $k=1.$",
    "context": "\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.\n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}. The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$",
    "full_context": "\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.\n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}. The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$\n\nIn addition to proving Theorem \\ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations. Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \\cite{bowen.non.commuting}. The basic strategy used in the proof of Theorem \\ref{t1} and the related results \\cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:\n\n\\begin{theorem}\\label{t2}\n For any $k\\in \\mathbb{N}$ and any $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ with $\\{x,y,x^y, xy^i: i\\leq k\\}$ monochromatic. \n\\end{theorem}\n\n\\begin{theorem}\\label{rational.moreira}\n Let $P\\subseteq\\pol$ be finite. For any finite coloring of $\\mathbb{N}$ there exist $x,y\\in\\mathbb{N}$ and a color class $C$ such that $\\{x,xy,x+p(y):p\\in P\\}\\subseteq C$.\n\\end{theorem}\n\n\\begin{prop}\\label{central*}\n For any finite set of polynomials $P\\subseteq\\pol$ and any finite coloring of $\\mathbb{N}$ the set $$\\{y: \\exists x \\textnormal{ with } \\{x,xy,x+p(y):p\\in P\\} \\textnormal{ monochromatic} \\} $$\n\n\\begin{proof}[Second proof of Corollary \\ref{all_syndetic}]\n Fix a finite set $P\\subseteq \\pol.$ By definition, for each $i$ there is a finite set $F_i\\subset \\mathbb{N}$ such that $S_i/F_i\\supseteq \\mathbb{N}.$ Let $k=\\max_{i\\leq r}|F_i|$ and consider the coloring $c:\\mathbb{N}\\rightarrow [k]^r$ where an $n\\in\\mathbb{N}$ is colored based on the $r$-tuple listing the minimal $f_i\\in F_i$ such that $f_in\\in S_i.$ Let $P'=\\{p/f: p\\in P, f\\in \\bigcup_{i\\leq r}F_i\\}$. By Theorem \\ref{rational.moreira}, there are integers $x',y$ such that $S=\\{x', x'y, x'+p'(y): p'\\in P'\\}$ is monochromatic according to the coloring $c.$ Now if $y\\in S_i$, by the definition of $c$ there is an $f_i$ such that $f_iS\\subseteq S_i.$ Consequently, setting $x=x'f_i$ is as desired. \n\\end{proof}\n\n\\begin{lemma}\\label{all thick}\n Suppose $k\\in\\mathbb{N}$ and $\\mathbb{N}=R\\cup B$ with both $R$ and $B$ multiplicatively thick. Then one of the colors contains a set $\\{x,y,xy,x+iy:i\\leq k\\}$.\n\\end{lemma}\n\\begin{proof}\nSuppose for the sake of contradiction that the lemma is false.\n Without loss of generality assume that $R$ is additively piecewise syndetic. Choose $N\\in\\mathbb{N}$ large enough depending on $k$.\n\n\\begin{lemma}\\label{log reduction}\n If any finite coloring of $\\mathbb{N}$ contains a monochromatic set $\\{x,y, x2^y, x+iy: i\\leq k\\}$ then any finite coloring contains a monochromatic set of the form $\\{x,y,x^y,xy^i: i\\leq k\\}.$\n\\end{lemma}\n\n\\begin{theorem}\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic. \n\\end{theorem}",
    "post_theorem_intro_text_len": 2146,
    "post_theorem_intro_text": "In addition to proving Theorem \\ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations.  Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \\cite{bowen.non.commuting}.  The basic strategy used in the proof of Theorem \\ref{t1} and the related results \\cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:\n\n\\begin{enumerate}\n    \\item First, show that the result holds when the color of the step size $y$ is ignored.  In this case, we are interested in finding monochromatic sets $\\{x,xy,x+iy: i\\leq k\\},$ which can be done either through Moreira's work \\cite{moreira2017monochromatic} or the first author's \\cite{alweiss2022monochromatic}.\n\n    \\item Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.\n\n    \\item Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well. \\end{enumerate}\n\nIn order to illustrate this technique we give two proofs of Theorem \\ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.  These are presented in the two subsections of Section \\ref{section +}.\n\nWe also show that the same basic approach can be adapted to prove the following exponential version of the result.\n\n\\begin{theorem}\\label{t2}\n    For any $k\\in \\mathbb{N}$ and any $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ with $\\{x,y,x^y, xy^i: i\\leq k\\}$ monochromatic. \n\\end{theorem}\n\nThe $k=1$ case of this result was proven by Sahasrabudhe \\cite{sahasrabudhe2018monochromatic}, even for arbitrary finite colorings.  However, the result seems to be new for $k>1.$  Along the way, we also prove that any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,x^y,xy^i: i\\leq k\\},$ which gives a new exponential version of Moreira's theorem.",
    "sketch": "The authors state that “the basic strategy used in the proof of Theorem \\ref{t1} … is essentially as follows:”\n\\begin{enumerate}\n\\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\\{x,xy,x+iy: i\\leq k\\}$, “which can be done either through Moreira's work … or the first author's …”.\n\\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”\n\\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”\n\\end{enumerate}\nThey further note that they “give two proofs of Theorem \\ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”",
    "expanded_sketch": "The authors state that “the basic strategy used in the proof of the main theorem … is essentially as follows:”\n\\begin{enumerate}\n\\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\\{x,xy,x+iy: i\\leq k\\}$, “which can be done either through Moreira's work … or the first author's …”.\n\\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”\n\\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”\n\\end{enumerate}\nThey further note that they “give two proofs of the main theorem based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”",
    "expanded_theorem": "\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic.,",
    "theorem_type": [
      "Universal–Existential"
    ],
    "mcq": {
      "question": "Let $k\\in\\mathbb{N}$, and let $\\mathbb{N}$ be colored with two colors (equivalently, let $c:\\mathbb{N}\\to\\{1,2\\}$ be any 2-coloring). Which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, xy,\\, x+iy: i\\le k\\}$ have the same color; equivalently, $\\{x,y,xy,x+y,x+2y,\\dots,x+ky\\}$ is monochromatic."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, xy,\\, x+iy: i\\le k\\}$ have the same color; equivalently, $\\{x,xy,x+y,x+2y,\\dots,x+ky\\}$ is monochromatic."
        },
        {
          "label": "C",
          "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, x+y,\\, x+2y,\\dots,x+ky\\}$ have the same color."
        },
        {
          "label": "D",
          "text": "For every color class $C$ of the 2-coloring, there exist $x,y\\in\\mathbb{N}$ such that $\\{x,\\, y,\\, xy,\\, x+iy: i\\le k\\}\\subseteq C$."
        },
        {
          "label": "E",
          "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, x^y,\\, xy^i: i\\le k\\}$ have the same color; equivalently, $\\{x,y,x^y,xy,xy^2,\\dots,xy^k\\}$ is monochromatic."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "control_of_y_term",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_multiplicative_term_xy",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "existential_color_class_replaced_by_every_color_class",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "log_reduction_conclusion_substituted_for_main_theorem",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only gives the coloring setup and asks which conclusion is valid; it does not state or strongly hint at the specific monochromatic configuration in the correct option."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a direct restatement in the stem. The item asks the learner to distinguish the actual theorem-strength conclusion from weaker, stronger, and altered alternatives."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to compare quantifiers and structural differences among the options, especially against the weaker true statement and the overly strong version. However, the item still mainly tests theorem recognition rather than substantial generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and varied: one is a weaker true statement, one improperly strengthens the quantifier, one omits a needed term, and one introduces an unrelated exponential pattern. These reflect realistic failure modes."
      },
      "total_score": 7,
      "overall_assessment": "A strong MCQ overall: it avoids answer leakage, uses plausible and diagnostically useful distractors, and is not tautological, though it leans more toward theorem identification than deep generative reasoning."
    }
  },
  {
    "id": "2512.09599v1",
    "paper_link": "http://arxiv.org/abs/2512.09599v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm: main}\nConsider \\eqref{eq: main} with random initial data \\eqref{eq: rmain}. Let $c_{n}$ in \\eqref{eq: rmain} be $\\langle n\\rangle^{-(\\frac{1}{2}+\\theta)},\\theta>0$. Let $T_{\\epsilon}\\approx \\epsilon^{-1}$. One has the following large deviation principle\n\\begin{equation}\\label{eq: estimatemain}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon\\ln \\mathbb{P}(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}|c_{n}|^{2}}.\n\\end{equation}",
      "start_pos": 30111,
      "end_pos": 30648,
      "label": "thm: main"
    },
    "ref_dict": {
      "eq: estimatemain": "\\begin{equation}\\label{eq: estimatemain}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon\\ln \\mathbb{P}(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}|c_{n}|^{2}}.\n\\end{equation}",
      "eq: rmain": "\\begin{equation}\\label{eq: rmain}\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}",
      "thm: main": "\\begin{thm}\\label{thm: main}\nConsider \\eqref{eq: main} with random initial data \\eqref{eq: rmain}. Let $c_{n}$ in \\eqref{eq: rmain} be $\\langle n\\rangle^{-(\\frac{1}{2}+\\theta)},\\theta>0$. Let $T_{\\epsilon}\\approx \\epsilon^{-1}$. One has the following large deviation principle\n\\begin{equation}\\label{eq: estimatemain}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon\\ln \\mathbb{P}(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}|c_{n}|^{2}}.\n\\end{equation}\t\n\\end{thm}",
      "eq: main": "\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\t\nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 688,
    "pre_theorem_intro_text": "\\subsection{Statement of results}\nConsider cubic NLS on 1d torus $\\mathbb{T}$,\n\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\t\nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}\nhere $\\epsilon>0$ is a parameter.\n\nWe will focus on the random initial data of type\n\\begin{equation}\\label{eq: rmain}\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}\nwhere $c_{n}=\\<n\\>^{-(\\frac{1}{2}+\\theta)}, \\theta>0$, and $g_{n}^{\\omega}$ be i.i.d standard complex Gaussian.\\\\\n\nNote that $u^{\\omega}_{0}\\in L_{x}^{2}$ almost surely, and it is well known that all $L^{2}$ initial data generates a global flow, \\cite{bourgain1993fourier}. Our main result is",
    "context": "\\subsection{Statement of results}\nConsider cubic NLS on 1d torus $\\mathbb{T}$,\n\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\ \nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}\nhere $\\epsilon>0$ is a parameter.\n\nWe will focus on the random initial data of type\n\\begin{equation}\\label{eq: rmain}\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}\nwhere $c_{n}=\\<n\\>^{-(\\frac{1}{2}+\\theta)}, \\theta>0$, and $g_{n}^{\\omega}$ be i.i.d standard complex Gaussian.\\\\\n\nNote that $u^{\\omega}_{0}\\in L_{x}^{2}$ almost surely, and it is well known that all $L^{2}$ initial data generates a global flow, \\cite{bourgain1993fourier}. Our main result is\n\n\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\t\nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}",
    "full_context": "\\subsection{Statement of results}\nConsider cubic NLS on 1d torus $\\mathbb{T}$,\n\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\ \nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}\nhere $\\epsilon>0$ is a parameter.\n\nWe will focus on the random initial data of type\n\\begin{equation}\\label{eq: rmain}\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}\nwhere $c_{n}=\\<n\\>^{-(\\frac{1}{2}+\\theta)}, \\theta>0$, and $g_{n}^{\\omega}$ be i.i.d standard complex Gaussian.\\\\\n\nNote that $u^{\\omega}_{0}\\in L_{x}^{2}$ almost surely, and it is well known that all $L^{2}$ initial data generates a global flow, \\cite{bourgain1993fourier}. Our main result is\n\n\\begin{equation}\\label{eq: main}\n\\begin{cases}\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\t\nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation}\n\nWe will focus on the random initial data of type\n\\begin{equation}\\label{eq: rmain}\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}\nwhere $c_{n}=\\<n\\>^{-(\\frac{1}{2}+\\theta)}, \\theta>0$, and $g_{n}^{\\omega}$ be i.i.d standard complex Gaussian.\\\\\n\nBoth \\cite{garrido2023large}, \\cite{liang2025random}, and this work actually work for time scale for $c\\epsilon^{-1}|\\ln \\epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \\cite{liang2025random}, they state the time scale as $O(\\epsilon^{-1}|\\ln \\epsilon|)$, however, it seems hard to cover time scale for $C\\epsilon^{-1}|\\ln \\epsilon|$ for $C$ large, see the line below (3.30) in \\cite{liang2025random}.\n\\end{rem}\n\n\\begin{lem}\\label{lem: modlinearldp}\nLet $z_{0}>0$, $T_{\\epsilon}=O(\\epsilon^{-1})$ and $u_{app}$ be defined as \\eqref{eq: uapp}, one has the large deviation principle,\n\\begin{equation}\\label{eq: modlinearldp}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon \\ln \\mathbb{P}(\\|u_{app}\\|_{L_{t,x}^{\\infty}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}c_{n}^{2}} \n\\end{equation}\n\nand \n\\begin{lem}\\label{lem: errorcontrol}\nLet $u_{0}$ be as in \\eqref{eq: u0}, and $u_{app}$ be as in \\eqref{eq: uapp}. Let $u$ be the associated solution to \\eqref{eq: main} with initial data $u_{0}$. Let $\\theta>0$ be fixed. Let $T_{\\epsilon}=O(\\epsilon^{-1})$. Let $C_{0}\\gg 1$. Then there exists $s>\\frac{1}{2}$, and $\\delta_{1}>0$, so that \n\\begin{equation}\\label{eq: errorcontrol}\n-\\ln\\mathbb{P}(\\|u-u_{app}\\|_{L_{t}^{\\infty}H^{s}([0,T_{\\epsilon}]\\times \\mathbb{T})}\\geq \\epsilon^{-\\frac{1}{2}+\\delta_{1}})\\geq C_{0}\\epsilon^{-1}\n\\end{equation}\n\nBy \\cite[Lemma 4.2]{garrido2023large} or \\cite[Lemma 4.2]{Tzvetkov2017quasi}, $\\eta$ is a standard complex Gaussian if and only $\\eta e^{it|\\eta|^{2}}$ is a standard complex Gaussian. Our proof for Lemma \\ref{lem: modlinearldp} actually yields to the same result\\footnote{We will not use this result in the proof, and strictly speaking those two results, though same, are not equivalent, but we feel it is good to record this estimate for linear flow here.} for the linear free flow.\n\\begin{lem}\\label{lem: ldpforlinearflow}\n Let $z_{0}>0$, $T_{\\epsilon}=O(\\epsilon^{-1})$ and $u_{0}$ be as in \\eqref{eq: u0}, one has the large deviation principle\n\\begin{equation}\\label{eq: linearldp}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon \\ln \\mathbb{P}(\\|e^{it\\Delta}u_{0}\\|_{L_{t,x}^{\\infty}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}c_{n}^{2}} \n\\end{equation}\n\\end{lem}\n\nIn this case, we we reduce \\eqref{eq: crucialreduce} to proving that for all $\\omega \\in F_\\epsilon - G_\\epsilon$, one has\n \\begin{equation}\\label{eq: I-I-II}\n \\sup_{0\\leq \\tau\\leq t} \\sum_{R_{n}(n_1,n_2,n_3)} \\frac{|g_{n_1}g_{n_2}h_{n_3}|}{\\<n_1\\>^{\\frac{1}{2}+\\theta }\\<n_2\\>^{\\frac{1}{2}+\\theta }} \n \\lesssim \\<n\\>^{-\\theta + \\delta_5} \\epsilon ^{-3/2-3\\delta_1}.\n \\end{equation}\n Using Cauchy-Schwarz inequality and \\eqref{eq: gepsic}, for all $\\omega \\in F_\\epsilon - G_\\epsilon$, we have\n \\begin{equation}\\label{eq: uniformI-I-II}\n \\begin{aligned}\n &\\sup_{0\\leq \\tau\\leq t} \\sum_{R_{n}(n_1,n_2,n_3)} \\frac{|g_{n_1}g_{n_2}h_{n_3}|}{\\<n_1\\>^{\\frac{1}{2}+\\theta }\\<n_2\\>^{\\frac{1}{2}+\\theta }}\\\\\n &\\lesssim \\|h\\|_{C_t[0,T]H^s} \\left(\\sum_{n_3} \\<n_3\\>^{-2s} \\left|\\sum_{R_{n_3,n}(n_1,n_2)} \\frac{\\epsilon ^{-1-2\\delta_1 } }{\\left\\langle n_1 \\right\\rangle ^{1/2+ \\theta -\\delta_4 } \\left\\langle n_2 \\right\\rangle ^{1/2 + \\theta -\\delta_4}}\\right|^2\\right)^\\frac{1}{2}\\\\\n &\\lesssim \\epsilon ^{-3/2 -3\\delta_1 } \\left(\\sum_{n_3} \\<n_3\\>^{-2s} \\left|\\sum_{R_{n_3,n}(n_1,n_2)} \\frac{1}{\\left\\langle n_1 \\right\\rangle ^{1/2+ \\theta -\\delta_4 } \\left\\langle n_2 \\right\\rangle ^{1/2 + \\theta -\\delta_4}}\\right|^2\\right)^\\frac{1}{2},\n \\end{aligned}\n \\end{equation}\n where for fixed $n,n_3$, we denote $R_{n_3,n}(n_1,n_2) := \\{(n_1,n_2)\\in\\mathbb{Z}^2:n=n_1-n_2+n_3,n_2\\neq n_1,n_3\\}$.\n Next, we aim to reduce \\eqref{eq: I-I-II} to following\n \\begin{equation}\\label{eq: mainreductionI-I-II}\n \\sum_{R_{n_3,n}(n_1,n_2)} \\frac{1}{\\left\\langle n_1 \\right\\rangle ^{1/2+ \\theta -\\delta_4 } \\left\\langle n_2 \\right\\rangle ^{1/2 + \\theta -\\delta_4}}\n \\lesssim \\left\\langle n-n_3 \\right\\rangle ^{-2\\theta +2\\delta_5} .\n \\end{equation}\n If \\eqref{eq: mainreductionI-I-II} holds, noting that\n \\begin{equation*}\n \\sum_{|n_3|\\ll |n|} \\<n_3\\>^{-2s} \\left\\langle n-n_3 \\right\\rangle ^{-4\\theta +4\\delta_5}\n \\lesssim \\<n\\>^{-4\\theta +4\\delta_5} \n \\end{equation*}\n and\n \\begin{equation*}\n \\sum_{|n_3|\\gg |n|} \\<n_3\\>^{-2s} \\left\\langle n-n_3 \\right\\rangle ^{-4\\theta +4\\delta_5}\n \\sim \\sum_{|n_3|\\gg |n|} \\<n_3\\>^{-2s-4\\theta +4\\delta_5} \n \\lesssim \\<n\\>^{-4\\theta +4\\delta_5} \n \\end{equation*}\n and\n \\begin{equation*}\n \\sum_{|n_3|\\sim |n|} \\<n_3\\>^{-2s} \\left\\langle n-n_3 \\right\\rangle ^{-4\\theta +4\\delta_5}\n \\sim \\<n\\>^{-2s} \\sum_{|n_3|\\sim |n|} \\<n_3-n\\>^{-4\\theta +4\\delta_5} \n \\lesssim \\<n\\>^{1-2s-4\\theta +4\\delta_5},\n \\end{equation*}\n then we have\n \\begin{equation*}\n \\sum_{n_3} \\<n_3\\>^{-2s} \\left\\langle n-n_3 \\right\\rangle ^{-4\\theta +4\\delta_5}\n \\lesssim \\<n\\>^{-4\\theta +4\\delta_5},\n \\end{equation*}\n which, along with \\eqref{eq: mainreductionI-I-II} and \\eqref{eq: uniformI-I-II}, yields \\eqref{eq: I-I-II}. So we reduce \\eqref{eq: I-I-II} to \\eqref{eq: mainreductionI-I-II} if we take \\fbox{$\\delta_5< \\theta $}. Similarly, \n noting that\n \\begin{equation*}\n \\sum_{|n_1|\\ll |n-n_3|} \\<n_1\\>^{\\delta_4-\\frac{1}{2}-\\theta } \\left\\langle n-n_3-n_1 \\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta}\n \\lesssim \\left\\langle n-n_3\\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta} \\sum_{|n_1|\\ll |n-n_3|} \\<n_1\\>^{\\delta_4-\\frac{1}{2}-\\theta }\n \\lesssim \\left\\langle n-n_3\\right\\rangle ^{2\\delta_4-2\\theta} \n \\end{equation*}\n and\n \\begin{equation*}\n \\sum_{|n_1|\\gg |n-n_3|} \\<n_1\\>^{\\delta_4-\\frac{1}{2}-\\theta } \\left\\langle n-n_3-n_1 \\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta}\n \\sim \\sum_{|n_1|\\gg |n-n_3|} \\<n_1\\>^{2\\delta_4-1-2\\theta } \n \\lesssim \\<n-n_3\\>^{-2\\theta +2\\delta_4} \n \\end{equation*}\n and\n \\begin{equation*}\n \\sum_{|n_1|\\sim |n-n_3|} \\<n_1\\>^{\\delta_4-\\frac{1}{2}-\\theta } \\left\\langle n-n_3-n_1 \\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta}\n \\sim \\left\\langle n-n_3\\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta} \\sum_{|n_1|\\sim |n-n_3|} \\left\\langle n-n_3-n_1 \\right\\rangle ^{\\delta_4-\\frac{1}{2}-\\theta}\n \\lesssim \\<n-n_3\\>^{-2\\theta +2\\delta_4} ,\n \\end{equation*}\n then we have \\eqref{eq: mainreductionI-I-II}\n if we take \\fbox{$\\delta_5\\leq \\delta_4$}.",
    "post_theorem_intro_text_len": 5128,
    "post_theorem_intro_text": "\\begin{rem}\nUpon finishing of this work, we notice a recent preprint of Liang-Wang, \\cite{liang2025random},  which handles Theorem \\ref{thm: main} in the case $\\theta>\\frac{1}{2}$. It should be noted that when $\\theta>\\frac{1}{2}$, the Fourier coefficient is in $l^{1}$, which forms an algebra. We handle all $\\theta>0$.\\\\\n\nBoth \\cite{garrido2023large}, \\cite{liang2025random}, and this work actually work for time scale for $c\\epsilon^{-1}|\\ln \\epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \\cite{liang2025random}, they state the time scale as $O(\\epsilon^{-1}|\\ln \\epsilon|)$, however, it seems hard to cover time scale for $C\\epsilon^{-1}|\\ln \\epsilon|$ for $C$ large, see the line below (3.30) in \\cite{liang2025random}.\n\\end{rem}\n\n\\subsection{Background and motivation}\nThis work is inspired by the pioneering work \\cite{garrido2023large}. Motivated by rogue wave phenomena, Garrido, Grande, Kurianski and Staffilani study \\eqref{eq: main}, with random initial data of form \\eqref{eq: rmain}, but for $c_{n}=ae^{-b|n|}$. They essentially\n\\footnote{They state the result for pointwise in $t\\leq T_{\\epsilon}\\sim \\epsilon^{-1}$. We take this chance to cover $\\|u\\|_{L_{t,x}^{\\infty}}$ rather than estimate pointwise in $t$.} obtained \\eqref{eq: estimatemain} for such types of random initial data. It has been remarked in \\cite[Remark 1.3]{garrido2023large}, \\textit{finding optimal family of coefficients (in terms of decay)}, rather than considering exponential decay $c_{n}$, remains an interesting open question.\n\nBy combining random data type techniques in \\cite{bourgain1994periodic,bourgain1996invariant} and some ideas in \\cite{compaan2021pointwise} (which also goes back to Bourgain's 1990s seminal work), we are able to extend the result in \\cite{garrido2023large} to coefficients with polynomial decay $\\langle n\\rangle^{-\\frac{1}{2}-\\theta}$, $\\theta>0$. Such constraint is optimal in the sense when $\n\\theta\\leq 0$, even the initial data will leave $L_{x}^{\\infty}$. It remains an interesting question to go beyond the time scale $T_{\\epsilon}$. We note that this $\\epsilon^{-1}$ time scale (neglecting logarithm corrections) is in some sense critical and we refer to \\cite[Remark 1.7]{garrido2023large} for more details.\nWe note that both current work and \\cite{garrido2023large},  can cover time scale of form $c\\frac{|\\ln \\epsilon|}{\\epsilon}$, where $c$ is some small universal number.\\\\\n\nThe overall proof scheme of current article follows \\cite{garrido2023large}, which contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part, (there is some subtle part here since one needs to first slightly perturb the free linear flow, we will discuss it later).\\\\\n\n Our main contributions in this work are\n\\begin{itemize}\n\\item We find a rather straightforward proof for the large deviation principle for the linear flow, which simplifies the arguments in \\cite{garrido2023large}, and more importantly, extends to random initial data of form $\\sum_{n}\\frac{g^\\omega _n}{\\<n\\>^{\\frac{1}{2}+\\theta}}e^{inx}$, $\\forall \\theta>0$. Our techniques for this part are crucial for our further analysis in the nonlinear smoothing part.\n\\item We adapt the $X^{s,b}$ analysis in \\cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \\cite{garrido2023large}, which gives the desired nonlinear smoothing\\footnote{On one hand, we will use the $X^{s,b}$ analysis in the local random data theory as a black box;  on the other hand, the long time analysis we use to get nonlinear smoothing does explicitly use computations of $X^{s,b}$ type, but without going to the explicit form of $X^{s,b}$ analysis.}.\n\\end{itemize}\n\nWe end this session by pointing out ever since the seminal work \\cite{bourgain1994periodic,bourgain1996invariant}, random initial data theory in nonlinear dispersive PDEs has become a very active research field. Many researchers contribute to this field and it is impossible to survey the field here. We simply mention that  in the perspective of  regularity in NLS model, the recent breakthrough \\cite{deng2022random} gives a quite complete answer. We refer to \\cite{deng2022random}, \\cite{garrido2023large} and reference therein for more related reference.\n\n\\subsection{Notation}\nMany terms $g_{n}^{\\omega}, u^{\\omega}, u_{0}^{\\omega}$ are random, and thus depend on $\\omega$. For simplicity, we will hide the implicit $\\omega$ and simply denote them as $g_{n}, u,u_{0}$.\\\\\n\nWe introduce $\\NNN_{1}, \\NNN_{2}$ as \n\\begin{equation}\\label{eq: n1}\n\\NNN_{1}(f_{1},f_{2},f_{3})=\\sum_{n_{2}\\neq n_{1}, n_{3}}\\hat{f}_{1}(n_{1})\\overline{\\hat{f}_{2}}(n_{2})\\hat{f}_{3}(n_{3})e^{i(n_{1}-n_{2}+n_{3})x},\n\\end{equation}\n\\begin{equation}\\label{eq: n_{2}}\n\\NNN_{2}(f_{1},f_{2},f_{3})=\\sum_{n}\\hat{f}_{1}(n)\\overline{\\hat{f}_{2}}(n)\\hat{f}_{3}(n)e^{inx}\t\n\\end{equation}\nand one has \n\\begin{equation}\\label{eq: algebra}\n|f|^{2}f-2\\frac{1}{2\\pi}(\\int |f|^{2})f=\\NNN_{1}(f,f,f)-\\NNN_{2}(f,f,f)\t\n\\end{equation}\nWe sometimes short $\\NNN_{i}(f,f,f)$ as $\\NNN_{i}(f)$.",
    "sketch": "The proof of Theorem~\\ref{thm: main} is described as following the scheme of \\cite{garrido2023large}: it “contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in \\cite{garrido2023large}” and extends to initial data $\\sum_{n}\\frac{g^\\omega _n}{\\<n\\>^{\\frac{1}{2}+\\theta}}e^{inx}$ for all $\\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in \\cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \\cite{garrido2023large}, which gives the desired nonlinear smoothing” for the Duhamel part.",
    "expanded_sketch": "The proof of the main theorem is described as following the scheme of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,”” and extends to initial data $\\sum_{n}\\frac{g^\\omega _n}{\\<n\\>^{\\frac{1}{2}+\\theta}}e^{inx}$ for all $\\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in Bourgain, “Periodic nonlinear Schrödinger equation and invariant measures,” and Bourgain, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation,” in the setting of long time analysis of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” which gives the desired nonlinear smoothing” for the Duhamel part.,",
    "expanded_theorem": "\\label{thm: main}\nConsider \\begin{equation}\\label{eq: main}\n\\begin{cases}\\niu_{t}+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\t\nu(0,x)=u_{0}(x)\n\\end{cases}\n\\end{equation} with random initial data \\begin{equation}\\label{eq: rmain}\\nu^{\\omega}_{0}(x)=\\sum_{n}c_{n}g_{n}^{\\omega}e^{inx}\n\\end{equation}. Let $c_{n}$ in the equation above be $\\langle n\\rangle^{-(\\frac{1}{2}+\\theta)},\\theta>0$. Let $T_{\\epsilon}\\approx \\epsilon^{-1}$. One has the following large deviation principle\n\\begin{equation}\\label{eq: estimatemain}\n\\lim_{\\epsilon\\rightarrow 0^{+}}\\epsilon\\ln \\mathbb{P}(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb{T})}>z_{0}\\epsilon^{-\\frac{1}{2}})=-\\frac{z_{0}^{2}}{\\sum_{n}|c_{n}|^{2}}.\n\\end{equation},",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\theta>0\\), \\(\\langle n\\rangle=(1+|n|^{2})^{1/2}\\), and for each \\(\\epsilon>0\\) consider the cubic nonlinear Schr\\\"odinger equation on the one-dimensional torus \\(\\mathbb T\\):\n\\[\n\\begin{cases}\niu_t+\\Delta u=\\epsilon^{2}|u|^{2}u,\\\\\nu(0,x)=u_0^{\\omega}(x),\n\\end{cases}\n\\]\nwith random initial data\n\\[\nu_0^{\\omega}(x)=\\sum_{n\\in\\mathbb Z} c_n g_n^{\\omega} e^{inx}, \\qquad c_n=\\langle n\\rangle^{-(\\frac12+\\theta)},\n\\]\nwhere \\(\\{g_n^{\\omega}\\}_{n\\in\\mathbb Z}\\) are i.i.d. standard complex Gaussian random variables. Let \\(u^{\\omega}\\) denote the corresponding solution, and let \\(T_{\\epsilon}\\) be comparable to \\(\\epsilon^{-1}\\). For a fixed \\(z_0>0\\), which large-deviation estimate holds for\n\\(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}\\) as \\(\\epsilon\\to 0^+\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\n\\lim_{\\epsilon\\to 0^+} \\epsilon\\,\\ln \\mathbb P\\Big(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}> z_0\\epsilon^{-1/2}\\Big)\n= -\\frac{z_0^2}{\\sum_{n\\in\\mathbb Z}|c_n|^2}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\n\\lim_{\\epsilon\\to 0^+} \\epsilon\\,\\ln \\mathbb P\\Big(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}> z_0\\epsilon^{-1/2}\\Big)\n= -\\frac{z_0^2}{\\sup_{n\\in\\mathbb Z}|c_n|^2}.\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\n\\limsup_{\\epsilon\\to 0^+} \\epsilon\\,\\ln \\mathbb P\\Big(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}> z_0\\epsilon^{-1/2}\\Big)\n\\le -\\frac{z_0^2}{\\sum_{n\\in\\mathbb Z}|c_n|^2}.\n\\]"
        },
        {
          "label": "D",
          "text": "\\[\n\\lim_{\\epsilon\\to 0^+} \\epsilon\\,\\ln \\mathbb P\\Big(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}> z_0\\epsilon^{-1/2}\\Big)\n= -\\frac{z_0^2}{2\\sum_{n\\in\\mathbb Z}|c_n|^2}.\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\n\\lim_{\\epsilon\\to 0^+} \\epsilon\\,\\ln \\mathbb P\\Big(\\|u^{\\omega}\\|_{L^{\\infty}_{t,x}([0,T_{\\epsilon}]\\times \\mathbb T)}> z_0\\Big)\n= -\\frac{z_0^2}{\\sum_{n\\in\\mathbb Z}|c_n|^2}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "Gaussian variance from full \\ell^2 mass \\sum_n |c_n|^2",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "drop the full limit identity to the corresponding upper large-deviation bound",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "exact rate constant inherited from the complex Gaussian tail",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical threshold scaling z_0\\epsilon^{-1/2} for the supremum norm",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the PDE, the random initial data, and the scaling regime, but not the exact large-deviation rate or the precise form of the limit."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: under the theorem's hypotheses, the student is asked to identify the exact conclusion. It mainly tests recognition of the stated result rather than deriving a new consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish between an exact limit, a weaker upper bound, an incorrect variance surrogate, and an overextended time-scale claim. However, the item still primarily rewards recall of the theorem's exact statement."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: a wrong Gaussian constant, a weaker-but-true style statement, confusion between total variance and single-mode size, and an invalid quantifier extension in the time scale. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-identification MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a specialized result rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.09683v1",
    "paper_link": "http://arxiv.org/abs/2512.09683v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
      "start_pos": 32731,
      "end_pos": 33011,
      "label": "thm:backbone"
    },
    "ref_dict": {
      "eq:p_e": "\\begin{equation}\\label{eq:p_e}\n\\P[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}",
      "thm:backbone": "\\begin{theorem}\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}\n\\end{theorem}",
      "eq:def-backbone": "\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\D}] \\text{ joining } \\varepsilon\\S^1 \\text{ and } \\frac{1}{2}\\S^1\\}.\n\\end{equation}",
      "eq:bb-proba": "\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
      "appendix": "\\begin{aligned}\nF(\\varepsilon)&\\le \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n\\\\\n&\\le  \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-\\frac{\\log2}{|\\log q_0|}(\\log\\log|\\log\\varepsilon|-\\log C')}+\\frac{C'}{\\log|\\log\\varepsilon|}\n\\le\\frac{C_2}{\\log|\\log\\varepsilon|}\n\\end{aligned}\n$$\nfor some positive constant $C_2$, as desired.\n\\end{proof}\n\n\\appendix\n\\section{Proof of Theorem~\\ref{thm-cr-mod-relation}}\n\\label{appendix}\n\nIn this appendix, we provide the proof of Theorem~\\ref{thm-cr-mod-relation}, which is implicit in~\\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\\hH:=\\{z\\in\\C:\\text{Im}(z)>0\\}$, and let $\\sm$ be the restriction of the $\\SLE_{8/3}$ loop measure on $\\C$ to the loops that are contained in $\\hH$ and surround $i$. According to the conformal restriction~\\cite{werner-loops}, if $\\Phi:\\D\\to\\hH$ is a conformal map with $\\Phi(0)=i$, then $\\sm=\\Phi_*\\SLE_{8/3,\\D}^{\\lp}$. Therefore, to prove Theorem~\\ref{thm-cr-mod-relation}, it suffices to show the following\n\\begin{theorem}\n\\label{thm-appendix}\nThere is a constant $C>0$ such that for any measurable function $f:\\R_+\\to\\R_+$ and $\\lambda\\ge0$, we have\n$$\n\\int f\\left(\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right)\\psi_\\eta'(i)^\\lambda\\sm(d\\eta)=\nC\\frac{\\sqrt{12\\lambda-1}}{\\sinh\\left(\\frac{\\pi}{3}\\sqrt{12\\lambda-1}\\right)}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\eta(2i\\tau)f(\\tau)d \\tau.\n$$\nHere $\\psi_\\eta$ is the conformal map from $\\hH$ to $D(\\eta)$ such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\\sqrt{\\frac{8}{3}}$. In the following we fix $\\gamma=\\sqrt{\\frac{8}{3}}$ and let $Q=\\frac{\\gamma}{2}+\\frac{2}{\\gamma}$. Let $\\P_{\\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\\hH$ with mean zero on the upper semi-circle $\\S^1\\cap\\hH$. \n\n\\begin{definition}[Liouville field on $\\hH$]\n\\label{def-LF-H}\nLet $(h,c)$ be sampled from $\\P_{\\hH}\\times[e^{-Qc}dc]$, and let $\\phi(z)=h(z)-2Q\\log|z|_++c$ (here $|z|_+:=\\max\\{|z|,1\\}$). Denote the law of $\\phi$ by $\\LF_{\\hH}$.\n\nFor $\\alpha\\in\\R$, let $\\LF_{\\hH}^{(\\alpha,i)}(d\\phi):=\\lim_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\alpha^2}{2}}e^{\\alpha\\phi_\\varepsilon(i)}\\LF_{\\hH}(d\\phi)$ (the limit exists in the vague topology, see~\\cite[Lemma 2.2]{ARS-FZZ}). Here $\\phi_\\varepsilon(i)$ is the average of $\\phi$ over the circle $\\partial B_\\varepsilon(i)$.\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_{\\hH}^{(\\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\\nu_{\\phi}:=\\lim\\limits_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\gamma^2}{4}}e^{\\frac{\\gamma}{2}\\phi_\\varepsilon(x)}dx$, where $\\phi_\\varepsilon(x)$ is the average of $\\phi$ over the semi-circle $\\partial B_\\varepsilon(x)\\cap\\hH$.\nThen denote $\\{\\LF_{\\hH}^{(\\alpha,i)}(\\ell)\\}_{\\ell>0}$ to be the disintegration of $\\LF_{\\hH}^{(\\alpha,i)}$ over $\\nu_\\phi(\\R)$; i.e.~$\\LF_{\\hH}^{(\\alpha,i)}=\\int_0^\\infty\\LF_{\\hH}^{(\\alpha,i)}(\\ell)d\\ell$. We refer readers to see e.g.~\\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\\cite[Lemma 2.7]{ARS-FZZ}, for $\\alpha>\\frac{\\gamma}{2}$,\nthere are constants $C_{\\alpha,\\gamma}>0$ such that $|\\LF_{\\hH}^{(\\alpha,i)}(\\ell)|=C_{\\alpha,\\gamma}\\ell^{\\frac{2}{\\gamma}(\\alpha-Q)-1}$ for any $\\ell>0$.\n\nWe also need the Liouville field on the annulus. For $\\tau>0$, let $\\mathcal{C}_\\tau=[0,\\tau]\\times[0,1]/\\mathord\\sim$ be the horizontal cylinder with modulus $\\tau$, where under $\\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\\in[0,\\tau]$. Let $\\P_{\\tau}$ be the free boundary GFF on $\\mathcal{C}_\\tau$\nwith zero mean on the circle $\\{\\frac{\\tau}{2}\\}\\times[0,1]/\\mathord\\sim$.\n\\begin{definition}[{\\cite[Definition 2.2]{ARS-Annulus}}]\n\\label{def-LF-annulus}\nLet $(h,c)$ be sampled from $\\P_\\tau\\times dc$. Then denote the law of $\\phi=h+c$ by $\\LF_\\tau$ (which is an infinite measure on $H^{-1}(\\mathcal{C}_\\tau)$).\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_\\tau$,\nwe can similarly define its boundary GMC measures $\\nu^1_\\phi, \\nu_\\phi^2$ on the two boundaries $\\partial_1\\mathcal{C}_\\tau=\\{0\\}\\times[0,1]\\mathord\\sim$ and $\\partial_2\\mathcal{C}_\\tau=\\{1\\}\\times[0,1]\\mathord\\sim$, respectively. We also let $\\{\\LF_\\tau(\\ell_1,\\ell_2)\\}_{\\ell_1,\\ell_2>0}$ be the disintegration of $\\LF_\\tau$ over $\\nu_\\phi^1(\\partial\\cC_1)$ and $\\nu_\\phi^2(\\partial\\cC_2)$, i.e.~$\\LF_\\tau=\\iint_{\\R_+^2}\\LF_\\tau(\\ell_1,\\ell_2)d\\ell_1d\\ell_2$. The following result from~\\cite{ARS-Annulus} gives the exact solvability of $|\\LF_\\tau(\\ell_1,\\ell_2)|$, which is based on~\\cite{wu-annulus}.\n\n\\begin{proposition}[{\\cite[Equation (3.6)]{ARS-Annulus}}]\n \\label{prop-LF-integrability}\nFor $\\tau>0$ and $y\\in(-1,\\frac{4}{\\gamma^2})$, we have\n$$\n\\iint_{\\R_+^2}\\ell_1e^{-\\ell_1}\\ell_2^{y}|\\LF_\\tau(\\ell_1,\\ell_2)|d\\ell_1d\\ell_2=\\frac{\\pi\\gamma y\\Gamma(1+y)}{2\\sin(\\frac{\\gamma^2}{4}\\pi y)}e^{\\frac{\\pi}{4}\\gamma^2\\tau y^2}.\n$$\n\\end{proposition}\n\nFor two domains $D,\\wt{D}\\subset\\C$ and a conformal map $g:D\\to\\wt{D}$, when $h$ is a distribution on $D$, we define $g\\bullet h:=h\\circ g^{-1}+Q\\log|(g^{-1})'|$ (which is a distribution on $\\wt D$). Let $\\Omega$ be the space of simple loops in $\\hH$ surrounding $i$, and $\\Conf(\\hH,i)$ be the group of conformal automorphisms of $\\hH$ that fix $i$. For $(\\phi,\\eta,g,\\theta)\\in H^{-1}(\\hH)\\times\\Omega\\times \\Conf(\\hH,i)\\times[0,1]$, define a measurable map $F$ by\n$$\nF(\\phi,\\eta,g,\\theta):=\\left((g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}, f_{\\eta}^\\theta\\bullet \\phi|_{\\hH\\backslash\\overline{D(\\eta)}},\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right).\n$$\nHere $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$, and\n$f_{\\eta}^\\theta:\\hH\\backslash\\overline{D(\\eta)}\\to\\mathcal{C}_\\tau$ is the conformal map such that $f_{\\eta}^\\theta(0)=\\theta i$ with $\\tau=\\Mod(\\hH\\backslash\\overline{D(\\eta)})$.\n\nLet $\\Haar_{(\\hH,i)}$ be the Haar measure on $\\Conf(\\hH,i)$ such that $|\\Haar_{(\\hH,i)}|=1$, and let $\\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.\nThe following proposition, which is essentially from~\\cite{ARS-Annulus}, describes the law of $\\LF_\\hH^{(\\gamma,i)}$ when cut by a simple loop sampled from $\\sm$.\n\n\\begin{proposition}\\label{prop-sle-weld}\nThere is a constant $C>0$ such that for any $\\ell_1>0$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\gamma,i)}(\\ell_1)\\times\\sm\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nHere $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\\begin{proof}\nBy~\\cite{ARS-Annulus}, the $\\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\\hH$ gives the Liouville field on $\\hH$~\\cite[Theorem 3.4]{ARS-FZZ}.\n\\end{proof}\n\nFor $\\alpha\\in\\R$, let $\\Delta_\\alpha=\\frac{\\alpha}{2}(Q-\\frac{\\alpha}{2})$. Define the measure $\\sm^\\alpha$ via $\\frac{d\\sm^\\alpha}{d\\sm}(\\eta)=\\psi_\\eta'(i)^{2\\Delta_\\alpha-2}$, where $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}. \n\n\\begin{proposition}\n \\label{prop-sle-reweight}\nThere is a constant $C>0$ such that for any $\\ell_1>0$ and $\\alpha\\in\\R$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_1)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nAs in Proposition~\\ref{prop-sle-weld}, we view the right side above as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\n\\begin{proof}\nFor $\\phi\\in H^{-1}(\\hH)$, $\\eta\\in\\Omega$ and $g\\in\\Conf(\\hH,i)$, let $\\phi_1=(g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}$. Recall that for $\\varepsilon>0$, $(\\phi_1)_\\varepsilon(i)$ is the average of $\\phi_1$ over $\\partial B_\\varepsilon(i)$.\nBy~\\cite[Lemma 4.8]{ARS-FZZ},\nas $\\varepsilon\\to0$, we have\n$$\n\\begin{aligned}\n&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi_1)_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n=~&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi\\circ\\psi_\\eta\\circ g^{-1})_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times|\\psi_\\eta'(i)|^{(\\alpha-\\gamma)Q}\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n\\to~& F_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right).\n\\end{aligned}"
    },
    "pre_theorem_intro_text_len": 2839,
    "pre_theorem_intro_text": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.",
    "context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.",
    "full_context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\P[\\Bac_\\varepsilon]$.\n\nLet $f$ be the conformal map from $D(\\ell_1)$ to $\\D$ with $f(0)=0$, $f'(0)>0$ and let $\\rho=f(\\wt\\ell_1)$. Note that $\\CR(\\rho,0)=\\frac{\\CR(\\wt\\ell_1,0)}{\\CR(\\ell_1,0)}$. \nThe main result of this section is the following exact law of $\\CR(\\rho,0)$, which is crucial to the final proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\\begin{proposition}\n\\label{prop-cr-rho}\nFor $\\lambda\\ge0$, we have\n\\begin{equation}\\label{eq:cr-rho}\n\\E[\\CR(\\rho,0)^\\lambda]=\\frac{\\sinh(\\frac{\\pi}{2}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}}-\\frac{2\\sqrt{3}\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right)}.\n\\end{equation}\nWhen $\\lambda\\in[0,\\frac{1}{12})$, the right side of~\\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\\ref{section-intro}.\n\\end{proposition}\n\nLet $\\nu_\\D$ be the law of the loop chosen from the counting measure over $\\{\\ell_i\\}_{i\\ge1}$; namely, $\\nu_\\D$ is such that $\\int F(\\eta)\\nu_\\D(d\\eta)=\\P\\left[\\sum_{i\\ge1}F(\\ell_i)\\right]$ for any bounded measurable function $F$. The following result, based on~\\cite{cfsx} and Theorem~\\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\\nu_\\D$.\n\\begin{lemma}\\label{lem-CR-BM-counting-ell}\nThere is a constant $C>0$ such that for $\\lambda\\ge0$, we have\n\\begin{equation}\n \\label{eq-CR-BM-counting-ell}\n \\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\pi\\sqrt{12\\lambda-1})}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right).\n \\end{equation}\nWhen $\\lambda\\in(0,\\frac{1}{12})$, the right side of~\\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.\n\\end{lemma}\n\\begin{proof}\nBy combining~\\cite[Lemma 5.3]{cfsx} with~\\cite[Theorem 1.3]{cfsx}, we have $\\frac{d\\nu_{\\D}}{d\\SLE^{\\lp}_{8/3,\\D}}(\\eta)=\\frac{C_1}{\\Mod(\\eta,\\S^1)}$ for some constant $C_1>0$.\nThen by Theorem~\\ref{thm-cr-mod-relation}, we find\n\\begin{equation}\\label{eq:cr-integral}\n\\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\n \\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau.\n\\end{equation}\nfor some constant $C>0$. \nNote that for $a>-\\frac{\\pi}{6}$, we have $\\int_0^\\infty e^{-a\\tau}\\eta(2i\\tau)d\\tau=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}$ (see e.g.~\\cite[Equation (A.4)]{ARS-Annulus}) and\n $$\n \\frac{d}{da}\\left[-\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{(2-\\sqrt{3})^2+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right)\\right]=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}.\n $$\nTherefore, we obtain\n $$\n \\int_0^\\infty e^{-a\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau=\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right).\n $$\nfor $a>-\\frac{\\pi}{6}$. Combined with the right side of~\\eqref{eq:cr-integral}, we conclude.\n\\end{proof}\n\nThe upper bound of $\\P[\\mathsf{Bac}_\\varepsilon]$ then relies on the following iterative inequality.\n\\begin{proposition}\n \\label{prop-upper-bound-backbone}\n Let $q_0=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n there is a constant $C_0>0$ such that for any $\\varepsilon\\in(0,10^{-4})$,\n $$\n \\P[\\Bac_\\varepsilon]\\le \\frac{C_0}{\\log|\\log\\varepsilon|}+q_0\\P[\\Bac_{5\\sqrt{\\varepsilon}}].\n $$\n\\end{proposition}\n\\begin{proof}\nNote that\n \\begin{align}\n \\P[\\Bac_\\varepsilon]&\\le\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &\\le \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &= \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset, \\Bac_\\varepsilon]\\label{eq:upper-1}\n \\end{align}\nHere the last line follows from that $(\\wt\\ell_1\\cap(5\\sqrt{\\varepsilon}\\D\\cup\\A_{1/2})=\\emptyset)\\cap\\Bac_\\varepsilon=\\emptyset$.\nBy Lemma~\\ref{lem-CR-BM-ell1}, Corollary~\\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that\n\\begin{align}\n\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]&\\le\\P[\\CR(\\ell_1,0)<20\\sqrt{\\varepsilon}]+\\P[\\CR(\\wt\\ell_1,0)<20\\sqrt{\\varepsilon}]\\nonumber\\\\\n & \\le C\\varepsilon^{\\frac{1}{8}}+C'\\frac{1}{\\log|\\log\\varepsilon|}\\le \\frac{C_0}{\\log|\\log\\varepsilon|}.\\label{eq:upper-2}\n\\end{align}\n\nLet $q_0:=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n\\begin{equation}\\label{eq:upper-3}\n\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset,\\Bac_\\varepsilon]\\le q_0\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset].\n\\end{equation}\nLet $g:D(\\wt\\ell_1)\\to\\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.\nBy Lemma~\\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\\in[0,s_1]}$ is independent of $B[s_1,\\tau_\\D]$, and has the same law (up to a time change) as $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$. Hence, if we denote $\\wt\\P$ to be the law of $(g(B_t))_{t\\in[0,s_1]}$ conditioned on the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$, then $\\wt\\P$ is still the same as (up to a time change) the law of $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$.\nBy Lemma~\\ref{lem-variant-Koebe-1/4}, \non the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$,\nwe have $\\overline{g(\\varepsilon\\D)}\\subset5\\sqrt{\\varepsilon}\\D$ and \n$g(\\frac{1}{2}\\D\\cap D(\\wt\\ell_1))\\supset \\frac{1}{2}\\D$. Then we find\n\\begin{equation}\\label{eq:upper-4}\n\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\le \\wt\\P[\\Bac_{5\\sqrt{\\varepsilon}}]=\\P[\\Bac_{5\\sqrt{\\varepsilon}}],\n\\end{equation}\nwhere the last equality follows from the rotational invariance of $B[0,\\tau_\\D]$. The result then follows from combining~\\eqref{eq:upper-1},~\\eqref{eq:upper-2},~\\eqref{eq:upper-3} and~\\eqref{eq:upper-4}.\n\\end{proof}\n\nNow we prove the upper bound for $\\P[\\Bac_\\varepsilon]$, finishing the proof of Theorem~\\ref{thm:backbone}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:backbone}, the upper bound]\nLet $\\varepsilon\\in(0,10^{-4})$ and $F(\\varepsilon):=\\P[\\Bac_\\varepsilon]$. Note that the function $x\\mapsto\\frac{1}{\\log|\\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\\frac{1}{\\log|\\log(25x)|}\\le\\frac{C}{\\log|\\log x|}$ for any $x\\in(0,10^{-4})$. Then for each $n\\in\\mathbbm{Z}_+$ with $25\\varepsilon^{\\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\\ref{prop-upper-bound-backbone} for $n$ times, we find\n\\begin{align}\n F(\\varepsilon)\n &\\le C_0\\sum\\limits_{k=0}^{n-1}\\frac{q_0^k}{\\log|\\log(5^{2-\\frac{1}{2^{k-1}}}\\varepsilon^{\\frac{1}{2^k}})|}+q_0^nF(5^{2-\\frac{1}{2^{n-1}}}\\varepsilon^{\\frac{1}{2^n}})\\nonumber\\\\\n &\\le \\frac{C_0}{1-q_0}\\frac{1}{\\log|\\log(25\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\le\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\nonumber\\\\\n &=\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n.\\label{eq-proof-upper-bound-backbone}\n\\end{align}",
    "post_theorem_intro_text_len": 7739,
    "post_theorem_intro_text": "\\begin{remark}\nThe same result also holds when $\\frac{1}{2}\\mathbbm{S}^1$ in~\\eqref{eq:def-backbone} is changed to any fixed $r\\mathbbm{S}^1$ with $r\\in(0,1)$ (and for sufficiently small $\\varepsilon$), except that the corresponding constants in~\\eqref{eq:bb-proba} will depend on $r$.\n\\end{remark}\n\nNote that the backbone event $\\Bac_\\varepsilon$ is closely related to the spatial distribution of the \\emph{cut points} of $B[0,\\tau_\\mathbbm{D}]$. Indeed, our proof of Theorem~\\ref{thm:backbone} is based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}. We mention that~\\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\\ref{thm:backbone} without relying on LQG.\n\nTheorem~\\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let\n\\[\n\\mathcal{B}:=\\{B_s:s\\in[0,\\tau_\\mathbbm{D}) \\text{ and there exists } \\varepsilon>0 \\text{ such that } (B_{s+u})_{0\\le u\\le\\varepsilon} \\text{ does not have a cut point}\\}.\n\\]\nThen Theorem~\\ref{thm:backbone} indicates that $\\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\\mathcal{B}$ with the gauge $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\\tau_\\mathbbm{D}]$ itself is $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$, see e.g.~\\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\\cite{BL}, we know that $\\mathcal{B}$ contains the set of double points of $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$, which has the Hausdorff gauge $r\\mapsto r^2(\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r})^2$~\\cite{LG87}.\n\n\\subsection{Outlook and discussions}\n\\label{section-discussion}\n\nHere we give several remarks and related questions before going into the proof.\n\n\\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\setlength{\\parsep}{0pt}\n\n\\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\\tau_\\mathbbm{D}]$ joining $\\varepsilon\\mathbbm{S}^1$ and $\\frac{1}{2}\\mathbbm{S}^1$ with $n\\ge2$. Note that for $n\\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\\tau_\\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\\tau_\\mathbbm{D}]$, $0$ and $B_{\\tau_\\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\\ge3$, see~\\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.\n\n\\item \nThere is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\\ge0}$ be the Brownian excursion on the upper half plane $\\mathbbm{H}$ from $0$ to $\\infty$, and define $R\\mathbbm{S}^+:=R\\mathbbm{S}^1\\cap\\mathbbm{H}$ for $R>0$. Then for $n\\ge1$, consider the asymptotic probability that \nthere exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\\mathbbm{S}^+$ and $R\\mathbbm{S}^+$ as $R\\to\\infty$.\nNote that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\\cite[Theorem 4]{brownian-beads}.\n\n\\item In our forthcoming work~\\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$ be as before, and let $\\mathcal{L}$ be an independent Brownian loop soup on $\\mathbbm{D}$ with intensity $\\frac{c}{2}$ for $c\\in(0,1]$. \nFor a subset $A\\subset\\ol\\mathbbm{D}$, denote $\\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\\mathcal{L}$ intersecting with $A$, \nand let $H:=\\mathcal{C}(B[0,\\tau_\\mathbbm{D}])$. \nHowever, unlike the Brownian motion case, \nthere is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$\nconnected to $\\frac{1}{2}\\mathbbm{S}^1$. Moreover, in~\\cite{cx} we will show the following\n\\begin{theorem}\nWe say $t\\in(0,\\tau_\\mathbbm{D}]$ is a cut time of $H$ if $\\mathcal{C}(B[0,t])\\cap \\mathcal{C}(B[t,\\tau_\\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$. \nThen for $c\\in(0,1)$,\n\\begin{equation}\\label{eq:p_e}\n\\mathbbm{P}[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}\nwhere $\\eta(2i\\tau):=e^{-\\frac{\\pi}{6}\\tau}\\prod_{n=1}^{\\infty}\\left(1-e^{-4\\pi n\\tau}\\right)$ is the Dedekind eta function.\n\\end{theorem}\nNote that the right side of~\\eqref{eq:p_e} tends to $1$ as $c\\uparrow1$, and tends to $0$ as $c\\downarrow0$.\n\n\\end{itemize}\n\n\\medskip\n\\noindent\\textbf{Organization of the paper.} In Section~\\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops. Such relation was implicitly established in~\\cite{ARS-Annulus}, and we will provide its proof in Appendix~\\ref{appendix} for completeness. Then in Section~\\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and solve it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\n\\medskip\n\\noindent\\textbf{Basic notations.}\nFor two compact sets $A,B\\subset\\mathbbm{C}$, let $\\mathrm{dist}(A,B):=\\inf\\{|a-b|:a\\in A,b\\in B\\}$.\nFor a simple loop $\\ell\\subset\\mathbbm{C}$, let $D(\\ell)$ be the bounded connected component of $\\mathbbm{C}\\backslash\\ell$. For a simple loop $\\ell\\subset\\mathbbm{C}$ with $0\\in D(\\ell)$, we denote the conformal radius of $D(\\ell)$ seen from 0 by $\\mathrm{CR}(\\ell,0)$, i.e. if $f:\\mathbbm{D}\\to D(\\ell)$ is a conformal map that fixes the origin, then $\\mathrm{CR}(\\ell,0)=|f'(0)|$.\n\nFor $0<r<1$, let $\\A_r$ be the standard annulus $\\{z\\in\\mathbbm{C}: r<|z|<1\\}$. For each annular domain $A\\subset\\mathbbm{C}$,\nthere exists a unique $\\tau>0$ such that $A$ and $\\A_{e^{-2\\pi\\tau}}$ are conformally equivalent. We call $\\tau$ the \\textit{modulus} of $A$ and write $\\mathrm{Mod}(A):=\\tau$. For two simple loops $\\eta_1,\\eta_2$ such that $\\overline{D(\\eta_1)}\\subset D(\\eta_2)$, we also write the modulus of the annular domain $D(\\eta_2)\\setminus\\overline{D(\\eta_1)}$ as $\\mathrm{Mod}(\\eta_1,\\eta_2)$ for simplicity.\n\nWe will frequently deal with elementary functions such as $\\sqrt{x},\\sinh(\\sqrt{x}),\\cosh(\\sqrt{x})$, and $\\tanh(\\sqrt{x})$. In the following, we view them as functions defined on $\\mathbbm{R}$ by taking $i\\sqrt{|x|},i\\sin(\\sqrt{|x|}),\\cos(\\sqrt{|x|})$, and $i\\tan(\\sqrt{|x|})$ for $x<0$, respectively.\n\n\\medskip\n\\noindent\\textbf{Acknowledgment.}\nWe are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.\nG.C. and Z.X.\\ were partially supported by National Key R\\&D Program of China (No.\\ 2023YFA1010700). G.C. was partially supported by National Key R\\&D Program of China (No. 2021YFA1002700).",
    "sketch": "Our proof of Theorem~\\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; Section~\\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.”",
    "expanded_sketch": "Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”",
    "expanded_theorem": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $(B_t)_{t\\ge 0}$ be planar Brownian motion started from $0$, let $\\mathbb D=\\{z\\in\\mathbb C:|z|<1\\}$, and let $\\tau_{\\mathbb D}$ be the first time that $B_t$ hits the unit circle $\\mathbb S^1=\\{z\\in\\mathbb C:|z|=1\\}$. For $\\varepsilon\\in(0,\\tfrac12)$, define the backbone event\n\\[\n\\mathrm{Bac}_\\varepsilon=\n\\Big\\{\\text{there exist two disjoint subpaths of }B[0,\\tau_{\\mathbb D}]\\text{ joining }\\varepsilon\\mathbb S^1\\text{ and }\\tfrac12\\mathbb S^1\\Big\\}.\n\\]\nEquivalently, there exist continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\\tau_{\\mathbb D}]$, such that $\\gamma^i(0)\\in \\varepsilon\\mathbb S^1$ and $\\gamma^i(1)\\in \\tfrac12\\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\\mathbb P[\\mathrm{Bac}_\\varepsilon]$ for small $\\varepsilon$?",
      "correct_choice": {
        "label": "A",
        "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\frac{C_1}{\\log|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\nC_1\\,\\frac{1}{|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le C_2\\,\\frac{1}{|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant $C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "D",
          "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac12)$,\n\\[\n\\frac{C_1}{\\log|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every $\\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\frac{C_1}{(\\log|\\log \\varepsilon|)^{1+\\delta}}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{(\\log|\\log \\varepsilon|)^{1-\\delta}}.\n\\]"
        }
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      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "iterated_log_scale",
          "template_used": "property_confusion"
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        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_lower_bound",
          "template_used": "weaker_true"
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          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "epsilon_range_(0,1/10)",
          "template_used": "boundary_range"
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          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "exact_iterated_log_order",
          "template_used": "wildcard"
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    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the event carefully but does not reveal the asymptotic form of the probability. The correct iterated-log scale is not stated or strongly hinted at in the setup."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: it asks which explicit probabilistic estimate holds, and the correct choice is the exact theorem statement up to constants. That makes it close to a direct restatement rather than a conceptually transformed application."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish exact truth from nearby alternatives (single log vs iterated log, two-sided vs one-sided bounds, validity range). However, the item mainly rewards recognition of the precise result rather than generating a conclusion from underlying principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are mathematically close to the true statement and reflect realistic confusions such as wrong decay scale, a weaker true statement, overextended parameter range, and an unjustified stronger upper bound."
      },
      "total_score": 5,
      "overall_assessment": "A solid recognition-based theorem-checking MCQ with high-quality distractors and no answer leakage, but it is largely a direct restatement of a known result and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.09683v1",
    "paper_link": "http://arxiv.org/abs/2512.09683v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
      "start_pos": 32731,
      "end_pos": 33011,
      "label": "thm:backbone"
    },
    "ref_dict": {
      "eq:p_e": "\\begin{equation}\\label{eq:p_e}\n\\P[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}",
      "thm:backbone": "\\begin{theorem}\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}\n\\end{theorem}",
      "eq:def-backbone": "\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\D}] \\text{ joining } \\varepsilon\\S^1 \\text{ and } \\frac{1}{2}\\S^1\\}.\n\\end{equation}",
      "eq:bb-proba": "\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
      "appendix": "\\begin{aligned}\nF(\\varepsilon)&\\le \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n\\\\\n&\\le  \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-\\frac{\\log2}{|\\log q_0|}(\\log\\log|\\log\\varepsilon|-\\log C')}+\\frac{C'}{\\log|\\log\\varepsilon|}\n\\le\\frac{C_2}{\\log|\\log\\varepsilon|}\n\\end{aligned}\n$$\nfor some positive constant $C_2$, as desired.\n\\end{proof}\n\n\\appendix\n\\section{Proof of Theorem~\\ref{thm-cr-mod-relation}}\n\\label{appendix}\n\nIn this appendix, we provide the proof of Theorem~\\ref{thm-cr-mod-relation}, which is implicit in~\\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\\hH:=\\{z\\in\\C:\\text{Im}(z)>0\\}$, and let $\\sm$ be the restriction of the $\\SLE_{8/3}$ loop measure on $\\C$ to the loops that are contained in $\\hH$ and surround $i$. According to the conformal restriction~\\cite{werner-loops}, if $\\Phi:\\D\\to\\hH$ is a conformal map with $\\Phi(0)=i$, then $\\sm=\\Phi_*\\SLE_{8/3,\\D}^{\\lp}$. Therefore, to prove Theorem~\\ref{thm-cr-mod-relation}, it suffices to show the following\n\\begin{theorem}\n\\label{thm-appendix}\nThere is a constant $C>0$ such that for any measurable function $f:\\R_+\\to\\R_+$ and $\\lambda\\ge0$, we have\n$$\n\\int f\\left(\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right)\\psi_\\eta'(i)^\\lambda\\sm(d\\eta)=\nC\\frac{\\sqrt{12\\lambda-1}}{\\sinh\\left(\\frac{\\pi}{3}\\sqrt{12\\lambda-1}\\right)}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\eta(2i\\tau)f(\\tau)d \\tau.\n$$\nHere $\\psi_\\eta$ is the conformal map from $\\hH$ to $D(\\eta)$ such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\\sqrt{\\frac{8}{3}}$. In the following we fix $\\gamma=\\sqrt{\\frac{8}{3}}$ and let $Q=\\frac{\\gamma}{2}+\\frac{2}{\\gamma}$. Let $\\P_{\\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\\hH$ with mean zero on the upper semi-circle $\\S^1\\cap\\hH$. \n\n\\begin{definition}[Liouville field on $\\hH$]\n\\label{def-LF-H}\nLet $(h,c)$ be sampled from $\\P_{\\hH}\\times[e^{-Qc}dc]$, and let $\\phi(z)=h(z)-2Q\\log|z|_++c$ (here $|z|_+:=\\max\\{|z|,1\\}$). Denote the law of $\\phi$ by $\\LF_{\\hH}$.\n\nFor $\\alpha\\in\\R$, let $\\LF_{\\hH}^{(\\alpha,i)}(d\\phi):=\\lim_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\alpha^2}{2}}e^{\\alpha\\phi_\\varepsilon(i)}\\LF_{\\hH}(d\\phi)$ (the limit exists in the vague topology, see~\\cite[Lemma 2.2]{ARS-FZZ}). Here $\\phi_\\varepsilon(i)$ is the average of $\\phi$ over the circle $\\partial B_\\varepsilon(i)$.\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_{\\hH}^{(\\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\\nu_{\\phi}:=\\lim\\limits_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\gamma^2}{4}}e^{\\frac{\\gamma}{2}\\phi_\\varepsilon(x)}dx$, where $\\phi_\\varepsilon(x)$ is the average of $\\phi$ over the semi-circle $\\partial B_\\varepsilon(x)\\cap\\hH$.\nThen denote $\\{\\LF_{\\hH}^{(\\alpha,i)}(\\ell)\\}_{\\ell>0}$ to be the disintegration of $\\LF_{\\hH}^{(\\alpha,i)}$ over $\\nu_\\phi(\\R)$; i.e.~$\\LF_{\\hH}^{(\\alpha,i)}=\\int_0^\\infty\\LF_{\\hH}^{(\\alpha,i)}(\\ell)d\\ell$. We refer readers to see e.g.~\\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\\cite[Lemma 2.7]{ARS-FZZ}, for $\\alpha>\\frac{\\gamma}{2}$,\nthere are constants $C_{\\alpha,\\gamma}>0$ such that $|\\LF_{\\hH}^{(\\alpha,i)}(\\ell)|=C_{\\alpha,\\gamma}\\ell^{\\frac{2}{\\gamma}(\\alpha-Q)-1}$ for any $\\ell>0$.\n\nWe also need the Liouville field on the annulus. For $\\tau>0$, let $\\mathcal{C}_\\tau=[0,\\tau]\\times[0,1]/\\mathord\\sim$ be the horizontal cylinder with modulus $\\tau$, where under $\\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\\in[0,\\tau]$. Let $\\P_{\\tau}$ be the free boundary GFF on $\\mathcal{C}_\\tau$\nwith zero mean on the circle $\\{\\frac{\\tau}{2}\\}\\times[0,1]/\\mathord\\sim$.\n\\begin{definition}[{\\cite[Definition 2.2]{ARS-Annulus}}]\n\\label{def-LF-annulus}\nLet $(h,c)$ be sampled from $\\P_\\tau\\times dc$. Then denote the law of $\\phi=h+c$ by $\\LF_\\tau$ (which is an infinite measure on $H^{-1}(\\mathcal{C}_\\tau)$).\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_\\tau$,\nwe can similarly define its boundary GMC measures $\\nu^1_\\phi, \\nu_\\phi^2$ on the two boundaries $\\partial_1\\mathcal{C}_\\tau=\\{0\\}\\times[0,1]\\mathord\\sim$ and $\\partial_2\\mathcal{C}_\\tau=\\{1\\}\\times[0,1]\\mathord\\sim$, respectively. We also let $\\{\\LF_\\tau(\\ell_1,\\ell_2)\\}_{\\ell_1,\\ell_2>0}$ be the disintegration of $\\LF_\\tau$ over $\\nu_\\phi^1(\\partial\\cC_1)$ and $\\nu_\\phi^2(\\partial\\cC_2)$, i.e.~$\\LF_\\tau=\\iint_{\\R_+^2}\\LF_\\tau(\\ell_1,\\ell_2)d\\ell_1d\\ell_2$. The following result from~\\cite{ARS-Annulus} gives the exact solvability of $|\\LF_\\tau(\\ell_1,\\ell_2)|$, which is based on~\\cite{wu-annulus}.\n\n\\begin{proposition}[{\\cite[Equation (3.6)]{ARS-Annulus}}]\n \\label{prop-LF-integrability}\nFor $\\tau>0$ and $y\\in(-1,\\frac{4}{\\gamma^2})$, we have\n$$\n\\iint_{\\R_+^2}\\ell_1e^{-\\ell_1}\\ell_2^{y}|\\LF_\\tau(\\ell_1,\\ell_2)|d\\ell_1d\\ell_2=\\frac{\\pi\\gamma y\\Gamma(1+y)}{2\\sin(\\frac{\\gamma^2}{4}\\pi y)}e^{\\frac{\\pi}{4}\\gamma^2\\tau y^2}.\n$$\n\\end{proposition}\n\nFor two domains $D,\\wt{D}\\subset\\C$ and a conformal map $g:D\\to\\wt{D}$, when $h$ is a distribution on $D$, we define $g\\bullet h:=h\\circ g^{-1}+Q\\log|(g^{-1})'|$ (which is a distribution on $\\wt D$). Let $\\Omega$ be the space of simple loops in $\\hH$ surrounding $i$, and $\\Conf(\\hH,i)$ be the group of conformal automorphisms of $\\hH$ that fix $i$. For $(\\phi,\\eta,g,\\theta)\\in H^{-1}(\\hH)\\times\\Omega\\times \\Conf(\\hH,i)\\times[0,1]$, define a measurable map $F$ by\n$$\nF(\\phi,\\eta,g,\\theta):=\\left((g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}, f_{\\eta}^\\theta\\bullet \\phi|_{\\hH\\backslash\\overline{D(\\eta)}},\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right).\n$$\nHere $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$, and\n$f_{\\eta}^\\theta:\\hH\\backslash\\overline{D(\\eta)}\\to\\mathcal{C}_\\tau$ is the conformal map such that $f_{\\eta}^\\theta(0)=\\theta i$ with $\\tau=\\Mod(\\hH\\backslash\\overline{D(\\eta)})$.\n\nLet $\\Haar_{(\\hH,i)}$ be the Haar measure on $\\Conf(\\hH,i)$ such that $|\\Haar_{(\\hH,i)}|=1$, and let $\\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.\nThe following proposition, which is essentially from~\\cite{ARS-Annulus}, describes the law of $\\LF_\\hH^{(\\gamma,i)}$ when cut by a simple loop sampled from $\\sm$.\n\n\\begin{proposition}\\label{prop-sle-weld}\nThere is a constant $C>0$ such that for any $\\ell_1>0$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\gamma,i)}(\\ell_1)\\times\\sm\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nHere $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\\begin{proof}\nBy~\\cite{ARS-Annulus}, the $\\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\\hH$ gives the Liouville field on $\\hH$~\\cite[Theorem 3.4]{ARS-FZZ}.\n\\end{proof}\n\nFor $\\alpha\\in\\R$, let $\\Delta_\\alpha=\\frac{\\alpha}{2}(Q-\\frac{\\alpha}{2})$. Define the measure $\\sm^\\alpha$ via $\\frac{d\\sm^\\alpha}{d\\sm}(\\eta)=\\psi_\\eta'(i)^{2\\Delta_\\alpha-2}$, where $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}. \n\n\\begin{proposition}\n \\label{prop-sle-reweight}\nThere is a constant $C>0$ such that for any $\\ell_1>0$ and $\\alpha\\in\\R$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_1)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nAs in Proposition~\\ref{prop-sle-weld}, we view the right side above as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\n\\begin{proof}\nFor $\\phi\\in H^{-1}(\\hH)$, $\\eta\\in\\Omega$ and $g\\in\\Conf(\\hH,i)$, let $\\phi_1=(g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}$. Recall that for $\\varepsilon>0$, $(\\phi_1)_\\varepsilon(i)$ is the average of $\\phi_1$ over $\\partial B_\\varepsilon(i)$.\nBy~\\cite[Lemma 4.8]{ARS-FZZ},\nas $\\varepsilon\\to0$, we have\n$$\n\\begin{aligned}\n&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi_1)_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n=~&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi\\circ\\psi_\\eta\\circ g^{-1})_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times|\\psi_\\eta'(i)|^{(\\alpha-\\gamma)Q}\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n\\to~& F_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right).\n\\end{aligned}"
    },
    "pre_theorem_intro_text_len": 2839,
    "pre_theorem_intro_text": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.",
    "context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.",
    "full_context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\P[\\Bac_\\varepsilon]$.\n\nLet $f$ be the conformal map from $D(\\ell_1)$ to $\\D$ with $f(0)=0$, $f'(0)>0$ and let $\\rho=f(\\wt\\ell_1)$. Note that $\\CR(\\rho,0)=\\frac{\\CR(\\wt\\ell_1,0)}{\\CR(\\ell_1,0)}$. \nThe main result of this section is the following exact law of $\\CR(\\rho,0)$, which is crucial to the final proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\\begin{proposition}\n\\label{prop-cr-rho}\nFor $\\lambda\\ge0$, we have\n\\begin{equation}\\label{eq:cr-rho}\n\\E[\\CR(\\rho,0)^\\lambda]=\\frac{\\sinh(\\frac{\\pi}{2}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}}-\\frac{2\\sqrt{3}\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right)}.\n\\end{equation}\nWhen $\\lambda\\in[0,\\frac{1}{12})$, the right side of~\\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\\ref{section-intro}.\n\\end{proposition}\n\nLet $\\nu_\\D$ be the law of the loop chosen from the counting measure over $\\{\\ell_i\\}_{i\\ge1}$; namely, $\\nu_\\D$ is such that $\\int F(\\eta)\\nu_\\D(d\\eta)=\\P\\left[\\sum_{i\\ge1}F(\\ell_i)\\right]$ for any bounded measurable function $F$. The following result, based on~\\cite{cfsx} and Theorem~\\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\\nu_\\D$.\n\\begin{lemma}\\label{lem-CR-BM-counting-ell}\nThere is a constant $C>0$ such that for $\\lambda\\ge0$, we have\n\\begin{equation}\n \\label{eq-CR-BM-counting-ell}\n \\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\pi\\sqrt{12\\lambda-1})}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right).\n \\end{equation}\nWhen $\\lambda\\in(0,\\frac{1}{12})$, the right side of~\\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.\n\\end{lemma}\n\\begin{proof}\nBy combining~\\cite[Lemma 5.3]{cfsx} with~\\cite[Theorem 1.3]{cfsx}, we have $\\frac{d\\nu_{\\D}}{d\\SLE^{\\lp}_{8/3,\\D}}(\\eta)=\\frac{C_1}{\\Mod(\\eta,\\S^1)}$ for some constant $C_1>0$.\nThen by Theorem~\\ref{thm-cr-mod-relation}, we find\n\\begin{equation}\\label{eq:cr-integral}\n\\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\n \\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau.\n\\end{equation}\nfor some constant $C>0$. \nNote that for $a>-\\frac{\\pi}{6}$, we have $\\int_0^\\infty e^{-a\\tau}\\eta(2i\\tau)d\\tau=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}$ (see e.g.~\\cite[Equation (A.4)]{ARS-Annulus}) and\n $$\n \\frac{d}{da}\\left[-\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{(2-\\sqrt{3})^2+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right)\\right]=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}.\n $$\nTherefore, we obtain\n $$\n \\int_0^\\infty e^{-a\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau=\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right).\n $$\nfor $a>-\\frac{\\pi}{6}$. Combined with the right side of~\\eqref{eq:cr-integral}, we conclude.\n\\end{proof}\n\nThe upper bound of $\\P[\\mathsf{Bac}_\\varepsilon]$ then relies on the following iterative inequality.\n\\begin{proposition}\n \\label{prop-upper-bound-backbone}\n Let $q_0=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n there is a constant $C_0>0$ such that for any $\\varepsilon\\in(0,10^{-4})$,\n $$\n \\P[\\Bac_\\varepsilon]\\le \\frac{C_0}{\\log|\\log\\varepsilon|}+q_0\\P[\\Bac_{5\\sqrt{\\varepsilon}}].\n $$\n\\end{proposition}\n\\begin{proof}\nNote that\n \\begin{align}\n \\P[\\Bac_\\varepsilon]&\\le\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &\\le \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &= \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset, \\Bac_\\varepsilon]\\label{eq:upper-1}\n \\end{align}\nHere the last line follows from that $(\\wt\\ell_1\\cap(5\\sqrt{\\varepsilon}\\D\\cup\\A_{1/2})=\\emptyset)\\cap\\Bac_\\varepsilon=\\emptyset$.\nBy Lemma~\\ref{lem-CR-BM-ell1}, Corollary~\\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that\n\\begin{align}\n\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]&\\le\\P[\\CR(\\ell_1,0)<20\\sqrt{\\varepsilon}]+\\P[\\CR(\\wt\\ell_1,0)<20\\sqrt{\\varepsilon}]\\nonumber\\\\\n & \\le C\\varepsilon^{\\frac{1}{8}}+C'\\frac{1}{\\log|\\log\\varepsilon|}\\le \\frac{C_0}{\\log|\\log\\varepsilon|}.\\label{eq:upper-2}\n\\end{align}\n\nLet $q_0:=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n\\begin{equation}\\label{eq:upper-3}\n\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset,\\Bac_\\varepsilon]\\le q_0\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset].\n\\end{equation}\nLet $g:D(\\wt\\ell_1)\\to\\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.\nBy Lemma~\\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\\in[0,s_1]}$ is independent of $B[s_1,\\tau_\\D]$, and has the same law (up to a time change) as $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$. Hence, if we denote $\\wt\\P$ to be the law of $(g(B_t))_{t\\in[0,s_1]}$ conditioned on the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$, then $\\wt\\P$ is still the same as (up to a time change) the law of $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$.\nBy Lemma~\\ref{lem-variant-Koebe-1/4}, \non the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$,\nwe have $\\overline{g(\\varepsilon\\D)}\\subset5\\sqrt{\\varepsilon}\\D$ and \n$g(\\frac{1}{2}\\D\\cap D(\\wt\\ell_1))\\supset \\frac{1}{2}\\D$. Then we find\n\\begin{equation}\\label{eq:upper-4}\n\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\le \\wt\\P[\\Bac_{5\\sqrt{\\varepsilon}}]=\\P[\\Bac_{5\\sqrt{\\varepsilon}}],\n\\end{equation}\nwhere the last equality follows from the rotational invariance of $B[0,\\tau_\\D]$. The result then follows from combining~\\eqref{eq:upper-1},~\\eqref{eq:upper-2},~\\eqref{eq:upper-3} and~\\eqref{eq:upper-4}.\n\\end{proof}\n\nNow we prove the upper bound for $\\P[\\Bac_\\varepsilon]$, finishing the proof of Theorem~\\ref{thm:backbone}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:backbone}, the upper bound]\nLet $\\varepsilon\\in(0,10^{-4})$ and $F(\\varepsilon):=\\P[\\Bac_\\varepsilon]$. Note that the function $x\\mapsto\\frac{1}{\\log|\\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\\frac{1}{\\log|\\log(25x)|}\\le\\frac{C}{\\log|\\log x|}$ for any $x\\in(0,10^{-4})$. Then for each $n\\in\\mathbbm{Z}_+$ with $25\\varepsilon^{\\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\\ref{prop-upper-bound-backbone} for $n$ times, we find\n\\begin{align}\n F(\\varepsilon)\n &\\le C_0\\sum\\limits_{k=0}^{n-1}\\frac{q_0^k}{\\log|\\log(5^{2-\\frac{1}{2^{k-1}}}\\varepsilon^{\\frac{1}{2^k}})|}+q_0^nF(5^{2-\\frac{1}{2^{n-1}}}\\varepsilon^{\\frac{1}{2^n}})\\nonumber\\\\\n &\\le \\frac{C_0}{1-q_0}\\frac{1}{\\log|\\log(25\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\le\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\nonumber\\\\\n &=\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n.\\label{eq-proof-upper-bound-backbone}\n\\end{align}",
    "post_theorem_intro_text_len": 7739,
    "post_theorem_intro_text": "\\begin{remark}\nThe same result also holds when $\\frac{1}{2}\\mathbbm{S}^1$ in~\\eqref{eq:def-backbone} is changed to any fixed $r\\mathbbm{S}^1$ with $r\\in(0,1)$ (and for sufficiently small $\\varepsilon$), except that the corresponding constants in~\\eqref{eq:bb-proba} will depend on $r$.\n\\end{remark}\n\nNote that the backbone event $\\Bac_\\varepsilon$ is closely related to the spatial distribution of the \\emph{cut points} of $B[0,\\tau_\\mathbbm{D}]$. Indeed, our proof of Theorem~\\ref{thm:backbone} is based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}. We mention that~\\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\\ref{thm:backbone} without relying on LQG.\n\nTheorem~\\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let\n\\[\n\\mathcal{B}:=\\{B_s:s\\in[0,\\tau_\\mathbbm{D}) \\text{ and there exists } \\varepsilon>0 \\text{ such that } (B_{s+u})_{0\\le u\\le\\varepsilon} \\text{ does not have a cut point}\\}.\n\\]\nThen Theorem~\\ref{thm:backbone} indicates that $\\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\\mathcal{B}$ with the gauge $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\\tau_\\mathbbm{D}]$ itself is $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$, see e.g.~\\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\\cite{BL}, we know that $\\mathcal{B}$ contains the set of double points of $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$, which has the Hausdorff gauge $r\\mapsto r^2(\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r})^2$~\\cite{LG87}.\n\n\\subsection{Outlook and discussions}\n\\label{section-discussion}\n\nHere we give several remarks and related questions before going into the proof.\n\n\\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\setlength{\\parsep}{0pt}\n\n\\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\\tau_\\mathbbm{D}]$ joining $\\varepsilon\\mathbbm{S}^1$ and $\\frac{1}{2}\\mathbbm{S}^1$ with $n\\ge2$. Note that for $n\\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\\tau_\\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\\tau_\\mathbbm{D}]$, $0$ and $B_{\\tau_\\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\\ge3$, see~\\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.\n\n\\item \nThere is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\\ge0}$ be the Brownian excursion on the upper half plane $\\mathbbm{H}$ from $0$ to $\\infty$, and define $R\\mathbbm{S}^+:=R\\mathbbm{S}^1\\cap\\mathbbm{H}$ for $R>0$. Then for $n\\ge1$, consider the asymptotic probability that \nthere exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\\mathbbm{S}^+$ and $R\\mathbbm{S}^+$ as $R\\to\\infty$.\nNote that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\\cite[Theorem 4]{brownian-beads}.\n\n\\item In our forthcoming work~\\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$ be as before, and let $\\mathcal{L}$ be an independent Brownian loop soup on $\\mathbbm{D}$ with intensity $\\frac{c}{2}$ for $c\\in(0,1]$. \nFor a subset $A\\subset\\ol\\mathbbm{D}$, denote $\\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\\mathcal{L}$ intersecting with $A$, \nand let $H:=\\mathcal{C}(B[0,\\tau_\\mathbbm{D}])$. \nHowever, unlike the Brownian motion case, \nthere is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$\nconnected to $\\frac{1}{2}\\mathbbm{S}^1$. Moreover, in~\\cite{cx} we will show the following\n\\begin{theorem}\nWe say $t\\in(0,\\tau_\\mathbbm{D}]$ is a cut time of $H$ if $\\mathcal{C}(B[0,t])\\cap \\mathcal{C}(B[t,\\tau_\\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$. \nThen for $c\\in(0,1)$,\n\\begin{equation}\\label{eq:p_e}\n\\mathbbm{P}[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}\nwhere $\\eta(2i\\tau):=e^{-\\frac{\\pi}{6}\\tau}\\prod_{n=1}^{\\infty}\\left(1-e^{-4\\pi n\\tau}\\right)$ is the Dedekind eta function.\n\\end{theorem}\nNote that the right side of~\\eqref{eq:p_e} tends to $1$ as $c\\uparrow1$, and tends to $0$ as $c\\downarrow0$.\n\n\\end{itemize}\n\n\\medskip\n\\noindent\\textbf{Organization of the paper.} In Section~\\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops. Such relation was implicitly established in~\\cite{ARS-Annulus}, and we will provide its proof in Appendix~\\ref{appendix} for completeness. Then in Section~\\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and solve it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\n\\medskip\n\\noindent\\textbf{Basic notations.}\nFor two compact sets $A,B\\subset\\mathbbm{C}$, let $\\mathrm{dist}(A,B):=\\inf\\{|a-b|:a\\in A,b\\in B\\}$.\nFor a simple loop $\\ell\\subset\\mathbbm{C}$, let $D(\\ell)$ be the bounded connected component of $\\mathbbm{C}\\backslash\\ell$. For a simple loop $\\ell\\subset\\mathbbm{C}$ with $0\\in D(\\ell)$, we denote the conformal radius of $D(\\ell)$ seen from 0 by $\\mathrm{CR}(\\ell,0)$, i.e. if $f:\\mathbbm{D}\\to D(\\ell)$ is a conformal map that fixes the origin, then $\\mathrm{CR}(\\ell,0)=|f'(0)|$.\n\nFor $0<r<1$, let $\\A_r$ be the standard annulus $\\{z\\in\\mathbbm{C}: r<|z|<1\\}$. For each annular domain $A\\subset\\mathbbm{C}$,\nthere exists a unique $\\tau>0$ such that $A$ and $\\A_{e^{-2\\pi\\tau}}$ are conformally equivalent. We call $\\tau$ the \\textit{modulus} of $A$ and write $\\mathrm{Mod}(A):=\\tau$. For two simple loops $\\eta_1,\\eta_2$ such that $\\overline{D(\\eta_1)}\\subset D(\\eta_2)$, we also write the modulus of the annular domain $D(\\eta_2)\\setminus\\overline{D(\\eta_1)}$ as $\\mathrm{Mod}(\\eta_1,\\eta_2)$ for simplicity.\n\nWe will frequently deal with elementary functions such as $\\sqrt{x},\\sinh(\\sqrt{x}),\\cosh(\\sqrt{x})$, and $\\tanh(\\sqrt{x})$. In the following, we view them as functions defined on $\\mathbbm{R}$ by taking $i\\sqrt{|x|},i\\sin(\\sqrt{|x|}),\\cos(\\sqrt{|x|})$, and $i\\tan(\\sqrt{|x|})$ for $x<0$, respectively.\n\n\\medskip\n\\noindent\\textbf{Acknowledgment.}\nWe are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.\nG.C. and Z.X.\\ were partially supported by National Key R\\&D Program of China (No.\\ 2023YFA1010700). G.C. was partially supported by National Key R\\&D Program of China (No. 2021YFA1002700).",
    "sketch": "Our proof of Theorem~\\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; Section~\\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.”",
    "expanded_sketch": "Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”",
    "expanded_theorem": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $(B_t)_{t\\ge 0}$ be planar Brownian motion started from $0$, let $\\mathbb D=\\{z\\in\\mathbb C:|z|<1\\}$, and let $\\tau_{\\mathbb D}$ be the first time that $B_t$ hits the unit circle $\\mathbb S^1=\\{z\\in\\mathbb C:|z|=1\\}$. For $\\varepsilon\\in(0,\\tfrac12)$, define the backbone event\n\\[\n\\mathrm{Bac}_\\varepsilon=\n\\Big\\{\\text{there exist two disjoint subpaths of }B[0,\\tau_{\\mathbb D}]\\text{ joining }\\varepsilon\\mathbb S^1\\text{ and }\\tfrac12\\mathbb S^1\\Big\\}.\n\\]\nEquivalently, there exist continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\\tau_{\\mathbb D}]$, such that $\\gamma^i(0)\\in \\varepsilon\\mathbb S^1$ and $\\gamma^i(1)\\in \\tfrac12\\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\\mathbb P[\\mathrm{Bac}_\\varepsilon]$ for small $\\varepsilon$?",
      "correct_choice": {
        "label": "A",
        "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\frac{C_1}{\\log|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\nC_1\\,\\frac{1}{|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le C_2\\,\\frac{1}{|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant $C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "D",
          "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac12)$,\n\\[\n\\frac{C_1}{\\log|\\log \\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log \\varepsilon|}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every $\\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\n\\[\n\\frac{C_1}{(\\log|\\log \\varepsilon|)^{1+\\delta}}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{(\\log|\\log \\varepsilon|)^{1-\\delta}}.\n\\]"
        }
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      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "iterated_log_scale",
          "template_used": "property_confusion"
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        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_lower_bound",
          "template_used": "weaker_true"
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          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "epsilon_range_(0,1/10)",
          "template_used": "boundary_range"
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          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "exact_iterated_log_order",
          "template_used": "wildcard"
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    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the event carefully but does not reveal the asymptotic form or explicitly hint at the iterated-log behavior. The correct answer is not leaked by wording in the question."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to theorem recall: the stem asks directly which quantitative estimate holds, and the correct option is essentially the theorem statement. However, the alternatives introduce competing nearby formulations, so it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact two-sided estimate from weaker, overextended, or slightly perturbed variants. Still, the item mainly tests recognition/recall of the precise result rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing log with iterated log, accepting a weaker true statement, overextending the epsilon range, or allowing flexible exponents. They are distinct and well-calibrated."
      },
      "total_score": 6,
      "overall_assessment": "A strong technical MCQ with no answer leakage and high-quality distractors, but it is somewhat theorem-recall-heavy and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.09782v1",
    "paper_link": "http://arxiv.org/abs/2512.09782v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "[Rigidity of lattices]\n    If $\\Gamma$ is a finitely generated group quasi-isometric to an irreducible lattice in a semisimple Lie group $G$, then there is a short exact sequence\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Lambda$ is a lattice in $G$, and $F$ is a finite group.",
      "start_pos": 6970,
      "end_pos": 7388,
      "label": null
    },
    "ref_dict": {
      "FLS": "\\begin{theorem}[Frigerio--Lafont--Sisto]\\label{FLS}\n    If $\\Gamma$ is a finitely generated group quasi-isometric to $\\pi_1(M) \\times \\bZ^d$, then there exist short exact sequences\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & \\bZ^d \\ar[r,\"j\"] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\pi_1(M') \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Gamma' \\le \\Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\\bZ^d)$ is contained in the center of $\\Gamma'$. In other words, $\\Gamma$ is virtually a central extension by $\\bZ^d$ of a finite extension of $\\pi_1(M')$.\n\\end{theorem}",
      "virtually nilcentral extension": "\\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}",
      "Theorem A": "\\begin{thmx}\\label{Theorem A}\n    Let $X \\ne \\bH^2$ be a negatively curved symmetric space. Let $\\Lambda$ be a non-uniform lattice in $\\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\\Gamma$ is a finitely generated group quasi-isometric to $\\Lambda \\times L$, then there exist short exact sequences\n    \\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\\end{thmx}"
    },
    "pre_theorem_intro_text_len": 1796,
    "pre_theorem_intro_text": "\\subsection{Background}\n\nOne theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry\" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \\cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \\cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \\cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.\n\nOne of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \\cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,",
    "context": "One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry\" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \\cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \\cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \\cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.\n\nOne of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \\cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,",
    "full_context": "One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry\" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \\cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \\cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \\cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.\n\nOne of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \\cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,\n\nOne of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \\cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,\n\nOne of the major breakthroughs leading to this general classification was the work of Schwartz \\cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.\n\n\\begin{theorem}[Schwartz]\\label{Schwartz-theorem}\n Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\\bH^2$. If $\\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\\Lambda$ in $\\Isom{X}$, then there exists a short exact sequence\n \\[\n \\begin{tikzcd}\n 1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda' \\ar[r] & 1.\n \\end{tikzcd}\n \\]\n where $\\Lambda' \\le \\Isom{X}$ is a non-uniform lattice commensurable to $\\Lambda$, and $F$ is a finite group.\n\\end{theorem}\n\n\\begin{theorem}[Frigerio--Lafont--Sisto]\\label{FLS}\n If $\\Gamma$ is a finitely generated group quasi-isometric to $\\pi_1(M) \\times \\bZ^d$, then there exist short exact sequences\n \\begin{equation*}\n \\begin{tikzcd}\n 1 \\ar[r] & \\bZ^d \\ar[r,\"j\"] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1\n \\end{tikzcd}\n \\end{equation*}\n \\begin{equation*}\n \\begin{tikzcd}\n 1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\pi_1(M') \\ar[r] & 1\n \\end{tikzcd}\n \\end{equation*}\n where $\\Gamma' \\le \\Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\\bZ^d)$ is contained in the center of $\\Gamma'$. In other words, $\\Gamma$ is virtually a central extension by $\\bZ^d$ of a finite extension of $\\pi_1(M')$.\n\\end{theorem}\n\n\\begin{thmx}\\label{Theorem A}\n Let $X \\ne \\bH^2$ be a negatively curved symmetric space. Let $\\Lambda$ be a non-uniform lattice in $\\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\\Gamma$ is a finitely generated group quasi-isometric to $\\Lambda \\times L$, then there exist short exact sequences\n \\begin{equation}\\label{virtually nilcentral extension}\n \\begin{tikzcd}\n 1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n \\end{tikzcd}\n \\end{equation}\n \\begin{equation*}\n \\begin{tikzcd}\n 1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n \\end{tikzcd}\n \\end{equation*}\n where $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\\end{thmx}\n\n\\begin{theorem}\nLet $X \\ne \\bH^2$ be a negatively curved symmetric space. Let $\\Lambda$ be a non-uniform lattice in $\\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\\Gamma$ is a finitely generated group quasi-isometric to $\\Lambda \\times L$, then there exist short exact sequences\n \\begin{equation}\n \\begin{tikzcd}\n 1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n \\tag{\\ref{virtually nilcentral extension}}\n \\end{tikzcd}\n \\end{equation}\n \\begin{equation*}\n \\begin{tikzcd}\n 1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n \\end{tikzcd}\n \\end{equation*}\n where $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\\end{theorem}\n\n\\begin{proof}\nBy Proposition \\ref{kernel qi to N}, $\\ker\\theta$ is quasi-isometric to $N$, where $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. In particular $\\ker\\theta$ has polynomial growth. Then by Gromov's polynomial growth theorem, $\\ker\\theta$ is virtually nilpotent. So $\\ker\\theta$ has a finite-index, hence finitely generated, nilpotent subgroup $K$. Such groups are virtually torsion-free, so take a finite-index torsion-free subgroup $K' \\le K$. Since $\\ker\\theta$ is finitely generated, it has only finitely many subgroups with the same index as $K'$. Let $L'$ denote their intersection. Then $L'$ is a finite-index characteristic subgroup of $\\ker\\theta$. Moreover, $L'$ is finitely generated, nilpotent, and torsion-free. A theorem of Malcev \\cite{malcev} then says that $L'$ embeds as a lattice in a simply connected nilpotent Lie group $N'$, known as the (real) Malcev completion of $L'$ (see also \\cite[Theorem 2.18]{raghunathan1972discrete}). Since $L'$ is characteristic in $\\ker\\theta$, it is normal in $\\Gamma$. By construction, $\\Gamma/L'$ is an extension of $\\Lambda_\\Gamma = \\Gamma/\\ker\\theta$ by the finite group $F = \\ker\\theta/L'$. By Corollary \\ref{commensurable}, $\\Lambda_\\Gamma$ is virtually a finite-index subgroup $\\Lambda' \\le \\Lambda$ which is also a non-uniform lattice in $\\text{Isom}(X)$. Let $\\Gamma' = \\theta^{-1}(\\Lambda')$ and $\\Delta = \\Gamma'/L'$. Then we obtain the desired short exact sequences.\n\nNow suppose we are in the situation of Theorem \\ref{Theorem A} and we have the following two short exact sequences.\n\\begin{equation}\\label{vne}\n\\begin{tikzcd}\n 1 \\ar[r] & N \\ar[r] & \\Gamma \\ar[r] & \\Delta \\ar[r] & 1\n\\end{tikzcd}\n\\end{equation}\n\\begin{equation}\\label{finite extension}\n\\begin{tikzcd}\n 1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda \\ar[r] & 1\n\\end{tikzcd}\n\\end{equation}\nwhere $N$ is a nilpotent lattice (we are now using $N$ to denote the lattice, not the ambient Lie group), $F$ is finite, and $\\Lambda$ is a non-uniform lattice in $\\text{Isom}(X)$, where $X \\ne \\bH^2$ is a negatively curved symmetric space. Since $N$ is finitely generated, nilpotent, and torsion-free, $Z(N/Z_i) = Z_{i+1}/Z_i$ is a finitely generated free abelian group. That is, $Z(N/Z_i) = \\bZ^{d_i}$ for some $d_i$, and so $\\Sigma(N) = \\max_i d_i$. We now focus on the situation when $X$ is quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\\text{Isom}(X)$ has sufficiently large dimension.\n\n\\begin{thmx}\\label{Theorem A}\n    Let $X \\ne \\bH^2$ be a negatively curved symmetric space. Let $\\Lambda$ be a non-uniform lattice in $\\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\\Gamma$ is a finitely generated group quasi-isometric to $\\Lambda \\times L$, then there exist short exact sequences\n    \\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\\end{thmx}\n\n\\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}",
    "post_theorem_intro_text_len": 7199,
    "post_theorem_intro_text": "One of the major breakthroughs leading to this general classification was the work of Schwartz \\cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.\n\n\\begin{theorem}[Schwartz]\\label{Schwartz-theorem}\n    Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\\mathbb{H}^2$. If $\\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\\Lambda$ in $\\textup{Isom}({X})$, then there exists a short exact sequence\n    \\[\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda' \\ar[r] & 1.\n    \\end{tikzcd}\n    \\]\n    where $\\Lambda' \\le \\textup{Isom}({X})$ is a non-uniform lattice commensurable to $\\Lambda$, and $F$ is a finite group.\n\\end{theorem}\n\nIn the case of $X = \\mathbb{H}^2$, non-uniform lattices in $\\Isom{\\mathbb{H}^2} = \\text{PSL}(2,\\mathbb{R})$ are virtually the fundamental groups of complete hyperbolic surfaces with finitely many punctures, and hence are virtually free. Any group quasi-isometric to a virtually free group is also virtually free, and every free group can be realized as a non-uniform lattice in $\\text{PSL}(2,\\mathbb{R})$, so quasi-isometric rigidity does hold in this case. However, the additional conclusion of commensurability fails. One reason for this failure is that commensurability preserves arithmeticity, and $\\text{PSL}(2,\\mathbb{R})$ contains both arithmetic and non-arithmetic non-uniform lattices. But since these lattices are virtually free, they are quasi-isometric to each other.\n\nFollowing Schwartz, lattices such as $\\Lambda$ and $\\Lambda'$ shall from now on be called \\textit{non-uniform rank one lattices}. Then an informal summary might be: any group quasi-isometric to a non-uniform rank one lattice is almost a commensurable non-uniform rank one lattice.\n\nIn an extensive study on high-dimensional graph manifolds \\cite{FLS}, Frigerio--Lafont--Sisto prove, among many other things, a quasi-isometric rigidity result for products of the form $\\pi_1(M) \\times \\mathbb{Z}^d$, where $M$ is a complete non-compact finite-volume hyperbolic $m$-manifold, $m\\ge3$.\n\n\\begin{theorem}[Frigerio--Lafont--Sisto]\\label{FLS}\n    If $\\Gamma$ is a finitely generated group quasi-isometric to $\\pi_1(M) \\times \\mathbb{Z}^d$, then there exist short exact sequences\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & \\mathbb{Z}^d \\ar[r,\"j\"] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\pi_1(M') \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Gamma' \\le \\Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\\mathbb{Z}^d)$ is contained in the center of $\\Gamma'$. In other words, $\\Gamma$ is virtually a central extension by $\\mathbb{Z}^d$ of a finite extension of $\\pi_1(M')$.\n\\end{theorem}\n\nObserve that the case $d = 0$ is covered by Schwartz' theorem, and indeed, the proof of this theorem applies many of the ideas and results from \\cite{Schwartz}.\n\n\\subsection{Main results}\n\nOur main contribution is a generalization of Theorem \\ref{FLS} to products $\\Lambda \\times L$, where $\\Lambda$ is a non-uniform rank one lattice and $L$ is a lattice in a simply connected nilpotent Lie group. From now on, a lattice such as $L$ is called a \\textit{nilpotent lattice}. Our first theorem says that up to finite noise, any group quasi-isometric to $\\Lambda \\times L$ is an extension of a non-uniform rank one lattice commensurable to $\\Lambda$ by a nilpotent lattice quasi-isometric to $L$.\n\n\\begin{thmx}\\label{Theorem A}\n    Let $X \\ne \\mathbb{H}^2$ be a negatively curved symmetric space. Let $\\Lambda$ be a non-uniform lattice in $\\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\\Gamma$ is a finitely generated group quasi-isometric to $\\Lambda \\times L$, then there exist short exact sequences\n    \\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\\end{thmx}\n\nThe general outline of the proof is similar to that of the proof of Theorem \\ref{FLS}. First, the quasi-isometry between $\\Gamma$ and $\\Lambda \\times L$ induces a quasi-action of $\\Gamma$ on $B \\times N$, where $B \\subset X$ is the neutered space associated to $\\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \\cite{KKL} guarantees that quasi-isometries $B \\times N \\to B \\times N$ project to quasi-isometries $B \\to B$, up to bounded error. Thus the quasi-action of $\\Gamma$ on $B\\times N$ induces a quasi-action of $\\Gamma$ on $B$. A key result in \\cite{Schwartz} is that quasi-isometries of the neutered space $B$ have finite distance (with respect to the sup norm) from isometries of $X$. Thus we obtain a homomorphism $\\theta : \\Gamma \\to \\text{Isom}(X)$, and we show that the image $\\text{im} \\hspace{.8mm} \\theta$ is a non-uniform lattice commensurable to $\\Lambda$. The action $\\theta : \\Gamma \\to \\text{Isom}(X)$ came from a quasi-action on $B$, which itself was coarsely projected from a quasi-action on $B \\times N$. Hence we are able to show that the kernel of $\\theta$ is quasi-isometric to $N$. Then $\\Gamma = \\text{im} \\hspace{.8mm} \\theta / \\ker \\theta$, so we pass to finite-index subgroups as necessary to obtain the desired short exact sequences.\n\nTheorem \\ref{FLS} asserts that $\\mathbb{Z}^d$  can be made central in the group extension. In our more general setting, however, the extension is by a nilpotent group. So we define the notion of a nilcentral extension, analogous to that of a central extension, and find conditions which are sufficient to guarantee that (\\ref{virtually nilcentral extension}) may be taken to be a nilcentral extension. Given a nilpotent group $G$ with upper central series $1 = Z_0 \\lhd Z_1 \\lhd \\dots \\lhd Z_n = G$, define \n\\[\n    \\Sigma(G) \\vcentcolon= \\max_i \\text{rank}(Z_{i+1}/Z_i).\n\\]\n\n\\begin{thmx}\\label{Theorem B}\n    Assume the hypotheses of Theorem \\ref{Theorem A} and let $L'$ be the nilpotent lattice obtained from the conclusion of the theorem. If $X$ is either quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\\dim\\textup{Isom}(X) > \\Sigma(L')$, then the group extension (\\ref{virtually nilcentral extension}) in the conclusion of Theorem \\ref{Theorem A} is virtually nilcentral.\n\\end{thmx}\n\nOur proof of this theorem relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T). The precise statement we apply is in \\cite{fisher2012strengthening}.",
    "sketch": "First, the quasi-isometry between $\\Gamma$ and $\\Lambda \\times L$ induces a quasi-action of $\\Gamma$ on $B \\times N$, where $B \\subset X$ is the neutered space associated to $\\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \\cite{KKL} implies that quasi-isometries $B \\times N \\to B \\times N$ project to quasi-isometries $B \\to B$ (up to bounded error), so the quasi-action on $B\\times N$ induces a quasi-action on $B$.\n\nA key result in \\cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\\theta: \\Gamma \\to \\Isom(X)$, and one shows $\\mathrm{im}\\,\\theta$ is a non-uniform lattice commensurable to $\\Lambda$.\n\nBecause $\\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\\times N$, one can show $\\ker\\theta$ is quasi-isometric to $N$. Since “$\\Gamma = \\mathrm{im}\\,\\theta / \\ker\\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences (as in Theorem~\\ref{Theorem A}).\n\nFor Theorem~\\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\\Sigma(L')$ and $\\dim\\Isom(X)$) ensuring that \\eqref{virtually nilcentral extension} is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”",
    "expanded_sketch": "First, the quasi-isometry between $\\Gamma$ and $\\Lambda \\times L$ induces a quasi-action of $\\Gamma$ on $B \\times N$, where $B \\subset X$ is the neutered space associated to $\\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \\cite{KKL} implies that quasi-isometries $B \\times N \\to B \\times N$ project to quasi-isometries $B \\to B$ (up to bounded error), so the quasi-action on $B\\times N$ induces a quasi-action on $B$.\n\nA key result in \\cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\\theta: \\Gamma \\to \\Isom(X)$, and one shows $\\mathrm{im}\\,\\theta$ is a non-uniform lattice commensurable to $\\Lambda$.\n\nBecause $\\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\\times N$, one can show $\\ker\\theta$ is quasi-isometric to $N$. Since “$\\Gamma = \\mathrm{im}\\,\\theta / \\ker\\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences, namely the conclusion that there exist short exact sequences\n\\begin{equation}\\label{virtually nilcentral extension}\n    \\begin{tikzcd}\n        1 \\ar[r] & L' \\ar[r] & \\Gamma' \\ar[r] & \\Delta \\ar[r] & 1,\n    \\end{tikzcd}\n    \\end{equation}\n\\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Delta \\ar[r] & \\Lambda' \\ar[r] & 1.\n    \\end{tikzcd}\n    \\end{equation*}\nwhere $\\Gamma' \\le \\Gamma$ and $\\Lambda' \\le \\Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\\Delta$ is a group, and $F$ is a finite group.\n\nFor Theorem~\\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\\Sigma(L')$ and $\\dim\\Isom(X)$) ensuring that the equation above is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”",
    "expanded_theorem": "[Rigidity of lattices]\n    If $\\Gamma$ is a finitely generated group quasi-isometric to an irreducible lattice in a semisimple Lie group $G$, then there is a short exact sequence\n    \\begin{equation*}\n    \\begin{tikzcd}\n        1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda \\ar[r] & 1\n    \\end{tikzcd}\n    \\end{equation*}\n    where $\\Lambda$ is a lattice in $G$, and $F$ is a finite group.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $G$ be a semisimple Lie group, and let an irreducible lattice in $G$ mean a discrete subgroup of finite covolume that is irreducible with respect to the semisimple decomposition of $G$. Suppose $\\Gamma$ is a finitely generated group that is quasi-isometric to such an irreducible lattice in $G$. Which of the following conclusions about $\\Gamma$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a short exact sequence\n\\[\n\\begin{tikzcd}\n1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda \\ar[r] & 1,\n\\end{tikzcd}\n\\]\nwhere $\\Lambda$ is a lattice in $G$ and $F$ is a finite group."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a short exact sequence\n\\[\n\\begin{tikzcd}\n1 \\ar[r] & F \\ar[r] & \\Gamma' \\ar[r] & \\Lambda \\ar[r] & 1,\n\\end{tikzcd}\n\\]\nwhere $\\Gamma' \\leq \\Gamma$ has finite index, $\\Lambda$ is an irreducible lattice in $G$, and $F$ is a finite group."
        },
        {
          "label": "C",
          "text": "$\\Gamma$ is virtually a finite extension of a lattice in $G$; equivalently, there exist a finite-index subgroup $\\Gamma' \\leq \\Gamma$, a lattice $\\Lambda \\leq G$, and a finite group $F$ fitting into a short exact sequence\n\\[\n\\begin{tikzcd}\n1 \\ar[r] & F \\ar[r] & \\Gamma' \\ar[r] & \\Lambda \\ar[r] & 1.\n\\end{tikzcd}\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a short exact sequence\n\\[\n\\begin{tikzcd}\n1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda \\ar[r] & 1,\n\\end{tikzcd}\n\\]\nwhere $\\Lambda$ is an irreducible lattice in $G$ commensurable with the given irreducible lattice, and $F$ is a finitely generated nilpotent group."
        },
        {
          "label": "E",
          "text": "There exists a short exact sequence\n\\[\n\\begin{tikzcd}\n1 \\ar[r] & F \\ar[r] & \\Gamma \\ar[r] & \\Lambda \\ar[r] & 1,\n\\end{tikzcd}\n\\]\nwhere $\\Lambda$ is a lattice in $G$ and $F$ is finite; moreover, one may always choose $\\Lambda$ to be commensurable to the original irreducible lattice in $G$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "need for exact conclusion on all of Gamma rather than only after passing to finite index in the product-lattice argument",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the requirement that the short exact sequence hold for \\Gamma itself, allowing passage to a finite-index subgroup",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "kernel is finite, not merely virtually nilpotent or finitely generated nilpotent",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "asserted commensurability to the original lattice, which is not stated in the main rigidity theorem",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states only the hypotheses and asks for the correct conclusion; it does not explicitly reveal the exact finite-extension conclusion or uniquely telegraph choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall/application of a quasi-isometric rigidity theorem: the hypotheses in the stem closely match the theorem, and the correct option states its conclusion almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle quantifiers and strength (exact sequence for all of Γ vs. after finite index, finite vs. nilpotent kernel, commensurability claims). Still, a student who knows the theorem can answer mainly by recognition rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically close, distinct, and plausible: one is a weaker true-looking variant, others alter the kernel type or add an unjustified commensurability claim. These reflect realistic failure modes in reading rigidity statements."
      },
      "total_score": 5,
      "overall_assessment": "A solid precision-testing theorem-recall MCQ with strong distractors, but it is quite close to a direct restatement of the underlying rigidity theorem rather than a deeply generative reasoning task."
    }
  },
  {
    "id": "2512.09839v1",
    "paper_link": "http://arxiv.org/abs/2512.09839v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\left\\vertV(w)\\right\\vert \\le a_0^2 \\left< w \\right>^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\Delta u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\left\\vertu(0)\\right\\vert = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\left\\vertu(w)\\right\\vert \\le \\exp\\pr{c_0 \\left\\vertw\\right\\vert^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\varepsilon \\in \\left( 0, \\frac N 2 \\right) $, there exists ${R}_0(N, a_0, c_0, \\varepsilon) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\left\\vertw_0\\right\\vert = R} \\left\\| u\\right\\|_{L^\\infty\\left( B_1(w_0) \\right) } \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\varepsilon}}.\n\\end{equation}",
      "start_pos": 21228,
      "end_pos": 22053,
      "label": "mainThm"
    },
    "ref_dict": {
      "mainThm": "\\begin{thm}[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\abs{V(w)} \\le a_0^2 \\innp{w}^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\LP u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\abs{u(0)} = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\eps \\in \\pr{0, \\frac N 2}$, there exists ${R}_0(N, a_0, c_0, \\eps) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\abs{w_0} = R} \\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\eps}}.\n\\end{equation}\n\\end{thm}",
      "ellipEq": "\\begin{equation}\n\\label{ellipEq}\n- \\LP u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}",
      "TransMaps": "\\begin{align}\n\\label{propApp}\n\\sup_{B_{\\tilde r} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{16 {\\tilde r}} {\\WT R}}^{\\tilde \\kappa} \\sup_{B_{\\WT T} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h},\n\\end{align}\nwhere\n\\begin{align*}\n\\tilde \\kappa\n:= \\kappa\\pr{\\WT R, \\WT S, M}\n= \\max \\set{6C_H C_1 R \\sqrt{\\log R}, 2^{13} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}}.\n\\end{align*}\nSince $R \\ge 2^{10}$ implies $\\frac R 8 > 1$, while $R \\ge R_2$ implies $\\sqrt{\\frac{98 \\cdot 16 R}{C_1 \\sqrt{\\log R}}} > 1$, then $r_0 := \\min\\set{\\frac R 8, \\sqrt{\\frac{98 \\cdot 16 R}{C_1 \\sqrt{\\log R}}}} > 1$.\nAssume that $R \\ge \\hat R_0$ and $r < r_0$.\nCombining \\eqref{ufComparison}, \\eqref{smallBallBound}, \\eqref{propApp}, \\eqref{scaleBounds}, and \\eqref{midBallBound} shows that\n\\begin{align*}\n\\frac 1{1 - c_b \\eps^2} \\sup_{B_r} \\abs{v}\n&\\ge \\sup_{B_r} \\abs{f}\n\\ge \\sup_{B_{\\tilde r} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{16 {\\tilde r}} {\\WT R}}^{\\tilde \\kappa} \\sup_{B_{\\WT T} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{r} {R}}^{2\\tilde \\kappa} \\frac {e^{- L}} {1 + c_b \\eps^2} .\n\\end{align*}\nSince $C_1 =  \\frac {a} {32} \\sqrt{\\frac{2 C_K }{\\ln 2}}$, $\\disp \\frac{1 - c_b \\eps^2}{1 + c_b \\eps^2} \\ge e^{- \\frac{c_d}{\\log R}}$, and $e < 4 \\le \\pr{2^{10}}^{\\frac 1 5} \\le \\pr{\\frac{R}{r}}^{\\frac 1 5}$, then \\eqref{propConc} follows, as required. \n\\end{proof}\n\n\\section{Transformation maps}\n\\label{TransMaps}\n\nIn this section, we introduce the transformation maps that serve as real-valued versions of the conformal maps $z \\mapsto z^\\al$.\nOnce the maps are defined, we show how they transform solutions to elliptic PDEs and how they transform balls.\nThese results will be used at the beginning and at the end of the proof of Theorem \\ref{mainThm}.\n\nGiven $z = (x,y) \\in \\R^2$ in Cartesian coordinates, the polar coordinates for $z$ are $(r, \\te) \\in \\R_+ \\times (-\\pi, \\pi]$, where $r = \\sqrt{x^2 + y^2}$ and $\\te = \\sgn(y) \\arccos\\pr{\\frac x r}$.\nLet $\\R^2_+ = \\set{z = (r,\\te) : r > 0, \\te \\in (- \\frac \\pi 2, \\frac \\pi 2)}$ denote the right half-plane.\nFor $\\al \\in (1, \\iny)$, we define $T_\\al : \\R^2 \\to \\R^2$ to be the map associated to the conformal transformation on $\\C$ given by $z \\mapsto z^\\al$. \nIn polar coordinates, $T_\\al$ is described as\n\\begin{equation}\n\\label{TalDefn}\nT_\\al(r, \\te) = (r^\\al, \\al \\te).\n\\end{equation}\nTo avoid continuity issues at $\\te = \\pi$, we restrict the domain and only consider $T_\\al : \\R^2_+ \\to \\R^2$.\n\n\\begin{lem}[Transformation of PDEs]\n\\label{transformEqLemma}\nIf $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq}, then with $v : \\R^2_+ \\to \\R$ and $W : \\R^2_+ \\to \\R$ defined by $v(z) = u(T_\\al(z))$ and $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z))$, it holds that\n$$-\\LP v + W v = 0 \\; \\text{ in } \\R^2_+.$$\n\\end{lem}\n\n\\begin{proof}\nLet $w := T_\\al(z)$ be given in polar coordinates by $\\pr{\\rho, \\vp}$.\nSince $u$ is defined on $\\R^2$, then in polar coordinates, $u(\\rho, \\vp)$ can be defined for all $\\rho \\in \\R^+$ and all $\\vp \\in \\R$ using periodicity.\nWe then see that $v(z) = v(r, \\te) = u(r^\\al, \\al \\te) = u(T_\\al(z))$ is well-defined on $\\R^2_+$. \nSince\n\\begin{align*}\n\\del_r v(r, \\te) &= \\al r^{\\al - 1} \\del_\\rho u(\\rho, \\vp) \\\\\n\\del_{r}^2 v(r, \\te) &= \\al^2 r^{2\\al - 2} \\del_{\\rho}^2 u(\\rho, \\vp) + \\al \\pr{\\al - 1} r^{\\al - 2} \\del_\\rho u(\\rho, \\vp) \\\\\n\\del_{\\te}^2 v(r, \\te) &= \\al^2 \\del_{\\vp}^2 u(\\rho, \\vp),\n\\end{align*}\nthen\n\\begin{align*}\n\\LP v(z)\n&= \\del_{r}^2 v(r, \\te) + \\frac 1 r \\del_{r} v(r, \\te) + \\frac 1 {r^2} \\del_{\\te}^2 v(r, \\te) \\\\\n&= \\al^2 r^{2\\al - 2} \\del_{\\rho}^2 u(\\rho, \\vp) + \\al \\pr{\\al - 1} r^{\\al - 2} \\del_\\rho u(\\rho, \\vp)\n+ \\al r^{\\al - 2} \\del_\\rho u(\\rho, \\vp)\n+ r^{-2} \\al^2 \\del_{\\vp}^2 u(\\rho, \\vp) \\\\\n&= \\al^2 r^{2\\al - 2} \\brac{\\del_\\rho^2 u(\\rho, \\vp) + \\rho^{-1} \\del_\\rho u(\\rho, \\vp) + \\rho^{-2}\\del_{\\vp}^2 u(\\rho, \\vp)} \\\\\n&= \\al^2 r^{2\\al - 2} \\LP u(w) \n=  \\al^2 r^{2\\al - 2} V(w) u(w),\n\\end{align*}\nwhere we have used \\eqref{ellipEq}.\nWith $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z)) = \\al^2 r^{2\\al - 2} V(w)$ as given, the conclusion follows.\n\\end{proof}\n\n\\begin{lem}[Ball containment]\n\\label{ballContainLemma}\nLet $z_0 = (r_0, 0)$ and set $\\disp \\tilde r = \\frac{r_0^{1 - \\al}}{2 \\sqrt 3 \\al}$.\nThere exists $r_\\al > 0$ so that whenever  with $r_0 \\ge r_\\al$, it holds that $T_\\al(B_{\\tilde r}(z_0)) \\su B_1(T_\\al(z_0))$ and $B_1(T_\\al(z_0)) \\su T_\\al\\pr{B_1(z_0)}$.\n\\end{lem}\n\n\\begin{proof}\nWith $\\tilde r$ as given, it can be shown that\n\\begin{equation}\n\\label{ballIn} \n\\begin{aligned}\nB_{\\tilde r}(z_0) \n&\\su \\set{r \\in [r_0 - \\tilde r, r_0 + \\tilde r], \\abs{\\te} \\le \\frac {2\\tilde r} {\\sqrt 3 r_0}} \\\\\n&= \\set{r \\in \\brac{r_0\\pr{1 -  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}, r_0\\pr{1 +  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}}, \\abs{\\te} \\le \\frac {1} {3 \\al r_0^\\al}}.\n\\end{aligned}\n\\end{equation}\nWith $w_0 = T_\\al(z_0)= \\pr{r_0^\\al, 0} = \\pr{\\rho_0, 0}$, it follows from \\eqref{ballIn} that\n\\begin{align*}\nT_\\al(B_{\\tilde r}(z_0)) \n&\\su \\set{r^\\al \\in \\brac{r_0^\\al\\pr{1 -  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}^\\al, r_0^\\al \\pr{1 +  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}^\\al}, \\abs{\\al \\te} \\le \\frac {\\al} {3 \\al r_0^\\al}} \\\\\n&= \\set{\\rho \\in \\brac{\\rho_0 \\pr{1 - \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al}}^\\al, \\rho_0 \\pr{1 + \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al}, \\abs{\\vp} \\le \\frac {1} {3 \\rho_0} }.\n\\end{align*}\nSince\n\\begin{align*}\n\\rho_0 \\pr{1 \\pm \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al\n&= \\rho_0 \\pm \\frac{1}{2 \\sqrt 3} + \\frac{(1 - \\frac 1 \\al)}{24} r_0^{- \\al} + \\ldots \n\\end{align*}\nthen there exists $r_1(\\al) \\gg 1$ so that whenever $r_0 \\ge r_1$, we have $\\rho_0 - \\frac 1 2 \\le \\rho_0 \\pr{1 - \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al$ and $\\rho_0 \\pr{1 + \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al \\le \\rho_0 +\\frac 1 2$.\nBecause,\n\\begin{align}\n&B_1(w_0)  \\supset \\set{\\rho \\in \\brac{\\rho_0 - \\frac 1 2, \\rho_0 + \\frac 1 2}, \\abs{\\vp} \\le \\frac 1 {3\\rho_0} },\n\\label{ballOf}\n\\end{align}",
      "harmonic": "\\begin{equation}\n\\label{uLower}\n\\inf_{\\abs{w_0} = R} \\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\eps}}.\n\\end{equation}\n\\end{thm}\n\nThe results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\be = \\frac{4 - 2N}{3} = \\frac 4 3 \\pr{1 - \\frac N 2} > 1 - \\frac N 2 + \\eps$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\nAs illustrated by the following example, Theorem \\ref{mainThm} is sharp (up to $\\eps$) for all $N \\in (0, 2)$.\nFix $N \\in (0, 2)$, then set $u(z) = \\exp\\pr{- \\abs{z}^{1 - \\frac N 2}}$.\nA computation shows that $u$ satisfies \\eqref{ellipEq} where\n$$V(z) := \\pr{1 - \\frac N 2}^2 \\pr{\\abs{z}^{1 - \\frac N 2} - 1}\\abs{z}^{-1 - \\frac N 2}$$ \nsatisfies $\\abs{V(z)} \\lesssim \\abs{z}^{-N}$.\nOn the other hand, for any $\\be > 0$, with $u(z) = \\abs{z}^{-\\be}$ on $\\abs{z} > 1$, we see that $\\disp \\LP u = \\be^2 \\abs{z}^{-\\be-2}$ and therefore $u$ satisfies \\eqref{ellipEq} on an exterior domain with $V(z) := \\be^2\\abs{z}^{-2}$.\nIn particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.\n\nTo prove Theorem \\ref{mainThm}, we use an iterative argument that is reminiscent of the one in \\cite{Dav14}, see also \\cite{LW14, DKW19, Dav20a, Dav25}.\nTo initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\be = 1$.\nThis result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \\cite{LMNN20}, and we formulate it in Theorem \\ref{LandisGrowth} below.\nThe iteration argument then relies on repeated applications of Proposition \\ref{InductiveProp} which is proved using the ideas from \\cite{LMNN20}.\nRoughly speaking, Proposition \\ref{InductiveProp} shows that if an estimate like \\eqref{est} holds with $\\be = \\be_0$, then for some $x_1$ with $\\abs{x_1} \\gg \\abs{x_0}$, another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and $\\be = \\be_1 \\in \\brac{1, \\be_0}$.\nWhen $\\be_0 = 1$, Proposition \\ref{InductiveProp} isn't useful, but when $\\be_0 > 1$, we can decrease the exponent, i.e., make $\\be_1 < \\be_0$.\nTherefore, to benefit from the iteration argument, we need to transform to a situation where $\\be_0 > 1$.\nWe observe that if $u$ is composed with the real-variable version of the conformal transformation $z \\mapsto z^\\al$, then the new function also satisfies a Schr\\\"odinger equation.\nBy choosing $\\al > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \\eqref{est} with $\\be > 1$.\nBy repeatedly applying Proposition \\ref{InductiveProp} to the transformed equation, we can make $\\be$ arbitrarily close to $1$.\nFinally, to reach the conclusion, we undo the change of variables.\n\nWe use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.\nGeneric constants are denoted by $c, C$ and may change from line to line without comment.\nSpecific constants will be indicated by subscripts.\n\nThe article is organized as follows.\nIn Section \\ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.\nThe content of this section is very similar to \\cite[Section 5]{LMNN20} and \\cite[Section 2]{Dav24}.\nThe iterative result described by Proposition \\ref{InductiveProp} is the content of Section \\ref{localProof}.\nProposition \\ref{InductiveProp} is a three-ball inequality for solutions to Schr\\\"odinger equations and its proof relies on the results from Section \\ref{harmonic}. In Section \\ref{TransMaps}, we introduce the real-valued versions of $z \\mapsto z^\\al$ and record some of their properties.\nIn particular, we show how solutions behave when they are composed with these transformations.\nFinally, the proof of Theorem \\ref{mainThm} is presented in Section \\ref{MainProof}.\n\n\\section{Decay properties of harmonic functions in punctured domains}\n\\label{harmonic}\n\nIn this section, we present and prove quantitative unique continuation results (in the form of three-ball inequalities) for harmonic functions in punctured domains.\nWe recall the following application of the Harnack inequality which appears in \\cite[Claim 5.2]{LMNN20} and is repeated in \\cite[Lemma 2.1]{Dav24}.\n\n\\begin{lem}[Harnack application]\n\\label{discBounds}\nLet $\\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane.\nAssume that $h$ is real-valued and harmonic in $\\R^2 \\setminus \\bigcup D_j$ and that for each index $j$, $h$ doesn't change sign in $5D_j \\setminus D_j$.\nThere exists an absolute constant $C_H \\ge 10$ for which\n\\begin{enumerate}\n\\item $\\disp \\max_{\\del \\pr{3 D_j}} \\abs{h} \\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}$\n\\item $\\disp \\max_{\\del \\pr{3 D_j}} \\abs{\\gr h} \\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nAn application of the Harnack inequality shows that there exists $C_H > 0$ so that for every $j$\n\\begin{align*}\n\\max_{\\del \\pr{3 D_j}} \\abs{h}\n&\\le \\sup_{4 D_j \\setminus 2 D_j} \\abs{h} \n\\le C_H \\inf_{4 D_j \\setminus 2 D_j} \\abs{h}\n\\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}.\n\\end{align*}\nFor each $z \\in \\del \\pr{3 D_j}$, since $h$ doesn't change signs in $B_2(z)$, an application of Cauchy's inequality as in \\cite[Lemma 1.11]{HL11} shows that $\\abs{\\gr h(z)} \\le \\abs{h(z)}$ and the conclusion follows.\n\\end{proof}\n\nNow we state and prove the main result of this section.\nThe following is a slight modification of \\cite[Lemma 2.1]{Dav24}, which resembles the result \\cite[Theorem 5.3]{LMNN20}.\n\n\\begin{prop}[Three-ball inequality for harmonic function in punctured domain]\n\\label{harmonicProp}\nLet $\\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane for which $0 \\notin \\bigcup 3 D_j$.\nFor some $R \\ge 2^{10}$, let $h$ be a harmonic function in $B_R \\setminus \\bigcup D_j$ with the property that for each index $j$, $h$ doesn't change sign in $\\pr{5D_j \\setminus D_j} \\cap B_R$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$ and some $M > \\log\\pr{16R}$, with $T := R - \\frac S {32}$, \nit holds that\n\\begin{equation}\n\\label{normalization}\n\\sup_{B_T \\setminus \\bigcup 3 D_j} \\abs{h}  \\ge e^{-M} \\sup_{B_R \\setminus \\bigcup 3 D_j} \\abs{h}.\n\\end{equation}",
      "localProof": "\\begin{align}\n\\label{j2j0Comp}\nC_Hm_{j_2} \\abs{z_{j_2}}^{-k} \n&\\ge \\sup_{\\del \\pr{3 D_{j_2}}} \\abs{h} \\abs{z_{j_2}}^{-k}\n\\ge \\abs{h(s z_{j_0})} \\abs{s z_{j_0}}^{-k}\n\\ge \\frac{m_{j_0}}{2} s^{-k} \\abs{z_{j_0}}^{-k}.\n\\end{align}\nSince $z_{j_0} \\in \\del \\pr{3 D_{j_0}}$ and $s z_{j_0} \\in \\del \\pr{3 D_{j_2}}$ where $j_0 \\ne j_2$, and the balls $\\set{D_j}$ are of unit radius and $100$-separated, then $\\abs{z_{j_0} - s z_{j_0}} \\ge 96$.\nAfter rearrangement, we see that $s^{-k} \\ge \\pr{1 - \\frac{96}{R}}^{-k}$.\nSince $10 \\le C_H$, $2C_HR \\le k$, and $\\frac{96}{R} < - \\log\\pr{1 - \\frac{96}{R}}$, then \n\\begin{align*}\n\\log\\pr{2C_H}\n&< 96 \\cdot 2 C_H\n\\le \\frac{96}{R} k\n< - k \\log\\pr{1 - \\frac{96}{R}}\n\\le - k \\log s,\n\\end{align*}\nfrom which it follows that $s^{-k} > 2C_H$.\nWe then conclude from \\eqref{j2j0Comp} that $m_{j_2} \\abs{z_{j_2}}^{-k} > m_{j_0} \\abs{z_{j_0}}^{-k}$ which contradicts \\eqref{j0Defn} and gives the desired contradiction.\nIn other words, \\eqref{contraBound} fails to hold and we see that\n\\begin{align*}\n\\sup_{B_r \\setminus \\bigcup 3 D_j} \\abs{h} \n&> \\pr{\\frac{16r} R}^{3k}\n= \\pr{\\frac {16r} R}^{3k} \\sup_{B_T \\setminus \\bigcup 3 D_j} \\abs{h},\n\\end{align*}\nwhich implies \\eqref{lowerBound} by our choice of $k$.\n\\end{proof}\n\n\\section{The iterative proposition}\n\\label{localProof}\n\nHere we present the proposition that is used repeatedly in the iteration argument.\nThis unique continuation result takes the form of a three-ball inequality for solutions to Schr\\\"odinger equations.\nThe techniques used to prove this theorem are very similar to those that appear in the proofs of \\cite[Theorem 2.2]{LMNN20} and \\cite[Theorem 1.2]{Dav24}.\n\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}",
      "InductiveProp": "\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}",
      "est": "\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}",
      "ePDE": "\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}",
      "MainProof": "\\begin{align}\n&B_1(w_0)  \\supset \\set{\\rho \\in \\brac{\\rho_0 - \\frac 1 2, \\rho_0 + \\frac 1 2}, \\abs{\\vp} \\le \\frac 1 {3\\rho_0} },\n\\label{ballOf}\n\\end{align}\nthen it follows that $T_\\al(B_{\\tilde r}(z_0))  \\su B_1(w_0)$.\n\nOn the other hand, with $T_\\al^{-1}(B_1(w_0)) = \\set{z : T_\\al(z) \\in B_1(w_0)}$, \\eqref{ballIn} (with $\\tilde r$ replaced by $1$) implies that \n\\begin{align*}\nT_\\al^{-1}(B_1(w_0))\n&\\su \\set{r^\\al \\in [\\rho_0 - 1, \\rho_0 + 1], \\abs{\\al \\te} \\le \\frac {2} {\\sqrt 3 \\rho_0}} \\\\\n&= \\set{r \\in \\brac{\\pr{r_0^\\al - 1}^{\\frac 1 \\al}, \\pr{r_0^\\al + 1}^{\\frac 1 \\al}}, \\abs{\\te} \\le \\frac {2} {\\sqrt 3 \\al r_0^\\al}}.\n\\end{align*}\nSince\n\\begin{align*}\n\\pr{r_0^\\al \\pm 1}^{\\frac 1 \\al}\n&= r_0 \\pr{1 \\pm \\frac 1 {r_0^\\al}}^{\\frac 1 \\al}\n= r_0 \\pm \\frac 1 {\\al r_0^{\\al-1}} - \\frac{1 - \\frac 1 \\al}{2 \\al r_0^{2\\al-1}}  + \\ldots \n\\end{align*}\nthen there exist $r_2(\\al) \\gg 1$ so that whenever $r_0 \\ge r_2$, we have $\\pr{r_0^\\al + 1}^{\\frac 1 \\al} \\le r_0 + \\frac 1 2$ and $\\pr{r_0^\\al - 1}^{\\frac 1 \\al} \\ge r_0 - \\frac 1 2$.\nIf $r_0 \\ge r_3 := \\pr{\\frac {2 \\sqrt 3} { \\al }}^{\\frac 1 {\\al - 1}}$, then\n\\begin{align*}\nT_\\al^{-1}(B_1(w_0))\n&\\su \\set{r \\in \\brac{r_0 - \\frac 1 2, r_0 + \\frac 1 2}, \\abs{\\te} \\le \\frac {1} {3 r_0}}\n\\su B_1(z_0),\n\\end{align*}\nwhere we have used \\eqref{ballOf}.\nIt follows that $B_1(w_0) \\su T^\\al(B_1(z_0))$.\nTo complete the proof, choose $r_\\al = \\max\\set{r_1, r_2, r_3}$.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{mainThm}}\n\\label{MainProof}\n\nIn this section, we prove the main result, Theorem \\ref{mainThm}.\nBefore proceeding to the rigorous details, we describe the main steps:\n\\begin{itemize}\n\\item Initialize the iterative argument by applying Theorem \\ref{LandisGrowth} to the solution function $u$.\n\\item Compose the solution with the transformation map given in \\eqref{TalDefn} to get a solution $v$ to a different Schr\\\"odinger equation, see Lemma \\ref{transformEqLemma}.\n\\item Apply the iterative result, Proposition \\ref{InductiveProp}, to $v$ many times to reduce the exponent.\n\\item Undo the change of variables to get the desired estimate for $u$.\n\\end{itemize}\nWe recall the following result originally proved in \\cite{LMNN20}, see also \\cite[Theorem 1.1]{Dav24} in the case where $N = 0$ for this formulation.\nThis estimate serves as the initialization step in our iteration argument.\n\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\nObserve that if $u$ satisfies the hypothesis of Theorem \\ref{mainThm}, then it also satisfies the hypotheses of Theorem \\ref{LandisGrowth}.\nWe now have all of the tools we need to prove Theorem \\ref{mainThm}.\n\n\\begin{proof}[Proof of Theorem \\ref{mainThm}]\n\nWith $N \\in (0, 2)$ as given, let $\\al =\\pr{1 - \\frac N 2}^{-1} = \\frac 2 {2-N} \\in (1, \\iny)$.\nSet $\\de = \\min\\set{\\frac{\\eps}{2 - N}, 1}$.\nWith $r_\\al$ from Lemma \\ref{ballContainLemma}, $\\overline{C}_0(a_0, c_0)$ and $\\overline{R}_0$ from Theorem \\ref{LandisGrowth}, and $\\hat R_0(a)$ from Proposition \\ref{InductiveProp}, define $\\bar{r}_1 \\ge \\max\\set{r_\\al, \\overline{R}_0^{\\frac 1 \\al}, \\hat R_0\\pr{a_0 \\al}}$ as small as possible so that each of the following conditions hold:\n\\begin{align}\n& r^{\\de} \\ge \\overline{C}_0 \\al^{\\frac 3 2} \\log^{\\frac 3 2} r\n\\label{rBig1} \\\\\n&r \\log r \\ge c_d \n\\label{rBig2} \\\\\n& r^{\\frac \\de {1 + \\de}} \\ge \\frac{3 a_0 \\al C_H}{2^{14} \\pr{5c_0 + 4} } \\sqrt{\\frac{C_K }{32\\ln 2}} \\sqrt{\\log r}\n\\label{rBig3} \\\\\n& r^{\\frac {\\de^2} 2 } \\ge \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5 }\\log r\n\\label{rBig4} \\\\\n& r^{\\frac {5\\de^2} 6 } \\ge \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5}\\log \\brac{\\frac{4\\al}{ \\sqrt 3}  \\pr{\\frac 3 2 r}^{\\al} }\n\\label{rBig5}.\n\\end{align}",
      "LandisGrowth": "\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 5108,
    "pre_theorem_intro_text": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.",
    "context": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\n\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\n\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}",
    "full_context": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\n\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\n\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\abs{V(z)} \\lesssim \\abs{z}^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\nThe results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\be = \\frac{4 - 2N}{3} = \\frac 4 3 \\pr{1 - \\frac N 2} > 1 - \\frac N 2 + \\eps$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}\n\nDefine $\\Om = B_R \\setminus \\pr{F_0 \\bigcup F_1}$ and $\\Om_1 = B_R \\setminus F_1$.\nAs shown in \\cite[\\S 3.1]{LMNN20}, there exists a universal constant $c_P$ (that depends on $c_s$) so that $\\Om$ has Poincar\\'e constant bounded above by $c_P \\rho^2$.\nIn particular, since $c_P \\rho^2 \\norm{W}_{L^\\iny(B_R)} \\le c_P \\rho^2 a^2$, then by choosing $\\rho \\le \\rho_0$, a universal constant, we can apply \\cite[Lemma 3.2]{LMNN20}.\nFor $\\eps \\ll 1$ to be defined later on, let \n\\begin{equation}\n\\label{rhoDef}\n\\rho = \\eps a^{-1}.\n\\end{equation}\nAn application of the arguments in \\cite[\\S 3.2]{LMNN20} then shows that there exists $\\phi : \\Om \\to \\R$ with the properties that\n\\begin{align}\n&\\LP \\phi - W \\phi = 0 \\text{ in } \\Om\n\\nonumber \\\\\n&\\phi -1 \\in W^{1,2}_0(\\Om)\n\\nonumber \\\\\n&\\norm{\\phi -1}_\\iny \\le c_b \\pr{\\rho a}^2 = c_b \\eps^2,\n\\label{vpuBound}\n\\end{align}\nwhere $c_b$ is a universal constant that depends on $c_P$, and we have used \\eqref{rhoDef}.\nBy extending $\\phi$ to equal $1$ across $F_0 \\bigcup F_1$, it is then shown in \\cite[Lemma 4.1]{LMNN20} that $\\disp f := \\frac v \\phi \\in W^{1,2}_{\\loc}(B_R)$ is a weak solution to the divergence-form equation\n$$\\di\\pr{\\phi^2 \\gr f} = 0 \\; \\text{ in } \\Om_1.$$\nMoreover, the bound in \\eqref{vpuBound} implies that for any $z \\in B_R$,\n\\begin{equation}\n\\label{ufComparison}\n\\pr{1 - c_b \\eps^2} \\abs{f(z)} \\le \\abs{v(z)} \\le \\pr{1 + c_b \\eps^2} \\abs{f(z)}.\n\\end{equation}\n\nWe introduce the Beltrami coefficient $\\mu$, defined as follows:\n\\begin{equation*}\n\\mu = \\left\\{ \\begin{array}{ll}\n\\frac{1 - \\phi^2}{1 + \\phi^2} \\frac{f_x + i f_y}{f_x - i f_y} & \\text{ in } \\Om_1 \\text{ when } \\, \\gr f \\ne 0 \\\\\n0 & \\text{ otherwise}\n\\end{array} \\right..\n\\end{equation*}\nSince $\\abs{\\mu} \\lesssim \\eps^2$, then as shown in \\cite{AIM09}, there exists a $K$-quasiconformal homeomorphism of the complex plane where $K \\le 1 + C_K \\eps^2$ and $C_K$ depends on $c_b$.\nThat is, there exists some $w \\in W^{1,2}_{\\loc}$ that satisfies the Beltrami equation $\\disp \\frac{\\del w}{\\del \\overline{z}} = \\mu \\frac{\\del w}{\\del z}$. \nIn fact, an application of the Riemann uniformization theorem shows that there exists a $K$-quasiconformal homeomorphism $g: B_R \\to B_R$ that is onto with $g(0) = 0$.\nMoreover, the function $h : = f \\circ g^{-1}$ is harmonic in $g(\\Om_1)$.\n\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\nNow assume that \\eqref{claim} holds for $k-1 \\in \\N$.\nDefine $S_k = \\frac 1 2 \\abs{z_{k-1}}$ and $R_k = \\abs{z_{k-1}}^{\\al_{k-1} +\\de}$. \nComparing with \\eqref{zkNorm}, we see that $\\abs{z_k} = R_k + S_k - 1$.\nSince $B_1(z_{k-1}) \\su B_{R_k - S_k}(z_k)$, then the inductive hypothesis shows that there exists $z_0 \\in \\overline{B}_{R_k - S_k}(z_k)$ so that\n\\begin{align*}\n\\abs{v(z_0)}\n\\ge \\exp\\pr{- \\abs{z_{k-1}}^{\\al_{k-1} + \\de}}\n= \\exp\\pr{- R_k}.\n\\end{align*}\nThe bound in \\eqref{uBound} shows that $\\disp \\abs{v(z)} = \\abs{u(T_\\al(z))} \\le \\exp\\pr{c_0 \\abs{z}^{\\al\\pr{1 - \\frac N 2}}} = \\exp\\pr{c_0 \\abs{z}}$ from which it follows that\n\\begin{align*}\n\\norm{v}_{L^\\iny\\pr{B_{R_k}(z_k)}}\n\\le \\exp\\pr{c_0 \\pr{\\abs{z_k} + R_k}}\n\\le \\exp\\pr{\\frac {5c_0} 2 R_k}.\n\\end{align*}\nLet $W : \\R^2_+ \\to \\R$ be given by $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z))= \\al^2 \\abs{w}^{2 - \\frac 2 \\al} V(w)$ so that by \\eqref{Vbound},\n\\begin{align*}\n\\abs{W(z)} = \\al^2 \\abs{w}^{2 - \\frac 2 \\al} \\abs{V(w)} \\le a_0^2 \\al^2 \\abs{w}^{2 - N - \\frac 2 \\al} = \\pr{a_0 \\al}^2.\n\\end{align*}\nAn application of Lemma \\ref{transformEqLemma} with assumption \\eqref{ellipEq} shows that\n\\begin{equation*}\n- \\LP v + W v = 0 \\; \\text{ in } \\R^2_+.\n\\end{equation*}\nAs $R_k \\ge \\bar{r}_1$ implies that $R_k \\ge \\hat R_0(a_0 \\al)$ and $S_k \\in \\pr{2^8, \\frac {R_k} 2}$, Proposition \\ref{InductiveProp} is applicable with $a = a_0 \\al$, $R= R_k$, $S= S_k$, $L = R_k$, and $c_1 = \\frac {5c_0} 2$.\nWith $r = 1$, Proposition \\ref{InductiveProp} shows that\n\\begin{align}\n\\label{vkBd}\n\\norm{v}_{L^\\iny(B_1(z_k))}\n&\\ge \\exp\\pr{-\\tau_k(R_k) \\log R_k},\n\\end{align}\nwhere\n\\begin{equation}\n\\label{taukDefn}\n\\begin{aligned}\n\\tau_k(R_k) &= \\max \\set{3 a_0 \\al C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R_k \\sqrt{\\log R_k}, 2^{15} \\pr{\\frac {5c_0} 2 + 1 + \\frac {c_d}{R_k \\log R_k}} R_k^{2 - \\frac 1 {\\al_{k-1} + \\de}}} \\\\\n&+\\frac 1 5 \\pr{R_k + \\frac{c_d}{\\log R_k}}.\n\\end{aligned}\n\\end{equation}\nBecause $R_k \\ge \\bar{r}_1$ and $\\al_{k - 1} > 1$, conditions \\eqref{rBig2} and \\eqref{rBig3} imply that\n\\begin{align}\n\\label{tauKBound}\n\\tau_k(R_k) &\\le \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5} R_k^{2 - \\frac 1 {\\al_{k-1} + \\de}} \n\\end{align}\nwhile condition \\eqref{rBig4} implies that\n$$ \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5} \\log R_k \\le R_k^{\\frac {\\de^2} 2}.$$\nSince $\\al > 1$ and $\\de \\le 1$ implies that $2 - \\frac{1}{\\al + \\de} + \\frac{\\de^2}2 < 2 - \\frac 1 \\al + \\de$, then combining these bounds shows that\n$$\\tau_k(R_k) \\log R_k \\le R_k^{2 - \\frac 1 {\\al_{k-1}} + \\de} = R_k^{\\al_{k} + \\de}.$$\nReturning to \\eqref{vkBd}, we conclude that\n\\begin{align*}\n\\norm{v}_{L^\\iny(B_1(z_k))}\n&\\ge \\exp\\pr{-R_k^{\\al_{k} + \\de}}\n\\ge \\exp\\pr{-\\abs{z_k}^{\\al_{k} + \\de}},\n\\end{align*}\nestablishing the claim given by \\eqref{claim}.",
    "post_theorem_intro_text_len": 4239,
    "post_theorem_intro_text": "The results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\beta = \\frac{4 - 2N}{3} = \\frac 4 3 \\left( 1 - \\frac N 2 \\right)  > 1 - \\frac N 2 + \\varepsilon$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\nAs illustrated by the following example, Theorem \\ref{mainThm} is sharp (up to $\\varepsilon$) for all $N \\in (0, 2)$.\nFix $N \\in (0, 2)$, then set $u(z) = \\exp\\pr{- \\left\\vertz\\right\\vert^{1 - \\frac N 2}}$.\nA computation shows that $u$ satisfies \\eqref{ellipEq} where\n$$V(z) := \\left( 1 - \\frac N 2 \\right) ^2 \\pr{\\left\\vertz\\right\\vert^{1 - \\frac N 2} - 1}\\left\\vertz\\right\\vert^{-1 - \\frac N 2}$$ \nsatisfies $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^{-N}$.\nOn the other hand, for any $\\beta > 0$, with $u(z) = \\left\\vertz\\right\\vert^{-\\beta}$ on $\\left\\vertz\\right\\vert > 1$, we see that $\\displaystyle \\Delta u = \\beta^2 \\left\\vertz\\right\\vert^{-\\beta-2}$ and therefore $u$ satisfies \\eqref{ellipEq} on an exterior domain with $V(z) := \\beta^2\\left\\vertz\\right\\vert^{-2}$.\nIn particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.\n\nTo prove Theorem \\ref{mainThm}, we use an iterative argument that is reminiscent of the one in \\cite{Dav14}, see also \\cite{LW14, DKW19, Dav20a, Dav25}.\nTo initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\beta = 1$.\nThis result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \\cite{LMNN20}, and we formulate it in Theorem \\ref{LandisGrowth} below.\nThe iteration argument then relies on repeated applications of Proposition \\ref{InductiveProp} which is proved using the ideas from \\cite{LMNN20}.\nRoughly speaking, Proposition \\ref{InductiveProp} shows that if an estimate like \\eqref{est} holds with $\\beta = \\be_0$, then for some $x_1$ with $\\left\\vertx_1\\right\\vert \\gg \\left\\vertx_0\\right\\vert$, another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and $\\beta = \\be_1 \\in \\left[1, \\be_0\\right]$.\nWhen $\\be_0 = 1$, Proposition \\ref{InductiveProp} isn't useful, but when $\\be_0 > 1$, we can decrease the exponent, i.e., make $\\be_1 < \\be_0$.\nTherefore, to benefit from the iteration argument, we need to transform to a situation where $\\be_0 > 1$.\nWe observe that if $u$ is composed with the real-variable version of the conformal transformation $z \\mapsto z^\\alpha$, then the new function also satisfies a Schr\\\"odinger equation.\nBy choosing $\\alpha > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \\eqref{est} with $\\beta > 1$.\nBy repeatedly applying Proposition \\ref{InductiveProp} to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.\nFinally, to reach the conclusion, we undo the change of variables.\n\nWe use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.\nGeneric constants are denoted by $c, C$ and may change from line to line without comment.\nSpecific constants will be indicated by subscripts.\n\nThe article is organized as follows.\nIn Section \\ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.\nThe content of this section is very similar to \\cite[Section 5]{LMNN20} and \\cite[Section 2]{Dav24}.\nThe iterative result described by Proposition \\ref{InductiveProp} is the content of Section \\ref{localProof}.\nProposition \\ref{InductiveProp} is a three-ball inequality for solutions to Schr\\\"odinger equations and its proof relies on the results from Section \\ref{harmonic}. In Section \\ref{TransMaps}, we introduce the real-valued versions of $z \\mapsto z^\\alpha$ and record some of their properties.\nIn particular, we show how solutions behave when they are composed with these transformations.\nFinally, the proof of Theorem \\ref{mainThm} is presented in Section \\ref{MainProof}.",
    "sketch": "To prove Theorem~\\ref{mainThm}, the paper uses “an iterative argument that is reminiscent of the one in \\cite{Dav14}.”\n\nKey steps (as described):\n\\begin{itemize}\n\\item \\textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\beta = 1$,” namely the real-valued planar Landis growth result proved in \\cite{LMNN20} and stated as Theorem~\\ref{LandisGrowth}.\n\\item \\textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of Proposition~\\ref{InductiveProp}.” Roughly, this proposition shows that if an estimate like \\eqref{est} holds with exponent $\\beta=\\be_0$, then for some $x_1$ with “$|x_1|\\gg |x_0|$,” another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and “$\\beta=\\be_1\\in[1,\\be_0]$.” Moreover, “when $\\be_0>1$, we can decrease the exponent, i.e., make $\\be_1<\\be_0$.”\n\\item \\textbf{Create a regime with $\\beta>1$ via a power map.} Since “when $\\be_0=1$, Proposition~\\ref{InductiveProp} isn’t useful,” they “transform to a situation where $\\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\\mapsto z^\\alpha$,” noting the composed function still “satisfies a Schr\\\"odinger equation.” Choosing “$\\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of \\eqref{est} “with $\\beta>1$.”\n\\item \\textbf{Iterate to push $\\beta$ down to $1$.} “By repeatedly applying Proposition~\\ref{InductiveProp} to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.”\n\\item \\textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.”\n\\end{itemize}\n",
    "expanded_sketch": "To prove the main theorem, the paper uses “an iterative argument that is reminiscent of the one in \\cite{Dav14}.”\n\nKey steps (as described):\n\\begin{itemize}\n\\item \\textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}\nwith $\\beta = 1$,” namely the real-valued planar Landis growth result proved in \\cite{LMNN20} and stated as\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\n\\item \\textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}\nRoughly, this proposition shows that if an estimate like the equation above holds with exponent $\\beta=\\be_0$, then for some $x_1$ with “$|x_1|\\gg |x_0|$,” another estimate like the equation above holds with $x_0$ replaced by $x_1$ and “$\\beta=\\be_1\\in[1,\\be_0]$.” Moreover, “when $\\be_0>1$, we can decrease the exponent, i.e., make $\\be_1<\\be_0$.”\n\n\\item \\textbf{Create a regime with $\\beta>1$ via a power map.} Since “when $\\be_0=1$, the iterative proposition above isn’t useful,” they “transform to a situation where $\\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\\mapsto z^\\alpha$,” noting the composed function still “satisfies a Schr\\\"odinger equation.” Choosing “$\\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of the equation above “with $\\beta>1$.”\n\n\\item \\textbf{Iterate to push $\\beta$ down to $1$.} “By repeatedly applying the iterative proposition above to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.”\n\n\\item \\textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.” This completes the proof of the main theorem.\n\\end{itemize}\n",
    "expanded_theorem": "[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\left\\vertV(w)\\right\\vert \\le a_0^2 \\left< w \\right>^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\Delta u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\left\\vertu(0)\\right\\vert = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\left\\vertu(w)\\right\\vert \\le \\exp\\pr{c_0 \\left\\vertw\\right\\vert^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\varepsilon \\in \\left( 0, \\frac N 2 \\right) $, there exists ${R}_0(N, a_0, c_0, \\varepsilon) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\left\\vertw_0\\right\\vert = R} \\left\\| u\\right\\|_{L^\\infty\\left( B_1(w_0) \\right) } \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\varepsilon}}.\n\\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $a_0\\ge 1$ and $N\\in(0,2)$. Suppose $V:\\mathbb{R}^2\\to\\mathbb{R}$ satisfies\n\\[\n|V(w)|\\le a_0^2\\langle w\\rangle^{-N}\\quad\\text{for all }w\\in\\mathbb{R}^2,\n\\]\nwhere $\\langle w\\rangle=\\sqrt{1+|w|^2}$. Let $u:\\mathbb{R}^2\\to\\mathbb{R}$ solve\n\\[\n-\\Delta u+Vu=0\\quad\\text{in }\\mathbb{R}^2,\n\\]\nwith\n\\[\n|u(0)|=1\n\\]\nand the growth bound\n\\[\n|u(w)|\\le \\exp\\big(c_0|w|^{1-N/2}\\big)\\quad\\text{for every }w\\in\\mathbb{R}^2.\n\\]\nHere $B_1(w_0)$ denotes the unit ball centered at $w_0$. Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\\varepsilon)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2}\\big).\n\\]"
        },
        {
          "label": "C",
          "text": "There exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R\\big).\n\\]"
        },
        {
          "label": "D",
          "text": "For every $\\varepsilon\\in(0,N/2)$ and every $R_0>0$, there exists $R\\ge R_0$ such that\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\n\\]"
        },
        {
          "label": "E",
          "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\\varepsilon)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|\\le R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "epsilon-loss in final exponent",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "replace sharp exponent $1-N/2+\\varepsilon$ by weaker linear exponent",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "eventual-for-all-$R$ conclusion weakened to infinitely-many-$R$ statement",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "sphere condition $|w_0|=R$ replaced by ball condition $|w_0|\\le R$",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the hypotheses and asks for the correct large-radius conclusion, but it does not explicitly reveal the exact bound, quantifier structure, or the infimum/supremum form of the answer."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: the assumptions are essentially those of a specific quantitative unique continuation result, and the correct option is the theorem’s conclusion. However, the presence of nearby alternatives with altered exponents and quantifiers makes it slightly more than a verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the sharp statement from stronger, weaker, or quantifier-tampered alternatives. Still, the item primarily tests recognition of the exact theorem statement rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one is an overstrong sharpness trap, one is a weaker true statement, one alters quantifier dependence, and one changes infimum to supremum. These reflect realistic mathematical failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-identification MCQ with high-quality distractors and no answer leakage, but it leans more toward precise theorem recall than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.09873v1",
    "paper_link": "http://arxiv.org/abs/2512.09873v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}",
      "start_pos": 32586,
      "end_pos": 33189,
      "label": "thm: main-ucp"
    },
    "ref_dict": {
      "eq: wave-eq-0": "\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T),\n\\end{equation}",
      "wave:nece:2": "\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation}",
      "def:classSc": "\\begin{definition}[Symmetric function pair]\\label{def:classSc}\nWe say \n$(f,g)$ is a symmetric  function pair if  there exists a decomposition pair $\\{(A_k, B_k),\\ k = 1,\\dots, K\\}$, a sequence of different numbers $\\{s_k\\}_{k= 1}^K$ such that \n\\begin{equation*}\n f=\\sum_{1\\leq k\\leq K}s_k\\chi_{A_k}, \\quad g=\\sum_{1\\leq k\\leq K}s_k\\chi_{B_k},\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all symmetric function pairs is denoted by $\\mathcal{S}^2_c$.\n\\end{definition}",
      "fig:GCCfail1": "\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times \\T$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times \\T$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{theorem}\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times \\T$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}\n\\end{theorem}\n\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}",
      "def:classS": "\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}",
      "eq:GCC-wave:0": "\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation}",
      "eq:GCC:measure": "\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}",
      "eq: wave-ob": "\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1(\\T)}+\\|u_1\\|^2_{L^2(\\T)}\\leq C\\int_G|\\partial_tu(t,x)|^2\\d t\\d x.\n\\end{equation}",
      "eq: wave-eq": "\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}",
      "def:GCC": "\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}",
      "def:weakGCC": "\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}",
      "fig:OSC-role": "\\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLe",
      "def:OSC": "\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}"
    },
    "pre_theorem_intro_text_len": 17122,
    "pre_theorem_intro_text": "Let $T > 0$. Let $G \\subset [0,T] \\times \\mathbb{T}$ be a spacetime measurable set with positive measure.\nConsider the {\\it observability problem} of the wave equation: whether there exists some $C= C(G)>0$ such that every solution $u$ to \n\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}),\n\\end{equation}\nsatisfies\n\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1({\\mathbb T})}+\\|u_1\\|^2_{L^2({\\mathbb T})}\\leq C\\int_G|\\partial_tu(t,x)|^2{\\,\\rm d} t{\\,\\rm d} x.\n\\end{equation}\n\nThanks to the classical Hilbert uniqueness method and an argument to raise regularity (see Appendix \\ref{sec: HUM-app} for details on this reduction), the observability of \\eqref{eq: wave-ob} is equivalent to the {\\it exact controllability} of\nthe controlled wave equation on $[0,T]\\times{\\mathbb T}$ with control force $f\\in L^2(G)$:\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}).\n\\end{equation}\n\nOur primary objective is to seek the sufficient and necessary geometric conditions on $G$ such that the unique continuation and observability hold.\n\n\\subsection{History and setting}\nFrom the early work of Russell \\cite{Russell-1}, the studies of the controllability and observability for wave equations have been central topics in control theory. \n\n\\subsubsection{Geometric control condition}\nIn the pioneering work of Rauch and Taylor \\cite{RT}, they first related the observability to a geometric condition on the damped region $\\omega$ and the rays of geometric optics in the boundaryless case. \nLater, in another pioneering work of Bardos-Lebeau-Rauch \\cite{BLR-gcc}, the well-known \\textit{Geometric control condition} is introduced: for an open observed region $\\omega$, every generalized ray should meet $\\omega$ in a finite time. \n\nSince then, it has become one of the most natural assumptions for the controlled waves. In the existing literature, the observation is most often made on cylindrical domains $G=(0,T)\\times \\omega$, with $\\omega$ being an open subset. Under suitable smooth conditions, it is well-known that GCC is sufficient, and depending on the domain, necessary for the observability in the cylindrical domains $(0,T)\\times \\omega$, see \\cite{BLR-gcc,BG-97}. When considering stabilization problems, GCC is also a useful condition for the exponential decay of energy. We refer to \\cite{RT,Haraux}. Otherwise, one may have logarithmic type of energy decay results \\cite{LR-97, Burq-98}. GCC also plays a role in practical issues, such as sensor designs, tomography techniques used for imaging bodies (see \\cite{RouLebeauAnalPDE2017} for example), etc. For a comprehensive reference of the numerical study, we refer to \\cite{zuazua-review} and its references therein. \n\nFinally, we give a very brief overview of the boundary control case. There are also fruitful results in this direction. For related GCC, we refer to \\cite{Lebeau-boundary}. In particular, for 1D wave equations, one can find many nonlinear results \\cite{Li-1,LY-2} and the references therein.\n\n\\subsubsection{Unique continuation}\nA qualitative version of observability is the unique continuation property.\nThe easiest way to ensure this property is to apply the analyticity based on Holmgren's theorem. Besides, H\\\"ormander's pseudo-convexity condition and Carleman estimates are powerful tools dealing with the unique continuation problem. In this direction, there is a large literature such as \\cite{RZ-98,Tataru,Hormander-92,Hormander-96} and more recent work includes \\cite{Laurent-Leautaud-2019, MS-21,FCL,Shao}. Here we point out that to construct the appropriate weight to apply Carleman estimates, it is crucial to understand the behavior across the suitable spacetime surfaces, which is more delicate than considering a cylindrical region.\n\n\\subsubsection{Our setting: spacetime measurable observable region}\nRecently, researchers started to focus on \n the spacetime setting \\cite{Castro-Cindea-Munch-2014, RouLebeauAnalPDE2017,Shao,PK}. Meanwhile, the study of the case where $G$ is a spacetime region is far from complete, even for open regions.  \n\n In this paper, we focus on the setting that $G\\subset[0,T]\\times{\\mathbb T}$ is measurable of positive measure. The following condition may be viewed as the natural analogue of the standard geometric control condition in this spacetime measurable setting. \n\n\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\nFrom now on, in this paper, \\blackhyperref{def:GCC}{({\\rm GCC})}  refers to the preceding definition with respect to \\eqref{eq:GCC-wave:0}. \nIt is well-known that under the standard setting, namely the observable region is a cylinder $[0, T]\\times \\omega$, GCC yields several important properties: weak observability and the necessity of observability. These results can be generalized to the spacetime \\blackhyperref{def:GCC}{({\\rm GCC})} for a spacetime measurable observable region. We put the former result in Section \\ref{sec: weak-ob-sec} and the later in Appendix \\ref{sec:app:3}.  \n\n\\subsection{A new symmetry condition}\n\nExtend the system $2\\pi-$periodically to $x\\in \\mathbb{R}$,  and introduce the null coordinate:\n\\begin{equation}\n    \\xi=x+t \\; \\textrm{ and } \\; \\eta=x-t. \\notag\n\\end{equation}\nUnder this new coordinate,  \n\\begin{equation*}\n  2  u_{\\xi}= u_{x}+ u_t \\; \\textrm{ and } \\;  2  u_{\\eta}= u_{x}- u_t\n\\end{equation*}\nand the wave equation \\eqref{eq: wave-eq} becomes\\footnote{We considered the equation on $U(\\xi, \\eta):= u(t, x)$ and $F(\\xi, \\eta)= f(t, x)$. For ease of notation, we still denote $U$ by $u$ and $F$ by $f$.  }\n\\begin{equation}\n    4 \\partial_{\\xi} \\partial_{\\eta} u=  f \\mathbf{1}_G. \\notag\n\\end{equation}\n\nFor any $\\xi_0\\in\\mathbb{R}$, we denote the  line $\\{(t,x):x+t=\\xi_0\\}$ by $L_{\\xi=\\xi_0}$,  and call it the $\\xi$-characteristic. Similarly, we denote the  line $\\{(t,x):x- t=\\eta_0\\}$ by $L_{\\eta=\\eta_0}$ and  call it $L_{\\eta=\\eta_0}$ the $\\eta$-characteristic. Thus the \\blackhyperref{def:GCC}{({\\rm GCC})} is equivalent to\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation} \nWe further define measurable cylinders for any measurable sets $A, B\\subset [0, 2\\pi]$ as\n\\begin{equation}\\label{eq: cylinder}\nL_{\\xi\\in A}:=\\bigcup_{\\xi_0\\in A}L_{\\xi=\\xi_0} \\; \\textrm{ and } \\; L_{\\eta\\in B}:=\\bigcup_{\\eta_0\\in B}L_{\\eta=\\eta_0}.\n\\end{equation}\n\nIntroduce the following symmetry condition:\n\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\mathbb{R}^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\mathbb{R}^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\vspace{2mm}\n\nWe call a pair $(A, B)$ trivial, if $|A|= |B|= 0$ or $|A|= |B|= 2\\pi$.  In principle, we are interested in {\\it non-trivial} pairs $(A, B)$. Since the observable symmetry condition is automatically satisfied for trivial pairs ($A, B$). \n\nThe observable symmetry condition is very important in our work, see Fig. \\ref{fig:OSC-role}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (192,38.4) .. controls (192,32.38) and (196.88,27.5) .. (202.9,27.5) -- (452.1,27.5) .. controls (458.12,27.5) and (463,32.38) .. (463,38.4) -- (463,71.1) .. controls (463,77.12) and (458.12,82) .. (452.1,82) -- (202.9,82) .. controls (196.88,82) and (192,77.12) .. (192,71.1) -- cycle ;\n\\draw   (72,158.7) .. controls (72,152.51) and (77.01,147.5) .. (83.2,147.5) -- (199.8,147.5) .. controls (205.99,147.5) and (211,152.51) .. (211,158.7) -- (211,192.3) .. controls (211,198.49) and (205.99,203.5) .. (199.8,203.5) -- (83.2,203.5) .. controls (77.01,203.5) and (72,198.49) .. (72,192.3) -- cycle ;\n\\draw   (254,151.3) .. controls (254,143.13) and (260.63,136.5) .. (268.8,136.5) -- (393.2,136.5) .. controls (401.37,136.5) and (408,143.13) .. (408,151.3) -- (408,195.7) .. controls (408,203.87) and (401.37,210.5) .. (393.2,210.5) -- (268.8,210.5) .. controls (260.63,210.5) and (254,203.87) .. (254,195.7) -- cycle ;\n\\draw   (436,158.7) .. controls (436,152.51) and (441.01,147.5) .. (447.2,147.5) -- (563.8,147.5) .. controls (569.99,147.5) and (575,152.51) .. (575,158.7) -- (575,192.3) .. controls (575,198.49) and (569.99,203.5) .. (563.8,203.5) -- (447.2,203.5) .. controls (441.01,203.5) and (436,198.49) .. (436,192.3) -- cycle ;\n\\draw    (226,82.5) -- (144.57,146.27) ;\n\\draw [shift={(143,147.5)}, rotate = 321.93] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (329,82.5) -- (329,134.5) ;\n\\draw [shift={(329,136.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (430,82.5) -- (509.44,146.25) ;\n\\draw [shift={(511,147.5)}, rotate = 218.75] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (219,47) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observable symmetry condition};\n\\draw (81,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Conservation law};\n\\draw (282,146.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\begin{minipage}[lt]{66.78pt}\\setlength\\topsep{0pt}\n\\begin{center}\nControllability \\\\Observability \\\\UCP\n\\end{center}\n\n\\end{minipage}};\n\\draw (455,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Symetric \\ \\ \\  class };\n\\draw (515,167.9) node [anchor=north west][inner sep=0.75pt]  [font=\\small]  {$\\mathcal{S}^{2}$};\n\\draw (128,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Noether}};\n\\draw (333,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize necessary}};\n\\draw (486,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize generate}};\n\n\\end{tikzpicture}\n\n    \\caption{The role of (OSC) in this paper}\n    \\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)=   \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\;  \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{remark}\n  This proposition can be understood as a variant of Noether’s theorem\\footnote{Noether’s theorem: ``Every continuous symmetry of the action (or the equations of motion, in a suitable sense) implies a conserved quantity.''} adapted to a controlled wave equation: a geometric symmetry of the forcing region yields a conserved functional of the solution. Indeed,   \nthe key hypothesis is that the region $G$ satisfies the  {\\it observable symmetry condition} for ($A, B$). The conclusion shows that the energy $I(t)$ is invariant in time, regardless of the force term. \n\\end{remark}\n\nThis new type of symmetry provides a {\\it necessary condition} for the controllability and observability of \\eqref{eq: wave-eq}, and even a {\\it necessary condition} for the unique continuation of  \\eqref{eq: wave-ob}. See Section \\ref{sec:symmetry} for the detailed construction of counterexamples.  Moreover, this construction gives rise to two classes of function pairs satisfying the symmetry condition, $\\mathcal{S}^2_c$ and $\\mathcal{S}^2$, in Definitions \\ref{def:classSc} and \\ref{def:classS}.\n\n\\begin{figure}[htp]\n    \\centering\n\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times {\\mathbb T}$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times {\\mathbb T}$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.",
    "context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}",
    "full_context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}\n\n\\begin{figure}[htp]\n \\centering\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times \\T$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n \\item[(1)] The observability inequality \\eqref{eq: wave-ob} holds on $G$;\n \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\n\\begin{proposition}[Unique continuation implies weak GCC]\nLet $T>0$ and $G\\in [0, T]\\times \\T$. Assume that the following unique continuation property holds\n$$\n\\mbox{ Let u be a solution of \\eqref{eq: wave-eq-0} and } u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0. \n$$\nThen $G$ satisfies the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}. \n\\end{proposition}\n\\begin{proof}\nWe argue by contradiction. Suppose that $G$ does not satisfy the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, there exists a subset $A\\subset \\T$ such that $\\mbox{meas}_\\R(A)>0$ and\n\\begin{align*}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0 \\quad \\mbox{ or } \\quad \n \\mbox{meas}_\\R(G\\cap L_{\\xi\\in A})=0.\n\\end{align*}\nWithout loss of generality, we only consider the case\n\\begin{align}\\label{equ-ucp-wgcc-1}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0. \n\\end{align}\nSince $A$ has positive measure, one can find a non-zero function $\\phi$ such that\n\\begin{align}\\label{equ-ucp-wgcc-2}\n \\int_A \\phi(x)\\d x=0, \\quad \\int_A|\\phi| \\d x=\\mbox{meas}_{\\R}(A)>0. \n\\end{align}\nSet \n$$\nu_{0x}=-u_1=\\chi_A \\phi.\n$$\nThis is possible since $\\int_\\T u_{0,x}\\d x =\\int_A\\phi \\d x =0$. Then we have\n$$\n\\partial_\\xi u|_{t=0}=u_{0x}+u_1=0, \\quad \\partial_\\eta u|_{t=0}=u_{0x}-u_1=2\\chi_A\\phi.\n$$\nIt follows that\n$$\n\\partial_\\xi u=0 \\quad \\mbox{ on } [0,T]\\times \\T\n$$\nand\n$$\n\\partial_\\eta u= 0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThus we have\n$$\nu_t=\\frac{1}{2}(\\partial_\\xi u-\\partial_\\eta u)=0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThanks to \\eqref{equ-ucp-wgcc-1}, we see $G\\subset L_{\\eta\\in \\T\\backslash A}$, and thus $u_t=0$ on $G$. By the UCP, we must have $\\partial_\\eta u|_{t=0}=2\\chi_A\\phi\\equiv 0$, which leads a contradiction with \\eqref{equ-ucp-wgcc-2}.\n\\end{proof}\n\n\\subsection{OSC and weak GCC imply unique continuation }\\label{sec: OSC+WGCC>UCP}\nIn this sequel, we generalize the result in Proposition \\ref{prop-ucp-S2}, which forms the following proposition.\n\\begin{proposition}[Weak GCC implies UCP up to $\\mathcal{S}^2$]\\label{prop-weakGCC-ucp-S2}\n Let $T>0$. Let $G\\in [0, T]\\times \\T$ satisfy the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}.\n If $u$ solves the wave equation \\eqref{eq: wave-eq-0} and $u_t = 0$ on $G$, then its initial state $(\\partial_\\xi u, \\partial_\\eta u)|_{t=0}$ belongs to the symmetry function pair class $\\mathcal{S}^2$.\n\\end{proposition}\n\\begin{proof}\nWe use the same notation in the proof of Proposition \\ref{prop-ucp-S2}. Recall that we have the relation \\eqref{equ-xi-eta-exchange}, namely \n\\begin{equation*} \\partial_{\\eta}u|_{\\{t= 0\\}\\cap L_{\\eta= \\eta_0}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)= \\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}\n\\end{equation*}\nup to sets with zero measure.\nNamely, the value of $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ determines almost every value of $\\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}$. Since $G$ satisfies the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, we know \n$$\n\\mbox{meas}_\\R(\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}})>0.\n$$\nIn other words, $\\partial_{\\xi}u|_{\\{t= 0\\}}$ equals to $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ at least on a positive measure subset set of $\\T$. Thus the set\n$$\n\\mathcal{U}_{(\\xi_0,\\eta_0)}:=\\{x_0\\in \\T: \\partial_{\\xi}u|_{\\{t= 0,x=x_0\\}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)\\}\n$$\nhas positive measure. Repeating this process for other point $(\\xi_0',\\eta_0')\\in [0,T]\\times \\T$, we shall find that, there exists a family sets $\\mathcal{U}_\\ell (\\ell\\in I)$, $I$ is a index set, such that\n\\begin{align}\\label{equ-Ul-129}\n \\mathcal{U}_\\ell\\cap \\mathcal{U}_{\\ell'}=\\emptyset \\mbox{ for } \\ell\\neq \\ell', \\quad \\mbox{meas}_\\R(\\mathcal{U}_\\ell)>0 \\mbox{ for } \\ell\\in I , \\quad \\T=\\cup_{\\ell\\in I}\\mathcal{U}_\\ell \n\\end{align} \nand $\\partial_{\\xi}u|_{\\{t= 0\\}}$ is a constant on each $\\mathcal{U}_\\ell$.\nOne can show that, under the restrictions \\eqref{equ-Ul-129}, the index set $I$ is at most countable, thus, after relabeling if necessary, $\\{\\mathcal{U}_\\ell, \\ell\\ge 1\\}$ is a weak decomposition of $\\T$. Moreover, there exists a sequence of different numbers $\\{s_\\ell\\}$ such that\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}}=s_\\ell \\mbox{ on } \\mathcal{U}_\\ell.\n$$\nNote that $\\partial_{\\xi}u|_{\\{t= 0\\}}\\in L^2(\\T)$, and $\\mathcal{U}_\\ell$ are disjoint for different $\\ell$, we find\n$$\n\\|\\partial_{\\xi}u|_{\\{t= 0\\}}\\|_{L^2(\\T)}^2=\\sum_\\ell |s_\\ell|^2\\mbox{meas}_\\R(\\mathcal{U}_\\ell)<\\infty.\n$$\nThus we obtain\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{U}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nSimilarly, we can find a weak decomposition $\\mathcal{V}_{\\ell}(\\ell\\geq 1)$ such that\n$$\n\\partial_{\\eta}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{V}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nIn summary, we find that $(\\partial_{\\xi}u|_{\\{t= 0\\}}, \\partial_{\\eta}u|_{\\{t= 0\\}})\\in \\mathcal{S}^2$.\n\\end{proof}\n\nThe following theorem gives a sufficient and necessary condition of the observable set for the transport equation. We only state the result for \\eqref{eq: transport-eq}, the reader easily figures out the the necessary modifications for \\eqref{eq: transport-eq-2}.\n\\begin{theorem}\\label{thm-tran-suff-nece}\nLet $T>0$ and $G$ be a measurable subset of $[0,T]\\times \\T$ with positive measure. Then the observability inequality\n\\begin{align}\\label{equ-tran-suff-necc-1}\n \\|u_0\\|^2_{L^2(\\T)}\\leq C\\int_G|u(t,x)|^2\\d x \\d t \n\\end{align}\nholds with a positive constant $C>0$ for all solutions to the transport equation \\eqref{eq: transport-eq} if and only if there exists a constant $c_0>0$ such that \n\\begin{equation}\\label{equ-tran-suff-necc-2}\n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s\\geq c_0 \\quad \\mbox{ for a.e. } x\\in\\T \n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe direction $\\eqref{equ-tran-suff-necc-2}\\Longrightarrow \\eqref{equ-tran-suff-necc-1}$ follows clearly from Proposition \\ref{prop: tran-eq-ob}. To show the inverse direction, namely $\\eqref{equ-tran-suff-necc-1}\\Longrightarrow \\eqref{equ-tran-suff-necc-2}$, we use the contradiction argument. Suppose that \\eqref{equ-tran-suff-necc-2} is not true, then for any $\\varepsilon>0$, there exists a set $E_\\varepsilon\\subset \\T$ with positive measure $|E_\\varepsilon|>0$ such that\n\\begin{align}\\label{equ-tran-suff-necc-3} \n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s<\\varepsilon \\quad \\mbox{ for a.e. } x\\in E_\\varepsilon. \n\\end{align}\nLet $u_{0\\varepsilon}(x)=1_{E_\\varepsilon}(x)$. Then $u_{0\\varepsilon}\\in L^2(\\T)$ and its support is contained in $E_\\varepsilon$. Then, similar to the proof in Proposition \\ref{prop: tran-eq-ob}, we have\n\\begin{align*}\n\\int_G|u(t,x)|^2\\d x \\d t =\\int_{\\T}|u_{0\\varepsilon}(x)|^2\\int_{\\T}\\mathbf{1}_G(t,x-t)\\d t\\d x\\leq \\varepsilon \\|u_{0\\varepsilon}\\|^2_{L^2(\\T)}. \n\\end{align*}\nTaking $\\varepsilon>0$ small enough, say $\\varepsilon=C/2$, we obtain a contradiction with \\eqref{equ-tran-suff-necc-1}. \n\\end{proof}\n\\subsection{GCC is necessary for the wave observability}\\label{sec:app:3}\n\\begin{corollary}\\label{wave-GCC-nece}\nIf the observability \\eqref{eq: wave-ob} holds on some observation set $G\\subset [0,T]\\times \\T$, then $G$ needs to satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}}.\n\\end{corollary}\n\\begin{proof}",
    "post_theorem_intro_text_len": 6409,
    "post_theorem_intro_text": "\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}\n\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+WGCC=UCP}\n\\end{figure}\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times {\\mathbb T}$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n    \\item[(1)] The observability inequality \\eqref{eq: wave-ob}  holds on $G$;\n    \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\nTo the best of our knowledge, this work provides the first necessary and sufficient geometric condition for observability of the wave equation on spacetime region. The observable symmetry condition plays a central role: it completes the classical GCC by capturing the additional geometric structure. This viewpoint also leads to necessary and sufficient conditions in a variety of other important geometric configurations, such as $[0, T]\\times \\omega$,  measurable Cartesian products  $E_t\\times F_{x}$, and general spacetime open sets.  A detailed comparison is presented in Section~\\ref{sec:example}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (178,130) -- (292,130) -- (292,170) -- (178,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (188,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:GCC}{{\\rm GCC}}};\n\\draw (439,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observability};\n\\draw (343,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:sharp:obser}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec-osc-nece}, Sec \\ref{sec:app:3}}};\n\n\\end{tikzpicture}\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+GCC=OB}\n\\end{figure}\n\n\\begin{remark}\nThe main theorem reveals two structural features of observability that distinguish the wave equation from other important models, for example, the heat equation,  \\cite{AEWZ,WWZZ}. First, the family of observable regions $\\mathcal{O}(T)$ depends intrinsically on $T$; non-observable sets arise from geometric obstructions that vary with $T$, leading to a genuinely time-dependent structure for $\\{\\mathcal{O}(T)\\}_{T>0}$.\nSecond, our characterization also suggests that no waiting time is needed for observability, indicating that, for wave equations, the spacetime geometry of the observation region may play a more decisive role than the duration of control. See Section~\\ref{sec:furthercomment}.\n\\end{remark}\n\nMoreover, we believe that the symmetry mechanism uncovered in this work may extend to other instances of wave equations and to a broader range of models, including coupled systems, semilinear equations, and geometric wave equations. These ideas may also provide insight for higher-dimensional problems, where geometric propagation and microlocal structures are considerably more intricate. In addition, the geometric perspectives here may prove useful in questions of sensor placement, optimization of observation regions, and numerical implementations of control and observability.\n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Nicolas Burq for valuable and useful discussions during the preparation of this manuscript.  Shengquan Xiang is partially supported by the NSFC under grants 12571474 and 12301562.   Ming Wang was partially supported by the NSFC under grants 12571260, 12171442 and 12171178.",
    "sketch": "A proof sketch for Theorem~\\ref{thm: main-ucp} is indicated as follows.\n\n- The key mechanism is the \\blackhyperref{def:OSC}{(OSC)} (\\emph{observable symmetry condition}): “Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}).”\n\n- Conceptual structure: “the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.”\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “\\blackhyperref{def:OSC}{{\\rm OSC}} + \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}” and “UCP,” with the relevant parts located in “Sec \\ref{sec: OSC+WGCC>UCP}” and “Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}.”",
    "expanded_sketch": "A proof sketch for the main theorem is indicated as follows.\n\n- The key mechanism is the following observable symmetry condition:\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nUnder the minimal assumption\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nunique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, where\n\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}\nIn particular, this class is generated precisely by the observable symmetry condition above.\n\n- Conceptual structure: the weak GCC condition stated above eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “the observable symmetry condition above + the weak GCC condition above” and “UCP,” with the relevant parts located later and in \\ref{sec: OSC+WGCC>UCP} and \\ref{sec:neosctoucp}, \\ref{sec: ucp-wgcc}.",
    "expanded_theorem": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nand does not obey\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nfor any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $T>0$ and let $G\\subset [0,T]\\times \\mathbb T$ be a measurable spacetime set, where $\\mathbb T$ is the one-dimensional torus. Which explicit geometric condition on $G$ is equivalent to the unique continuation property (UCP) on $G$ for the wave setting?",
      "correct_choice": {
        "label": "A",
        "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0,$$ where $\\Delta$ denotes symmetric difference."
      },
      "choices": [
        {
          "label": "B",
          "text": "$G$ satisfies the geometric control condition in the strong sense that there exists a constant $c_0>0$ such that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds\\ge c_0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds\\ge c_0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"
        },
        {
          "label": "C",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$."
        },
        {
          "label": "D",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any pair $(A,B)$ of measurable subsets of $\\mathbb T$, including the trivial pair; equivalently, for every measurable pair $(A,B)$ one has $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)>0.$$"
        },
        {
          "label": "E",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ at least one of the two integrals $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds$ or $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds$ is strictly positive, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "weak-vs-strong GCC lower bound",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the no-OSC condition",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "restriction to non-trivial symmetry pairs",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "both characteristic directions required in weak GCC",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It only asks for an equivalent characterization of UCP, and the answer must be identified by comparing several closely related formulations."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall question: the correct choice appears to restate the equivalence theorem with only nearby perturbations in the distractors. It is not a pure tautology because the alternatives differ in meaningful quantifiers and hypotheses."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish weak vs. strong GCC, both-vs.-one visibility, and non-trivial-vs.-all pairs in the symmetry condition. However, the task mainly tests precise recall/recognition of the theorem rather than deep generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one is too strong, one is insufficient, one overstates the symmetry exclusion, and one weakens the geometric condition incorrectly. These reflect realistic failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it leans more toward precise recall of a known equivalence than genuine generative reasoning."
    }
  },
  {
    "id": "2512.09873v1",
    "paper_link": "http://arxiv.org/abs/2512.09873v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}",
      "start_pos": 32586,
      "end_pos": 33189,
      "label": "thm: main-ucp"
    },
    "ref_dict": {
      "eq: wave-eq-0": "\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T),\n\\end{equation}",
      "wave:nece:2": "\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation}",
      "def:classSc": "\\begin{definition}[Symmetric function pair]\\label{def:classSc}\nWe say \n$(f,g)$ is a symmetric  function pair if  there exists a decomposition pair $\\{(A_k, B_k),\\ k = 1,\\dots, K\\}$, a sequence of different numbers $\\{s_k\\}_{k= 1}^K$ such that \n\\begin{equation*}\n f=\\sum_{1\\leq k\\leq K}s_k\\chi_{A_k}, \\quad g=\\sum_{1\\leq k\\leq K}s_k\\chi_{B_k},\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all symmetric function pairs is denoted by $\\mathcal{S}^2_c$.\n\\end{definition}",
      "fig:GCCfail1": "\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times \\T$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times \\T$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{theorem}\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times \\T$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}\n\\end{theorem}\n\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}",
      "def:classS": "\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}",
      "eq:GCC-wave:0": "\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation}",
      "eq:GCC:measure": "\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}",
      "eq: wave-ob": "\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1(\\T)}+\\|u_1\\|^2_{L^2(\\T)}\\leq C\\int_G|\\partial_tu(t,x)|^2\\d t\\d x.\n\\end{equation}",
      "eq: wave-eq": "\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}",
      "def:GCC": "\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}",
      "def:weakGCC": "\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}",
      "fig:OSC-role": "\\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLe",
      "def:OSC": "\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}"
    },
    "pre_theorem_intro_text_len": 17122,
    "pre_theorem_intro_text": "Let $T > 0$. Let $G \\subset [0,T] \\times \\mathbb{T}$ be a spacetime measurable set with positive measure.\nConsider the {\\it observability problem} of the wave equation: whether there exists some $C= C(G)>0$ such that every solution $u$ to \n\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}),\n\\end{equation}\nsatisfies\n\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1({\\mathbb T})}+\\|u_1\\|^2_{L^2({\\mathbb T})}\\leq C\\int_G|\\partial_tu(t,x)|^2{\\,\\rm d} t{\\,\\rm d} x.\n\\end{equation}\n\nThanks to the classical Hilbert uniqueness method and an argument to raise regularity (see Appendix \\ref{sec: HUM-app} for details on this reduction), the observability of \\eqref{eq: wave-ob} is equivalent to the {\\it exact controllability} of\nthe controlled wave equation on $[0,T]\\times{\\mathbb T}$ with control force $f\\in L^2(G)$:\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}).\n\\end{equation}\n\nOur primary objective is to seek the sufficient and necessary geometric conditions on $G$ such that the unique continuation and observability hold.\n\n\\subsection{History and setting}\nFrom the early work of Russell \\cite{Russell-1}, the studies of the controllability and observability for wave equations have been central topics in control theory. \n\n\\subsubsection{Geometric control condition}\nIn the pioneering work of Rauch and Taylor \\cite{RT}, they first related the observability to a geometric condition on the damped region $\\omega$ and the rays of geometric optics in the boundaryless case. \nLater, in another pioneering work of Bardos-Lebeau-Rauch \\cite{BLR-gcc}, the well-known \\textit{Geometric control condition} is introduced: for an open observed region $\\omega$, every generalized ray should meet $\\omega$ in a finite time. \n\nSince then, it has become one of the most natural assumptions for the controlled waves. In the existing literature, the observation is most often made on cylindrical domains $G=(0,T)\\times \\omega$, with $\\omega$ being an open subset. Under suitable smooth conditions, it is well-known that GCC is sufficient, and depending on the domain, necessary for the observability in the cylindrical domains $(0,T)\\times \\omega$, see \\cite{BLR-gcc,BG-97}. When considering stabilization problems, GCC is also a useful condition for the exponential decay of energy. We refer to \\cite{RT,Haraux}. Otherwise, one may have logarithmic type of energy decay results \\cite{LR-97, Burq-98}. GCC also plays a role in practical issues, such as sensor designs, tomography techniques used for imaging bodies (see \\cite{RouLebeauAnalPDE2017} for example), etc. For a comprehensive reference of the numerical study, we refer to \\cite{zuazua-review} and its references therein. \n\nFinally, we give a very brief overview of the boundary control case. There are also fruitful results in this direction. For related GCC, we refer to \\cite{Lebeau-boundary}. In particular, for 1D wave equations, one can find many nonlinear results \\cite{Li-1,LY-2} and the references therein.\n\n\\subsubsection{Unique continuation}\nA qualitative version of observability is the unique continuation property.\nThe easiest way to ensure this property is to apply the analyticity based on Holmgren's theorem. Besides, H\\\"ormander's pseudo-convexity condition and Carleman estimates are powerful tools dealing with the unique continuation problem. In this direction, there is a large literature such as \\cite{RZ-98,Tataru,Hormander-92,Hormander-96} and more recent work includes \\cite{Laurent-Leautaud-2019, MS-21,FCL,Shao}. Here we point out that to construct the appropriate weight to apply Carleman estimates, it is crucial to understand the behavior across the suitable spacetime surfaces, which is more delicate than considering a cylindrical region.\n\n\\subsubsection{Our setting: spacetime measurable observable region}\nRecently, researchers started to focus on \n the spacetime setting \\cite{Castro-Cindea-Munch-2014, RouLebeauAnalPDE2017,Shao,PK}. Meanwhile, the study of the case where $G$ is a spacetime region is far from complete, even for open regions.  \n\n In this paper, we focus on the setting that $G\\subset[0,T]\\times{\\mathbb T}$ is measurable of positive measure. The following condition may be viewed as the natural analogue of the standard geometric control condition in this spacetime measurable setting. \n\n\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\nFrom now on, in this paper, \\blackhyperref{def:GCC}{({\\rm GCC})}  refers to the preceding definition with respect to \\eqref{eq:GCC-wave:0}. \nIt is well-known that under the standard setting, namely the observable region is a cylinder $[0, T]\\times \\omega$, GCC yields several important properties: weak observability and the necessity of observability. These results can be generalized to the spacetime \\blackhyperref{def:GCC}{({\\rm GCC})} for a spacetime measurable observable region. We put the former result in Section \\ref{sec: weak-ob-sec} and the later in Appendix \\ref{sec:app:3}.  \n\n\\subsection{A new symmetry condition}\n\nExtend the system $2\\pi-$periodically to $x\\in \\mathbb{R}$,  and introduce the null coordinate:\n\\begin{equation}\n    \\xi=x+t \\; \\textrm{ and } \\; \\eta=x-t. \\notag\n\\end{equation}\nUnder this new coordinate,  \n\\begin{equation*}\n  2  u_{\\xi}= u_{x}+ u_t \\; \\textrm{ and } \\;  2  u_{\\eta}= u_{x}- u_t\n\\end{equation*}\nand the wave equation \\eqref{eq: wave-eq} becomes\\footnote{We considered the equation on $U(\\xi, \\eta):= u(t, x)$ and $F(\\xi, \\eta)= f(t, x)$. For ease of notation, we still denote $U$ by $u$ and $F$ by $f$.  }\n\\begin{equation}\n    4 \\partial_{\\xi} \\partial_{\\eta} u=  f \\mathbf{1}_G. \\notag\n\\end{equation}\n\nFor any $\\xi_0\\in\\mathbb{R}$, we denote the  line $\\{(t,x):x+t=\\xi_0\\}$ by $L_{\\xi=\\xi_0}$,  and call it the $\\xi$-characteristic. Similarly, we denote the  line $\\{(t,x):x- t=\\eta_0\\}$ by $L_{\\eta=\\eta_0}$ and  call it $L_{\\eta=\\eta_0}$ the $\\eta$-characteristic. Thus the \\blackhyperref{def:GCC}{({\\rm GCC})} is equivalent to\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation} \nWe further define measurable cylinders for any measurable sets $A, B\\subset [0, 2\\pi]$ as\n\\begin{equation}\\label{eq: cylinder}\nL_{\\xi\\in A}:=\\bigcup_{\\xi_0\\in A}L_{\\xi=\\xi_0} \\; \\textrm{ and } \\; L_{\\eta\\in B}:=\\bigcup_{\\eta_0\\in B}L_{\\eta=\\eta_0}.\n\\end{equation}\n\nIntroduce the following symmetry condition:\n\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\mathbb{R}^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\mathbb{R}^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\vspace{2mm}\n\nWe call a pair $(A, B)$ trivial, if $|A|= |B|= 0$ or $|A|= |B|= 2\\pi$.  In principle, we are interested in {\\it non-trivial} pairs $(A, B)$. Since the observable symmetry condition is automatically satisfied for trivial pairs ($A, B$). \n\nThe observable symmetry condition is very important in our work, see Fig. \\ref{fig:OSC-role}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (192,38.4) .. controls (192,32.38) and (196.88,27.5) .. (202.9,27.5) -- (452.1,27.5) .. controls (458.12,27.5) and (463,32.38) .. (463,38.4) -- (463,71.1) .. controls (463,77.12) and (458.12,82) .. (452.1,82) -- (202.9,82) .. controls (196.88,82) and (192,77.12) .. (192,71.1) -- cycle ;\n\\draw   (72,158.7) .. controls (72,152.51) and (77.01,147.5) .. (83.2,147.5) -- (199.8,147.5) .. controls (205.99,147.5) and (211,152.51) .. (211,158.7) -- (211,192.3) .. controls (211,198.49) and (205.99,203.5) .. (199.8,203.5) -- (83.2,203.5) .. controls (77.01,203.5) and (72,198.49) .. (72,192.3) -- cycle ;\n\\draw   (254,151.3) .. controls (254,143.13) and (260.63,136.5) .. (268.8,136.5) -- (393.2,136.5) .. controls (401.37,136.5) and (408,143.13) .. (408,151.3) -- (408,195.7) .. controls (408,203.87) and (401.37,210.5) .. (393.2,210.5) -- (268.8,210.5) .. controls (260.63,210.5) and (254,203.87) .. (254,195.7) -- cycle ;\n\\draw   (436,158.7) .. controls (436,152.51) and (441.01,147.5) .. (447.2,147.5) -- (563.8,147.5) .. controls (569.99,147.5) and (575,152.51) .. (575,158.7) -- (575,192.3) .. controls (575,198.49) and (569.99,203.5) .. (563.8,203.5) -- (447.2,203.5) .. controls (441.01,203.5) and (436,198.49) .. (436,192.3) -- cycle ;\n\\draw    (226,82.5) -- (144.57,146.27) ;\n\\draw [shift={(143,147.5)}, rotate = 321.93] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (329,82.5) -- (329,134.5) ;\n\\draw [shift={(329,136.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (430,82.5) -- (509.44,146.25) ;\n\\draw [shift={(511,147.5)}, rotate = 218.75] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (219,47) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observable symmetry condition};\n\\draw (81,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Conservation law};\n\\draw (282,146.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\begin{minipage}[lt]{66.78pt}\\setlength\\topsep{0pt}\n\\begin{center}\nControllability \\\\Observability \\\\UCP\n\\end{center}\n\n\\end{minipage}};\n\\draw (455,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Symetric \\ \\ \\  class };\n\\draw (515,167.9) node [anchor=north west][inner sep=0.75pt]  [font=\\small]  {$\\mathcal{S}^{2}$};\n\\draw (128,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Noether}};\n\\draw (333,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize necessary}};\n\\draw (486,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize generate}};\n\n\\end{tikzpicture}\n\n    \\caption{The role of (OSC) in this paper}\n    \\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)=   \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\;  \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{remark}\n  This proposition can be understood as a variant of Noether’s theorem\\footnote{Noether’s theorem: ``Every continuous symmetry of the action (or the equations of motion, in a suitable sense) implies a conserved quantity.''} adapted to a controlled wave equation: a geometric symmetry of the forcing region yields a conserved functional of the solution. Indeed,   \nthe key hypothesis is that the region $G$ satisfies the  {\\it observable symmetry condition} for ($A, B$). The conclusion shows that the energy $I(t)$ is invariant in time, regardless of the force term. \n\\end{remark}\n\nThis new type of symmetry provides a {\\it necessary condition} for the controllability and observability of \\eqref{eq: wave-eq}, and even a {\\it necessary condition} for the unique continuation of  \\eqref{eq: wave-ob}. See Section \\ref{sec:symmetry} for the detailed construction of counterexamples.  Moreover, this construction gives rise to two classes of function pairs satisfying the symmetry condition, $\\mathcal{S}^2_c$ and $\\mathcal{S}^2$, in Definitions \\ref{def:classSc} and \\ref{def:classS}.\n\n\\begin{figure}[htp]\n    \\centering\n\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times {\\mathbb T}$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times {\\mathbb T}$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.",
    "context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}",
    "full_context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}\n\n\\begin{figure}[htp]\n \\centering\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times \\T$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n \\item[(1)] The observability inequality \\eqref{eq: wave-ob} holds on $G$;\n \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\n\\begin{proposition}[Unique continuation implies weak GCC]\nLet $T>0$ and $G\\in [0, T]\\times \\T$. Assume that the following unique continuation property holds\n$$\n\\mbox{ Let u be a solution of \\eqref{eq: wave-eq-0} and } u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0. \n$$\nThen $G$ satisfies the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}. \n\\end{proposition}\n\\begin{proof}\nWe argue by contradiction. Suppose that $G$ does not satisfy the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, there exists a subset $A\\subset \\T$ such that $\\mbox{meas}_\\R(A)>0$ and\n\\begin{align*}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0 \\quad \\mbox{ or } \\quad \n \\mbox{meas}_\\R(G\\cap L_{\\xi\\in A})=0.\n\\end{align*}\nWithout loss of generality, we only consider the case\n\\begin{align}\\label{equ-ucp-wgcc-1}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0. \n\\end{align}\nSince $A$ has positive measure, one can find a non-zero function $\\phi$ such that\n\\begin{align}\\label{equ-ucp-wgcc-2}\n \\int_A \\phi(x)\\d x=0, \\quad \\int_A|\\phi| \\d x=\\mbox{meas}_{\\R}(A)>0. \n\\end{align}\nSet \n$$\nu_{0x}=-u_1=\\chi_A \\phi.\n$$\nThis is possible since $\\int_\\T u_{0,x}\\d x =\\int_A\\phi \\d x =0$. Then we have\n$$\n\\partial_\\xi u|_{t=0}=u_{0x}+u_1=0, \\quad \\partial_\\eta u|_{t=0}=u_{0x}-u_1=2\\chi_A\\phi.\n$$\nIt follows that\n$$\n\\partial_\\xi u=0 \\quad \\mbox{ on } [0,T]\\times \\T\n$$\nand\n$$\n\\partial_\\eta u= 0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThus we have\n$$\nu_t=\\frac{1}{2}(\\partial_\\xi u-\\partial_\\eta u)=0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThanks to \\eqref{equ-ucp-wgcc-1}, we see $G\\subset L_{\\eta\\in \\T\\backslash A}$, and thus $u_t=0$ on $G$. By the UCP, we must have $\\partial_\\eta u|_{t=0}=2\\chi_A\\phi\\equiv 0$, which leads a contradiction with \\eqref{equ-ucp-wgcc-2}.\n\\end{proof}\n\n\\subsection{OSC and weak GCC imply unique continuation }\\label{sec: OSC+WGCC>UCP}\nIn this sequel, we generalize the result in Proposition \\ref{prop-ucp-S2}, which forms the following proposition.\n\\begin{proposition}[Weak GCC implies UCP up to $\\mathcal{S}^2$]\\label{prop-weakGCC-ucp-S2}\n Let $T>0$. Let $G\\in [0, T]\\times \\T$ satisfy the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}.\n If $u$ solves the wave equation \\eqref{eq: wave-eq-0} and $u_t = 0$ on $G$, then its initial state $(\\partial_\\xi u, \\partial_\\eta u)|_{t=0}$ belongs to the symmetry function pair class $\\mathcal{S}^2$.\n\\end{proposition}\n\\begin{proof}\nWe use the same notation in the proof of Proposition \\ref{prop-ucp-S2}. Recall that we have the relation \\eqref{equ-xi-eta-exchange}, namely \n\\begin{equation*} \\partial_{\\eta}u|_{\\{t= 0\\}\\cap L_{\\eta= \\eta_0}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)= \\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}\n\\end{equation*}\nup to sets with zero measure.\nNamely, the value of $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ determines almost every value of $\\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}$. Since $G$ satisfies the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, we know \n$$\n\\mbox{meas}_\\R(\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}})>0.\n$$\nIn other words, $\\partial_{\\xi}u|_{\\{t= 0\\}}$ equals to $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ at least on a positive measure subset set of $\\T$. Thus the set\n$$\n\\mathcal{U}_{(\\xi_0,\\eta_0)}:=\\{x_0\\in \\T: \\partial_{\\xi}u|_{\\{t= 0,x=x_0\\}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)\\}\n$$\nhas positive measure. Repeating this process for other point $(\\xi_0',\\eta_0')\\in [0,T]\\times \\T$, we shall find that, there exists a family sets $\\mathcal{U}_\\ell (\\ell\\in I)$, $I$ is a index set, such that\n\\begin{align}\\label{equ-Ul-129}\n \\mathcal{U}_\\ell\\cap \\mathcal{U}_{\\ell'}=\\emptyset \\mbox{ for } \\ell\\neq \\ell', \\quad \\mbox{meas}_\\R(\\mathcal{U}_\\ell)>0 \\mbox{ for } \\ell\\in I , \\quad \\T=\\cup_{\\ell\\in I}\\mathcal{U}_\\ell \n\\end{align} \nand $\\partial_{\\xi}u|_{\\{t= 0\\}}$ is a constant on each $\\mathcal{U}_\\ell$.\nOne can show that, under the restrictions \\eqref{equ-Ul-129}, the index set $I$ is at most countable, thus, after relabeling if necessary, $\\{\\mathcal{U}_\\ell, \\ell\\ge 1\\}$ is a weak decomposition of $\\T$. Moreover, there exists a sequence of different numbers $\\{s_\\ell\\}$ such that\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}}=s_\\ell \\mbox{ on } \\mathcal{U}_\\ell.\n$$\nNote that $\\partial_{\\xi}u|_{\\{t= 0\\}}\\in L^2(\\T)$, and $\\mathcal{U}_\\ell$ are disjoint for different $\\ell$, we find\n$$\n\\|\\partial_{\\xi}u|_{\\{t= 0\\}}\\|_{L^2(\\T)}^2=\\sum_\\ell |s_\\ell|^2\\mbox{meas}_\\R(\\mathcal{U}_\\ell)<\\infty.\n$$\nThus we obtain\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{U}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nSimilarly, we can find a weak decomposition $\\mathcal{V}_{\\ell}(\\ell\\geq 1)$ such that\n$$\n\\partial_{\\eta}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{V}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nIn summary, we find that $(\\partial_{\\xi}u|_{\\{t= 0\\}}, \\partial_{\\eta}u|_{\\{t= 0\\}})\\in \\mathcal{S}^2$.\n\\end{proof}\n\nThe following theorem gives a sufficient and necessary condition of the observable set for the transport equation. We only state the result for \\eqref{eq: transport-eq}, the reader easily figures out the the necessary modifications for \\eqref{eq: transport-eq-2}.\n\\begin{theorem}\\label{thm-tran-suff-nece}\nLet $T>0$ and $G$ be a measurable subset of $[0,T]\\times \\T$ with positive measure. Then the observability inequality\n\\begin{align}\\label{equ-tran-suff-necc-1}\n \\|u_0\\|^2_{L^2(\\T)}\\leq C\\int_G|u(t,x)|^2\\d x \\d t \n\\end{align}\nholds with a positive constant $C>0$ for all solutions to the transport equation \\eqref{eq: transport-eq} if and only if there exists a constant $c_0>0$ such that \n\\begin{equation}\\label{equ-tran-suff-necc-2}\n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s\\geq c_0 \\quad \\mbox{ for a.e. } x\\in\\T \n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe direction $\\eqref{equ-tran-suff-necc-2}\\Longrightarrow \\eqref{equ-tran-suff-necc-1}$ follows clearly from Proposition \\ref{prop: tran-eq-ob}. To show the inverse direction, namely $\\eqref{equ-tran-suff-necc-1}\\Longrightarrow \\eqref{equ-tran-suff-necc-2}$, we use the contradiction argument. Suppose that \\eqref{equ-tran-suff-necc-2} is not true, then for any $\\varepsilon>0$, there exists a set $E_\\varepsilon\\subset \\T$ with positive measure $|E_\\varepsilon|>0$ such that\n\\begin{align}\\label{equ-tran-suff-necc-3} \n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s<\\varepsilon \\quad \\mbox{ for a.e. } x\\in E_\\varepsilon. \n\\end{align}\nLet $u_{0\\varepsilon}(x)=1_{E_\\varepsilon}(x)$. Then $u_{0\\varepsilon}\\in L^2(\\T)$ and its support is contained in $E_\\varepsilon$. Then, similar to the proof in Proposition \\ref{prop: tran-eq-ob}, we have\n\\begin{align*}\n\\int_G|u(t,x)|^2\\d x \\d t =\\int_{\\T}|u_{0\\varepsilon}(x)|^2\\int_{\\T}\\mathbf{1}_G(t,x-t)\\d t\\d x\\leq \\varepsilon \\|u_{0\\varepsilon}\\|^2_{L^2(\\T)}. \n\\end{align*}\nTaking $\\varepsilon>0$ small enough, say $\\varepsilon=C/2$, we obtain a contradiction with \\eqref{equ-tran-suff-necc-1}. \n\\end{proof}\n\\subsection{GCC is necessary for the wave observability}\\label{sec:app:3}\n\\begin{corollary}\\label{wave-GCC-nece}\nIf the observability \\eqref{eq: wave-ob} holds on some observation set $G\\subset [0,T]\\times \\T$, then $G$ needs to satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}}.\n\\end{corollary}\n\\begin{proof}",
    "post_theorem_intro_text_len": 6409,
    "post_theorem_intro_text": "\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}\n\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+WGCC=UCP}\n\\end{figure}\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times {\\mathbb T}$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n    \\item[(1)] The observability inequality \\eqref{eq: wave-ob}  holds on $G$;\n    \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\nTo the best of our knowledge, this work provides the first necessary and sufficient geometric condition for observability of the wave equation on spacetime region. The observable symmetry condition plays a central role: it completes the classical GCC by capturing the additional geometric structure. This viewpoint also leads to necessary and sufficient conditions in a variety of other important geometric configurations, such as $[0, T]\\times \\omega$,  measurable Cartesian products  $E_t\\times F_{x}$, and general spacetime open sets.  A detailed comparison is presented in Section~\\ref{sec:example}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (178,130) -- (292,130) -- (292,170) -- (178,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (188,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:GCC}{{\\rm GCC}}};\n\\draw (439,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observability};\n\\draw (343,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:sharp:obser}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec-osc-nece}, Sec \\ref{sec:app:3}}};\n\n\\end{tikzpicture}\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+GCC=OB}\n\\end{figure}\n\n\\begin{remark}\nThe main theorem reveals two structural features of observability that distinguish the wave equation from other important models, for example, the heat equation,  \\cite{AEWZ,WWZZ}. First, the family of observable regions $\\mathcal{O}(T)$ depends intrinsically on $T$; non-observable sets arise from geometric obstructions that vary with $T$, leading to a genuinely time-dependent structure for $\\{\\mathcal{O}(T)\\}_{T>0}$.\nSecond, our characterization also suggests that no waiting time is needed for observability, indicating that, for wave equations, the spacetime geometry of the observation region may play a more decisive role than the duration of control. See Section~\\ref{sec:furthercomment}.\n\\end{remark}\n\nMoreover, we believe that the symmetry mechanism uncovered in this work may extend to other instances of wave equations and to a broader range of models, including coupled systems, semilinear equations, and geometric wave equations. These ideas may also provide insight for higher-dimensional problems, where geometric propagation and microlocal structures are considerably more intricate. In addition, the geometric perspectives here may prove useful in questions of sensor placement, optimization of observation regions, and numerical implementations of control and observability.\n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Nicolas Burq for valuable and useful discussions during the preparation of this manuscript.  Shengquan Xiang is partially supported by the NSFC under grants 12571474 and 12301562.   Ming Wang was partially supported by the NSFC under grants 12571260, 12171442 and 12171178.",
    "sketch": "A proof sketch for Theorem~\\ref{thm: main-ucp} is indicated as follows.\n\n- The key mechanism is the \\blackhyperref{def:OSC}{(OSC)} (\\emph{observable symmetry condition}): “Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}).”\n\n- Conceptual structure: “the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.”\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “\\blackhyperref{def:OSC}{{\\rm OSC}} + \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}” and “UCP,” with the relevant parts located in “Sec \\ref{sec: OSC+WGCC>UCP}” and “Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}.”",
    "expanded_sketch": "A proof sketch for the main theorem is indicated as follows.\n\n- The key mechanism is the following observable symmetry condition:\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nUnder the minimal assumption\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nunique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, where\n\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}\nIn particular, this class is generated precisely by the observable symmetry condition above.\n\n- Conceptual structure: the weak GCC condition stated above eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “the observable symmetry condition above + the weak GCC condition above” and “UCP,” with the relevant parts located later and in \\ref{sec: OSC+WGCC>UCP} and \\ref{sec:neosctoucp}, \\ref{sec: ucp-wgcc}.",
    "expanded_theorem": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nand does not obey\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nfor any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $T>0$ and let $G\\subset [0,T]\\times \\mathbb T$ be a measurable spacetime set, where $\\mathbb T$ is the one-dimensional torus. Which explicit geometric condition on $G$ is equivalent to the unique continuation property (UCP) on $G$ for the wave setting?",
      "correct_choice": {
        "label": "A",
        "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0,$$ where $\\Delta$ denotes symmetric difference."
      },
      "choices": [
        {
          "label": "B",
          "text": "$G$ satisfies the geometric control condition in the strong sense that there exists a constant $c_0>0$ such that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds\\ge c_0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds\\ge c_0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"
        },
        {
          "label": "C",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$."
        },
        {
          "label": "D",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any pair $(A,B)$ of measurable subsets of $\\mathbb T$, including the trivial pair; equivalently, for every measurable pair $(A,B)$ one has $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)>0.$$"
        },
        {
          "label": "E",
          "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ at least one of the two integrals $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds$ or $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds$ is strictly positive, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "weak-vs-strong GCC lower bound",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the no-OSC condition",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "restriction to non-trivial symmetry pairs",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "both characteristic directions required in weak GCC",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct condition; it only asks for the geometric characterization equivalent to UCP. No explicit wording in the stem singles out choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially a direct theorem-recall question: it asks for the exact condition equivalent to UCP, and the correct option states that equivalence almost verbatim rather than requiring application in a new setting."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle strengthenings/weakenings (strong vs weak GCC, both directions vs one, omission or overstrengthening of the symmetry clause). However, solving it is still largely a matter of recalling or recognizing the exact theorem statement rather than generating a new conclusion."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target natural failure modes: replacing weak by strong GCC, dropping the no-OSC requirement, making the symmetry prohibition too strong, or weakening the two-ray requirement to one. They are distinct and credible."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors and little answer leakage, but it is fairly tautological and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.10177v1",
    "paper_link": "http://arxiv.org/abs/2512.10177v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theoremA",
      "content": "[Theorem~\\ref{thm:trees-cycles}]\nAll trees and all cycles are realizable as Bell coloring graphs.",
      "start_pos": 16743,
      "end_pos": 16871,
      "label": null
    },
    "ref_dict": {
      "fig:Bell3-K13": "\\begin{tikzpicture}[\n    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\\small},\n    edge label/.style={midway, fill=white, font=\\footnotesize, inner sep=1.5pt},\n    part label/.style={font=\\scriptsize, color=black, align=center},\n    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},\n    scale=1.2\n]\n\n\\coordinate (P1) at (0, -2, 0);\n\\coordinate (P2) at (2, 1, 0);\n\\coordinate (P3) at (-2, 1, 0);\n\\coordinate (P4) at (0, 0, 3);\n\n\\draw (P1) -- (P2) node[edge label, pos=0.53] {$v_3$};\n\\draw (P1) -- (P3) node[edge label, pos=0.55] {$v_2$}; \n\\draw (P2) -- (P3) node[edge label] {$v_1$}; \n\\draw (P4) -- (P1) node[edge label, pos=0.5] {$v_1$};\n\\draw (P4) -- (P2) node[edge label, pos=0.4] {$v_2$};\n\\draw (P4) -- (P3) node[edge label, pos=0.6] {$v_3$};\n\n\\node[vertex] at (P1) {$P_1$};\n\\node[part label, below left=0.3cm of P1] {$\\{\\{u\\}, \\{v_2, v_3\\}, \\{v_1\\} \\}$};\n\n\\node[vertex] at (P2) {$P_2$};\n\\node[part label, below=0.4cm of P2] {$\\{\\{u\\}, \\{v_1, v_3\\}, \\{v_2\\} \\}$};\n\n\\node[vertex] at (P3) {$P_3$};\n\\node[part label, below right=0.3cm of P3] {$\\{\\{u\\}, \\{v_1, v_2\\}, \\{v_3\\} \\}$};\n\n\\node[vertex] at (P4) {$P_4$};\n\\node[part label, above=0.4cm of P4] {$\\{\\{u\\}, \\{v_1, v_2, v_3\\}, \\emptyset\\}$};\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_3(K_{1,3})\\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\\{u, v_1, v_2, v_3\\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K13}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}[\n    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\\small},\n    edge label/.style={midway, fill=white, font=\\footnotesize, inner sep=1.5pt},\n    part label/.style={font=\\scriptsize, color=black, align=center},\n    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},\n    scale=1.2\n]\n\n\\coordinate (P1) at (0, -2, 0);\n\\coordinate (P2) at (2, 1, 0);\n\\coordinate (P3) at (-2, 1, 0);\n\\coordinate (P4) at (0, 0, 3);\n\n\\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};\n\\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$}; \n\\draw (P2) -- (P3) node[edge label] {$w$}; \n\\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};\n\\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};\n\\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};\n\n\\node[vertex] at (P1) {$P_1$};\n\\node[part label, below left=0.3cm of P1] {$\\{\\{w, v_1\\}, \\{v_2\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P2) {$P_2$};\n\\node[part label, below=0.4cm of P2] {$\\{\\{w, v_2\\}, \\{v_1\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P3) {$P_3$};\n\\node[part label, below right=0.3cm of P3] {$\\{\\{w, v_3\\}, \\{v_1\\}, \\{v_2\\}, \\emptyset \\}$};\n\n\\node[vertex] at (P4) {$P_4$};\n\\node[part label, above=0.4cm of P4] {$\\{\\{w\\}, \\{v_1\\}, \\{v_2\\}, \\{v_3\\}\\}$};\n\n\\end{tikzpicture}",
      "thm:trees-cycles": "\\begin{theorem}\\label{thm:trees-cycles}\n    All trees and all cycle graphs are realizable as Bell coloring graphs.\n\\end{theorem}",
      "thm:clique_classification": "\\begin{theorem}\\label{thm:clique_classification}\nFor any clique $\\mathcal K=\\{P_1,\\dots,P_m\\}$ with $m\\ge 1$ in $\\B_k(G)$, exactly one of the following holds:\n\\begin{enumerate}\n    \\item $\\mathcal K=\\{P_1,P_2,P_3\\}$ is a cyclic $T$-triangle;\n    \\item $\\mathcal K=\\{P_1,P_2,P_3\\}$  is a radial $T$-triangle;\n    \\item $\\mathcal K=\\{P_1,P_2,P_3,P_4\\}$  is a split $T$-tetrahedron; \n    \\item $\\mathcal K=\\{P_1,P_2,P_3,P_4\\}$ is a fused $T$-tetrahedron; or \n    \\item $\\mathcal K=\\{P_1,\\dots,P_m\\}$ is an $S$-clique.\n\\end{enumerate} \n\\end{theorem}",
      "fig:Bell3-K3-plus-K1": "\\begin{tikzpicture}[\n    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\\small},\n    edge label/.style={midway, fill=white, font=\\footnotesize, inner sep=1.5pt},\n    part label/.style={font=\\scriptsize, color=black, align=center},\n    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},\n    scale=1.2\n]\n\n\\coordinate (P1) at (0, -2, 0);\n\\coordinate (P2) at (2, 1, 0);\n\\coordinate (P3) at (-2, 1, 0);\n\\coordinate (P4) at (0, 0, 3);\n\n\\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};\n\\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$}; \n\\draw (P2) -- (P3) node[edge label] {$w$}; \n\\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};\n\\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};\n\\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};\n\n\\node[vertex] at (P1) {$P_1$};\n\\node[part label, below left=0.3cm of P1] {$\\{\\{w, v_1\\}, \\{v_2\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P2) {$P_2$};\n\\node[part label, below=0.4cm of P2] {$\\{\\{w, v_2\\}, \\{v_1\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P3) {$P_3$};\n\\node[part label, below right=0.3cm of P3] {$\\{\\{w, v_3\\}, \\{v_1\\}, \\{v_2\\}, \\emptyset \\}$};\n\n\\node[vertex] at (P4) {$P_4$};\n\\node[part label, above=0.4cm of P4] {$\\{\\{w\\}, \\{v_1\\}, \\{v_2\\}, \\{v_3\\}\\}$};\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_4(K_{3}\\sqcup K_1)\\cong K_4$. The graph $K_3\\sqcup K_1$ has vertex set $\\{v_1, v_2, v_3, w\\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K3-plus-K1}\n\\end{figure}\n\nOur work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\\B_k(G\\sqcup K_k) \\cong \\mathcal{C}_k(G)$ for any graph $G$ and integer $k\\ge 1$ \\cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \\emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \\emph{only} cycles that are realizable as coloring graphs \\cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following: \n\n\\begin{theoremA}[Theorem~\\ref{thm:trees-cycles}]\nAll trees and all cycles are realizable as Bell coloring graphs.\n\\end{theoremA}\n\nWe prove Theorem~\\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs. \n\nThe papers \\cites{BFHRS16, ABFR17} examine forbidden subgraphs in standard coloring graphs. To treat the analogous question for Bell coloring graphs, we first obtain a structural description of their cliques (see Theorem~\\ref{thm:clique_classification}), showing that every clique belongs to one of two explicit families. This classification allows us to construct an infinite family of forbidden induced subgraphs of Bell coloring graphs.\n\n\\begin{theoremA}[Theorem~\\ref{thm:forbidden}]\nThe graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\\ge 6$ is an infinite family of forbidden induced subgraphs.\n\\end{theoremA}\n\nAnother natural question is whether the structure of the reconfiguration graph uniquely determines the base graph. This is known for standard coloring graphs; in particular, if $T$ is a tree, $\\mathcal{C}_3(T)$ uniquely determines $T$ (\\cite{AKL25}*{Theorem 1.1} or \\cite{BBHvdHHP25}*{Theorem 1.2}). Having lost the color labels, $\\mathcal{B}_3(T)$ has less symmetry than the $3$-coloring graph $\\mathcal{C}_3(T)$. Despite this compression, our next result guarantees that a tree $T$ can be reconstructed uniquely from its Bell $3$-coloring graph. Let $\\mathsf{Trees}$ and $\\mathsf{Graphs}$ denote the sets of isomorphism classes of all (finite) trees and graphs, respectively. \n\n\\begin{theoremA}[Theorem~\\ref{thm:tree-reconstruction}]\nThe map $\\mathsf{Trees} \\to \\mathsf{Graphs}$ given by $T\\mapsto \\mathcal{B}_3(T)$ is injective.\n\\end{theoremA}\n\nWe also find it useful to work with a multigraph variant, the \\emph{Bell coloring multigraph}, denoted $\\BellMulti{k}{G}$, which uses multiple edges to encode how many vertices are responsible for each adjacency. If $P$ and $P'$ are two stable partitions of $G$, then in the Bell coloring graph, $P$ and $P'$ are adjacent whenever there exists a vertex $v\\in V(G)$ with $P-v = P'-v$. Thus, multiple vertices may witness the same adjacency, but $\\B_k(G)$ still records only a single edge between $P$ and $P'$. In the multigraph version, we instead add a separate parallel edge between $P$ and $P'$ for each distinct vertex $v$ such that $P-v = P'-v$. \n\nThe additional information means that Bell coloring multigraphs serve as a more refined graph invariant. For example, Figures~\\ref{fig:Bell3-K13}~and~\\ref{fig:Bell3-K3-plus-K1} show that $\\B_3(K_{1,3})\\cong \\B_4(K_3\\sqcup K_1)$. However, each edge incident to the partition $P_4$ in the latter graph is witnessed by two vertices, so \n$\\BellMulti{3}{K_{1,3}}\\not\\cong\\BellMulti{4}{K_3\\sqcup K_{1}}$. \n\nYet even a Bell coloring multigraph cannot serve as a complete graph invariant. For instance, $\\BellMulti{3}{K_{1,3}}\\cong \\BellMulti{2}{\\overline{K}_3}$. This ambiguity arises because the center vertex in $K_{1,3}$ is adjacent to every other vertex. When a vertex $v$ is adjacent to every vertex in $G-v$, we call $v$ a \\emph{universal vertex}. \nIf $G$ has a universal vertex $w$, then $\\B_{k+1}(G) \\cong \\B_k(G-w)$ for every $k\\in\\mathbb{N}$ due to the natural bijection between stable $k$-partitions of $G-w$ and stable $(k+1)$-partitions of $G$ in which $\\{w\\}$ forms an additional singleton part.\n\nTo handle this, let $U(G)$ denote the set of universal vertices of $G$. We define the \\emph{core} of $G$, denoted $G^{\\circ}$, as the induced subgraph $G[V(G)\\setminus U(G)]$. We define the equivalence relation $\\sim_{\\text{uni}}$ on $\\mathsf{Graphs}$ as follows: $G\\sim_{\\text{uni}} G'$ if their cores are isomorphic, $G^{\\circ} \\cong (G')^{\\circ}$.\nThe equivalence classes of graphs under this relation form the set $\\mathsf{Graphs}^{\\circ} = \\mathsf{Graphs}/\\!\\sim_{\\text{uni}}$.\n\nOur final result shows that a base graph of order $n$ can be reconstructed from its Bell $n$-coloring multigraph up to universal vertices:\n\n\\begin{theoremA}[Theorem~\\ref{thm:general-reconstruction-multigraph:intro}]\nThe map $\\mathsf{Graphs}^{\\circ} \\to \\mathsf{Multigraphs}$ given by $G\\mapsto \\BellMultiIntro{|V(G)|}{G}$  is injective.\n\\end{theoremA}\n\n\\textbf{Outline of the paper.} Section~\\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\\ge 6$. Section~\\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell\n$3$-coloring graphs. Section~\\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\n\\subsection{Notation and conventions} Let $\\mathbb{N}=\\{1, 2, 3, \\dots\\}$ denote the set of positive integers. For sets $X$ and $Y$, we use the notation $X\\setminus Y=\\{x\\in X \\ | \\ x\\notin Y\\}$ for the set difference and $X\\cup Y$ for set union; in the special case when $Y=\\{v\\}$ is a singleton, we also denote $X\\setminus\\{v\\}$ by $X-v$ and $X\\cup\\{v\\}$ by $X+v$. Given a graph $G$, we let $\\overline{G}$ denote its complement. We write $G\\sqcup H$ for the disjoint union of graphs $G$ and $H$. \n\n\\begin{defn}\\label{def:stable-partition}\nA \\emph{stable $k$-partition} of a graph $G$ is a multiset $P=\\{V_1,\\dots,V_k\\}$ of independent sets that partition $V(G)$, where we allow some of the parts $V_i$ to be empty. \n\\end{defn}\n\nLet $\\cP_k(G)$ denote the set of all stable $k$-partitions of $G$. For $v\\in V(G)$, we write\n\\[\nP-v \\colonequals \\{V_i-v : 1\\leq i\\leq k\\},\n\\]\nviewed as a multiset of $k$ subsets.\n\nA Bell $k$-coloring graph $\\B_k(G)$ is the graph whose vertex set is $\\cP_k(G)$. Two vertices $P_1, P_2 \\in \\cP_k(G)$ are adjacent if and only if $P_1-v=P_2-v$ for some $v\\in V(G)$. \n\nThroughout, partitions $P, Q, R \\in \\cP_k(G)$ are treated as \\emph{unordered multisets} of parts. We write $P \\sim Q$ to denote adjacency in $\\B_k(G)$. To specify the vertex responsible for the adjacency, we use the following notation:\n\\[\nP \\sim_v Q \\iff P \\sim Q \\text{ and } P-v = Q-v.\n\\]\nWe say that a vertex $v\\in V(G)$ is \\emph{responsible} for an edge $PQ$ in $\\B_k(G)$ if $P\\sim_v Q$. Equivalently, the edge $PQ$ is \\emph{realized} by $v$. The condition $P-v=Q-v$ implies that $P$ and $Q$ agree on the structure of the partition when restricted to $V(G)\\setminus\\{v\\}$. \n\nWe record the following useful fact, which we use several times in the paper: if $P_1\\sim_v P_2$ and $P_2\\sim_v P_3$, then $P_1$ and $P_3$ agree on $V(G)\\setminus\\{v\\}$, so $P_1\\sim_v P_3$.\n\n\\subsection{Example Bell coloring graphs} We begin with a few illustrative examples.\n\n\\begin{ex} Consider $\\B_2(\\overline{K}_2)$ where $V(\\overline{K}_2)=\\{1,2\\}$. The stable 2-partitions are $R_1 = \\{\\{1,2\\}, \\emptyset\\}$ and $R_2 = \\{\\{1\\}, \\{2\\}\\}$.\nWe have $R_1-1 = \\{\\{2\\}, \\emptyset\\}$ and $R_2-1 = \\{\\emptyset, \\{2\\}\\}$, so $R_1 \\sim_1 R_2$. Similarly $R_1 \\sim_2 R_2$. Thus $\\B_2(\\overline{K}_2) \\cong K_2$.\n\\end{ex}\n\n\\begin{ex}\\label{ex:K3}\nConsider $G=K_3 \\sqcup K_1$ on vertices $\\{1,2,3,4\\}$, where $\\{1,2,3\\}$ form the $K_3$ and the vertex $4$ is isolated. We examine $\\B_3(G)$. Since $\\{1,2,3\\}$ is a clique, any stable 3-partition must place them in different parts. The vertex $4$ can join any part.\n\\begin{align*}\nP_1 &= \\{\\{1,4\\}, \\{2\\}, \\{3\\}\\}, \\\\\nP_2 &= \\{\\{1\\}, \\{2,4\\}, \\{3\\}\\}, \\\\\nP_3 &= \\{\\{1\\}, \\{2\\}, \\{3,4\\}\\}.\n\\end{align*}\nThese are the only stable 3-partitions of $G$. We check adjacencies: $P_1-4 = P_2-4 = P_3-4 = \\{\\{1\\}, \\{2\\}, \\{3\\}\\}$. Thus, we have $P_1 \\sim_4 P_2$, $P_2 \\sim_4 P_3$, and $P_3 \\sim_4 P_1$, so $\\B_3(G) \\cong K_3$.\n\\end{ex}\n\n\\begin{ex}\\label{ex:P4}\nConsider the path graph $G$ on vertices $\\{1,2,3,4\\}$ with edges 12, 23, 34. The stable $3$-partitions are:\n\\[\n\\begin{aligned}\nR_1 &= \\{\\{1,3\\}, \\{2\\}, \\{4\\}\\}, & R_2 &= \\{\\{1,4\\}, \\{2\\}, \\{3\\}\\},\\\\\nR_3 &= \\{\\{1\\}, \\{2,4\\}, \\{3\\}\\}, & R_4 &= \\{\\{1,3\\}, \\{2,4\\}, \\emptyset\\}.\n\\end{aligned}\n\\]\nA direct check shows that $R_1 \\sim_1 R_2$, $R_2 \\sim_4 R_3$, $R_3 \\sim_1 R_4$, and $R_4 \\sim_4 R_1$. The graph $\\B_3(P_4)$ is the cycle $C_4$; see Figure~\\ref{fig:B3_P4}.\n\\end{ex}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{tikzpicture}[\n        partition/.style={rectangle, draw, rounded corners, thick, inner sep=5pt, align=center, fill=white},\n        edge label/.style={midway, fill=white, font=\\small, inner sep=1.5pt},\n        scale=1.1\n    ]\n\n    \\node[partition] (R1) at (0, 2) {$\\{1,3\\}, \\{2\\}, \\{4\\}$};\n\n    \\node[partition] (R2) at (5, 2) {$\\{1,4\\}, \\{2\\}, \\{3\\}$};\n\n    \\node[partition] (R3) at (5, 0) {$\\{1\\}, \\{2,4\\}, \\{3\\}$};\n\n    \\node[partition] (R4) at (0, 0) {$\\{1,3\\}, \\{2,4\\}, \\emptyset$};\n\n    \\draw[thick] (R1) -- (R2) node[edge label] {$1$};\n\n    \\draw[thick] (R2) -- (R3) node[edge label] {$4$};\n\n    \\draw[thick] (R3) -- (R4) node[edge label] {$1, 3$};\n\n    \\draw[thick] (R4) -- (R1) node[edge label] {$2, 4$};\n\n    \\end{tikzpicture}",
      "thm:general-reconstruction-multigraph:intro": "\\begin{theorem}\\label{thm:general-reconstruction-multigraph:intro}\nSuppose $G_1$ and $G_2$ are two graphs of orders $n_1$ and $n_2$, respectively. If $\\BellMulti{n_1}{G_1} \\cong \\BellMulti{n_2}{G_2}$, then $G_1$ and $G_2$ have isomorphic cores.\n\\end{theorem}",
      "thm:forbidden": "\\begin{theorem}\\label{thm:forbidden}\nThe graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\\ge 6$ is an infinite family of forbidden induced subgraphs.\n\\end{theorem}",
      "thm:tree-reconstruction": "\\begin{theorem}\\label{thm:tree-reconstruction}\nFor trees $T_1$ and $T_2$, we have $\\B_3(T_1)\\cong \\B_3(T_2)$ if and only if $T_1\\cong T_2$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6181,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nGraph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \\cite{JT95}, the themed collection \\cite{BW15}, and the recent text \\cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \\emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \\cites{HSTT24, BBHvdHHP25, AKL25}.\n\nIn this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\\in\\mathbb{N}$, we define the \\emph{Bell $k$-coloring graph} $\\mathcal{B}_k(G)$ as follows.\n\\begin{itemize}\n    \\item The vertices of $\\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \\emph{stable} $k$-partitions.\n    \\item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\\in V(G)$, formalized as $P_1-v = P_2-v$.\n\\end{itemize}\nThe notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\\ref{fig:Bell3-K13}~and~\\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.\n\nThe name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \\cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \\cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \\cite{Haa12}, \\cite{FM14}) address Bell coloring graphs.\n\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}[\n    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\\small},\n    edge label/.style={midway, fill=white, font=\\footnotesize, inner sep=1.5pt},\n    part label/.style={font=\\scriptsize, color=black, align=center},\n    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},\n    scale=1.2\n]\n\n\\coordinate (P1) at (0, -2, 0);\n\\coordinate (P2) at (2, 1, 0);\n\\coordinate (P3) at (-2, 1, 0);\n\\coordinate (P4) at (0, 0, 3);\n\n\\draw (P1) -- (P2) node[edge label, pos=0.53] {$v_3$};\n\\draw (P1) -- (P3) node[edge label, pos=0.55] {$v_2$}; \n\\draw (P2) -- (P3) node[edge label] {$v_1$}; \n\\draw (P4) -- (P1) node[edge label, pos=0.5] {$v_1$};\n\\draw (P4) -- (P2) node[edge label, pos=0.4] {$v_2$};\n\\draw (P4) -- (P3) node[edge label, pos=0.6] {$v_3$};\n\n\\node[vertex] at (P1) {$P_1$};\n\\node[part label, below left=0.3cm of P1] {$\\{\\{u\\}, \\{v_2, v_3\\}, \\{v_1\\} \\}$};\n\n\\node[vertex] at (P2) {$P_2$};\n\\node[part label, below=0.4cm of P2] {$\\{\\{u\\}, \\{v_1, v_3\\}, \\{v_2\\} \\}$};\n\n\\node[vertex] at (P3) {$P_3$};\n\\node[part label, below right=0.3cm of P3] {$\\{\\{u\\}, \\{v_1, v_2\\}, \\{v_3\\} \\}$};\n\n\\node[vertex] at (P4) {$P_4$};\n\\node[part label, above=0.4cm of P4] {$\\{\\{u\\}, \\{v_1, v_2, v_3\\}, \\emptyset\\}$};\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_3(K_{1,3})\\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\\{u, v_1, v_2, v_3\\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K13}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}[\n    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\\small},\n    edge label/.style={midway, fill=white, font=\\footnotesize, inner sep=1.5pt},\n    part label/.style={font=\\scriptsize, color=black, align=center},\n    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},\n    scale=1.2\n]\n\n\\coordinate (P1) at (0, -2, 0);\n\\coordinate (P2) at (2, 1, 0);\n\\coordinate (P3) at (-2, 1, 0);\n\\coordinate (P4) at (0, 0, 3);\n\n\\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};\n\\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$}; \n\\draw (P2) -- (P3) node[edge label] {$w$}; \n\\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};\n\\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};\n\\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};\n\n\\node[vertex] at (P1) {$P_1$};\n\\node[part label, below left=0.3cm of P1] {$\\{\\{w, v_1\\}, \\{v_2\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P2) {$P_2$};\n\\node[part label, below=0.4cm of P2] {$\\{\\{w, v_2\\}, \\{v_1\\}, \\{v_3\\}, \\emptyset\\}$};\n\n\\node[vertex] at (P3) {$P_3$};\n\\node[part label, below right=0.3cm of P3] {$\\{\\{w, v_3\\}, \\{v_1\\}, \\{v_2\\}, \\emptyset \\}$};\n\n\\node[vertex] at (P4) {$P_4$};\n\\node[part label, above=0.4cm of P4] {$\\{\\{w\\}, \\{v_1\\}, \\{v_2\\}, \\{v_3\\}\\}$};\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_4(K_{3}\\sqcup K_1)\\cong K_4$. The graph $K_3\\sqcup K_1$ has vertex set $\\{v_1, v_2, v_3, w\\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K3-plus-K1}\n\\end{figure}\n\nOur work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\\B_k(G\\sqcup K_k) \\cong \\mathcal{C}_k(G)$ for any graph $G$ and integer $k\\ge 1$ \\cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \\emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \\emph{only} cycles that are realizable as coloring graphs \\cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:",
    "context": "Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \\cite{JT95}, the themed collection \\cite{BW15}, and the recent text \\cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \\emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \\cites{HSTT24, BBHvdHHP25, AKL25}.\n\nIn this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\\in\\mathbb{N}$, we define the \\emph{Bell $k$-coloring graph} $\\mathcal{B}_k(G)$ as follows.\n\\begin{itemize}\n \\item The vertices of $\\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \\emph{stable} $k$-partitions.\n \\item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\\in V(G)$, formalized as $P_1-v = P_2-v$.\n\\end{itemize}\nThe notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\\ref{fig:Bell3-K13}~and~\\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.\n\nThe name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \\cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \\cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \\cite{Haa12}, \\cite{FM14}) address Bell coloring graphs.\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_3(K_{1,3})\\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\\{u, v_1, v_2, v_3\\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K13}\n\\end{figure}\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_4(K_{3}\\sqcup K_1)\\cong K_4$. The graph $K_3\\sqcup K_1$ has vertex set $\\{v_1, v_2, v_3, w\\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K3-plus-K1}\n\\end{figure}\n\nOur work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\\B_k(G\\sqcup K_k) \\cong \\mathcal{C}_k(G)$ for any graph $G$ and integer $k\\ge 1$ \\cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \\emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \\emph{only} cycles that are realizable as coloring graphs \\cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:",
    "full_context": "Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \\cite{JT95}, the themed collection \\cite{BW15}, and the recent text \\cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \\emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \\cites{HSTT24, BBHvdHHP25, AKL25}.\n\nIn this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\\in\\mathbb{N}$, we define the \\emph{Bell $k$-coloring graph} $\\mathcal{B}_k(G)$ as follows.\n\\begin{itemize}\n \\item The vertices of $\\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \\emph{stable} $k$-partitions.\n \\item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\\in V(G)$, formalized as $P_1-v = P_2-v$.\n\\end{itemize}\nThe notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\\ref{fig:Bell3-K13}~and~\\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.\n\nThe name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \\cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \\cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \\cite{Haa12}, \\cite{FM14}) address Bell coloring graphs.\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_3(K_{1,3})\\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\\{u, v_1, v_2, v_3\\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K13}\n\\end{figure}\n\n\\end{tikzpicture}\n\\caption{Illustration of $\\mathcal{B}_4(K_{3}\\sqcup K_1)\\cong K_4$. The graph $K_3\\sqcup K_1$ has vertex set $\\{v_1, v_2, v_3, w\\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.} \n\\label{fig:Bell3-K3-plus-K1}\n\\end{figure}\n\nOur work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\\B_k(G\\sqcup K_k) \\cong \\mathcal{C}_k(G)$ for any graph $G$ and integer $k\\ge 1$ \\cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \\emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \\emph{only} cycles that are realizable as coloring graphs \\cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:\n\nOur work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\\B_k(G\\sqcup K_k) \\cong \\mathcal{C}_k(G)$ for any graph $G$ and integer $k\\ge 1$ \\cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \\emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \\emph{only} cycles that are realizable as coloring graphs \\cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:\n\nWe prove Theorem~\\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs.\n\n\\textbf{Outline of the paper.} Section~\\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\\ge 6$. Section~\\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell\n$3$-coloring graphs. Section~\\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.\n\nIn this section, we introduce the matching reconfiguration graph and establish its connection to Bell coloring graphs. As a consequence, we obtain Theorem~\\ref{thm:trees-cycles}, stating that all trees and cycle graphs are realizable as Bell coloring graphs.\n\nThe isomorphism in Proposition~\\ref{prop:matchings} enables us to realize all cycles and trees as Bell coloring graphs. We focus on the case where $n=|V(G)|=2k+1$. More formally, if $n=2k+1$ then we call $\\mathcal{M}_k(G)$ the \\emph{near-perfect matching graph} of $G$. The following lemma gives three equivalent conditions for two near-perfect matchings to be adjacent in a matching graph, which helps us reason about matching graphs that are cycles. In particular, it shows that near-perfect matchings are adjacent if and only if the unmatched vertices $u$ and $v$ have a common neighbor $w$ such that the symmetric difference of the matching is $\\{uw,uv\\}$.\n\nHaving shown that all cycles are realizable as near-perfect matching graphs and hence Bell coloring graphs, we turn to the task of realizing all trees as Bell coloring graphs. To do this, we first describe a procedure for joining near-perfect matching graphs to create larger near-perfect matching graphs. This joining construction relies on a particular property of odd-order graphs. For a graph $G$ on $n=2k+1$ vertices, a vertex $v$ is \\emph{uniquely unmatched} if the graph $G\\setminus v$ has a unique perfect matching. For two such graphs $H_1$ and $H_2$, Figure~\\ref{fig:npm} depicts a near-perfect matching of these graphs joined on their uniquely unmatched vertices. We show that the near-perfect matching graph of this joined graph is exactly the joined near-perfect matching graphs of $H_1$ and $H_2$.\n\n\\begin{theorem}\\label{thm:trees-cycles}\n All trees and all cycle graphs are realizable as Bell coloring graphs.\n\\end{theorem}\n\nThis section proves Theorem~\\ref{thm:tree-reconstruction} through a sequence of lemmas that yield an algorithmic procedure for reconstructing a tree given its Bell 3-coloring graph. First, we prove that three categories of trees can be differentiated based on the degree sequence of their Bell coloring graphs, and then we show that within each category, other Bell coloring graph features determine the specific tree. As we can manually verify the result for all trees of order at most $5$, we assume $n\\geq 6$ throughout this section.\n\n\\begin{theorem}\\label{thm:tree-reconstruction}\nFor trees $T_1$ and $T_2$, we have $\\B_3(T_1)\\cong \\B_3(T_2)$ if and only if $T_1\\cong T_2$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:trees-cycles}\n    All trees and all cycle graphs are realizable as Bell coloring graphs.\n\\end{theorem}",
    "post_theorem_intro_text_len": 5164,
    "post_theorem_intro_text": "We prove Theorem~\\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs. \n\nThe papers \\cites{BFHRS16, ABFR17} examine forbidden subgraphs in standard coloring graphs. To treat the analogous question for Bell coloring graphs, we first obtain a structural description of their cliques (see Theorem~\\ref{thm:clique_classification}), showing that every clique belongs to one of two explicit families. This classification allows us to construct an infinite family of forbidden induced subgraphs of Bell coloring graphs.\n\n\\begin{theoremA}[Theorem~\\ref{thm:forbidden}]\nThe graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\\ge 6$ is an infinite family of forbidden induced subgraphs.\n\\end{theoremA}\n\nAnother natural question is whether the structure of the reconfiguration graph uniquely determines the base graph. This is known for standard coloring graphs; in particular, if $T$ is a tree, $\\mathcal{C}_3(T)$ uniquely determines $T$ (\\cite{AKL25}*{Theorem 1.1} or \\cite{BBHvdHHP25}*{Theorem 1.2}). Having lost the color labels, $\\mathcal{B}_3(T)$ has less symmetry than the $3$-coloring graph $\\mathcal{C}_3(T)$. Despite this compression, our next result guarantees that a tree $T$ can be reconstructed uniquely from its Bell $3$-coloring graph. Let $\\mathsf{Trees}$ and $\\mathsf{Graphs}$ denote the sets of isomorphism classes of all (finite) trees and graphs, respectively. \n\n\\begin{theoremA}[Theorem~\\ref{thm:tree-reconstruction}]\nThe map $\\mathsf{Trees} \\to \\mathsf{Graphs}$ given by $T\\mapsto \\mathcal{B}_3(T)$ is injective.\n\\end{theoremA}\n\nWe also find it useful to work with a multigraph variant, the \\emph{Bell coloring multigraph}, denoted $\\BellMulti{k}{G}$, which uses multiple edges to encode how many vertices are responsible for each adjacency. If $P$ and $P'$ are two stable partitions of $G$, then in the Bell coloring graph, $P$ and $P'$ are adjacent whenever there exists a vertex $v\\in V(G)$ with $P-v = P'-v$. Thus, multiple vertices may witness the same adjacency, but $\\B_k(G)$ still records only a single edge between $P$ and $P'$. In the multigraph version, we instead add a separate parallel edge between $P$ and $P'$ for each distinct vertex $v$ such that $P-v = P'-v$. \n\nThe additional information means that Bell coloring multigraphs serve as a more refined graph invariant. For example, Figures~\\ref{fig:Bell3-K13}~and~\\ref{fig:Bell3-K3-plus-K1} show that $\\B_3(K_{1,3})\\cong \\B_4(K_3\\sqcup K_1)$. However, each edge incident to the partition $P_4$ in the latter graph is witnessed by two vertices, so \n$\\BellMulti{3}{K_{1,3}}\\not\\cong\\BellMulti{4}{K_3\\sqcup K_{1}}$. \n\nYet even a Bell coloring multigraph cannot serve as a complete graph invariant. For instance, $\\BellMulti{3}{K_{1,3}}\\cong \\BellMulti{2}{\\overline{K}_3}$. This ambiguity arises because the center vertex in $K_{1,3}$ is adjacent to every other vertex. When a vertex $v$ is adjacent to every vertex in $G-v$, we call $v$ a \\emph{universal vertex}. \nIf $G$ has a universal vertex $w$, then $\\B_{k+1}(G) \\cong \\B_k(G-w)$ for every $k\\in\\mathbb{N}$ due to the natural bijection between stable $k$-partitions of $G-w$ and stable $(k+1)$-partitions of $G$ in which $\\{w\\}$ forms an additional singleton part.\n\nTo handle this, let $U(G)$ denote the set of universal vertices of $G$. We define the \\emph{core} of $G$, denoted $G^{\\circ}$, as the induced subgraph $G[V(G)\\setminus U(G)]$. We define the equivalence relation $\\sim_{\\text{uni}}$ on $\\mathsf{Graphs}$ as follows: $G\\sim_{\\text{uni}} G'$ if their cores are isomorphic, $G^{\\circ} \\cong (G')^{\\circ}$.\nThe equivalence classes of graphs under this relation form the set $\\mathsf{Graphs}^{\\circ} = \\mathsf{Graphs}/\\!\\sim_{\\text{uni}}$.\n\nOur final result shows that a base graph of order $n$ can be reconstructed from its Bell $n$-coloring multigraph up to universal vertices:\n\n\\begin{theoremA}[Theorem~\\ref{thm:general-reconstruction-multigraph:intro}]\nThe map $\\mathsf{Graphs}^{\\circ} \\to \\mathsf{Multigraphs}$ given by $G\\mapsto \\BellMultiIntro{|V(G)|}{G}$  is injective.\n\\end{theoremA}\n\n\\textbf{Outline of the paper.} Section~\\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\\ge 6$. Section~\\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell\n$3$-coloring graphs. Section~\\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.",
    "sketch": "We prove Theorem~\\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs. In particular, Section~\\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\begin{theorem}\\label{thm:trees-cycles}\n    All trees and all cycle graphs are realizable as Bell coloring graphs.\n\\end{theorem}\nAll trees and all cycles are realizable as Bell coloring graphs.",
    "theorem_type": [
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(H\\) be a graph that is either a tree or a cycle graph. A graph is said to be realizable as a Bell coloring graph if there exist a graph \\(G\\) and an integer \\(k\\ge 1\\) such that \\(H\\cong \\mathcal{B}_k(G)\\), where \\(\\mathcal{B}_k(G)\\) is the graph whose vertices are the stable \\(k\\)-partitions of \\(V(G)\\) (that is, partitions of \\(V(G)\\) into \\(k\\) independent sets, some possibly empty), and where two such partitions \\(P_1\\) and \\(P_2\\) are adjacent when they differ only in the placement of a single vertex, equivalently \\(P_1-v=P_2-v\\) for some vertex \\(v\\in V(G)\\). Which statement holds for every such \\(H\\)?",
      "correct_choice": {
        "label": "A",
        "text": "Every tree and every cycle graph is realizable as a Bell coloring graph; equivalently, for each such graph \\(H\\), there exist a graph \\(G\\) and an integer \\(k\\ge 1\\) with \\(H\\cong \\mathcal{B}_k(G)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "Every tree is realizable as a Bell coloring graph, but among cycle graphs only the even cycles are realizable; equivalently, for each tree or even cycle graph \\(H\\), there exist a graph \\(G\\) and an integer \\(k\\ge 1\\) with \\(H\\cong \\mathcal{B}_k(G)\\)."
        },
        {
          "label": "C",
          "text": "Every tree is realizable as a Bell coloring graph; equivalently, for each tree \\(H\\), there exist a graph \\(G\\) and an integer \\(k\\ge 1\\) with \\(H\\cong \\mathcal{B}_k(G)\\)."
        },
        {
          "label": "D",
          "text": "For every graph \\(H\\) that is either a tree or a cycle graph, there exists an integer \\(k\\ge 1\\) such that for every such \\(H\\) one can find a graph \\(G\\) with \\(H\\cong \\mathcal{B}_k(G)\\)."
        },
        {
          "label": "E",
          "text": "Every tree and every cycle graph is realizable as a Bell coloring graph; moreover, for each such graph \\(H\\), one may always choose \\(G=H\\) and some integer \\(k\\ge 1\\) so that \\(H\\cong \\mathcal{B}_k(H)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "all cycles vs only even cycles",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the cycle-graph conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of k on H",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "existence of some realizing graph G replaced by G=H",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It defines the Bell coloring graph notion and asks which global statement is true, without embedding the theorem’s conclusion."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct choice is essentially a direct restatement of the target theorem: that every tree and every cycle graph is realizable as a Bell coloring graph. The item mainly tests recall of that exact statement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options vary by strength, quantifiers, and special cases (e.g., weaker true statement, parity restriction, fixed-k trap, stronger G=H claim). However, if the theorem is known, the correct answer is immediate rather than generated through substantial reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: B introduces a plausible even-cycle restriction, C is a weaker true statement, D tests quantifier dependence, and E is an overstrong realization claim. These reflect common failure modes."
      },
      "total_score": 5,
      "overall_assessment": "Low leakage and high-quality distractors, but the item is close to a theorem-recall question and thus only weakly tests generative mathematical reasoning."
    }
  },
  {
    "id": "2512.10202v1",
    "paper_link": "http://arxiv.org/abs/2512.10202v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{mainthm1} Let $\\blam\\in\\P_{\\ell,n}$ and $\\mathfrak{s},\\mathfrak{t}\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{MM}}(\\fm_{\\mathfrak{s}\\mathfrak{t}})=\n\\begin{cases}\nq^{\\sum\\limits_{k=1}^n(k-k(\\t_{\\downarrow\\leq k}))}\\bigg(\\prod\\limits_{j=1}^n\\prod\\limits_{i=l(\\mathfrak{t},j)+1}^{\\ell}(-Q_i)\\bigg), & \\mbox{if\\ \\ }\\mathfrak{s}=\\mathfrak{t}, \\\\\n0, & \\mbox{if\\ \\ }\\mathfrak{s}\\neq \\mathfrak{t}.\n\\end{cases}$$",
      "start_pos": 16873,
      "end_pos": 17210,
      "label": "mainthm1"
    },
    "ref_dict": {
      "mainthm1": "\\begin{thm}\\label{mainthm1} Let $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{MM}}(\\fm_{\\s\\t})=\n\\begin{cases}\nq^{\\sum\\limits_{k=1}^n(k-k(\\t_{\\downarrow\\leq k}))}\\bigg(\\prod\\limits_{j=1}^n\\prod\\limits_{i=l(\\t,j)+1}^{\\ell}(-Q_i)\\bigg), & \\mbox{if\\ \\ }\\s=\\t, \\\\\n0, & \\mbox{if\\ \\ }\\s\\neq \\t.\n\\end{cases}$$\n\\end{thm}",
      "degenerate": "\\begin{dfn}\\label{degenerate} The degenerate cyclotomic Hecke algebra $H_{\\ell,n}=H_{\\ell,n}^R(\\bu)$ of type $G(\\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1},s_2,\\dots,s_{n-1},L_1,L_2,\\dots,L_n$ and the following defining relations:\n\\begin{align}\n\t&(L_1-u_1)(L_1-u_2)\\cdots(L_1-u_\\ell)=0;\\\\\n\\label{degquadraticrela}\t&s_{i}^{2}=1,\\quad \\forall\\,1\\leq i\\leq n-1;\\\\\n\\label{degbraidrela1}\t&s_{i}s_{j}=s_{j}s_{i},\\ \\ \\forall\\,1\\leq i<j-1<n-1;\\\\\n\\label{degbraidrela2}\t&s_is_{i+1}s_i=s_{i+1}s_is_{i+1},\\ \\ \\forall\\,1\\leq i<n-1;\\\\\n\\label{deg-L-comm1} &L_iL_k=L_kL_i,\\ \\  s_iL_l=L_ls_i,\\ \\ 1\\leq i<n,\\ 1\\leq k,l\\leq n,\\ l\\neq i,i+1;\\\\\n\\label{deg-L-comm2} &L_{i+1}=s_iL_is_i+s_i,\\ \\ 1\\leq i<n .\n\\end{align}\n\\end{dfn}",
      "maincor12": "\\begin{cor}\\label{maincor12}\nLet $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$\n\\begin{enumerate}\n  \\item If $n(\\s)= n(\\t),$ then\nfor each $(\\fp,\\fq)\\in \\Std^2(\\P_{\\ell,n-1})$ with $(\\fp,\\fq)\\rhd (\\s_{\\downarrow\\leq (n-1)},\\t_{\\downarrow\\leq (n-1)}),$ we have\n$$\\begin{aligned}\n&c_{\\t,n}+\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\n\\\\\\u_{\\downarrow\\leq (n-1)}=\\s_{\\downarrow\\leq (n-1)}\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\t_{\\downarrow\\leq (n-1)}}}\nb_{\\u\\v}^{\\s\\t}c_{\\v,n}\n=q^{n-n(\\t)}\\Biggr(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\Biggr),\\\\\n&\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\\\\\n\\u_{\\downarrow\\leq (n-1)}=\\fp\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\fq}} b_{\\u\\v}^{\\s\\t} c_{\\v,n}=\nq^{n-n(\\t)}\\Biggr(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\Biggr) b_{\\fp\\fq}^{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}.\n\\end{aligned} $$\n  \\item If $n(\\s)\\neq n(\\t),$ then for each $(\\fp,\\fq)\\in \\Std^2(\\P_{\\ell,n-1})$\n  with $(\\fp,\\fq)\\rhd (\\s_{\\downarrow\\leq (n-1)},\\t_{\\downarrow\\leq (n-1)}),$ we have\n$$\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\\\\\n\\u_{\\downarrow\\leq (n-1)}=\\fp\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\fq}} b_{\\u\\v}^{\\s\\t} c_{\\v,n}=0. $$\n\\end{enumerate}\n\\end{cor}",
      "maincor13": "\\begin{cor}\\label{maincor13}\nLet $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then we have\n\\begin{align*}\n\\delta_{\\s\\t}d_\\t+\\sum_{\\substack{\\u\\in\\te{Std}(\\P_{\\ell,n})\\\\(\\u,\\u)\\rhd (\\s,\\t)}}a_{\\u\\u}^{\\s\\t}d_\\u=\\delta_{\\s\\t} c_\\t,\\quad\n\\delta_{\\s\\t}c_\\t+\\sum_{\\substack{\\u\\in\\te{Std}(\\P_{\\ell,n})\\\\(\\u,\\u)\\rhd (\\s,\\t)}}b_{\\u\\u}^{\\s\\t}c_\\u=\\delta_{\\s\\t} d_\\t.\n\\end{align*}\n\\end{cor}",
      "mainthm2": "\\begin{thm}\\label{mainthm2} Let $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{BK}}(m_{\\s\\t})=\n\\begin{cases}\n1, & \\mbox{if\\ \\,}\\s=\\t\\te{\\ \\,and\\ \\,} l(\\t,i)=1,\\ \\forall\\,1\\leq i\\leq n, \\\\\n0, & \\mbox{\\text{otherwise}}.\n\\end{cases}$$\n\\end{thm}",
      "MainProp1": "\\begin{prop}\\label{MainProp1}\nLet $\\blam\\in\\mathscr{P}_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then\n$$\\ve_n(\\fm_{\\s\\t})=\\begin{cases}\nq^{n-n(\\t)}\\bigg(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\bigg)\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}},\n& \\text{if\\ \\ $n(\\s)=n(\\t),$}\\\\\n0, & \\text{if\\ \\ $n(\\s)\\neq n(\\t)$.}\n\\end{cases} $$\n\\end{prop}",
      "maincor11": "\\begin{cor}\\label{maincor11}\nLet $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$\n\\begin{enumerate}\n  \\item If $n(\\s)= n(\\t),$ then\nfor each $(\\fp,\\fq)\\in \\Std^2(\\P_{\\ell,n-1})$ with $(\\fp,\\fq)\\rhd (\\s_{\\downarrow\\leq (n-1)},\\t_{\\downarrow\\leq (n-1)}),$ we have\n$$\\begin{aligned}\n&q^{n-n(\\t)}\\Biggr(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\Biggr)+\n\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\n\\\\\\u_{\\downarrow\\leq (n-1)}=\\s_{\\downarrow\\leq (n-1)}\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\t_{\\downarrow\\leq (n-1)}}}\na_{\\u\\v}^{\\s\\t}q^{n-n(\\u)}\\Biggr(\\prod\\limits_{i=l(\\u,n)+1}^{\\ell}(-Q_i)\\Biggr)=c_{\\t,n},\\\\\n&\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\\\\\n\\u_{\\downarrow\\leq (n-1)}=\\fp\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\fq}} a_{\\u\\v}^{\\s\\t}q^{n-n(\\u)}\n\\Biggr(\\prod\\limits_{i=l(\\u,n)+1}^{\\ell}(-Q_i)\\Biggr)=c_{\\t,n} a_{\\fp\\fq}^{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}.\n\\end{aligned} $$\n  \\item If $n(\\s)\\neq n(\\t),$ then for each $(\\fp,\\fq)\\in \\Std^2(\\P_{\\ell,n-1})$\n  with $(\\fp,\\fq)\\rhd (\\s_{\\downarrow\\leq (n-1)},\\t_{\\downarrow\\leq (n-1)}),$ we have\n$$\\sum_{\\substack{(\\u,\\v)\\in\\te{Std}^2(\\P_{\\ell,n})\\\\(\\u,\\v)\\rhd (\\s,\\t)\\\\\n\\u_{\\downarrow\\leq (n-1)}=\\fp\\\\\n\\v_{\\downarrow\\leq (n-1)}=\\fq}} a_{\\u\\v}^{\\s\\t}q^{n-n(\\u)}\n\\Biggr(\\prod\\limits_{i=l(\\u,n)+1}^{\\ell}(-Q_i)\\Biggr)=0. $$\n\\end{enumerate}\n\\end{cor}"
    },
    "pre_theorem_intro_text_len": 6861,
    "pre_theorem_intro_text": "Let $R$ be an integral domain, $1\\neq q\\in R^\\times$ and ${\\mathbf Q}:=(Q_{1},Q_2,\\dots, Q_{\\ell})\\in R^\\ell$. The non-degenerate cyclotomic Hecke algebras $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ of type $G(\\ell,1,n)$ with Hecke parameter $q$ and cyclotomic parameters $Q_{1},Q_2,\\dots, Q_{\\ell}$ were first introduced in \\cite[Definition 3.1]{AK1}, \\cite[Definition 4.1]{BM:cyc} and \\cite[before Proposition 3.2]{C} as certain deformations of the group ring $R[W_{\\ell,n}]$ of the complex reflection group $W_{\\ell,n}$ of type $G(\\ell,1,n)$. By definition, $\\mathscr{H}_{\\ell,n}=\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ is the unital associative $R$-algebra with generators $T_0,T_1,\\dots,T_{n-1}$ and the following defining relations:\n\\begin{align}\n&(T_0-Q_1)(T_0-Q_2)\\cdots(T_0-Q_\\ell)=0;\\\\\n&T_0T_1T_0T_1=T_1T_0T_1T_0;\\\\\n\\label{nondegquadraticrela}&(T_i-q)(T_i+1)=0,\\quad \\forall\\,1\\leq i\\leq n-1;\\\\\n\\label{nondegbraidrela1}&T_iT_j=T_jT_i,\\ \\ \\forall\\,1\\leq i<j-1<n-1;\\\\\n\\label{nondegbraidrela2}&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\\ \\ \\forall\\,1\\leq i<n-1.\n\\end{align}\nIf $\\ell=1$ or $\\ell=2$, then the cyclotomic Hecke algebra $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ becomes the Iwahori-Hecke algebra of type $A_{n-1}$ or type $B_n$ respectively.\n\nThe symmetric group $\\Sym_n$ on $\\{1,2,\\dots,n\\}$ is generated by $\\{\\sigma_i:=(i,i+1)\\,|\\,1\\leq i<n\\}$. A word $w=\\sigma_{i_{1}}\\sigma_{i_{2}}\\cdots\\sigma_{i_{k}}$, where $1\\leq i_1,i_2,\\dots,i_k\\leq n-1$, is called a reduced expression of $w$ if $k$ is minimal; in this case we say that $w$ has length $k$ and we write $\\ell(w)=k$. Note that $T_w:=T_{i_1}T_{i_2}\\cdots T_{i_k}$ depends only on $w$ but not on the choice of the reduced expression of $w$ because of the braid relations holding in $\\HH_{\\ell,n}$.\n\nThe $R$-subalgebra of $\\mathscr{H}_{\\ell,n}$ generated by $T_1,\\dots,T_{n-1}$ is isomorphic to the Iwahori-Hecke algebra of type $A_{n-1}$, which has a standard basis $\\{T_w\\,|\\, w\\in\\Sym_n\\}$. We set $$\n\\mL_{1}=T_{0},\\quad \\mL_{k+1}=q^{-1}T_{k}L_{k}T_{k},\\ \\ \\forall\\,1\\leq k\\leq n-1 .\n$$\nThese elements commute with each other, and are called the {\\bf Jucys-Murphy operators} of $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$.\n\n\\begin{lem}\\label{AK basis}\\te{(}Ariki--Koike \\cite[(3.10)]{AK1}\\te{)}\nThe algebra $\\mathscr{H}_{\\ell,n}$ is free as an $R$-module with basis\n$$\\{\\mL_1^{c_1}\\mL_2^{c_2}\\cdots \\mL_n^{c_n}T_w\\mid\n       w\\in\\Sym_n,\\ 0\\leq c_i\\leq \\ell-1,\\ \\forall\\,1\\leq i\\leq n\\}.$$\nIn particular, the $R$-rank of $\\mathscr{H}_{\\ell,n}$ is equal to $\\ell^nn!$.\n\\end{lem}\n\n\\begin{dfn}\\label{ssf} For any $w\\in \\Sym_n$ and integers $0\\leq c_1,c_2,\\dots,c_n<\\ell,$ we define\n$$\n\\tau_{R}^{\\rm{MM}}(\\mL_1^{c_1}\\mL_2^{c_2}\\cdots \\mL_n^{c_n}T_w):=\\begin{cases}\n1, &\\text{if\\ \\,$w=1$\\ and\\ $c_1=c_2=\\cdots=c_n=0$},\\\\\n0, &\\text{otherwise.}\n\\end{cases}\n$$\nWe extend $\\tau_{R}^{\\rm{MM}}$ linearly to an $R$-linear function on $\\HH_{\\ell,n}$.\n\\end{dfn}\nOne can show directly that the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ is symmetric, i.e., $\\tau_{R}^{\\rm{MM}}(xy)=\\tau_{R}^{\\rm{MM}}(yx)$, $\\forall\\,x,y\\in \\HH_{\\ell,n}$, see \\cite{MM} and \\cite{HHL}. We call $\\tau_R^{\\rm{MM}}$ the {\\bf{standard symmetrizing form}} on $\\HH_{\\ell,n}$. If furthermore, $(Q_{1},Q_2,\\dots, Q_{\\ell})\\in (R^\\times)^\\ell$, then by a result of Malle-Mathas \\cite{MM}, the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ on $\\HH_{\\ell,n}$ is non-degenerate, which makes $\\HH_{\\ell,n}$ into a symmetric algebra over $R$.\n\nWe use $\\P_{\\ell,n}$ to denote the set of $\\ell$-partitions of $n$. For each $\\blam\\in\\P_{\\ell,n}$, we use $\\Std(\\blam)$ to denote the set of standard $\\blam$-tableaux. Each $\\mathfrak{t}\\in\\Std(\\blam)$ can be viewed as a bijective map\n$\\mathfrak{t}: [\\blam]\\rightarrow\\{1,2,\\dots,n\\}$ from the Young diagram $[\\blam]:=\\{(r,c,l)\\mid 1\\leq l\\leq\\ell,\\ r\\geq 1,\\ 1\\leq c\\leq\\lam_r^{(l)}\\}$ to $\\{1,2,\\dots,n\\}$.\n\nIn \\cite{DJM}, Dipper-James-Mathas constructed a cellular basis $\\{\\fm_{\\mathfrak{s}\\mathfrak{t}}\\,|\\, \\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$\nin the sense of \\cite{GL}. This basis generalizes the well-known Murphy basis for the Iwahori-Hecke algebra associated to $\\Sym_n$ (i.e., the level one case). We call it {\\bf the Murphy basis for the non-degenerate cyclotomic Hecke algebra $\\HH_{\\ell,n}$}. When $\\HH_{\\ell,n}$ is semisimple, Mathas also constructed a seminormal basis $\\{\\mf_{\\mathfrak{s}\\mathfrak{t}}\\,|\\,\\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$. We refer the readers to Sections 2.1 and 3.2 for their precise definitions.\n\nA critical challenge in the study of $\\HH_{\\ell,n}$ lies in characterizing the transition matrix between the Murphy basis and the seminormal basis. The explicit transformation between the Murphy basis and the seminormal basis—whose entries encode essential structural correlations—lacks a systematic description. This gap persists partly due to the absence of precise computations of trace forms on\n the Murphy basis, as the values of the trace form on seminormal basis elements can be easily expressed in terms of the gamma coefficients and the Schur functions. Without such exact values, the transition matrix remains elusive, obstructing efforts to unify the combinatorial (Murphy basis) and representation-theoretic (seminormal basis) perspectives of Hecke algebras.\n\nAnother pressing problem concerns the integral basis of the Hecke algebra's cocenter. In \\cite{HS25}, the first author and Lei Shi have constructed an integral basis for the cocenter of $\\HH_{\\ell,n}$. However, it is very hard to apply that integral basis of the cocenter because it is given by a subset of Bremke-Malle's standard basis $\\{T_w\\}$ whose definition depends on the choice of the reduced expression of $w$ and is not directly related to the cellular structure of $\\HH_{\\ell,n}$. We really expect an integral basis for the cocenter of $\\HH_{\\ell,n}$ which is directly related to the cellular structure of $\\HH_{\\ell,n}$. The Murphy basis, with its inherent integral and recursive properties, offer a promising avenue for constructing such a basis—but this requires precise knowledge of trace forms on the Murphy basis to ensure integrality and linear independence.\n\n\\begin{dfn}\\label{keydefn} Let $k\\in\\mathbb N,\\ \\blam=(\\lambda^{(1)},\\dots,\\lambda^{(\\ell)})\\in\\P_{\\ell,k}$ and $\\mathfrak{t}\\in\\Std(\\blam)$.\nLet $1\\leq k(\\mathfrak{t})\\leq k$ be the unique integer such that\n\\begin{equation*}\\label{ntDef}\\mathfrak{t}^{-1}(k)=(\\mathfrak{t}^{\\blam})^{-1}(k(\\mathfrak{t})).\n\\end{equation*}\nFor each integer $1\\leq j\\leq n$, we let $r(\\mathfrak{t},j),\\,c(\\mathfrak{t},j)$ and $l(\\mathfrak{t},j)$ be the integers such that\n\\begin{equation*}\\label{coordinate}\n\\mathfrak{t}^{-1}(j)=(r(\\mathfrak{t},j),c(\\mathfrak{t},j),l(\\mathfrak{t},j))\\in [\\blam].\n\\end{equation*}\n\\end{dfn}\n\nThe following theorem is the first main result of this paper.",
    "context": "Let $R$ be an integral domain, $1\\neq q\\in R^\\times$ and ${\\mathbf Q}:=(Q_{1},Q_2,\\dots, Q_{\\ell})\\in R^\\ell$. The non-degenerate cyclotomic Hecke algebras $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ of type $G(\\ell,1,n)$ with Hecke parameter $q$ and cyclotomic parameters $Q_{1},Q_2,\\dots, Q_{\\ell}$ were first introduced in \\cite[Definition 3.1]{AK1}, \\cite[Definition 4.1]{BM:cyc} and \\cite[before Proposition 3.2]{C} as certain deformations of the group ring $R[W_{\\ell,n}]$ of the complex reflection group $W_{\\ell,n}$ of type $G(\\ell,1,n)$. By definition, $\\mathscr{H}_{\\ell,n}=\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ is the unital associative $R$-algebra with generators $T_0,T_1,\\dots,T_{n-1}$ and the following defining relations:\n\\begin{align}\n&(T_0-Q_1)(T_0-Q_2)\\cdots(T_0-Q_\\ell)=0;\\\\\n&T_0T_1T_0T_1=T_1T_0T_1T_0;\\\\\n\\label{nondegquadraticrela}&(T_i-q)(T_i+1)=0,\\quad \\forall\\,1\\leq i\\leq n-1;\\\\\n\\label{nondegbraidrela1}&T_iT_j=T_jT_i,\\ \\ \\forall\\,1\\leq i<j-1<n-1;\\\\\n\\label{nondegbraidrela2}&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\\ \\ \\forall\\,1\\leq i<n-1.\n\\end{align}\nIf $\\ell=1$ or $\\ell=2$, then the cyclotomic Hecke algebra $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ becomes the Iwahori-Hecke algebra of type $A_{n-1}$ or type $B_n$ respectively.\n\n\\begin{dfn}\\label{ssf} For any $w\\in \\Sym_n$ and integers $0\\leq c_1,c_2,\\dots,c_n<\\ell,$ we define\n$$\n\\tau_{R}^{\\rm{MM}}(\\mL_1^{c_1}\\mL_2^{c_2}\\cdots \\mL_n^{c_n}T_w):=\\begin{cases}\n1, &\\text{if\\ \\,$w=1$\\ and\\ $c_1=c_2=\\cdots=c_n=0$},\\\\\n0, &\\text{otherwise.}\n\\end{cases}\n$$\nWe extend $\\tau_{R}^{\\rm{MM}}$ linearly to an $R$-linear function on $\\HH_{\\ell,n}$.\n\\end{dfn}\nOne can show directly that the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ is symmetric, i.e., $\\tau_{R}^{\\rm{MM}}(xy)=\\tau_{R}^{\\rm{MM}}(yx)$, $\\forall\\,x,y\\in \\HH_{\\ell,n}$, see \\cite{MM} and \\cite{HHL}. We call $\\tau_R^{\\rm{MM}}$ the {\\bf{standard symmetrizing form}} on $\\HH_{\\ell,n}$. If furthermore, $(Q_{1},Q_2,\\dots, Q_{\\ell})\\in (R^\\times)^\\ell$, then by a result of Malle-Mathas \\cite{MM}, the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ on $\\HH_{\\ell,n}$ is non-degenerate, which makes $\\HH_{\\ell,n}$ into a symmetric algebra over $R$.\n\nWe use $\\P_{\\ell,n}$ to denote the set of $\\ell$-partitions of $n$. For each $\\blam\\in\\P_{\\ell,n}$, we use $\\Std(\\blam)$ to denote the set of standard $\\blam$-tableaux. Each $\\mathfrak{t}\\in\\Std(\\blam)$ can be viewed as a bijective map\n$\\mathfrak{t}: [\\blam]\\rightarrow\\{1,2,\\dots,n\\}$ from the Young diagram $[\\blam]:=\\{(r,c,l)\\mid 1\\leq l\\leq\\ell,\\ r\\geq 1,\\ 1\\leq c\\leq\\lam_r^{(l)}\\}$ to $\\{1,2,\\dots,n\\}$.\n\nIn \\cite{DJM}, Dipper-James-Mathas constructed a cellular basis $\\{\\fm_{\\mathfrak{s}\\mathfrak{t}}\\,|\\, \\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$\nin the sense of \\cite{GL}. This basis generalizes the well-known Murphy basis for the Iwahori-Hecke algebra associated to $\\Sym_n$ (i.e., the level one case). We call it {\\bf the Murphy basis for the non-degenerate cyclotomic Hecke algebra $\\HH_{\\ell,n}$}. When $\\HH_{\\ell,n}$ is semisimple, Mathas also constructed a seminormal basis $\\{\\mf_{\\mathfrak{s}\\mathfrak{t}}\\,|\\,\\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$. We refer the readers to Sections 2.1 and 3.2 for their precise definitions.\n\n\\begin{dfn}\\label{keydefn} Let $k\\in\\mathbb N,\\ \\blam=(\\lambda^{(1)},\\dots,\\lambda^{(\\ell)})\\in\\P_{\\ell,k}$ and $\\mathfrak{t}\\in\\Std(\\blam)$.\nLet $1\\leq k(\\mathfrak{t})\\leq k$ be the unique integer such that\n\\begin{equation*}\\label{ntDef}\\mathfrak{t}^{-1}(k)=(\\mathfrak{t}^{\\blam})^{-1}(k(\\mathfrak{t})).\n\\end{equation*}\nFor each integer $1\\leq j\\leq n$, we let $r(\\mathfrak{t},j),\\,c(\\mathfrak{t},j)$ and $l(\\mathfrak{t},j)$ be the integers such that\n\\begin{equation*}\\label{coordinate}\n\\mathfrak{t}^{-1}(j)=(r(\\mathfrak{t},j),c(\\mathfrak{t},j),l(\\mathfrak{t},j))\\in [\\blam].\n\\end{equation*}\n\\end{dfn}\n\nThe following theorem is the first main result of this paper.",
    "full_context": "Let $R$ be an integral domain, $1\\neq q\\in R^\\times$ and ${\\mathbf Q}:=(Q_{1},Q_2,\\dots, Q_{\\ell})\\in R^\\ell$. The non-degenerate cyclotomic Hecke algebras $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ of type $G(\\ell,1,n)$ with Hecke parameter $q$ and cyclotomic parameters $Q_{1},Q_2,\\dots, Q_{\\ell}$ were first introduced in \\cite[Definition 3.1]{AK1}, \\cite[Definition 4.1]{BM:cyc} and \\cite[before Proposition 3.2]{C} as certain deformations of the group ring $R[W_{\\ell,n}]$ of the complex reflection group $W_{\\ell,n}$ of type $G(\\ell,1,n)$. By definition, $\\mathscr{H}_{\\ell,n}=\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ is the unital associative $R$-algebra with generators $T_0,T_1,\\dots,T_{n-1}$ and the following defining relations:\n\\begin{align}\n&(T_0-Q_1)(T_0-Q_2)\\cdots(T_0-Q_\\ell)=0;\\\\\n&T_0T_1T_0T_1=T_1T_0T_1T_0;\\\\\n\\label{nondegquadraticrela}&(T_i-q)(T_i+1)=0,\\quad \\forall\\,1\\leq i\\leq n-1;\\\\\n\\label{nondegbraidrela1}&T_iT_j=T_jT_i,\\ \\ \\forall\\,1\\leq i<j-1<n-1;\\\\\n\\label{nondegbraidrela2}&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\\ \\ \\forall\\,1\\leq i<n-1.\n\\end{align}\nIf $\\ell=1$ or $\\ell=2$, then the cyclotomic Hecke algebra $\\HH_{\\ell,n}^R(q;\\mathbf{Q})$ becomes the Iwahori-Hecke algebra of type $A_{n-1}$ or type $B_n$ respectively.\n\n\\begin{dfn}\\label{ssf} For any $w\\in \\Sym_n$ and integers $0\\leq c_1,c_2,\\dots,c_n<\\ell,$ we define\n$$\n\\tau_{R}^{\\rm{MM}}(\\mL_1^{c_1}\\mL_2^{c_2}\\cdots \\mL_n^{c_n}T_w):=\\begin{cases}\n1, &\\text{if\\ \\,$w=1$\\ and\\ $c_1=c_2=\\cdots=c_n=0$},\\\\\n0, &\\text{otherwise.}\n\\end{cases}\n$$\nWe extend $\\tau_{R}^{\\rm{MM}}$ linearly to an $R$-linear function on $\\HH_{\\ell,n}$.\n\\end{dfn}\nOne can show directly that the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ is symmetric, i.e., $\\tau_{R}^{\\rm{MM}}(xy)=\\tau_{R}^{\\rm{MM}}(yx)$, $\\forall\\,x,y\\in \\HH_{\\ell,n}$, see \\cite{MM} and \\cite{HHL}. We call $\\tau_R^{\\rm{MM}}$ the {\\bf{standard symmetrizing form}} on $\\HH_{\\ell,n}$. If furthermore, $(Q_{1},Q_2,\\dots, Q_{\\ell})\\in (R^\\times)^\\ell$, then by a result of Malle-Mathas \\cite{MM}, the $R$-linear function $\\tau_{R}^{\\rm{MM}}$ on $\\HH_{\\ell,n}$ is non-degenerate, which makes $\\HH_{\\ell,n}$ into a symmetric algebra over $R$.\n\nWe use $\\P_{\\ell,n}$ to denote the set of $\\ell$-partitions of $n$. For each $\\blam\\in\\P_{\\ell,n}$, we use $\\Std(\\blam)$ to denote the set of standard $\\blam$-tableaux. Each $\\mathfrak{t}\\in\\Std(\\blam)$ can be viewed as a bijective map\n$\\mathfrak{t}: [\\blam]\\rightarrow\\{1,2,\\dots,n\\}$ from the Young diagram $[\\blam]:=\\{(r,c,l)\\mid 1\\leq l\\leq\\ell,\\ r\\geq 1,\\ 1\\leq c\\leq\\lam_r^{(l)}\\}$ to $\\{1,2,\\dots,n\\}$.\n\nIn \\cite{DJM}, Dipper-James-Mathas constructed a cellular basis $\\{\\fm_{\\mathfrak{s}\\mathfrak{t}}\\,|\\, \\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$\nin the sense of \\cite{GL}. This basis generalizes the well-known Murphy basis for the Iwahori-Hecke algebra associated to $\\Sym_n$ (i.e., the level one case). We call it {\\bf the Murphy basis for the non-degenerate cyclotomic Hecke algebra $\\HH_{\\ell,n}$}. When $\\HH_{\\ell,n}$ is semisimple, Mathas also constructed a seminormal basis $\\{\\mf_{\\mathfrak{s}\\mathfrak{t}}\\,|\\,\\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $\\HH_{\\ell,n}$. We refer the readers to Sections 2.1 and 3.2 for their precise definitions.\n\n\\begin{dfn}\\label{keydefn} Let $k\\in\\mathbb N,\\ \\blam=(\\lambda^{(1)},\\dots,\\lambda^{(\\ell)})\\in\\P_{\\ell,k}$ and $\\mathfrak{t}\\in\\Std(\\blam)$.\nLet $1\\leq k(\\mathfrak{t})\\leq k$ be the unique integer such that\n\\begin{equation*}\\label{ntDef}\\mathfrak{t}^{-1}(k)=(\\mathfrak{t}^{\\blam})^{-1}(k(\\mathfrak{t})).\n\\end{equation*}\nFor each integer $1\\leq j\\leq n$, we let $r(\\mathfrak{t},j),\\,c(\\mathfrak{t},j)$ and $l(\\mathfrak{t},j)$ be the integers such that\n\\begin{equation*}\\label{coordinate}\n\\mathfrak{t}^{-1}(j)=(r(\\mathfrak{t},j),c(\\mathfrak{t},j),l(\\mathfrak{t},j))\\in [\\blam].\n\\end{equation*}\n\\end{dfn}\n\nThe following theorem is the first main result of this paper.\n\nThe following theorem is the first main result of this paper.\n\nLet $\\bu=(u_1,u_2,\\dots,u_\\ell)\\in R^{\\ell}$. The degenerate cyclotomic Hecke algebra $H_{\\ell,n}=H_{\\ell,n}^R(\\bu)$ of type $G(\\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1} ,\\dots,s_{n-1},L_1, \\dots,L_n$ which satisfy certain relations, see Definition \\ref{degenerate}. In \\cite{AMR}, Ariki, Mathas and Rui constructed a cellular basis $\\{m_{\\s\\t}\\,|\\, \\s,\\t\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $H_{\\ell,n}$. This basis generalizes the well-known Murphy basis for the symmetric group algebra $R[\\Sym_n]$ (i.e., the level one case). We call it {\\bf the Murphy basis for the degenerate cyclotomic Hecke algebra $H_{\\ell,n}$}. Brundan-Kleshchev \\cite{BK08} introduced a symmetrizing form $\\tau_R^{\\te{BK}}$ on $H_{\\ell,n}$ which makes it into a symmetric algebra over $R$. We refer the readers to Section 2.2 for their precise definitions. The following theorem is the second main result of this paper.\n\n\\begin{prop}\\label{nondegvemst}\nLet $\\blam\\in\\mathscr{P}_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam)$. Suppose that either $n(\\s)=n$ or $n(\\t)=n$. Then\n$$\\ve_n(\\fm_{\\s\\t})=\\begin{cases} \\bigg(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\bigg)\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}, &\\text{if\\ \\ $n(\\s)=n(\\t),$}\\\\\n0, &\\text{if\\ \\ $n(\\s)\\neq n(\\t).$}\\end{cases}$$\n\\end{prop}\n\n\\begin{prop}\\label{nondegvemst2}\nLet $\\blam\\in\\mathscr{P}_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam)$. Suppose that $n(\\s)\\neq n\\neq n(\\t)$. Then\n$$\\ve_n(\\fm_{\\s\\t})=\\begin{cases} q^{n-n(\\t)}\\bigg(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\bigg)\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}, &\\text{if\\ \\,$n(\\s)=n(\\t),$}\\\\\n0, &\\text{if\\ \\,$n(\\s)\\neq n(\\t).$}\\end{cases}$$\n\\end{prop}\n\n\\smallskip\\noindent\n{\\it Case 2.} Suppose $n(\\s)\\neq n=n(\\t)$. Then $d(\\s)\\neq d(\\s_{\\downarrow\\leq (n-1)})$ and $d(\\t)=d(\\t_{\\downarrow\\leq (n-1)}).$\nThen for each $w\\in\\Sym_\\blam$ such that $\\sigma_{n-1}\\not\\leq w,$\nusing (\\ref{decompositionofdt}), (\\ref{deg-L-comm1}) and Lemma \\ref{deg-L-comm3} (1) we deduce that\n\\begin{align}\\label{degcompu3}\nd(\\s)^*w u_\\blam^+ d(\\t)\n&=d(\\s_{\\downarrow\\leq (n-1)})^*\\gamma_{n-1,n(\\s)} w\n\\Biggr(\\prod_{s=2}^{\\ell}\\prod_{k=1}^{\\fa_s}(L_k-u_s)\\Biggr)d(\\t_{\\downarrow\\leq (n-1)})\\\\\n\\notag &=\\varpi_{\\s,\\t,w}^{(2,1)} s_{n-1}\\varpi^{(2,2)}_{\\s,\\t,w} + \\rp^{(2)}_{\\s,\\t,w},\n\\end{align}\nwhere\n$$\\begin{aligned}\n\\varpi_{\\s,\\t,w}^{(2,1)}&=\nd(\\s_{\\downarrow\\leq (n-1)})^*\n\\Biggr(\\prod\\limits_{s=\\ell_0+1}^{\\ell}(L_{n-1}-u_s)\\Biggr)\\in H_{\\ell,n-1},\\\\\n\\varpi_{\\s,\\t,w}^{(2,2)}&=\\gamma_{n-2,n(\\s)}w\n\\Biggr(\\prod\\limits_{s=2}^{\\ell}\\prod\\limits_{\\substack{1\\leq k\\leq \\fa_s\\\\k\\neq n}}(L_k-u_s)\\Biggr) d(\\t_{\\downarrow\\leq (n-1)})\\in H_{\\ell,n-1},\\end{aligned}$$ and $$\\begin{aligned}\n\\rp^{(2)}_{\\s,\\t,w}&=\n\\begin{cases}\\begin{matrix}\nd(\\s_{\\downarrow\\leq (n-1)})^* \\bigg(\\sum\\limits_{i=\\ell_0+1}^{\\ell}\n\\bigg( \\prod\\limits_{\\ell_0+1\\leq s<i}(L_{n-1}-u_s)\n\\prod\\limits_{i< t\\leq \\ell}(L_{n}-u_t)\\bigg)\\bigg) \\\\\n\\times \\gamma_{n-2,n(\\s)}w\\bigg(\\prod\\limits_{s=2}^{\\ell}\\prod\\limits_{\\substack{1\\leq k\\leq \\fa_s\\\\k\\neq n}}(L_k-u_s)\\bigg)d(\\t_{\\downarrow\\leq (n-1)}),\\end{matrix} & \\mbox{if\\ \\ } \\ell_0<\\ell,\\\\\n 0, & \\mbox{if\\ \\ } \\ell_0=\\ell.\n\\end{cases}\n\\end{aligned}$$\nAs in Case 1, using (\\ref{easyobservationdeg}) we see that in this case,\n\\begin{equation}\\label{Case2Vendeg1}\n\\e_n\\bigl({d(\\s)}^*w u_\\blam^+ {d(\\t)}\\bigr)=\\rp_{\\ell-1}^{(n)}(\\rp^{(2)}_{\\s,\\t,w})=0.\n\\end{equation}\nOn the other hand, for each $w\\in\\Sym_\\blam$ such that $\\sigma_{n-1} \\leq w,$\nusing (\\ref{decompositionofdt})-(\\ref{decompositionofwlamr}), (\\ref{degbraidrela1}), (\\ref{deg-L-comm1}), Lemma \\ref{keytoolsdeg} (1) and Lemma \\ref{deg-L-comm3} (1)\nand noticing that $b_{\\ell_0,r_{\\ell_0}}\\geq a_{\\ell_0,r_{\\ell_0}-1}+1>n(\\s)$, we have\n\\begin{align*}\nd(\\s)^*w u_\\blam^+ d(\\t)\n& = d(\\s_{\\downarrow\\leq (n-1)})^*\\gamma_{n-1,n(\\s)}\n\\Biggr(\\prod_{i=1}^{\\ell_0-1} w_{i}\\Biggr)\\Biggr(\\prod_{j=1}^{r_{\\ell_0}-1}w_{\\ell_0,j}\\Biggr)\n\\beta_{b_{\\ell_0,r_{\\ell_0},n-1}}w_{\\ell_0,r_{\\ell_0}}^{(1)} \\\\\n\\notag&\\quad\\ \\times \\Biggr(\\prod_{s=2}^{\\ell}\\prod_{k=1}^{\\fa_s}(L_k-Q_s)\\Biggr)d(\\t_{\\downarrow\\leq (n-1)})\\\\\n&= d(\\s_{\\downarrow\\leq (n-1)})^*\\gamma_{n-1,n(\\s)}\\beta_{b_{\\ell_0,r_{\\ell_0},n-1}}\n\\Biggr(\\prod_{i=1}^{\\ell_0-1} w_{i}\\Biggr)\\Biggr(\\prod_{j=1}^{r_{\\ell_0}-1}w_{\\ell_0,j}\\Biggr)w_{\\ell_0,r_{\\ell_0}}^{(1)}\\\\\n\\notag&\\quad\\ \\times\n\\Biggr(\\prod_{s=2}^{\\ell}\\prod_{k=1}^{\\fa_s}(L_k-u_s)\\Biggr)d(\\t_{\\downarrow\\leq (n-1)})\\\\\n&= d(\\s_{\\downarrow\\leq (n-1)})^*\\beta_{b_{\\ell_0,r_{\\ell_0}}-1,n-2}s_{n-1}\\gamma_{n-2,n(\\s)}\n\\Biggr(\\prod_{i=1}^{\\ell_0-1} w_{i}\\Biggr)\\Biggr(\\prod_{j=1}^{r_{\\ell_0}-1}w_{\\ell_0,j}\\Biggr) w_{\\ell_0,r_{\\ell_0}}^{(1)}\\\\\n\\notag&\\quad\\ \\times\n\\Biggr(\\prod_{s=2}^{\\ell}\\prod_{k=1}^{\\fa_s}(L_k-u_s)\\Biggr)d(\\t_{\\downarrow\\leq (n-1)})\\\\\n&= \\varpi_{\\s,\\t,w}^{(3,1)} s_{n-1}\\varpi^{(3,2)}_{\\s,\\t,w} + \\rp^{(3)}_{\\s,\\t,w},\n\\end{align*}\nwhere\n$$\\begin{aligned}\n\\varpi_{\\s,\\t,w}^{(3,1)}&=\nd(\\s_{\\downarrow\\leq (n-1)})^*\\beta_{b_{\\ell_0,r_{\\ell_0}}-1,n-2}\n\\Biggr(\\prod\\limits_{s=l(\\blam,n)+1}^{\\ell}(L_{n-1}-u_s)\\Biggr)\\in H_{\\ell,n-1},\\\\\n\\varpi_{\\s,\\t,w}^{(3,2)}&=\\gamma_{n-2,n(\\s)}\n\\Biggr(\\prod_{i=1}^{\\ell_0-1} w_{i}\\Biggr)\\Biggr(\\prod_{j=1}^{r_{\\ell_0}-1}w_{\\ell_0,j}\\Biggr) w_{\\ell_0,r_{\\ell_0}}^{(1)}\n\\Biggr(\\prod\\limits_{s=2}^{\\ell}\\prod\\limits_{\\substack{1\\leq k\\leq \\fa_s\\\\k\\neq n}}(L_k-u_s)\\Biggr) d(\\t_{\\downarrow\\leq (n-1)})\\in H_{\\ell,n-1},\\end{aligned}$$ and $$\\begin{aligned}\n\\rp^{(3)}_{\\s,\\t,w} =\n\\begin{cases}\\begin{matrix}\nd(\\s_{\\downarrow\\leq (n-1)})^* \\beta_{b_{\\ell_0,r_{\\ell_0}}-1,n-2}\n \\bigg(\\sum\\limits_{i=\\ell_0+1}^{\\ell}\\bigg( \\prod\\limits_{\\ell_0+1\\leq s<i}(L_{n-1}-u_s)\\\\\n\\times \\prod\\limits_{i< t\\leq \\ell}(L_{n}-u_t)\\bigg)\\bigg)\n\\gamma_{n-2,n(\\s)}\\bigg(\\prod\\limits_{i=1}^{\\ell_0-1} w_{i}\\bigg)\\bigg(\\prod\\limits_{j=1}^{r_{\\ell_0}-1}w_{\\ell_0,j}\\bigg) w_{\\ell_0,r_{\\ell_0}}^{(1)}\\\\\n\\times \\bigg(\\prod\\limits_{s=2}^{\\ell}\\prod\\limits_{\\substack{1\\leq k\\leq \\fa_s\\\\k\\neq n}}(L_k-u_s)\\bigg)d(\\t_{\\downarrow\\leq (n-1)}),\\end{matrix} & \\mbox{if\\ \\ }\\ell_0<\\ell,\\\\\n 0, & \\mbox{if\\ \\ } \\ell_0=\\ell.\n\\end{cases}\n\\end{aligned}$$\nThen still using (\\ref{easyobservationdeg}) we see that in this case, \\begin{equation}\\label{Case2Vendeg2}\n\\e_n\\bigr(d(\\s)^*w u_\\blam^+ d(\\t)\\bigr)=\\rp_{\\ell-1}^{(n)}(\\rp^{(3)}_{\\s,\\t,w})=0.\n\\end{equation}\nTherefore, combining (\\ref{Case2Vendeg1}) and (\\ref{Case2Vendeg2}), we get\n$\\e_n(m_{\\s\\t}) =0.$",
    "post_theorem_intro_text_len": 3414,
    "post_theorem_intro_text": "Let $\\mathbf{u}=(u_1,u_2,\\dots,u_\\ell)\\in R^{\\ell}$. The degenerate cyclotomic Hecke algebra $H_{\\ell,n}=H_{\\ell,n}^R(\\mathbf{u})$ of type $G(\\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1} ,\\dots,s_{n-1},L_1, \\dots,L_n$ which satisfy certain relations, see Definition \\ref{degenerate}. In \\cite{AMR}, Ariki, Mathas and Rui constructed a cellular basis $\\{m_{\\mathfrak{s}\\mathfrak{t}}\\,|\\, \\mathfrak{s},\\mathfrak{t}\\in\\Std(\\blam),\\ \\blam\\in\\P_{\\ell,n}\\}$ for $H_{\\ell,n}$. This basis generalizes the well-known Murphy basis for the symmetric group algebra $R[\\Sym_n]$ (i.e., the level one case). We call it {\\bf the Murphy basis for the degenerate cyclotomic Hecke algebra $H_{\\ell,n}$}. Brundan-Kleshchev \\cite{BK08} introduced a symmetrizing form $\\tau_R^{\\te{BK}}$ on $H_{\\ell,n}$ which makes it into a symmetric algebra over $R$. We refer the readers to Section 2.2 for their precise definitions. The following theorem is the second main result of this paper.\n\n\\begin{thm}\\label{mainthm2} Let $\\blam\\in\\P_{\\ell,n}$ and $\\mathfrak{s},\\mathfrak{t}\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{BK}}(m_{\\mathfrak{s}\\mathfrak{t}})=\n\\begin{cases}\n1, & \\mbox{if\\ \\,}\\mathfrak{s}=\\mathfrak{t}\\te{\\ \\,and\\ \\,} l(\\mathfrak{t},i)=1,\\ \\forall\\,1\\leq i\\leq n, \\\\\n0, & \\mbox{\\text{otherwise}}.\n\\end{cases}$$\n\\end{thm}\n\nThe main idea of the proof of Theorems \\ref{mainthm1} and \\ref{mainthm2} comes from \\cite{HLL},\nwhere we identify the standard symmetrizing form on $\\HH_{\\ell,n}$ (resp., on $H_{\\ell,n}$) with the composition of a series of descending map $\\ve_k: \\HH_{\\ell,k}\\rightarrow\\HH_{\\ell,k-1}$  (resp., $\\e_k: H_{\\ell,k}\\rightarrow H_{\\ell,k-1}$),\nwhere each descending map $\\ve_k$ (resp., $\\e_k$) is given by the cyclotomic Mackey decomposition of $\\HH_{\\ell,n}$ (resp., of $H_{\\ell,n}$).\nInstead of computing the value of the standard symmetrizing form via its original definition, we shall show that\neach $\\ve_n(\\fm_{\\mathfrak{s}\\mathfrak{t}})$ (resp., $\\e_n(m_{\\mathfrak{s}\\mathfrak{t}})$) is zero or a scalar multiple of\n$\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$ (resp., of $m_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$),\nand we shall precisely compute this scalar.\n\nThe content of the paper is organised as follows.\nIn Section 2, we recall some preliminary results on the non-degenerate cyclotomic Hecke algebra and the degenerate cyclotomic Hecke algebra of type $G(\\ell,1,n)$.\nIn Section 3, we use the cyclotomic Mackey decomposition of the non-degenerate cyclotomic Hecke algebra of type $G(\\ell,1,n)$\nto show in Proposition \\ref{MainProp1}  that each $\\ve_n(\\fm_{\\mathfrak{s}\\mathfrak{t}})$ is zero or a scalar multiple of $\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$, and thus get a proof of the first main result Theorem \\ref{mainthm1} of this paper.\nThen we apply Proposition \\ref{MainProp1} and Theorem \\ref{mainthm1}\nto derive some equations in Corollaries \\ref{maincor11}, \\ref{maincor12} and \\ref{maincor13}  on the coefficients\nwhich occur in the transition matrix between the Murphy basis elements and the seminormal basis elements. In Section 4, we generalize all the results obtained in Section 3 to the degenerate cyclotomic Hecke algebra of type $G(\\ell,1,n)$.\n\n\\bigskip\\bigskip\n\\centerline{Acknowledgements}\n\\bigskip\n\nThe research was support by the National Natural Science Foundation of China (No. 12431002).\n\\bigskip\n\n\\bigskip",
    "sketch": "The post-theorem introduction says that “The main idea of the proof of Theorems \\ref{mainthm1} and \\ref{mainthm2} comes from \\cite{HLL},” where they “identify the standard symmetrizing form on $\\HH_{\\ell,n}$ … with the composition of a series of descending map $\\ve_k: \\HH_{\\ell,k}\\rightarrow\\HH_{\\ell,k-1}$ … where each descending map $\\ve_k$ … is given by the cyclotomic Mackey decomposition.”\n\n“For Theorem \\ref{mainthm1}” they avoid the original definition of the symmetrizing form and instead “show that each $\\ve_n(\\fm_{\\mathfrak{s}\\mathfrak{t}})$ is zero or a scalar multiple of $\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$, and … precisely compute this scalar.” Concretely, in Section 3 they “use the cyclotomic Mackey decomposition … to show in Proposition \\ref{MainProp1} that each $\\ve_n(\\fm_{\\mathfrak{s}\\mathfrak{t}})$ is zero or a scalar multiple of $\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$, and thus get a proof of … Theorem \\ref{mainthm1}.”",
    "expanded_sketch": "The post-theorem introduction says that “The main idea of the proof of Theorems mainthm1 and \\begin{thm}\\label{mainthm2} Let $\\blam\\in\\P_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{BK}}(m_{\\s\\t})=\n\\begin{cases}\n1, & \\mbox{if\\ \\,}\\s=\\t\\te{\\ \\,and\\ \\,} l(\\t,i)=1,\\ \\forall\\,1\\leq i\\leq n, \\\\\n0, & \\mbox{\\text{otherwise}}.\n\\end{cases}$$\n\\end{thm} comes from \\cite{HLL},” where they “identify the standard symmetrizing form on $\\HH_{\\ell,n}$ … with the composition of a series of descending map $\\ve_k: \\HH_{\\ell,k}\\rightarrow\\HH_{\\ell,k-1}$ … where each descending map $\\ve_k$ … is given by the cyclotomic Mackey decomposition.”\n\n“In establishing the main theorem,” they avoid the original definition of the symmetrizing form and instead “show that each $\\ve_n(\\fm_{\\mathfrak{s}\\mathfrak{t}})$ is zero or a scalar multiple of $\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}}$, and … precisely compute this scalar.” Concretely, they “use the cyclotomic Mackey decomposition … to show the following proposition.\n\n\\begin{prop}\\label{MainProp1}\nLet $\\blam\\in\\mathscr{P}_{\\ell,n}$ and $\\s,\\t\\in\\te{Std}(\\blam).$ Then\n$$\\ve_n(\\fm_{\\s\\t})=\\begin{cases}\nq^{n-n(\\t)}\\bigg(\\prod\\limits_{i=l(\\t,n)+1}^{\\ell}(-Q_i)\\bigg)\\fm_{\\s_{\\downarrow\\leq (n-1)}\\t_{\\downarrow\\leq (n-1)}},\n& \\text{if\\ \\ $n(\\s)=n(\\t),$}\\\\\n0, & \\text{if\\ \\ $n(\\s)\\neq n(\\t)$.}\n\\end{cases} $$\n\\end{prop}\n\nThey then deduce from this proposition a proof of the main theorem.”",
    "expanded_theorem": "\\label{mainthm1} Let $\\blam\\in\\P_{\\ell,n}$ and $\\mathfrak{s},\\mathfrak{t}\\in\\te{Std}(\\blam).$ Then\n$$\\tau_R^{\\te{MM}}(\\fm_{\\mathfrak{s}\\mathfrak{t}})=\n\\begin{cases}\nq^{\\sum\\limits_{k=1}^n(k-k(\\t_{\\downarrow\\leq k}))}\\bigg(\\prod\\limits_{j=1}^n\\prod\\limits_{i=l(\\mathfrak{t},j)+1}^{\\ell}(-Q_i)\\bigg), & \\mbox{if\\ \\ }\\mathfrak{s}=\\mathfrak{t}, \\\\\n0, & \\mbox{if\\ \\ }\\mathfrak{s}\\neq \\mathfrak{t}.\n\\end{cases}$$,",
    "theorem_type": [
      "Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $R$ be an integral domain, let $1\\neq q\\in R^\\times$, let $\\mathbf Q=(Q_1,\\dots,Q_\\ell)\\in R^\\ell$, and let $\\mathscr H_{\\ell,n}=\\mathscr H_{\\ell,n}^R(q;\\mathbf Q)$ be the non-degenerate cyclotomic Hecke algebra of type $G(\\ell,1,n)$. Let $\\tau_R^{\\mathrm{MM}}$ be the $R$-linear map on $\\mathscr H_{\\ell,n}$ determined by $\\tau_R^{\\mathrm{MM}}(\\mathcal L_1^{c_1}\\cdots \\mathcal L_n^{c_n}T_w)=1$ if $w=1$ and $c_1=\\cdots=c_n=0$, and $0$ otherwise, extended linearly. For each $\\lambda\\in\\mathscr P_{\\ell,n}$, let $\\mathrm{Std}(\\lambda)$ be the set of standard $\\lambda$-tableaux, and let $\\{\\fm_{\\mathfrak u\\mathfrak v}\\mid \\mathfrak u,\\mathfrak v\\in\\mathrm{Std}(\\lambda),\\ \\lambda\\in\\mathscr P_{\\ell,n}\\}$ be the Murphy basis of $\\mathscr H_{\\ell,n}$. For a standard tableau $\\mathfrak t$, write $\\mathfrak t^{-1}(j)=(r(\\mathfrak t,j),c(\\mathfrak t,j),l(\\mathfrak t,j))$, so $l(\\mathfrak t,j)$ is the component index containing $j$. For $1\\le k\\le n$, let $\\mathfrak t_{\\downarrow\\le k}$ be the restriction of $\\mathfrak t$ to the entries $1,\\dots,k$, and let $k(\\mathfrak t_{\\downarrow\\le k})$ be the unique integer such that the box containing $k$ in $\\mathfrak t_{\\downarrow\\le k}$ is the same box that contains $k(\\mathfrak t_{\\downarrow\\le k})$ in the initial standard tableau of the same shape. With the usual convention that an empty product equals $1$, which statement holds for every $\\lambda\\in\\mathscr P_{\\ell,n}$ and every $\\mathfrak s,\\mathfrak t\\in\\mathrm{Std}(\\lambda)$?",
      "correct_choice": {
        "label": "A",
        "text": "$\\tau_R^{\\mathrm{MM}}(\\fm_{\\mathfrak s\\mathfrak t})=\\begin{cases}q^{\\sum_{k=1}^n\\bigl(k-k(\\mathfrak t_{\\downarrow\\le k})\\bigr)}\\left(\\prod_{j=1}^n\\prod_{i=l(\\mathfrak t,j)+1}^{\\ell}(-Q_i)\\right),&\\text{if }\\mathfrak s=\\mathfrak t,\\\\0,&\\text{if }\\mathfrak s\\ne\\mathfrak t.\\end{cases}$"
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\tau_R^{\\mathrm{MM}}(\\fm_{\\mathfrak s\\mathfrak t})=\\begin{cases}q^{\\sum_{k=1}^n\\bigl(k-k(\\mathfrak t_{\\downarrow\\le k})\\bigr)}\\left(\\prod_{j=1}^n\\prod_{i=l(\\mathfrak t,j)}^{\\ell}(-Q_i)\\right),&\\text{if }\\mathfrak s=\\mathfrak t,\\\\0,&\\text{if }\\mathfrak s\\ne\\mathfrak t.\\end{cases}$"
        },
        {
          "label": "C",
          "text": "$\\tau_R^{\\mathrm{MM}}(\\fm_{\\mathfrak s\\mathfrak t})=0\\quad\\text{if }\\mathfrak s\\ne\\mathfrak t.$"
        },
        {
          "label": "D",
          "text": "$\\tau_R^{\\mathrm{MM}}(\\fm_{\\mathfrak s\\mathfrak t})=\\begin{cases}q^{\\sum_{k=1}^n\\bigl(k-k(\\mathfrak t_{\\downarrow\\le k})\\bigr)}\\left(\\prod_{j=1}^n\\prod_{i=l(\\mathfrak t,j)+1}^{\\ell}(-Q_i)\\right),&\\text{if }n(\\mathfrak s)=n(\\mathfrak t),\\\\0,&\\text{if }n(\\mathfrak s)\\ne n(\\mathfrak t).\\end{cases}$"
        },
        {
          "label": "E",
          "text": "$\\tau_R^{\\mathrm{MM}}(\\fm_{\\mathfrak s\\mathfrak t})=\\begin{cases}q^{\\sum_{k=1}^n\\bigl(k-k(\\mathfrak s_{\\downarrow\\le k})\\bigr)}\\left(\\prod_{j=1}^n\\prod_{i=l(\\mathfrak s,j)+1}^{\\ell}(-Q_i)\\right),&\\text{if }\\mathfrak s=\\mathfrak t,\\\\0,&\\text{if }\\mathfrak s\\ne\\mathfrak t.\\end{cases}$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "component-product lower bound i=l(\\mathfrak t,j)+1",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped explicit diagonal scalar formula",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "equality criterion \\mathfrak s=\\mathfrak t replaced by n(\\mathfrak s)=n(\\mathfrak t)",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "dependence on terminal tableau switched from \\mathfrak t to \\mathfrak s in the scalar",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem introduces notation and definitions but does not state or strongly hint at the full explicit formula. The correct answer is not leaked directly from the wording."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially asking for the exact explicit formula of a stated result. This is a direct theorem-recall/restatement question rather than a problem requiring a new conclusion from premises."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact formula from close variants, especially the weaker true statement and the subtle indexing changes. However, the task mainly tests precise recall/recognition rather than genuinely generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one changes an index bound, one gives only a weaker true statement, one alters the equality condition, and one swaps dependence from t to s. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recognition item with strong distractors and no answer leakage, but it is mostly theorem recall and thus weak on non-tautological, generative reasoning."
    }
  },
  {
    "id": "2512.10455v1",
    "paper_link": "http://arxiv.org/abs/2512.10455v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "bigthm",
      "content": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.",
      "start_pos": 22363,
      "end_pos": 22882,
      "label": "thm:charac-Markov"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 4061,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n  \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n  \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n  Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n  $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n  the following are equivalent. \n  \\begin{enumerate}\n    \\item $X_0$ is completable by a cycle of rational curves. \n    \\item $\\overline \\kappa (X_0) = 0$. \n    \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n  \\end{enumerate}\n Or\n \\begin{enumerate}\n   \\item $X_0$ is completable by a tree of rational curves. \n   \\item $\\overline \\kappa (X_0) = - \\infty$.\n   \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms. \n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.",
    "context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.",
    "full_context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\C$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\n\\begin{rmq}\\label{rmq:}\n If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n is a curve of degree 3 in $\\P^2$ with a nodal singularity, then $X_0 := \\P^2 \\setminus C$ is a smooth affine surface\n completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n elementary.\n\\end{rmq}\n\nLet $X_0$ be a normal affine surface, we write $K[X_0]$ for its ring of regular functions. A $K$-valuation over\n$K[X_0]$ is a map $v : K[X_0] \\rightarrow \\R \\cup \\left\\{ \\infty \\right\\}$ such that \n\\begin{enumerate}\n \\item $v_{|K^\\times} = 0$. \n \\item $v(0) = \\infty$.\n \\item $\\forall P,Q \\in K[X_0], \\quad v(PQ) = v(P) + v (Q)$. \n \\item $\\forall P,Q \\in K[X_0], \\quad v(P+Q) \\geq \\min \\left( v(P), v(Q) \\right)$.\n\\end{enumerate}\nWe give two examples that will be useful for this note. First we have \\emph{divisorial} valuations. Let $X$ be a\ncompletion of $X_0$ and $E$ a prime divisor at infinity, then $\\ord_E$ the order of vanishing along $E$ is a valuations\nover $K[X_0]$. The second example is as follows. Let $E,F$ be two divisors at infinity in $X$ intersecting at a point\n$p$, then we can find local coordinates $(x,y)$ at $p$ such that $x = 0 $ is a local equation of $E$ and $y = 0$ is a\nlocal equation of $F$. For $\\alpha, \\beta > 0$ we define $v_{\\alpha,\\beta}$ as follows. The completion of the local ring\nat $p$ with respect to its maximal ideal is $K \\left[ [ x,y ] \\right]$ the ring of formal power series in $x,y$. Define \n\\begin{equation}\n v_{\\alpha,\\beta} \\left( \\sum_{i,j} a_{ij} x^i y^j \\right) = \\min \\left( \\alpha i + \\beta j : a_{ij} \\neq 0 \\right).\n \\label{<+label+>}\n\\end{equation}\nThis defines a valuation over $K[X_0]$ because every $P \\in K[X_0]$ is in the fraction field of $\\OO_{X,p}$ the local\nring at $p$ and this fraction field embeds into $K [ [ x ,y ] ]$. We have two possibilities, if $\\alpha / \\beta \\in \\Q$,\nthen $v_{\\alpha,\\beta} = \\ord_G$ is also divisorial but we need to blow up $p$ and infinitely near points to make $G$\nappear. Otherwise $v_{\\alpha,\\beta}$ is \\emph{irrational}.\n\n\\section{Proof of Theorem \\ref{thm:charac-Markov}}\\label{sec:}\nLet $X_0$ be a smooth affine surface completable by a cycle of rational curves and assume that $\\Aut (X_0)$ is not elementary.\n\\begin{prop}\\label{prop:vanishing-self-intersection}\n For any $[v] \\in \\cC_\\infty$, we have $Z_v^2 = 0$.\n\\end{prop}\n\\begin{proof}\n Let $f,g $ be two loxodromic automorphisms generating a non-elementary subgroup. Write $v_f^\\pm, v_g^\\pm$ for the\n eigenvaluations of $f$ and $g$ respectively. Recall that they are all different by Proposition\n \\ref{prop:different-eigenvaluations}. From Proposition \\ref{prop:eigenvaluations-divisor}, we have that \n \\begin{equation}\n Z_{v^\\pm_f}^2 = Z_{v^\\pm_g}^2 = 0.\n \\label{<+label+>}\n \\end{equation}\n Let $Y$ be a cyclic completion of $X_0$ such that the centers of these four eigenvaluations are distinct. Recall that\n they are irrational valuations so their centers is always a satellite point. Let\n $p = E \\cap F = c_Y (v_f^+)$. We look at the monomial valuations centered at $p$. The function \n \\begin{equation}\n \\phi: s \\in [0, +\\infty) \\mapsto Z_{v_{1,s}}^2 \n \\label{<+label+>}\n \\end{equation}\n is a polynomial map of $s$ of degree $\\leq 2$ by Lemma \\ref{lemme:self-intersection-irrational}. We have that $v_f^+ = v_{1, s_0}$ for some $s_0 > 0$. Now, we know\n that $f^n_* [v_g^+] \\rightarrow [v_f^+]$ by Theorem \\ref{thm:dynamics-loxodromic-automorphisms}. This implies that after a finite number of steps we have up to\n renormalisation\n \\begin{equation}\n f^n_* v_g^+ = v_{1, s_n}\n \\label{<+label+>}\n \\end{equation}\nfor some $s_n > 0$ with $s_n \\neq s_0$ and $s_n \\rightarrow s_0$. But since $Z_{v_g^+}^2 = 0$ and this is invariant by the action of $f$\nwe have that $Z_{v_{1,s_n}}^2 = 0$ so the function $\\phi$ is zero. Since for any valuation $[v] \\in \\cC_\\infty \\setminus\n\\left\\{ [v_f^-] \\right\\}, f^n_* [v] \\rightarrow [v_f^+]$ we have that $Z_v^2 = 0$.\n\\end{proof}\nThis implies by Proposition \\ref{prop:nef-valuation} that for every $v \\in \\cC_\\infty, Z_v$ is nef.\n\\begin{cor}\\label{cor:intersection-positive}\n If $E,F$ are different prime divisors at infinity of a cyclic completion $X$ of $X_0$, then \n \\begin{equation}\n \\hat E \\cdot \\hat F > 0.\n \\label{<+label+>}\n \\end{equation}\n\\end{cor}\n\\begin{proof}\n We have by Proposition \\ref{prop:vanishing-self-intersection} that $(\\hat E)^2 = (\\hat F)^2 = 0$ so that they are nef\n and effective divisors by Proposition \\ref{prop:nef-valuation}. This implies that $\\hat E \\cdot \\hat F \\geq 0$.\n If the intersection number is zero, then by the Hodge index theorem we must have that $\\hat E = \\hat F$ but this is a\n contradiction.\n\\end{proof}\n\n\\begin{prop}\\label{prop:trivial-k-plus-delta}\n Let $X_0$ be a smooth affine surface completable by a cycle of rational curves with a non-elementary automorphism\n group, then the class $K+\\Delta \\in \\cNS_{cyc} (X_0)$ is equal to $0$.\n\\end{prop}\n\\begin{proof}\nRecall by Proposition \\ref{prop:Cartier-canonical-class} that $(K+ \\Delta) \\in \\Cartier_{cyc}(X_0)_\\R$ and it is defined\n by $K_X + \\Delta_X$ for any cyclic completion $X$ of $X_0$. Furthermore, it is fixed by $\\Aut (X_0)$. We show that\n \\begin{equation}\n \\forall v \\in \\cC_\\infty, \\quad Z_v \\cdot (K+\\Delta) = 0.\n \\end{equation}\n Let $f,g \\in \\Aut (X_0)$ be two loxodromic automorphisms not sharing a common iterate with their eigenvaluations\n $v_h^\\pm$ for $h = f,g$. We have that \n \\begin{equation}\n Z_{v_h^\\pm} \\cdot (K+\\Delta) = (h^{\\pm 1})^* \\left( Z_{v_h^\\pm} \\cdot (K+\\Delta) \\right) = \\lambda \\left(h^{\\pm 1}\\right) Z_{v_h^\\pm}\n \\cdot (K+\\Delta).\n \\label{<+label+>}\n \\end{equation}\n This implies that $Z_{v_h^\\pm} \\cdot (K+\\Delta) = 0$. Now, let $X$ be a cyclic completion of $X_0$ and let $p = E \\cap F = c_X (v_f^+)$. The function \n \\begin{equation}\n \\phi: s > 0 \\mapsto Z_{v_{1,s}} \\cdot (K+\\Delta)\n \\label{<+label+>}\n \\end{equation}\n is a polynomial of degree at most 1 because we have \n \\begin{equation}\n Z_{v_{1,s}} \\cdot (K+\\Delta) = (Z_{v_{1,s}})_X \\cdot (K+\\Delta)_X = (\\hat E + s \\hat F) \\cdot (K_X + \\Delta_X).\n \\label{<+label+>}\n \\end{equation}\n Now we have that $v_f^+ = v_{1,s_0}$ for some $s_0 > 0$. As in the proof of Proposition\n \\ref{prop:vanishing-self-intersection}, for $n$ large enough, $f^n_* v_g^+ = v_{1,s_n}$ and we have $\\phi(s_n) = 0$.\n So that the function $\\phi$ is 0. Now, for any $[v] \\in \\cC_\\infty \\setminus \\left\\{ [v_{-,f}] \\right\\}$ we have that\n $f^n_* [v] \\rightarrow [v_{+, f}]$ so it will belong to $[v_E, v_F[$ after finitely many iterations. So that we get\n $Z_v \\cdot (K+\\Delta) = (f^n)_* (Z_v \\cdot (K+\\Delta)) = Z_{f^n_* v} \\cdot (K+\\Delta) = 0 $.",
    "post_theorem_intro_text_len": 3141,
    "post_theorem_intro_text": "\\begin{rmq}\\label{rmq:}\n  If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n  is a curve of degree 3 in $\\mathbf{P}^2$ with a nodal singularity, then $X_0 := \\mathbf{P}^2 \\setminus C$ is a smooth affine surface\n  completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n  $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n  $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n  elementary.\n\\end{rmq}\n\n\\subsection{Idea of the proof}\\label{subsec:}\nThe proof goes as follows. We use the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}. In the\ncase of an affine surface completable by a cycle of rational curves, the space of valuations centered at infinity of\n$X_0$ contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle. This circle is homeomorphic to the\ncompletion of the inverse limits of the dual graphs of cyclic completions. If $\\Aut(X_0)$ is non-elementary, then its\naction on this circle has very large orbits. This imposes strong constraints on the intersection form on the space of\ndivisors supported at infinity. We then conclude that $X_0$ has to be the complement of a triangle of lines in a smooth\ncubic surface unless $X_0 = \\G_m^2$.\n\n\\subsection{Study of endomorphisms of affine surfaces of Markov type}\\label{subsec:endomorphisms-intro}\nWe fully classify the dynamics of cubic affine surfaces of Markov type in characteristic zero by showing that they do not have\nnon-invertible dominant endomorphisms. \n\n\\begin{bigthm}\\label{thm:endomorphisms-affine-surface}\n  Let $K$ be a field of characteristic zero. If $X_0$ is a smooth cubic affine surface of Markov type over $K$ and $f$ is a\n  dominant endomorphism of $X_0$, then $f$ is an automorphism.\n\\end{bigthm}\nTo show this result we can assume that $K = \\mathbf{C}$, then using valuative techniques and the geometry of the space of\nvaluations centered at infinity we show that $f$ must be proper and unramified. Thus, it is a covering of the complex\nmanifold $X_0 (\\mathbf{C})$ which is simply connected (see eg \\cite[Lemma 3.10]{cantatHolomorphicDynamicsPainleve2007}). Hence\n$f$ must be a homeomorphism and therefore an automorphism.\n\nNotice that this is not true for singular cubic affine surfaces. Indeed, let $\\sC \\subset \\mathbf{A}^3$ be the Cayley cubic defined by \n\\begin{equation}\n  x^2 + y^2 + z^2 = xyz + 4.\n  \\label{<+label+>}\n\\end{equation}\nIt is the quotient of $\\G_m^2$ by the involution $(u,v) \\mapsto (u^{-1}, v^{-1})$. The Cayley cubic has four singular\npoints which are orbifold singularities of order 2. It has many endomorphisms, namely every monomial endomorphism of\n$\\G_m^2$ descends to an endomorphism of $\\sC$. The proof fails for this surface because it is not simply connected as an\norbifold. Indeed, its orbifold fundamental group is not trivial.\n\n\\subsection*{Acknowledgements} I thank Serge Cantat and Matteo Ruggiero for related discussions on this problem.",
    "sketch": "The proof of Theorem~\\ref{thm:charac-Markov} is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"",
    "expanded_sketch": "The proof of the main theorem is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"",
    "expanded_theorem": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.",
    "theorem_type": [
      "Classification or Bijection",
      "Implication"
    ],
    "mcq": {
      "question": "Let $K$ be an algebraically closed field, and let $X_0$ be a smooth affine surface over $K$. Assume that $\\operatorname{Aut}(X_0)$ is non-elementary, meaning that it contains two loxodromic automorphisms with no common iterates, where an automorphism $f$ is called loxodromic if its dynamical degree satisfies $\\lambda(f)>1$, with\n\\[\n\\lambda(f)=\\lim_{n\\to\\infty}\\bigl(((f^n)^*H)\\cdot H\\bigr)^{1/n}\n\\]\nfor an ample divisor $H$ on any smooth projective completion of $X_0$. Also assume that $X_0$ is completable by a cycle of rational curves, i.e. there exists a smooth projective surface $X$ containing $X_0$ as a Zariski dense open subset such that the boundary $X\\setminus X_0$ is a cycle of rational curves. A cubic affine surface of Markov type means the complement $S\\setminus \\Delta$, where $S\\subset \\mathbf P^3$ is a smooth cubic surface and $\\Delta$ is a triangle of lines on $S$. Under these hypotheses, which conclusion about $X_0$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is a cubic affine surface of Markov type (that is, $X_0\\cong S\\setminus \\Delta$ for a smooth cubic surface $S\\subset \\mathbf P^3$ and a triangle of lines $\\Delta\\subset S$). The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
      },
      "choices": [
        {
          "label": "B",
          "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is the complement of a cycle of rational curves in some smooth del Pezzo surface. In particular, every smooth affine surface with non-elementary automorphism group that is completable by a cycle of rational curves arises from a del Pezzo surface, and the distinction between the two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        },
        {
          "label": "C",
          "text": "Under these hypotheses, $X_0$ is either isomorphic to $\\mathbf G_m^2$ or is a cubic affine surface of Markov type."
        },
        {
          "label": "D",
          "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is the complement $S\\setminus \\Delta$ of a triangle of lines $\\Delta$ in a cubic surface $S\\subset \\mathbf P^3$; moreover, the cubic surface $S$ need not be smooth. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        },
        {
          "label": "E",
          "text": "There are exactly two mutually exclusive possibilities: either, after replacing $X_0$ by a finite étale cover, one has $X_0\\cong \\mathbf G_m^2$, or, after such a cover, $X_0$ becomes a cubic affine surface of Markov type. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "precise geometric classification as smooth cubic with triangle of lines",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "drops mutual exclusivity and the criterion via invertible regular functions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "smoothness of the cubic surface",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "classification of X_0 itself versus only after finite étale cover",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the classification, without giving away the cubic/torus dichotomy or the criterion via invertible functions."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the hypotheses in the stem closely match the classification theorem, and the correct choice is basically the theorem statement itself."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact theorem from nearby variants (weaker true statement, weakened hypothesis, overgeneralization), but the main task is recognition/recollection rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful. They reflect common failure modes: overgeneralizing to arbitrary del Pezzo surfaces, selecting a weaker true dichotomy, weakening the hypothesis from non-elementary to one loxodromic automorphism, or confusing the geometric model."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is quite tautological and tests recall more than generative mathematical reasoning."
    }
  },
  {
    "id": "2512.10455v1",
    "paper_link": "http://arxiv.org/abs/2512.10455v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "bigthm",
      "content": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.",
      "start_pos": 22363,
      "end_pos": 22882,
      "label": "thm:charac-Markov"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 4061,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n  \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n  \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n  Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n  $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n  the following are equivalent. \n  \\begin{enumerate}\n    \\item $X_0$ is completable by a cycle of rational curves. \n    \\item $\\overline \\kappa (X_0) = 0$. \n    \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n  \\end{enumerate}\n Or\n \\begin{enumerate}\n   \\item $X_0$ is completable by a tree of rational curves. \n   \\item $\\overline \\kappa (X_0) = - \\infty$.\n   \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms. \n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.",
    "context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.",
    "full_context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\C$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\n\\begin{rmq}\\label{rmq:}\n If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n is a curve of degree 3 in $\\P^2$ with a nodal singularity, then $X_0 := \\P^2 \\setminus C$ is a smooth affine surface\n completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n elementary.\n\\end{rmq}\n\nLet $X_0$ be a normal affine surface, we write $K[X_0]$ for its ring of regular functions. A $K$-valuation over\n$K[X_0]$ is a map $v : K[X_0] \\rightarrow \\R \\cup \\left\\{ \\infty \\right\\}$ such that \n\\begin{enumerate}\n \\item $v_{|K^\\times} = 0$. \n \\item $v(0) = \\infty$.\n \\item $\\forall P,Q \\in K[X_0], \\quad v(PQ) = v(P) + v (Q)$. \n \\item $\\forall P,Q \\in K[X_0], \\quad v(P+Q) \\geq \\min \\left( v(P), v(Q) \\right)$.\n\\end{enumerate}\nWe give two examples that will be useful for this note. First we have \\emph{divisorial} valuations. Let $X$ be a\ncompletion of $X_0$ and $E$ a prime divisor at infinity, then $\\ord_E$ the order of vanishing along $E$ is a valuations\nover $K[X_0]$. The second example is as follows. Let $E,F$ be two divisors at infinity in $X$ intersecting at a point\n$p$, then we can find local coordinates $(x,y)$ at $p$ such that $x = 0 $ is a local equation of $E$ and $y = 0$ is a\nlocal equation of $F$. For $\\alpha, \\beta > 0$ we define $v_{\\alpha,\\beta}$ as follows. The completion of the local ring\nat $p$ with respect to its maximal ideal is $K \\left[ [ x,y ] \\right]$ the ring of formal power series in $x,y$. Define \n\\begin{equation}\n v_{\\alpha,\\beta} \\left( \\sum_{i,j} a_{ij} x^i y^j \\right) = \\min \\left( \\alpha i + \\beta j : a_{ij} \\neq 0 \\right).\n \\label{<+label+>}\n\\end{equation}\nThis defines a valuation over $K[X_0]$ because every $P \\in K[X_0]$ is in the fraction field of $\\OO_{X,p}$ the local\nring at $p$ and this fraction field embeds into $K [ [ x ,y ] ]$. We have two possibilities, if $\\alpha / \\beta \\in \\Q$,\nthen $v_{\\alpha,\\beta} = \\ord_G$ is also divisorial but we need to blow up $p$ and infinitely near points to make $G$\nappear. Otherwise $v_{\\alpha,\\beta}$ is \\emph{irrational}.\n\n\\section{Proof of Theorem \\ref{thm:charac-Markov}}\\label{sec:}\nLet $X_0$ be a smooth affine surface completable by a cycle of rational curves and assume that $\\Aut (X_0)$ is not elementary.\n\\begin{prop}\\label{prop:vanishing-self-intersection}\n For any $[v] \\in \\cC_\\infty$, we have $Z_v^2 = 0$.\n\\end{prop}\n\\begin{proof}\n Let $f,g $ be two loxodromic automorphisms generating a non-elementary subgroup. Write $v_f^\\pm, v_g^\\pm$ for the\n eigenvaluations of $f$ and $g$ respectively. Recall that they are all different by Proposition\n \\ref{prop:different-eigenvaluations}. From Proposition \\ref{prop:eigenvaluations-divisor}, we have that \n \\begin{equation}\n Z_{v^\\pm_f}^2 = Z_{v^\\pm_g}^2 = 0.\n \\label{<+label+>}\n \\end{equation}\n Let $Y$ be a cyclic completion of $X_0$ such that the centers of these four eigenvaluations are distinct. Recall that\n they are irrational valuations so their centers is always a satellite point. Let\n $p = E \\cap F = c_Y (v_f^+)$. We look at the monomial valuations centered at $p$. The function \n \\begin{equation}\n \\phi: s \\in [0, +\\infty) \\mapsto Z_{v_{1,s}}^2 \n \\label{<+label+>}\n \\end{equation}\n is a polynomial map of $s$ of degree $\\leq 2$ by Lemma \\ref{lemme:self-intersection-irrational}. We have that $v_f^+ = v_{1, s_0}$ for some $s_0 > 0$. Now, we know\n that $f^n_* [v_g^+] \\rightarrow [v_f^+]$ by Theorem \\ref{thm:dynamics-loxodromic-automorphisms}. This implies that after a finite number of steps we have up to\n renormalisation\n \\begin{equation}\n f^n_* v_g^+ = v_{1, s_n}\n \\label{<+label+>}\n \\end{equation}\nfor some $s_n > 0$ with $s_n \\neq s_0$ and $s_n \\rightarrow s_0$. But since $Z_{v_g^+}^2 = 0$ and this is invariant by the action of $f$\nwe have that $Z_{v_{1,s_n}}^2 = 0$ so the function $\\phi$ is zero. Since for any valuation $[v] \\in \\cC_\\infty \\setminus\n\\left\\{ [v_f^-] \\right\\}, f^n_* [v] \\rightarrow [v_f^+]$ we have that $Z_v^2 = 0$.\n\\end{proof}\nThis implies by Proposition \\ref{prop:nef-valuation} that for every $v \\in \\cC_\\infty, Z_v$ is nef.\n\\begin{cor}\\label{cor:intersection-positive}\n If $E,F$ are different prime divisors at infinity of a cyclic completion $X$ of $X_0$, then \n \\begin{equation}\n \\hat E \\cdot \\hat F > 0.\n \\label{<+label+>}\n \\end{equation}\n\\end{cor}\n\\begin{proof}\n We have by Proposition \\ref{prop:vanishing-self-intersection} that $(\\hat E)^2 = (\\hat F)^2 = 0$ so that they are nef\n and effective divisors by Proposition \\ref{prop:nef-valuation}. This implies that $\\hat E \\cdot \\hat F \\geq 0$.\n If the intersection number is zero, then by the Hodge index theorem we must have that $\\hat E = \\hat F$ but this is a\n contradiction.\n\\end{proof}\n\n\\begin{prop}\\label{prop:trivial-k-plus-delta}\n Let $X_0$ be a smooth affine surface completable by a cycle of rational curves with a non-elementary automorphism\n group, then the class $K+\\Delta \\in \\cNS_{cyc} (X_0)$ is equal to $0$.\n\\end{prop}\n\\begin{proof}\nRecall by Proposition \\ref{prop:Cartier-canonical-class} that $(K+ \\Delta) \\in \\Cartier_{cyc}(X_0)_\\R$ and it is defined\n by $K_X + \\Delta_X$ for any cyclic completion $X$ of $X_0$. Furthermore, it is fixed by $\\Aut (X_0)$. We show that\n \\begin{equation}\n \\forall v \\in \\cC_\\infty, \\quad Z_v \\cdot (K+\\Delta) = 0.\n \\end{equation}\n Let $f,g \\in \\Aut (X_0)$ be two loxodromic automorphisms not sharing a common iterate with their eigenvaluations\n $v_h^\\pm$ for $h = f,g$. We have that \n \\begin{equation}\n Z_{v_h^\\pm} \\cdot (K+\\Delta) = (h^{\\pm 1})^* \\left( Z_{v_h^\\pm} \\cdot (K+\\Delta) \\right) = \\lambda \\left(h^{\\pm 1}\\right) Z_{v_h^\\pm}\n \\cdot (K+\\Delta).\n \\label{<+label+>}\n \\end{equation}\n This implies that $Z_{v_h^\\pm} \\cdot (K+\\Delta) = 0$. Now, let $X$ be a cyclic completion of $X_0$ and let $p = E \\cap F = c_X (v_f^+)$. The function \n \\begin{equation}\n \\phi: s > 0 \\mapsto Z_{v_{1,s}} \\cdot (K+\\Delta)\n \\label{<+label+>}\n \\end{equation}\n is a polynomial of degree at most 1 because we have \n \\begin{equation}\n Z_{v_{1,s}} \\cdot (K+\\Delta) = (Z_{v_{1,s}})_X \\cdot (K+\\Delta)_X = (\\hat E + s \\hat F) \\cdot (K_X + \\Delta_X).\n \\label{<+label+>}\n \\end{equation}\n Now we have that $v_f^+ = v_{1,s_0}$ for some $s_0 > 0$. As in the proof of Proposition\n \\ref{prop:vanishing-self-intersection}, for $n$ large enough, $f^n_* v_g^+ = v_{1,s_n}$ and we have $\\phi(s_n) = 0$.\n So that the function $\\phi$ is 0. Now, for any $[v] \\in \\cC_\\infty \\setminus \\left\\{ [v_{-,f}] \\right\\}$ we have that\n $f^n_* [v] \\rightarrow [v_{+, f}]$ so it will belong to $[v_E, v_F[$ after finitely many iterations. So that we get\n $Z_v \\cdot (K+\\Delta) = (f^n)_* (Z_v \\cdot (K+\\Delta)) = Z_{f^n_* v} \\cdot (K+\\Delta) = 0 $.",
    "post_theorem_intro_text_len": 3141,
    "post_theorem_intro_text": "\\begin{rmq}\\label{rmq:}\n  If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n  is a curve of degree 3 in $\\mathbf{P}^2$ with a nodal singularity, then $X_0 := \\mathbf{P}^2 \\setminus C$ is a smooth affine surface\n  completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n  $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n  $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n  elementary.\n\\end{rmq}\n\n\\subsection{Idea of the proof}\\label{subsec:}\nThe proof goes as follows. We use the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}. In the\ncase of an affine surface completable by a cycle of rational curves, the space of valuations centered at infinity of\n$X_0$ contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle. This circle is homeomorphic to the\ncompletion of the inverse limits of the dual graphs of cyclic completions. If $\\Aut(X_0)$ is non-elementary, then its\naction on this circle has very large orbits. This imposes strong constraints on the intersection form on the space of\ndivisors supported at infinity. We then conclude that $X_0$ has to be the complement of a triangle of lines in a smooth\ncubic surface unless $X_0 = \\G_m^2$.\n\n\\subsection{Study of endomorphisms of affine surfaces of Markov type}\\label{subsec:endomorphisms-intro}\nWe fully classify the dynamics of cubic affine surfaces of Markov type in characteristic zero by showing that they do not have\nnon-invertible dominant endomorphisms. \n\n\\begin{bigthm}\\label{thm:endomorphisms-affine-surface}\n  Let $K$ be a field of characteristic zero. If $X_0$ is a smooth cubic affine surface of Markov type over $K$ and $f$ is a\n  dominant endomorphism of $X_0$, then $f$ is an automorphism.\n\\end{bigthm}\nTo show this result we can assume that $K = \\mathbf{C}$, then using valuative techniques and the geometry of the space of\nvaluations centered at infinity we show that $f$ must be proper and unramified. Thus, it is a covering of the complex\nmanifold $X_0 (\\mathbf{C})$ which is simply connected (see eg \\cite[Lemma 3.10]{cantatHolomorphicDynamicsPainleve2007}). Hence\n$f$ must be a homeomorphism and therefore an automorphism.\n\nNotice that this is not true for singular cubic affine surfaces. Indeed, let $\\sC \\subset \\mathbf{A}^3$ be the Cayley cubic defined by \n\\begin{equation}\n  x^2 + y^2 + z^2 = xyz + 4.\n  \\label{<+label+>}\n\\end{equation}\nIt is the quotient of $\\G_m^2$ by the involution $(u,v) \\mapsto (u^{-1}, v^{-1})$. The Cayley cubic has four singular\npoints which are orbifold singularities of order 2. It has many endomorphisms, namely every monomial endomorphism of\n$\\G_m^2$ descends to an endomorphism of $\\sC$. The proof fails for this surface because it is not simply connected as an\norbifold. Indeed, its orbifold fundamental group is not trivial.\n\n\\subsection*{Acknowledgements} I thank Serge Cantat and Matteo Ruggiero for related discussions on this problem.",
    "sketch": "The proof of Theorem~\\ref{thm:charac-Markov} is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"",
    "expanded_sketch": "The proof of the main theorem is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"",
    "expanded_theorem": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.",
    "theorem_type": [
      "Classification or Bijection",
      "Implication"
    ],
    "mcq": {
      "question": "Let $K$ be an algebraically closed field, and let $X_0$ be a smooth affine surface over $K$. Assume that $\\operatorname{Aut}(X_0)$ is non-elementary, meaning that it contains two loxodromic automorphisms with no common iterates, where an automorphism $f$ is called loxodromic if its dynamical degree satisfies $\\lambda(f)>1$, with\n\\[\n\\lambda(f)=\\lim_{n\\to\\infty}\\bigl(((f^n)^*H)\\cdot H\\bigr)^{1/n}\n\\]\nfor an ample divisor $H$ on any smooth projective completion of $X_0$. Also assume that $X_0$ is completable by a cycle of rational curves, i.e. there exists a smooth projective surface $X$ containing $X_0$ as a Zariski dense open subset such that the boundary $X\\setminus X_0$ is a cycle of rational curves. A cubic affine surface of Markov type means the complement $S\\setminus \\Delta$, where $S\\subset \\mathbf P^3$ is a smooth cubic surface and $\\Delta$ is a triangle of lines on $S$. Under these hypotheses, which conclusion about $X_0$ holds?",
      "correct_choice": {
        "label": "A",
        "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is a cubic affine surface of Markov type (that is, $X_0\\cong S\\setminus \\Delta$ for a smooth cubic surface $S\\subset \\mathbf P^3$ and a triangle of lines $\\Delta\\subset S$). The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
      },
      "choices": [
        {
          "label": "B",
          "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is the complement of a cycle of rational curves in some smooth del Pezzo surface. In particular, every smooth affine surface with non-elementary automorphism group that is completable by a cycle of rational curves arises from a del Pezzo surface, and the distinction between the two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        },
        {
          "label": "C",
          "text": "Under these hypotheses, $X_0$ is either isomorphic to $\\mathbf G_m^2$ or is a cubic affine surface of Markov type."
        },
        {
          "label": "D",
          "text": "There are exactly two mutually exclusive possibilities: either $X_0\\cong \\mathbf G_m^2$, or $X_0$ is the complement $S\\setminus \\Delta$ of a triangle of lines $\\Delta$ in a cubic surface $S\\subset \\mathbf P^3$; moreover, the cubic surface $S$ need not be smooth. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        },
        {
          "label": "E",
          "text": "There are exactly two mutually exclusive possibilities: either, after replacing $X_0$ by a finite étale cover, one has $X_0\\cong \\mathbf G_m^2$, or, after such a cover, $X_0$ becomes a cubic affine surface of Markov type. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "precise geometric classification as smooth cubic with triangle of lines",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "drops mutual exclusivity and the criterion via invertible regular functions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "smoothness of the cubic surface",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "classification of X_0 itself versus only after finite étale cover",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and definitions but does not state or strongly hint at the exact classification conclusion. The correct bifurcation and the criterion via invertible regular functions are not leaked."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall/classification question: the stem lists the assumptions and asks for the conclusion. However, it is not a pure tautology because the options include weaker, overgeneralized, and subtly altered variants that must be distinguished."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to identify the strongest correct conclusion and reject nearby variants such as a weaker true statement, a smoothness error, and a finite-étale-cover modification. Still, the item mainly tests precise recall of the theorem rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: overgeneralization to del Pezzo surfaces, dropping smoothness, weakening the conclusion, and changing the quantifier to 'after finite étale cover.' They are distinct and nontrivial."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-classification MCQ with strong distractors and little answer leakage, but it mainly tests accurate recall of a specific result rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.10491v1",
    "paper_link": "http://arxiv.org/abs/2512.10491v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\partial W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map.",
      "start_pos": 16222,
      "end_pos": 16859,
      "label": "thm: A"
    },
    "ref_dict": {
      "ex: neglinebdl2": "\\begin{example}\\label{ex: neglinebdl2}\nFollowing Example \\ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\\pi: D(E) \\rightarrow B$ now serves as a complement-exact subdomain. Assume that\n\\begin{itemize} \n\\item $[\\ow] = c_1(B)$ and $[\\ow]$ is primitive in $\\Ho_2(B; \\Z);$\n\\item $B$ is simply-connected.\n\\end{itemize}\nThen $\\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \\ref{thm: A}, where $S(E)$ is the corresponding circle bundle. \n\nFor a more concrete example, one takes the blow-up $B : = \\CP^2 \\# k \\overline{\\CP}^2$ of $\\CP^2$ at $k$ generic points, with $0 < k < 9$. This admits a symplectic form $\\ow_k$ whose Poincar\\'e dual is given by $3H - E_1 - \\cdots -E_k$ where $H$ is the class of a line in $\\CP^2$ and $E_i$ is the exceptional curve derived from $i$-th blow-up. In particular $[\\ow_k] = c_1(B)$ is primitive in $H_2(B;\\Z)$ and $B$ is simply-connected.  \n\\end{example}",
      "lem: actionorbits": "\\begin{lemma}\\label{lem: actionorbits}\nFor 1-orbits $x$ in the region \\emph{(I)} and \\emph{(II)}, we have $\\mathcal{F}(x) \\leq 0$. In the other regions, we have $\\mathcal{F}(x) > 0$.\n\\end{lemma}",
      "cor: algebraisom": "\\begin{corollary}\\label{cor: algebraisom}\n    Under the assumptions of Theorem \\ref{thm: nonnegandW}, we have a canonical $\\Lda$-algebra isomorphism\n    \\[\n    \\SH^*_{ \\leq 0}(V) \\cong \\SH^*(W).\n    \\]\n\\end{corollary}",
      "thm: A": "\\begin{theorem}\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\p W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map. \n\\end{theorem}",
      "ex: resolution": "\\begin{example}\\label{ex: resolution}\n    Let $X$ be a $\\Q$-factorial variety with a unique singularity at the origin $O \\in X$ and assume that the complement $X \\setminus \\{O\\}$ admits an exact K\\\"ahler form $\\ow_X$. Then by \\cite[Lemma 3.2]{McRi23}, any resolution $\\pi: Y \\rightarrow X$ admits a K\\\"ahler form $\\ow_Y$ such that $\\pi^* \\ow_X = \\ow_Y$. In particular, $Y$ is a convex symplectic domain, and a neighborhood of $\\pi^{-1}(\\{O\\}) \\subset Y$ serves as a complement-exact subdomain.\n\\end{example}",
      "rem: Weinattach": "\\begin{remark}\\label{rem: Weinattach}\n    Given a convex symplectic domain $W$ with $\\pi_1(\\p W) = 0$ and $c_1(W)|_{\\pi_2(W)} = 0$, attach the Weinstein handle $\\mathcal{H}$ to $W$ along the boundary $\\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \\ref{thm: A}. As an analogue of the seminal result of Cieliebak \\cite{Ci02}, it is likely that the transfer map $\\Phi: \\SH^*(V) \\rightarrow \\SH^*(W)$ gives rise to an algebra isomorphism when the handle $\\mathcal{H}$ is subcritical. \n\\end{remark}",
      "thm: nonnegandW": "\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}",
      "fig: tranfer_Ham": "\\begin{overpic}[width=0.5\\linewidth]{transf_Ham.pdf}\n    \\put(85, 3){$r_W$}\n    \\put(-3, 77){$B$}\n    \\put(24, 3){$1$}\n    \\put(48, 3){$A$}\n    \\put(30, 50){$\\kappa$} \n    \\put(210, 3){$r_V$}\n    \\put(145, 3){$A+1+P$}\n    \\put(170, 100){$\\frac{1}{2}\\kappa$}\n    \\end{overpic}\n    \\caption{Transfer-admissible Hamiltonians}\n    \\label{fig: tranfer_Ham}\n    \\vspace{12pt}\n    \\centering\n   \\begin{overpic}[width=0.5\\linewidth]{cutoff2.pdf}\n    \\put(90, 3){$r_W$}\n    \\put(24, 3){$1$}\n    \\put(48, 3){$A$}\n    \\put(43, 62){$1$}    \n    \\put(215, 3){$r_V$}\n    \\put(145, 3){$A+1+P$}\n    \\put(162, 104){$1$}\n   \\end{overpic}",
      "rem: usualactionfctl": "\\begin{remark}\\label{rem: usualactionfctl}\nFor an admissible Hamiltonian $H$, a standard action functional $\\mathcal{A}_H : \\widetilde{\\mathcal{L}_0 V} \\rightarrow \\R$ is given by\n$$\n\\mathcal{A}_H(x, v) =  -\\int_{D^2} v^*\\ow + \\int_{S^1} H(x(t)) dt.\n$$\nThe class $(x, v)$ is a critical point of $\\mathcal{A}_H$ if $x$ is a $1$-periodic Hamiltonian orbit of $H$. Here, we use the convention $\\ow(\\cdot, X_H)  = d H$ for the definition of the Hamiltonian vector field $X_H$.\n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 2181,
    "pre_theorem_intro_text": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}. \n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.",
    "context": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}.\n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}",
    "full_context": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}.\n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\p W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\nThe $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \\cite[Section 4.1]{BeRi20} and \\cite[Section 5]{Rit}.\n\n\\begin{example}\\label{ex: neglinebdl2}\nFollowing Example \\ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\\pi: D(E) \\rightarrow B$ now serves as a complement-exact subdomain. Assume that\n\\begin{itemize} \n\\item $[\\ow] = c_1(B)$ and $[\\ow]$ is primitive in $\\Ho_2(B; \\Z);$\n\\item $B$ is simply-connected.\n\\end{itemize}\nThen $\\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \\ref{thm: A}, where $S(E)$ is the corresponding circle bundle.\n\n\\subsubsection{Admissible Hamiltonians} \\label{sec: admHam} Let $(V, \\ow, \\lda)$ be a convex symplectic domain. We assume that the first Chern class $c_1(V)$ vanishes on the second homotopy group $\\pi_2(V)$ \nA Hamiltonian $H: \\widehat V \\rightarrow \\R$ is called \\emph{admissible} if \n\\begin{itemize}\n\\item there exists $r_0 \\in [1, \\infty)$ such that $H$ depends only on the radial coordinate $r$ for $r \\geq r_0$, say $H(z) = h(r)$, and $h'(r) \\geq 0$;\n\\item $h(r) = \\kappa r + b$ for sufficiently large $r$ where the \\emph{slope} $\\kappa$ is not the period of the Reeb orbit on the contact boundary $(\\p V, \\alpha)$.\n\\end{itemize}\n\n\\subsubsection{Ring structure} The symplectic cohomology $\\SH^*(V)$ admits a ring structure by the pair-of-pants product. \nLet $\\mathcal{S}$ be the Riemann sphere with two positive punctures and one negative puncture; this means that $\\mathcal{S}$ admits a parametrization $[0, \\infty) \\times S^1$ and $(-\\infty, 0] \\times S^1$, respectively, near the punctures. Given three admissible Hamiltonians $H_1, H_2, H_3$ on $\\widehat V$, we take an $\\mathcal{S}$-parametrized Hamiltonian $H_{\\mathcal{S}}$ which coincides with $H_1, H_2$ near the positive punctures and with $H_3$ near the negative puncture. We likewise take an $\\mathcal{S}$-parameterized admissible almost complex structure $J_{\\mathcal{S}}$. We define a product\n\\[\n\\HF^k(H_1) \\otimes \\HF^{\\ell}(H_2) \\rightarrow \\HF^{k+\\ell}(H_3), \\quad x_1 \\otimes x_2 \\mapsto x_3\n\\]\nby counting the solutions $u: \\mathcal{S} \\rightarrow \\widehat V$ of the Floer equation\n\\begin{equation} \\label{eq: Floereq}\n (du - X_{H_{\\mathcal{S}}} \\otimes \\beta)^{0, 1} = 0 \n\\end{equation} \nwhich converges to $x_1, x_2$ at the positive punctures and converges to $x_3$ at the negative puncture. Here, $\\beta \\in \\Omega^1(\\mathcal{S})$ is a one-form that agrees with $dt$ near the punctures (where $t$ denotes the $S^1$-coordinates of the parameterizations). As is well-known, the product is compatible with the continuation maps and hence induces a product on the symplectic cohomology as\n\\[\n\\SH^k(V) \\otimes \\SH^{\\ell}(V) \\rightarrow \\SH^{k+\\ell}(V).\n\\]\nWith this product, the symplectic cohomology $\\SH^*(V)$ is now a unital $\\Z$-graded algebra over the Novikov field $\\Lda_V$. For more details on the construction of the ring structure, we refer the reader to \\cite[Section 6]{Rit} and \\cite[Section 4.7]{BeRi20}.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Transfer maps} Transfer-admissible Hamiltonians $H: \\widehat V \\rightarrow \\R$ form a cofinal family of admissible Hamiltonians. The symplectic cohomology $\\SH^*(V; \\Lda_V)$ is the direct limit of Hamiltonian Floer cohomology $\\HF^*(H)$ along the slope $\\kappa$ of transfer-admissible Hamiltonians $H$. Consequently, the action functional $\\mathcal{F}$ defines a natural map into the quotient\n\\begin{equation}\\label{eq: fulltononpositive}\n \\SH^*(V; \\Lda_V) \\rightarrow \\SH_{\\leq 0}^*(V; \\Lda_V), \n\\end{equation}\nand by the discussion in Section \\ref{sec: ring on nonneg}, this map preserves the ring structure. Now, the composition with the identification $\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W; \\Lda_W)$ by Corollary \\ref{cor: algebraisom} yields an algebra homomorphism \n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W, \\Lda_W).\n$$\nover $\\Lda_V = \\Lda_W$.\n\\begin{proof}[Proof of Theorem \\ref{thm: A}]\nIt only remains to explain about the commutative diagram\n\\[\n\\begin{tikzcd}\n \\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\nFor a transfer-admissible Hamiltonian $H^0$ with the slope $\\kappa$ (and hence $\\frac{1}{2}\\kappa$) sufficiently small, 1-periodic orbits of $H^0$ are precisely those of constant orbits corresponding to Morse critical points on $W$ and $V$ respectively. Moreover, the standard Floer theory, e.g. \\cite{Fl88}, tells us that they recover the quantum cohomology $\\QH^*(V)$ and $\\QH^*(W)$ including the ring structure \\cite[Lemma 13]{Ri14}. Now the restriction of the Hamiltonian $H^0: \\widehat{V} \\rightarrow \\R$ to $H^0_W: \\widehat{W}\\rightarrow \\R$, as in the proof of Theorem \\ref{thm: nonnegandW}, corresponds to the natural restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$. In other words, we have a commutative diagram:\n\\[\n\\begin{tikzcd}\n \\SH^*(V) \\arrow{r} & \\SH_{\\leq 0}^*(V) \\\\\n \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\nHere, the upper horizontal map is the homomorphism in \\eqref{eq: fulltononpositive}, and the canonical $c^*$-map is defined by the inclusion; see \\cite[Section 5]{Rit}. Now, applying the identification $\\SH^*_{\\leq 0}(V) \\cong \\SH^*(W)$ in Corollary \\ref{cor: algebraisom}, we obtain the desired diagram.\n\\end{proof}\n\n\\begin{remark}\\label{rem: Weinattach}\n Given a convex symplectic domain $W$ with $\\pi_1(\\p W) = 0$ and $c_1(W)|_{\\pi_2(W)} = 0$, attach the Weinstein handle $\\mathcal{H}$ to $W$ along the boundary $\\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \\ref{thm: A}. As an analogue of the seminal result of Cieliebak \\cite{Ci02}, it is likely that the transfer map $\\Phi: \\SH^*(V) \\rightarrow \\SH^*(W)$ gives rise to an algebra isomorphism when the handle $\\mathcal{H}$ is subcritical. \n\\end{remark}\n\n\\begin{corollary}\\label{cor: algebraisom}\n    Under the assumptions of Theorem \\ref{thm: nonnegandW}, we have a canonical $\\Lda$-algebra isomorphism\n    \\[\n    \\SH^*_{ \\leq 0}(V) \\cong \\SH^*(W).\n    \\]\n\\end{corollary}\n\n\\begin{theorem}\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\p W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map. \n\\end{theorem}",
    "post_theorem_intro_text_len": 3010,
    "post_theorem_intro_text": "The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \\cite[Section 4.1]{BeRi20} and \\cite[Section 5]{Rit}.\n\nFor construction, we basically follow McLean \\cite[Section 10.2]{Mclean_extension} and Ritter \\cite[Section 9]{Rit} where transfer maps are constructed in symplectic (co)homology for exact domains. An essential ingredient is to use a special type of admissible Hamiltonians which is adapted to the embedding $W \\hookrightarrow V$, see Figure \\ref{fig: tranfer_Ham}, and also properly interacts with an action filtration on the symplectic (co)homology. In particular, a careful estimate of the action values of the generators of the corresponding Hamiltonian Floer cochain complex shows that the part $\\SH^*_{\\leq 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$ of the subdomain $W$. Then the natural quotient map $\\SH^*(V) \\rightarrow \\SH^*_{\\leq 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V) \\rightarrow \\SH^*(W)$. This was shown to be a ring homomorphism \\cite[Section 10.2]{Mclean_extension} and more generally compatible with TQFT operations \\cite[Section 9]{Rit} for exact domains.\n\nThe main technical point of the current non-exact setup is that the standard action functional, as in Remark \\ref{rem: usualactionfctl}, now depends not only on loops but on their capping disks, and this makes the action value estimates delicate. To handle this, we introduce a new action functional, inspired by the work of McLean--Ritter \\cite{McRi23},\nwhich is designed to deal with non-exact convex symplectic domains. The basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes to the action values, and moreover, they can be effectively understood by Reeb periods of the contact boundary.\n\nIn Section \\ref{sec: newfiltration}, we define an action functional $\\mathcal{F}$ and give the relevant action value estimates in Lemma \\ref{lem: actionorbits}. In particular, we show that the non-positive part $\\SH^*_{\\leq 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to the symplectic cohomology $\\SH^*(W)$ of the subdomain $W$; see Theorem \\ref{thm: nonnegandW}. It is also shown in Corollary \\ref{cor: algebraisom} that the correspondence is compatible with the respective ring structure. The commutative diagram in Theorem \\ref{thm: A} then follows from a standard argument.\n\nThere are various interesting examples of convex symplectic domains with complement-exact subdomains for potential applications. We consider negative line bundles in Example \\ref{ex: neglinebdl2} and resolutions of isolated singularities in Example \\ref{ex: resolution}. For a given convex domain $W$, attaching an exact cobordism along the boundary produces a larger convex domain $V$, and $W$ is complement-exact in $V$; See Remark \\ref{rem: Weinattach}.",
    "sketch": "For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\\SH^*_{\\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$.” Then “the natural quotient map $\\SH^*(V)\\to \\SH^*_{\\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V)\\to \\SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”\n\nIn the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\\SH^*_{\\le 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to $\\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” “The commutative diagram in Theorem~\\ref{thm: A} then follows from a standard argument.”",
    "expanded_sketch": "For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\\SH^*_{\\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$.” Then “the natural quotient map $\\SH^*(V)\\to \\SH^*_{\\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V)\\to \\SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”\n\nIn the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\\SH^*_{\\le 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to $\\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” The commutative diagram in establishing the main theorem then follows from a standard argument.”",
    "expanded_theorem": "\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\partial W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map.",
    "theorem_type": [
      "Existence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold $(V,\\omega)$ that is exact near $\\partial V$ and whose Liouville vector field points outward along $\\partial V$. Let $W\\subset V$ be a codimension-zero subdomain that is complement-exact, i.e. $\\omega$ is exact on $V\\setminus W$. Assume moreover that $c_1(V)|_{\\pi_2(V)}=0$ and $\\pi_1(\\partial W)=0$. Write $\\Lambda_V$ and $\\Lambda_W$ for the Novikov fields associated to $V$ and $W$, $\\SH^*$ for symplectic cohomology, $\\QH^*$ for quantum cohomology, $i^*:\\QH^*(V)\\to\\QH^*(W)$ for the restriction map induced by the inclusion $i:W\\hookrightarrow V$, and $c^*$ for the canonical map from quantum cohomology to symplectic cohomology. Under these assumptions, which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
      },
      "choices": [
        {
          "label": "B",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra isomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\xrightarrow{\\cong}\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "C",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "D",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(W;\\Lambda_W)\\to\\SH^*(V;\\Lambda_V)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(W) \\arrow{r}{\\Phi} & \\SH^*(V) \\\\\n\\QH^*(W) \\arrow{r}{i_*} \\arrow{u}{c^*} & \\QH^*(V) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "E",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes, and moreover $\\Phi$ is induced by restricting Hamiltonians from $V$ to $W$ on the full symplectic cohomology complex, without passing to any action filtration or quotient."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "quotient-to-homomorphism versus isomorphism",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "algebra structure preservation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "direction of transfer and induced map on quantum cohomology",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "need for action filtration and quotient by positive-action generators",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 1,
        "justification": "The stem does not state the correct conclusion outright, but it preloads nearly all objects appearing in the true answer (the maps i^* and c^*, the Novikov-field identification context, and the focus on an existence statement), which strongly hints that the correct option will be a transfer-map theorem with a commutative diagram."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to theorem restatement/recognition: the task is essentially to identify the precise formal statement of a known result under the listed hypotheses. The nearby variants prevent it from being completely tautological, but it is still largely a reformulation of the theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish the exact theorem from weaker true or stronger false variants (mere existence vs commutative diagram vs isomorphism vs extra positive-action claim). However, the problem mainly tests precise recall/discrimination rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: one is a weaker true statement, one is an unjustified strengthening to an isomorphism, and others add plausible but false extra properties about positive-action summands or construction details. These reflect common theorem-distortion failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-constructed theorem-discrimination MCQ with strong distractors, but it leans heavily on recognizing the exact theorem statement and gives substantial structural hints in the stem rather than demanding deeper generative reasoning."
    }
  },
  {
    "id": "2512.10491v1",
    "paper_link": "http://arxiv.org/abs/2512.10491v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\partial W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map.",
      "start_pos": 16222,
      "end_pos": 16859,
      "label": "thm: A"
    },
    "ref_dict": {
      "ex: neglinebdl2": "\\begin{example}\\label{ex: neglinebdl2}\nFollowing Example \\ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\\pi: D(E) \\rightarrow B$ now serves as a complement-exact subdomain. Assume that\n\\begin{itemize} \n\\item $[\\ow] = c_1(B)$ and $[\\ow]$ is primitive in $\\Ho_2(B; \\Z);$\n\\item $B$ is simply-connected.\n\\end{itemize}\nThen $\\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \\ref{thm: A}, where $S(E)$ is the corresponding circle bundle. \n\nFor a more concrete example, one takes the blow-up $B : = \\CP^2 \\# k \\overline{\\CP}^2$ of $\\CP^2$ at $k$ generic points, with $0 < k < 9$. This admits a symplectic form $\\ow_k$ whose Poincar\\'e dual is given by $3H - E_1 - \\cdots -E_k$ where $H$ is the class of a line in $\\CP^2$ and $E_i$ is the exceptional curve derived from $i$-th blow-up. In particular $[\\ow_k] = c_1(B)$ is primitive in $H_2(B;\\Z)$ and $B$ is simply-connected.  \n\\end{example}",
      "lem: actionorbits": "\\begin{lemma}\\label{lem: actionorbits}\nFor 1-orbits $x$ in the region \\emph{(I)} and \\emph{(II)}, we have $\\mathcal{F}(x) \\leq 0$. In the other regions, we have $\\mathcal{F}(x) > 0$.\n\\end{lemma}",
      "cor: algebraisom": "\\begin{corollary}\\label{cor: algebraisom}\n    Under the assumptions of Theorem \\ref{thm: nonnegandW}, we have a canonical $\\Lda$-algebra isomorphism\n    \\[\n    \\SH^*_{ \\leq 0}(V) \\cong \\SH^*(W).\n    \\]\n\\end{corollary}",
      "thm: A": "\\begin{theorem}\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\p W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map. \n\\end{theorem}",
      "ex: resolution": "\\begin{example}\\label{ex: resolution}\n    Let $X$ be a $\\Q$-factorial variety with a unique singularity at the origin $O \\in X$ and assume that the complement $X \\setminus \\{O\\}$ admits an exact K\\\"ahler form $\\ow_X$. Then by \\cite[Lemma 3.2]{McRi23}, any resolution $\\pi: Y \\rightarrow X$ admits a K\\\"ahler form $\\ow_Y$ such that $\\pi^* \\ow_X = \\ow_Y$. In particular, $Y$ is a convex symplectic domain, and a neighborhood of $\\pi^{-1}(\\{O\\}) \\subset Y$ serves as a complement-exact subdomain.\n\\end{example}",
      "rem: Weinattach": "\\begin{remark}\\label{rem: Weinattach}\n    Given a convex symplectic domain $W$ with $\\pi_1(\\p W) = 0$ and $c_1(W)|_{\\pi_2(W)} = 0$, attach the Weinstein handle $\\mathcal{H}$ to $W$ along the boundary $\\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \\ref{thm: A}. As an analogue of the seminal result of Cieliebak \\cite{Ci02}, it is likely that the transfer map $\\Phi: \\SH^*(V) \\rightarrow \\SH^*(W)$ gives rise to an algebra isomorphism when the handle $\\mathcal{H}$ is subcritical. \n\\end{remark}",
      "thm: nonnegandW": "\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}",
      "fig: tranfer_Ham": "\\begin{overpic}[width=0.5\\linewidth]{transf_Ham.pdf}\n    \\put(85, 3){$r_W$}\n    \\put(-3, 77){$B$}\n    \\put(24, 3){$1$}\n    \\put(48, 3){$A$}\n    \\put(30, 50){$\\kappa$} \n    \\put(210, 3){$r_V$}\n    \\put(145, 3){$A+1+P$}\n    \\put(170, 100){$\\frac{1}{2}\\kappa$}\n    \\end{overpic}\n    \\caption{Transfer-admissible Hamiltonians}\n    \\label{fig: tranfer_Ham}\n    \\vspace{12pt}\n    \\centering\n   \\begin{overpic}[width=0.5\\linewidth]{cutoff2.pdf}\n    \\put(90, 3){$r_W$}\n    \\put(24, 3){$1$}\n    \\put(48, 3){$A$}\n    \\put(43, 62){$1$}    \n    \\put(215, 3){$r_V$}\n    \\put(145, 3){$A+1+P$}\n    \\put(162, 104){$1$}\n   \\end{overpic}",
      "rem: usualactionfctl": "\\begin{remark}\\label{rem: usualactionfctl}\nFor an admissible Hamiltonian $H$, a standard action functional $\\mathcal{A}_H : \\widetilde{\\mathcal{L}_0 V} \\rightarrow \\R$ is given by\n$$\n\\mathcal{A}_H(x, v) =  -\\int_{D^2} v^*\\ow + \\int_{S^1} H(x(t)) dt.\n$$\nThe class $(x, v)$ is a critical point of $\\mathcal{A}_H$ if $x$ is a $1$-periodic Hamiltonian orbit of $H$. Here, we use the convention $\\ow(\\cdot, X_H)  = d H$ for the definition of the Hamiltonian vector field $X_H$.\n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 2181,
    "pre_theorem_intro_text": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}. \n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.",
    "context": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}.\n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}",
    "full_context": "Transfer maps in Floer theory, introduced by Viterbo \\cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \\cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \\cite{FHW, GH18}.\n\nRoughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \\hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.\nTo our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and \nthe purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.\n\nLet $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\\partial V$ and the Liouville vector field points outward along $\\partial V$. Under some additional assumptions depending on the context,\nthe symplectic cohomology $\\SH^*(V)$ is defined as a $\\Z$-graded algebra over the Novikov field $\\Lda_{V}$. A standard construction of $\\SH^*(V)$ is briefly reviewed in Section \\ref{sec: symcohomology}; we refer the reader to \\cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\partial W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}\n\nLet $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \\setminus W$; we call it a \\emph{complement-exact} subdomain. When the boundary $\\p W$ is simply-connected, the Novikov field $\\Lda_W$ associated with $W$ can be canonically identified with the one $\\Lda_V$ of the ambient domain $V$; see Theorem \\ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\\SH^*(V)$ to $\\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$ on the quantum cohomology rings.\n\nThe $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \\cite[Section 4.1]{BeRi20} and \\cite[Section 5]{Rit}.\n\n\\begin{example}\\label{ex: neglinebdl2}\nFollowing Example \\ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\\pi: D(E) \\rightarrow B$ now serves as a complement-exact subdomain. Assume that\n\\begin{itemize} \n\\item $[\\ow] = c_1(B)$ and $[\\ow]$ is primitive in $\\Ho_2(B; \\Z);$\n\\item $B$ is simply-connected.\n\\end{itemize}\nThen $\\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \\ref{thm: A}, where $S(E)$ is the corresponding circle bundle.\n\n\\subsubsection{Admissible Hamiltonians} \\label{sec: admHam} Let $(V, \\ow, \\lda)$ be a convex symplectic domain. We assume that the first Chern class $c_1(V)$ vanishes on the second homotopy group $\\pi_2(V)$ \nA Hamiltonian $H: \\widehat V \\rightarrow \\R$ is called \\emph{admissible} if \n\\begin{itemize}\n\\item there exists $r_0 \\in [1, \\infty)$ such that $H$ depends only on the radial coordinate $r$ for $r \\geq r_0$, say $H(z) = h(r)$, and $h'(r) \\geq 0$;\n\\item $h(r) = \\kappa r + b$ for sufficiently large $r$ where the \\emph{slope} $\\kappa$ is not the period of the Reeb orbit on the contact boundary $(\\p V, \\alpha)$.\n\\end{itemize}\n\n\\subsubsection{Ring structure} The symplectic cohomology $\\SH^*(V)$ admits a ring structure by the pair-of-pants product. \nLet $\\mathcal{S}$ be the Riemann sphere with two positive punctures and one negative puncture; this means that $\\mathcal{S}$ admits a parametrization $[0, \\infty) \\times S^1$ and $(-\\infty, 0] \\times S^1$, respectively, near the punctures. Given three admissible Hamiltonians $H_1, H_2, H_3$ on $\\widehat V$, we take an $\\mathcal{S}$-parametrized Hamiltonian $H_{\\mathcal{S}}$ which coincides with $H_1, H_2$ near the positive punctures and with $H_3$ near the negative puncture. We likewise take an $\\mathcal{S}$-parameterized admissible almost complex structure $J_{\\mathcal{S}}$. We define a product\n\\[\n\\HF^k(H_1) \\otimes \\HF^{\\ell}(H_2) \\rightarrow \\HF^{k+\\ell}(H_3), \\quad x_1 \\otimes x_2 \\mapsto x_3\n\\]\nby counting the solutions $u: \\mathcal{S} \\rightarrow \\widehat V$ of the Floer equation\n\\begin{equation} \\label{eq: Floereq}\n (du - X_{H_{\\mathcal{S}}} \\otimes \\beta)^{0, 1} = 0 \n\\end{equation} \nwhich converges to $x_1, x_2$ at the positive punctures and converges to $x_3$ at the negative puncture. Here, $\\beta \\in \\Omega^1(\\mathcal{S})$ is a one-form that agrees with $dt$ near the punctures (where $t$ denotes the $S^1$-coordinates of the parameterizations). As is well-known, the product is compatible with the continuation maps and hence induces a product on the symplectic cohomology as\n\\[\n\\SH^k(V) \\otimes \\SH^{\\ell}(V) \\rightarrow \\SH^{k+\\ell}(V).\n\\]\nWith this product, the symplectic cohomology $\\SH^*(V)$ is now a unital $\\Z$-graded algebra over the Novikov field $\\Lda_V$. For more details on the construction of the ring structure, we refer the reader to \\cite[Section 6]{Rit} and \\cite[Section 4.7]{BeRi20}.\n\n\\begin{theorem}\\label{thm: nonnegandW}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\\pi_1(\\p W) = 0$ and $c_1(V)|_{\\pi_2(V)} = 0$. Then\n\\begin{enumerate}\n\\item The inclusion $i: W \\rightarrow V$ induces an isomorphism between the Novikov fields $\\Lda_W$ and $\\Lda_V$.\n\\item Under the above identification $\\Lda_V \\cong \\Lda_W$, we have a canonical vector space isomorphism \n$$\n\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W;\\Lda_W).\n$$\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Transfer maps} Transfer-admissible Hamiltonians $H: \\widehat V \\rightarrow \\R$ form a cofinal family of admissible Hamiltonians. The symplectic cohomology $\\SH^*(V; \\Lda_V)$ is the direct limit of Hamiltonian Floer cohomology $\\HF^*(H)$ along the slope $\\kappa$ of transfer-admissible Hamiltonians $H$. Consequently, the action functional $\\mathcal{F}$ defines a natural map into the quotient\n\\begin{equation}\\label{eq: fulltononpositive}\n \\SH^*(V; \\Lda_V) \\rightarrow \\SH_{\\leq 0}^*(V; \\Lda_V), \n\\end{equation}\nand by the discussion in Section \\ref{sec: ring on nonneg}, this map preserves the ring structure. Now, the composition with the identification $\\SH^*_{\\leq 0}(V; \\Lda_V) \\cong \\SH^*(W; \\Lda_W)$ by Corollary \\ref{cor: algebraisom} yields an algebra homomorphism \n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W, \\Lda_W).\n$$\nover $\\Lda_V = \\Lda_W$.\n\\begin{proof}[Proof of Theorem \\ref{thm: A}]\nIt only remains to explain about the commutative diagram\n\\[\n\\begin{tikzcd}\n \\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\nFor a transfer-admissible Hamiltonian $H^0$ with the slope $\\kappa$ (and hence $\\frac{1}{2}\\kappa$) sufficiently small, 1-periodic orbits of $H^0$ are precisely those of constant orbits corresponding to Morse critical points on $W$ and $V$ respectively. Moreover, the standard Floer theory, e.g. \\cite{Fl88}, tells us that they recover the quantum cohomology $\\QH^*(V)$ and $\\QH^*(W)$ including the ring structure \\cite[Lemma 13]{Ri14}. Now the restriction of the Hamiltonian $H^0: \\widehat{V} \\rightarrow \\R$ to $H^0_W: \\widehat{W}\\rightarrow \\R$, as in the proof of Theorem \\ref{thm: nonnegandW}, corresponds to the natural restriction map $i^*: \\QH^*(V) \\rightarrow \\QH^*(W)$. In other words, we have a commutative diagram:\n\\[\n\\begin{tikzcd}\n \\SH^*(V) \\arrow{r} & \\SH_{\\leq 0}^*(V) \\\\\n \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\nHere, the upper horizontal map is the homomorphism in \\eqref{eq: fulltononpositive}, and the canonical $c^*$-map is defined by the inclusion; see \\cite[Section 5]{Rit}. Now, applying the identification $\\SH^*_{\\leq 0}(V) \\cong \\SH^*(W)$ in Corollary \\ref{cor: algebraisom}, we obtain the desired diagram.\n\\end{proof}\n\n\\begin{remark}\\label{rem: Weinattach}\n Given a convex symplectic domain $W$ with $\\pi_1(\\p W) = 0$ and $c_1(W)|_{\\pi_2(W)} = 0$, attach the Weinstein handle $\\mathcal{H}$ to $W$ along the boundary $\\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \\ref{thm: A}. As an analogue of the seminal result of Cieliebak \\cite{Ci02}, it is likely that the transfer map $\\Phi: \\SH^*(V) \\rightarrow \\SH^*(W)$ gives rise to an algebra isomorphism when the handle $\\mathcal{H}$ is subcritical. \n\\end{remark}\n\n\\begin{corollary}\\label{cor: algebraisom}\n    Under the assumptions of Theorem \\ref{thm: nonnegandW}, we have a canonical $\\Lda$-algebra isomorphism\n    \\[\n    \\SH^*_{ \\leq 0}(V) \\cong \\SH^*(W).\n    \\]\n\\end{corollary}\n\n\\begin{theorem}\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\p W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map. \n\\end{theorem}",
    "post_theorem_intro_text_len": 3010,
    "post_theorem_intro_text": "The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \\cite[Section 4.1]{BeRi20} and \\cite[Section 5]{Rit}.\n\nFor construction, we basically follow McLean \\cite[Section 10.2]{Mclean_extension} and Ritter \\cite[Section 9]{Rit} where transfer maps are constructed in symplectic (co)homology for exact domains. An essential ingredient is to use a special type of admissible Hamiltonians which is adapted to the embedding $W \\hookrightarrow V$, see Figure \\ref{fig: tranfer_Ham}, and also properly interacts with an action filtration on the symplectic (co)homology. In particular, a careful estimate of the action values of the generators of the corresponding Hamiltonian Floer cochain complex shows that the part $\\SH^*_{\\leq 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$ of the subdomain $W$. Then the natural quotient map $\\SH^*(V) \\rightarrow \\SH^*_{\\leq 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V) \\rightarrow \\SH^*(W)$. This was shown to be a ring homomorphism \\cite[Section 10.2]{Mclean_extension} and more generally compatible with TQFT operations \\cite[Section 9]{Rit} for exact domains.\n\nThe main technical point of the current non-exact setup is that the standard action functional, as in Remark \\ref{rem: usualactionfctl}, now depends not only on loops but on their capping disks, and this makes the action value estimates delicate. To handle this, we introduce a new action functional, inspired by the work of McLean--Ritter \\cite{McRi23},\nwhich is designed to deal with non-exact convex symplectic domains. The basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes to the action values, and moreover, they can be effectively understood by Reeb periods of the contact boundary.\n\nIn Section \\ref{sec: newfiltration}, we define an action functional $\\mathcal{F}$ and give the relevant action value estimates in Lemma \\ref{lem: actionorbits}. In particular, we show that the non-positive part $\\SH^*_{\\leq 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to the symplectic cohomology $\\SH^*(W)$ of the subdomain $W$; see Theorem \\ref{thm: nonnegandW}. It is also shown in Corollary \\ref{cor: algebraisom} that the correspondence is compatible with the respective ring structure. The commutative diagram in Theorem \\ref{thm: A} then follows from a standard argument.\n\nThere are various interesting examples of convex symplectic domains with complement-exact subdomains for potential applications. We consider negative line bundles in Example \\ref{ex: neglinebdl2} and resolutions of isolated singularities in Example \\ref{ex: resolution}. For a given convex domain $W$, attaching an exact cobordism along the boundary produces a larger convex domain $V$, and $W$ is complement-exact in $V$; See Remark \\ref{rem: Weinattach}.",
    "sketch": "For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\\SH^*_{\\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$.” Then “the natural quotient map $\\SH^*(V)\\to \\SH^*_{\\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V)\\to \\SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”\n\nIn the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\\SH^*_{\\le 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to $\\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” “The commutative diagram in Theorem~\\ref{thm: A} then follows from a standard argument.”",
    "expanded_sketch": "For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\\SH^*_{\\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\\SH^*(W)$.” Then “the natural quotient map $\\SH^*(V)\\to \\SH^*_{\\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\\SH^*(V)\\to \\SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”\n\nIn the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\\SH^*_{\\le 0}(V)$ with respect to the action filtration induced by $\\mathcal{F}$ again corresponds to $\\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” The commutative diagram in establishing the main theorem then follows from a standard argument.”",
    "expanded_theorem": "\\label{thm: A}\nLet $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\\pi_2(V)} = 0$ and $\\pi_1(\\partial W) = 0$. There exists a $\\Z$-graded algebra homomorphism\n$$\n\\Phi: \\SH^*(V; \\Lda_V) \\rightarrow \\SH^*(W; \\Lda_W)\n$$\nunder a canonical identification $\\Lda_V = \\Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.\n\\[\\begin{tikzcd}\n        \\SH^*(V) \\arrow{r}{\\Phi}  & \\SH^*(W)  \\\\\n        \\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*}& \\QH^*(W) \\arrow[swap]{u}{c^*}\n    \\end{tikzcd}\n\\]\nwhere the vertical maps are the canonical $c^*$-map.",
    "theorem_type": [
      "Existence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold $(V,\\omega)$ that is exact near $\\partial V$ and whose Liouville vector field points outward along $\\partial V$. Let $W\\subset V$ be a codimension-zero subdomain that is complement-exact, i.e. $\\omega$ is exact on $V\\setminus W$. Assume moreover that $c_1(V)|_{\\pi_2(V)}=0$ and $\\pi_1(\\partial W)=0$. Write $\\Lambda_V$ and $\\Lambda_W$ for the Novikov fields associated to $V$ and $W$, $\\SH^*$ for symplectic cohomology, $\\QH^*$ for quantum cohomology, $i^*:\\QH^*(V)\\to\\QH^*(W)$ for the restriction map induced by the inclusion $i:W\\hookrightarrow V$, and $c^*$ for the canonical map from quantum cohomology to symplectic cohomology. Under these assumptions, which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
      },
      "choices": [
        {
          "label": "B",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra isomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\xrightarrow{\\cong}\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "C",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "D",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(W;\\Lambda_W)\\to\\SH^*(V;\\Lambda_V)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(W) \\arrow{r}{\\Phi} & \\SH^*(V) \\\\\n\\QH^*(W) \\arrow{r}{i_*} \\arrow{u}{c^*} & \\QH^*(V) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes."
        },
        {
          "label": "E",
          "text": "Under the canonical identification $\\Lambda_V=\\Lambda_W$, there exists a $\\mathbb{Z}$-graded algebra homomorphism\n$$\n\\Phi:\\SH^*(V;\\Lambda_V)\\to\\SH^*(W;\\Lambda_W)\n$$\nsuch that the diagram\n\\[\n\\begin{tikzcd}\n\\SH^*(V) \\arrow{r}{\\Phi} & \\SH^*(W) \\\\\n\\QH^*(V) \\arrow{r}{i^*} \\arrow{u}{c^*} & \\QH^*(W) \\arrow[swap]{u}{c^*}\n\\end{tikzcd}\n\\]\ncommutes, and moreover $\\Phi$ is induced by restricting Hamiltonians from $V$ to $W$ on the full symplectic cohomology complex, without passing to any action filtration or quotient."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "quotient-to-homomorphism versus isomorphism",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "algebra structure preservation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "direction of transfer and induced map on quantum cohomology",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "need for action filtration and quotient by positive-action generators",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct choice. It sets up hypotheses and asks for the valid conclusion, without giving away the direction, strength, or algebraic properties of the map."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall item: the stem lists the assumptions and asks which conclusion holds. The options do introduce nearby variants (isomorphism, wrong direction, weaker structure, extra unsupported claim), so it is not a pure verbatim restatement, but it is still close to one."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the solver must distinguish between existence vs. isomorphism, algebra homomorphism vs. plain homomorphism, and correct functorial direction. However, the task mainly tests recognition of the precise theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and reflect natural failure modes: overstrengthening to an isomorphism, reversing the transfer direction, and adding an unjustified chain-level claim. But choice C is a weakly true statement if A is true, so the options are not cleanly exclusive, which weakens the distractor set."
      },
      "total_score": 5,
      "overall_assessment": "Moderate-quality MCQ: it avoids answer leakage and has mostly plausible distractors, but it is still close to theorem restatement and is weakened by overlap between the correct answer and the weaker option C."
    }
  },
  {
    "id": "2512.10820v1",
    "paper_link": "http://arxiv.org/abs/2512.10820v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in ${\\bf C}^n$ with $\\partial M\\in C^2$. Assume that   the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\partial M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\overline M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\overline M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.",
      "start_pos": 10767,
      "end_pos": 11596,
      "label": "vbint-"
    },
    "ref_dict": {
      "d2=0": "\\begin{prop}\\label{d2=0}\nLet $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\\om \\otimes e$ and $\\om\\in L^2_{loc}(U)$ and $d\\om\\in L^1_{loc}(U)$. If $A$ is admissible, then $\\hat\\om=(dA)A^{-1}+A\\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\\Om A^{-1}=\\hat\\Om$. \n\\end{prop}",
      "vbint-": "\\begin{thm}\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in $\\cc^n$ with $\\pd M\\in C^2$. Assume that   the Levi form of $\\pd M$ on $T^{(1,0)}\\pd M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\pd M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\ov M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Om^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\ov M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.\n\\end{thm}",
      "vbint": "\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that    the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$.    Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 2656,
    "pre_theorem_intro_text": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain  with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle   over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the  spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying,  for any   $f\\in C_{(k)}^\\infty(\\overline M)$ and     $s\\in C^\\infty(\\overline M,E)$,  \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$     The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n  Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad  D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are  the connection and curvature forms.  \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad  \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types,  $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields  \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it  integrability problem} studied in this paper is to find a frame   $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy    $\\dbar  A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a   holomorphic vector bundle structure.    By (\\ref{hatomom})-(\\ref{om02}),  the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered,   the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity  for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.",
    "context": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying, for any $f\\in C_{(k)}^\\infty(\\overline M)$ and $s\\in C^\\infty(\\overline M,E)$, \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$ The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are the connection and curvature forms. \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types, $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it integrability problem} studied in this paper is to find a frame $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy $\\dbar A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a holomorphic vector bundle structure. By (\\ref{hatomom})-(\\ref{om02}), the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.",
    "full_context": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying, for any $f\\in C_{(k)}^\\infty(\\overline M)$ and $s\\in C^\\infty(\\overline M,E)$, \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$ The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are the connection and curvature forms. \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types, $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it integrability problem} studied in this paper is to find a frame $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy $\\dbar A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a holomorphic vector bundle structure. By (\\ref{hatomom})-(\\ref{om02}), the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.\n\n\\begin{thm}[convex configuration estimate]\\label{conv-est} Let $r>0$ and $1\\leq q\\leq n-1$.\nLet $(D^1,D_{r_2}^2)$ be an $(n-q)$-convex configuration. Let $D^{12}_r=D^1\\cap D^2_r$.\nThe homotopy operators $H_q, H_{q+1}$ in \\rta{hf-c} satisfies\n\\aln{}\\|H_s\\var\\|_{\\Lambda^{r+1/2}(D^{1 2}_{(1-\\theta)r_2})}&\\leq \\f{C_r( \\nabla\\rho^1,\\nabla^2\\rho^1)}{\\theta^{3r+2r_0}}\\|\\var\\|_{\\Lambda^r(D^{1 2}_{r_2})},\\quad \\forall\\theta\\in(0,1). \n\\end{align*}\nMoreover, $C_r(\\nabla\\rho^1,\\nabla^2\\rho^1)$ is stable under small $C^2$ perturbations of $\\rho^1$.\n\\end{thm}\n\\begin{proof}Recall from \\re{hq1}-\\re{nhq2-} that $H_s =H^{(1)}_s + H_s^{(2)}$ with \n\\aln{} \n H^{(1)}_s f&:=R_{ D^{2}; s-1}^0 \\cL E_{D^{12}} f+R_{U^1;s-1 }^{01}[\\db,\\cL E_{D^{12}}] f,\\\\\nH^{(2)}_sf&=L_{1^+;{s-1}\n}^{01} \\cL E_{D^{12}} f +L_{2;s-1}^{02} f+L_{12;s-1}^{012}f.\n\\end{align*}\nTake a cut-off function $\\chi\\in C^\\infty_0(D^{2}_{(1-\\theta/3)r_2})$ with $\\chi=1$ on $D^{2}_{(1-\\theta/2)r_2}$, and decompose\n$\n H^{(1)}_s f= H^{(1)}_s(\\chi f)+ H^{(1)}_s( (1-\\chi)f).\n$\nThen the estimate in~\\cite{gong-shi-nn} says that \n$$\n\\|H_s^{(1)}(\\chi \\var)\\|_{\\Lambda^{r+1/2}(D^{1 2}_{r_2})}\\leq C_r \\|\\chi\\var\\|_{\\Lambda^r(D^{1 2}_{r_2})}.\n$$\nWe can choose $\\chi$ such that $\\|\\chi\\|_{C^r}\\leq {C_r}{\\theta^{-r-1}}$. Thus, \n$$\n\\|\\chi\\var\\|_{\\Lambda^r(D^{12}_{r_2})}\\leq {C_r }{\\theta^{-r-r_0}}\\|\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$$\nSince $(1-\\chi)\\var$ vanishes on $D_{(1-\\theta/2)r}^{12}$ and\n\\ga{}\\label{giest}\n|g^i(z,\\zeta)\\cdot(\\zeta-z)|\\geq |\\zeta-z|^2/C,\\quad z\\in D^{1}\\cap D^2,\\zeta\\in D^2\\setminus D^1, i=0,1,2,\n\\end{gather}\nthen we can obtain $$\n\\|H_s^{(1)}((1-\\chi) \\var)\\|_{\\Lambda^{r+1/2}(D^{12}_{(1-\\theta)r_2})}\\leq C_r\\theta^{-2r-r_0} \\|(1-\\chi)\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$$\nThe $H_s^{(2)}$ is given by boundary integrals on $S^1_+, S^2$ and $ S^{12}$ respectively. The estimate \\re{giest} also yields \n$\\|H_s^{(2)}\\var\\|_{\\Lambda^{r+1/2}(D^{12}_{(1-\\theta)r_2})}\\leq C_r \\theta^{-2r-r_0} \\|\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$\nWe have obtained the desired estimate of $H_s$. \n\\end{proof}\nWe now derive estimates for the concave case. We still have $H_s=H_s^{(1)}+H_s^{(2)}$, where $H_s^{(1)}$ has the same form as in the convex case. However, \n$$\nH_{s}^{(2)}f=\\sum_{i=1}^2L_{i3,s}^{0i3}f- L^{01}_{1^+,s-1}\\cL Ef +L_{12;s-1}^{123}f-\\hat T_{ D^{23}_{(1-\\theta)r}, s}L^{23}_{12;s} f.\n$$\nBy \\re{g3theta}, we have $\n\\RE\\{g^3(z,\\zeta)\\cdot(\\zeta-z)\\} \\geq\\theta r_3^2$ for $z\\in D^{23}_{(1-\\theta)r}$ and $\\zeta\\in S^{13}\\cup S^{23}\\cup S^{12}\\subset\\pd D^{123}_r$. Some kernels in $H^{(2)}_s$ involve first-order $z$-derivatives of $\\nabla\\rho^1(z)$.\nAs in the convex case, we can obtain \n$$\\|H^{(2)}_sf\\|_{\\Lambda^{r+1/2}(D^{123}_{(1-\\theta)r})}\\leq {C_r(\\nabla\\rho^1 )}{\\theta^{-r-r_0}}\\|\\rho^1\\|_{\\Lambda^{3/2}}\\|f\\|_{\\Lambda^\\e}.\n$$The kernels of $H^{(1)}f$ involve the second-order $z$-derivatives $ \\nabla\\rho^2(z)$.\nThen by ~\\cite{gong-shi-nn}*{Thm. 5.12}, we have \n$$\\|H^{(1)}_sf\n\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}\\leq {C_r }{\\theta^{-r-r_0}}\\left(\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\e}+ \\|f\\|_{\\Lambda^r}\\right).\n$$Therefore, we have obtained part $(a)$ below.\n\\begin{thm}[concave configuration estimate]\\label{concave-est} Let $\\e>0,r>0$ and $1\\leq q\\leq n-3$. \nAssume that $\\rho^1\\in C^{2+\\e}$. The following hold.\n\\bpp\n\\item The operators $H_q, H_{q+1}$ in Theorem~$\\ref{cchf}$ satisfy\n\\al{}\n\\label{hqfr12}\n\\|H_sf\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}&\\leq \\f{C_{r,\\e}}{\\theta^{3r+r_0}} (\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\epsilon({D_{r}^{123}})}+ \\|f\\|_{\\Lambda^r(D^{123}_{r})})\n\\end{align}\nfor $0\\leq\\delta\\leq1/2$ and $0<\\theta<1$. Furthermore, $C_{r,\\e}=C_{r,\\e}(\\nabla\\rho^1,\\nabla^2\\rho^1)$ is stable under small $C^2$ perturbations of $\\rho^1$;\n the same estimate holds for the $\\db$ solution operator $H_q$ in \\rta{cchf-closed}.\n \\item \nThe homotopy formula \\rea{tsqf+-cv} in Theorem~$\\ref{cchf}$ holds for $f\\in \\Lambda^{\\e}_{(0,q)}(D^{123})$ when $\\db f\\in \\Lambda^{\\e}_{(0,q)}(D^{123})$, and the $\\db$-solution operator $H_q$ in \\rta{cchf-closed} is valid for all $\\db$-closed $f\\in\\Lambda^{\\e}_{(0,q)}(D^{123})$ and satisfies \\rea{hqfr12}.\n\\epp\n\\end{thm}\n\\begin{proof}We verify $(b)$ using $(a)$. We know that $f=\\db H_qf+H_{q+1}\\db f$ when $f\\in C^{1+\\e}$. Let us write $H_q$ as $H_{D^{123}_r}$ to indicate the dependent on the domain. Now assume $f\\in \\Lambda^r_{(0,q)}(D^{123})$ with $\\db f\\in\\Lambda^r$. We find a sequence $f_j\\in C^{\\infty}_{(0,q)}(D^{123})$ such that $f_j $ and $\\db f_j $ converge to $f,\\db f$ in $\\Lambda^{r'}(D^{123})$ as $j\\to\\infty$ for any $r'<r$. Furthermore, $$\\|f_j\\|_{\\Lambda^{r}(D^{123})}\\leq C_r\\|f\\|_{\\Lambda^{r}(D^{123})}, \\quad \\|\\db f_j\\|_{\\Lambda^{r}(D^{123})}\\leq C_r\\|\\db f\\|_{\\Lambda^{r}(D^{123})}.$$\nThen $H_qf=\\lim H_qf_j$ and $H_{q+1}\\db f=\\lim H_{q+1}\\db f_j$ gives us \\rea{tsqf+-cv}.\nAssume now $f$ is $\\db$-closed. Set $L_t(z)=z+tie_{q+2}$ with $t>0$, where $e_{q+2}$ is the $(q+2)$-th unit vector of $\\cc^n$. Fix $\\theta\\in(0,1/2)$. When $0<t<t_\\theta$, we have $L_t(D^{123}_{(1-\\theta)r})\\subset D^{123}_r$. Using a mollifier, we find $\\db$-closed $f_j\\in C^\\infty(D^{123}_{(1-\\theta)r})$ such that $f_j$ converges to $f$ in $\\Lambda^{r'}(D^{123}_{(1-\\theta)r})$-norm for any $r'<r$, whereas $\\|f_j\\|_{\\Lambda^{r}(D^{123}_{(1-\\theta)r})}\\leq C_r\\|f\\|_{\\Lambda^{r}(D^{123}_{r})}$. We have $f_j=\\db H_{D^{123}_{(1-\\theta')r},q}f_j$ on $D^{123}_{(1-\\theta)r}$ for $0<\\theta'<\\theta/2$.\nLetting $j\\to\\infty$, we obtain $f=\\db H_{D^{123}_{(1-\\theta')r},q}f$ on $D^{123}_{(1-\\theta)r}$ and\n\\eq{Hth''}\n\\|H_{D^{123}_{(1-\\theta')r},q}f\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}\\leq \\f{C_{r,\\e}}{\\theta^{3r+r_0}} (\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\epsilon }+ \\|f\\|_{\\Lambda^r }).\n\\eeq\nBy the Arzel\\`{a}-Ascoli theorem, $H_{D^{123}_{(1-\\theta'_j)r},q}f$ converges to $H_{D^{123}_{r},q}f$ in $\\Lambda^{r'+\\delta}(D^{123}_{(1-\\theta)r})$ norm for a sequence $\\theta'_j\\to0$ for any $r'<r$. Then \\re{Hth''} yield\n\\re{hqfr12} for $H_q$.\n\\end{proof}\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\\section{Integrability of complex vector bundles}\\label{sec:NM}\nIn Section 2, we have seen that \\rt{vbint-} follows from the following result.\n\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$. Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$ of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm} \n\\begin{proof}The proof is based on a KAM-type iteration method. Such an approach was developed by Webster~\\ci{MR999729} for an interior version of Newlander-Nirenberg theorem. It was also used in~\\cites{MR2742034,MR2829316} to establish the integrability of CR vector bundles on strictly pseudoconvex hypersurfaces in $\\cc^n$ for $n\\geq4$.",
    "post_theorem_intro_text_len": 5320,
    "post_theorem_intro_text": "Our theorem concerns the integrability of $(E,D)$ in the H\\\"older-Zygmund spaces $\\Lambda^r$, which is the standard H\\\"older spaces $C^r$ when $r$ is not an integer; see Section 5 for definition. The theorem also treats  a Koszul connection $D$ that maps the space $ Lip(M,E)$ of Lipschitz sections  \n into  $ L^\\infty_{(1)}( M,E)$, the space  of $E$-valued $1$-forms of class $L^\\infty$ on $M$. Such a connection extends trivially to a Koszul connection on $E$ over $\\overline M$.   The theory of $L^\\infty$ connections, which is of independent interest, is required  even when the original connections are $C^\\infty$ for the (strictly) $3$-concave case. Theorem~\\ref{vbint-}  is reduced to Theorem~\\ref{vbint}, and the reduction requires a careful discussion in Section~\\ref{sec:int-formal} of   $L^\\infty$ connections on Lipschitz bundles  and of  {\\it admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; see Proposition~\\ref{d2=0}.\n\n\\medskip\n\nWe now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. \n We   apply homotopy formulas to the study of (\\ref{dbAA}). \nThe use of integral representations to establish regularity of $\\overline\\partial$-solutions on strongly pseudoconvex domains in ${\\bf C}^n$ has a long history. Sup-norm estimate for $\\dbar$-solutions to $(0,1)$-forms was proved by  Grauert and Lieb~\\cite{MR273057} and  Henkin~\\cite{MR0249660}.   Kerzman~\\cite{MR0281944} obtained  $L^p$ and $C^{\\beta}$ estimates of $\\dbar$-solutions for $(0,1)$-forms and all $\\beta<1/2$.\n  Lieb~~\\cite{MR283235} obtained the $L^\\infty$ and the $C^\\beta$ estimates of $\\dbar$-solutions for $(0,q)$-forms.  \nHenkin and Romanov~\\cite{MR0293121}  proved the sharp $C^{1/2}$ estimate of \n$\\dbar$-solutions for continuous $(0,1)$-forms. \n\nWe recall several important developments on integral representation formulas for  $\\overline\\partial$   that lead to derivative estimates on strongly pseudoconvex domains $D\\subset{\\bf C}^n$. When $\\partial D$ is of class $ C^{k+2}$,   Siu~\\cite{MR330515} showed that the Henkin solution operator satisfies a $C^{k+1/2}$ estimate  for  $\\dbar$-closed $(0,1)$ forms of class $C^k$, and for  $\\dbar$-closed $(0,q)$-forms   with $q\\geq1$,  Lieb and Range~\\cite{MR597825} constructed a new  $\\dbar$ solution operator   and established  the same estimate. When $\\partial D$ is only $ C^2$,  a homotopy formula was constructed  in~\\cite{MR3961327} using the Stein extension, producing   homotopy operators with $\\Lambda^{r+1/2}$ estimate for $r>1$.    Shi and Yao~\\cites{MR4688544,MR4861589} later developed   a homotopy formula   employing the Rychkov extension and obtained $\\Lambda^{r+1/2}$ estimates for all $r>0$, as well as $H^{s+1/2,p}$ estimates  for $1<p<\\infty$ and $s>1/p$ when $\\partial D\\in C^2$; they also showed that the  estimates hold  for all $s\\in{\\bf R}$ when $\\partial D$ is sufficiently smooth.  Very recently, Yao~~\\cite{yaoc2} further reduced the boundary smoothness assumption to $ C^2$ for all $s\\in{\\bf R}$. We also refer the reader to references cited in~\\cites{MR3961327,MR4688544,MR4861589,yaoc2}.\n\nWe also note that   Range and Siu~~\\cite{MR338450} proved the $C^\\beta$ estimate for all $\\beta<1/2$ for $\\dbar$-solutions for continuous $(0,q)$-forms on the (real) transversal intersection of\n  strictly pseudoconvex domains. Whether the optimal gain of $1/2$  derivative   estimates  holds for $\\dbar$-solutions on such transversal intersections remains an open problem.\n  Higher-order derivative estimates in this setting were obtained by Brinkmann~~\\cite{Br84}, Michel~~\\cite{MR928297}, and Michel-Perotti~~\\cite{MR1038709}.\n Peters~~\\cite{MR1135535}  constructed a homotopy formula for the weakly transversal intersection of strictly pseudoconvex domains and established higher-order estimates, though with a loss of derivatives.\n It would be interesting to determine whether the results mentioned above can be applied to the integrability problem for intersections of strictly pseudoconvex domains. We also remark that the integrability of almost CR vector bundles on strongly pseudoconvex hypersurfaces in $\\mathbb C^n$ with $n\\geq4$ was proved by Webster~~\\cite{MR1128608}, and sharper regularity results in the finitely smooth case were obtained by Gong-Webster~\\cites{MR2742034,MR2829316}.\nIt remains an open question whether Theorem~\\ref{vbint-} holds if $\\partial M$ has two negative Levi eigenvalues. This appears to be related to the unsettled embedding problem for strongly pseudoconvex local CR structures in ${\\bf R}^5$, investigated by Webster~\\cite{MR0995504} and Gong-Webster~~\\cite{MR2868966}.\n\nWe organize the paper as follows. In Section 2, we examine the formal integrability condition in detail for connections on Lipschitz complex vector bundles and introduce the notion of admissible frame changes. Sections 3 and 4 reformulate the local homotopy formulas in~\\cites{MR986248,MR4866351} \n for shrinking domains. In Section 5, we recall  estimates established in \\cites{MR986248,gong-shi-nn} and derive  new estimates on shrinking domains. Finally, in Section 6, we complete the proof of Theorem~\\ref{vbint-}  by establishing Theorem~\\ref{vbint}.\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}",
    "sketch": "Theorem~\\ref{vbint-} is reduced to Theorem~\\ref{vbint}, and the reduction requires a careful discussion in Section~\\ref{sec:int-formal} of $L^\\infty$ connections on Lipschitz bundles and of \\textit{admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; see Proposition~\\ref{d2=0}.\n\n“We now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\\ref{dbAA}).”\n\nThe proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and “Finally, in Section 6, we complete the proof of Theorem~\\ref{vbint-} by establishing Theorem~\\ref{vbint}.”",
    "expanded_sketch": "To prove the main theorem, we reduce it to the following theorem.\n\n\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that    the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$.    Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm}\n\nThe reduction requires a careful discussion of $L^\\infty$ connections on Lipschitz bundles and of \\textit{admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; for this we use the following proposition.\n\n\\begin{prop}\\label{d2=0}\nLet $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\\om \\otimes e$ and $\\om\\in L^2_{loc}(U)$ and $d\\om\\in L^1_{loc}(U)$. If $A$ is admissible, then $\\hat\\om=(dA)A^{-1}+A\\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\\Om A^{-1}=\\hat\\Om$. \n\\end{prop}\n\n“We now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\\ref{dbAA}).”\n\nThe proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and finally we complete the proof of the main theorem by establishing the theorem above.",
    "expanded_theorem": "\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in ${\\bf C}^n$ with $\\partial M\\in C^2$. Assume that   the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\partial M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\overline M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\overline M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.",
    "theorem_type": [
      "Existence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $0<r\\le \\infty$, and let $M\\subset \\mathbb C^n$ be a domain with $\\partial M\\in C^2$. Fix $\\zeta_0\\in \\partial M$, and assume that the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive eigenvalues or at least three negative eigenvalues at $\\zeta_0$. Let $E$ be a complex vector bundle over $\\overline M$ of class $Lip$ (respectively of class $\\Lambda^{r+1}$). Suppose $D$ is a connection with mapping property\n$$\nD: Lip(M,E)\\to L_{(1)}^\\infty(M,E)\n\\qquad \\text{(respectively } D:\\Lambda^{r+1}(M,E)\\to \\Lambda^r_{(1)}(M,E)\\text{)},\n$$\nand let $\\Omega=D^2$ be its curvature. Assume that the $(0,2)$-component of the curvature vanishes on $M$ in the sense of distributions, i.e. $\\Omega^{(0,2)}=0$. A holomorphic vector bundle structure here means that locally one can choose bundle frames in which the $(0,1)$-part of the connection vanishes (equivalently, in local frames with connection matrix $\\omega$, one can solve $\\bar\\partial A + A\\omega=0$ for an invertible change-of-frame matrix $A$). Under these hypotheses, which conclusion holds near $\\zeta_0$?",
      "correct_choice": {
        "label": "A",
        "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\\Lambda^{r+1}$ structure on $E|_U$ it is of class $\\Lambda^{r+1/2}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1}$, and relative to the original $\\Lambda^{r+1}$ structure on $E|_U$ it is of class $\\Lambda^{r+1}$."
        },
        {
          "label": "C",
          "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure."
        },
        {
          "label": "D",
          "text": "There exists an open neighborhood $U\\subset \\overline M$ of $\\zeta_0$ and an invertible change-of-frame matrix $A$ on $U$ such that $\\bar\\partial A+A\\omega=0$ on $U\\setminus \\partial M$, where $A$ is of class $Lip$ in the $Lip$ case and of class $\\Lambda^{r+1}$ in the $\\Lambda^{r+1}$ case; hence $E|_U$ carries a holomorphic vector bundle structure with the same regularity as the original bundle structure."
        },
        {
          "label": "E",
          "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure, provided the Levi form at $\\zeta_0$ has either $n-1$ positive eigenvalues or at least two negative eigenvalues. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\\Lambda^{r+1}$ structure on $E|_U$ it is of class $\\Lambda^{r+1/2}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "half-derivative regularity gain/loss",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped boundary-regularity conclusion for the induced holomorphic structure",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "regularity of the gauge matrix and induced structure",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "Levi-signature threshold: at least three negative eigenvalues",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and definitions but does not explicitly state the conclusion. The correct option is not leaked directly; the examinee must know or infer the precise local consequence and regularity statement."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to a theorem-recall question: it presents the exact hypotheses and asks for the conclusion. However, it is not a pure verbatim restatement because the choices vary in strength, regularity, and Levi-signature threshold."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish the exact conclusion from stronger, weaker, or slightly altered variants, especially the half-derivative regularity loss and the 'at least three negative eigenvalues' condition. Still, the pressure is mainly on precise theorem recall rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they test overclaiming regularity, accepting a weaker-but-true statement, confusing gauge regularity with induced holomorphic regularity, and misremembering the Levi-signature hypothesis."
      },
      "total_score": 6,
      "overall_assessment": "A solid but theorem-centric MCQ. It avoids direct answer leakage and has strong distractors, but it mainly tests precise recall of a stated result rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.10845v1",
    "paper_link": "http://arxiv.org/abs/2512.10845v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.",
      "start_pos": 62389,
      "end_pos": 62637,
      "label": "Thm for E"
    },
    "ref_dict": {
      "Thm for E": "\\begin{theorem}\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.\n\\end{theorem}",
      "lemma abc": "\\begin{lemma}\\label{lemma abc}\n  Assume the curvature $\\Theta$ of $h$ is positive on every fiber $\\mathcal{X}_t$. For $1\\leq k\\leq m$, the following are equivalent. \n\\begin{enumerate}[label=\\Alph*.]\n    \\item\\label{A} The curvature $\\Theta$ has at least $n+k$ positive eigenvalues at every point in $\\mathcal{X}$.\n    \\item\\label{B} The horizontal component $\\Theta_\\mathcal{H}$ has at least $k$ positive eigenvalues at every point in $\\mathcal{X}$.\n    \\item\\label{C}  There exists a positive $\\beta\\in C^{\\infty}(\\mathcal{X}, p^*(\\wedge^{1,1}T^*Y))$ such that the sum of any $m-k+1$ eigenvalues of $\\Theta_\\mathcal{H}$ with respect to $\\beta$ is positive.\n\\end{enumerate}\nMoreover, when $k=1$, statement C can be rephrased as $\\Theta_\\mathcal{H}\\wedge \\beta^{m-1}>0$ or equivalently $\\Theta^{n+1}\\wedge \\beta^{m-1}>0$.\n\\end{lemma}",
      "Thm 1": "\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 4709,
    "pre_theorem_intro_text": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness.  RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of  Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be  $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and  $v=\\sum_j v_j\\partial/\\partial t_j$. \n\n\\begin{definition} \n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$  (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n    \\item\\label{a} $\\Theta$ is positive on every fiber. \n    \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n    \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity,  hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.",
    "context": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and $v=\\sum_j v_j\\partial/\\partial t_j$.\n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n \\item\\label{a} $\\Theta$ is positive on every fiber. \n \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.",
    "full_context": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and $v=\\sum_j v_j\\partial/\\partial t_j$.\n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n \\item\\label{a} $\\Theta$ is positive on every fiber. \n \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.\n\nIn this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.\n\nAnother motivation for establishing Theorem \\ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \\cite[Theorem 1.4]{YangCamb} and \\cite[Theorem 1.3]{YangForum}, for a compact K\\\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.\n\\begin{enumerate}\n \\item The holomorphic tangent bundle $TX$ is uniformly RC-positive.\n \\item The exterior power $\\wedge^p TX$ is RC-positive for $1\\leq p\\leq n$.\n \\end{enumerate}\n One can ask if the converse is true (\\cite[Problem 4.15]{YangForum}). A partial converse is proved in \\cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\\wedge^p TX}(-1)$ is RC-positive for $1\\leq p \\leq n$. So, Theorem \\ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.\n\nFor the proof of Theorem \\ref{Thm for E}, instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \\ref{Thm for E}\nas a special case. We consider a proper holomorphic surjection $p:\\mathcal{X}^{n+m}\\to Y^m$ between two complex manifolds with $\\mathcal{X}$ K\\\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\\mathcal{X}_t$ for $t\\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\\mathcal{X}$. Let $$V_t=H^0(\\mathcal{X}_t, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}).$$ We assume that $\\dim V_t$ is independent of $t\\in Y$. So, the direct image of the sheaf of sections of $L\\otimes K_{\\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\\in Y$, \n\\begin{equation}\\label{metric}\n H(u,u):=\\int_{\\mathcal{X}_t}h(u,u). \n\\end{equation}\nWe extend the metric $h$ to act on sections $u$ of $L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\\mathcal{X}_t$. In terms of local coordinates, if $u=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\\mathcal{X}_t}$, then $h(u,u)=c_n u' \\wedge \\overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show \n\\begin{theorem}\\label{Thm 1}\n If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\n\nNow, we consider the fibration $p:P(E^*)\\to X$, and we assume $X$ is K\\\"ahler to make sure $P(E^*)$ is K\\\"ahler (see \\cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \\ref{Thm 1}, we have Theorem \\ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \\ref{Thm 1} is associated with the direct image of $L\\otimes K_{\\mathcal{X}/Y}$. In the present situation, the relative canonical bundle\n$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\\otimes p^*\\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\\otimes \\det E$. On the other hand, if we choose $O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).\n\nWe choose a coordinate system $(t_1,\\ldots,t_m)$ around the point $t_0$ in $Y$ such that $v_0=\\partial/\\partial t_1$ at $t_0$. Consider a fixed $u_0\\neq 0$ in $V_{t_0}$. A standard argument allows us to extend $u_0$ to a local holomorphic section $u$ of $V$ such that $D'u=0$ at $t_0$ and $u(t_0)=u_0\\neq0$. A straightforward computation gives\n\\begin{equation}\\label{standard} \\partial \\bar{\\partial} H(u,u)=- H(\\Theta^V u,u) \\text{ at } t_0. \n\\end{equation}\nOn the other hand, if we let $\\mathbf{u}$ be a representative of $u$ and write $\\mathbf{u}=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ some local frame of $L$, then \n\\begin{equation}\n H(u,u)=p_*(c_n u'\\wedge \\overline{u'} e^{-\\phi}) \n\\end{equation}\nwhere $e^{-\\phi}=h(e,e)$ and $c_n=i^{n^2}$. According to \\cite[Proposition 4.2]{Berndtsson09}, we can choose a representative $\\mathbf{u}$ such that in $\\bar{\\partial}\\mathbf{u}= \\sum \\eta_j\\wedge dt_j$, the $\\eta_j$ is primitive on $\\mathcal{X}_{t_0}$. Moreover, \n$\\partial^\\phi u'=0$ at $t_0$. After using such a representative, we obtain\n\\begin{equation}\\label{4.4}\n\\partial\\bar{\\partial}H(u,u)\n = \n -c_n p_* ( u'\\wedge\\overline{u'}\\wedge \\partial\\bar{\\partial} \\phi e^{-\\phi})\n +\n (-1)^n c_np_* (\\bar{\\partial}u'\\wedge \\overline{\\bar{\\partial}u'}e^{-\\phi}) \\text{ at } t_0. \n\\end{equation}\nWe apply the above $(1,1)$-form to the tangent vector $v_0=\\partial/\\partial t_1$ and get \n\\begin{equation}\\label{4.6}\n\\partial\\bar{\\partial}H(u,u)(v_0,\\bar{v}_0)\n = \n -c_n p_* ( u'\\wedge\\overline{u'}\\wedge \\partial\\bar{\\partial} \\phi e^{-\\phi})(v_0,\\bar{v}_0)\n +\n (-1)^n c_np_* (\\bar{\\partial}u'\\wedge \\overline{\\bar{\\partial}u'}e^{-\\phi}) (v_0,\\bar{v}_0)\\text{ at } t_0.\n\\end{equation}\nBecause $\\bar{\\partial}\\mathbf{u}= \\sum \\eta_j\\wedge dt_j$ and $\\mathbf{u}=u'\\otimes e$, we see $\\sum \\eta_j\\wedge dt_j=\\bar{\\partial}\\mathbf{u}=\\bar{\\partial}u'\\otimes e $. If we write $\\eta_j=\\eta_j'\\otimes e$, then $\\bar{\\partial}u'=\\sum \\eta_j'\\wedge dt_j$. So the last term in (\\ref{4.6}) is equal to \\begin{equation}\\label{4.7}\n (-1)^n c_n\\int_{\\mathcal{X}_{t_0}}(-1)^{n}\\sum \\eta'_j\\wedge \n\\overline{\\eta'}_k \\wedge dt_j\\wedge d\\bar{t}_k e^{-\\phi}(v_0,\\bar{v}_0)= c_n \\int_{\\mathcal{X}_{t_0}} \\eta'_1\\wedge \n\\overline{\\eta'}_1 e^{-\\phi}\\leq 0;\n\\end{equation}\nthe last inequality is by the fact that the $\\eta_1$ is primitive on $\\mathcal{X}_{t_0}$.\n\nThe case we care about most in this paper is when $k=1$ in Lemma \\ref{lemma abc} because it corresponds to weak RC-positivity. The difficulty in proving a theorem like Theorem \\ref{Thm for E} or Theorem \\ref{Thm 1} for weak RC-positivity is that the $\\beta$ in Lemma \\ref{lemma abc} is on $\\mathcal{X}$, so it does not quite fit into Berndtsson's computation, especially formula (\\ref{4.11}).\nSo, we raise the question:\n\\begin{question}\n Is it possible to choose $\\beta$ in Lemma \\ref{lemma abc} so that $\\beta=p^*\\alpha$ for some Hermitian metric $\\alpha$ on $Y$? \n\\end{question}\nThis question is somewhat bold because if it is possible to choose $\\beta=p^*\\alpha$, then we can use \\cite[Theorem 4]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $S^k\\otimes \\det E$ has positive mean curvature for $k\\geq 0$. Moreover, it is even possible to use \\cite[Theorem 5]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $E$ has positive mean curvature. Since positive mean curvature implies RC-positivity (\\cite[Theorem 3.6]{YangCamb}), this would mean that RC-positivity, weak RC-positivity, and mean curvature positivity are all equivalent. Such an equivalence is conjectured for tangent bundle $TX$ in\n\\cite[Problems 4.15 and 4.17]{YangForum}.",
    "post_theorem_intro_text_len": 5285,
    "post_theorem_intro_text": "Another motivation for establishing Theorem \\ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \\cite[Theorem 1.4]{YangCamb} and \\cite[Theorem 1.3]{YangForum}, for a compact K\\\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.\n\\begin{enumerate}\n     \\item The holomorphic tangent bundle $TX$ is uniformly RC-positive.\n     \\item The exterior power $\\wedge^p TX$ is RC-positive for $1\\leq p\\leq n$.\n \\end{enumerate}\n One can ask if the converse is true (\\cite[Problem 4.15]{YangForum}). A partial converse is proved in \\cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\\wedge^p TX}(-1)$ is RC-positive for $1\\leq p \\leq n$. So, Theorem \\ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.\n\nWe also prove a lemma (Lemma \\ref{lemma abc} in Section \\ref{section ?}) and discuss how a variant of this lemma might lead to a solution to the original question of Yang, namely, weak RC-positivity of $E$ implying RC positivity of $E$.\n\nFor the proof of Theorem \\ref{Thm for E}, instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \\ref{Thm for E}\nas a special case. We consider a proper holomorphic surjection $p:\\mathcal{X}^{n+m}\\to Y^m$ between two complex manifolds with $\\mathcal{X}$ K\\\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\\mathcal{X}_t$ for $t\\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\\mathcal{X}$. Let $$V_t=H^0(\\mathcal{X}_t, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}).$$ We assume that $\\dim V_t$ is independent of $t\\in Y$. So, the direct image of the sheaf of sections of $L\\otimes K_{\\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\\in Y$, \n\\begin{equation}\\label{metric}\n  H(u,u):=\\int_{\\mathcal{X}_t}h(u,u).  \n\\end{equation}\nWe extend the metric $h$ to act on sections $u$ of $L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\\mathcal{X}_t$. In terms of local coordinates, if $u=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\\mathcal{X}_t}$, then $h(u,u)=c_n u' \\wedge \\overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show \n\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\n\nActually, the precise statement we prove in Theorem \\ref{Thm 1} is: For a fixed point $t_0 \\in Y$, if the curvature $\\Theta$ of $h$ is positive on the fiber $\\mathcal{X}_{t_0}$ , and  there exists a nonzero tangent vector $v\\in T_{t_0}Y$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t_0,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t_0,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive at $t_0$.\n\nNow, we consider the fibration $p:P(E^*)\\to X$, and we assume $X$ is K\\\"ahler to make sure $P(E^*)$ is K\\\"ahler (see \\cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \\ref{Thm 1}, we have Theorem \\ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \\ref{Thm 1} is associated with the direct image of $L\\otimes K_{\\mathcal{X}/Y}$. In the present situation, the relative canonical bundle\n$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\\otimes p^*\\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\\otimes \\det E$. On the other hand, if we choose $O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large). \n\nThe proof of Theorem \\ref{Thm 1} is an adaptation of \\cite[Section 3]{wu2025mean}, but we still include the details for completeness  (the original argument is due to Berndtsson in \\cite[Section 4]{Berndtsson09} and \\cite[Section 2]{BoMathz}. See also \\cite{CampanaCaoMihai}).\n\nThis paper is organized as follows. In Section \\ref{section prelim}, we give a local expression for the assumption (uniform weak RC-positivity) in Theorem \\ref{Thm 1} which will be used in the proof of the main theorem. In Section \\ref{section proof}, we prove Theorem \\ref{Thm 1}. In Section \\ref{section ?}, we discuss\na characterization of weak RC-positivity and its possible application.\n\nI would like to thank Shin-ichi Matsumura for bringing to my attention the question of RC-positivity and weak RC-positivity. I am grateful to L\\'aszl\\'o Lempert, Siarhei Finski, and Xiaokui Yang for their interest in the paper. Thanks are also due to the Erd\\H{o}s Center, Budapest for the support.",
    "sketch": "To prove Theorem~\\ref{Thm for E}, the text says that “instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem~\\ref{Thm for E}$ as a special case.” Concretely, one considers a proper holomorphic submersion $p:\\mathcal{X}^{n+m}\\to Y^m$ with $\\mathcal{X}$ K\\\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\\mathcal{X}$, and the direct-image bundle $V$ with fibers\n\\[\nV_t=H^0(\\mathcal{X}_t,\\, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}),\n\\]\nendowed with the “naturally defined Hermitian metric”\n\\[\nH(u,u):=\\int_{\\mathcal{X}_t} h(u,u).\\tag{\\ref{metric}}\n\\]\nUnder the hypothesis (in Theorem~\\ref{Thm 1}) that “the curvature $\\Theta$ of $h$ is positive on every fiber” and that for each $t\\in Y$ there is a nonzero $v\\in T_tY$ with “$\\Theta(\\widetilde v,\\overline{\\widetilde v})|_{(t,z)}>0$ for any lift $\\widetilde v$ of $v$,” one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”\n\nThen Theorem~\\ref{Thm for E} is obtained by applying Theorem~\\ref{Thm 1} to $p:P(E^*)\\to X$ (with $X$ K\\\"ahler so $P(E^*)$ is K\\\"ahler). In this specialization, one uses that\n\\[\nK_{P(E^*)/X}\\simeq O_{P(E^*)}(-r)\\otimes p^*\\det E.\n\\]\nChoosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\\otimes\\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”\n\nFinally, the proof of Theorem~\\ref{Thm 1} is described as “an adaptation of \\cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).",
    "expanded_sketch": "To prove the main theorem, the text says that “instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains the main theorem as a special case.” Concretely, one considers a proper holomorphic submersion $p:\\mathcal{X}^{n+m}\\to Y^m$ with $\\mathcal{X}$ K\\\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\\mathcal{X}$, and the direct-image bundle $V$ with fibers\n\\[\nV_t=H^0(\\mathcal{X}_t,\\, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}),\n\\]\nendowed with the “naturally defined Hermitian metric”\n\\[\nH(u,u):=\\int_{\\mathcal{X}_t} h(u,u).\\tag{\\ref{metric}}\n\\]\nWe first prove the following theorem.\n\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\nUnder these hypotheses, one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”\n\nThen the main theorem is obtained by applying the preceding theorem to $p:P(E^*)\\to X$ (with $X$ K\\\"ahler so $P(E^*)$ is K\\\"ahler). In this specialization, one uses that\n\\[\nK_{P(E^*)/X}\\simeq O_{P(E^*)}(-r)\\otimes p^*\\det E.\n\\]\nChoosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\\otimes\\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”\n\nFinally, the proof of the preceding theorem is described as “an adaptation of \\cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).",
    "expanded_theorem": "\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let $X$ be a compact K\\\"ahler manifold and let $E\\to X$ be a holomorphic vector bundle. Write $P(E^*)$ for the projectivized dual bundle and $O_{P(E^*)}(1)$ for its tautological line bundle. Say that $E$ is uniformly weakly RC-positive if there exists a Hermitian metric $h$ on $O_{P(E^*)}(1)$ whose curvature form $\\Theta$ satisfies: (i) $\\Theta$ is positive on every fiber of $P(E^*)\\to X$; and (ii) for every $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$ such that for every point $[\\zeta]\\in P(E_t^*)$ and every lift $\\widetilde v\\in T_{(t,[\\zeta])}P(E^*)$ of $v$, one has $\\Theta(\\widetilde v,\\overline{\\widetilde v})>0$. Also, a holomorphic vector bundle $F\\to X$ is uniformly RC-positive if it admits a Hermitian metric $H$ such that for every $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$ with $H(\\Theta^H u,u)(v,\\bar v)>0$ for every nonzero $u\\in F_t$. Here $S^kE$ denotes the $k$th symmetric power of $E$, and $\\det E$ its determinant line bundle.\n\nWhich statement holds for every uniformly weakly RC-positive holomorphic vector bundle $E$ over a compact K\\\"ahler manifold $X$?",
      "correct_choice": {
        "label": "A",
        "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large integers $k$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large integers $k$."
        },
        {
          "label": "C",
          "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is uniformly RC-positive."
        },
        {
          "label": "D",
          "text": "For every integer $k\\ge 0$, both $S^kE\\otimes \\det E$ and $S^kE$ are uniformly RC-positive."
        },
        {
          "label": "E",
          "text": "There exists an integer $k_0\\ge 0$, depending only on $\\operatorname{rank}(E)$, such that for every integer $k\\ge k_0$, both $S^kE\\otimes \\det E$ and $S^kE$ are uniformly RC-positive."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "output-strength of direct-image theorem",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "large-$k$ conclusion for $S^kE$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "necessity of the large-$k$ absorption step for $K^{-1}_{P(E^*)/X}$",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "non-uniform dependence of the large-$k$ threshold",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only definitions and hypotheses; it does not reveal the conclusion about symmetric powers or determinant twists. The correct option is not signposted by wording in the stem."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a restatement of a definition. It asks for a substantive theorem-level consequence involving symmetric powers, determinant twisting, and an eventual-large-k conclusion, so the respondent must distinguish among competing claims."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some reasoning about theorem strength, quantifiers, and positivity notions, especially to separate 'for all k' from 'for sufficiently large k' and 'RC-positive' from 'uniformly RC-positive.' However, it is still largely theorem recall/discrimination rather than deep derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they vary the positivity strength, remove the large-k caveat, or introduce an unjustified uniform threshold depending only on rank. These reflect common overgeneralization and quantifier errors."
      },
      "total_score": 7,
      "overall_assessment": "A strong MCQ with no answer leakage and well-designed distractors. It tests careful discrimination of a nontrivial theorem, though it leans more toward precise recall than fully generative reasoning."
    }
  },
  {
    "id": "2512.11051v1",
    "paper_link": "http://arxiv.org/abs/2512.11051v1",
    "theorems_cnt": 6,
    "theorem": {
      "env_name": "ltheorem",
      "content": "[Nonstandard/Standard CLT] \\label{thm:CLT}\nIf $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,\n$(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)$ as $t\\to\\infty$,\nwhere $\\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.\n\nIf $I_v=0$, then there exists $\\sigma^2\\ge0$, typically nonzero, such that\n$v$ satisfies a standard CLT. That is,\n$t^{-1/2}v_t \\to_d N(0,\\sigma^2)$ as $t\\to\\infty$.",
      "start_pos": 9684,
      "end_pos": 10139,
      "label": "thm:CLT"
    },
    "ref_dict": {
      "thm:CLT": "\\begin{ltheorem}[Nonstandard/Standard CLT] \\label{thm:CLT}\nIf $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,\n$(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)$ as $t\\to\\infty$,\nwhere $\\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.\n\nIf $I_v=0$, then there exists $\\sigma^2\\ge0$, typically nonzero, such that\n$v$ satisfies a standard CLT. That is,\n$t^{-1/2}v_t \\to_d N(0,\\sigma^2)$ as $t\\to\\infty$.\n\\end{ltheorem}",
      "thm:decay": "\\begin{ltheorem} \\label{thm:decay}\n Let $v,\\,w:M\\to\\R$ be sufficiently smooth observables. Then there is a constant\n$C>0$ such that \n$$\n\\left|\\int_M v \\cdot (w\\circ g_t)\\,d\\mu-\\int_Mv\\,d\\mu\\int_Mw\\,d\\mu\\right|\n\\leq C t^{-1}\n\\quad\\text{for all $t>0$}.\n$$\n\\end{ltheorem}",
      "thm:decaymap": "\\begin{thm} \\label{thm:decaymap}\nFor all H\\\"older observables $v,w:\\Sigma\\to\\R$, there is a constant\n$C>0$ such that \n$$\n\\left|\\int_{\\Sigma} v \\cdot (w\\circ g^n)\\,d\\mu_{\\Sigma}-\\int_{\\Sigma}v\\,d\\mu_{\\Sigma}\\int_{\\Sigma}w\\,d\\mu_{\\Sigma}\\right|\n\\leq C n^{-1}\n\\quad\\text{for all $n\\ge1$}.\n$$\nMoreover, for all H\\\"older observables $v,w:\\Sigma\\to\\R$  supported in $\\Sigma_0$,\n\\[\n\\int_{\\Sigma} v \\cdot (w\\circ g^n)\\,d\\mu_{\\Sigma}-\\int_{\\Sigma}v\\,d\\mu_{\\Sigma}\\int_{\\Sigma}w\\,d\\mu_{\\Sigma}\\sim \\bar\\tau\\sigma_R^2\nn^{-1} \\int_\\Sigma v\\,d\\mu_\\Sigma\\, \\int_\\Sigma w\\,d\\mu_\\Sigma \\quad\\text{for all $n\\ge1$}.\n\\]\n\\end{thm}",
      "eq:alpha": "\\begin{equation} \\label{eq:alpha}\n\\alpha_0=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,0)\\,ds\\,d\\theta, \\qquad\n\\alpha_\\pi=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,\\pi)\\,ds\\,d\\theta.\n\\end{equation}",
      "thm:WIP": "\\begin{ltheorem}[Nonstandard WIP] \\label{thm:WIP}\nIf $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,\n$W_n \\to_w W$ in $C[0,1]$ as $n\\to\\infty$,\nwhere $W$ is a Brownian motion with variance $\\sigma_v^2$. \n\\end{ltheorem}",
      "app:C": "\\label{app:C}\n\nFor convenience, we list the Chernov axioms as stated in~\\cite[Section 3]{LMM24}. A more in-depth discussion of the axioms, and where they differ from treatments in~\\cite{Balint-Toth,Ch",
      "fig:surface-intro": "\\begin{figure}[hbt!]\n\\centering\n\\def\\svgwidth{11cm}\n   \\input{nonpositive_curvature-intro.pdf_tex}\n\\caption{Surface\nwith flat cylinder $\\cC$.\nThe region $\\cR$ between the two curves $\\beta_\\pm$ is a surface of revolution\nwith profile $\\xi$.}\n\\label{fig:surface-intro}\n\\end{figure}"
    },
    "pre_theorem_intro_text_len": 4302,
    "pre_theorem_intro_text": "\\label{sec-intro}\n\nIt is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.\nErgodicity with respect to volume was proved for surfaces in~\\cite{Hopf39} and for general dimension in~\\cite{Anosov67}. The Bernoulli property was shown in~\\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\\cite{DenkerPhilipp84,Ratner73}.\nIn major breakthroughs, exponential decay of correlations was established in~\\cite{Dolgopyat98} for surfaces and~\\cite{Liverani04} in general dimension.\n\nIn contrast, there are few results on statistical properties beyond the Bernoulli property for \ngeodesic flows on nonpositively curved manifolds when there exist points of zero curvature.\nIn~\\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.\nIn particular, we considered a class of closed surfaces with one flat geodesic\nand proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP\n(with standard normalisation).\nIn this paper, we provide examples of geodesic flows on nonpositively curved surfaces\nthat satisfy a CLT and WIP with \\emph{nonstandard normalisation}.\n\nOur approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\\cite{ChernovMarkarian}.\nGeneral results on statistical limit laws and decay of correlations seem infeasible,\nbut it is possible to analyse interesting classes of examples.\nMoreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.\nOn the other hand, the prerequisites for the study of dispersing billiards are firmly established. \nFor geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.\nA case in point is the smoothness of the foliations induced by the Green bundles and the necessity\nto extend existing results such as those in~\\cite{GerberNitica99,GerberWilkinson99}, see Section~\\ref{sec:GW}.\n\n\\vspace{1ex}\nThe class of geodesic flows that we study in this paper is as follows.\nLet $r\\in[5,\\infty)$ and fix $0<L<\\ve_0$.\nDefine ${\\mathcal R}$ to be the surface of revolution\nwith profile~$\\xi$ given by\n$\\xi(s)=1$ for $|s|\\leq L$ and $\\xi(s)=1+(|s|-L)^r$ for $L\\leq |s|\\leq \\ve_0$.\nWe form \na closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\\beta_\\pm$ bounding ${\\mathcal R}$,\nsee Figure~\\ref{fig:surface-intro}.\nWe call $S$ a \\emph{surface with flat cylinder ${\\mathcal C}=[-L,L]\\times{\\mathbb S}^1$}, where\n${\\mathbb S}^1={\\mathbb R}/(2\\pi{\\mathbb Z})$.\nAway from~${\\mathcal C}$ the surface $S$ has negative curvature.\n\n\\begin{figure}[hbt!]\n\\centering\n\\def\\svgwidth{11cm}\n   \\input{nonpositive_curvature-intro.pdf_tex}\n\\caption{Surface\nwith flat cylinder ${\\mathcal C}$.\nThe region ${\\mathcal R}$ between the two curves $\\beta_\\pm$ is a surface of revolution\nwith profile $\\xi$.}\n\\label{fig:surface-intro}\n\\end{figure}\n\nLet $g_t:M\\to M$ denote the geodesic flow on the unit tangent bundle\n$M=T^1S$, and let $\\mu$ denote the normalised Riemannian volume on the \nthree-dimensional manifold $M$. Then $\\mu$ is an ergodic $g_t$-invariant\nprobability measure \\cite{Pesin77b}. \n\nThe flat cylinder ${\\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.\nLet $v:M\\to{\\mathbb R}$ be H\\\"older with $\\int_M v\\,d\\mu=0$. We define $I_v=\\alpha_0^2+\\alpha_\\pi^2$ where\n$\\alpha_0$ and $\\alpha_\\pi$ are the averages of $v$ over these two families of periodic orbits.\n(The notation derives from the explicit formulas~\\eqref{eq:alpha} given in Section~\\ref{sec:WIP}.)\nFor $t>0$, define $v_t=\\int_0^t v\\circ g_s\\,ds:M\\to{\\mathbb R}$.\nOur first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.",
    "context": "It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.\nErgodicity with respect to volume was proved for surfaces in~\\cite{Hopf39} and for general dimension in~\\cite{Anosov67}. The Bernoulli property was shown in~\\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\\cite{DenkerPhilipp84,Ratner73}.\nIn major breakthroughs, exponential decay of correlations was established in~\\cite{Dolgopyat98} for surfaces and~\\cite{Liverani04} in general dimension.\n\nIn contrast, there are few results on statistical properties beyond the Bernoulli property for \ngeodesic flows on nonpositively curved manifolds when there exist points of zero curvature.\nIn~\\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.\nIn particular, we considered a class of closed surfaces with one flat geodesic\nand proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP\n(with standard normalisation).\nIn this paper, we provide examples of geodesic flows on nonpositively curved surfaces\nthat satisfy a CLT and WIP with \\emph{nonstandard normalisation}.\n\nOur approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\\cite{ChernovMarkarian}.\nGeneral results on statistical limit laws and decay of correlations seem infeasible,\nbut it is possible to analyse interesting classes of examples.\nMoreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.\nOn the other hand, the prerequisites for the study of dispersing billiards are firmly established. \nFor geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.\nA case in point is the smoothness of the foliations induced by the Green bundles and the necessity\nto extend existing results such as those in~\\cite{GerberNitica99,GerberWilkinson99}, see Section~\\ref{sec:GW}.\n\n\\vspace{1ex}\nThe class of geodesic flows that we study in this paper is as follows.\nLet $r\\in[5,\\infty)$ and fix $0<L<\\ve_0$.\nDefine ${\\mathcal R}$ to be the surface of revolution\nwith profile~$\\xi$ given by\n$\\xi(s)=1$ for $|s|\\leq L$ and $\\xi(s)=1+(|s|-L)^r$ for $L\\leq |s|\\leq \\ve_0$.\nWe form \na closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\\beta_\\pm$ bounding ${\\mathcal R}$,\nsee Figure~\\ref{fig:surface-intro}.\nWe call $S$ a \\emph{surface with flat cylinder ${\\mathcal C}=[-L,L]\\times{\\mathbb S}^1$}, where\n${\\mathbb S}^1={\\mathbb R}/(2\\pi{\\mathbb Z})$.\nAway from~${\\mathcal C}$ the surface $S$ has negative curvature.\n\nLet $g_t:M\\to M$ denote the geodesic flow on the unit tangent bundle\n$M=T^1S$, and let $\\mu$ denote the normalised Riemannian volume on the \nthree-dimensional manifold $M$. Then $\\mu$ is an ergodic $g_t$-invariant\nprobability measure \\cite{Pesin77b}.\n\nThe flat cylinder ${\\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.\nLet $v:M\\to{\\mathbb R}$ be H\\\"older with $\\int_M v\\,d\\mu=0$. We define $I_v=\\alpha_0^2+\\alpha_\\pi^2$ where\n$\\alpha_0$ and $\\alpha_\\pi$ are the averages of $v$ over these two families of periodic orbits.\n(The notation derives from the explicit formulas~\\eqref{eq:alpha} given in Section~\\ref{sec:WIP}.)\nFor $t>0$, define $v_t=\\int_0^t v\\circ g_s\\,ds:M\\to{\\mathbb R}$.\nOur first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.\n\n\\begin{equation} \\label{eq:alpha}\n\\alpha_0=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,0)\\,ds\\,d\\theta, \\qquad\n\\alpha_\\pi=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,\\pi)\\,ds\\,d\\theta.\n\\end{equation}\n\n\\begin{figure}[hbt!]\n\\centering\n\\def\\svgwidth{11cm}\n   \\input{nonpositive_curvature-intro.pdf_tex}\n\\caption{Surface\nwith flat cylinder $\\cC$.\nThe region $\\cR$ between the two curves $\\beta_\\pm$ is a surface of revolution\nwith profile $\\xi$.}\n\\label{fig:surface-intro}\n\\end{figure}",
    "full_context": "It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.\nErgodicity with respect to volume was proved for surfaces in~\\cite{Hopf39} and for general dimension in~\\cite{Anosov67}. The Bernoulli property was shown in~\\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\\cite{DenkerPhilipp84,Ratner73}.\nIn major breakthroughs, exponential decay of correlations was established in~\\cite{Dolgopyat98} for surfaces and~\\cite{Liverani04} in general dimension.\n\nIn contrast, there are few results on statistical properties beyond the Bernoulli property for \ngeodesic flows on nonpositively curved manifolds when there exist points of zero curvature.\nIn~\\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.\nIn particular, we considered a class of closed surfaces with one flat geodesic\nand proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP\n(with standard normalisation).\nIn this paper, we provide examples of geodesic flows on nonpositively curved surfaces\nthat satisfy a CLT and WIP with \\emph{nonstandard normalisation}.\n\nOur approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\\cite{ChernovMarkarian}.\nGeneral results on statistical limit laws and decay of correlations seem infeasible,\nbut it is possible to analyse interesting classes of examples.\nMoreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.\nOn the other hand, the prerequisites for the study of dispersing billiards are firmly established. \nFor geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.\nA case in point is the smoothness of the foliations induced by the Green bundles and the necessity\nto extend existing results such as those in~\\cite{GerberNitica99,GerberWilkinson99}, see Section~\\ref{sec:GW}.\n\n\\vspace{1ex}\nThe class of geodesic flows that we study in this paper is as follows.\nLet $r\\in[5,\\infty)$ and fix $0<L<\\ve_0$.\nDefine ${\\mathcal R}$ to be the surface of revolution\nwith profile~$\\xi$ given by\n$\\xi(s)=1$ for $|s|\\leq L$ and $\\xi(s)=1+(|s|-L)^r$ for $L\\leq |s|\\leq \\ve_0$.\nWe form \na closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\\beta_\\pm$ bounding ${\\mathcal R}$,\nsee Figure~\\ref{fig:surface-intro}.\nWe call $S$ a \\emph{surface with flat cylinder ${\\mathcal C}=[-L,L]\\times{\\mathbb S}^1$}, where\n${\\mathbb S}^1={\\mathbb R}/(2\\pi{\\mathbb Z})$.\nAway from~${\\mathcal C}$ the surface $S$ has negative curvature.\n\nLet $g_t:M\\to M$ denote the geodesic flow on the unit tangent bundle\n$M=T^1S$, and let $\\mu$ denote the normalised Riemannian volume on the \nthree-dimensional manifold $M$. Then $\\mu$ is an ergodic $g_t$-invariant\nprobability measure \\cite{Pesin77b}.\n\nThe flat cylinder ${\\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.\nLet $v:M\\to{\\mathbb R}$ be H\\\"older with $\\int_M v\\,d\\mu=0$. We define $I_v=\\alpha_0^2+\\alpha_\\pi^2$ where\n$\\alpha_0$ and $\\alpha_\\pi$ are the averages of $v$ over these two families of periodic orbits.\n(The notation derives from the explicit formulas~\\eqref{eq:alpha} given in Section~\\ref{sec:WIP}.)\nFor $t>0$, define $v_t=\\int_0^t v\\circ g_s\\,ds:M\\to{\\mathbb R}$.\nOur first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.\n\n\\begin{equation} \\label{eq:alpha}\n\\alpha_0=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,0)\\,ds\\,d\\theta, \\qquad\n\\alpha_\\pi=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,\\pi)\\,ds\\,d\\theta.\n\\end{equation}\n\n\\begin{figure}[hbt!]\n\\centering\n\\def\\svgwidth{11cm}\n   \\input{nonpositive_curvature-intro.pdf_tex}\n\\caption{Surface\nwith flat cylinder $\\cC$.\nThe region $\\cR$ between the two curves $\\beta_\\pm$ is a surface of revolution\nwith profile $\\xi$.}\n\\label{fig:surface-intro}\n\\end{figure}\n\nThe flat cylinder $\\cC$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.\nLet $v:M\\to\\R$ be H\\\"older with $\\int_M v\\,d\\mu=0$. We define $I_v=\\alpha_0^2+\\alpha_\\pi^2$ where\n$\\alpha_0$ and $\\alpha_\\pi$ are the averages of $v$ over these two families of periodic orbits.\n(The notation derives from the explicit formulas~\\eqref{eq:alpha} given in Section~\\ref{sec:WIP}.)\nFor $t>0$, define $v_t=\\int_0^t v\\circ g_s\\,ds:M\\to\\R$.\nOur first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.\n\n\\begin{rmk} As in~\\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\\\"older observables $v$ with $\\int_M v\\,d\\mu=I_v=0$. See for example the discussion in~\\cite[End of Section~4]{HM07}.\n\n\\begin{ltheorem}[Nonstandard WIP] \\label{thm:WIP}\nIf $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,\n$W_n \\to_w W$ in $C[0,1]$ as $n\\to\\infty$,\nwhere $W$ is a Brownian motion with variance $\\sigma_v^2$. \n\\end{ltheorem}\n\n\\begin{rmk} \\label{rmk:lower}\n The upper bound in Theorem~\\ref{thm:decay} is sharp in the sense that if $v$ is H\\\"older with $\\int_Mv\\,d\\mu=0$ and $I_v\\neq0$, then \n\\[\nt\\int_M v \\cdot (v\\circ g_t)\\,d\\mu\\not\\to0\n\\quad\\text{as $t\\to\\infty$}.\n\\]\nThis is a consequence of the nonstandard limit law in Theorem~\\ref{thm:CLT}.\nSee~\\cite[Corollary~1.3]{BalintGouezel06} or~\\cite[Proposition~9.14]{BBM19}.\n\\end{rmk}\n\n\\begin{lemma}\\label{lem:hyperbolicity}\nLet $r\\ge 5$, $\\omega\\in(1,r/2)$. The following are true:\n\\begin{description}\n\\item[{\\rm (1)}] Lebesgue almost every $x\\in\\Sigma_0$ has an LSM/LUM $W^{s/u}_x$ for $f$.\n\\item[{\\rm (2)}] \nFor all $x,\\overline{x}\\in \\Sigma_{\\rm in}$,\n\\begin{enumerate}\n\\item[{\\rm (a)}] There is a continuous function $a:\\Sigma_{\\rm in}\\to\\R$ such that\n$E^u_x$ is spanned by $\\begin{bmatrix}a(x) \\\\ 1\\end{bmatrix}$.\nMoreover, \n$|a(x)-a(\\bar x)|\\ll |\\log d(x,\\overline{x})|^{-\\omega}$. \n\\item[{\\rm (b)}] There is a $C^{1+{\\rm Lip}}$ function $\\Theta$ such that\n$W^u_x$ is locally the graph $\\{(\\Theta(\\psi),\\psi)\\}$ of $\\Theta$.\n\\end{enumerate}\n\\item[{\\rm (3)} Growth bounds:] {\\color{white} a}\n\\begin{enumerate}\n\\item[{\\rm (a)}] If $x\\in\\mfD_n^>$, then $\\|df|_{E^u_x}\\|\\approx n^{3-\\frac{2}{r}}$.\n\\item[{\\rm (b)}] If $x\\in\\mfD_n^<$, then $\\|df|_{E^u_x}\\|\\approx n^3$.\n\\end{enumerate}\n\\item[{\\rm (4)} Distortion bounds:] If $x,\\overline{x}\\in\\mfD_n^>$ or $x,\\overline{x}\\in\\mfD_n^<$,\nwith $\\overline{x}\\in W^u_x$, then\n$$\n\\big|\\log \\|df|_{E^u_x}\\|-\\log \\|df|_{E^u_{\\overline x}}\\|\\big| \\ll\nd(fx, f \\overline{x})^\\frac13.\n$$\n\\item[{\\rm (5)} Jacobian of holonomies:]\nIf $x,\\overline{x}\\in\\mfD_n^>$ or $x,\\overline{x}\\in\\mfD_n^<$,\nwith $\\overline{x}\\in W^s_x$, then\n\\[\n\\big|\\log \\|df|_{E^u_x}\\|-\\log \\|df|_{E^u_{\\overline{x}}}\\|\\big|\\ll |\\log d(x,\\overline{x})|^{-\\omega}.\n\\]\n\\end{description}\n\\end{lemma}\n\n\\begin{thm} \\label{thm:JR}\nLet $\\{\\alpha_i:i\\in I\\}\\subset\\R$ be a list of the values attained by $J$.\nLet $\\sigma_i\\ge0$ be constants, $i\\in I$, such that \n$\\sum_{i\\in I}\\sigma_i^2\\in(0,\\infty)$.\nAssume that there are constants $C>0$, $\\eps\\in(0,1)$ such that\n\\begin{equation} \\label{eq:JR1}\n \\sum_{i\\in I}|\\mu_\\Delta(R_0=n,\\,J=\\alpha_i)-2\\sigma_i^2 n^{-3}|\\le Cn^{-(3+\\eps)}\n\\quad\\text{for all $n\\ge1$},\n\\end{equation}\nand\n\\begin{equation} \\label{eq:JR2}\n\\mu_\\Delta(R_0=k,\\,R_0\\circ f_\\Delta^n=\\ell)\\le Ck^{-(2+\\eps)}\\ell^{-(2+\\eps)}\n\\quad\\text{for all $k,\\ell,n\\ge1$}.\n\\end{equation}\nThen $JR_0$ satisfies a nonstandard CLT with variance \n$\\sigma_J^2=\\sum_{i\\in I}\\alpha_i^2\\sigma_i^2\\ge0$. That is,\n$(n\\log n)^{-1/2}\\sum_{j=0}^{n-1}(JR_0)\\circ f_\\Delta^j\\to_d N(0,\\sigma_J^2)$.\n\\end{thm}\n\n\\begin{cor} \\label{cor:RCLT}\n$R:\\Delta\\to\\Z^+$ satisfies a nonstandard CLT \non $(\\Delta,f_\\Delta,\\mu_\\Delta)$. That is, \n$(n\\log n)^{-1/2}(\\sum_{j=0}^{n-1}R\\circ f_\\Delta^j-n\\bar R)\\to_d N(0,\\sigma_R^2)$ where\n$\\bar R=\\int_\\Delta R\\,d\\mu_\\Delta$.\n\\end{cor}\n\nIn this section, we consider decay of correlations for the geodesic flow $g_t:M\\to M$ and the global Poincar\\'e map $g:\\Sigma\\to\\Sigma$.\nThe general setup is the same as in~\\cite[Section~7]{LMM24}.\nWe recall the basics here.\\footnote{We caution that for example $\\tau$ here was called $\\sigma$ in~\\cite{LMM24}.}\n\\begin{enumerate}[$\\bullet$]\n\\item The flow $g_t$ and map $g$ are related by the identity\n$g=g_h$ where $h:\\Sigma\\to\\R^+$ is a H\\\"older roof function with $0<\\inf h<\\sup h<\\infty$.\n\\item \nThe first return maps $g:\\Sigma\\to\\Sigma$ and $f:\\Sigma_0\\to\\Sigma_0$ are related by $f=g^R$ where $R:\\Sigma_0\\to\\Z^+$ is a discrete first return time.\n\\item \nThe uniformly hyperbolic first return map $f:\\Sigma_0\\to\\Sigma_0$ is modelled by a Young tower with exponential tails~\\cite{Young98}: there is a subset $Y\\subset\\Sigma_0$ with product structure and bounded distortion, a probability measure $\\mu_Y$ on $Y$, and a ``good'' return time $\\tau:Y\\to\\Z^+$ constant along stable leaves with $\\mu_Y(\\tau>n)=O(e^{-cn})$ for some $c>0$, such that $f^\\tau:Y\\to Y$ is measure-preserving. Moreover, quotienting along stable leaves yields a full-branch Gibbs-Markov map $F:Z\\to Z$ as in Section~\\ref{sec:CLT}, and $\\tau$ is constant on the associated partition elements.\n\\item \nThe global Poincar\\'e map $g:\\Sigma\\to\\Sigma$ is modelled by a subexponential Young tower over $g^{\\varphi_*}=(g^R)^\\tau=f^\\tau:Y\\to Y$ where the return time $\\varphi_*:Y\\to\\Z^+$ is given by $\\varphi_*=\\sum_{\\ell=0}^{\\tau-1}R\\circ f^\\ell$.\n\\item \nWe can regard $f^\\tau:Y\\to Y$ as a (non-first return) Poincar\\'e map for the flow $g_t$ with roof function $\\varphi=h_{\\varphi_*}=\\sum_{\\ell=0}^{\\varphi_*-1}h\\circ g^\\ell$.\n\\end{enumerate}\n\n\\begin{equation} \\label{eq:alpha}\n\\alpha_0=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,0)\\,ds\\,d\\theta, \\qquad\n\\alpha_\\pi=\\frac{1}{2L}\\int_{\\bbS^1}\\int_{-L}^L v(s,\\theta,\\pi)\\,ds\\,d\\theta.\n\\end{equation}\n\n\\begin{ltheorem}[Nonstandard/Standard CLT] \\label{thm:CLT}\nIf $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,\n$(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)$ as $t\\to\\infty$,\nwhere $\\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.\n\nIf $I_v=0$, then there exists $\\sigma^2\\ge0$, typically nonzero, such that\n$v$ satisfies a standard CLT. That is,\n$t^{-1/2}v_t \\to_d N(0,\\sigma^2)$ as $t\\to\\infty$.\n\\end{ltheorem}\n\n\\begin{ltheorem} \\label{thm:decay}\n Let $v,\\,w:M\\to\\R$ be sufficiently smooth observables. Then there is a constant\n$C>0$ such that \n$$\n\\left|\\int_M v \\cdot (w\\circ g_t)\\,d\\mu-\\int_Mv\\,d\\mu\\int_Mw\\,d\\mu\\right|\n\\leq C t^{-1}\n\\quad\\text{for all $t>0$}.\n$$\n\\end{ltheorem}",
    "post_theorem_intro_text_len": 4924,
    "post_theorem_intro_text": "\\begin{rmk} As in~\\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\\\"older observables $v$ with $\\int_M v\\,d\\mu=I_v=0$. See for example the discussion in~\\cite[End of Section~4]{HM07}.\n\nWe note that the vanishing of $\\int_Mv\\,d\\mu$ is ``without loss of generality'' since one can always centre the observable $v$ by considering $v-\\int_Mv\\,d\\mu$.\nThe vanishing of $I_v$ is ``codimension two'' since $\\alpha_0=\\alpha_\\pi=0$\nmay occur in generic two-parameter families. In contrast, the degenerate case $\\sigma^2=0$ \nrequires infinitely many coincidences beyond the constraint $I_v=0$.\n\\end{rmk}\n\nIn the case $I_v>0$, we also prove a nonstandard WIP (which implies the CLT).\nDefine the sequence of continuous functions \n\\[\nW_n\\in C[0,1], \\qquad W_n(t)=(n\\log n)^{-1/2}v_{nt},\\quad n\\ge1.\n\\]\nSince each function $W_n$ depends on the initial condition in the probability space $(M,\\mu)$, we\ncan regard $W_n$ as a random element in the metric space $(C[0,1],\\|\\;\\|_\\infty)$.\n\n\\begin{ltheorem}[Nonstandard WIP] \\label{thm:WIP}\nIf $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,\n$W_n \\to_w W$ in $C[0,1]$ as $n\\to\\infty$,\nwhere $W$ is a Brownian motion with variance $\\sigma_v^2$. \n\\end{ltheorem}\n\nThere are only a few situations in deterministic dynamical systems where \nthe nonstandard CLT/WIP is known to hold.\nThe first example was for intermittent maps $f$ satisfying $f(x)=x+bx^{3/2}$ at a neutral fixed point $x=0$; see~\\cite{Gouezel04} for the nonstandard CLT and~\\cite{DedeckerMerlevede09} for the nonstandard WIP.\nFor various billiard examples, namely the infinite horizon Lorentz gas, Bunimovich stadium and billiards with cusps, we refer to~\\cite{BalintChernovDolgopyat11,BalintGouezel06,LMMsub,SzaszVarju07}.\n\nOur proofs of Theorems~\\ref{thm:CLT} and~\\ref{thm:WIP} closely follows abstract arguments in~\\cite{BalintChernovDolgopyat11} and~\\cite{LMMsub}, using the latter to reduce to a nonstandard CLT for a simpler class of observables, and using the former to establish this simplified nonstandard CLT.\n\nOur final main result concerns the rate of decay of correlations for the flow $g_t$.\n\n\\begin{ltheorem} \\label{thm:decay}\n Let $v,\\,w:M\\to{\\mathbb R}$ be sufficiently smooth observables. Then there is a constant\n$C>0$ such that \n$$\n\\left|\\int_M v \\cdot (w\\circ g_t)\\,d\\mu-\\int_Mv\\,d\\mu\\int_Mw\\,d\\mu\\right|\n\\leq C t^{-1}\n\\quad\\text{for all $t>0$}.\n$$\n\\end{ltheorem}\n\n\\begin{rmk} An observable is ``sufficiently smooth'' if it is $C^k$ in the flow direction for some $k\\ge0$ that depends only on the surface $S$. In \nparticular, if we choose the surface $S$ to be $C^\\infty$ away from ${\\mathcal C}$, then any observable that is $C^\\infty$ on $M$ and flat near $\\{s=\\pm L\\}$ is sufficiently smooth.\n\nThe analogous result for the infinite horizon Lorentz gas and Bunimovich stadia  was obtained in~\\cite{BBM19}.\n\\end{rmk}\n\n\\begin{rmk} \\label{rmk:lower}\n The upper bound in Theorem~\\ref{thm:decay} is sharp in the sense that if $v$ is H\\\"older with $\\int_Mv\\,d\\mu=0$ and $I_v\\neq0$,  then \n\\[\nt\\int_M v \\cdot (v\\circ g_t)\\,d\\mu\\not\\to0\n\\quad\\text{as $t\\to\\infty$}.\n\\]\nThis is a consequence of the nonstandard limit law in Theorem~\\ref{thm:CLT}.\nSee~\\cite[Corollary~1.3]{BalintGouezel06} or~\\cite[Proposition~9.14]{BBM19}.\n\\end{rmk}\n\nThe remainder of this paper is organised as follows.\nIn Section \\ref{sec:surfaces}, we review known facts about the geometry and dynamics of \ngeodesic flows on nonpositively curved surfaces, with special attention to surfaces with flat cylinder.\nIn Section \\ref{sec:R-dynamics}, we make a systematic study of the dynamics of the geodesic flow on the surface of revolution ${\\mathcal R}$.\nIn Section~\\ref{sec:GW}, we prove sufficient regularity of the stable/unstable foliations\ninduced by the Green bundles, partly extending~\\cite{GerberWilkinson99}.\nIn Section~\\ref{sec:f}, we construct a uniformly hyperbolic first return map $f:\\Sigma_0\\to\\Sigma_0$ satisfying the Chernov axioms~\\cite{Chernov-1999}, showing that $f$ is modelled by a Young tower with exponential tails~\\cite{Young98}.\n\nIn Section~\\ref{sec:CLT}, we recall and slightly extend an abstract result on the nonstandard CLT due to~\\cite{BalintChernovDolgopyat11}.\nIn Section~\\ref{sec:WIP}, we prove Theorems~\\ref{thm:CLT} and~\\ref{thm:WIP}.\nFinally, in Section~\\ref{sec:decay} we prove Theorem~\\ref{thm:decay} as well as a related result, Theorem~\\ref{thm:decaymap}, for a suitable global Poincar\\'e map $g$.\nThe Chernov axioms are stated for convenience in Appendix~\\ref{app:C}.\n\n\\vspace{-2ex}\n\\paragraph{Notation}\nWe use ``big O'' and $\\ll$ notation interchangeably, writing $a_n=O(b_n)$ or $a_n\\ll b_n$\nif there are constants $C>0$, $n_0\\ge1$ such that\n$a_n\\le Cb_n$ for all $n\\ge n_0$.\nWe write $a_n\\approx b_n$ if $a_n\\ll b_n$ and $b_n\\ll a_n$, and \n$a_n\\sim b_n$ if $a_n/b_n\\to1$.",
    "sketch": "Our proofs of Theorems~\\ref{thm:CLT} and~\\ref{thm:WIP} \\emph{closely follows abstract arguments} in~\\cite{BalintChernovDolgopyat11} and~\\cite{LMMsub}: using~\\cite{LMMsub} to \\emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\\cite{BalintChernovDolgopyat11} to \\emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove a nonstandard WIP (which implies the CLT), formulated via $W_n(t)=(n\\log n)^{-1/2}v_{nt}$ and convergence $W_n\\to_w W$ in $C[0,1]$ to Brownian motion with variance $\\sigma_v^2$.",
    "expanded_sketch": "Our proofs of To prove the main theorem,  and\n\\begin{ltheorem}[Nonstandard WIP] \\label{thm:WIP}\nIf $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,\n$W_n \\to_w W$ in $C[0,1]$ as $n\\to\\infty$,\nwhere $W$ is a Brownian motion with variance $\\sigma_v^2$. \n\\end{ltheorem}\n\\emph{closely follows abstract arguments} in~\\cite{BalintChernovDolgopyat11} and~\\cite{LMMsub}: using~\\cite{LMMsub} to \\emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\\cite{BalintChernovDolgopyat11} to \\emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove the preceding theorem (which implies the CLT), formulated via $W_n(t)=(n\\log n)^{-1/2}v_{nt}$ and convergence $W_n\\to_w W$ in $C[0,1]$ to Brownian motion with variance $\\sigma_v^2$.",
    "expanded_theorem": "[Nonstandard/Standard CLT] \\label{thm:CLT}\nIf $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,\n$(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)$ as $t\\to\\infty$,\nwhere $\\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.\n\nIf $I_v=0$, then there exists $\\sigma^2\\ge0$, typically nonzero, such that\n$v$ satisfies a standard CLT. That is,\n$t^{-1/2}v_t \\to_d N(0,\\sigma^2)$ as $t\\to\\infty$.",
    "theorem_type": [
      "Implication",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $S$ be a closed $C^4$ Riemannian surface of nonpositive curvature obtained by attaching negatively curved boundary pieces to a flat cylinder ${\\mathcal C}=[-L,L]\\times \\mathbb S^1$, where $0<L<\\varepsilon_0$, $r\\in[5,\\infty)$, and the cylinder comes from a surface of revolution with profile $\\xi(s)=1$ for $|s|\\le L$ and $\\xi(s)=1+(|s|-L)^r$ for $L\\le |s|\\le \\varepsilon_0$. Let $g_t:M\\to M$ be the geodesic flow on the unit tangent bundle $M=T^1S$, let $\\mu$ be the normalised Riemannian volume on $M$, and let $v:M\\to\\mathbb R$ be a H\\\"older observable with $\\int_M v\\,d\\mu=0$. For $t>0$, define\n\\[\nv_t=\\int_0^t v\\circ g_s\\,ds.\n\\]\nThe flat cylinder is foliated by two families of periodic geodesics (clockwise and counterclockwise), and define\n\\[\n\\alpha_0=\\frac1{2L}\\int_{\\mathbb S^1}\\int_{-L}^L v(s,\\theta,0)\\,ds\\,d\\theta,\n\\qquad\n\\alpha_\\pi=\\frac1{2L}\\int_{\\mathbb S^1}\\int_{-L}^L v(s,\\theta,\\pi)\\,ds\\,d\\theta,\n\\]\nwith\n\\[\nI_v=\\alpha_0^2+\\alpha_\\pi^2.\n\\]\nWhich conclusion about the asymptotic distribution of $v_t$ holds under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem: \n\\[\n(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)\\quad\\text{as }t\\to\\infty,\n\\]\nwhere $\\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\\sigma^2\\ge 0$, typically nonzero, such that $v$ satisfies the standard central limit theorem:\n\\[\nt^{-1/2}v_t \\to_d N(0,\\sigma^2)\\quad\\text{as }t\\to\\infty.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:\n\\[\n(t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)\\quad\\text{as }t\\to\\infty,\n\\]\nwhere $\\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\\sigma^2\\ge 0$ such that\n\\[\n(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma^2)\\quad\\text{as }t\\to\\infty.\n\\]"
        },
        {
          "label": "C",
          "text": "If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:\n\\[\n(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)\\quad\\text{as }t\\to\\infty,\n\\]\nfor some variance $\\sigma_v^2>0$. If $I_v=0$, then there exists $\\sigma^2\\ge 0$ such that\n\\[\nt^{-1/2}v_t \\to_d N(0,\\sigma^2)\\quad\\text{as }t\\to\\infty.\n\\]"
        },
        {
          "label": "D",
          "text": "There exists a positive constant $b>0$, independent of the surface $S$ and the observable $v$, such that if $I_v>0$, then\n\\[\n(t\\log t)^{-1/2}v_t \\to_d N(0,b I_v)\\quad\\text{as }t\\to\\infty.\n\\]\nIf $I_v=0$, then $v$ still satisfies the standard central limit theorem\n\\[\nt^{-1/2}v_t \\to_d N(0,\\sigma^2)\\quad\\text{as }t\\to\\infty\n\\]\nfor some $\\sigma^2\\ge 0$."
        },
        {
          "label": "E",
          "text": "If $I_v>0$, then $v$ satisfies the stronger functional limit law\n\\[\n(t\\log t)^{-1/2}v_t \\to_d N(0,\\sigma_v^2)\\quad\\text{as }t\\to\\infty,\n\\]\nwith $\\sigma_v^2=b_SI_v>0$, and if $I_v=0$, then necessarily $\\sigma^2=0$, so that\n\\[\nt^{-1/2}v_t \\to_d 0\\quad\\text{as }t\\to\\infty.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "normalisation split between the cases $I_v>0$ and $I_v=0$",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "explicit identification $\\sigma_v^2=b_S I_v$ with $b_S$ depending only on $S$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of the variance constant on the surface $S$",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "the $I_v=0$ case allows arbitrary $\\sigma^2\\ge 0$, typically nonzero",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or the correct normalization/variance formula. It introduces the geometric setup and the quantity I_v, but the actual limit law must still be selected from the options."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the hypotheses are spelled out in full and the task is to identify the theorem’s conclusion. It is very close to a restatement rather than an application or inference problem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some discrimination required among subtly different conclusions (normalization, dependence on S, and the I_v=0 case), especially because one distractor is a weaker true-looking statement. However, the item mainly tests precise recall of a known result rather than genuine generative reasoning from the given data."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: swapping the standard/nonstandard scalings, weakening the variance identification, asserting incorrect uniformity in constants, or forcing degeneracy in the I_v=0 case."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed theorem-recall MCQ with strong distractors and no direct answer leakage, but it is largely tautological and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.11246v1",
    "paper_link": "http://arxiv.org/abs/2512.11246v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$.  The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.",
      "start_pos": 8094,
      "end_pos": 8289,
      "label": "t:mainthm1"
    },
    "ref_dict": {
      "c:OT_PCF": "\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}"
    },
    "pre_theorem_intro_text_len": 1275,
    "pre_theorem_intro_text": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}.  A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}.  In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense.  Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton.  It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}).  In this work we confirm some aspects of this conjecture.  \n\nThe first main result is to establish the global existence of the flow:",
    "context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}",
    "full_context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nThe first main result is to establish the global existence of the flow:\n\nThe expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is\n\\begin{align}\\label{e:normalized equation}\n \\omega^{a,b}_h(t) = \\sum_{i=1}^s \\sqrt{-1}((1-e^{-t})\\frac{3}{4}+e^{-t}a_i)\\frac{1}{(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i + \\sqrt{-1}e^{-t}b_i\\operatorname{Im}w_i dz_i\\wedge d\\bar{z}_i.\n\\end{align}\nLater, we shall denote the time-dependent normalized model metric starting with $\\omega^{a,b}_h$ as $\\omega^{a,b}_h(t)$. We shall write $\\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\\leq i\\leq s$. \nObserve that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\\leq C$. Moreover, it follows that\n\\begin{align*}\n \\frac{\\omega_h^{a,b}(t)}{t+1}\\to \\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i,\n\\end{align*}\nwhich can be considered as a degenerate metric on $X(K, U)$. As explained in \\cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:\n\\begin{thm} \\label{t:homogOT} (\\cite{fusi2024pluriclosed}) Let $\\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\\omega_0$ converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{thm}\n\n\\noindent It is reasonable to conjecture that this behavior holds in the general case:\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nWe shall prove the first part of the conjecture in Section \\ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.\n\\begin{prop}\\label{p:GH convergence argument}\n Let $\\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that \n \\begin{itemize}\n \\item $C^{-1}\\leq \\operatorname{tr}_{\\omega(t)}\\omega_h(t)\\leq C$,\n \\item $\\lim\\limits_{t\\to\\infty}\\operatorname{tr}_{\\omega(t)}\\omega_h(t) = 1$ for all $x\\in X(K,U)$.\n \\end{itemize}\n Then, we have\n \\begin{align*}\n (X(K,U),\\omega(t))\\to (\\mathbb{T}^s,d(K, U)),\n\\end{align*}\nto the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.\n\\end{prop}\n\\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \\cite{fusi2024pluriclosed}, \\cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:\n \\begin{align*}\n F:X(K,U)\\to \\mathbb{T}^s\n \\end{align*}\n where the fiber is diffeomorphic to $\\mathbb{T}^{3s}$. In particular, $E_\\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\\{z\\}\\times \\mathbb{C}^s$ is dense in the $\\mathbb{T}^{3s}$ fiber (See Section 2 of \\cite{VerbiOT}), the degenerate metric $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$ will induce a metric on the base $\\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\\mathbb{T}^s$ induced from $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$, which is flat.\n\nSecondly, when $s = 1$, then $E_\\mathbb{C}$ and $E_\\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\\dot{\\phi}$.\n\\begin{lemma}\\label{l:potential derivative lower bound}\n On Inoue surface $S_M$, for the potential $\\phi$, there exists a constant $C>0$, such that:\n \\begin{align*}\n \\dot{\\phi}\\geq -C.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Note $s=1$. Choose a large constant $\\Lambda>0$, consider the quantity $\\dot{\\phi} + \\phi+\\frac{1}{\\Lambda}\\phi$. Then\n \\begin{align*}\n (\\frac{\\partial}{\\partial t} - \\Delta)(\\dot{\\phi}+\\phi+\\frac{1}{\\Lambda}\\phi) = 1 +\\frac{1}{\\Lambda}\\dot{\\phi}+ \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C} - \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{H}}g^0_\\mathbb{H} - (\\frac{1}{\\Lambda}(1-e^{-t})+1)\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}.\n \\end{align*}\n Now, for $A>0$, a very large constant, such that $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi < -A$. By \\eqref{e:potential eq}, notice that in this case $s=1$, we know:\n \\begin{align*}\n \\dot{\\phi} +\\phi = \\log\\frac{\\det g_\\mathbb{C}/c_1}{\\det e^{-t}h_\\mathbb{C}} - \\log\\frac{\\det g_\\mathbb{H}}{\\det \\frac{3}{4}h_\\mathbb{H}} = -\\log(c_1e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}) + \\log (\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}).\n \\end{align*}\n By Corollary \\ref{c:Inoue C0}, in particular $|\\phi|\\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\\dot{\\phi}\\leq -A'$, and \n \\begin{align*}\n \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}e^{-t}c_1 \\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}c_0e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C},\n \\end{align*}\n for some constant $c_0$, only depends on the initial metric.\n By choosing $\\Lambda$ large, such that $1-\\frac{C_0}{\\Lambda}>\\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\\frac{C_0}{\\Lambda}-A} \\leq \\frac{1}{\\Lambda}$. Thus, choose an $A = \\frac{\\Lambda}{2}$ large enough, such that $\\dot{\\phi}(t)\\geq -\\frac{1}{2}A$, for $t\\in[0,T]$, where $T$ is large enough. Then for the first time $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi = -A$, we have:\n \\begin{align*}\n (\\frac{\\partial}{\\partial t}-\\Delta)(\\dot{\\phi} + \\phi +\\frac{1}{\\Lambda}\\phi)&\\geq (\\frac{1}{\\Lambda } - \\frac{C_1e^{-t}}{\\Lambda}-c_0e^{\\frac{C_0}{\\Lambda}-A})e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C} + 1+ \\frac{1}{\\Lambda}\\dot{\\phi}\\\\\n &>0.\n \\end{align*}\n By the maximum principle, the result follows.\n\\end{proof}\nCombine with the potential derivative upper bound (Lemma \\ref{l:potential derivative upper bound}), we have the following metric lower bound.\n\\begin{thm}\\label{t:Inoue metric lower bound}\n On an Inoue surface $S_M$, for generalized K\\\"ahler metric $\\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\\omega(t)$with initial data $\\gw_0$ will satisfy\n \\begin{align*}\n \\omega(t)\\geq C\\omega_h(t),\n \\end{align*}\n where $\\omega_h(t)$ is the model flow.\n\\end{thm}\n\\begin{proof}\n By Lemma \\ref{l:potential derivative upper bound} and Lemma \\ref{l:potential derivative lower bound}, we have that:\n \\begin{align*}\n -C\\leq\\dot{\\phi} +\\phi \\leq C\n \\end{align*}\n for some constant $C$. Now, from the equation \\eqref{e:potential eq} and $s=1$, we have:\n \\begin{align*}\n -C\\leq \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}/\\operatorname{tr}_{g_\\mathbb{C}}e^{-t}h_\\mathbb{C}\\leq C.\n \\end{align*}\n Note that in the normalized pluriclosed flow model case, \\eqref{e:normalized equation}, the $E_\\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\\mathbb{H}$ part will be equivalent to $h_\\mathbb{H}$. Thus, by Corollary \\ref{c:lower z}, we have the desired lower bound for $E_\\mathbb{C}$.\n\\end{proof}\n\\begin{proof} [Proof of Theorem \\ref{t:mainthm2}]\nCombine Lemma \\ref{l:C0 bound}, Corollary \\ref{c:imporved C0}, Lemma \\ref{l:potential derivative upper bound}, Corollary \\ref{c:Inoue C0}, Lemma \\ref{l:potential derivative lower bound}, and Theorem \\ref{t:Inoue metric lower bound}, the results follow.\n\\end{proof}",
    "post_theorem_intro_text_len": 1301,
    "post_theorem_intro_text": "\\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \\cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\\\"ahler-Ricci flow (GKRF) \\cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\\\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\\`ere type \\cite{StreetsSTB}.\n\n\\begin{thm} \\label{t:mainthm2} For generalized K\\\"ahler-Ricci flow on an Oeljeklaus-Toma manifold\n\\begin{enumerate}\n\\item The scalar potential $\\phi$ satisfies\n\\begin{align*}\n    - C \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item Assuming there exists a Tricerri-type metric in $[\\gw_0]$, we have\n\\begin{align*}\n    - C e^{-t}(1 + t) \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:\n\\begin{align*}\n    -C\\leq \\dot{\\phi}\\leq C.\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, we have:\n\\begin{align*}\n        \\omega(t)\\geq C\\omega_h(t),\n\\end{align*}\nwhere $\\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.\n\\end{enumerate}\n\\end{thm}",
    "sketch": "The proof of Theorem~\\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”",
    "expanded_sketch": "The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”",
    "expanded_theorem": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.,",
    "theorem_type": [
      "Universal",
      "Existence"
    ],
    "mcq": {
      "question": "Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\\omega_0$ satisfies $\\partial\\bar\\partial\\omega_0=0$)?",
      "correct_choice": {
        "label": "A",
        "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\\infty)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\\to\\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$."
        },
        {
          "label": "C",
          "text": "The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$."
        },
        {
          "label": "D",
          "text": "There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$."
        },
        {
          "label": "E",
          "text": "For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the global-in-time conclusion $[0,\\infty)$ to mere short-time existence on a maximal interval",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replaces infinite-time existence by a uniform manifold-dependent finite existence time",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the global-in-time conclusion outright; it only asks which existence statement is valid. The equivalence defining pluriclosed is background, not a hint toward the correct option."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: it asks for the correct existence statement attached to a specific geometric setting. However, it is not a pure verbatim restatement because the options vary in maximal-time behavior, quantifiers, and added convergence claims."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish global existence from merely local existence and from stronger unsupported claims. Still, the question mainly tests recognition of the theorem rather than substantial derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: finite-time maximal existence, merely local existence, incorrect quantifier dependence, and an overstrong convergence/uniqueness statement. They reflect common failure modes in interpreting PDE existence theorems."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward recall of a known result than genuine generative reasoning."
    }
  },
  {
    "id": "2512.11246v1",
    "paper_link": "http://arxiv.org/abs/2512.11246v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$.  The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.",
      "start_pos": 8094,
      "end_pos": 8289,
      "label": "t:mainthm1"
    },
    "ref_dict": {
      "c:OT_PCF": "\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}"
    },
    "pre_theorem_intro_text_len": 1275,
    "pre_theorem_intro_text": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}.  A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}.  In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense.  Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton.  It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}).  In this work we confirm some aspects of this conjecture.  \n\nThe first main result is to establish the global existence of the flow:",
    "context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}",
    "full_context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nThe first main result is to establish the global existence of the flow:\n\nThe expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is\n\\begin{align}\\label{e:normalized equation}\n \\omega^{a,b}_h(t) = \\sum_{i=1}^s \\sqrt{-1}((1-e^{-t})\\frac{3}{4}+e^{-t}a_i)\\frac{1}{(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i + \\sqrt{-1}e^{-t}b_i\\operatorname{Im}w_i dz_i\\wedge d\\bar{z}_i.\n\\end{align}\nLater, we shall denote the time-dependent normalized model metric starting with $\\omega^{a,b}_h$ as $\\omega^{a,b}_h(t)$. We shall write $\\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\\leq i\\leq s$. \nObserve that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\\leq C$. Moreover, it follows that\n\\begin{align*}\n \\frac{\\omega_h^{a,b}(t)}{t+1}\\to \\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i,\n\\end{align*}\nwhich can be considered as a degenerate metric on $X(K, U)$. As explained in \\cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:\n\\begin{thm} \\label{t:homogOT} (\\cite{fusi2024pluriclosed}) Let $\\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\\omega_0$ converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{thm}\n\n\\noindent It is reasonable to conjecture that this behavior holds in the general case:\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nWe shall prove the first part of the conjecture in Section \\ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.\n\\begin{prop}\\label{p:GH convergence argument}\n Let $\\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that \n \\begin{itemize}\n \\item $C^{-1}\\leq \\operatorname{tr}_{\\omega(t)}\\omega_h(t)\\leq C$,\n \\item $\\lim\\limits_{t\\to\\infty}\\operatorname{tr}_{\\omega(t)}\\omega_h(t) = 1$ for all $x\\in X(K,U)$.\n \\end{itemize}\n Then, we have\n \\begin{align*}\n (X(K,U),\\omega(t))\\to (\\mathbb{T}^s,d(K, U)),\n\\end{align*}\nto the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.\n\\end{prop}\n\\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \\cite{fusi2024pluriclosed}, \\cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:\n \\begin{align*}\n F:X(K,U)\\to \\mathbb{T}^s\n \\end{align*}\n where the fiber is diffeomorphic to $\\mathbb{T}^{3s}$. In particular, $E_\\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\\{z\\}\\times \\mathbb{C}^s$ is dense in the $\\mathbb{T}^{3s}$ fiber (See Section 2 of \\cite{VerbiOT}), the degenerate metric $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$ will induce a metric on the base $\\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\\mathbb{T}^s$ induced from $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$, which is flat.\n\nSecondly, when $s = 1$, then $E_\\mathbb{C}$ and $E_\\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\\dot{\\phi}$.\n\\begin{lemma}\\label{l:potential derivative lower bound}\n On Inoue surface $S_M$, for the potential $\\phi$, there exists a constant $C>0$, such that:\n \\begin{align*}\n \\dot{\\phi}\\geq -C.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Note $s=1$. Choose a large constant $\\Lambda>0$, consider the quantity $\\dot{\\phi} + \\phi+\\frac{1}{\\Lambda}\\phi$. Then\n \\begin{align*}\n (\\frac{\\partial}{\\partial t} - \\Delta)(\\dot{\\phi}+\\phi+\\frac{1}{\\Lambda}\\phi) = 1 +\\frac{1}{\\Lambda}\\dot{\\phi}+ \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C} - \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{H}}g^0_\\mathbb{H} - (\\frac{1}{\\Lambda}(1-e^{-t})+1)\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}.\n \\end{align*}\n Now, for $A>0$, a very large constant, such that $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi < -A$. By \\eqref{e:potential eq}, notice that in this case $s=1$, we know:\n \\begin{align*}\n \\dot{\\phi} +\\phi = \\log\\frac{\\det g_\\mathbb{C}/c_1}{\\det e^{-t}h_\\mathbb{C}} - \\log\\frac{\\det g_\\mathbb{H}}{\\det \\frac{3}{4}h_\\mathbb{H}} = -\\log(c_1e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}) + \\log (\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}).\n \\end{align*}\n By Corollary \\ref{c:Inoue C0}, in particular $|\\phi|\\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\\dot{\\phi}\\leq -A'$, and \n \\begin{align*}\n \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}e^{-t}c_1 \\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}c_0e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C},\n \\end{align*}\n for some constant $c_0$, only depends on the initial metric.\n By choosing $\\Lambda$ large, such that $1-\\frac{C_0}{\\Lambda}>\\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\\frac{C_0}{\\Lambda}-A} \\leq \\frac{1}{\\Lambda}$. Thus, choose an $A = \\frac{\\Lambda}{2}$ large enough, such that $\\dot{\\phi}(t)\\geq -\\frac{1}{2}A$, for $t\\in[0,T]$, where $T$ is large enough. Then for the first time $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi = -A$, we have:\n \\begin{align*}\n (\\frac{\\partial}{\\partial t}-\\Delta)(\\dot{\\phi} + \\phi +\\frac{1}{\\Lambda}\\phi)&\\geq (\\frac{1}{\\Lambda } - \\frac{C_1e^{-t}}{\\Lambda}-c_0e^{\\frac{C_0}{\\Lambda}-A})e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C} + 1+ \\frac{1}{\\Lambda}\\dot{\\phi}\\\\\n &>0.\n \\end{align*}\n By the maximum principle, the result follows.\n\\end{proof}\nCombine with the potential derivative upper bound (Lemma \\ref{l:potential derivative upper bound}), we have the following metric lower bound.\n\\begin{thm}\\label{t:Inoue metric lower bound}\n On an Inoue surface $S_M$, for generalized K\\\"ahler metric $\\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\\omega(t)$with initial data $\\gw_0$ will satisfy\n \\begin{align*}\n \\omega(t)\\geq C\\omega_h(t),\n \\end{align*}\n where $\\omega_h(t)$ is the model flow.\n\\end{thm}\n\\begin{proof}\n By Lemma \\ref{l:potential derivative upper bound} and Lemma \\ref{l:potential derivative lower bound}, we have that:\n \\begin{align*}\n -C\\leq\\dot{\\phi} +\\phi \\leq C\n \\end{align*}\n for some constant $C$. Now, from the equation \\eqref{e:potential eq} and $s=1$, we have:\n \\begin{align*}\n -C\\leq \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}/\\operatorname{tr}_{g_\\mathbb{C}}e^{-t}h_\\mathbb{C}\\leq C.\n \\end{align*}\n Note that in the normalized pluriclosed flow model case, \\eqref{e:normalized equation}, the $E_\\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\\mathbb{H}$ part will be equivalent to $h_\\mathbb{H}$. Thus, by Corollary \\ref{c:lower z}, we have the desired lower bound for $E_\\mathbb{C}$.\n\\end{proof}\n\\begin{proof} [Proof of Theorem \\ref{t:mainthm2}]\nCombine Lemma \\ref{l:C0 bound}, Corollary \\ref{c:imporved C0}, Lemma \\ref{l:potential derivative upper bound}, Corollary \\ref{c:Inoue C0}, Lemma \\ref{l:potential derivative lower bound}, and Theorem \\ref{t:Inoue metric lower bound}, the results follow.\n\\end{proof}",
    "post_theorem_intro_text_len": 1301,
    "post_theorem_intro_text": "\\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \\cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\\\"ahler-Ricci flow (GKRF) \\cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\\\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\\`ere type \\cite{StreetsSTB}.\n\n\\begin{thm} \\label{t:mainthm2} For generalized K\\\"ahler-Ricci flow on an Oeljeklaus-Toma manifold\n\\begin{enumerate}\n\\item The scalar potential $\\phi$ satisfies\n\\begin{align*}\n    - C \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item Assuming there exists a Tricerri-type metric in $[\\gw_0]$, we have\n\\begin{align*}\n    - C e^{-t}(1 + t) \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:\n\\begin{align*}\n    -C\\leq \\dot{\\phi}\\leq C.\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, we have:\n\\begin{align*}\n        \\omega(t)\\geq C\\omega_h(t),\n\\end{align*}\nwhere $\\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.\n\\end{enumerate}\n\\end{thm}",
    "sketch": "The proof of Theorem~\\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”",
    "expanded_sketch": "The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”",
    "expanded_theorem": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.,",
    "theorem_type": [
      "Universal",
      "Existence"
    ],
    "mcq": {
      "question": "Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\\omega_0$ satisfies $\\partial\\bar\\partial\\omega_0=0$)?",
      "correct_choice": {
        "label": "A",
        "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\\infty)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\\to\\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$."
        },
        {
          "label": "C",
          "text": "The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$."
        },
        {
          "label": "D",
          "text": "There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$."
        },
        {
          "label": "E",
          "text": "For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the global-in-time conclusion $[0,\\infty)$ to mere short-time existence on a maximal interval",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replaces infinite-time existence by a uniform manifold-dependent finite existence time",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct conclusion, and it does not contain strong linguistic cues that single out choice A. The correct answer must be identified from the mathematical content of the options."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a direct theorem-recall question: choice A is essentially the theorem-level statement. However, the presence of stronger, weaker, and proof-confusion alternatives makes it more than a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact asserted result from stronger claims (B, E) and weaker or altered existence statements (C, D). Still, the question primarily tests recognition of the theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: B is an overstrengthening, C is a weaker true statement, D alters the time range in a subtle way, and E confuses proof machinery with the theorem statement. These reflect realistic failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it mainly tests theorem recall/precision rather than deeper generative reasoning."
    }
  },
  {
    "id": "2512.11294v1",
    "paper_link": "http://arxiv.org/abs/2512.11294v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main_theorem}\n\tLet $u\\in C(0,T;L^2(\\Omega,\\mathbb{R}^N))\\cap L^q(0,T;W^{1,q}(\\Omega,\\mathbb{R}^N))\\cap L^\\infty(\\Omega_T)$ be a weak solution to (\\ref{eq}) in $\\Omega_T$. \n\tThere exist constants $\\ep_0=\\ep_0(\\mathit{data})$ and $c=c(\\mathit{data} , \\|a\\|_{L^\\infty(\\Om_T)})$ such that for every $Q_{4r}(z_0)\\subset\\Om_T$ with $r\\in (0,1)$ and $\\epsilon\\in(0,\\ep_0)$,\n\t\\begin{align*}\n    \\begin{split}\n        &\\iint_{Q_{r}(z_0)}H(z,|\\nabla u|)^{1+\\epsilon}\\,dz\\\\\n\t&\\le c\\left(\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^p+\\|a\\|_{L^\\infty(\\Om_T)}\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^q +1\\right)^{1+\\frac{q\\epsilon}{p}}\\\\\nint_{Q_{4r}(z_0)}H(z,|F|)^{1+\\epsilon} \\,dz\\right)^{1+\\frac{q}{2}}\\,.\n    \\end{split}\n\\end{align*}",
      "start_pos": 48270,
      "end_pos": 49009,
      "label": "main_theorem"
    },
    "ref_dict": {
      "lemma_decay": "\\begin{lemma}\\label{lemma_decay}\n   Suppose \\ref{p1} and \\ref{p2}. Then there exists a constant $\\cc=\\cc(\\data)\\ge K^\\frac{1}{p}$\n   such that\n   \\[\n   \\rho\\le \\cc \\la^{-1}\\,.\n   \\]\n\\end{lemma}",
      "p_est_vitali": "\\begin{proposition}\\label{p_est_vitali} \nSuppose \\ref{p1} and \\ref{p2}.\n\tThere exist constants $c=c(\\data)$ and $\\theta_0=\\theta_0(n,p)\\in(0,1)$, such that for any $\\theta\\in(\\theta_0,1)$ we have\n\t\\begin{align*}\n\t\t\t\\begin{split}\n\t\t\t    \\iint_{Q_{2\\cv\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\n\t\t\t&\\le c\\La^{1-\\theta}\\iint_{Q_{2\\rho}^\\la(z_0)\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz\\\\\n            &\\qquad+c\\iint_{Q_{2\\rho}^\\la(z_0)\\cap \\Theta(c^{-1}\\La)}H(z,|F|) \\,dz\\,.\n\t\t\t\\end{split}\n\t\\end{align*}\n    Here, $\\kappa$ is defined in \\eqref{K_and_kappa} and it appears in \\ref{p4}.\n \\end{proposition}",
      "vitali_lem": "\\begin{lemma}\\label{vitali_lem}\n    Let $\\mathcal{F}$ be defined as above and $\\kappa$ is in \\eqref{K_and_kappa}, Then there exists a pairwise disjoint countable subcollection $\\mathcal{G}$ of $\\mathcal{F}$ such that for any $U(w)\\in \\mathcal{F}$, there exists $U(v)\\in \\mathcal{G}$ with\n    \\[\n    U(w)\\subset \\kappa U(v)\\,.\n    \\]\n\\end{lemma}",
      "ssec:sol": "\\begin{lemma}\\label{sobolev_lem}\n\tLet $B_{\\rho}(x_0)\\subset\\RR^n$, $\\sig,s,r\\in[1,\\infty)$ and $\\vartheta\\in(0,1)$ such that \n\t\\[\n\t\t-\\frac{n}{\\sig}\\le \\vartheta\\left(1-\\frac{n}{s}\\right)-(1-\\vartheta)\\frac{n}{r}\\,.\n\t\\]\n\tThen for every $v\\in W^{1,s}(B_{\\rho}(x_0))$ there exists a constant $c=c(n,\\sig)$ such that\n\t\\[\nnt_{B_{\\rho}(x_0)}\\frac{|v|^\\sig}{\\rho^\\sig}\\,dx\nnt_{B_{\\rho}(x_0)}\\frac{|v|^r}{\\rho^r}\\,dx\\right)^\\frac{(1-\\vartheta)\\sig}{r}\\,.\n\t\\]\n\\end{lemma}\n\n\\subsection{Solutions}\\label{ssec:sol}\nThe notion of weak solutions to \\eqref{eq} is defined as follows.\n\\begin{definition}\tA map $u:\\Om_T\\longrightarrow\\RR^N$ satisfying\n \\[\n\t\t\tu\\in C(0,T;L^2(\\Om,\\RR^N))\\cap L^q(0,T;W^{1,q}(\\Om,\\RR^N))\n\t\\]\nis a weak solution to \\eqref{eq}, if \nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\RR^N)$ it holds\n\\begin{align*}\n\\begin{split}\n    &\\iint_{\\Om_T}(-u\\cdot\\varphi_t+\\mA(z,\\na u)\\cdot \\na \\varphi)\\,dz\\\\\n    &=\\iint_{\\Om_T}(|F|^{p-2}F\\cdot\\na \\varphi+a(z)|F|^{p-2}F\\cdot \\na\\varphi)\\,dz\\,.\n\\end{split}\n\\end{align*}\n\\end{definition}\n\n We note that to our best knowledge that despite the existence of weak solutions to~\\eqref{eq} is expected in the full scope of our current inhnomogeneous in time and space study~\\eqref{range_pq}, it is not yet established. We refer to~\\cite{cgzg,C-b} for existence to problems with more general growth, but covering probably not sharp regularity with respect to the time variable, and \\cite{bgs-discontinuous} for the existence to related problems, that do not fully embrace our case, but remarkably relax demanded time regularity.  At this place we shall stress that in the steady case, due to~\\cite{eslemi}, if for $u$ being a weak solution or a local minimizer to related variational functionals, $H(x,\\nabla u)\\in L^1$, $a\\in C^{0,\\alpha}$ and $p$ and $q$ satisfy some closeness condition governed by $\\alpha$, then $u\\in L^q$. Surprisingly, a counterpart of such a result in a parabolic setting is still missing. Nonetheless, we expect it holds true and we continue our work for $L^q(0,T;W^{1,q}(\\Om,\\RR^N))$-solutions. As for the parabolic approaches under the natural energy regime  $H(z,\\nabla u)\\in L^1$, let us refer to~\\cite{KIM2025110738},\n\n\\section{Energy estimates}\\label{sec:energy-estimates}\nIn this section, we provide energy estimates. The first estimate is the Caccioppoli inequality.\n\\begin{lemma}\\label{caccio_lem}\n\tLet $u$ be a weak solution to \\eqref{eq} in $\\Omega_T$. Then for every $Q_{R,\\ell}(z_0)\\subset\\Omega_T$, with $R,\\ell>0$, and for $r\\in [R/2,R)$ and $\\tau^2\\in[\\ell^2/2^2,\\ell^2)$, there exists a constant $c=c(n,p,q,\\nu,L)$, such that\n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{r,\\tau}(z_0)}(|\\na u|^p+a(z)|\\na u|^q)\\,dz\\\\\nint_{Q_{R,\\ell}(z_0)}\\left(\\frac{|u-u_{Q_{R,\\ell}(z_0)}|^p}{(R-r)^p}+a(z)\\frac{|u-u_{Q_{R,\\ell}(z_0)}|^q}{(R-r)^q}\\right)\\,dz\\\\\nint_{Q_{R,\\ell}(z_0)} (|F|^p+a(z)|F|^q)  \\,dz\\,.\n\t\t\\end{split}\n\t\\end{align*}\n\\end{lemma}",
      "la_comp_lem": "\\begin{lemma}\\label{la_comp_lem}\n    Suppose $U(w), U(v)\\in \\mathcal{F}$ with $U(w)\\cap U(v)\\ne \\emptyset$ and $l_w\\le 2l_v$. Then for $\\cc\\ge 2$ defined in Lemma~\\ref{p_decay_lem}, we have\n    \\[\n    \\lav\\le (2(1+\\ca+10\\cc\\ca))^\\frac{1}{p}\\law\\,.\n    \\]\n    Moreover, if $U(v)$ is $(p,q)$-intrinsic, then we also have\n    \\[\n    \\law\\le (2(1+\\ca+10\\cc\\ca))^\\frac{1}{p}\\lav\\,.\n    \\]\n\\end{lemma}",
      "stopping time argument": "\\begin{split}\n             \\iint_{G_{2\\cv\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\n         &\\le 2\\kappa^{n+2}c\\La^{1-\\theta}\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Psi((4c)^{-1/\\theta}\\La)}H(z,|\\na u|)^{\\theta }\\,dz\\\\\n         &\\qquad+2\\kappa^{n+2}c\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Theta((4c)^{-1}\\La)}H(z,|F|) \\,dz\\,.\n         \\end{split}\n     \\end{align*}\n     Replacing the constant $c$ above with $(4\\kappa^{n+2}c)^\\frac{1}{\\theta_0}$, the proof is completed.\n\\end{proof}\n\n\\section{Proof of the main result}\\label{sec:main-proof}\n\\subsection{Stopping time argument}\\label{stopping time argument}\nIn this section, we prove that conditions \\ref{p1}-\\ref{p2} and \\ref{q1}-\\ref{q2} are satisfied under our regime. First of all, for any $\\rho>0$ and $\\hbar>1$, we denote the constant in Lemma~\\ref{caccio_lem} for $r=\\rho$, $R=2\\rho$, $\\tau^2=\\hbar^{2-p}\\rho^2$ and $\\ell^2=\\hbar^{2-p}(2\\rho)^2$ by\n\\begin{align}\\label{def_const_caccio}\n    \\ccc=\\ccc(n,p,q,\\nu,L)\\,.\n\\end{align}\nLet $r\\in(0,1)$ and suppose $Q_{4r}(z_0)\\subset\\Om_T$. We define $\\la_0$ and $\\La_0$ as\n\\begin{align}\\label{def_la}\n\t\t\\begin{split}\n\t\t    \\la^p_0\nint_{Q_{4r}(z_0)} H(z,|F|)\\,dz \\right)^\\frac{p}{2} +1\n\t\t\\end{split}",
      "q_reverse_lem": "\\begin{lemma}\\label{q_reverse_lem}\nSuppose \\ref{q1} and \\ref{q2}.\n\tThere exist constants $c=c(n,N,p,q,\\nu,L,\\ca)$ and $\\theta_0=\\theta_0(n,p,q)\\in(0,1)$ such that for any $\\theta\\in(\\theta_0,1)$ there holds\n\t\\begin{align*}\nint_{G_{\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\nint_{G_{2\\rho}^\\la(z_0)}H(z,|F|) \\,dz\\,.\n\t\\end{align*}\t\n\\end{lemma}",
      "eq": "\\begin{align}\\label{eq}\n\tu_t-\\dv\\mA(z,\\na u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}",
      "range_pq": "\\begin{align} \\label{range_pq}\n2\\le p <\\infty, \\quad p<q\\le p+\\alpha\\,.    \n\\end{align}",
      "p_reverse_lem": "\\begin{lemma}\\label{p_reverse_lem}\nSuppose \\ref{p1} and \\ref{p2}.\n\tThere exist constants $c=c(\\data)$ and $\\theta_0=\\theta_0(n,p)\\in(0,1)$ such that for any $\\theta\\in(\\theta_0,1)$ there holds\n\t\\begin{align*}\nint_{Q_{\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\nint_{Q_{2\\rho}^\\la(z_0)}H(z,|F|)\\,dz\\,.\n\t\\end{align*}\t\n\\end{lemma}",
      "main_theorem": "\\begin{theorem}\\label{main_theorem}\n\tLet $u\\in C(0,T;L^2(\\Om,\\RR^N))\\cap L^q(0,T;W^{1,q}(\\Om,\\RR^N))\\cap L^\\infty(\\Omega_T)$ be a weak solution to \\eqref{eq} in $\\Omega_T$. \n\tThere exist constants $\\ep_0=\\ep_0(\\mathit{data})$ and $c=c(\\mathit{data} , \\|a\\|_{L^\\infty(\\Om_T)})$ such that for every $Q_{4r}(z_0)\\subset\\Om_T$ with $r\\in (0,1)$ and $\\ep\\in(0,\\ep_0)$,\n\t\\begin{align*}\n    \\begin{split}\n        &\\iint_{Q_{r}(z_0)}H(z,|\\na u|)^{1+\\epsilon}\\,dz\\\\\n\t&\\le c\\left(\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^p+\\|a\\|_{L^\\infty(\\Om_T)}\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^q +1\\right)^{1+\\frac{q\\epsilon}{p}}\\\\\nint_{Q_{4r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\right)^{1+\\frac{q}{2}}\\,.\n    \\end{split}\n\\end{align*} \n\\end{theorem}",
      "q_est_vitali": "\\begin{proposition}\n    \\label{q_est_vitali}\nSuppose \\ref{q1} and \\ref{q2}.\n\tThere exist constants $\\theta_0=\\theta_0(n,p,q)\\in(0,1)$ and $c=c(n,N,p,q,\\nu,L,\\ca)$  such that for any $\\theta\\in(\\theta_0,1)$ we have\n\t\\begin{align*}\n\t\t\t\\begin{split}\n\t\t\t    \\iint_{G_{2\\cv\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\n\t\t\t&\\le c\\La^{1-\\theta}\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz \\\\\n            &\\qquad+c\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Theta(c^{-1}\\La)}H(z,|F|)\\,dz\\,.\n\t\t\t\\end{split}\n\t\\end{align*}\n    Here, $\\kappa$ is defined in \\eqref{K_and_kappa} and it appears in \\ref{q5},\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 4397,
    "pre_theorem_intro_text": "Parabolic equations with double-phase growth represent a natural parabolic counterpart of models describing materials with heterogeneous hardening, composite media, or diffusion processes with switching intensities. Mathematical description of them have attracted considerable attention over the past decade~\\cite{mira,C-b,H-book}. Even for strongly nonlinear problems, one typically expects solutions to exhibit regularity beyond that guaranteed by mere membership in the energy space; see, for instance,~\\cite{ElMe,Str,KiLe-Duke}. Recently, there has been remarkable progress in the regularity theory for parabolic double-phase problems, such as \\[u_t-\\dv(|\\nabla u|^{p-2}\\nabla u+a(x,t)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\\,. \\] In a short time, it has led to several deep and influential regularity results including~\\cite{WMS,WS,sen2025lipschitzregularityparabolicdouble,Sen2025,MR4926910,K-CZ,KKM,KIM2025110738, buskr,ssy}. Some results in a more refined framework are also available, \n see \\cite{oh2025gradient,MR4926910,MR4774133,Sen2025}. Nevertheless, the theory is still far from being fully understood.  \n\n In this work, we focus on gradient higher integrability, which plays a decisive role in the analysis of finer regularity of weak solutions. Solutions to double phase problems are expected be regular provided the closeness condition on the exponents is controlled by the regularity of the weight $a$, which broadens under a priori knowledge about the regularity of $u$. This is observed in the elliptic case~\\cite{comi-bdd,bacomi-cv,ok-na}, but its parabolic counterpart remains largely unexplored. We study the regularity to parabolic double phase problems for a priori bounded solutions, where the evolution brings deep complications absent in the elliptic situation. Although related results are available for problems of similar type~\\cite{BoDu-nonst,DiBFr,KiLe-Duke,KKM}, the structure of the system considered here prevents an application of the existing techniques. What is more, once we finished the first draft of our manuscript we learned about~\\cite{kim2025boundedsolutionsinterpolativegap} where the method cannot embrace the presence of the non-zero right-hand side.   Henceforth, we deliver a substantially new approach. \n\nLet us present in detail our result. We shall consider weak solutions to the parabolic double-phase system\n\\begin{align}\\label{eq}\n\tu_t-\\dv\\mathcal{A}(z,\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}\n where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here, \nwe assume $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the following structure assumptions: there exist constants $0<\\nu\\le L<\\infty$ such that\n\t\\begin{align}\\label{str}\n\t    \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge \\nu(|\\xi|^p+a(z)|\\xi|^q)\n        \\quad\\text{and}\\quad\n\t\t\t|\\mathcal{A}(z,\\xi)|\\le L(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}),\n\t\\end{align}\nfor almost every $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$. Throughout the rest of the paper, we use the notation for $z\\in\\Omega_T$ and $s\\geq 0$ \\begin{equation}\n    \\label{H-def}\nH(z,s)=s^p+a(z) s^q\\,.\n\\end{equation}\n\nWe focus of gradient higher integrability of the solutions in the spirit of \\cite{KiLe-Duke}. We prove it for a priori bounded $u$, $a \\geq 0$ and $a\\in \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$ and \n\\begin{align} \\label{range_pq}\n2\\le p <\\infty, \\quad p<q\\le p+\\alpha\\,.    \n\\end{align}\nHere $a\\in  \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exist constants $c_a>0$, such that\n\\begin{align}\n\t\\label{def_holder}\n\t\t|a(x,t)-a(y,s)|\\le c_a\\left(\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\\right)\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \n\nThe main result of this paper is the following higher integrability estimate for the gradient of a weak solution to (\\ref{eq}) assuming that the forcing term $H(z,|F|)\\in L^\\gamma$,\nwhere $\\gamma=\\tfrac{n+p}{p}$. We denote the constant $c=c(\\mathit{data})$ if $c$ depends on the following\n  \\begin{align*}\n  \\begin{split}\n      \\mathit{data}=&(n,N,p,q,\\nu,L,c_a,\\|u\\|_{L^\\infty(\\Om_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Om_T)})\\,.\n  \\end{split}\n\\end{align*} Now we state our main result.",
    "context": "Parabolic equations with double-phase growth represent a natural parabolic counterpart of models describing materials with heterogeneous hardening, composite media, or diffusion processes with switching intensities. Mathematical description of them have attracted considerable attention over the past decade~\\cite{mira,C-b,H-book}. Even for strongly nonlinear problems, one typically expects solutions to exhibit regularity beyond that guaranteed by mere membership in the energy space; see, for instance,~\\cite{ElMe,Str,KiLe-Duke}. Recently, there has been remarkable progress in the regularity theory for parabolic double-phase problems, such as \\[u_t-\\dv(|\\nabla u|^{p-2}\\nabla u+a(x,t)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\\,. \\] In a short time, it has led to several deep and influential regularity results including~\\cite{WMS,WS,sen2025lipschitzregularityparabolicdouble,Sen2025,MR4926910,K-CZ,KKM,KIM2025110738, buskr,ssy}. Some results in a more refined framework are also available, \n see \\cite{oh2025gradient,MR4926910,MR4774133,Sen2025}. Nevertheless, the theory is still far from being fully understood.\n\nIn this work, we focus on gradient higher integrability, which plays a decisive role in the analysis of finer regularity of weak solutions. Solutions to double phase problems are expected be regular provided the closeness condition on the exponents is controlled by the regularity of the weight $a$, which broadens under a priori knowledge about the regularity of $u$. This is observed in the elliptic case~\\cite{comi-bdd,bacomi-cv,ok-na}, but its parabolic counterpart remains largely unexplored. We study the regularity to parabolic double phase problems for a priori bounded solutions, where the evolution brings deep complications absent in the elliptic situation. Although related results are available for problems of similar type~\\cite{BoDu-nonst,DiBFr,KiLe-Duke,KKM}, the structure of the system considered here prevents an application of the existing techniques. What is more, once we finished the first draft of our manuscript we learned about~\\cite{kim2025boundedsolutionsinterpolativegap} where the method cannot embrace the presence of the non-zero right-hand side. Henceforth, we deliver a substantially new approach.\n\nLet us present in detail our result. We shall consider weak solutions to the parabolic double-phase system\n\\begin{align}\\label{eq}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}\n where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here, \nwe assume $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the following structure assumptions: there exist constants $0<\\nu\\le L<\\infty$ such that\n \\begin{align}\\label{str}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge \\nu(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad\n |\\mathcal{A}(z,\\xi)|\\le L(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}),\n \\end{align}\nfor almost every $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$. Throughout the rest of the paper, we use the notation for $z\\in\\Omega_T$ and $s\\geq 0$ \\begin{equation}\n \\label{H-def}\nH(z,s)=s^p+a(z) s^q\\,.\n\\end{equation}\n\nWe focus of gradient higher integrability of the solutions in the spirit of \\cite{KiLe-Duke}. We prove it for a priori bounded $u$, $a \\geq 0$ and $a\\in \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$ and \n\\begin{align} \\label{range_pq}\n2\\le p <\\infty, \\quad p<q\\le p+\\alpha\\,. \n\\end{align}\nHere $a\\in \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exist constants $c_a>0$, such that\n\\begin{align}\n \\label{def_holder}\n |a(x,t)-a(y,s)|\\le c_a\\left(\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\\right)\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$.\n\nThe main result of this paper is the following higher integrability estimate for the gradient of a weak solution to (\\ref{eq}) assuming that the forcing term $H(z,|F|)\\in L^\\gamma$,\nwhere $\\gamma=\\tfrac{n+p}{p}$. We denote the constant $c=c(\\mathit{data})$ if $c$ depends on the following\n \\begin{align*}\n \\begin{split}\n \\mathit{data}=&(n,N,p,q,\\nu,L,c_a,\\|u\\|_{L^\\infty(\\Om_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Om_T)})\\,.\n \\end{split}\n\\end{align*} Now we state our main result.\n\n\\begin{align}\\label{eq}\n\tu_t-\\dv\\mA(z,\\na u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}",
    "full_context": "Parabolic equations with double-phase growth represent a natural parabolic counterpart of models describing materials with heterogeneous hardening, composite media, or diffusion processes with switching intensities. Mathematical description of them have attracted considerable attention over the past decade~\\cite{mira,C-b,H-book}. Even for strongly nonlinear problems, one typically expects solutions to exhibit regularity beyond that guaranteed by mere membership in the energy space; see, for instance,~\\cite{ElMe,Str,KiLe-Duke}. Recently, there has been remarkable progress in the regularity theory for parabolic double-phase problems, such as \\[u_t-\\dv(|\\nabla u|^{p-2}\\nabla u+a(x,t)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\\,. \\] In a short time, it has led to several deep and influential regularity results including~\\cite{WMS,WS,sen2025lipschitzregularityparabolicdouble,Sen2025,MR4926910,K-CZ,KKM,KIM2025110738, buskr,ssy}. Some results in a more refined framework are also available, \n see \\cite{oh2025gradient,MR4926910,MR4774133,Sen2025}. Nevertheless, the theory is still far from being fully understood.\n\nIn this work, we focus on gradient higher integrability, which plays a decisive role in the analysis of finer regularity of weak solutions. Solutions to double phase problems are expected be regular provided the closeness condition on the exponents is controlled by the regularity of the weight $a$, which broadens under a priori knowledge about the regularity of $u$. This is observed in the elliptic case~\\cite{comi-bdd,bacomi-cv,ok-na}, but its parabolic counterpart remains largely unexplored. We study the regularity to parabolic double phase problems for a priori bounded solutions, where the evolution brings deep complications absent in the elliptic situation. Although related results are available for problems of similar type~\\cite{BoDu-nonst,DiBFr,KiLe-Duke,KKM}, the structure of the system considered here prevents an application of the existing techniques. What is more, once we finished the first draft of our manuscript we learned about~\\cite{kim2025boundedsolutionsinterpolativegap} where the method cannot embrace the presence of the non-zero right-hand side. Henceforth, we deliver a substantially new approach.\n\nLet us present in detail our result. We shall consider weak solutions to the parabolic double-phase system\n\\begin{align}\\label{eq}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}\n where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here, \nwe assume $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the following structure assumptions: there exist constants $0<\\nu\\le L<\\infty$ such that\n \\begin{align}\\label{str}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge \\nu(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad\n |\\mathcal{A}(z,\\xi)|\\le L(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}),\n \\end{align}\nfor almost every $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$. Throughout the rest of the paper, we use the notation for $z\\in\\Omega_T$ and $s\\geq 0$ \\begin{equation}\n \\label{H-def}\nH(z,s)=s^p+a(z) s^q\\,.\n\\end{equation}\n\nWe focus of gradient higher integrability of the solutions in the spirit of \\cite{KiLe-Duke}. We prove it for a priori bounded $u$, $a \\geq 0$ and $a\\in \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$ and \n\\begin{align} \\label{range_pq}\n2\\le p <\\infty, \\quad p<q\\le p+\\alpha\\,. \n\\end{align}\nHere $a\\in \\mathcal{C}^{\\alpha,\\frac\\alpha 2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exist constants $c_a>0$, such that\n\\begin{align}\n \\label{def_holder}\n |a(x,t)-a(y,s)|\\le c_a\\left(\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\\right)\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$.\n\nThe main result of this paper is the following higher integrability estimate for the gradient of a weak solution to (\\ref{eq}) assuming that the forcing term $H(z,|F|)\\in L^\\gamma$,\nwhere $\\gamma=\\tfrac{n+p}{p}$. We denote the constant $c=c(\\mathit{data})$ if $c$ depends on the following\n \\begin{align*}\n \\begin{split}\n \\mathit{data}=&(n,N,p,q,\\nu,L,c_a,\\|u\\|_{L^\\infty(\\Om_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Om_T)})\\,.\n \\end{split}\n\\end{align*} Now we state our main result.\n\n\\begin{align}\\label{eq}\n\tu_t-\\dv\\mA(z,\\na u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}\n\n\\subsection{Proof of Theorem~\\ref{main_theorem}}\nRecall we have set $r\\le r_1<r_2\\le 2r$ with $r\\in(0,1)$.\nFor selected subcollection $\\mathcal{G}$ in Lemma~\\ref{vitali_lem}, we simply write\n\\[\n \\mathcal{G}=\\{ \\mQ_i \\}_{1\\le i \\le \\infty}\\,,\n\\]\nwhere $\\mQ_i=\\mQ(w_i)$ for some $w_i\\in \\Psi(\\La,r_1)$. Depending on the intrinsic geometry of $\\mQ_i\\in \\mathcal{G}$, it follows by Proposition~\\ref{p_est_vitali} and Proposition~\\ref{q_est_vitali} that for any $i\\in \\mathbb{N}$\n\\begin{align*}\n \\begin{split}\n \\iint_{\\cv\\mQ _{i}}H(z,|\\na u|)\\,dz\n &\\le c\\La^{1-\\theta}\\iint_{\\mQ _i\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad + c\\iint_{U_i\\cap \\Theta(c^{-1} \\la) } H(z,|F|)\\,dz\\,,\n \\end{split}\n\\end{align*}\nwhere $c=c(\\data)>1$ is a constant and fixed $\\theta\\in (0,1)$.\nEmploying disjointedness and covering property in Lemma~\\ref{vitali_lem}, we get\n\\begin{align*}\n \\begin{split}\n \\iint_{\\Psi(\\La,r_1)}H(z,|\\na u|)\\,dz\n&\\le\\sum_{i=1}^\\infty\\iint_{\\cv\\mQ _{i}}H(z,|\\na u|)\\,dz\\\\\n&\\le c\\La^{1-\\theta}\\sum_{i=1}^\\infty\\iint_{\\mQ _i\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz\\\\\n&\\qquad +c \\sum_{i=1}^\\infty\\iint_{\\mQ _i\\cap \\Theta(c^{-1}\\La)}H(z,|F|)\\,dz\\\\\n&\\le c\\La^{1-\\theta}\\iint_{\\Psi(c^{-1}\\La,r_2)}H(z,|\\na u|)^\\theta\\,dz\\\\\n&\\qquad+ c\\iint_{\\Theta(c^{-1}\\La,r_2)}H(z,|F|)\\,dz\\,.\n \\end{split}\n\\end{align*}\nOn the other hand, since we have\n\\[\n\\iint_{\\Psi(c^{-1}\\La,r_1)\\setminus \\Psi(\\La,r_1)}H(z,|\\na u|)\\,dz\n\\le\\La^{1-\\theta}\\iint_{\\Psi(c^{-1}\\La,r_2)}H(z,|\\na u|)^{\\theta}\\,dz\\,,\n\\]\nwe deduce that\n\\begin{align}\\label{covering_est_1}\n\\begin{split}\n \\iint_{\\Psi(c^{-1}\\La,r_1)}H(z,|\\na u|)\\,dz\n &\\le c\\La^{1-\\theta}\\iint_{\\Psi(c^{-1}\\La,r_2)}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad +c \\iint_{\\Theta(c^{-1}\\La,r_2)}H(z,|F|)\\,dz\\,,\n\\end{split}\n\\end{align}\nwhere $c=c(data)>1$ is a constant.\nWe now take $k\\in\\mathbb N$ and consider\n\\[ \nH(z,|\\na u|)_k=\\min\\{H(z,|\\na u|),k\\}\n\\]\nand\n\\[\n\\Psi_k(\\La,\\rho)=\\{z\\in Q_{\\rho}(z_0):H(z,|\\na u(z)|)_k>\\La\\}\\,.\n\\]\nIt is easy to see that if $\\La>k$, then $\\Psi_k(\\La,\\rho)=\\emptyset$, and if $\\La\\le k$, then $\\Psi_k(\\La,\\rho)=\\Psi(\\La,\\rho)$. Therefore, along with these notations, \\eqref{covering_est_1} becomes\n\\begin{align*}\n \\begin{split}\n \\iint_{\\Psi_k(c^{-1}\\La,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\n &\\le c\\La^{1-\\theta}\\iint_{\\Psi_k(c^{-1}\\La,r_2)}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad +c \\iint_{\\Theta(c^{-1}\\La,r_2)}H(z,|F|)\\,dz\n \\end{split}\n\\end{align*}\nfor any \n\\[\n\\La>(4\\ccc)^{q}\\left(\\frac{8\\cv r}{r_2-r_1}\\right)^{\\frac{q^2}{p}+\\frac{q(n+2)}{2} }\\La_0\\,.\n\\]\nDenoting\n\\[\n \\La_1= c^{-1}(4\\ccc)^q\\left(\\frac{8\\cv r}{r_2-r_1}\\right)^{\\frac{q^2}{p}+\\frac{q(n+2)}{2}}\\La_0\\,,\n\\]\nfor any $\\La>\\La_1$ we obtain\n\\begin{align*}\n \\begin{split}\n &\\iint_{\\Psi_k(\\La,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\le c\\La^{1-\\theta}\\iint_{\\Psi_k(\\La,r_2)}H(z,|\\na u|)^\\theta\\,dz+c\\iint_{\\Theta(\\La,2r)}H(z,|F|)\\,dz\\,.\n \\end{split}\n\\end{align*}\n\nLet $\\ep\\in(0,1)$ will be determined later. We multiply the inequality above by $\\La^{\\ep-1}$ and integrate over $(\\La_1,\\infty)$. Then, we get\n\\begin{align*}\n \\begin{split}\n \\mathrm{I}&=\\int_{\\La_1}^{\\infty}\\La^{\\ep-1}\\iint_{\\Psi_k(\\La,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\,d\\La\\\\\n &\\le c\\int_{\\La_1}^{\\infty}\\La^{\\ep-\\theta}\\iint_{\\Psi_k(\\La,r_2)}H(z,|\\na u|)^\\theta\\,dz\\,d\\La + c \\int_{\\La_1}^{\\infty}\\La^{\\ep-1} \\iint_{\\Theta(\\La,2r)}H(z,|F|)\\,dz \\, d\\La \\\\\n &= \\mathrm{II} + \\mathrm{III}\\,.\n \\end{split}\n\\end{align*}\nApplying Fubini's theorem to $\\mathrm{I}$, it follows that\n\\begin{align*}\n \\begin{split}\n \\mathrm{I}\n &=\\frac{1}{\\ep}\\iint_{\\Psi_k(\\La_1,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad-\\frac{1}{\\ep}\\La_1^\\ep\\iint_{\\Psi_k(\\La_1,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\,.\n \\end{split}\n\\end{align*}\nMeanwhile, since we have\n\\begin{align*}\n \\begin{split}\n &\\iint_{Q_{r_1}(z_0)\\setminus \\Psi_k(\\La_1,r_1)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\le \\La_1^{\\ep}\\iint_{Q_{2r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\,,\n \\end{split}\n\\end{align*}\nwe obtain\n\\begin{align*}\n \\begin{split}\n \\mathrm{I}\\ge& \\frac{1}{\\ep}\\iint_{Q_{r_1}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad-\\frac{2}{\\ep}\\La_1^\\ep\\iint_{Q_{2r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\,.\n \\end{split}\n\\end{align*}\nSimilarly, by Fubini's theorem, we have\n\\[\n \\mathrm{II}\n \\le\\frac{1}{1-\\theta+\\ep}\\iint_{Q_{r_2}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta \\,dz\n\\]\nand\n\\[\n\\mathrm{III}\\le \\frac{1}{\\ep}\\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\,.\n\\]\nCombining the estimates above, we obtain\n\\begin{align*}\n \\begin{split}\n &\\iint_{Q_{r_1}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\le \\frac{c\\ep}{1-\\theta+\\ep}\\iint_{Q_{r_2}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta \\,dz\\\\\n &\\qquad+c\\La_1^\\ep\\iint_{Q_{2r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz \\\\\n &\\qquad + c\\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\n \\end{split}\n\\end{align*}\nfor $c=c(\\data)$ and $\\theta=\\theta(\\data)\\in(0,1)$. We choose $\\ep_0=\\ep_0(\\data)\\in(0,1)$ so that for any $\\ep\\in(0,\\ep_0)$,\n\\[\n \\frac{c\\ep}{1-\\theta+\\ep}\\le\\frac{1}{2}\\,.\n\\]\nThen, there holds\n\\begin{align*}\n \\begin{split}\n &\\iint_{Q_{r_1}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\le \\frac{1}{2}\\iint_{Q_{r_2}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta \\,dz\\\\\n &\\qquad+c\\La_1^\\ep\\iint_{Q_{2r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz \\\\\n &\\qquad+ c\\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\,.\n \\end{split}\n\\end{align*}\nRecalling $r\\le r_1<r_2\\le 2r$, we conclude from Lemma~\\ref{tech_lem} that\n\\begin{align*}\n \\begin{split}\n &\\iint_{Q_{r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta+\\ep}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\le c\\La_0^\\ep\\iint_{Q_{2r}(z_0)}\\left(H(z,|\\na u|)_k\\right)^{1-\\theta}H(z,|\\na u|)^\\theta\\,dz\\\\\n &\\qquad+ c\\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\,.\n \\end{split}\n\\end{align*}\nLetting $k$ go to $\\infty$ and recalling \\eqref{def_la} and $\\eqref{def_La}$, we have \n\\begin{align*}\n \\begin{split}\nint_{Q_{r}(z_0)}H(z,|\\na u|)^{1+\\epsilon}\\,dz\\\\\n &\\le c\\left(\\frac{\\|u\\|^p_{\\infty}}{r^p} +\\|a\\|_{\\infty} \\frac{\\|u\\|^q_{\\infty}}{r^q} + \nint_{Q_{4r}(z_0)}H(z,|F|)\\,dz \\right)^\\frac{p}{2} +1\\right)^{\\frac{q\\epsilon}{p}}\\\\\nint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\,,\n \\end{split}\n\\end{align*}\nwhere we abbreviated $\\| \\cdot \\|_{\\infty}= \\| \\cdot \\|_{L^\\infty(\\Om_T)}$.\nFinally, applying Lemma~\\ref{caccio_lem} with $Q_{2r}(z_0)$ and $Q_{4r}(z_0)$, we obtain\n\\begin{align*}\n \\begin{split}\nint_{Q_{r}(z_0)}H(z,|\\na u|)^{1+\\epsilon}\\,dz\\\\\n &\\le c\\left(\\frac{\\|u\\|^p_{\\infty}}{r^p} +\\|a\\|_{\\infty} \\frac{\\|u\\|^q_{\\infty}}{r^q} + \nint_{Q_{4r}(z_0)}H(z,|F|)\\,dz \\right)^\\frac{p}{2} +1\\right)^{\\frac{q\\epsilon}{p}}\\\\\nint_{Q_{4r}(z_0)}H(z,|F|)\\,dz \\right)\\\\\nint_{Q_{2r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\,,\n \\end{split}\n\\end{align*}\nwhere $c=c(\\data,\\|a\\|_{L^\\infty(\\Om_T)})$.\nSince $p\\ge2$, we apply Young's inequality to the term with exponent $2$ on the right hand side in order to have\n\\begin{align*}\n \\begin{split}\nint_{Q_{r}(z_0)}H(z,|\\na u|)^{1+\\epsilon}\\,dz\n &\\le c\\left(\\frac{\\|u\\|^p_{\\infty}}{r^p} +\\|a\\|_{\\infty}\\frac{\\|u\\|^q_{\\infty}}{r^q} +1\\right)^{1+\\frac{q\\epsilon}{p}} \\\\\nint_{Q_{4r}(z_0)}H(z,|F|)^{1+\\ep} \\,dz\\right)^{1+\\frac{q}{2}}\\,.\n \\end{split}\n\\end{align*}\nThe proof is completed. \\qed",
    "post_theorem_intro_text_len": 4426,
    "post_theorem_intro_text": "For the rest of the paper, whenever we say that $u$ is a weak solution of (\\ref{eq}) in $\\Omega_T$, we assume that $u\\in C(0,T;L^2(\\Omega,\\mathbb{R}^N))\\cap L^q(0,T;W^{1,q}(\\Omega,\\mathbb{R}^N))\\cap L^\\infty(\\Omega_T)$. See Section~\\ref{ssec:sol} for more comments.\n\n\\begin{remark}[Sharpness]\\rm \n    The question on the optimality of a range of parameters for multiple regularity results, was a topic studied in depth on elliptic problems~\\cite{Zhikov86,eslemi}, providing that under a priori boundedness assumption~(\\ref{range_pq}) is actually sharp, cf.~\\cite{comi,comi-bdd,bacomi-cv,fomami,bcdfm}. From this point of view, the current paper is a natural spin-off of~\\cite{WMS}, where  gradient higher integrability of the solutions to~(\\ref{eq}) was proven under no extra a priori regularity of a solution itself.\n\\end{remark}\n\n{\\bf Methods.} The main idea is typical for the double phase problems -- we use the phase separation to regions when the contribution of $a$ is small or dominating. As much as employing the exit time reasoning is expected, a key point is to exploit a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument, where the stopping time depends on the energy of $\\nabla u$ and of $F$. We face several new challenges in the proof of reverse H\\\"older-type result (in Section~\\ref{sec:rev-hold}) comparing to the previous analysis. In fact, the phase separation in  \\cite{KKM}, i.e., in the case of non a priori bounded solutions, relied on the  comparison of two values $\\lambda^p$ and $a(z_0)\\lambda^q$ by using just the energy of the gradient of the solutions. In contrast, the phase analysis here is formulated on integrated quantities, allowing us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa. Then the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems.   We shall point out that the analysis requires delicate interplay between the homogeneity of the scaling factors of cubes and their radii.\n\nA major challenge is obtaining a variant of reverse H\\\"older inequality within the intrinsic and locally varying scaling, that forms the analytic backbone for the higher integrability proved in Theorem~\\ref{main_theorem}. We prove the reversed H\\\"older inequality  adapted to the intrinsic geometry distinguishing {\\it $p$-intrinsic case}, when our problem is a perturbed $p$-Laplace evolution and a perturbation is controlled by  $\\|u\\|_{L^\\infty}$. Namely, for some $K=K(\\mathit{data})>1$ and relevant cube $Q_\\rho$ it holds\n\\[K\\geq (\\|u\\|_{\\infty}/\\rho)^{q-p}\\sup_{cQ_{\\rho}} a\\,.\\]\nOn the other hand, when this condition fails, we deal with the {\\it $(p,q)$-intrinsic case}, when our problem resembles the $q$-Laplace problem. The reversed H\\\"older inequality is provided  for $p$-phase in Lemma~\\ref{p_reverse_lem} and for $(p,q)$-phase in Lemma~\\ref{q_reverse_lem}. \n\nThis separation is effective in the presence of $F$ under the sharp range of phase parameters~(\\ref{range_pq}). The proof is possible due to Lemma~\\ref{lemma_decay} yielding a key decay property enabling the datum $F$ to be nonzero.\n\nThe arguments are closed with use of the Vitali covering theorem, see Lemma~\\ref{vitali_lem}, and consequences of the reverse H\\\"older inequality provided for  $p$-phase in Proposition~\\ref{p_est_vitali} and for $(p,q)$-phase in Proposition~\\ref{q_est_vitali}. \\textcolor{black}{Let us stress that since intrinsic geometries may vary from point to point, the standard Vitali covering lemma cannot be directly applied. The required comparability of the scaling factors is ensured by the Hölder continuity of the coefficient $a$ in Lemma~\\ref{la_comp_lem}, which allows us to establish a modified Vitali covering lemma with a scaled covering constant.} The admissible range $q\\le p+\\alpha$ is precisely what guarantees that the oscillation of $a(\\cdot)$ over the cylinders is below the threshold needed for the proof to be valid, see Section~\\ref{stopping time argument}.\\newline\n\n{\\bf Organization. } In Section~\\ref{sec:prelim} we present notation and basic definitions. Section~\\ref{sec:energy-estimates} provides a priori estimates, while Section~\\ref{sec:rev-hold} is devoted to the reverse H\\\"older estimate. The main proof is concluded in Section~\\ref{sec:main-proof}.\\newline",
    "sketch": "{\\bf Methods.} The proof of Theorem~\\ref{main_theorem} follows the typical double phase strategy: one uses \\emph{phase separation} into regions where the contribution of $a$ is small or dominating. A key ingredient is an \\emph{exit time} (stopping time) argument, exploiting “a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument,” where “the stopping time depends on the energy of $\\nabla u$ and of $F$.”\n\nCompared to earlier work, the phase analysis is “formulated on integrated quantities,” which “allow[s] us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa.” Then “the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems,” requiring a “delicate interplay between the homogeneity of the scaling factors of cubes and their radii.”\n\nThe analytic backbone is a \\emph{reverse H\\\"older inequality} “within the intrinsic and locally varying scaling,” adapted to the intrinsic geometry and split into:\n\\begin{itemize}\n\\item the \\emph{$p$-intrinsic case}, when the problem is a perturbed $p$-Laplace evolution and the perturbation is controlled by $\\|u\\|_{L^\\infty}$; for some $K>1$ and relevant cube $Q_\\rho$ one has\n\\[K\\ge (\\|u\\|_{\\infty}/\\rho)^{q-p}\\sup_{cQ_{\\rho}} a\\,;\\]\n\\item the \\emph{$(p,q)$-intrinsic case}, when the above condition fails and the problem resembles the $q$-Laplace problem.\n\\end{itemize}\nThe corresponding reverse H\\\"older inequalities are proved “for $p$-phase in Lemma~\\ref{p_reverse_lem} and for $(p,q)$-phase in Lemma~\\ref{q_reverse_lem}.”\n\nThis separation “is effective in the presence of $F$ under the sharp range of phase parameters~(\\ref{range_pq}),” and “is possible due to Lemma~\\ref{lemma_decay} yielding a key decay property enabling the datum $F$ to be nonzero.”\n\nFinally, “the arguments are closed with use of the Vitali covering theorem,” via Lemma~\\ref{vitali_lem} and the consequences of the reverse H\\\"older inequality in Proposition~\\ref{p_est_vitali} (for $p$-phase) and Proposition~\\ref{q_est_vitali} (for $(p,q)$-phase). Since “intrinsic geometries may vary from point to point,” a standard Vitali lemma cannot be applied directly; “comparability of the scaling factors is ensured by the H\\\"older continuity of the coefficient $a$ in Lemma~\\ref{la_comp_lem},” yielding a modified Vitali covering. The range “$q\\le p+\\alpha$” is stated to guarantee the needed small oscillation of $a(\\cdot)$ over cylinders for the stopping-time/Vitali argument to work.",
    "expanded_sketch": "{\\bf Methods.} To prove the main theorem, one follows the typical double phase strategy: one uses \\emph{phase separation} into regions where the contribution of $a$ is small or dominating. A key ingredient is an \\emph{exit time} (stopping time) argument, exploiting “a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument,” where “the stopping time depends on the energy of $\\nabla u$ and of $F$.”\n\nCompared to earlier work, the phase analysis is “formulated on integrated quantities,” which “allow[s] us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa.” Then “the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems,” requiring a “delicate interplay between the homogeneity of the scaling factors of cubes and their radii.”\n\nThe analytic backbone is a \\emph{reverse H\\\"older inequality} “within the intrinsic and locally varying scaling,” adapted to the intrinsic geometry and split into:\n\\begin{itemize}\n\\item the \\emph{$p$-intrinsic case}, when the problem is a perturbed $p$-Laplace evolution and the perturbation is controlled by $\\|u\\|_{L^\\infty}$; for some $K>1$ and relevant cube $Q_\\rho$ one has\n\\[K\\ge (\\|u\\|_{\\infty}/\\rho)^{q-p}\\sup_{cQ_{\\rho}} a\\,;\\]\n\\item the \\emph{$(p,q)$-intrinsic case}, when the above condition fails and the problem resembles the $q$-Laplace problem.\n\\end{itemize}\nWe first record the corresponding reverse H\\\"older inequalities.\n\\begin{lemma}\\label{p_reverse_lem}\nSuppose \\ref{p1} and \\ref{p2}.\n\\tThere exist constants $c=c(\\data)$ and $\\theta_0=\\theta_0(n,p)\\in(0,1)$ such that for any $\\theta\\in(\\theta_0,1)$ there holds\n\\t\\begin{align*}\nint_{Q_{\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\nint_{Q_{2\\rho}^\\la(z_0)}H(z,|F|)\\,dz\\,.\n\\t\\end{align*}\\t\n\\end{lemma}\n\\begin{lemma}\\label{q_reverse_lem}\nSuppose \\ref{q1} and \\ref{q2}.\n\\tThere exist constants $c=c(n,N,p,q,\\nu,L,\\ca)$ and $\\theta_0=\\theta_0(n,p,q)\\in(0,1)$ such that for any $\\theta\\in(\\theta_0,1)$ there holds\n\\t\\begin{align*}\nint_{G_{\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\nint_{G_{2\\rho}^\\la(z_0)}H(z,|F|) \\,dz\\,.\n\\t\\end{align*}\\t\n\\end{lemma}\n\nThis separation “is effective in the presence of $F$ under the sharp range of phase parameters\n\\begin{align} \\label{range_pq}\n2\\le p <\\infty, \\quad p<q\\le p+\\alpha\\,.    \n\\end{align}\n,” and “is possible due to” the following decay property, which yields a key decay estimate enabling the datum $F$ to be nonzero.\n\\begin{lemma}\\label{lemma_decay}\n   Suppose \\ref{p1} and \\ref{p2}. Then there exists a constant $\\cc=\\cc(\\data)\\ge K^\\frac{1}{p}$\n   such that\n   \\[\n   \\rho\\le \\cc \\la^{-1}\\,.\n   \\]\n\\end{lemma}\n\nFinally, “the arguments are closed with use of the Vitali covering theorem,” via the following Vitali-type selection.\n\\begin{lemma}\\label{vitali_lem}\n    Let $\\mathcal{F}$ be defined as above and $\\kappa$ is in \\eqref{K_and_kappa}, Then there exists a pairwise disjoint countable subcollection $\\mathcal{G}$ of $\\mathcal{F}$ such that for any $U(w)\\in \\mathcal{F}$, there exists $U(v)\\in \\mathcal{G}$ with\n    \\[\n    U(w)\\subset \\kappa U(v)\\,.\n    \\]\n\\end{lemma}\nTogether with the consequences of the reverse H\\\"older inequalities encoded in the Vitali-estimate propositions for the $p$-phase and $(p,q)$-phase:\n\\begin{proposition}\\label{p_est_vitali} \nSuppose \\ref{p1} and \\ref{p2}.\n\\tThere exist constants $c=c(\\data)$ and $\\theta_0=\\theta_0(n,p)\\in(0,1)$, such that for any $\\theta\\in(\\theta_0,1)$ we have\n\\t\\begin{align*}\n\\t\\t\\t\\begin{split}\n\\t\\t\\t    \\iint_{Q_{2\\cv\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\n\\t\\t\\t&\\le c\\La^{1-\\theta}\\iint_{Q_{2\\rho}^\\la(z_0)\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz\\\\\n            &\\qquad+c\\iint_{Q_{2\\rho}^\\la(z_0)\\cap \\Theta(c^{-1}\\La)}H(z,|F|) \\,dz\\,.\n\\t\\t\\t\\end{split}\n\\t\\end{align*}\n    Here, $\\kappa$ is defined in \\eqref{K_and_kappa} and it appears in \\ref{p4}.\n \\end{proposition}\n\\begin{proposition}\n    \\label{q_est_vitali}\nSuppose \\ref{q1} and \\ref{q2}.\n\\tThere exist constants $\\theta_0=\\theta_0(n,p,q)\\in(0,1)$ and $c=c(n,N,p,q,\\nu,L,\\ca)$  such that for any $\\theta\\in(\\theta_0,1)$ we have\n\\t\\begin{align*}\n\\t\\t\\t\\begin{split}\n\\t\\t\\t    \\iint_{G_{2\\cv\\rho}^\\la(z_0)}H(z,|\\na u|)\\,dz\n\\t\\t\\t&\\le c\\La^{1-\\theta}\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Psi(c^{-1}\\La)}H(z,|\\na u|)^\\theta\\,dz \\\\\n            &\\qquad+c\\iint_{G_{2\\rho}^\\la(z_0)\\cap \\Theta(c^{-1}\\La)}H(z,|F|)\\,dz\\,.\n\\t\\t\\t\\end{split}\n\\t\\end{align*}\n    Here, $\\kappa$ is defined in \\eqref{K_and_kappa} and it appears in \\ref{q5},\n\\end{proposition}\nSince “intrinsic geometries may vary from point to point,” a standard Vitali lemma cannot be applied directly; “comparability of the scaling factors is ensured” by the following comparability statement.\n\\begin{lemma}\\label{la_comp_lem}\n    Suppose $U(w), U(v)\\in \\mathcal{F}$ with $U(w)\\cap U(v)\\ne \\emptyset$ and $l_w\\le 2l_v$. Then for $\\cc\\ge 2$ defined in Lemma~\\ref{p_decay_lem}, we have\n    \\[\n    \\lav\\le (2(1+\\ca+10\\cc\\ca))^\\frac{1}{p}\\law\\,.\n    \\]\n    Moreover, if $U(v)$ is $(p,q)$-intrinsic, then we also have\n    \\[\n    \\law\\le (2(1+\\ca+10\\cc\\ca))^\\frac{1}{p}\\lav\\,.\n    \\]\n\\end{lemma}\nThe range “$q\\le p+\\alpha$” in \\eqref{range_pq} is stated to guarantee the needed small oscillation of $a(\\cdot)$ over cylinders for the stopping-time/Vitali argument to work.",
    "expanded_theorem": "\\label{main_theorem}\n\tLet $u\\in C(0,T;L^2(\\Omega,\\mathbb{R}^N))\\cap L^q(0,T;W^{1,q}(\\Omega,\\mathbb{R}^N))\\cap L^\\infty(\\Omega_T)$ be a weak solution to\n\t\\begin{align}\\label{eq}\n\\tu_t-\\dv\\mA(z,\\na u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\\quad \\text{in $\\Omega_T=\\Omega\\times(0,T)$\\,,}\n\\end{align}\n\tin $\\Omega_T$. \n\tThere exist constants $\\ep_0=\\ep_0(\\mathit{data})$ and $c=c(\\mathit{data} , \\|a\\|_{L^\\infty(\\Om_T)})$ such that for every $Q_{4r}(z_0)\\subset\\Om_T$ with $r\\in (0,1)$ and $\\epsilon\\in(0,\\ep_0)$,\n\t\\begin{align*}\n    \\begin{split}\n        &\\iint_{Q_{r}(z_0)}H(z,|\\nabla u|)^{1+\\epsilon}\\,dz\\\\\n\t&\\le c\\left(\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^p+\\|a\\|_{L^\\infty(\\Om_T)}\\left( \\frac{\\|u\\|_{L^\\infty(\\Om_T)}}{r} \\right)^q +1\\right)^{1+\\frac{q\\epsilon}{p}}\\\\\nint_{Q_{4r}(z_0)}H(z,|F|)^{1+\\epsilon} \\,dz\\right)^{1+\\frac{q}{2}}\\,.\n    \\end{split}\n\\end{align*},",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb R^n\\) be a bounded domain with \\(n\\ge 2\\), let \\(T>0\\), and write \\(\\Omega_T=\\Omega\\times(0,T)\\). Fix \\(N\\ge 1\\). Assume \\(a\\ge 0\\) and \\(a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)\\) for some \\(\\alpha\\in(0,1]\\), meaning that for some \\(c_a>0\\),\n\\[\n|a(x,t)-a(y,s)|\\le c_a\\max\\{|x-y|^\\alpha,|t-s|^{\\alpha/2}\\}\n\\]\nfor all relevant points, and assume\n\\[\n2\\le p<\\infty,\\qquad p<q\\le p+\\alpha.\n\\]\nDefine\n\\[\nH(z,s)=s^p+a(z)s^q .\n\\]\nLet \\(\\mathcal A:\\Omega_T\\times\\mathbb R^{Nn}\\to\\mathbb R^{Nn}\\) be a Carath\\'eodory vector field such that for some \\(0<\\nu\\le L<\\infty\\),\n\\[\n\\mathcal A(z,\\xi)\\cdot \\xi\\ge \\nu\\big(|\\xi|^p+a(z)|\\xi|^q\\big),\n\\qquad\n|\\mathcal A(z,\\xi)|\\le L\\big(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}\\big)\n\\]\nfor a.e. \\(z\\in\\Omega_T\\) and every \\(\\xi\\in\\mathbb R^{Nn}\\). Assume also that \\(H(z,|F|)\\in L^\\gamma(\\Omega_T)\\) with \\(\\gamma=(n+p)/p\\), and that\n\\[\nu\\in C(0,T;L^2(\\Omega,\\mathbb R^N))\\cap L^q(0,T;W^{1,q}(\\Omega,\\mathbb R^N))\\cap L^\\infty(\\Omega_T)\n\\]\nis a weak solution of\n\\[\nu_t-\\operatorname{div}\\mathcal A(z,\\nabla \\nu)=-\\operatorname{div}\\big(|F|^{p-2}F+a(z)|F|^{q-2}F\\big)\n\\quad\\text{in }\\Omega_T.\n\\]\nFor a parabolic cylinder \\(Q_{4r}(z_0)\\subset \\Omega_T\\) with \\(r\\in(0,1)\\), which quantitative estimate holds for \\(H(z,|\\nabla \\nu|)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exist constants \\(\\epsilon_0=\\epsilon_0(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})>0\\) and \\(c=c(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)},\\|a\\|_{L^\\infty(\\Omega_T)})\\) such that for every \\(\\epsilon\\in(0,\\epsilon_0)\\),\n\\[\n\\iint_{Q_r(z_0)} H(z,|\\nabla \\nu|)^{1+\\epsilon}\\,dz\n\\le c\\left(\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^p+\\|a\\|_{L^\\infty(\\Omega_T)}\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^q+1\\right)^{1+\\frac{q\\epsilon}{p}}\n+ c\\left(\\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\\epsilon}\\,dz\\right)^{1+\\frac q2}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist constants \\(\\epsilon_0=\\epsilon_0(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})>0\\) and \\(c=c(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)},\\|a\\|_{L^\\infty(\\Omega_T)})\\) such that for every \\(\\epsilon\\in(0,\\epsilon_0)\\),\n\\[\n\\iint_{Q_r(z_0)} H(z,|\\nabla \\nu|)^{1+\\epsilon}\\,dz\n\\le c\\left(\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^p+\\|a\\|_{L^\\infty(\\Omega_T)}\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^q+1\\right)^{1+\\frac{q\\epsilon}{p}}\n+ c\\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\\epsilon}\\,dz.\n\\]"
        },
        {
          "label": "C",
          "text": "There exist constants \\(\\epsilon_0=\\epsilon_0(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})>0\\) and \\(c=c(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)},\\|a\\|_{L^\\infty(\\Omega_T)})\\) such that for every \\(\\epsilon\\in(0,\\epsilon_0)\\),\n\\[\n\\iint_{Q_r(z_0)} H(z,|\\nabla \\nu|)^{1+\\epsilon}\\,dz\n\\le c\\left(\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^p+\\|a\\|_{L^\\infty(\\Omega_T)}\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^q+1\\right)^{1+\\frac{q\\epsilon}{p}}\n+ c\\left(\\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\\epsilon}\\,dz\\right).\n\\]"
        },
        {
          "label": "D",
          "text": "There exist constants \\(\\epsilon_0=\\epsilon_0(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})>0\\) and \\(c=c(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)},\\|a\\|_{L^\\infty(\\Omega_T)})\\) such that for every \\(\\epsilon\\in(0,\\epsilon_0)\\) and every parabolic cylinder \\(Q_{4r}(z_0)\\subset \\Omega_T\\) with \\(r>0\\),\n\\[\n\\iint_{Q_r(z_0)} H(z,|\\nabla \\nu|)^{1+\\epsilon}\\,dz\n\\le c\\left(\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^p+\\|a\\|_{L^\\infty(\\Omega_T)}\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^q+1\\right)^{1+\\frac{q\\epsilon}{p}}\n+ c\\left(\\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\\epsilon}\\,dz\\right)^{1+\\frac q2}.\n\\]"
        },
        {
          "label": "E",
          "text": "There exist constants \\(\\epsilon_0=\\epsilon_0(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})>0\\) and \\(c=c(n,N,p,q,\\nu,L,c_a,\\|\\nu\\|_{L^\\infty(\\Omega_T)},\\|H(z,|F|)\\|_{L^\\gamma(\\Omega_T)})\\) such that for every \\(\\epsilon\\in(0,\\epsilon_0)\\),\n\\[\n\\iint_{Q_r(z_0)} H(z,|\\nabla \\nu|)^{1+\\epsilon}\\,dz\n\\le c\\left(\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^p+\\|a\\|_{L^\\infty(\\Omega_T)}\\left(\\frac{\\|\\nu\\|_{L^\\infty(\\Omega_T)}}{r}\\right)^q+1\\right)^{1+\\frac{q\\epsilon}{p}}\n+ c\\left(\\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\\epsilon}\\,dz\\right)^{1+\\frac q2}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "forcing-term exponent after Young inequality",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "drop the stronger outer exponent on the forcing term",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "radius restriction r\\in(0,1) tied to intrinsic scaling and exit-time geometry",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of the constant on \\|a\\|_{L^\\infty} introduced in the final Caccioppoli step",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or quote the exact conclusion. It gives the full hypothesis set of a theorem, but the distinguishing features of the right answer (such as the precise forcing-term exponent and dependence structure of constants) are not stated in the stem."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem lists the hypotheses and asks for the guaranteed estimate. The task is mainly to recognize the exact stated conclusion rather than infer among genuinely different mathematical outcomes."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare subtle differences in exponents, radius restrictions, and parameter dependence, but the item mostly rewards memory/recognition of the theorem statement rather than generative mathematical reasoning from the assumptions."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically close to the target statement: they alter the forcing-term exponent, weaken the estimate, change the radius range, or tamper with dependence of constants. These align with realistic failure modes in recalling technical regularity results."
      },
      "total_score": 5,
      "overall_assessment": "Technically well-constructed with strong distractors and little direct answer leakage, but it is largely a theorem-restatement/recognition question rather than a strong test of generative reasoning."
    }
  },
  {
    "id": "2512.11432v1",
    "paper_link": "http://arxiv.org/abs/2512.11432v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theo",
      "content": "Let $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:\n\\[\n\\frac{\\partial^{\\alpha +\\beta}f}{\\partial x_1^\\alpha \\partial x_2^\\beta}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha,\\beta \\in\\mathbb{N}.\n\\]",
      "start_pos": 6484,
      "end_pos": 6754,
      "label": null
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 2627,
    "pre_theorem_intro_text": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable  functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated  Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\  In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets.  \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?}  \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz?  \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example:  \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there.  \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring.  \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin.  \nOur first main result states that any such function must vanish to infinite order.",
    "context": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\ In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets. \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring. \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin. \nOur first main result states that any such function must vanish to infinite order.",
    "full_context": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\ In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets. \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring. \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin. \nOur first main result states that any such function must vanish to infinite order.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe then formulate a general geometric criterion for flatness, valid for arbitrary \ncollections of smooth arcs tangent at a point with unbounded curvature or \ninfinitely varying radius of osculation. \nThis leads to a \\emph{degenerate \\L{}ojasiewicz inequality} adapted to situations \nwhere the classical inequality necessarily fails.\n\n\\begin{theo}\nLet $\\Gamma = \\bigcup\\limits_{n \\ge 1} \\gamma_n$ be a union of smooth embedded arcs \nmeeting at a common point $p$ with curvature tending to $\\infty$ or oscillating \nwithout bound. \nIf $f \\in \\mathcal{E}^\\infty(\\Omega)$ vanishes on $\\Gamma$, then $f$ satisfies a flatness \nestimate of the form\n\\[\n|f(x)| \\le C_N\\, d(x,\\Gamma)^N,\n\\qquad \\text{for all } N \\ge 1 \\text{ and all } x \\text{ near } p,\n\\]\nfor suitable constants $C_N>0$. \nIn particular, $f$ is flat at $p$.\n\\end{theo}\n\n\\begin{rem}\nIn this case, for any system of generators $g_1,\\ldots,g_p$ of $I$, \nthe functions $\\sum\\limits_{j=1}^p g_j^2$ and $\\sum\\limits_{j=1}^p |g_j|$ both satisfy \nthe {\\L}ojasiewicz inequality with respect to $Z(I)$.\n\\end{rem}\nIt should be noted that the property of an ideal to be {\\L}ojasiewicz is a \nlocal one: if an ideal is {\\L}ojasiewicz on an open set $U$, then the induced \nideal on any smaller open subset is also {\\L}ojasiewicz.\\\\\nAny analytic ideal is {\\L}ojasiewicz, as this follows from the fundamental \n{\\L}ojasiewicz inequality for analytic functions \\cite{Lojasiewicz1965}. Moreover, any finitely \ngenerated ideal that is closed in $\\mathcal{E}^\\infty(\\Omega)$ is also a \n{\\L}ojasiewicz ideal, see Corollary 4.4, Ch V, \\cite{Tougeron1972}. Indeed, by Whitney's spectral theorem, a smooth function \nbelongs to a closed finitely generated ideal $I=(f_1,\\ldots,f_k)$ if and only if, \nfor every point $x\\in\\Omega$, its Taylor expansion at $x$ lies in the ideal \ngenerated by the Taylor expansions $T_x f_1,\\ldots,T_x f_k$ in the formal power \nseries algebra $\\mathbb{R}[[X-x]]$. \nHowever, it should be emphasized that there exist {\\L}ojasiewicz ideals \nwhich are not closed, in dimension $n >1$, see Examples 4.8, Ch V, \\cite{Tougeron1972}.\n\\subsection{Smooth points of the locus of zeros}\n\\begin{defn}[Smooth point]\nLet $E \\subset \\mathbb{R}^n$ be closed set. \nA point $x \\in E$ is said to be smooth if, in a neighborhood of $x$, \nthe set $E$ coincides with a $k$-dimensional embedded submanifold of $\\mathbb{R}^n$, \nwhere $k$ is its local dimension at $x$ and may vary with $x$.\n\\end{defn}\n\\begin{Ex}\n The \\emph{Hawaiian earring}, \n defined as the union of circles \n\\[\n\\bigcup_{n=1}^{\\infty} \\left\\{(x,y)\\in \\mathbb{R}^2 : \\left(x-\\frac{1}{n}\\right)^2 + y^2 = \\frac{1}{n^2} \\right\\},\n\\] \nis smooth at every point except the origin, where infinitely many circles accumulate. \nHence, the set of smooth points of the Hawaiian earring is dense in $E= \\bigcup_{n=1}^{\\infty} \\left\\{(x,y)\\in \\mathbb{R}^2 : \\left(x-\\frac{1}{n}\\right)^2 + y^2 = \\frac{1}{n^2} \\right\\}$.\n\n\\begin{theo}\nLet $f:\\mathbb{R}^2\\to\\mathbb{R}$ be a $C^\\infty$ function. \nAssume that $f$ vanishes on the Hawaiian earring\n\\[\n\\mathcal H=\\bigcup_{n\\ge1} C_n, \\qquad \nC_n=\\left\\{(x,y):\\left(x-\\tfrac1n\\right)^2+y^2=\\tfrac1{n^2}\\right\\}.\n\\]\nThen $f$ is flat at the origin: \n\\[\n\\frac{\\partial^{\\alpha_1 +\\alpha_2 f}}{\\partial x_1^{\\alpha_1} \\partial x^{\\alpha_2}_2}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha =(\\alpha_1,\\alpha_2) \\in\\N^2.\n\\]\n\\end{theo}\n\nThus there is no nonzero homogeneous term in the Taylor expansion of $f$ at $(0,0)$, \nso all derivatives of $f$ at the origin vanish. \nHence $f$ is flat at $(0,0)$.\n\\section{Łojasiewicz Inequality and the Hawaiian Earring}\n\\begin{theo}\nLet $f \\in C^\\infty(\\mathbb{R}^2)$ and assume its zero set is the Hawaiian earring\n\\[\n\\mathcal H:= \\bigcup_{k=1}^{\\infty} \\left\\{ (x,y) \\in \\mathbb{R}^2 : \\left(x - \\frac{1}{k}\\right)^2 + y^2 = \\frac{1}{k^2} \\right\\}.\n\\]\nThen $f$ is flat at the origin and does \\emph{not} satisfy a Łojasiewicz inequality with respect to $H$ at $0$; that is, there exist no constants $C>0$ and $\\theta>0$ such that\n\\[\n|f(x)| \\ge C \\, \\mathrm{dist}(x,H)^\\theta\n\\]\nfor all $x$ in a neighborhood of $0$.\n\\end{theo}\n{\\bf{Proof.}}\nSince $f \\in C^\\infty$ vanishes on the Hawaiian earring, which has tangent directions at $0$ dense in the unit circle $S^1$, Theorem 4 implies that $f$ is flat at the origin:\n\\[\nD^\\alpha f(0) = 0 \\quad \\text{for all multi-indices } \\alpha.\n\\]\nLet $x \\in \\mathbb{R}^2$ be close to $0$. Consider the sequence of circles in $H$ with radii $r_k = 1/k$ and centers $c_k = (1/k,0)$. \nChoosing $k \\sim 1/\\|x\\|$, the circle $C_k$ closest to $x$ has radius $r_k \\sim \\|x\\|$. \nHence, for points near $0$,\n\\[\n\\mathrm{dist}(x,H) \\sim \\|x\\|.\n\\]\nAssume, for contradiction, that there exist constants $C>0$ and $\\theta>0$ such that\n\\[\n|f(x)| \\ge C \\, \\mathrm{dist}(x,H)^\\theta\n\\]\nfor all $x$ near $0$.\n\nChoosing $N > \\theta$ gives\n\\[\n|f(x)| \\ll \\mathrm{dist}(x,H)^\\theta\n\\]\nfor $x$ sufficiently close to $0$, which contradicts the assumed Łojasiewicz inequality. \nTherefore, the ideal generated by\n the function $f$ in the the ring $\\mathcal{E}^\\infty(\\R^2)$ is not a \n Lojasiewicz ideal.\n\\section{{Jet-Determination of Smooth Functions via Families of Arcs}}\n\\begin{theo}\nLet $\\mathcal A = \\{\\gamma_i\\}_{i \\in I}$ be a family of smooth arcs\n\\[\n\\gamma_i : (-\\varepsilon,\\varepsilon) \\to \\mathbb{R}^n, \\qquad \\gamma_i(0)=0, \\quad \\gamma_i \\not\\equiv \\text{constant}.\n\\]\nFor each arc, denote its $m$-jet at $0$ by\n\\[\nj^m \\gamma_i(0) = (\\gamma_i'(0), \\gamma_i''(0), \\dots, \\gamma_i^{(m)}(0)).\n\\]\nThe following assertions are equivalent:\n\\begin{enumerate}\n\\item[(A)] \\textbf{Jet–Determination Property:} For every function $f \\in C^\\infty(\\mathbb{R}^n)$,\n\\[\nf(\\gamma_i(s)) \\equiv 0 \\ \\text{for all } i \\quad \\Longrightarrow \\quad D^\\alpha f(0)=0 \\ \\text{for all multi-indices } \\alpha.\n\\]\n\\item[(B)] \\textbf{Jet–Nondegeneracy Condition:} For every integer $m \\ge 1$, there exists no nonzero homogeneous polynomial $P_m$ of degree $m$ such that\n\\[\nP_m(j^m \\gamma_i(0))=0 \\qquad \\text{for all } i \\in I.\n\\]\n\\end{enumerate}\n\\end{theo}\n{\\bf{Proof.}}\n\\textbf{(A) $\\implies$ (B).} \\\\Suppose (A) holds and, for contradiction, that (B) fails for some $m$. \nThen there exists a nonzero homogeneous polynomial $P_m$ of degree $m$ such that\n\\[\nP_m(j^m \\gamma_i(0)) = 0 \\quad \\text{for all } i.\n\\]\n\nThen $E_n$ is jet-determining at $0$: if $f \\in C^\\infty(\\mathbb{R}^n)$ vanishes on $E_n$, then $f$ is flat at the origin:\n\\[\nD^\\alpha f(0) = 0 \\quad \\text{for all multi-indices } \\alpha.\n\\]\n\\end{cor}",
    "post_theorem_intro_text_len": 1547,
    "post_theorem_intro_text": "The proof relies on purely geometric considerations: any nonzero term of the \nTaylor expansion of $f$ at $(0,0)$ would define an algebraic curve of finite \norder, and no such curve can contain infinitely many smooth arcs with tangency \nand curvature behavior comparable to the circles $C_n$.  \nThus the geometry of the Hawaiian earring forces infinite-order vanishing.\n\nWe then formulate a general geometric criterion for flatness, valid for arbitrary \ncollections of smooth arcs tangent at a point with unbounded curvature or \ninfinitely varying radius of osculation.  \nThis leads to a \\emph{degenerate \\L{}ojasiewicz inequality} adapted to situations \nwhere the classical inequality necessarily fails.\n\n\\begin{theo}\nLet $\\Gamma = \\bigcup\\limits_{n \\ge 1} \\gamma_n$ be a union of smooth embedded arcs \nmeeting at a common point $p$ with curvature tending to $\\infty$ or oscillating \nwithout bound.  \nIf $f \\in \\mathcal{E}^\\infty(\\Omega)$ vanishes on $\\Gamma$, then $f$ satisfies a flatness \nestimate of the form\n\\[\n|f(x)| \\le C_N\\, d(x,\\Gamma)^N,\n\\qquad \\text{for all } N \\ge 1 \\text{ and all } x \\text{ near } p,\n\\]\nfor suitable constants $C_N>0$.  \nIn particular, $f$ is flat at $p$.\n\\end{theo}\n\nThis result demonstrates that certain pathological smooth zero sets necessarily \nforce infinite-order degeneracy, and therefore cannot arise from \\L{}ojasiewicz \nideals.  \nIn particular, the ideal generated by a function defining the Hawaiian earring is \nnever \\L{}ojasiewicz, despite the fact that its zero set is smooth away from a \nsingle point.",
    "sketch": "The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.\n\nThe introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \\L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\\le C_N d(x,\\Gamma)^N$) in settings “where the classical inequality necessarily fails.”",
    "expanded_sketch": "The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.\n\nThe introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \\L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\\le C_N d(x,\\Gamma)^N$) in settings “where the classical inequality necessarily fails.”",
    "expanded_theorem": "Let $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:\n\\[\n\\frac{\\partial^{\\alpha +\\beta}f}{\\partial x_1^\\alpha \\partial x_2^\\beta}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha,\\beta \\in\\mathbb{N}.\n\\],",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let\n\\[\n\\mathcal H:= \\bigcup_{k=1}^{\\infty} \\left\\{ (x,y) \\in \\mathbb{R}^2 : \\left(x-\\frac{1}{k}\\right)^2+y^2=\\frac{1}{k^2} \\right\\}\n\\]\nbe the Hawaiian earring, and let \\(Z(f)=\\{(x,y)\\in\\mathbb R^2: f(x,y)=0\\}\\) denote the zero set of a smooth function \\(f\\in \\mathcal E^{\\infty}(\\mathbb R^2)=C^{\\infty}(\\mathbb R^2)\\). Which statement holds for every smooth function \\(f\\) such that \\(Z(f)=\\mathcal H\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The function \\(f\\) is flat at the origin; equivalently, for every pair of integers \\(\\alpha,\\beta\\ge 0\\),\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^{\\alpha}\\partial x_2^{\\beta}}(0,0)=0.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an integer \\(N\\ge 1\\) such that \\(f\\) vanishes to order at least \\(N\\) at the origin; equivalently,\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^{\\alpha}\\partial x_2^{\\beta}}(0,0)=0\n\\qquad\\text{for all }\\alpha,\\beta\\in\\mathbb N\\text{ with }\\alpha+\\beta\\le N,\n\\]\nbut not necessarily for all orders."
        },
        {
          "label": "C",
          "text": "For every pair of integers \\(\\alpha,\\beta\\ge 0\\), if \\(\\alpha+\\beta=1\\) then\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^{\\alpha}\\partial x_2^{\\beta}}(0,0)=0.\n\\]"
        },
        {
          "label": "D",
          "text": "There exist constants \\(C>0\\) and \\(\\theta>0\\) such that, for all \\((x,y)\\) sufficiently close to \\((0,0)\\),\n\\[\n|f(x,y)|\\ge C\\,\\mathrm{dist}\\bigl((x,y),\\mathcal H\\bigr)^{\\theta}.\n\\]"
        },
        {
          "label": "E",
          "text": "The function \\(f\\) is real-analytic in a neighborhood of the origin; equivalently, if its Taylor expansion at \\((0,0)\\) has any nonzero term, then that finite-order algebraic curve contains the arcs of \\(\\mathcal H\\) near the origin."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "infinite-order vanishing replaced by finite-order vanishing",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped all higher-order derivative vanishing, keeping only first-order consequences",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "classical Lojasiewicz lower bound in a setting where only degenerate flatness estimates are compatible",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "finite-order algebraic-curve obstruction turned into analyticity near the origin",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state or strongly hint that flatness at the origin is the required conclusion. It only specifies the Hawaiian earring as the zero set and asks for a universal consequence."
      },
      "TAS": {
        "score": 1,
        "justification": "The question is close to a theorem-recall item: the correct choice is essentially the main conclusion associated with this zero-set configuration. However, it is not completely tautological because the options include weaker true and plausible false alternatives that must be distinguished."
      },
      "GPS": {
        "score": 2,
        "justification": "The item demands substantial reasoning or theorem-level understanding: one must infer infinite-order vanishing from the geometry of the Hawaiian earring and reject tempting alternatives such as finite-order vanishing, mere first-order vanishing, a Łojasiewicz bound, or analyticity."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful. B is a near-miss, C is a weaker true statement, D reflects a common confusion with analytic/geometric inequalities, and E exploits a plausible but false analyticity intuition."
      },
      "total_score": 7,
      "overall_assessment": "A strong MCQ with little answer leakage and excellent distractors. Its main weakness is that the correct option closely mirrors the underlying theorem, so it leans somewhat toward theorem recognition rather than fully novel inference."
    }
  },
  {
    "id": "2512.11523v1",
    "paper_link": "http://arxiv.org/abs/2512.11523v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]",
      "start_pos": 11715,
      "end_pos": 11923,
      "label": "quantization of MA energy"
    },
    "ref_dict": {
      "epsilon twisted local holomorphic morse": "\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}",
      "Pointwise Semi-Classical OT": "\\begin{thm}\\label{Pointwise Semi-Classical OT} There exists $C>0$ and $p_1\\in \\N$ such that for any $p\\geq p_1$, any $x\\in X$, and any $v\\in (A^p)_x$,  one can find a global holomorphic section $a=a_{p,x,v}\\in H^0(X,A^p)$ with\n\\begin{equation}\\label{OT bound}\n    a(x)=v,\\quad |v|_{(h_0^A)^p}\\leq \\sup_X|a|_{(h_0^A)^p}\\leq \\left(1+\\frac{C}{p}\\right)|v|_{(h_0^A)^p},\n\\end{equation}\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}",
      "convergence of Bergman measures": "\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}",
      "Bergman kernel comparison": "\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}",
      "Bergman Measures, Positivity of Direct Images, and Equilibrium": "\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}",
      "quantization of MA energy": "\\begin{thm}\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]\n\\end{thm}",
      "MA envelope": "\\begin{equation}\\label{MA envelope}\n\t\tP_\\theta(f):=usc(\\sup\\{u\\in \\PSH(X,\\theta):u\\leq f\\})\n\t\\end{equation}",
      "pitfall for finite energy potentials": "\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}",
      "Hilbert norm": "\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}",
      "volume of a line bundle": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}",
      "finite energy space and weak geodesic": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}",
      "on diagonal Bergman kernel": "\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5031,
    "pre_theorem_intro_text": "Given an ample line bundle \\(L\\) over a complex projective manifold \\(X\\), endowed with the Kähler form \\(\\omega\\) polarized by \\(L\\), a central theme in Kähler geometry—first articulated by Yau \\cite[p.~139]{Yau}—is the approximation of transcendental geometric objects on \\((X,\\omega)\\) by asymptotic algebraic data coming from the powers \\(L^k:=L^{\\otimes k}\\). This is the philosophy of \\emph{Kähler quantization}.\n\nThe initial proposal is to approximate the infinite-dimensional space\n\\[\n   \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished  subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nDonaldson later proposed that not only the metrics but also the \\emph{geometry} of \\(\\mathcal{H}_\\omega\\) should be captured by the finite-dimensional geometries of \\(\\mathcal{H}_k\\) \\cite[p.~483]{DonaldsonI}. This principle has since been realized in various settings and has had deep impacts on problems concerning canonical metrics and stability; see, for example, \\cite{PS,SongZelditchqu,CS,BerndtssonProb,DLR,DonaldsonI,DonaldsonScalII,RTZ,Zhang} among a rapidly expanding body of literature. We refer to \\cite{Mama} for a detailed exposition of some classical developments in this direction.\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\)  only holds in a weak sense.  The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to  Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive.  Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.  \n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with  the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written  as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding  a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n    E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the  quantization of Monge--Amp\\`ere energy in adjoint twisting.",
    "context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}",
    "full_context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}\n\nRescaling $w=\\sqrt{k}z$ on the standard shrinking polydisc gives the uniform approximation\n\\[\nk\\Phi_\\epsilon(w/\\sqrt{k})\n=\\Phi_\\epsilon^0(w)+\\rho_k+\\epsilon\\widetilde\\rho_k,\n\\qquad \n\\rho_k,\\widetilde\\rho_k\\to0.\n\\]\nRepeating the submean-value argument used for Theorem~\\ref{Berman's local Morse inequality} then yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\le\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\frac{n!dV_n}{(2\\pi)^n\\prod_{j=1}^n\\int_0^{\\log k} e^{-(\\lambda_j+\\epsilon)r^2}r\\,dr}.\n\\]\nSince the quadratic model satisfies $\\lambda_j+\\epsilon\\ge\\epsilon>0$, the Gaussian integrals in the denominator behave exactly as in the proof of Theorem~\\ref{Berman's local Morse inequality}, with $\\lambda_j$ replaced by $\\lambda_j+\\epsilon$. \nUsing the identity\n\\[\n(\\theta+\\epsilon\\omega)^n\n= \\frac{n!}{\\pi^n}\\!\\left(\\prod_{j=1}^n (\\lambda_j+\\epsilon)\\right)dV_n,\n\\]\nthe same computation as in the untwisted case yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\;\\le\\;\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\Biggl(\\prod_{j=1}^n\n \\frac{1}{1-e^{-(\\lambda_j+\\epsilon)(\\log k)^2}}\\Biggr)\n(\\theta+\\epsilon\\omega)^n(x)\\leq {(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n(x),\n\\]\nsince $\\lambda_j(x)+\\epsilon\\ge\\epsilon$ for all $j$ and $x\\in X$. \\eqref{epsilon twisted local holomorphic morse} then follows from extremal characterization \\eqref{extremal for twisted Bergman measure}.\n\\end{proof}\n\\section{Proof of Theorem \\ref{quantization of MA energy} and Corollary \\ref{convergence of Bergman measures}}\nRecall that $X$ is a compact projective manifold of dimension $n$, $L$ be a big and semipositive line bundle over $X$ with fixed smooth metric $h_0^L$ on $L$ with Chern curvature form $\\theta:=c_1(L,h^L_0)\\geq 0$. Let us first deduce Corollary \\ref{convergence of Bergman measures} from Theorem \\ref{quantization of MA energy}.\n\\begin{proof}[Proof of Corollary \\ref{convergence of Bergman measures}] For $f\\in C^0(X)$ and $t\\in \\R$, $u+tf$ is bounded and thus is admissible. We set \n\\[\nf_k(t):=E_k(u+tf),\\quad g(t):=E_\\theta(P_\\theta(u+tf)).\n\\]\nNotice that $u+tf\\geq P_\\theta(u+tf)\\geq m:=\\inf_X(u+tf)>-\\infty$ by definition of envelope and thus, $P_\\theta(u+tf)\\in \\PSH(X,\\theta)\\cap L^\\infty$. By Theorem \\ref{quantization of MA energy}, we obtain that for any $t\\in \\R$,\n\\[\n\\liminf_{k\\to\\infty}f_k(t)\\geq \\lim E_k(P_\\theta(u+tf))=g(t),\\quad \\lim_{k\\to\\infty}f_k(0)=\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u)=g(0).\n\\]\nBy Lemma \\ref{variational formula for QMA along affine path}, $f_k(t)$ is concave and is differentiable at $t=0$:\n\\[\nf_k'(0)=\\int_X f\\beta_{L^k,u}.\n\\]\nOn the other hand, by Lemma \\ref{variation formula of MA energy}, $g(t)$ is differentiable at $t=0$ and\n\\[\ng'(0)=\\int_Xf\\frac{\\theta_u^n}{\\vol(\\{\\theta\\})}.\n\\]\nThe result follows from the elementary lemma proved in \\cite[Lemma 3.1]{BBWN}.\n\\end{proof}\n\\begin{rmk}\n In fact, the arguments in Theorem \\ref{upper bound estimate} below shows that $\\lim_{k\\to\\infty}f_k(t)=g(t)$.\n\\end{rmk}\n\nWe now begin with the upper bound in Theorem \\ref{quantization of MA energy} which holds for any finite-energy potentials.\n\\begin{thm}\\label{upper bound estimate} For $u\\in \\E^1(X,\\theta)$, $\\limsup_{k\\to\\infty}E_k(u)\\leq E_\\theta(u)$.\n\\end{thm}\n\\begin{proof} The proof goes exactly as in \\cite[Theorem 3.5]{BF14} and \\cite[Theorem A]{BB10}. Since $u$ is upper semi-continuous, we can find a sequence $\\phi_j\\in C^\\infty(X)$ such that $\\phi_j\\downarrow u$. Clearly, $\\phi_j$ is admissible and $E_k(u)\\leq E_k(\\phi_j)$ by Lemma \\ref{properties of Hilb and QMA} (ii). By Corollary \\ref{Quantization of MOnge-Ampere energy for smooth functions}, \n\\[\n\\limsup_{k\\to\\infty}E_k(u)\\leq \\lim_{k\\to\\infty}E_k(\\phi_j)=E_\\theta(P_\\theta(\\phi_j)),\\quad \\forall j\\in \\N.\n\\]\nOn the other hand, by definition of envelope, $u\\leq P_\\theta(\\phi_j)\\leq \\phi_j$ and thus $P_\\theta(\\phi_j)\\downarrow u$. Then\n\\[\n\\lim_{k\\to\\infty}E_k(u)\\leq\\lim_{j\\to\\infty}E_\\theta(P_\\theta(\\phi_j))=E_\\theta(u),\n\\]\nby continuity of Monge--Amp\\`ere energy along decreasing sequences. \n\\end{proof}\n\\begin{rmk} The proof actually shows that Theorem \\ref{upper bound estimate} holds for any big and nef line bundle $L$.\n\\end{rmk}\n\\begin{proof}[Proof of Theorem \\ref{quantization of MA energy}] It remains to prove that for any $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\begin{equation}\\label{lower bound on energy}\n \\liminf_{{k\\to\\infty}}E_k(u)\\geq E_\\theta(u).\n\\end{equation} \nFollowing \\cite{BF14}, we consider the following functional on $\\E^1(X,\\theta)$:\n\\[\n\\mathcal{F}_k(u):=E_k(u)-E_\\theta(u),\\quad u\\in \\E^1(X,\\theta).\n\\]\nBy Lemma \\ref{Properties on Energy and E1} (iii) and Lemma \\ref{properties of Hilb and QMA} (ii), $\\F_k$ is continuous along decreasing sequences in $\\E^1(X,\\theta)$. Also, by definition \\eqref{MA energy} and Lemma \\ref{properties of Hilb and QMA} (iii), $\\F_k(u+C)=\\F_k(u)$ for $C\\in \\R$. For fixed $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we may assume that $u\\leq 0$ by addition of a constant. By Lemma \\ref{minimal singular/finite energy geodesic} (iii), $\\dot{u}_0\\in L^\\infty(X)$.\n\nNow, for fixed $\\epsilon\\in \\Q^+$ and an positive line bundle $(A,h_0^A)$ with curvature $\\omega$, we set $\\omega_\\epsilon:=\\theta+\\epsilon\\omega$. By Theorem \\ref{Bergman kernel comparison} and \\eqref{epsilon twisted local holomorphic morse}, \n\\[\n\\frac{1}{N_k}B_{L^k,0}\\leq N_k^{-1}\\left(1+\\frac\n{C}{\\epsilon k}\\right)^2B_{k,\\epsilon}\\leq \\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n\n\\]\nAs $\\dot{u}_0\\leq 0$, we obtain\n\\begin{equation}\\label{key estimate}\n\\F_k(u)\\geq \\int_X\\dot{u}_0\\left(\\left(1+\\frac{C}{\\epsilon k}\\right)^2\\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n-\\frac{1}{\\vol L}\\theta^n\\right). \n\\end{equation}\nTaking $k\\to \\infty$, we see that for fixed $\\epsilon\\in \\Q^+$, since $\\rho_k,\\td{\\rho}_k\\to 0$ and from \\eqref{volume of a line bundle} (since $L$ is big):\n\\[\n1+\\frac{C}{\\epsilon k}\\to 1,\\quad \\frac{k^n/n!}{N_k}\\to \\frac{1}{\\vol L},\\quad\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\to 1.\n\\]\nAs a result, we obtain for fixed $\\epsilon\\in \\Q^+$,\n\\[\n\\liminf_{k\\to\\infty}\\F_k(u)\\geq \\frac{1}{\\vol L}\\int_X \\dot{u}_0\\left(\\omega_\\epsilon^n-\\theta^n\\right)=\\frac{1}{\\vol L}\\sum_{j=1}^n\\int_X\\dot{u}_0\\theta^{n-j}\\wedge (\\epsilon \\omega)^j.\n\\]\nTaking $\\epsilon\\to 0+$, we obtain the result.\n\\end{proof}\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any general sufficient condition.\n\\end{rmk}\n\\printbibliography",
    "post_theorem_intro_text_len": 6334,
    "post_theorem_intro_text": "A result such as Theorem~\\ref{quantization of MA energy} was raised as an open problem in Berman–Freixas i Montplet \\cite[Remark~3.6]{BF14}, and our theorem provides a partial answer in the big and semipositive setting.\nWhen $L$ is ample and no adjoint twisting is present, Donaldson \\cite{DonaldsonScalII} first established the convergence for smooth Kähler potentials; this was later extended to continuous potentials by Berman–Boucksom \\cite[Theorem~A]{BB10} and to all finite-energy potentials by Darvas–Lu–Rubinstein \\cite{DLR}.\nIn the adjoint setting, Berman–Freixas i Montplet \\cite[Theorem~3.5]{BF14} proved the quantization of energy for finite-energy potentials, and subsequently Darvas–Xia \\cite{DX22} obtained the result for arbitrary twisting.\nIn the big, untwisted case, Berman–Boucksom \\cite[Theorem~A]{BB10} showed convergence for all continuous metrics.\nMore recently, Darvas–Xia \\cite{DX24} proved a “partial quantization’’ theorem for continuous metrics under arbitrary twisting on a pseudoeffective line bundle.\n\nTheorem \\ref{quantization of MA energy} has an immediate application to quantization of non-pluripolar measure. For an admissible $u$, we fix a basis $\\{S_1,\\dots,S_{N_k}\\}$ of $H^0(X,L^k\\otimes K_X)$ with $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Then we define \\textit{Bergman measure} and its normalization by\n\t\\begin{equation}\\label{Bergman measure}\n\tB_{L^k,u}(x)=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_u^L)^k}=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_0^L)^k}e^{-ku(x)},\\quad \\beta_{L^k,u}=\\frac{1}{N_k}B_{L^k,u}.\n\t\\end{equation}\nCoupling with  a standard trick in Berman--Boucksom--Witt Nystr\\\"om \\cite{BBWN}, Theorem \\ref{quantization of MA energy} yields the following weak convergence of normalized Bergman measures to non-pluripolar measures\n\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}\nOur approach refines the arguments in \\cite{BF14} as follows. Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in \\mathbb Q^+$. We prove a comparison Theorem for Bergman measure to $L$ and to small ample twist $L\\otimes A^\\varepsilon$ with almost optimal asymptotic bound (Theorem \\ref{Bergman kernel comparison}). This allows us to apply Berman's local Morse inequality \\cite[Proposition 2.5]{BF14} to the twisted bundle with explicit dependence on $\\varepsilon$, see  \\eqref{epsilon twisted local holomorphic morse}. Combining this with the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures, and the convexity of the quantized Monge–Ampère energy along weak geodesics,  we can carry out the proof of Theorem \\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}. However, as we will explain in Remark \\ref{pitfall for finite energy potentials}, our method does not presently extend to arbitrary finite-energy potentials, due to integrability issues for initial tangent vectors of weak geodesics. \n\nAs this work was being finalized, Siarhei Finski kindly informed us of his related work \\cite{FinskiBig}, in which he quantizes the Darvas–Mabuchi $d_p$ distances on big line bundles and for potentials of the type $P_\\theta(f)$ (defined below \\ref{MA envelope}), where $f$ is continuous. In particular, for $p=1$ and $\\theta$ semipositive, his result implies the version of Theorem~\\ref{quantization of MA energy} for such potentials,  a subset of the class $\\PSH(X,\\theta)\\cap L^\\infty$. Our setting and approach differ from his in two main aspects. His quantization is formulated using the untwisted section spaces $H^0(X,L^k)$, whereas we work in the adjoint setting. This allows us to apply the Berndtsson–P\\u{a}un positivity theorem directly. Additionally, he reduces the problem to the ample case via passing to certain birational models over $X$, while our reduction is by twisting $L^k\\otimes K_X$ by an ample line bundle on $X$ as explained above.\n\n\\subsection*{Organization}\nWe briefly describe the structure of the paper.\n\nIn Section~2 we collect the necessary background from pluripotential theory,  following \\cite{BEGZ,BBGZ,DDL18,DDL18L1}. We also recall the basic variational formulas for \\(E_\\theta\\).\n\nSection~3 is devoted to Bergman measures and quantized Monge--Amp\\`ere energies in the adjoint setting. We prove basic properties of $B_{L^k,u}$ and $E_k(u)$ and discuss their variational formula. Next, we recall the positivity of direct images à la Berndtsson–P\\u{a}un \\cite{BerndtssonPositivity1,BerndtssonPositivity2,BP08,PT18} and the relation with convexity of $E_k$ along weak geodesics. Finally, we recall Berman’s local holomorphic Morse inequalities and the convergence of Bergman measures at equilibrium, adapting the arguments of \\cite{Ber09,BB10} to our adjoint framework.\n\nSection 4 is the new input in our argument.  We prove the the comparison Theorem  (Theorem \\ref{Bergman kernel comparison}).    This follows from choosing peak sections with asymptotically optimal $L^\\infty$-control from a pointwise semi-classical Ohsawa--Takegoshi type extension theorem for powers of an ample line bundle (Theorem \\ref{Pointwise Semi-Classical OT}). We also discuss the relation of Theorem \\ref{Pointwise Semi-Classical OT} with recent work of Finski \\cite[Theorem 1.10]{Finski1}. We also present a  local Morse inequality in ample twisting setting.\n\nFinally, in Section~5 we combine these ingredients to prove Theorems~\\ref{quantization of MA energy} and Corollary  \\ref{convergence of Bergman measures} and conclude with obstacles to aribtrary finite-energy potentials in Remark \\ref{pitfall for finite energy potentials}.\n\\subsection*{Acknowledgment}  This work was supported by NSF Grant DMS-2405274 and the Hauptman Fellowship. The author is deeply grateful to his advisor, Tamás Darvas, for his constant support and guidance. The author also wishes to thank Siarhei Finski for his enlightening lecture and discussions at the summer school at the Rényi Institute, Budapest, and for kindly sharing his preprint \\cite{FinskiBig} along with related insights.",
    "sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound (Theorem~\\ref{Bergman kernel comparison}); (ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$ (see \\eqref{epsilon twisted local holomorphic morse}); (iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to “carry out the proof of Theorem~\\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}.” The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics” (Remark~\\ref{pitfall for finite energy potentials}).",
    "expanded_sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound. We first prove the following theorem. \n\n\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}\n\n(ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$, namely\n\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}\n\n(iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to carry out the proof of the main theorem for bounded potentials following the standard argument as in Berman–Boucksom, Theorem A in \\cite[Theorem A]{BB10} and Berman–Fujita, Theorem 3.5 in \\cite[Theorem 3.5]{BF14}.\n\nThe method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics.” More precisely:\n\n\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}",
    "expanded_theorem": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Implication"
    ],
    "mcq": {
      "question": "Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and choose a smooth Hermitian metric $h_0^L$ on $L$ with semipositive Chern curvature $\\theta:=c_1(L,h_0^L)$. For $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$, where $\\PSH(X,\\theta)$ denotes the $\\theta$-plurisubharmonic functions (so $\\theta+dd^c u\\ge 0$ in the sense of currents), define for each $k\\in\\mathbb N$ the Hilbert norm on $H^0(X,L^k\\otimes K_X)$ by\n\\[\n(s_1,s_2)_{\\mathrm{Hilb}_k(u)}:=\\int_X \\langle s_1,s_2\\rangle_{(h_0^L)^k}e^{-ku},\n\\]\nand the quantized Monge--Amp\\`ere energy by\n\\[\nE_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det \\mathrm{Hilb}_k(u)}{\\det \\mathrm{Hilb}_k(0)}\\right),\n\\]\nwhere $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Let $E_\\theta(u)$ denote the Monge--Amp\\`ere (Aubin--Yau) energy on $\\PSH(X,\\theta)$, i.e. the primitive of the non-pluripolar Monge--Amp\\`ere operator. Under these assumptions, which statement about $E_k(u)$ holds as $k\\to\\infty$?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u),\n\\]\nand in fact equality holds only when the semipositive form \\(\\theta\\) is K\\\"ahler."
        },
        {
          "label": "C",
          "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u).\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)\\), there exists a subsequence \\(k_j\\to\\infty\\) such that\n\\[\n\\lim_{j\\to\\infty} E_{k_j}(u)=E_\\theta(u),\n\\]\nand the convergence is uniform in \\(u\\) over bounded subsets of \\(\\PSH(X,\\theta)\\cap L^\\infty(X)\\).\n\\]"
        },
        {
          "label": "E",
          "text": "If \\(L\\) is big and semipositive, then for any \\(u\\in \\PSH(X,\\theta)\\cap \\mathcal E^1(X,\\theta)\\), one has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "semipositive-nonample equality case",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "full limit equality weakened to upper-bound asymptotics",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "pointwise convergence upgraded to uniform-in-u subsequential convergence",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "boundedness hypothesis replaced by finite-energy class",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and setup but does not reveal the asymptotic conclusion. The correct convergence statement is not stated or strongly hinted at."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is essentially the standard theorem statement in this setting, so the item is close to theorem recall. However, it is not a pure restatement because the alternatives include stronger, weaker, and misdirected variants."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact theorem from nearby false or weaker claims, especially regarding boundedness, identification of the limit, and uniform rates. Still, the item primarily tests recognition of the precise statement rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common errors: overgeneralizing the class of potentials, settling for a weaker convergence claim, asserting an unjustified uniform O(1/k) rate, or confusing the limiting energy functional."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.11523v1",
    "paper_link": "http://arxiv.org/abs/2512.11523v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]",
      "start_pos": 11715,
      "end_pos": 11923,
      "label": "quantization of MA energy"
    },
    "ref_dict": {
      "epsilon twisted local holomorphic morse": "\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}",
      "Pointwise Semi-Classical OT": "\\begin{thm}\\label{Pointwise Semi-Classical OT} There exists $C>0$ and $p_1\\in \\N$ such that for any $p\\geq p_1$, any $x\\in X$, and any $v\\in (A^p)_x$,  one can find a global holomorphic section $a=a_{p,x,v}\\in H^0(X,A^p)$ with\n\\begin{equation}\\label{OT bound}\n    a(x)=v,\\quad |v|_{(h_0^A)^p}\\leq \\sup_X|a|_{(h_0^A)^p}\\leq \\left(1+\\frac{C}{p}\\right)|v|_{(h_0^A)^p},\n\\end{equation}\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}",
      "convergence of Bergman measures": "\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}",
      "Bergman kernel comparison": "\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}",
      "Bergman Measures, Positivity of Direct Images, and Equilibrium": "\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}",
      "quantization of MA energy": "\\begin{thm}\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]\n\\end{thm}",
      "MA envelope": "\\begin{equation}\\label{MA envelope}\n\t\tP_\\theta(f):=usc(\\sup\\{u\\in \\PSH(X,\\theta):u\\leq f\\})\n\t\\end{equation}",
      "pitfall for finite energy potentials": "\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}",
      "Hilbert norm": "\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}",
      "volume of a line bundle": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}",
      "finite energy space and weak geodesic": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}",
      "on diagonal Bergman kernel": "\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 5031,
    "pre_theorem_intro_text": "Given an ample line bundle \\(L\\) over a complex projective manifold \\(X\\), endowed with the Kähler form \\(\\omega\\) polarized by \\(L\\), a central theme in Kähler geometry—first articulated by Yau \\cite[p.~139]{Yau}—is the approximation of transcendental geometric objects on \\((X,\\omega)\\) by asymptotic algebraic data coming from the powers \\(L^k:=L^{\\otimes k}\\). This is the philosophy of \\emph{Kähler quantization}.\n\nThe initial proposal is to approximate the infinite-dimensional space\n\\[\n   \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished  subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nDonaldson later proposed that not only the metrics but also the \\emph{geometry} of \\(\\mathcal{H}_\\omega\\) should be captured by the finite-dimensional geometries of \\(\\mathcal{H}_k\\) \\cite[p.~483]{DonaldsonI}. This principle has since been realized in various settings and has had deep impacts on problems concerning canonical metrics and stability; see, for example, \\cite{PS,SongZelditchqu,CS,BerndtssonProb,DLR,DonaldsonI,DonaldsonScalII,RTZ,Zhang} among a rapidly expanding body of literature. We refer to \\cite{Mama} for a detailed exposition of some classical developments in this direction.\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\)  only holds in a weak sense.  The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to  Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive.  Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.  \n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with  the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written  as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding  a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n    E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the  quantization of Monge--Amp\\`ere energy in adjoint twisting.",
    "context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}",
    "full_context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}\n\nRescaling $w=\\sqrt{k}z$ on the standard shrinking polydisc gives the uniform approximation\n\\[\nk\\Phi_\\epsilon(w/\\sqrt{k})\n=\\Phi_\\epsilon^0(w)+\\rho_k+\\epsilon\\widetilde\\rho_k,\n\\qquad \n\\rho_k,\\widetilde\\rho_k\\to0.\n\\]\nRepeating the submean-value argument used for Theorem~\\ref{Berman's local Morse inequality} then yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\le\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\frac{n!dV_n}{(2\\pi)^n\\prod_{j=1}^n\\int_0^{\\log k} e^{-(\\lambda_j+\\epsilon)r^2}r\\,dr}.\n\\]\nSince the quadratic model satisfies $\\lambda_j+\\epsilon\\ge\\epsilon>0$, the Gaussian integrals in the denominator behave exactly as in the proof of Theorem~\\ref{Berman's local Morse inequality}, with $\\lambda_j$ replaced by $\\lambda_j+\\epsilon$. \nUsing the identity\n\\[\n(\\theta+\\epsilon\\omega)^n\n= \\frac{n!}{\\pi^n}\\!\\left(\\prod_{j=1}^n (\\lambda_j+\\epsilon)\\right)dV_n,\n\\]\nthe same computation as in the untwisted case yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\;\\le\\;\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\Biggl(\\prod_{j=1}^n\n \\frac{1}{1-e^{-(\\lambda_j+\\epsilon)(\\log k)^2}}\\Biggr)\n(\\theta+\\epsilon\\omega)^n(x)\\leq {(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n(x),\n\\]\nsince $\\lambda_j(x)+\\epsilon\\ge\\epsilon$ for all $j$ and $x\\in X$. \\eqref{epsilon twisted local holomorphic morse} then follows from extremal characterization \\eqref{extremal for twisted Bergman measure}.\n\\end{proof}\n\\section{Proof of Theorem \\ref{quantization of MA energy} and Corollary \\ref{convergence of Bergman measures}}\nRecall that $X$ is a compact projective manifold of dimension $n$, $L$ be a big and semipositive line bundle over $X$ with fixed smooth metric $h_0^L$ on $L$ with Chern curvature form $\\theta:=c_1(L,h^L_0)\\geq 0$. Let us first deduce Corollary \\ref{convergence of Bergman measures} from Theorem \\ref{quantization of MA energy}.\n\\begin{proof}[Proof of Corollary \\ref{convergence of Bergman measures}] For $f\\in C^0(X)$ and $t\\in \\R$, $u+tf$ is bounded and thus is admissible. We set \n\\[\nf_k(t):=E_k(u+tf),\\quad g(t):=E_\\theta(P_\\theta(u+tf)).\n\\]\nNotice that $u+tf\\geq P_\\theta(u+tf)\\geq m:=\\inf_X(u+tf)>-\\infty$ by definition of envelope and thus, $P_\\theta(u+tf)\\in \\PSH(X,\\theta)\\cap L^\\infty$. By Theorem \\ref{quantization of MA energy}, we obtain that for any $t\\in \\R$,\n\\[\n\\liminf_{k\\to\\infty}f_k(t)\\geq \\lim E_k(P_\\theta(u+tf))=g(t),\\quad \\lim_{k\\to\\infty}f_k(0)=\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u)=g(0).\n\\]\nBy Lemma \\ref{variational formula for QMA along affine path}, $f_k(t)$ is concave and is differentiable at $t=0$:\n\\[\nf_k'(0)=\\int_X f\\beta_{L^k,u}.\n\\]\nOn the other hand, by Lemma \\ref{variation formula of MA energy}, $g(t)$ is differentiable at $t=0$ and\n\\[\ng'(0)=\\int_Xf\\frac{\\theta_u^n}{\\vol(\\{\\theta\\})}.\n\\]\nThe result follows from the elementary lemma proved in \\cite[Lemma 3.1]{BBWN}.\n\\end{proof}\n\\begin{rmk}\n In fact, the arguments in Theorem \\ref{upper bound estimate} below shows that $\\lim_{k\\to\\infty}f_k(t)=g(t)$.\n\\end{rmk}\n\nWe now begin with the upper bound in Theorem \\ref{quantization of MA energy} which holds for any finite-energy potentials.\n\\begin{thm}\\label{upper bound estimate} For $u\\in \\E^1(X,\\theta)$, $\\limsup_{k\\to\\infty}E_k(u)\\leq E_\\theta(u)$.\n\\end{thm}\n\\begin{proof} The proof goes exactly as in \\cite[Theorem 3.5]{BF14} and \\cite[Theorem A]{BB10}. Since $u$ is upper semi-continuous, we can find a sequence $\\phi_j\\in C^\\infty(X)$ such that $\\phi_j\\downarrow u$. Clearly, $\\phi_j$ is admissible and $E_k(u)\\leq E_k(\\phi_j)$ by Lemma \\ref{properties of Hilb and QMA} (ii). By Corollary \\ref{Quantization of MOnge-Ampere energy for smooth functions}, \n\\[\n\\limsup_{k\\to\\infty}E_k(u)\\leq \\lim_{k\\to\\infty}E_k(\\phi_j)=E_\\theta(P_\\theta(\\phi_j)),\\quad \\forall j\\in \\N.\n\\]\nOn the other hand, by definition of envelope, $u\\leq P_\\theta(\\phi_j)\\leq \\phi_j$ and thus $P_\\theta(\\phi_j)\\downarrow u$. Then\n\\[\n\\lim_{k\\to\\infty}E_k(u)\\leq\\lim_{j\\to\\infty}E_\\theta(P_\\theta(\\phi_j))=E_\\theta(u),\n\\]\nby continuity of Monge--Amp\\`ere energy along decreasing sequences. \n\\end{proof}\n\\begin{rmk} The proof actually shows that Theorem \\ref{upper bound estimate} holds for any big and nef line bundle $L$.\n\\end{rmk}\n\\begin{proof}[Proof of Theorem \\ref{quantization of MA energy}] It remains to prove that for any $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\begin{equation}\\label{lower bound on energy}\n \\liminf_{{k\\to\\infty}}E_k(u)\\geq E_\\theta(u).\n\\end{equation} \nFollowing \\cite{BF14}, we consider the following functional on $\\E^1(X,\\theta)$:\n\\[\n\\mathcal{F}_k(u):=E_k(u)-E_\\theta(u),\\quad u\\in \\E^1(X,\\theta).\n\\]\nBy Lemma \\ref{Properties on Energy and E1} (iii) and Lemma \\ref{properties of Hilb and QMA} (ii), $\\F_k$ is continuous along decreasing sequences in $\\E^1(X,\\theta)$. Also, by definition \\eqref{MA energy} and Lemma \\ref{properties of Hilb and QMA} (iii), $\\F_k(u+C)=\\F_k(u)$ for $C\\in \\R$. For fixed $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we may assume that $u\\leq 0$ by addition of a constant. By Lemma \\ref{minimal singular/finite energy geodesic} (iii), $\\dot{u}_0\\in L^\\infty(X)$.\n\nNow, for fixed $\\epsilon\\in \\Q^+$ and an positive line bundle $(A,h_0^A)$ with curvature $\\omega$, we set $\\omega_\\epsilon:=\\theta+\\epsilon\\omega$. By Theorem \\ref{Bergman kernel comparison} and \\eqref{epsilon twisted local holomorphic morse}, \n\\[\n\\frac{1}{N_k}B_{L^k,0}\\leq N_k^{-1}\\left(1+\\frac\n{C}{\\epsilon k}\\right)^2B_{k,\\epsilon}\\leq \\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n\n\\]\nAs $\\dot{u}_0\\leq 0$, we obtain\n\\begin{equation}\\label{key estimate}\n\\F_k(u)\\geq \\int_X\\dot{u}_0\\left(\\left(1+\\frac{C}{\\epsilon k}\\right)^2\\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n-\\frac{1}{\\vol L}\\theta^n\\right). \n\\end{equation}\nTaking $k\\to \\infty$, we see that for fixed $\\epsilon\\in \\Q^+$, since $\\rho_k,\\td{\\rho}_k\\to 0$ and from \\eqref{volume of a line bundle} (since $L$ is big):\n\\[\n1+\\frac{C}{\\epsilon k}\\to 1,\\quad \\frac{k^n/n!}{N_k}\\to \\frac{1}{\\vol L},\\quad\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\to 1.\n\\]\nAs a result, we obtain for fixed $\\epsilon\\in \\Q^+$,\n\\[\n\\liminf_{k\\to\\infty}\\F_k(u)\\geq \\frac{1}{\\vol L}\\int_X \\dot{u}_0\\left(\\omega_\\epsilon^n-\\theta^n\\right)=\\frac{1}{\\vol L}\\sum_{j=1}^n\\int_X\\dot{u}_0\\theta^{n-j}\\wedge (\\epsilon \\omega)^j.\n\\]\nTaking $\\epsilon\\to 0+$, we obtain the result.\n\\end{proof}\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any general sufficient condition.\n\\end{rmk}\n\\printbibliography",
    "post_theorem_intro_text_len": 6334,
    "post_theorem_intro_text": "A result such as Theorem~\\ref{quantization of MA energy} was raised as an open problem in Berman–Freixas i Montplet \\cite[Remark~3.6]{BF14}, and our theorem provides a partial answer in the big and semipositive setting.\nWhen $L$ is ample and no adjoint twisting is present, Donaldson \\cite{DonaldsonScalII} first established the convergence for smooth Kähler potentials; this was later extended to continuous potentials by Berman–Boucksom \\cite[Theorem~A]{BB10} and to all finite-energy potentials by Darvas–Lu–Rubinstein \\cite{DLR}.\nIn the adjoint setting, Berman–Freixas i Montplet \\cite[Theorem~3.5]{BF14} proved the quantization of energy for finite-energy potentials, and subsequently Darvas–Xia \\cite{DX22} obtained the result for arbitrary twisting.\nIn the big, untwisted case, Berman–Boucksom \\cite[Theorem~A]{BB10} showed convergence for all continuous metrics.\nMore recently, Darvas–Xia \\cite{DX24} proved a “partial quantization’’ theorem for continuous metrics under arbitrary twisting on a pseudoeffective line bundle.\n\nTheorem \\ref{quantization of MA energy} has an immediate application to quantization of non-pluripolar measure. For an admissible $u$, we fix a basis $\\{S_1,\\dots,S_{N_k}\\}$ of $H^0(X,L^k\\otimes K_X)$ with $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Then we define \\textit{Bergman measure} and its normalization by\n\t\\begin{equation}\\label{Bergman measure}\n\tB_{L^k,u}(x)=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_u^L)^k}=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_0^L)^k}e^{-ku(x)},\\quad \\beta_{L^k,u}=\\frac{1}{N_k}B_{L^k,u}.\n\t\\end{equation}\nCoupling with  a standard trick in Berman--Boucksom--Witt Nystr\\\"om \\cite{BBWN}, Theorem \\ref{quantization of MA energy} yields the following weak convergence of normalized Bergman measures to non-pluripolar measures\n\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}\nOur approach refines the arguments in \\cite{BF14} as follows. Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in \\mathbb Q^+$. We prove a comparison Theorem for Bergman measure to $L$ and to small ample twist $L\\otimes A^\\varepsilon$ with almost optimal asymptotic bound (Theorem \\ref{Bergman kernel comparison}). This allows us to apply Berman's local Morse inequality \\cite[Proposition 2.5]{BF14} to the twisted bundle with explicit dependence on $\\varepsilon$, see  \\eqref{epsilon twisted local holomorphic morse}. Combining this with the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures, and the convexity of the quantized Monge–Ampère energy along weak geodesics,  we can carry out the proof of Theorem \\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}. However, as we will explain in Remark \\ref{pitfall for finite energy potentials}, our method does not presently extend to arbitrary finite-energy potentials, due to integrability issues for initial tangent vectors of weak geodesics. \n\nAs this work was being finalized, Siarhei Finski kindly informed us of his related work \\cite{FinskiBig}, in which he quantizes the Darvas–Mabuchi $d_p$ distances on big line bundles and for potentials of the type $P_\\theta(f)$ (defined below \\ref{MA envelope}), where $f$ is continuous. In particular, for $p=1$ and $\\theta$ semipositive, his result implies the version of Theorem~\\ref{quantization of MA energy} for such potentials,  a subset of the class $\\PSH(X,\\theta)\\cap L^\\infty$. Our setting and approach differ from his in two main aspects. His quantization is formulated using the untwisted section spaces $H^0(X,L^k)$, whereas we work in the adjoint setting. This allows us to apply the Berndtsson–P\\u{a}un positivity theorem directly. Additionally, he reduces the problem to the ample case via passing to certain birational models over $X$, while our reduction is by twisting $L^k\\otimes K_X$ by an ample line bundle on $X$ as explained above.\n\n\\subsection*{Organization}\nWe briefly describe the structure of the paper.\n\nIn Section~2 we collect the necessary background from pluripotential theory,  following \\cite{BEGZ,BBGZ,DDL18,DDL18L1}. We also recall the basic variational formulas for \\(E_\\theta\\).\n\nSection~3 is devoted to Bergman measures and quantized Monge--Amp\\`ere energies in the adjoint setting. We prove basic properties of $B_{L^k,u}$ and $E_k(u)$ and discuss their variational formula. Next, we recall the positivity of direct images à la Berndtsson–P\\u{a}un \\cite{BerndtssonPositivity1,BerndtssonPositivity2,BP08,PT18} and the relation with convexity of $E_k$ along weak geodesics. Finally, we recall Berman’s local holomorphic Morse inequalities and the convergence of Bergman measures at equilibrium, adapting the arguments of \\cite{Ber09,BB10} to our adjoint framework.\n\nSection 4 is the new input in our argument.  We prove the the comparison Theorem  (Theorem \\ref{Bergman kernel comparison}).    This follows from choosing peak sections with asymptotically optimal $L^\\infty$-control from a pointwise semi-classical Ohsawa--Takegoshi type extension theorem for powers of an ample line bundle (Theorem \\ref{Pointwise Semi-Classical OT}). We also discuss the relation of Theorem \\ref{Pointwise Semi-Classical OT} with recent work of Finski \\cite[Theorem 1.10]{Finski1}. We also present a  local Morse inequality in ample twisting setting.\n\nFinally, in Section~5 we combine these ingredients to prove Theorems~\\ref{quantization of MA energy} and Corollary  \\ref{convergence of Bergman measures} and conclude with obstacles to aribtrary finite-energy potentials in Remark \\ref{pitfall for finite energy potentials}.\n\\subsection*{Acknowledgment}  This work was supported by NSF Grant DMS-2405274 and the Hauptman Fellowship. The author is deeply grateful to his advisor, Tamás Darvas, for his constant support and guidance. The author also wishes to thank Siarhei Finski for his enlightening lecture and discussions at the summer school at the Rényi Institute, Budapest, and for kindly sharing his preprint \\cite{FinskiBig} along with related insights.",
    "sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound (Theorem~\\ref{Bergman kernel comparison}); (ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$ (see \\eqref{epsilon twisted local holomorphic morse}); (iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to “carry out the proof of Theorem~\\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}.” The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics” (Remark~\\ref{pitfall for finite energy potentials}).",
    "expanded_sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound. We first prove the following theorem. \n\n\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}\n\n(ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$, namely\n\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}\n\n(iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to carry out the proof of the main theorem for bounded potentials following the standard argument as in Berman–Boucksom, Theorem A in \\cite[Theorem A]{BB10} and Berman–Fujita, Theorem 3.5 in \\cite[Theorem 3.5]{BF14}.\n\nThe method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics.” More precisely:\n\n\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}",
    "expanded_theorem": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Implication"
    ],
    "mcq": {
      "question": "Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and choose a smooth Hermitian metric $h_0^L$ on $L$ with semipositive Chern curvature $\\theta:=c_1(L,h_0^L)$. For $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$, where $\\PSH(X,\\theta)$ denotes the $\\theta$-plurisubharmonic functions (so $\\theta+dd^c u\\ge 0$ in the sense of currents), define for each $k\\in\\mathbb N$ the Hilbert norm on $H^0(X,L^k\\otimes K_X)$ by\n\\[\n(s_1,s_2)_{\\mathrm{Hilb}_k(u)}:=\\int_X \\langle s_1,s_2\\rangle_{(h_0^L)^k}e^{-ku},\n\\]\nand the quantized Monge--Amp\\`ere energy by\n\\[\nE_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det \\mathrm{Hilb}_k(u)}{\\det \\mathrm{Hilb}_k(0)}\\right),\n\\]\nwhere $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Let $E_\\theta(u)$ denote the Monge--Amp\\`ere (Aubin--Yau) energy on $\\PSH(X,\\theta)$, i.e. the primitive of the non-pluripolar Monge--Amp\\`ere operator. Under these assumptions, which statement about $E_k(u)$ holds as $k\\to\\infty$?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u),\n\\]\nand in fact equality holds only when the semipositive form \\(\\theta\\) is K\\\"ahler."
        },
        {
          "label": "C",
          "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u).\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)\\), there exists a subsequence \\(k_j\\to\\infty\\) such that\n\\[\n\\lim_{j\\to\\infty} E_{k_j}(u)=E_\\theta(u),\n\\]\nand the convergence is uniform in \\(u\\) over bounded subsets of \\(\\PSH(X,\\theta)\\cap L^\\infty(X)\\).\n\\]"
        },
        {
          "label": "E",
          "text": "If \\(L\\) is big and semipositive, then for any \\(u\\in \\PSH(X,\\theta)\\cap \\mathcal E^1(X,\\theta)\\), one has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "semipositive-nonample equality case",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "full limit equality weakened to upper-bound asymptotics",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "pointwise convergence upgraded to uniform-in-u subsequential convergence",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "boundedness hypothesis replaced by finite-energy class",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and hypotheses but does not explicitly reveal the asymptotic conclusion. There is no direct answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the exact theorem statement, so it mostly tests recall of the known conclusion rather than deriving a new consequence. The presence of nearby variants prevents it from being a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact convergence statement from stronger or weaker variants, but this is limited. Moreover, choice C is also true if A is true, so the question does not cleanly force a uniquely reasoned selection."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and reflect common theorem-tampering patterns (extra Kahler assumption, weaker limsup statement, stronger uniformity, larger function class). However, C is a genuinely true weaker statement, so the distractor set is ambiguous rather than cleanly single-answer."
      },
      "total_score": 5,
      "overall_assessment": "Moderate-quality MCQ: no answer leakage and some plausible variants, but it is close to theorem recall and is weakened by ambiguity because a weaker true option is also present."
    }
  },
  {
    "id": "2512.11595v1",
    "paper_link": "http://arxiv.org/abs/2512.11595v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}",
      "start_pos": 21673,
      "end_pos": 22123,
      "label": "th:cfs"
    },
    "ref_dict": {
      "fig:FareyPoly": "\\begin{tikzpicture}[\n    level 1/.style = {sibling distance=6cm},\n    level 2/.style = {sibling distance=3cm},\n    level 3/.style = {sibling distance=1.5cm},\n    level distance          = 1cm,\n    edge from parent/.style = {draw},\n    scale=1.2\n    ]\n\n    \\node {$\\frac{x}{x+1}$}\n    child{\n      node {$\\frac{x}{x+2}$}\n      child{\n        node {$\\frac{x}{x+3}$}\n        child{\n          node {$\\frac{x}{x+4}$}\n        }\n        child{\n          node {$\\frac{x+3}{x+4}$}\n        }\n      }\n      child{\n        node {$\\frac{x+2}{x+3}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x}{x^2+4x+2}$}\n        }\n      }\n    }\n    child{\n      node {$\\frac{x+1}{x+2}$}\n      child{\n        node {$\\frac{x^2+x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+2}$}\n        }\n      }\n      child{\n        node {$\\frac{x^2+2x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+3}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+3}$}\n        }\n      }\n    };\n\\end{tikzpicture}\n\n        \\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n    \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n    \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n    \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.\n\n\\begin{theorem}\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}\n\\end{theorem}\nNote that from (1) in Theorem \\ref{th:cfs} it follows that the rational functions in the tree don't intersect on $(0,\\infty)$.\n\n\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\nWe begin \\Cref{sec:cfs} with a derivation of this tree. For Theorem \\ref{th:cfs} we use the close relationship between the tree and a family of continued fractions explained in Section \\ref{sec:cfresults}. Theorem \\ref{th:dense} is proven in \\Cref{sec:dense} by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$; comments are included regarding the possibility of expanding this result to $\\alpha \\in (-1,0)$. A similar tree derived from the theory of backwards continued fractions is presented in \\Cref{sec:other trees}.\nWe will see a similar behaviour in view of Theorem \\ref{th:cfs} and a very different one in view of Theorem \\ref{th:dense} for this tree.\n\n\\section{Forward Farey Tree}\\label{sec:cfs}\n\\subsection{Background and Definition}\nThe \\textit{Gauss map} $T:[0,1) \\mapsto [0,1)$, given by\n\\[T(t) = \\begin{cases}\n    \\frac{1}{t} - \\floor{\\frac{1}{t}} & t\\neq 0\\\\\n    0 & t=0,\n\\end{cases}\\]\nmay be used to generate the \\textit{regular continued fraction} expansion of $t$:\n\\[ t = \\cfrac{1}{a_1+\\cfrac{1}{a_2+\\cfrac{1}{a_3+\\ddots}}},\\]\nwhere $a_i =a(T^{i-1}(t))= \\floor{1/T^{i-1}(t)}$. This map is the `sped-up' version of the \\textit{Farey map} $F$, given by\n\\[ F(t) = \\begin{cases}\n    F_0(t) = \\frac{1-t}{t} & x \\geq 1/2\\\\\n    F_1(t) = \\frac{t}{1-t} & x \\leq 1/2\n\\end{cases}\\]\nwith the observation that \n\\begin{equation}\\label{eqn: RCF functional}T(t) = F_0 \\circ F_1 ^{a(t)-1} (t).\\end{equation}\nNote that $a(t)-1=\\min(n\\in\\mathbb{N}_0 : T^n(t)\\in [\\frac{1}{2},1])$ i.e. it is first hitting time of $x$ to the interval $[\\frac{1}{2},1]$ (possibly zero) and $T$ can be understood as the composition of $F_0$ and the induced transformation on this interval (where a return time of zero is allowed).\n\nA generalization of regular continued fractions, $N$-continued fractions, allow for all numerators to be some fixed $N \\in \\mathbb{Z}\\setminus\\{0\\}$. These were first studied in  \\cite{AW11,BGRKWY208} and later in many other papers such as \\cite{CK,DO18,DKW13,JKN,KL} . Most papers study positive values of $N$ and in the remainder of this section we will also take $N>0$. For negative $N$, there is a relation with the backward continued fractions in Section \\ref{sec:other trees}. One of the key differences with regular continued fractions is that there is not a unique continued fraction for every $t\\in(0,1)$. For some studied algorithms generating $N$-continued fractions it is proven that there are quadratic irrationals without periodic expansion (and even rationals with aperiodic expansions), see \\cite{KL}. For the greedy $N$-continued fractions, which we have here, there is strong numeric evidence that there are quadratic irrationals with an aperiodic expansion (see \\cite{DKW13}), though this problem is still open. The greedy $N$-continued fractions can be generated with a similar map as the Gauss map namely by \\[ T_N(t) = \\begin{cases}\n   \\left(\\frac{N}{t}-N\\right) - \\floor{\\frac{N}{t}-N} & t \\neq 0\\\\\n     0 & t=0,\n\\end{cases}\\]\nwhich generates\n\\[ t= \\cfrac{N}{N+(a_1-1)+\\cfrac{N}{N+(a_2-1)+\\cfrac{N}{N+(a_3-1)+\\ddots}}},\\]\nwhere now $a_i=a(N,t)= \\floor{N/T_N^{i-1}(t)-N}+1$. Then $T_N(x)$ is a sped-up version of the associated $N$-Farey map\n\\[F_N(t) = \\begin{cases}\n    F_{N,0}(t) = \\frac{N(1-t)}{t} &  t \\geq \\frac{N}{N+1} \\\\\n    F_{N,1}(t) = \\frac{Nt}{N-t} & t \\leq \\frac{N}{N+1}\n\\end{cases}\\]\nwith the observation that\n\\begin{equation}\\label{eqn: N-CF functional}T_N(t) = F_{N,0} \\circ F_{N,1}^{a(N,t)-1}(t).\\end{equation}\nSee Figure \\ref{fig:Fareymap} (right) for an example of $N=2$. It is important to point out that this definition of $a_i=a(N,t)$ is not typical: more typical, e.g. in the above references for N-continued fractions, is that $a(N,t)=\\floor{N/t}$ and $T_N(t)$ is equivalent but without the `$-N$' both inside and outside the integer part function. Our definition here differs only by an integer constant for $a$, is identical for $T_N$, gives a similar presentation between \\Cref{eqn: RCF functional} and (\\ref{eqn: N-CF functional}), and is what we will now generalize. \n\nNote that from a dynamical point of view, there is no necessity of taking $N\\in\\mathbb{Z}^+$ in the definition of the map $F_N(t)$. To this end, let us define a map $F_x:[0,1]\\rightarrow[0,1]$ with $x\\in(0,\\infty)$\\footnote{note that for other values of $x$, we have that $\\frac{x}{x+1}$ is not in $[0,1]$  and the map cannot be used to generate continued fractions.} as\n\\begin{equation}\\label{eq:F_x}\n  F_x(t)=\n\\begin{cases}\nF_{x,0}(t)=\\frac{x(1-t)}{t}   & t \\geq \\frac{x}{x+1}\\\\[1ex]\nF_{x,1}(t)=\\frac{xt}{x-t}, & t \\leq \\frac{x}{x+1},\n\\end{cases}  \n\\end{equation}\nsee Figure \\ref{fig:Fareymap} (left) for $x=1/3$.\n\\begin{figure}[ht]\n\t\t\\centering\n\n\t\t\\subfigure{\\begin{tikzpicture}[scale=5]\n\t\t\t\t\\draw[white] (-0.25,0)--(1,0);\n\t\t\t\t\\draw(0,0)node[below]{\\small $0$}--(1,0)node[below]{\\small $1$}--(1,1)--(0,1)node[left]{\\small $1$}--(0,0);\n\n\t\t\t\t\\draw[thick, blue, smooth, samples =20, domain=1/4:1] plot(\\x,{1/(3*\\x)-1/3});\n\t\t\t\t\\draw[thick,blue, smooth, samples =20, domain=0:1/4] plot(\\x,{\\x/(1-3*\\x)});\n\n\t\t\t\t\\draw[dotted](1/4,0)node[below]{\\small $\\frac{1}{4}$}--(1/4,1);\n\n\t\t\\end{tikzpicture}",
      "th:cfs": "\\begin{theorem}\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}\n\\end{theorem}",
      "th:dense": "\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3793,
    "pre_theorem_intro_text": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\begin{figure}[ht]\n    \\begin{center}\n\n\\begin{tikzpicture}[\n    level 1/.style = {sibling distance=6cm},\n    level 2/.style = {sibling distance=3cm},\n    level 3/.style = {sibling distance=1.5cm},\n    level distance          = 1cm,\n    edge from parent/.style = {draw},\n    scale=1.2\n    ]\n\n    \\node {$\\frac{x}{x+1}$}\n    child{\n      node {$\\frac{x}{x+2}$}\n      child{\n        node {$\\frac{x}{x+3}$}\n        child{\n          node {$\\frac{x}{x+4}$}\n        }\n        child{\n          node {$\\frac{x+3}{x+4}$}\n        }\n      }\n      child{\n        node {$\\frac{x+2}{x+3}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x}{x^2+4x+2}$}\n        }\n      }\n    }\n    child{\n      node {$\\frac{x+1}{x+2}$}\n      child{\n        node {$\\frac{x^2+x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+2}$}\n        }\n      }\n      child{\n        node {$\\frac{x^2+2x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+3}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+3}$}\n        }\n      }\n    };\n\\end{tikzpicture}\n\n        \\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n    \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n    \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n    \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.",
    "context": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.",
    "full_context": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.\n\n\\begin{theorem}\\label{th:dense}\n For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\n\\section{Forward Farey Tree}\\label{sec:cfs}\n\\subsection{Background and Definition}\nThe \\textit{Gauss map} $T:[0,1) \\mapsto [0,1)$, given by\n\\[T(t) = \\begin{cases}\n \\frac{1}{t} - \\floor{\\frac{1}{t}} & t\\neq 0\\\\\n 0 & t=0,\n\\end{cases}\\]\nmay be used to generate the \\textit{regular continued fraction} expansion of $t$:\n\\[ t = \\cfrac{1}{a_1+\\cfrac{1}{a_2+\\cfrac{1}{a_3+\\ddots}}},\\]\nwhere $a_i =a(T^{i-1}(t))= \\floor{1/T^{i-1}(t)}$. This map is the `sped-up' version of the \\textit{Farey map} $F$, given by\n\\[ F(t) = \\begin{cases}\n F_0(t) = \\frac{1-t}{t} & x \\geq 1/2\\\\\n F_1(t) = \\frac{t}{1-t} & x \\leq 1/2\n\\end{cases}\\]\nwith the observation that \n\\begin{equation}\\label{eqn: RCF functional}T(t) = F_0 \\circ F_1 ^{a(t)-1} (t).\\end{equation}\nNote that $a(t)-1=\\min(n\\in\\mathbb{N}_0 : T^n(t)\\in [\\frac{1}{2},1])$ i.e. it is first hitting time of $x$ to the interval $[\\frac{1}{2},1]$ (possibly zero) and $T$ can be understood as the composition of $F_0$ and the induced transformation on this interval (where a return time of zero is allowed).\n\nA generalization of regular continued fractions, $N$-continued fractions, allow for all numerators to be some fixed $N \\in \\mathbb{Z}\\setminus\\{0\\}$. These were first studied in \\cite{AW11,BGRKWY208} and later in many other papers such as \\cite{CK,DO18,DKW13,JKN,KL} . Most papers study positive values of $N$ and in the remainder of this section we will also take $N>0$. For negative $N$, there is a relation with the backward continued fractions in Section \\ref{sec:other trees}. One of the key differences with regular continued fractions is that there is not a unique continued fraction for every $t\\in(0,1)$. For some studied algorithms generating $N$-continued fractions it is proven that there are quadratic irrationals without periodic expansion (and even rationals with aperiodic expansions), see \\cite{KL}. For the greedy $N$-continued fractions, which we have here, there is strong numeric evidence that there are quadratic irrationals with an aperiodic expansion (see \\cite{DKW13}), though this problem is still open. The greedy $N$-continued fractions can be generated with a similar map as the Gauss map namely by \\[ T_N(t) = \\begin{cases}\n \\left(\\frac{N}{t}-N\\right) - \\floor{\\frac{N}{t}-N} & t \\neq 0\\\\\n 0 & t=0,\n\\end{cases}\\]\nwhich generates\n\\[ t= \\cfrac{N}{N+(a_1-1)+\\cfrac{N}{N+(a_2-1)+\\cfrac{N}{N+(a_3-1)+\\ddots}}},\\]\nwhere now $a_i=a(N,t)= \\floor{N/T_N^{i-1}(t)-N}+1$. Then $T_N(x)$ is a sped-up version of the associated $N$-Farey map\n\\[F_N(t) = \\begin{cases}\n F_{N,0}(t) = \\frac{N(1-t)}{t} & t \\geq \\frac{N}{N+1} \\\\\n F_{N,1}(t) = \\frac{Nt}{N-t} & t \\leq \\frac{N}{N+1}\n\\end{cases}\\]\nwith the observation that\n\\begin{equation}\\label{eqn: N-CF functional}T_N(t) = F_{N,0} \\circ F_{N,1}^{a(N,t)-1}(t).\\end{equation}\nSee Figure \\ref{fig:Fareymap} (right) for an example of $N=2$. It is important to point out that this definition of $a_i=a(N,t)$ is not typical: more typical, e.g. in the above references for N-continued fractions, is that $a(N,t)=\\floor{N/t}$ and $T_N(t)$ is equivalent but without the `$-N$' both inside and outside the integer part function. Our definition here differs only by an integer constant for $a$, is identical for $T_N$, gives a similar presentation between \\Cref{eqn: RCF functional} and (\\ref{eqn: N-CF functional}), and is what we will now generalize.\n\n\\begin{proposition}\\label{prop: elementary vertex properties without proof}\n At every vertex $[a_1,\\ldots,a_k]_x=p/q$, we have:\n \\begin{itemize}\n \\item $p,q$ are polynomials of the same degree.\n \\item All coefficients of each are positive integers, except the constant term of $p(x)$ which may be zero.\n \\item Term-by-term, the coefficients of $q$ are strictly larger than those of $p$, except for the leading coefficients which are both one.\n \\end{itemize}\n \\begin{proof}\n All properties are preserved by both $\\Phi_0$ and $\\Phi_1$, and are true for the root vertex $p(x)=x$, $q(x)=x+1$.\n \\end{proof}\n\\end{proposition}\n\\begin{proposition}\\label{prop: increasing functions}\n For every vertex $v=[a_1,\\ldots,a_k]_x$, we have both\n \\begin{align}\n v'(x)&> 0 &\\textrm{(wherever defined)}\\\\\n xv'(x) &\\leq v(x) & \\textrm{(with equality if and only if $x=0$ and $v(0)=0$)}\n \\end{align}\n \\begin{proof}\n We induct on $k$, the length of the expansion of $v$. For $k=1$ we have \n \\[v=\\frac{x}{x+a_1-1}\\]\n (where $a_1 \\geq 2$) for which both claims may be trivially verified. Now assume both properties hold for $v$, and for some $a \\in \\mathbb{Z}^+$ set \n \\[V = [a,v]_x = \\begin{cases}\\frac{x}{x+a-1+v} & a \\geq 2 \\textrm{ or } v(0) \\neq 0\\\\ \\frac{1}{1+v/x} & a=1 \\textrm{ and } v(0)=0. \n \\end{cases}\\]\n Note that by \\Cref{prop: elementary vertex properties without proof} $x=0$ is at most a simple root of $v$, so the second case may be properly defined even for $x=0$, and if we establish $v'>0$, then $xv'$ also has at most a simple root at $x=0$.\n In the first case, we compute\n \\begin{align*}\n V'&=\\frac{a-1+v-xv'}{(x+a-1+v)^2}.\\\\\n \\intertext{Since we have assumed $xv'<v$, the first claim now follows.}\n xV'-V&= \\frac{x((a-1)+v-xv')-x(x+a-1+v)}{(x+a-1+v)^2}\\\\\n &=-x^2\\frac{v'+1}{(x+a-1+v)^2}.\\\\\n \\end{align*}\n Which is negative unless $x=0$, in which case it follows that $V(0)=0$ as well.\n\n\\section{Backward Farey Tree}\\label{sec:other trees}\nThe results we have on the Farey polynomial tree raise the question of how general these results are: there are many other different families of continued fraction algorithms. In this section we take the backward continued fractions as a starting point and compare it with Theorem \\ref{th:cfs} and Theorem \\ref{th:dense}. \nJust as for regular continued fractions, the backward continued fractions appears in different fields of mathematics. There are many nice papers studying backwards continued fractions, but we want to highlight \\cite{MO19,MO20} where backwards (and regular) continued fractions are used to draw connections between graph theory, combinatorics, number theory, and algebra. \nFor backward continued fractions, a parametrised family is already studied in the form of $ 1/(u(1-x)) \\mod 1$ where $u\\in(0,4)$, introduced and studied in \\cite{GH96a,GH96b} and more recently \\cite{LS20, LS22}. In contrast to those papers, in this article we will make the branches full again by shifting the digit set in order to get a better comparison with the Farey polynomial tree. Of course, instead of having $ 1/(u(1-x)) \\mod 1$, we can also put the parameter in the numerator. This leads to the fast map \n\\begin{equation}\n T_x(t)=\\frac{x}{1-t}- x -\\left\\lfloor \\frac{x}{1-t}-x \\right\\rfloor.\n\\end{equation}\nThe continued fractions are of the form \n\\[\nt=1-\\cfrac{x}{x + a_1-\\cfrac{x}{x+ a_2 -\\ddots}}\n\\]\nwhere $a_i\\in\\mathbb{Z^+}$, given by\n\\[a(t)=1+\\floor{\\frac{x}{1-t}-x}.\\] Note that, when taking $x\\in \\mathbb{Z^+}$, you essentially get $N$-continued fractions with a negative value for $N$. \nThe slow map is given by\n\\begin{equation}\\label{eq:F_x backwards}\n F_x(t)=\n\\begin{cases}\n\\dfrac{xt}{1-t} & t\\in [0,\\frac{1}{x+1})\\\\[1ex]\n1-\\displaystyle\\frac{x(t-1)}{1-t-x}, & t\\in [\\frac{1}{x+1},1],\n\\end{cases} \n\\end{equation}\n see Figure \\ref{fig:Fareymapbackward}.\n\n\\begin{theorem}\\label{th:backwardcf}\n For a fixed value of $x$, we consider the function which maps the backward tree to the set of values given by evaluating every vertex at this value of $x$.\n \\begin{enumerate}\n \\item Setting a value $x\\geq 1$ injectively maps the backward tree to a dense subset of $(0,1)$. \\\\For $x\\in(0,1)$, the mapping is injective and maps to a subset of $(0,1)$ but the image is not dense.\n \\item Setting a value $x \\geq 1$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n \\end{enumerate}\n\\end{theorem}",
    "post_theorem_intro_text_len": 1118,
    "post_theorem_intro_text": "Note that from (1) in Theorem \\ref{th:cfs} it follows that the rational functions in the tree don't intersect on $(0,\\infty)$.\n\n\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\nWe begin \\Cref{sec:cfs} with a derivation of this tree. For Theorem \\ref{th:cfs} we use the close relationship between the tree and a family of continued fractions explained in Section \\ref{sec:cfresults}. Theorem \\ref{th:dense} is proven in \\Cref{sec:dense} by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$; comments are included regarding the possibility of expanding this result to $\\alpha \\in (-1,0)$. A similar tree derived from the theory of backwards continued fractions is presented in \\Cref{sec:other trees}.\nWe will see a similar behaviour in view of Theorem \\ref{th:cfs} and a very different one in view of Theorem \\ref{th:dense} for this tree.",
    "sketch": "For Theorem~\\ref{th:cfs} the introduction says it is proved by “us[ing] the close relationship between the tree and a family of continued fractions explained in Section~\\ref{sec:cfresults}.”\n\n(No further proof structure for Theorem~\\ref{th:cfs} is described here; the only other proof outline given is for Theorem~\\ref{th:dense}, which is “proven … by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$.”)",
    "expanded_sketch": "To prove the main theorem, the introduction says it is proved by “us[ing] the close relationship between the tree and a family of continued fractions explained later.”\n\n(No further proof structure for the main theorem is described here; the only other proof outline given is for \\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}, which is “proven … by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$.”)",
    "expanded_theorem": "\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Consider the Farey polynomial tree whose vertices are rational functions in a variable $X$, with root $\\dfrac{X}{X+1}$, and where each vertex $\\dfrac{p(X)}{q(X)}$ has two children given by\n\\[\n\\Phi_0\\!\\left(\\frac{p(X)}{q(X)}\\right)=\n\\begin{cases}\n\\dfrac{Xq(X)}{Xq(X)+p(X)} & \\text{if } p(0)\\neq 0,\\\\[1ex]\n\\dfrac{q(X)}{q(X)+p(X)/X} & \\text{if } p(0)=0,\n\\end{cases}\n\\qquad\n\\Phi_1\\!\\left(\\frac{p(X)}{q(X)}\\right)=\n\\begin{cases}\n\\dfrac{Xp(X)}{Xq(X)+p(X)} & \\text{if } p(0)\\neq 0,\\\\[1ex]\n\\dfrac{p(X)}{q(X)+p(X)/X} & \\text{if } p(0)=0.\n\\end{cases}\n\\]\nFor a fixed real number $x$, evaluate every vertex at $X=x$, thereby obtaining a map from the vertices of the tree to real numbers. Which statement holds about this evaluation map?",
      "correct_choice": {
        "label": "A",
        "text": "For every fixed $x>0$, the evaluation map is injective and its image is a dense subset of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every fixed $x\\ge 1$, the evaluation map is injective and its image is a dense subset of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."
        },
        {
          "label": "C",
          "text": "For every fixed $x\\in\\mathbb{Z}^+$, the evaluation map is injective and its image is a dense subset of $(0,1)$, and in fact it is a bijection onto $\\mathbb{Q}\\cap(0,1)$."
        },
        {
          "label": "D",
          "text": "For every fixed $x>0$, the evaluation map is injective and its image is all of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."
        },
        {
          "label": "E",
          "text": "For every fixed $x>0$, the evaluation map has dense image in $(0,1)$, and it is injective if and only if $x\\in\\mathbb{Z}^+$. Moreover, in that case the map is a bijection onto $\\mathbb{Q}\\cap(0,1)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "parameter range $x>0$ replaced by $x\\ge 1$",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "restricts the universal quantifier from all $x>0$ to integer parameters only",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dense image strengthened to surjective image $(0,1)$",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "injectivity made dependent on integrality rather than holding for all $x>0$",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the construction and asks for the correct global property of the evaluation map. It does not explicitly state or strongly hint at the correct conclusion."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-identification item: the correct option is essentially the full theorem statement, while the others are nearby variants. It is not a direct restatement of the stem, but it still leans heavily on recall/recognition rather than independent derivation."
      },
      "GPS": {
        "score": 1,
        "justification": "The student must compare subtle logical differences among the options (all x>0 vs integers, dense image vs surjective image, unconditional injectivity vs conditional injectivity). That creates moderate reasoning pressure, but the option structure still makes it more recognition-based than genuinely generative."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: they vary quantifier range, strength of image statement, and dependence of injectivity on integrality. These reflect common theorem-misremembering and over/under-strengthening errors."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with little answer leakage and strong distractors, though it tests nuanced recall/comparison more than fully generative reasoning."
    }
  },
  {
    "id": "2512.11601v1",
    "paper_link": "http://arxiv.org/abs/2512.11601v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.",
      "start_pos": 103245,
      "end_pos": 103752,
      "label": "thm:K2"
    },
    "ref_dict": {
      "cor:behave": "\\begin{corollary}\\label{cor:behave}\n  For the sequences defined by \\eqref{eq:mexdef}, \n  there exist a bounded function $\\lambda_\\ell:\\mathbb{N}\\to\\mathbb{Z}$ such that, for all $n$,   \n  \\[\n   a_n=\\lfloor (n+\\ell)\\phi \\rfloor +\\lambda_\\ell(n).\n \\]\n So, in particular, \\(b_n=\\lfloor (n+\\ell)\\phi^2 \\rfloor + \\lambda_\\ell(n) +1\\).\n  \\end{corollary}",
      "thm:K2": "\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}",
      "thm:rec_car": "\\begin{theorem}\\label{thm:rec_car}\n  Let $\\ell\\ge 1$ and define recursively $(a_n,b_n)_{n\\ge 0}$ by\n\\begin{equation}\\label{eq:mexdef}\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}),\\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right. \n\\end{equation} \nThe set of non-terminal $\\mathcal{P}$-positions of $K^\\ell$ is $\\{(a_n,b_n)\\mid n\\ge 0\\} \\cup \\{(b_n,a_n)\\mid n\\ge 0\\} $.\n\\end{theorem}",
      "eq:Ppos": "\\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}",
      "def:fg": "\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}",
      "ssec:autom_proof": "\\begin{array}{c|llll}\n    &g_n & \\mu(g_n) & \\mu^2(g_n) & \\mu^3(g_n) \\\\\n    \\hline\n 0&1& 0& 1& 10\\\\\n 1&0& 1& 10& 011\\\\\n 2&1& 10& 011& 11010\\\\\n 3&1& 01& 110& 01011\\\\\n 4&0& 11& 010& 11011\\\\\n 5&1& 0& 11& 010\\\\\n  \\end{array}\n\\]\n  \\caption{The different $3$-types when applying a $\\varphi$-morphism to $g$.}\n  \\label{tab:types}\n\\end{table}\n\nNow we associate each symbol $g_n$ with its $3$-type and get the infinite word\n\\[  \n  01|2|31|45|2|35|4|31|45|4|35|45|\\cdots\n\\]\nwhere the vertical bars indicate the factorization of the Fibonacci word $ab|a|ab|ab|a|\\cdots$ with factors $\\sigma(a)=ab$ and $\\sigma(b)=a$, the images of the letters by the Fibonacci morphism. From this, we deduce the morphism $\\mu$ given in Example~\\ref{exa:mu}: the image of the $n$th letter has to be the $n$th block in the factorization. So $0\\mapsto 01$, then $1\\mapsto 2$ and so on. From the morphism, it is a routine to obtain the DFA in Figure~\\ref{fig:automG}, see Definition~\\ref{def:dfao}.\n\nFinally, one may define a coding~$\\rho$ to recover the original sequence where each type is mapped to the first element of the tuple. So, $0,2,3,5\\mapsto 1$ and $1,4\\mapsto 0$. This can equivalently be performed by comparing the morphic word and the original sequence\n\\begin{align*}\n  &01231452354314543545\\cdots\\\\\n  &10110011110100101101\\cdots\n\\end{align*}\nand matching corresponding elements.\n\\subsection{An automatic proof for $K^2$}\\label{ssec:autom_proof}\n\nNow that $g$ has been defined by a DFA fed with Fibonacci representations (and stored as {\\tt G.txt} in {\\tt Word Automata Lib}), the set of $\\mathcal{P}$-positions \\eqref{eq:Ppos} obtained by Komak et al. can be described by the following commands:\n\\begin{verbatim}\ndef pposK2_asym \"?msd_fib (a+b<=1) | En,x $phin(n,x) & a+1=x+G[n]\n                                                     & b=x+n+G[n]\":\ndef pposK2 \"?msd_fib $pposK2_asym(a,b) | $pposK2_asym(b,a) \":\n\\end{verbatim}\nThe resulting automaton has $27$ states.\n\nFor the sake of presentation, let us recall some basics on combinatorial games.\n\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}\nIt is a well-known property in graph theory that every acyclic graph has a unique kernel, i.e., a stable and absorbing set of vertices. In a take-away game where tokens are removed from piles, the corresponding game graph is acyclic: it is not possible to visit twice the same position during a game.\n\\begin{proposition}[Folklore]\\label{pro:kernel}\n  The sets of $\\mathcal{P}$- and $\\mathcal{N}$-positions of an\n  impartial acyclic game are uniquely determined by the following two\n  properties:\n\\begin{enumerate}\n\\item Every move from a $\\mathcal{P}$-position leads to an\n  $\\mathcal{N}$-position; equivalently there is no move between two $\\mathcal{P}$-positions (stability property of $\\mathcal{P}(G)$).\n  \\item From every $\\mathcal{N}$-position, there exists a move leading to a $\\mathcal{P}$-position (absorbing property of $\\mathcal{P}(G)$).\n\\end{enumerate}\n\\end{proposition}\n\nAs observed in \\cite{MRRW}, it is then a routine test to check stability and absorbing property:\n\\begin{verbatim}\neval stableK2 \"?msd_fib Ap,q,r,s (($pposK2(p,q) & $pposK2(r,s)\n     & p>=r & q>=s & p+q>2) => ((p=r & q=s) | (p>r & q>s & p+s!=q+r)))\":\n\neval absorbingK2 \"?msd_fib Ap,q (~$pposK2(p,q) => Ex,y\n     (x<=p & y<=q & $pposK2(x,y) & (p+y=q+x | p=x | q=y))) \":\n\\end{verbatim}\nThese two commands return {\\tt TRUE} resulting in the following result, \\cite[Thm.~3]{Komak}. From Proposition~\\ref{pro:kernel}, we immediately get an alternative proof of Theorem~\\ref{thm:K2}.\n\n\\section{Towards an algebraic characterization for $K^\\ell$}\\label{sec:newGame}\n\nIn \\cite{Komak}, Komak et al. chose $\\{(x,y)\\mid x+y\\le 2\\}$ as set of terminal positions. It is natural to consider a parameterized version where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. The corresponding game is denoted by $K^\\ell$. Note that the game $K^0$ is the classic Wythoff game. In this section, we obtain new results concerning the games $K^1$ and~$K^3$.\n\n\\subsection{The $K^1$-game}\nUp to our knowledge, the game $K^1$ does not seem to have been studied in the literature. Computing the first $\\mathcal{P}$-positions of $K^1$ and writing their Fibonacci expansions permitted us to state a conjecture that can be tested with {\\tt Walnut}. This result is quite similar to the famous characterization of the $\\mathcal{P}$-positions of Wythoff game obtained by Fraenkel \\cite{Fraenkel3}.\n\\begin{table}[h!t]\n  \\[\n    \\begin{array}{c|r|c|r}\n      a_n & \\rep_F(a_n) & b_n & \\rep_F(b_n) \\\\\n      \\hline\n 0&\\varepsilon  & 1 &1 \\\\\n 2&10 & 4 &101 \\\\\n 3&100 & 6 & 1001 \\\\\n 5&1000 & 9 & 10001 \\\\\n 7&1010 & 12 & 10101 \\\\\n 8&10000 & 14 & 100001 \\\\\n 10&10010 & 17 &100101 \\\\\n 11&10100 & 19 & 101001 \\\\\n 13&100000 & 22 & 1000001 \\\\\n 15&100010 & 25 & 1000101 \\\\\n  \\end{array}",
      "def:stab": "\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}",
      "thm:carK1": "\\begin{theorem}\\label{thm:carK1}\n  A position $(a,b)$, with $a\\le b$, is a $\\mathcal{P}$-position of the game $K^1$, with a set of terminal positions being $\\{(x,y)\\mid x+y\\le 1\\}$ and Wythoff's moves, if and only if $\\rep_F(a)$ ends with $0$ and $\\rep_F(b)=\\rep_F(a)1$. \n\\end{theorem}",
      "tab:gh": "\\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}"
    },
    "pre_theorem_intro_text_len": 3799,
    "pre_theorem_intro_text": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nWhen the numeration system has good properties (i.e., it is an {\\em addable} system as discussed in \\cite{ShallitBook}), the formalism of first-order logic can be used to express certain properties of combinatorial games. We have recently exploited this approach and used {\\tt Walnut} to reprove classical results in combinatorial games theory, as well as to obtain new ones \\cite{MRRW}. {\\tt Walnut} is a free, open-source prover for first-order statements dealing with automatic sequences \\cite{Mousavi,ShallitBook}. \n\nWe assume that the reader is sufficiently familiar with the use of {\\tt Walnut}. For references on combinatorial game theory, we refer to \\cite{RigoLNM} and \\cite{MRRW}.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game. \n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n  \\centering\n  \\[\n    \\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:",
    "context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:",
    "full_context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n \\end{array}\n \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nThe first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k3}. From now on, we take this convention: in the indexing of the $P$-positions, we do not take terminal positions into account. Thus, $(a_0,b_0)$ represents the first non-terminal $\\mathcal{P}$-position. Due to the symmetry of the game, we only consider sequences where $a_n<b_n$. \n\\begin{table}[h!t]\n\\[\n \\begin{array}{c|cccccccccccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n& 4 & 5 & 6 & 7 & 9 & 11 & 13 & 15 & 16 & 18 & 19 & 21 & 22 & 24 & 25 & 27 &\n 29 & 30 \\\\\n b_n & 8 & 10 & 12 & 14 & 17 & 20 & 23 & 26 & 28 & 31 & 33 & 36 & 38 & 41 & 43 & 46\n & 49 & 51 & \\\\\n \\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^3$, $(n\\ge 0)$.}\n \\label{tab:k3}\n\\end{table}\n\\begin{theorem}\\label{thm:fibaut2}\nThere exists a Fibonacci-automatic function $g_3:\\mathbb{N}\\to\\{0,1,2\\}$ such that, for all $n\\ge 0$, the $n$th non-terminal $\\mathcal{P}$-position $(a_n,b_n)$ of $K^3$ is given by \n\\begin{equation}\\label{eq:K3}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_3(n+1)-1 \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_3(n+1)+1.\n\\end{equation}\n\\end{theorem}\nThe first few values of $g_3(n)$ are\n\\[\n 1221011221211111111121121121211211112111121111111211112\\cdots\n\\]\nUsing the heuristic described in Section~\\ref{ssec:heuristic}, assuming that the above infinite word is generated by a $\\varphi$-morphism, we obtain the morphism \n\\begin{align*}\n \\mu_3:\\;&\n0 \\mapsto 01,\n1 \\mapsto 2,\n2 \\mapsto 34,\n3 \\mapsto 56,\n4 \\mapsto 7,\n5 \\mapsto 78,\\\\ \n& 6 \\mapsto 9,\n7 \\mapsto ab,\n8 \\mapsto a,\n9 \\mapsto 56,\na \\mapsto ab, \nb \\mapsto 7\n\\end{align*}\nand the coding\n\\[\\rho_3:0,3,5,6,8,a,b\\mapsto 1,\\quad 1,2,7,9\\mapsto 2,\\quad 4\\mapsto 0.\\]\nWe thus define $g_3(n)$ as the $n$th symbol in $\\rho_3(\\mu_3^\\omega(1))$. Note that to produce enough value of $g_3(n)$, we have to compute sufficiently many $\\mathcal{P}$-positions of $W^3$.\n\n\\subsection{Even further}\nFor the game~$K^4$, we have applied the same procedure. The first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k4}.\n\\begin{table}[h!t]\n\\[\n\\begin{array}{c|cccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n &5 & 6 & 7 & 8 & 9 & 11 & 13 & 15 & 17 & 19 & 20 & 22 & 23 & 25 & 26 & 28 & 29 & 31 \\\\\n b_n & 10 & 12 & 14 & 16 & 18 & 21 & 24 & 27 & 30 & 33 & 35 & 38 & 40 & 43 & 45 & 48 & 50 & 53 \\\\\n\\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^4$, $(n\\ge 0)$.}\n \\label{tab:k4}\n\\end{table}\nTo get a long enough prefix, we have computed the first $600$ $\\mathcal{P}$-positions $(a_n,b_n)$ with $a_n\\le b_n$ (up to $a_n$ close to $1000$). From this, we conjectured that\n\\begin{equation}\\label{eq:K4}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_4(n+1) \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_4(n+1)+3,\n\\end{equation}\nwhere the first few values of $g_4(n)$ are\n\\[\n 12210001112111111010110110111111111111111111\\cdots.\n\\]\nHaving the first $600$ values of $g_4(n)$, we apply the heuristic of Section~\\ref{ssec:heuristic}. Note that here, to be able to discern types correctly, we have to consider $4$-types (there are distinct $4$-types whose restrictions to $3$-types are equal). Since we have to take fourth iterations, this explains why we need a longer prefix. We obtain $18$ different $4$-types and the heuristic gives the following morphisms:\n\\begin{align*}\n \\mu_4:\\;&0 \\mapsto 01,\\ 1 \\mapsto 2,\\ 2 \\mapsto 34,\\ 3 \\mapsto 56,\\ 4 \\mapsto 7,\\ 5 \\mapsto 89,\\ 6 \\mapsto a,\\ 7 \\mapsto bc,\\ 8 \\mapsto dc,\\\\\n &9 \\mapsto d,\\ a \\mapsto ef,\\ b \\mapsto ef,\\ c \\mapsto e,\\ d \\mapsto 7g,\\ e \\mapsto eh,\\ f \\mapsto b,\\ g \\mapsto d,\\ h \\mapsto b \n\\end{align*}\n\\[\n \\rho_4: 0,3,7,8,9,b,c,d,e,h \\mapsto 1,\\ 1,2,a \\mapsto 2,\\ 4,5,6,f,g \\mapsto 0\n\\]\nWith $g_4(n)$ being the $n$th symbol of $\\rho_4(\\mu_4^\\omega(1))$, we have a $18$-state DFAO. Formulas \\eqref{eq:K4} translates in {\\tt Walnut} into\n\\begin{verbatim}\ndef pposK4_asym \"?msd_fib (a+b<=4) | (En,x $phin(n+2,x) & (a=x+K4[n+1])\n & (b=x+K4[n+1]+n+5))\":\n\\end{verbatim}\nand we then follow exactly the same proof to get the result.\n\nLet us begin with some classical and well-known results on Wythoff game before moving on to the games discussed in this article. We recall the {\\em recursive definition} of the $\\mathcal{P}$-positions of Wythoff game~$K^0$, with our convention:\n\\[\n (a_0,b_0)=(1,2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0\\}),\\\\\nb_n =a_n+n+1,\\\\\n \\end{array}\n \\right.\n\\]\nwhere $\\mex$ stands for {\\em minimum excluded value}, i.e. the least non-negative integer not in the set. In other words, $\\mex S=\\min(\\mathbb{N}\\setminus S)$. So $a_1=\\mex\\{0,1,2\\}=3$ and $b_1=a_1+2=5$, then $a_2=\\mex\\{0,1,2,3,5\\}=4$ and $b_2=a_2+3=7$ and so on.\n\nLarsson characterized \\cite[Thm.~1.1]{block} the set of $\\mathcal{P}$-positions of $W^2$, a variant of Wythoff game where one option may be blocked, see Figure~\\ref{fig:w2w3}. We provide an automatic proof of this result. \n\\begin{theorem}\nThe set of $\\mathcal{P}$-positions of $W^2$ is given by\n\\[\n \\mathcal{R}_2=\\{(0,0)\\}\\cup \\{ \\{n,2n+1\\}\\mid n\\ge 0\\}\\cup \\{\\{2\\lfloor n\\phi\\rfloor+2,2\\lfloor n\\phi^2\\rfloor+2\\}\\mid n\\ge 0\\}.\n\\] \n\\end{theorem}",
    "post_theorem_intro_text_len": 6231,
    "post_theorem_intro_text": "The authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n  b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by   \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nIt is well-known that the set of $\\mathcal{P}$-positions of the Wythoff game are coded by the Fibonacci word~$\\mathbf{f}=abaababa\\cdots$, see \\cite{Vision,Duchene}. If we start indexing letters of~$\\mathbf{f}$ with $1$, the indices of the letters $a$ (resp. $b$) in~$\\mathbf{f}$ correspond to the sequence $(a_n)_{n\\ge 0}$ (resp. $(b_n)_{n\\ge 0}$). Using the recursive definition from the previous section, we obtain  similar results with $K^1,K^2,K^3$: their $\\mathcal{P}$-positions are encoded by morphic words obtained by $\\varphi$-morphisms and codings. \n\nIn Section~\\ref{sec:3}, we obtain automatic proofs of Larsson's result about $W^2$ and $W^3$. Interestingly, the automata arising in the computations are much larger than those for $K^2$.\n\nIn Section~\\ref{sec:redundant}, we are concerned with redundant moves of the games discussed so far. A move is considered {\\em redundant} if, upon its removal from the rule-set, then the set of $\\mathcal{P}$-positions remains unchanged. It is a classical question in combinatorial game theory: do distinct rule-sets yield the same set of $\\mathcal{P}$-positions? In particular, can certain moves be added or removed without altering the $\\mathcal{P}$-positions?\n\n\\begin{remark}\n  A {\\tt Jupyter} notebook recording all the {\\tt Walnut} computations is available (it has been produced using Ollinger's {\\tt Walnut-Kernel}. \n  The {\\tt Mathematica} code and the various {\\tt Walnut} files are also available online\\footnote{at {\\tt https://hdl.handle.net/2268/338482}, you may also download a standalone Wolfram Player.}. In particular, the package permits us to compute $\\mathcal{P}$-positions and $\\varphi$-morphisms.\n\\end{remark}",
    "sketch": "To prove Theorem~\\ref{thm:K2}, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set \\eqref{eq:Ppos} has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).",
    "expanded_sketch": "To prove the main theorem, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\n is both stable and absorbing (see \\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set in the equation above has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).",
    "expanded_theorem": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is defined as follows: \\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}",
    "theorem_type": [
      "Classification or Bijection",
      "Equivalence"
    ],
    "mcq": {
      "question": "Consider the Wythoff-type game on positions (x,y) ∈ ℕ² in which a move from (x,y) consists of replacing it by (x-n,y), (x,y-n), or (x-n,y-n), where n is any positive integer allowed by the coordinates. The terminal positions are exactly those with x+y ≤ 2, and under normal play a P-position means a losing position. Let φ=(1+√5)/2, and define g:ℕ→ℕ by g(0)=1, g(1)=0, and for n≥0,\n\n g(n)=1-g(m) if floor(nφ)=floor(m(φ+1))+1 for some m≥0, and g(n)=1 otherwise.\n\nWhich statement gives exactly the set of P-positions of this game?",
      "correct_choice": {
        "label": "A",
        "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥0 }."
      },
      "choices": [
        {
          "label": "B",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n), ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)) : n≥0 }."
        },
        {
          "label": "C",
          "text": "Every position of the form\n\n(⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n))\n\nwith n≥0 is a P-position."
        },
        {
          "label": "D",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥1 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥1 }."
        },
        {
          "label": "E",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 }."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "offset-by-1 needed for stability against terminal region and moves within the candidate set",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped the converse exactness/union description, keeping only one included family of P-positions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "boundary index n=0",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "symmetry of coordinates required for absorbing all non-candidate positions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the correct option; it only defines the game, the function g, and asks for the exact characterization of the \u0012cal P-position set."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: it asks for the exact explicit description of the \u0012cal P-positions, with answer choices that are small variants of the target formula. It is not a pure verbatim restatement, but it is only a mild reformulation."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish exactness from a stronger set, a weaker subset, and a boundary-case omission, especially regarding symmetry and the n=0 case. However, the item mainly tests recognition of a known formula rather than substantial generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and target plausible failure modes: missing the -1 offset, dropping the symmetric family, excluding the boundary case n=0, and replacing union by intersection."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and generally strong distractors, but it leans heavily toward theorem recognition rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.11601v1",
    "paper_link": "http://arxiv.org/abs/2512.11601v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.",
      "start_pos": 103245,
      "end_pos": 103752,
      "label": "thm:K2"
    },
    "ref_dict": {
      "cor:behave": "\\begin{corollary}\\label{cor:behave}\n  For the sequences defined by \\eqref{eq:mexdef}, \n  there exist a bounded function $\\lambda_\\ell:\\mathbb{N}\\to\\mathbb{Z}$ such that, for all $n$,   \n  \\[\n   a_n=\\lfloor (n+\\ell)\\phi \\rfloor +\\lambda_\\ell(n).\n \\]\n So, in particular, \\(b_n=\\lfloor (n+\\ell)\\phi^2 \\rfloor + \\lambda_\\ell(n) +1\\).\n  \\end{corollary}",
      "thm:K2": "\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}",
      "thm:rec_car": "\\begin{theorem}\\label{thm:rec_car}\n  Let $\\ell\\ge 1$ and define recursively $(a_n,b_n)_{n\\ge 0}$ by\n\\begin{equation}\\label{eq:mexdef}\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}),\\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right. \n\\end{equation} \nThe set of non-terminal $\\mathcal{P}$-positions of $K^\\ell$ is $\\{(a_n,b_n)\\mid n\\ge 0\\} \\cup \\{(b_n,a_n)\\mid n\\ge 0\\} $.\n\\end{theorem}",
      "eq:Ppos": "\\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}",
      "def:fg": "\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}",
      "ssec:autom_proof": "\\begin{array}{c|llll}\n    &g_n & \\mu(g_n) & \\mu^2(g_n) & \\mu^3(g_n) \\\\\n    \\hline\n 0&1& 0& 1& 10\\\\\n 1&0& 1& 10& 011\\\\\n 2&1& 10& 011& 11010\\\\\n 3&1& 01& 110& 01011\\\\\n 4&0& 11& 010& 11011\\\\\n 5&1& 0& 11& 010\\\\\n  \\end{array}\n\\]\n  \\caption{The different $3$-types when applying a $\\varphi$-morphism to $g$.}\n  \\label{tab:types}\n\\end{table}\n\nNow we associate each symbol $g_n$ with its $3$-type and get the infinite word\n\\[  \n  01|2|31|45|2|35|4|31|45|4|35|45|\\cdots\n\\]\nwhere the vertical bars indicate the factorization of the Fibonacci word $ab|a|ab|ab|a|\\cdots$ with factors $\\sigma(a)=ab$ and $\\sigma(b)=a$, the images of the letters by the Fibonacci morphism. From this, we deduce the morphism $\\mu$ given in Example~\\ref{exa:mu}: the image of the $n$th letter has to be the $n$th block in the factorization. So $0\\mapsto 01$, then $1\\mapsto 2$ and so on. From the morphism, it is a routine to obtain the DFA in Figure~\\ref{fig:automG}, see Definition~\\ref{def:dfao}.\n\nFinally, one may define a coding~$\\rho$ to recover the original sequence where each type is mapped to the first element of the tuple. So, $0,2,3,5\\mapsto 1$ and $1,4\\mapsto 0$. This can equivalently be performed by comparing the morphic word and the original sequence\n\\begin{align*}\n  &01231452354314543545\\cdots\\\\\n  &10110011110100101101\\cdots\n\\end{align*}\nand matching corresponding elements.\n\\subsection{An automatic proof for $K^2$}\\label{ssec:autom_proof}\n\nNow that $g$ has been defined by a DFA fed with Fibonacci representations (and stored as {\\tt G.txt} in {\\tt Word Automata Lib}), the set of $\\mathcal{P}$-positions \\eqref{eq:Ppos} obtained by Komak et al. can be described by the following commands:\n\\begin{verbatim}\ndef pposK2_asym \"?msd_fib (a+b<=1) | En,x $phin(n,x) & a+1=x+G[n]\n                                                     & b=x+n+G[n]\":\ndef pposK2 \"?msd_fib $pposK2_asym(a,b) | $pposK2_asym(b,a) \":\n\\end{verbatim}\nThe resulting automaton has $27$ states.\n\nFor the sake of presentation, let us recall some basics on combinatorial games.\n\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}\nIt is a well-known property in graph theory that every acyclic graph has a unique kernel, i.e., a stable and absorbing set of vertices. In a take-away game where tokens are removed from piles, the corresponding game graph is acyclic: it is not possible to visit twice the same position during a game.\n\\begin{proposition}[Folklore]\\label{pro:kernel}\n  The sets of $\\mathcal{P}$- and $\\mathcal{N}$-positions of an\n  impartial acyclic game are uniquely determined by the following two\n  properties:\n\\begin{enumerate}\n\\item Every move from a $\\mathcal{P}$-position leads to an\n  $\\mathcal{N}$-position; equivalently there is no move between two $\\mathcal{P}$-positions (stability property of $\\mathcal{P}(G)$).\n  \\item From every $\\mathcal{N}$-position, there exists a move leading to a $\\mathcal{P}$-position (absorbing property of $\\mathcal{P}(G)$).\n\\end{enumerate}\n\\end{proposition}\n\nAs observed in \\cite{MRRW}, it is then a routine test to check stability and absorbing property:\n\\begin{verbatim}\neval stableK2 \"?msd_fib Ap,q,r,s (($pposK2(p,q) & $pposK2(r,s)\n     & p>=r & q>=s & p+q>2) => ((p=r & q=s) | (p>r & q>s & p+s!=q+r)))\":\n\neval absorbingK2 \"?msd_fib Ap,q (~$pposK2(p,q) => Ex,y\n     (x<=p & y<=q & $pposK2(x,y) & (p+y=q+x | p=x | q=y))) \":\n\\end{verbatim}\nThese two commands return {\\tt TRUE} resulting in the following result, \\cite[Thm.~3]{Komak}. From Proposition~\\ref{pro:kernel}, we immediately get an alternative proof of Theorem~\\ref{thm:K2}.\n\n\\section{Towards an algebraic characterization for $K^\\ell$}\\label{sec:newGame}\n\nIn \\cite{Komak}, Komak et al. chose $\\{(x,y)\\mid x+y\\le 2\\}$ as set of terminal positions. It is natural to consider a parameterized version where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. The corresponding game is denoted by $K^\\ell$. Note that the game $K^0$ is the classic Wythoff game. In this section, we obtain new results concerning the games $K^1$ and~$K^3$.\n\n\\subsection{The $K^1$-game}\nUp to our knowledge, the game $K^1$ does not seem to have been studied in the literature. Computing the first $\\mathcal{P}$-positions of $K^1$ and writing their Fibonacci expansions permitted us to state a conjecture that can be tested with {\\tt Walnut}. This result is quite similar to the famous characterization of the $\\mathcal{P}$-positions of Wythoff game obtained by Fraenkel \\cite{Fraenkel3}.\n\\begin{table}[h!t]\n  \\[\n    \\begin{array}{c|r|c|r}\n      a_n & \\rep_F(a_n) & b_n & \\rep_F(b_n) \\\\\n      \\hline\n 0&\\varepsilon  & 1 &1 \\\\\n 2&10 & 4 &101 \\\\\n 3&100 & 6 & 1001 \\\\\n 5&1000 & 9 & 10001 \\\\\n 7&1010 & 12 & 10101 \\\\\n 8&10000 & 14 & 100001 \\\\\n 10&10010 & 17 &100101 \\\\\n 11&10100 & 19 & 101001 \\\\\n 13&100000 & 22 & 1000001 \\\\\n 15&100010 & 25 & 1000101 \\\\\n  \\end{array}",
      "def:stab": "\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}",
      "thm:carK1": "\\begin{theorem}\\label{thm:carK1}\n  A position $(a,b)$, with $a\\le b$, is a $\\mathcal{P}$-position of the game $K^1$, with a set of terminal positions being $\\{(x,y)\\mid x+y\\le 1\\}$ and Wythoff's moves, if and only if $\\rep_F(a)$ ends with $0$ and $\\rep_F(b)=\\rep_F(a)1$. \n\\end{theorem}",
      "tab:gh": "\\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}"
    },
    "pre_theorem_intro_text_len": 3799,
    "pre_theorem_intro_text": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nWhen the numeration system has good properties (i.e., it is an {\\em addable} system as discussed in \\cite{ShallitBook}), the formalism of first-order logic can be used to express certain properties of combinatorial games. We have recently exploited this approach and used {\\tt Walnut} to reprove classical results in combinatorial games theory, as well as to obtain new ones \\cite{MRRW}. {\\tt Walnut} is a free, open-source prover for first-order statements dealing with automatic sequences \\cite{Mousavi,ShallitBook}. \n\nWe assume that the reader is sufficiently familiar with the use of {\\tt Walnut}. For references on combinatorial game theory, we refer to \\cite{RigoLNM} and \\cite{MRRW}.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game. \n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n  \\centering\n  \\[\n    \\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:",
    "context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:",
    "full_context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n \\end{array}\n \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nThe first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k3}. From now on, we take this convention: in the indexing of the $P$-positions, we do not take terminal positions into account. Thus, $(a_0,b_0)$ represents the first non-terminal $\\mathcal{P}$-position. Due to the symmetry of the game, we only consider sequences where $a_n<b_n$. \n\\begin{table}[h!t]\n\\[\n \\begin{array}{c|cccccccccccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n& 4 & 5 & 6 & 7 & 9 & 11 & 13 & 15 & 16 & 18 & 19 & 21 & 22 & 24 & 25 & 27 &\n 29 & 30 \\\\\n b_n & 8 & 10 & 12 & 14 & 17 & 20 & 23 & 26 & 28 & 31 & 33 & 36 & 38 & 41 & 43 & 46\n & 49 & 51 & \\\\\n \\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^3$, $(n\\ge 0)$.}\n \\label{tab:k3}\n\\end{table}\n\\begin{theorem}\\label{thm:fibaut2}\nThere exists a Fibonacci-automatic function $g_3:\\mathbb{N}\\to\\{0,1,2\\}$ such that, for all $n\\ge 0$, the $n$th non-terminal $\\mathcal{P}$-position $(a_n,b_n)$ of $K^3$ is given by \n\\begin{equation}\\label{eq:K3}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_3(n+1)-1 \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_3(n+1)+1.\n\\end{equation}\n\\end{theorem}\nThe first few values of $g_3(n)$ are\n\\[\n 1221011221211111111121121121211211112111121111111211112\\cdots\n\\]\nUsing the heuristic described in Section~\\ref{ssec:heuristic}, assuming that the above infinite word is generated by a $\\varphi$-morphism, we obtain the morphism \n\\begin{align*}\n \\mu_3:\\;&\n0 \\mapsto 01,\n1 \\mapsto 2,\n2 \\mapsto 34,\n3 \\mapsto 56,\n4 \\mapsto 7,\n5 \\mapsto 78,\\\\ \n& 6 \\mapsto 9,\n7 \\mapsto ab,\n8 \\mapsto a,\n9 \\mapsto 56,\na \\mapsto ab, \nb \\mapsto 7\n\\end{align*}\nand the coding\n\\[\\rho_3:0,3,5,6,8,a,b\\mapsto 1,\\quad 1,2,7,9\\mapsto 2,\\quad 4\\mapsto 0.\\]\nWe thus define $g_3(n)$ as the $n$th symbol in $\\rho_3(\\mu_3^\\omega(1))$. Note that to produce enough value of $g_3(n)$, we have to compute sufficiently many $\\mathcal{P}$-positions of $W^3$.\n\n\\subsection{Even further}\nFor the game~$K^4$, we have applied the same procedure. The first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k4}.\n\\begin{table}[h!t]\n\\[\n\\begin{array}{c|cccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n &5 & 6 & 7 & 8 & 9 & 11 & 13 & 15 & 17 & 19 & 20 & 22 & 23 & 25 & 26 & 28 & 29 & 31 \\\\\n b_n & 10 & 12 & 14 & 16 & 18 & 21 & 24 & 27 & 30 & 33 & 35 & 38 & 40 & 43 & 45 & 48 & 50 & 53 \\\\\n\\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^4$, $(n\\ge 0)$.}\n \\label{tab:k4}\n\\end{table}\nTo get a long enough prefix, we have computed the first $600$ $\\mathcal{P}$-positions $(a_n,b_n)$ with $a_n\\le b_n$ (up to $a_n$ close to $1000$). From this, we conjectured that\n\\begin{equation}\\label{eq:K4}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_4(n+1) \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_4(n+1)+3,\n\\end{equation}\nwhere the first few values of $g_4(n)$ are\n\\[\n 12210001112111111010110110111111111111111111\\cdots.\n\\]\nHaving the first $600$ values of $g_4(n)$, we apply the heuristic of Section~\\ref{ssec:heuristic}. Note that here, to be able to discern types correctly, we have to consider $4$-types (there are distinct $4$-types whose restrictions to $3$-types are equal). Since we have to take fourth iterations, this explains why we need a longer prefix. We obtain $18$ different $4$-types and the heuristic gives the following morphisms:\n\\begin{align*}\n \\mu_4:\\;&0 \\mapsto 01,\\ 1 \\mapsto 2,\\ 2 \\mapsto 34,\\ 3 \\mapsto 56,\\ 4 \\mapsto 7,\\ 5 \\mapsto 89,\\ 6 \\mapsto a,\\ 7 \\mapsto bc,\\ 8 \\mapsto dc,\\\\\n &9 \\mapsto d,\\ a \\mapsto ef,\\ b \\mapsto ef,\\ c \\mapsto e,\\ d \\mapsto 7g,\\ e \\mapsto eh,\\ f \\mapsto b,\\ g \\mapsto d,\\ h \\mapsto b \n\\end{align*}\n\\[\n \\rho_4: 0,3,7,8,9,b,c,d,e,h \\mapsto 1,\\ 1,2,a \\mapsto 2,\\ 4,5,6,f,g \\mapsto 0\n\\]\nWith $g_4(n)$ being the $n$th symbol of $\\rho_4(\\mu_4^\\omega(1))$, we have a $18$-state DFAO. Formulas \\eqref{eq:K4} translates in {\\tt Walnut} into\n\\begin{verbatim}\ndef pposK4_asym \"?msd_fib (a+b<=4) | (En,x $phin(n+2,x) & (a=x+K4[n+1])\n & (b=x+K4[n+1]+n+5))\":\n\\end{verbatim}\nand we then follow exactly the same proof to get the result.\n\nLet us begin with some classical and well-known results on Wythoff game before moving on to the games discussed in this article. We recall the {\\em recursive definition} of the $\\mathcal{P}$-positions of Wythoff game~$K^0$, with our convention:\n\\[\n (a_0,b_0)=(1,2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0\\}),\\\\\nb_n =a_n+n+1,\\\\\n \\end{array}\n \\right.\n\\]\nwhere $\\mex$ stands for {\\em minimum excluded value}, i.e. the least non-negative integer not in the set. In other words, $\\mex S=\\min(\\mathbb{N}\\setminus S)$. So $a_1=\\mex\\{0,1,2\\}=3$ and $b_1=a_1+2=5$, then $a_2=\\mex\\{0,1,2,3,5\\}=4$ and $b_2=a_2+3=7$ and so on.\n\nLarsson characterized \\cite[Thm.~1.1]{block} the set of $\\mathcal{P}$-positions of $W^2$, a variant of Wythoff game where one option may be blocked, see Figure~\\ref{fig:w2w3}. We provide an automatic proof of this result. \n\\begin{theorem}\nThe set of $\\mathcal{P}$-positions of $W^2$ is given by\n\\[\n \\mathcal{R}_2=\\{(0,0)\\}\\cup \\{ \\{n,2n+1\\}\\mid n\\ge 0\\}\\cup \\{\\{2\\lfloor n\\phi\\rfloor+2,2\\lfloor n\\phi^2\\rfloor+2\\}\\mid n\\ge 0\\}.\n\\] \n\\end{theorem}",
    "post_theorem_intro_text_len": 6231,
    "post_theorem_intro_text": "The authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n  b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by   \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nIt is well-known that the set of $\\mathcal{P}$-positions of the Wythoff game are coded by the Fibonacci word~$\\mathbf{f}=abaababa\\cdots$, see \\cite{Vision,Duchene}. If we start indexing letters of~$\\mathbf{f}$ with $1$, the indices of the letters $a$ (resp. $b$) in~$\\mathbf{f}$ correspond to the sequence $(a_n)_{n\\ge 0}$ (resp. $(b_n)_{n\\ge 0}$). Using the recursive definition from the previous section, we obtain  similar results with $K^1,K^2,K^3$: their $\\mathcal{P}$-positions are encoded by morphic words obtained by $\\varphi$-morphisms and codings. \n\nIn Section~\\ref{sec:3}, we obtain automatic proofs of Larsson's result about $W^2$ and $W^3$. Interestingly, the automata arising in the computations are much larger than those for $K^2$.\n\nIn Section~\\ref{sec:redundant}, we are concerned with redundant moves of the games discussed so far. A move is considered {\\em redundant} if, upon its removal from the rule-set, then the set of $\\mathcal{P}$-positions remains unchanged. It is a classical question in combinatorial game theory: do distinct rule-sets yield the same set of $\\mathcal{P}$-positions? In particular, can certain moves be added or removed without altering the $\\mathcal{P}$-positions?\n\n\\begin{remark}\n  A {\\tt Jupyter} notebook recording all the {\\tt Walnut} computations is available (it has been produced using Ollinger's {\\tt Walnut-Kernel}. \n  The {\\tt Mathematica} code and the various {\\tt Walnut} files are also available online\\footnote{at {\\tt https://hdl.handle.net/2268/338482}, you may also download a standalone Wolfram Player.}. In particular, the package permits us to compute $\\mathcal{P}$-positions and $\\varphi$-morphisms.\n\\end{remark}",
    "sketch": "To prove Theorem~\\ref{thm:K2}, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set \\eqref{eq:Ppos} has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).",
    "expanded_sketch": "To prove the main theorem, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\n is both stable and absorbing (see \\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set in the equation above has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).",
    "expanded_theorem": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is defined as follows: \\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}",
    "theorem_type": [
      "Classification or Bijection",
      "Equivalence"
    ],
    "mcq": {
      "question": "Consider the Wythoff-type game on positions (x,y) ∈ ℕ² in which a move from (x,y) consists of replacing it by (x-n,y), (x,y-n), or (x-n,y-n), where n is any positive integer allowed by the coordinates. The terminal positions are exactly those with x+y ≤ 2, and under normal play a P-position means a losing position. Let φ=(1+√5)/2, and define g:ℕ→ℕ by g(0)=1, g(1)=0, and for n≥0,\n\n g(n)=1-g(m) if floor(nφ)=floor(m(φ+1))+1 for some m≥0, and g(n)=1 otherwise.\n\nWhich statement gives exactly the set of P-positions of this game?",
      "correct_choice": {
        "label": "A",
        "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥0 }."
      },
      "choices": [
        {
          "label": "B",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n), ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)) : n≥0 }."
        },
        {
          "label": "C",
          "text": "Every position of the form\n\n(⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n))\n\nwith n≥0 is a P-position."
        },
        {
          "label": "D",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥1 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥1 }."
        },
        {
          "label": "E",
          "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 }."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "offset-by-1 needed for stability against terminal region and moves within the candidate set",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped the converse exactness/union description, keeping only one included family of P-positions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "boundary index n=0",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "symmetry of coordinates required for absorbing all non-candidate positions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the correct option or uniquely reveal choice A. It gives the game and the auxiliary function g, but the exact P-position description still has to be identified from the alternatives."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-statement recognition question: the task is essentially to pick the exact characterization of the P-positions. However, it is not a pure restatement, since the options differ in meaningful ways such as symmetry, boundary indexing, and offset terms."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish an exact characterization from weaker true statements and near-miss formulas. Still, the item mainly rewards theorem recall/recognition rather than substantial generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: dropping the converse, missing symmetry, mishandling the n=0 boundary case, and omitting the offset adjustment. They are distinct and well aligned with common mistakes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it leans toward theorem recognition rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.11658v1",
    "paper_link": "http://arxiv.org/abs/2512.11658v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$.",
      "start_pos": 42697,
      "end_pos": 43112,
      "label": "thm-main"
    },
    "ref_dict": {
      "rem: non-complete": "\\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}",
      "thm-main": "\\begin{thm} \\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$. \n\\end{thm}",
      "thm-klsph": "\\begin{thm}[Kleiner--Leeb] \\label{thm-klsph}\nLet $B$ be a spherical building, let $C\\subset B$ be a closed, convex, spherical subset. Then there exists an apartment $A\\subset B$ which contains $C$.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 699,
    "pre_theorem_intro_text": "Euclidean buildings were introduced by  Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics.  Their metric structure played a central role in super-rigidity results starting with  \\cite{KleinerLeeb1}, \\cite{GS}.  We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to  \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and  verified  in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In  this note we provide a proof of this result:",
    "context": "Euclidean buildings were introduced by Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics. Their metric structure played a central role in super-rigidity results starting with \\cite{KleinerLeeb1}, \\cite{GS}. We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:",
    "full_context": "Euclidean buildings were introduced by Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics. Their metric structure played a central role in super-rigidity results starting with \\cite{KleinerLeeb1}, \\cite{GS}. We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:\n\nAll Euclidean buildings in this note, in particular, in Theorem \\ref{thm-main}, are assumed to be complete, cf. Remark \\ref{rem: non-complete} below for a comment about possible generalizations.\n\n\\begin{rem} \\label{rem: non-complete}\n Here is the only place in this paper, where we discuss non-complete buildings, we refer to \\cite{Parreau}, \\cite{Str} for their discussion.\n In any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$. Such $C$ cannot be contained in an apartment. However, an extension of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to \n $\\Lambda$-buildings (cf. \\cite{Schwer}) seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n\\begin{cor}\nLet $C\\subset B$ be a closed, convex, spherical subset in a spherical building $B$. Let $x\\in C$ be arbitrary and let $A_x\\subset \\Sigma _x B$ be an apartment in the space of directions such that $\\Sigma _xC\\subset A_x$. Then there exists an apartment $A\\subset B$ which contains $C$ and such that $\\Sigma _xA=A_x$.\n\\end{cor}\n\n\\begin{cor} \\label{cor: cone}\nLet $C$ be a flat, closed, convex subset in a Euclidean building $X$. Assume that $C$ is a Euclidean cone with its tip at the point $x$. Let $A_x$ be an apartment in the tangent cone $T_xX$, which contains $T_xC$. Then $C$ is contained in an apartment $A$, such that $T_xA=A_x$.\n\\end{cor}\n\n\\begin{cor}\nLet $F\\subset X$ be a convex subset isometric to a Euclidean space in a Euclidean building $X$. Then for every $x\\in F$ the tangent space $T_x(P(F))$ at $x$ to the parallel set $P(F)$ of $F$ coincides with the parallel set $P(T_xF)$ of the flat $T_xF$ in the building $T_xX$. \n\\end{cor}\n\n\\begin{lem} \\label{lem: final}\n Let $X$ be a CAT(0) space. Let $Y_1,Y_2 $ be closed, convex subsets of $X$ isometric to convex subsets of some Euclidean spaces. Let $x\\in Y_1\\cap Y_2$ be a point and assume that\n \\begin{enumerate} \n \\item $T:={\\operatorname{Log}}_x (Y_1) \\cup {\\operatorname{Log}}_x (Y_2) \\subset T_xX \\,$ is a convex subset of $T_xX$,\n \\item ${\\operatorname{Log}}_x (Y_1) \\cap {\\operatorname{Log}}_x (Y_2) ={\\operatorname{Log}}_x (Y_1\\cap Y_2)\\,.$\n \\end{enumerate}\n Then $Y:=Y_1\\cup Y_2$ is a convex subset of $X$ isometric to $T$. \n\\end{lem}\n\nChoose an arbitrary interior point $x$ of the facet $\\mathcal F$. Then $T_x \\mathcal P$ is a $k$-dimensional, flat halfspace in the tangent space $T_xX$. By Corollary \\ref{cor: cone}, we find an apartment $A^x$ in the (infinitesimal) Euclidean building $T_xX$ with $T_x \\mathcal P \\subset A^x$.\n\n\\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n\\begin{thm} \\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$. \n\\end{thm}",
    "post_theorem_intro_text_len": 2889,
    "post_theorem_intro_text": "All Euclidean buildings in this note, in particular, in Theorem \\ref{thm-main}, are assumed to be complete, cf. Remark \\ref{rem: non-complete} below for a comment about possible generalizations.\n\nThe analogue of the first part of this result  for spherical buildings is contained in   \\cite{KleinerLeeb1}, see Theorem \\ref{thm-klsph} below. \nIn some special cases,  Theorem \\ref{thm-main} is contained in \\cite{KleinerLeeb1} as well.\n\nA different proof of the spherical theorem  was obtained later in \\cite[Proposition 1.3]{BL} in a slightly more general setting.  Moreover, this approach also covered the case of Theorem \\ref{thm-main} for discrete Euclidean buildings, \\cite[Remark 1.4]{BL}. \n\nFor some convex subsets in discrete Euclidean buildings,  Theorem \\ref{thm-main} is presented in \\cite[Theorem 11.53]{Abram}. The case of convex triangles $C$ in Bruhat--Tits buildings appears in \\cite[Exercise 2.3.10.3]{Rousseau}.  Several  special cases of  Theorem \\ref{thm-main} have been obtained and used in \\cite{BM}.\n\n \\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n  In Section \\ref{sec: sph} we follow \\cite{KleinerLeeb1}, in order to  reprove and  to extend the spherical version of our main theorem.   One major difference between the spherical and the Euclidean case is  the  finiteness of the simplicial decomposition of spherical apartments, which allows for a choice of a potential enlargement $A$  of the convex subset $C$.  In Section 4,  we prove Theorem \\ref{thm-main} by enlarging the convex set step by step.  In order to make this enlargement procedure  successful, we deal with the case of convex polyhedra $C$ first, \n  and then deduce the result for  general convex subsets  by an appropriate approximation. \n\n{\\bf Acknowledgments.} Alexander Lytchak is grateful to the authors of \\cite{BM} for their interest in this topic, especially  to Christine Breiner, for several related questions, which gave rise to this project.    Raphael Appenzeller would like to thank Isobel Davies for discussions. Raphael Appenzeller and Auguste H\\'ebert acknowledge support from the Procope project\n\"Buildings, galleries and beyond\" and would like to thank its members, St\\'ephane Gaussent, Zijun Li, Bianca Marchionna, Paul Philippe and Petra Schwer for helpful discussions.",
    "sketch": "In Section 4, the authors \"prove Theorem \\ref{thm-main} by enlarging the convex set step by step.\" To make this enlargement procedure work, they \"deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation.\" They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on \"the finiteness of the simplicial decomposition of spherical apartments,\" which in the spherical setting \"allows for a choice of a potential enlargement $A$ of the convex subset $C$.\"",
    "expanded_sketch": "Next, the authors prove the main theorem by enlarging the convex set step by step. To make this enlargement procedure work, they deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation. They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on the finiteness of the simplicial decomposition of spherical apartments, which in the spherical setting allows for a choice of a potential enlargement $A$ of the convex subset $C$.",
    "expanded_theorem": "\\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $X$ be a complete Euclidean building, and let $C\\subset X$ be a convex subset that is isometric to a convex subset of some Euclidean space. For a point $x\\in C$, let $T_xX$ denote the tangent cone at $x$ (the infinitesimal Euclidean building at $x$), and let $T_xC\\subset T_xX$ denote the tangent cone of $C$ at $x$. Suppose that, for some chosen $x\\in C$, an apartment $A^x\\subset T_xX$ satisfies $T_xC\\subset A^x$. Which conclusion holds under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, for every $x\\in C$ and every apartment $A^x\\subset T_xX$ with $T_xC\\subset A^x$, the same apartment $A$ can be chosen so that $T_xA=A^x$."
        },
        {
          "label": "C",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$."
        },
        {
          "label": "D",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$ provided that $C$ is a convex polyhedron. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$ in this polyhedral case."
        },
        {
          "label": "E",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then every apartment $A\\subset X$ containing $C$ satisfies $T_xA=A^x$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the global apartment on the chosen tangent apartment and point",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the prescribed tangent-space matching condition $T_xA=A^x$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "approximation step from convex polyhedra to arbitrary convex Euclidean subsets",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "existential choice of apartment containing $C$ replaced by a uniqueness-type condition on all such apartments",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or uniquely signal choice A. It states only the hypotheses and asks which existence statement is valid."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a multiple-choice restatement of a theorem: the correct answer is the theorem's conclusion almost verbatim, including the tangent-apartment refinement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the distractors vary by quantifier strength, locality vs globality, and uniqueness vs existence, and choice C is a weaker true statement. Still, the item mainly tests theorem recall rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well-targeted: B overstates uniqueness, C is a weaker true statement, D imposes an overly strong simultaneous condition, and E weakens global containment to a local one."
      },
      "total_score": 5,
      "overall_assessment": "Good distractor design, but the item is largely theorem-recall and close to tautological rather than a strong test of generative reasoning."
    }
  },
  {
    "id": "2512.11658v1",
    "paper_link": "http://arxiv.org/abs/2512.11658v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$.",
      "start_pos": 42697,
      "end_pos": 43112,
      "label": "thm-main"
    },
    "ref_dict": {
      "rem: non-complete": "\\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}",
      "thm-main": "\\begin{thm} \\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$. \n\\end{thm}",
      "thm-klsph": "\\begin{thm}[Kleiner--Leeb] \\label{thm-klsph}\nLet $B$ be a spherical building, let $C\\subset B$ be a closed, convex, spherical subset. Then there exists an apartment $A\\subset B$ which contains $C$.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 699,
    "pre_theorem_intro_text": "Euclidean buildings were introduced by  Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics.  Their metric structure played a central role in super-rigidity results starting with  \\cite{KleinerLeeb1}, \\cite{GS}.  We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to  \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and  verified  in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In  this note we provide a proof of this result:",
    "context": "Euclidean buildings were introduced by Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics. Their metric structure played a central role in super-rigidity results starting with \\cite{KleinerLeeb1}, \\cite{GS}. We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:",
    "full_context": "Euclidean buildings were introduced by Jacques Tits in \\cite{Tits} and play an important role in different areas of mathematics. Their metric structure played a central role in super-rigidity results starting with \\cite{KleinerLeeb1}, \\cite{GS}. We refer to \\cite{Linus},\n \\cite{KLM} for further investigations of their metric properties and to \\cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:\n\nThe following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \\cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:\n\nAll Euclidean buildings in this note, in particular, in Theorem \\ref{thm-main}, are assumed to be complete, cf. Remark \\ref{rem: non-complete} below for a comment about possible generalizations.\n\n\\begin{rem} \\label{rem: non-complete}\n Here is the only place in this paper, where we discuss non-complete buildings, we refer to \\cite{Parreau}, \\cite{Str} for their discussion.\n In any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$. Such $C$ cannot be contained in an apartment. However, an extension of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to \n $\\Lambda$-buildings (cf. \\cite{Schwer}) seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n\\begin{cor}\nLet $C\\subset B$ be a closed, convex, spherical subset in a spherical building $B$. Let $x\\in C$ be arbitrary and let $A_x\\subset \\Sigma _x B$ be an apartment in the space of directions such that $\\Sigma _xC\\subset A_x$. Then there exists an apartment $A\\subset B$ which contains $C$ and such that $\\Sigma _xA=A_x$.\n\\end{cor}\n\n\\begin{cor} \\label{cor: cone}\nLet $C$ be a flat, closed, convex subset in a Euclidean building $X$. Assume that $C$ is a Euclidean cone with its tip at the point $x$. Let $A_x$ be an apartment in the tangent cone $T_xX$, which contains $T_xC$. Then $C$ is contained in an apartment $A$, such that $T_xA=A_x$.\n\\end{cor}\n\n\\begin{cor}\nLet $F\\subset X$ be a convex subset isometric to a Euclidean space in a Euclidean building $X$. Then for every $x\\in F$ the tangent space $T_x(P(F))$ at $x$ to the parallel set $P(F)$ of $F$ coincides with the parallel set $P(T_xF)$ of the flat $T_xF$ in the building $T_xX$. \n\\end{cor}\n\n\\begin{lem} \\label{lem: final}\n Let $X$ be a CAT(0) space. Let $Y_1,Y_2 $ be closed, convex subsets of $X$ isometric to convex subsets of some Euclidean spaces. Let $x\\in Y_1\\cap Y_2$ be a point and assume that\n \\begin{enumerate} \n \\item $T:={\\operatorname{Log}}_x (Y_1) \\cup {\\operatorname{Log}}_x (Y_2) \\subset T_xX \\,$ is a convex subset of $T_xX$,\n \\item ${\\operatorname{Log}}_x (Y_1) \\cap {\\operatorname{Log}}_x (Y_2) ={\\operatorname{Log}}_x (Y_1\\cap Y_2)\\,.$\n \\end{enumerate}\n Then $Y:=Y_1\\cup Y_2$ is a convex subset of $X$ isometric to $T$. \n\\end{lem}\n\nChoose an arbitrary interior point $x$ of the facet $\\mathcal F$. Then $T_x \\mathcal P$ is a $k$-dimensional, flat halfspace in the tangent space $T_xX$. By Corollary \\ref{cor: cone}, we find an apartment $A^x$ in the (infinitesimal) Euclidean building $T_xX$ with $T_x \\mathcal P \\subset A^x$.\n\n\\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n\\begin{thm} \\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$. \n\\end{thm}",
    "post_theorem_intro_text_len": 2889,
    "post_theorem_intro_text": "All Euclidean buildings in this note, in particular, in Theorem \\ref{thm-main}, are assumed to be complete, cf. Remark \\ref{rem: non-complete} below for a comment about possible generalizations.\n\nThe analogue of the first part of this result  for spherical buildings is contained in   \\cite{KleinerLeeb1}, see Theorem \\ref{thm-klsph} below. \nIn some special cases,  Theorem \\ref{thm-main} is contained in \\cite{KleinerLeeb1} as well.\n\nA different proof of the spherical theorem  was obtained later in \\cite[Proposition 1.3]{BL} in a slightly more general setting.  Moreover, this approach also covered the case of Theorem \\ref{thm-main} for discrete Euclidean buildings, \\cite[Remark 1.4]{BL}. \n\nFor some convex subsets in discrete Euclidean buildings,  Theorem \\ref{thm-main} is presented in \\cite[Theorem 11.53]{Abram}. The case of convex triangles $C$ in Bruhat--Tits buildings appears in \\cite[Exercise 2.3.10.3]{Rousseau}.  Several  special cases of  Theorem \\ref{thm-main} have been obtained and used in \\cite{BM}.\n\n \\begin{rem} \\label{rem: non-complete}\n\tHere is the only place in this paper, where we discuss non-complete buildings, we refer to  \\cite{Parreau}, \\cite{Str} for their discussion.\n\tIn any non-complete Euclidean building, Theorem \\ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\\gamma:[0,1) \\to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. \tHowever, an extension  of Theorem \\ref{thm-main} to non-complete Euclidean buildings and to  \n\t $\\Lambda$-buildings (cf. \\cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work. \n\\end{rem}\n\n  In Section \\ref{sec: sph} we follow \\cite{KleinerLeeb1}, in order to  reprove and  to extend the spherical version of our main theorem.   One major difference between the spherical and the Euclidean case is  the  finiteness of the simplicial decomposition of spherical apartments, which allows for a choice of a potential enlargement $A$  of the convex subset $C$.  In Section 4,  we prove Theorem \\ref{thm-main} by enlarging the convex set step by step.  In order to make this enlargement procedure  successful, we deal with the case of convex polyhedra $C$ first, \n  and then deduce the result for  general convex subsets  by an appropriate approximation. \n\n{\\bf Acknowledgments.} Alexander Lytchak is grateful to the authors of \\cite{BM} for their interest in this topic, especially  to Christine Breiner, for several related questions, which gave rise to this project.    Raphael Appenzeller would like to thank Isobel Davies for discussions. Raphael Appenzeller and Auguste H\\'ebert acknowledge support from the Procope project\n\"Buildings, galleries and beyond\" and would like to thank its members, St\\'ephane Gaussent, Zijun Li, Bianca Marchionna, Paul Philippe and Petra Schwer for helpful discussions.",
    "sketch": "In Section 4, the authors \"prove Theorem \\ref{thm-main} by enlarging the convex set step by step.\" To make this enlargement procedure work, they \"deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation.\" They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on \"the finiteness of the simplicial decomposition of spherical apartments,\" which in the spherical setting \"allows for a choice of a potential enlargement $A$ of the convex subset $C$.\"",
    "expanded_sketch": "Next, the authors prove the main theorem by enlarging the convex set step by step. To make this enlargement procedure work, they deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation. They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on the finiteness of the simplicial decomposition of spherical apartments, which in the spherical setting allows for a choice of a potential enlargement $A$ of the convex subset $C$.",
    "expanded_theorem": "\\label{thm-main}\n\tLet $X$ be a Euclidean  building.  Let $C\\subset X$ be a convex subset of $X$\n\twhich is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.  \n\n\tMoreover, if for some $x\\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let $X$ be a complete Euclidean building, and let $C\\subset X$ be a convex subset that is isometric to a convex subset of some Euclidean space. For a point $x\\in C$, let $T_xX$ denote the tangent cone at $x$ (the infinitesimal Euclidean building at $x$), and let $T_xC\\subset T_xX$ denote the tangent cone of $C$ at $x$. Suppose that, for some chosen $x\\in C$, an apartment $A^x\\subset T_xX$ satisfies $T_xC\\subset A^x$. Which conclusion holds under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, for every $x\\in C$ and every apartment $A^x\\subset T_xX$ with $T_xC\\subset A^x$, the same apartment $A$ can be chosen so that $T_xA=A^x$."
        },
        {
          "label": "C",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$."
        },
        {
          "label": "D",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$ provided that $C$ is a convex polyhedron. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$ in this polyhedral case."
        },
        {
          "label": "E",
          "text": "There exists an apartment $A\\subset X$ such that $C\\subset A$. Moreover, if $x\\in C$ and an apartment $A^x\\subset T_xX$ is chosen with $T_xC\\subset A^x$, then every apartment $A\\subset X$ containing $C$ satisfies $T_xA=A^x$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the global apartment on the chosen tangent apartment and point",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the prescribed tangent-space matching condition $T_xA=A^x$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "approximation step from convex polyhedra to arbitrary convex Euclidean subsets",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "existential choice of apartment containing $C$ replaced by a uniqueness-type condition on all such apartments",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states hypotheses only and does not explicitly reveal the correct conclusion. Although the mention of a chosen tangent apartment foreshadows a conclusion involving matching tangent data, several options exploit that same feature, so the correct answer is not leaked."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct option is the precise theorem conclusion under the stated hypotheses. The task is largely to recognize the exact statement rather than derive a new consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish the exact existential conclusion from stronger quantifier claims, weaker true statements, and false uniqueness/conditional variants. But the item still mainly tests recognition of the theorem's exact wording rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: strengthening existential to universal dependence, dropping part of the conclusion, adding an unnecessary polyhedral restriction, and replacing existence by uniqueness-type behavior. They are distinct and well-aligned with likely misunderstandings."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors, but it is fairly tautological and only moderately probes reasoning beyond recall of the exact statement."
    }
  },
  {
    "id": "2512.12077v1",
    "paper_link": "http://arxiv.org/abs/2512.12077v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.",
      "start_pos": 9002,
      "end_pos": 9105,
      "label": "thm:square"
    },
    "ref_dict": {
      "corollary:lt": "\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}",
      "thm:square": "\\begin{theorem}\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1922,
    "pre_theorem_intro_text": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.",
    "context": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.",
    "full_context": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.\n\nThe exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential.\n\nWhile previous work dealt with the worst-case behavior of $\\rm{LT}(s)$, \\cref{thm:square} implies the best possible bound for its average-case behavior. \n\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}\n\n\\begin{lemma}\\label{lem:pt_decomposition} Let $s\\in\\{0,1\\}^\\omega$ be uniformly random. Let $\\cev q_t(w;v)=q_t(\\rev(w);\\rev(v))$.\nThen for all $v\\in\\{0,1\\}^*$,\n\\begin{equation}\\label{eq:pt_as_qt}\n\\PP[v\\in B_n(s)]=\\cev q_n(\\emptyword;v)+\\frac 12\\sum_{t=0}^{n-1}\\sum_{w\\in \\{0,1\\}^*}\\Big(\\delta_t(1w1,w)\\cev q_{n-t-1}(w1;v)+\\delta_t(0w0,w)\\cev q_{n-t-1}(w0;v)\\Big).\n\\end{equation}\nIn particular, we have\n\\begin{equation}\\label{eq:pt_as_qt_beta}\n\\PP[\\emptyword\\in B_n(s)]=q_n(\\emptyword)+\\sum_{t=0}^{n-1}\\sum_{w1\\in \\Sigma_2^{(01)}}\\delta_t(1w1,w) q_{n-t-1}(\\rev(w1)).\n\\end{equation}\n\\end{lemma}\nRoughly speaking, \\eqref{eq:pt_as_qt_beta} captures the fact that any successful buffer thread $(w_j)_j$ has a critical first time $n-t-1$ at word $w_{n-t-1}=1w$ where it diverges from the greedy algorithm.\n\\begin{proof}\nWe prove \\eqref{eq:pt_as_qt} by induction on $n$. By construction, $\\PP[v\\in B_0]=\\bbone_{v=\\emptyword}=q_0(\\emptyword;\\rev(v))$. Observe that for any word $v1$ ending in $1$, we have\n\\begin{align*}\n\\PP[v1\\in B_{n+1}]&=\\PP[s(n+1)=1]\\PP[(v\\in B_n)\\cup (1v1\\in B_n)]+\\PP[s(n+1)=0]\\PP[0v1\\in B_n]\\\\\n&=\\frac 12\\Big(\\PP[v\\in B_n]+\\delta_n(1v1,v)+\\PP[0v1\\in B_n]\\Big).\n\\end{align*}\nObserve that $q_n$ satisfies a similar recurrence: for each initializiation of the form $1v'$, we can condition on the first bit of $s$ to get\n\\begin{align*}\nq_n(w;1v')&=\\PP[s(1)=1]\\PP[\\greedy(s[2,n],v')=w]+\\PP[s(1)=0]\\PP[\\greedy(s[2,n],1v'0)=w]\\\\\n&=\\frac 12(q_{n-1}(w;v')+q_{n-1}(w;1v'0)).\n\\end{align*}\nReversing the words, we obtain\n\\[\n\\cev q_n(w;v1)=\\frac 12(\\cev q_{n-1}(w;v)+\\cev q_{n-1}(w;0v1)).\n\\]\nDenote the RHS of~\\eqref{eq:pt_as_qt} by $Q_n(v)$. Since the above is a linear recurrence relation, $Q_n$ obeys the same equation, plus the added terms at time $n+1$:\n\\[\nQ_{n+1}(v1)=\\frac 12(Q_n(v)+Q_n(0v1))+\\frac 12\\sum_{w\\in \\{0,1\\}^*}\\Big(\\delta_n(1w1,w)\\cev q_{0}(w1;v1)+\\delta_n(0w0,w)\\cev q_{0}(w0;v1)\\Big)\n\\]\nRecall that $\\cev q_0(w;v)=\\bbone_{w=v}$, so the summation evaluates to $\\frac 12\\delta_n(1v1,v)$. Thus by induction, $\\PP[v1\\in B_{n+1}]=Q_{n+1}(v1)$. By symmetry, $\\PP[v0\\in B_{n+1}]=Q_{n+1}(v0)$, showing \\eqref{eq:pt_as_qt} holds for all nonempty words $v$ at time $n+1$.\n\nWe prove that $X^*_1$ has the desired statistics and $X^*_2$ is a negligible perturbation. First, consider $X^*_1$. Since $X,Z$ are independent, we immediately have $\\Ex[(X-1)\\bbone_{Z=0}]=\\frac12 (k-1)$, and $\\Ex[(X-1-k)^2\\bbone_{Z=0}]=\\frac 12(2k+1)$. Next we control the second term in $X_1^*$; both technical claims below are proven in Section~\\ref{section:numericalclaims}.\n\\begin{claim}\\label{claim:Ybounds} (Proved in~\\cref{section:numericalclaims}.) With $X,Y,Z$ as defined above, $\\Ex[(Y-Z)\\bbone_{Z>0}]=\\frac 12(k-3+2^{-k})$ and $\\Ex[(Y-Z-k)^2\\bbone_{Z>0}]<4k+9$.\n\\end{claim}\nTogether, we have\n\\begin{align*}\n\\Ex[X^*_1-k]&=\\frac 12(k-1)+\\frac 12(k-3+2^{-k})-k=-2+2^{-k-1},\\quad\\text{and}\\\\\n\\Ex[(X^*_1-k)^2]&=\\Ex[(X^*_1-k)^2\\bbone_{Z=0}]+\\Ex[(X^*_1-k)^2\\bbone_{Z>0}]<\\frac 12(2k+1)+\\frac 12(4k+9)=3k+5.\n\\end{align*}\nNext, we control $X^*_2$. Because $\\PP[X=0]\n=2^{-k}$, it is simple to check that $\\Ex[\\max(Z,1)\\bbone_{X=0}]=\\frac 32\\cdot 2^{-k}$ and $\\Ex[\\max(Z,1)^2\\bbone_{X=0}]=\\frac 72\\cdot 2^{-k}$, which are very small. The next claim bounds the $2C'$ term.\n\\begin{claim}\\label{claim:C'bounds}(Proved in~\\cref{section:numericalclaims}.) With $M,Z,C'$ as defined above, $\\Ex[C']\\leq 2(7/8)^{k'}$ and $\\Ex[(C')^2]\\leq 6(7/8)^{k'}$. \n\\end{claim}\nWe conclude that $\\Ex[X^*_2],\\Ex[(X^*_2)^2]=O((7/8)^k)$. Therefore,\n\\[\n\\Ex[X^*-k]=\\Ex[X^*_1-k]+\\Ex[X^*_2]=-2+O((7/8)^k),\n\\]\nand similarly by the Minkowski inequality,\n\\[\n\\Ex[(X^*-k)^2]\\leq \\Big(\\sqrt{\\Ex[(X^*_1-k)^2]}+\\sqrt{\\Ex[(X^*_2)^2]}\\Big)^2=3k+5+o(1)\n\\]\nThis proves~\\eqref{eq:algorithmstep2}. To obtain tail bounds~\\eqref{eq:algorithmstep3} and~\\eqref{eq:algorithmstep4}, first note that \n\\begin{equation}\\label{eq:T*expection}\nT^*\\leq T_3= (k+X)+(Z+1)+(X-1+Y),\n\\end{equation}\nand thus $\\Ex[T^*]\\leq 4k$. By considering cases, we have\n\\[\\PP[T^* \\geq (4+5\\eps)k]\\leq \\PP[X\\geq (1+\\eps)k]+\\PP[Y\\geq (1+2\\eps)k\\mid X<(1+\\eps)k]+\\PP[Z\\geq \\eps k]\\]\nBy the definition of $Z$, we have $\\PP[Z\\geq \\eps k]=2^{-\\eps k}$. For the other two terms, recall that $X\\sim \\rm{NB}(k)$, and the conditional distribution $\\big(Y\\mid X< (1+\\eps)k\\big)$ is stochastically dominated by $\\rm{NB}((1+\\eps)k)$. Both terms can be bounded by Proposition \\ref{prop:nbchernoff}, so \n\\[\n\\PP[T^*\\geq (4+5\\eps)k]\\leq 3e^{-\\Omega(\\eps^2k/(1+\\eps))},\n\\]\nwhich gives~\\eqref{eq:algorithmstep3}. We obtain tail bounds on $X^*$ similarly. Using~\\eqref{eq:X*expression}, we concentrate $X^*$ by bounding $X,Y$, and $Z$, noting that $|2C'-Z|\\leq |Z|$:\n\\begin{multline*}\n\\PP[|X^*-k|\\geq 3\\eps k]\\leq \\PP[Z\\geq \\eps k]+\\PP[|X-k-1|\\geq \\eps k]+\n\\PP\\big[|Y-k|\\geq 2\\eps k\\ \\big|\\ |X-k-1|<\\eps k\\big].\n\\end{multline*}\nVerifying that each of these is bounded by $e^{-\\Omega(\\eps^2k/(1+\\eps))}$, we conclude~\\eqref{eq:algorithmstep4}.\n\\end{proof}\n\n\\begin{problem}\\label{prob:ternary} Is it true that $|\\cS_3(n)|=(9-o(1))^n$? \n\\end{problem}\n\\noindent This problem is closely related to the extremal problem of determining whether $\\rm{LT}(3,n)=(\\frac 12-o(1))n$. The following one directional implication follows from work of Bukh and Zhou.\n\\begin{theorem}[{\\cite[Theorem 4]{bukhzhou}}]\\label{prop:alphatwins} If $\\cS_k(n)\\leq (b_k)^{2n+o(n)}$ and there exists $\\alpha_k\\in (2/k,1)$ such that\n\\[\n\\alpha_k\\log\\Big(\\frac{b_k}{\\alpha_kk}\\Big)+(1-\\alpha_k)\\log \\Big(\\frac{k-1}{k(1-\\alpha_k)}\\Big)<0,\n\\]\nthen $\\rm{LT}(k,n)< \\frac 12\\alpha_kn$ for sufficiently large $n$. In particular, if $b_k<k$, then we can take $\\alpha_k<1$.\n\\end{theorem}\n\\noindent This can be obtained by a union bound over all ``leftmost\" embeddings of shuffle squares in a random word $s$. Though we do not offer a conjecture on Problem~\\ref{prob:ternary}, numerical evidence suggests that $|\\cS_3(n)|=o(9^n)$.",
    "post_theorem_intro_text_len": 3114,
    "post_theorem_intro_text": "The exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential. \n\nClosely related to the shuffle square problem is the ``twins'' problem introduced by Axenovich, Person, and Puzynina~\\cite{Axenovich}. For a word $s\\in [k]^*$, let $\\rm{LT}(s)$ be the maximum semi-length of a subword of $s$ that is a shuffle square, so $s\\in \\cS_k(n)$ if and only if $\\rm{LT}(s)=n$. Equivalently, $\\rm{LT}(s)$ is the largest $m$ such that $s$ contains ``twins'' of length $m$, i.e. disjoint identical subwords of length $m$. Axenovich, Person, and Puzynina proved that for any $s\\in \\{0,1\\}^{2n}$, we have $\\rm{LT}(s)=(1-o(1))n$. More progress has been made on twins for various alphabets~\\cite{basu,bukhzhou}, and the twins problem is related to regularity lemmas for words, the longest common subsequence problem \\cite{bukhhogenson,bukhma,ChvatalSankoff,He-Li}, and codes for correcting deletion errors \\cite{Guruswami-He-Li}.\n\nWhile previous work dealt with the worst-case behavior of $\\rm{LT}(s)$, \\cref{thm:square} implies the best possible bound for its average-case behavior. \n\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}\n\n\\begin{proof} Let $s\\in \\{0,1\\}^{2n}$ be uniformly random among words with an even number of $0$'s and $1$'s. If $\\rm{LT}(s)=n$, then by deleting one or two bits, we immediately have $\\rm{LT}(s[1,2n-1])=n-1$ and $\\rm{LT}(s[1,2n-2])\\geq n-2$. Thus by \\cref{thm:square},\n\\[\n\\PP[\\rm{LT}(s[1,2n-2])\\geq n-2]\\geq \\PP[\\rm{LT}(s[1,2n-1])=n-1]\\geq \\PP[s\\in \\cS_2(n)]=1-o(1).\n\\]\nMoreover, $s[1,2n-1]$ is uniformly random in $\\{0,1\\}^{2n-1}$ and $s[1,2n-2]$ is uniformly random in $\\{0,1\\}^{2n-2}$, implying the corollary.\n\\end{proof}\n\nIn \\cite{HHNT}, it was conjectured that almost every binary word can be made into a shuffle square by deleting at most three bits - \\cref{corollary:lt} shows that it actually suffices to delete two (which is best possible).\n\n\\par{\\bf Organization.} In Section~\\ref{section:preliminaries}, we introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$. We define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior. In Section~\\ref{section:buffersetanalysis}, we analyze the evolution of the set of all buffers to show that a random word $s$ can ``absorb\" small defects in an almost-perfect partition. In Section~\\ref{section:boostedgreedy}, we give an improved algorithm which produces a shuffle square with constant probability. This boosted greedy algorithm stores local information about the word and occasionally backtracks for an improvement. Together with the buffer set analysis, this proves~\\cref{thm:square}. In Section~\\ref{section:conclusion}, we discuss the shuffle square problem on larger alphabets, and we provide a boosted greedy algorithm for $k$-ary words which gives a new lower bound for $|\\cS_k(n)|$.",
    "sketch": "In the \".\\par{\\bf Organization.}\" paragraph, the authors outline how they prove Theorem~\\ref{thm:square}: (1) they \"introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$\"; (2) they \"define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior\"; (3) they \"analyze the evolution of the set of all buffers to show that a random word $s$ can \\`\\`absorb\\\" small defects in an almost-perfect partition\"; (4) they \"give an improved algorithm which produces a shuffle square with constant probability\"—a \"boosted greedy algorithm\" that \"stores local information about the word and occasionally backtracks for an improvement\"; (5) \"[t]ogether with the buffer set analysis, this proves~\\cref{thm:square}.\"",
    "expanded_sketch": "In the \".\\par{\\bf Organization.}\" paragraph, the authors outline how they prove the main theorem: (1) they \"introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$\"; (2) they \"define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior\"; (3) they \"analyze the evolution of the set of all buffers to show that a random word $s$ can \\`\\`absorb\\\" small defects in an almost-perfect partition\"; (4) they \"give an improved algorithm which produces a shuffle square with constant probability\"—a \"boosted greedy algorithm\" that \"stores local information about the word and occasionally backtracks for an improvement\"; (5) \"[t]ogether with the buffer set analysis, this proves the main theorem.\"",
    "expanded_theorem": "\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $[2]^*$ be the set of all finite binary words over $\\{0,1\\}$. A word $s'\\in[2]^*$ is a subword of $s\\in[2]^*$ if $s'$ is a subsequence of $s$ (not necessarily contiguous). A binary word $s$ of length $2n$ is called a shuffle square if its positions can be partitioned into two disjoint subwords that are identical as words. Let $\\mathcal S_2(n)$ denote the set of shuffle squares of semi-length $n$, i.e. binary shuffle squares of length $2n$. Which asymptotic estimate holds for $|\\mathcal S_2(n)|$ as $n\\to\\infty$?",
      "correct_choice": {
        "label": "A",
        "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-o\\!ig(n^{-1/15}\\big)\\right)2^{2n}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}+o\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."
        },
        {
          "label": "C",
          "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-o(1)\\right)2^{2n}$."
        },
        {
          "label": "D",
          "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-O\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."
        },
        {
          "label": "E",
          "text": "$|\\mathcal S_2(n)|\\geq \\left(\\frac{1}{2}-o\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "defect-absorption yields a deficit from $1/2$, not an excess",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "explicit polynomial error term $o(n^{-1/15})$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "non-effective little-o error replaced by stronger big-O control",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "exact asymptotic equality replaced by one-sided lower bound at the same scale",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the definition of binary shuffle squares and asks for the asymptotic count; it does not reveal the specific asymptotic form or strongly hint at the correct sign/error term."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a direct theorem recall question: it asks which asymptotic statement holds, and the correct option is essentially the theorem's conclusion. The nearby variants add some discrimination, but the structure is still largely a restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish the exact result from a weaker true statement and from stronger/false variants. However, the question mainly tests recall of a specific asymptotic theorem rather than generating or deriving a conclusion from principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are well-designed: one flips the sign of the correction, one gives a weaker true asymptotic, one offers an unjustifiably stronger rate, and one confuses the density ceiling. These are plausible and mathematically distinct failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it only moderately avoids tautology and does limited work in testing genuinely generative reasoning."
    }
  },
  {
    "id": "2512.12195v1",
    "paper_link": "http://arxiv.org/abs/2512.12195v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence \\eqref{eq:Serre-SS} for the fibration\n$\\Omega^3_0G_2\\to B\\mathcal{G}_k\\to B\\G$, and let\n$M=H^*(\\Omega^3_0G_2;\\mathbb{F}_2)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(B\\G)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(B\\G)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(B\\G)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $G_2$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0G_2;\\mathbb{F}_2)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0G_2)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(B\\G)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(B\\G)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(B\\G).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\mathbb{F}_2$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(B\\G)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\mathbb{F}_2)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.",
      "start_pos": 10636,
      "end_pos": 13335,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:Serre-SS": "\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}",
      "thm:main": "\\begin{theorem}[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence \\eqref{eq:Serre-SS} for the fibration\n$\\Omega^3_0\\G\\to B\\mathcal{G}_k\\to \\BG$, and let\n$M=H^*(\\Omega^3_0\\G;\\Ftwo)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(\\BG)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(\\BG)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(\\BG)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $\\G$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0\\G;\\Ftwo)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0\\G)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(\\BG)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\Ftwo$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(\\BG)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(\\BG).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\Ftwo$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(\\BG)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\Ftwo)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5880,
    "pre_theorem_intro_text": "\\subsection{Gauge groups and their homotopy types}\n\nLet $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle.  The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$.  When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}.  In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nThe problem of determining the homotopy types of these gauge groups\nfor specific Lie groups $G$ has received sustained attention; see\nfor example \\cite{KonoSU2,TheriaultSp2,KishimotoTheriaultTsutayaG2}.  In many cases\nthe classification is governed by the order of a fundamental Samelson\nproduct in~$G$.\n\nThe case $G=G_2$ is particularly interesting because of the exceptional\nnature of $G_2$.  Kishimoto, Theriault and Tsutaya proved that the\nclassification for $G_2$--gauge groups reduces to determining the\norder of the Samelson product\n\\[\n\\langle i_3,1\\rangle\\in [\\Sigma^3G_2,G_2],\n\\]\nwhere $i_3\\colon S^3\\to G_2$ is the inclusion of the bottom cell and\n$1$ denotes the identity map~\\cite{KishimotoTheriaultTsutayaG2}.\n\nVery recently, Kameko~\\cite{KamekoG2gauge} resolved this issue at the prime $2$ and\ncompleted the $p$--local classification of $G_2$--gauge groups.  He\nproved that the order of $\\langle i_3,1\\rangle$ is exactly $84$\n\\cite[Theorem~1.1]{KamekoG2gauge}.\nConsequently, the $p$--local homotopy type of $\\mathcal{G}_k(G_2)$ depends only on\nthe greatest common divisor $(k,84)$.  Thus the homotopy types of the\ngauge groups are completely understood up to $p$--localization.  It is\nnatural to ask how far this information is reflected in the\n(co)homology of their classifying spaces.\n\n\\subsection{Cohomology of $B\\mathcal{G}_k$ and the Steenrod algebra}\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}.  The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nBy contrast, there seems to be no published description of the full\nmod~$2$ cohomology ring $H^*(B\\mathcal{G}_k;\\mathbb{F}_2)$ for any\nnontrivial $k$, nor of its structure as an unstable $\\mathcal{A}$--module.\nOur aim here is to give a precise structural description of\n$H^*(B\\mathcal{G}_k;\\mathbb{F}_2)$ in low degrees, in terms of the evaluation\nfibration and the first nontrivial cohomology of the triple loop space\n$\\Omega^3_0G_2$.\n\n\\subsection{Main result: a low-degree description}\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$.  Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra.  The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}.  This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$.  Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees.  For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$.  Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.",
    "context": "Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle. The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$. When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}. In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}. The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}. This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$. Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees. For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}",
    "full_context": "Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle. The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$. When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}. In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}. The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}. This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$. Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees. For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\n\nIn this paper we begin a complementary study of the mod~$2$\ncohomology of the classifying spaces $B\\mathcal{G}_k(G_2)$. Our goal\nis to understand the structure of $H^*(B\\mathcal{G}_k;\\Ftwo)$ as an\nunstable module over the mod~$2$ Steenrod algebra $\\A$ in a low range\nof degrees. Using the evaluation fibration\n\\[\n\\Omega_0^3 G_2 \\longrightarrow B\\mathcal{G}_k \\xrightarrow{\\;\\mathrm{ev}\\;} BG_2\n\\]\ntogether with Serre and Eilenberg--Moore spectral sequences, we\nstudy the Serre spectral sequence\n\\[\nH^s(BG_2;H^t(\\Omega^3_0G_2)) \\Longrightarrow H^{s+t}(B\\mathcal{G}_k)\n\\]\nin total degree $\\le 10$. A careful analysis of the homotopy groups\nof $G_2$ shows that\n\\[\nH^j(\\Omega^3_0G_2;\\Ftwo)=0\\quad\\text{for }1\\le j\\le 4,\n\\qquad\nH^5(\\Omega^3_0G_2;\\Ftwo)\\neq 0,\n\\]\nso the first positive-degree generator of the fibre cohomology occurs\nin degree $5$. As a consequence, there is a distinguished class\n\\[\nu_5\\in H^5(\\Omega^3_0G_2;\\Ftwo)\n\\]\nwhose only possible Serre differential in total degree $\\le 10$ is a\n$d_6$--differential\n\\[\nd_6(u_5) = \\varepsilon(k)\\,x_6\n\\]\nfrom $u_5$ to the degree-$6$ generator $x_6\\in H^6(BG_2;\\Ftwo)$, for a\ncertain scalar $\\varepsilon(k)\\in\\Ftwo$ depending on the bundle\nclass~$k$. \n\\end{abstract}\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\Ftwo)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(\\BG)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon \\G \\longrightarrow \\Omega^3_0 \\G,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0\\G;\\Ftwo)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\Ftwo$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\item In total degree $j\\le 10$ the differentials\n \\[\n d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n \\]\n with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n the only page on which a differential with source in $E_r^{0,5}(k)$\n can be nonzero is $r=6$. After choosing a generator\n $x_6\\in H^6(\\BG)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n there is a uniquely determined scalar $\\varepsilon(k)\\in\\Ftwo$ such that\n \\[\n d_6(u_5) = \\varepsilon(k)\\, x_6\n \\]\n in $E_6^{6,0}(k)$.\n\n\\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(\\BG)$--submodule\n generated by $u_5$. In total degree $j\\le 10$ all differentials\n $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n \\[\n d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n \\]\n with $s+5\\le 10$ satisfies\n \\[\n d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n \\qquad\\text{for }x\\in H^s(\\BG).\n \\]\n In particular, within $U$ the entire effect of the Serre\n differentials in total degree $\\le 10$ is determined by the single\n scalar $\\varepsilon(k)\\in\\Ftwo$.\n\n\\begin{remark}\nTheorem~\\ref{thm:main} should be viewed as a structural reduction. It\ndoes not claim an explicit formula for $\\varepsilon(k)$, nor does it\nassert that $\\varepsilon(k)$ is nontrivial. Kameko's result shows\nthat the $2$--local homotopy type of $\\mathcal{G}_k(\\G)$ depends on\n$k\\bmod 4$ through the $2$--primary order of the Samelson product\n$\\langle i_3,1\\rangle$ \\cite{KishimotoTheriaultTsutayaG2,\nKamekoG2gauge}. It is natural to expect that $\\varepsilon(k)$ is a\nfunction of this $2$--primary data; however, determining\n$\\varepsilon(k)$ explicitly requires detailed knowledge of\n$H^*(\\Omega^3_0\\G)$ and of the connecting map $\\partial_k$ on mod~$2$\ncohomology, and we leave this as an open problem.\n\\end{remark}\n\n\\item[(ii)] Lemma~\\ref{lem:full-bidegree-vanish} only analyzes the differentials\nwith source in the $H^*(BG_2)$--submodule\n\\[\nU \\;=\\; H^*(BG_2)\\cdot u_5 \\;\\subset\\; E_2^{*,*}(k).\n\\]\nWe do not claim that $M^j=H^j(\\Omega^3_0G_2;\\F_2)$ vanishes for $j\\ge 6$,\nand additional fibre generators in these degrees could in principle\nsupport nontrivial differentials in the Serre spectral sequence.\nSuch differentials, however, do not change the description of the\npossible $d_6$--differentials on $U$ in total degree $\\le 10$, which is\nall that is needed for Theorem~\\ref{thm:main}. A determination of $M$\nin higher degrees would require an explicit computation via\nEilenberg--Moore spectral sequences, and lies beyond the scope of the\npresent paper.\n\\end{itemize}\n\\end{remark}\n\n\\begin{note}\\label{rem:epsilon-zero}\nThe case $k=0$ admits a particularly simple description. The class\n$0\\in\\pi_4(BG_2)$ corresponds to the trivial principal\n$G_2$--bundle $S^4\\times G_2\\to S^4$, and by Gottlieb's theorem \\cite{GottliebGauge}, we have\n\\[\nB\\mathcal{G}_0 \\;\\simeq\\; \\mathrm{Map}_0(S^4,BG_2),\n\\]\nwhere $\\mathrm{Map}_0(S^4,BG_2)$ denotes the component containing the\nconstant map. The evaluation map $\\mathrm{ev}\\colon \\mathrm{Map}_0(S^4,BG_2)\\to BG_2$\nadmits a section $s\\colon BG_2\\to \\mathrm{Map}_0(S^4,BG_2)$ defined by sending\n$y\\in BG_2$ to the constant map at $y$. The existence of this section\nimplies that the homomorphism\n\\[\n\\mathrm{ev}^*\\colon H^*(BG_2)\\longrightarrow H^*(B\\mathcal{G}_0)\n\\]\nis injective. In the Serre spectral sequence for the evaluation fibration\nwith $k=0$, the class $x_6\\in H^6(BG_2)\\cong E_2^{6,0}$ must therefore\nsurvive to $E_\\infty$ and cannot lie in the image of any differential.\nIn particular, $x_6$ cannot be hit by $d_6(u_5)$, forcing $d_6(u_5)=0$.\nThus, in our notation, $\\varepsilon(0)=0$.\n\\end{note}",
    "post_theorem_intro_text_len": 1320,
    "post_theorem_intro_text": "\\begin{remark}\nTheorem~\\ref{thm:main} should be viewed as a structural reduction.  It\ndoes not claim an explicit formula for $\\varepsilon(k)$, nor does it\nassert that $\\varepsilon(k)$ is nontrivial.  Kameko's result shows\nthat the $2$--local homotopy type of $\\mathcal{G}_k(G_2)$ depends on\n$k\\bmod 4$ through the $2$--primary order of the Samelson product\n$\\langle i_3,1\\rangle$ \\cite{KishimotoTheriaultTsutayaG2,\nKamekoG2gauge}.  It is natural to expect that $\\varepsilon(k)$ is a\nfunction of this $2$--primary data; however, determining\n$\\varepsilon(k)$ explicitly requires detailed knowledge of\n$H^*(\\Omega^3_0G_2)$ and of the connecting map $\\partial_k$ on mod~$2$\ncohomology, and we leave this as an open problem.\n\\end{remark}\n\n\\medskip\n\nThe paper is organised as follows.  In Section~\\ref{sec:prelim} we\nreview the necessary background on gauge groups, Samelson products,\nthe mod~$2$ cohomology of $B\\G$ and the Steenrod algebra.  In\nSection~\\ref{sec:SS-setup} we describe the evaluation fibration and\nits Serre spectral sequence, and we recall the relationship between\nthe connecting map and the Samelson product. Finally, Section~\\ref{sec:low}\ncontains the proof of Theorem~\\ref{thm:main}, including a detailed\nanalysis of the bidegrees for $j\\le 10$ and the vanishing of\n$H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.",
    "sketch": "The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Section~\\ref{sec:SS-setup} “describe[s] the evaluation fibration and its Serre spectral sequence,” and “recall[s] the relationship between the connecting map and the Samelson product.” Then Section~\\ref{sec:low} “contains the proof of Theorem~\\ref{thm:main}, including a detailed analysis of the bidegrees for $j\\le 10$ and the vanishing of $H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.”",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Next the authors describe the evaluation fibration and its Serre spectral sequence, and recall the relationship between the connecting map and the Samelson product. Then later they establish the main theorem, including a detailed analysis of the bidegrees for $j\\le 10$ and the vanishing of $H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.",
    "expanded_theorem": "[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nfor the fibration\n$\\Omega^3_0G_2\\to B\\mathcal{G}_k\\to B\\G$, and let\n$M=H^*(\\Omega^3_0G_2;\\mathbb{F}_2)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(B\\G)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(B\\G)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(B\\G)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $G_2$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0G_2;\\mathbb{F}_2)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0G_2)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(B\\G)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(B\\G)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(B\\G).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\mathbb{F}_2$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(B\\G)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\mathbb{F}_2)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.",
    "theorem_type": [
      "Universal",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Working 2-locally and with mod-2 cohomology, let \\(k\\in\\mathbb Z\\), let \\(\\mathcal G_k=\\mathcal G_k(G_2)\\) denote the gauge group of the principal \\(G_2\\)-bundle over \\(S^4\\) classified by \\(k\\in \\pi_4(BG_2)\\cong \\mathbb Z\\), and consider the evaluation fibration \\(\\Omega_0^3G_2\\to B\\mathcal G_k\\to BG_2\\). Let\n\\[\nE_2^{s,t}(k)\\cong H^s\\!\\left(BG_2;H^t(\\Omega_0^3G_2;\\mathbb F_2)\\right)\\Longrightarrow H^{s+t}(B\\mathcal G_k;\\mathbb F_2)\n\\]\nbe its Serre spectral sequence, write \\(M=H^*(\\Omega_0^3G_2;\\mathbb F_2)\\), and for \\(N\\ge 0\\) write \\(M^{\\le N}=\\bigoplus_{j\\le N}M^j\\). Which statement holds for every such integer \\(k\\)?",
      "correct_choice": {
        "label": "A",
        "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon(k)\\). Finally, for each \\(j\\le 10\\), the spectral sequence induces a finite filtration of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) whose associated graded contains, as a direct summand, the image of \\(U\\) in the total-degree-\\(j\\) part of \\(E_\\infty\\); this summand is explicitly computable from \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\) together with \\(\\varepsilon(k)\\), while the remaining extension term depends only on \\(M^{\\le j}\\) and the multiplicative structure of the Serre spectral sequence. Equivalently, in total degree \\(j\\le 10\\), the dependence of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) on the bundle class \\(k\\) that is visible through the first nontrivial fibre class \\(u_5\\) is completely encoded by the single differential \\(d_6(u_5)=\\varepsilon(k)x_6\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 5\\) and \\(M^6\\neq 0\\), so there exists a nonzero class \\(u_6\\in M^6\\cong E_2^{0,6}(k)\\), and \\(u_6\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,6}(k)\\) can be nonzero is \\(r=7\\). After choosing generators \\(x_7\\in H^7(BG_2;\\mathbb F_2)\\cong E_2^{7,0}(k)\\) and \\(u_6\\in M^6\\cong E_2^{0,6}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_7(u_6)=\\varepsilon(k)x_7\\) in \\(E_7^{7,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_6\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 6\\) vanish on \\(U\\), and every differential \\(d_7:E_7^{s,6}(k)\\to E_7^{s+7,0}(k)\\) with \\(s+6\\le 10\\) satisfies \\(d_7(x\\cdot u_6)=x\\cdot d_7(u_6)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon(k)\\)."
        },
        {
          "label": "C",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\)."
        },
        {
          "label": "D",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon\\in \\mathbb F_2\\), independent of \\(k\\), such that \\(d_6(u_5)=\\varepsilon x_6\\) in \\(E_6^{6,0}(k)\\) for every integer \\(k\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon\\), and therefore is independent of the bundle class \\(k\\)."
        },
        {
          "label": "E",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a uniquely determined scalar \\(\\varepsilon(k)\\in\\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Finally, for each \\(j\\le 10\\), the spectral sequence induces a finite filtration of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) whose associated graded is completely and explicitly determined by \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\) together with \\(\\varepsilon(k)\\); in particular, there are no additional extension terms depending on higher fibre classes or on the multiplicative structure of the Serre spectral sequence."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "bidegree-vanish",
          "tampered_component": "first_nonzero_fibre_degree",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "bidegree-vanish",
          "tampered_component": "drops the claims about the submodule U and the associated-graded direct-summand/extension-term conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence_of_epsilon_on_k",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "existence_of_remaining_extension_term",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives setup and notation but does not state the correct conclusion or uniquely signal choice A. There is no explicit answer leakage, though the structure suggests a theorem-selection task."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not a direct restatement of the stem, but it largely asks the examinee to recognize the strongest theorem-like formulation among close variants. It is more theorem recall/identification than an independently generated conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare subtle claims about the first possible differential, dependence on k, and extension control. However, the task is mostly discriminating among prewritten statements rather than producing or deriving the conclusion from scratch."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and target real mathematical failure modes: wrong differential page (B), wrong parameter dependence (D), and overclaiming control of extensions (E). But choice C appears to be a weaker true statement, creating ambiguity and weakening distractor quality; also the correct choice is unusually comprehensive/long."
      },
      "total_score": 5,
      "overall_assessment": "A technically sophisticated MCQ with little answer leakage and reasonably plausible distractors, but it is weakened by ambiguity because a weaker true option appears among the distractors, and the task leans more toward theorem recognition than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.12195v1",
    "paper_link": "http://arxiv.org/abs/2512.12195v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence \\eqref{eq:Serre-SS} for the fibration\n$\\Omega^3_0G_2\\to B\\mathcal{G}_k\\to B\\G$, and let\n$M=H^*(\\Omega^3_0G_2;\\mathbb{F}_2)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(B\\G)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(B\\G)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(B\\G)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $G_2$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0G_2;\\mathbb{F}_2)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0G_2)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(B\\G)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(B\\G)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(B\\G).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\mathbb{F}_2$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(B\\G)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\mathbb{F}_2)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.",
      "start_pos": 10636,
      "end_pos": 13335,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:Serre-SS": "\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}",
      "thm:main": "\\begin{theorem}[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence \\eqref{eq:Serre-SS} for the fibration\n$\\Omega^3_0\\G\\to B\\mathcal{G}_k\\to \\BG$, and let\n$M=H^*(\\Omega^3_0\\G;\\Ftwo)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(\\BG)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(\\BG)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(\\BG)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $\\G$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0\\G;\\Ftwo)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0\\G)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(\\BG)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\Ftwo$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(\\BG)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(\\BG).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\Ftwo$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(\\BG)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\Ftwo)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5880,
    "pre_theorem_intro_text": "\\subsection{Gauge groups and their homotopy types}\n\nLet $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle.  The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$.  When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}.  In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nThe problem of determining the homotopy types of these gauge groups\nfor specific Lie groups $G$ has received sustained attention; see\nfor example \\cite{KonoSU2,TheriaultSp2,KishimotoTheriaultTsutayaG2}.  In many cases\nthe classification is governed by the order of a fundamental Samelson\nproduct in~$G$.\n\nThe case $G=G_2$ is particularly interesting because of the exceptional\nnature of $G_2$.  Kishimoto, Theriault and Tsutaya proved that the\nclassification for $G_2$--gauge groups reduces to determining the\norder of the Samelson product\n\\[\n\\langle i_3,1\\rangle\\in [\\Sigma^3G_2,G_2],\n\\]\nwhere $i_3\\colon S^3\\to G_2$ is the inclusion of the bottom cell and\n$1$ denotes the identity map~\\cite{KishimotoTheriaultTsutayaG2}.\n\nVery recently, Kameko~\\cite{KamekoG2gauge} resolved this issue at the prime $2$ and\ncompleted the $p$--local classification of $G_2$--gauge groups.  He\nproved that the order of $\\langle i_3,1\\rangle$ is exactly $84$\n\\cite[Theorem~1.1]{KamekoG2gauge}.\nConsequently, the $p$--local homotopy type of $\\mathcal{G}_k(G_2)$ depends only on\nthe greatest common divisor $(k,84)$.  Thus the homotopy types of the\ngauge groups are completely understood up to $p$--localization.  It is\nnatural to ask how far this information is reflected in the\n(co)homology of their classifying spaces.\n\n\\subsection{Cohomology of $B\\mathcal{G}_k$ and the Steenrod algebra}\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}.  The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nBy contrast, there seems to be no published description of the full\nmod~$2$ cohomology ring $H^*(B\\mathcal{G}_k;\\mathbb{F}_2)$ for any\nnontrivial $k$, nor of its structure as an unstable $\\mathcal{A}$--module.\nOur aim here is to give a precise structural description of\n$H^*(B\\mathcal{G}_k;\\mathbb{F}_2)$ in low degrees, in terms of the evaluation\nfibration and the first nontrivial cohomology of the triple loop space\n$\\Omega^3_0G_2$.\n\n\\subsection{Main result: a low-degree description}\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$.  Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra.  The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}.  This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$.  Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees.  For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$.  Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.",
    "context": "Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle. The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$. When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}. In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}. The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}. This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$. Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees. For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}",
    "full_context": "Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW\ncomplex, and let $P\\to X$ be a principal $G$--bundle. The \\emph{gauge\ngroup} $\\mathcal{G}(P)$ is the topological group of $G$--equivariant\nbundle automorphisms of $P$ that cover the identity on $X$. When $X$\nis finite, Crabb--Sutherland showed that there are only finitely many\nhomotopy types among the gauge groups $\\mathcal{G}(P)$ as $P$ ranges\nover all principal $G$--bundles over $X$~\\cite{CrabbSutherland}. In\nthe fundamental case $X=S^4$ and $G$ simply connected and simple,\nisomorphism classes of $G$--bundles are classified by\n$\\pi_4(BG)\\cong \\mathbb{Z}$, and we write $\\mathcal{G}_k(G)$ for the\ngauge group of the bundle classified by $k\\in \\mathbb{Z}$.\n\nGottlieb identified the classifying space of $\\mathcal{G}_k$ with a\ncomponent of a mapping space~\\cite{GottliebGauge}: specifically,\n$B\\mathcal{G}_k$ is homotopy equivalent to $\\mathrm{Map}_k(S^4,BG)$,\nthe connected component of the mapping space $\\mathrm{Map}(S^4,BG)$\ncontaining the classifying map $k\\colon S^4\\to BG$.\nThis identification yields a fundamental evaluation fibration\n\\[\n\\Omega^3_0 G \\;\\longrightarrow\\; B\\mathcal{G}_k\n\\xrightarrow{\\;\\mathrm{ev}\\;} BG.\n\\]\nFor $G=G_2$, this expresses $B\\mathcal{G}_k$ as a twisted\nmapping space over the base $B\\G$.\n\nThe mod~$2$ cohomology of $B\\G$ is classically known: Borel showed that\n\\[\nH^*(B\\G;\\mathbb{F}_2) \\cong \\mathbb{F}_2[x_4,x_6,x_7]\n\\]\nis a polynomial algebra on generators in degrees $4,6,7$\n\\cite{BorelCohomology}. The action of the Steenrod algebra $\\mathcal{A}$ on\n$H^*(B\\G)$ can be described explicitly via Wu formulas, and from the\nviewpoint of the hit problem one is interested in minimal\ngenerating sets for the unstable $\\mathcal{A}$--module $\\mathbb{F}_2[x_4,x_6,x_7]$.\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\mathbb{F}_2)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(B\\G;H^t(\\Omega^3_0G_2))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(B\\G)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon G_2 \\longrightarrow \\Omega^3_0 G_2,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nThe cohomology of the fibre $\\Omega^3_0 G$ can in principle be\ncomputed by iterated Eilenberg--Moore spectral sequences applied to\nthe path-loop fibrations of $BG$ and $G$.\nChoi and Yoon conjectured that, for any simply connected finite\n$H$--space $X$, the Eilenberg--Moore spectral sequences converging\nto the mod $p$ homology of $\\Omega^2 X$ and $\\Omega^3 X$ collapse at\n$E_2$~\\cite{ChoiYoonEMSS}. This conjecture was subsequently proved by\nLin~\\cite{LinEMSS}; in particular, for any compact simple Lie group\n$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing\n$H_*(\\Omega^2 G;\\mathbb F_p)$ and $H_*(\\Omega^3 G;\\mathbb F_p)$\ncollapse at $E_2$. Combined with the classical description of\n$H^*(B\\G)$, this implies that $H^*(\\Omega^3_0G_2)$ is of finite type in\neach degree and that its generators occur in bounded degrees. For our\npurposes, however, we only need to know the very first nontrivial\ncohomology group of $\\Omega^3_0G_2$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0G_2;\\mathbb{F}_2)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\n\nIn this paper we begin a complementary study of the mod~$2$\ncohomology of the classifying spaces $B\\mathcal{G}_k(G_2)$. Our goal\nis to understand the structure of $H^*(B\\mathcal{G}_k;\\Ftwo)$ as an\nunstable module over the mod~$2$ Steenrod algebra $\\A$ in a low range\nof degrees. Using the evaluation fibration\n\\[\n\\Omega_0^3 G_2 \\longrightarrow B\\mathcal{G}_k \\xrightarrow{\\;\\mathrm{ev}\\;} BG_2\n\\]\ntogether with Serre and Eilenberg--Moore spectral sequences, we\nstudy the Serre spectral sequence\n\\[\nH^s(BG_2;H^t(\\Omega^3_0G_2)) \\Longrightarrow H^{s+t}(B\\mathcal{G}_k)\n\\]\nin total degree $\\le 10$. A careful analysis of the homotopy groups\nof $G_2$ shows that\n\\[\nH^j(\\Omega^3_0G_2;\\Ftwo)=0\\quad\\text{for }1\\le j\\le 4,\n\\qquad\nH^5(\\Omega^3_0G_2;\\Ftwo)\\neq 0,\n\\]\nso the first positive-degree generator of the fibre cohomology occurs\nin degree $5$. As a consequence, there is a distinguished class\n\\[\nu_5\\in H^5(\\Omega^3_0G_2;\\Ftwo)\n\\]\nwhose only possible Serre differential in total degree $\\le 10$ is a\n$d_6$--differential\n\\[\nd_6(u_5) = \\varepsilon(k)\\,x_6\n\\]\nfrom $u_5$ to the degree-$6$ generator $x_6\\in H^6(BG_2;\\Ftwo)$, for a\ncertain scalar $\\varepsilon(k)\\in\\Ftwo$ depending on the bundle\nclass~$k$. \n\\end{abstract}\n\nWe work $2$--locally throughout and write $H^*(-)$ for\n$H^*(-;\\Ftwo)$. Consider the Serre spectral sequence of the\nevaluation fibration\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nThe $E_2$--term is a module over $H^*(\\BG)$ and an unstable module\nover the Steenrod algebra. The twisting of this spectral sequence\ndepends on $k$ through the connecting map\n\\[\n\\partial_k\\colon \\G \\longrightarrow \\Omega^3_0 \\G,\n\\]\nwhich is the triple adjoint of the Samelson product $\\langle\nk\\cdot i_3,1\\rangle$ by work of\nKishimoto--Theriault--Tsutaya~\\cite{KishimotoTheriaultTsutayaG2}.\nKameko's calculation of the $2$--primary order of $\\langle i_3,1\\rangle$\nimplies that, after localization at $2$, the map $\\partial_k$ depends\nonly on $k\\bmod 4$.\n\nLet\n\\[\nM \\;=\\; H^*(\\Omega^3_0\\G;\\Ftwo)\n\\]\nand write $M^{\\le N}=\\bigoplus_{j\\le N}M^j$ for its truncation in\ndegrees $\\le N$. Our main theorem describes the Serre spectral\nsequence \\eqref{eq:Serre-SS} in total degree $\\le 10$ and singles out\na distinguished scalar $\\varepsilon(k)\\in\\Ftwo$ which already\ncaptures the effect of $k$ on the first nontrivial fibre class.\n\n\\item In total degree $j\\le 10$ the differentials\n \\[\n d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n \\]\n with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n the only page on which a differential with source in $E_r^{0,5}(k)$\n can be nonzero is $r=6$. After choosing a generator\n $x_6\\in H^6(\\BG)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n there is a uniquely determined scalar $\\varepsilon(k)\\in\\Ftwo$ such that\n \\[\n d_6(u_5) = \\varepsilon(k)\\, x_6\n \\]\n in $E_6^{6,0}(k)$.\n\n\\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(\\BG)$--submodule\n generated by $u_5$. In total degree $j\\le 10$ all differentials\n $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n \\[\n d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n \\]\n with $s+5\\le 10$ satisfies\n \\[\n d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n \\qquad\\text{for }x\\in H^s(\\BG).\n \\]\n In particular, within $U$ the entire effect of the Serre\n differentials in total degree $\\le 10$ is determined by the single\n scalar $\\varepsilon(k)\\in\\Ftwo$.\n\n\\begin{remark}\nTheorem~\\ref{thm:main} should be viewed as a structural reduction. It\ndoes not claim an explicit formula for $\\varepsilon(k)$, nor does it\nassert that $\\varepsilon(k)$ is nontrivial. Kameko's result shows\nthat the $2$--local homotopy type of $\\mathcal{G}_k(\\G)$ depends on\n$k\\bmod 4$ through the $2$--primary order of the Samelson product\n$\\langle i_3,1\\rangle$ \\cite{KishimotoTheriaultTsutayaG2,\nKamekoG2gauge}. It is natural to expect that $\\varepsilon(k)$ is a\nfunction of this $2$--primary data; however, determining\n$\\varepsilon(k)$ explicitly requires detailed knowledge of\n$H^*(\\Omega^3_0\\G)$ and of the connecting map $\\partial_k$ on mod~$2$\ncohomology, and we leave this as an open problem.\n\\end{remark}\n\n\\item[(ii)] Lemma~\\ref{lem:full-bidegree-vanish} only analyzes the differentials\nwith source in the $H^*(BG_2)$--submodule\n\\[\nU \\;=\\; H^*(BG_2)\\cdot u_5 \\;\\subset\\; E_2^{*,*}(k).\n\\]\nWe do not claim that $M^j=H^j(\\Omega^3_0G_2;\\F_2)$ vanishes for $j\\ge 6$,\nand additional fibre generators in these degrees could in principle\nsupport nontrivial differentials in the Serre spectral sequence.\nSuch differentials, however, do not change the description of the\npossible $d_6$--differentials on $U$ in total degree $\\le 10$, which is\nall that is needed for Theorem~\\ref{thm:main}. A determination of $M$\nin higher degrees would require an explicit computation via\nEilenberg--Moore spectral sequences, and lies beyond the scope of the\npresent paper.\n\\end{itemize}\n\\end{remark}\n\n\\begin{note}\\label{rem:epsilon-zero}\nThe case $k=0$ admits a particularly simple description. The class\n$0\\in\\pi_4(BG_2)$ corresponds to the trivial principal\n$G_2$--bundle $S^4\\times G_2\\to S^4$, and by Gottlieb's theorem \\cite{GottliebGauge}, we have\n\\[\nB\\mathcal{G}_0 \\;\\simeq\\; \\mathrm{Map}_0(S^4,BG_2),\n\\]\nwhere $\\mathrm{Map}_0(S^4,BG_2)$ denotes the component containing the\nconstant map. The evaluation map $\\mathrm{ev}\\colon \\mathrm{Map}_0(S^4,BG_2)\\to BG_2$\nadmits a section $s\\colon BG_2\\to \\mathrm{Map}_0(S^4,BG_2)$ defined by sending\n$y\\in BG_2$ to the constant map at $y$. The existence of this section\nimplies that the homomorphism\n\\[\n\\mathrm{ev}^*\\colon H^*(BG_2)\\longrightarrow H^*(B\\mathcal{G}_0)\n\\]\nis injective. In the Serre spectral sequence for the evaluation fibration\nwith $k=0$, the class $x_6\\in H^6(BG_2)\\cong E_2^{6,0}$ must therefore\nsurvive to $E_\\infty$ and cannot lie in the image of any differential.\nIn particular, $x_6$ cannot be hit by $d_6(u_5)$, forcing $d_6(u_5)=0$.\nThus, in our notation, $\\varepsilon(0)=0$.\n\\end{note}",
    "post_theorem_intro_text_len": 1320,
    "post_theorem_intro_text": "\\begin{remark}\nTheorem~\\ref{thm:main} should be viewed as a structural reduction.  It\ndoes not claim an explicit formula for $\\varepsilon(k)$, nor does it\nassert that $\\varepsilon(k)$ is nontrivial.  Kameko's result shows\nthat the $2$--local homotopy type of $\\mathcal{G}_k(G_2)$ depends on\n$k\\bmod 4$ through the $2$--primary order of the Samelson product\n$\\langle i_3,1\\rangle$ \\cite{KishimotoTheriaultTsutayaG2,\nKamekoG2gauge}.  It is natural to expect that $\\varepsilon(k)$ is a\nfunction of this $2$--primary data; however, determining\n$\\varepsilon(k)$ explicitly requires detailed knowledge of\n$H^*(\\Omega^3_0G_2)$ and of the connecting map $\\partial_k$ on mod~$2$\ncohomology, and we leave this as an open problem.\n\\end{remark}\n\n\\medskip\n\nThe paper is organised as follows.  In Section~\\ref{sec:prelim} we\nreview the necessary background on gauge groups, Samelson products,\nthe mod~$2$ cohomology of $B\\G$ and the Steenrod algebra.  In\nSection~\\ref{sec:SS-setup} we describe the evaluation fibration and\nits Serre spectral sequence, and we recall the relationship between\nthe connecting map and the Samelson product. Finally, Section~\\ref{sec:low}\ncontains the proof of Theorem~\\ref{thm:main}, including a detailed\nanalysis of the bidegrees for $j\\le 10$ and the vanishing of\n$H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.",
    "sketch": "The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Section~\\ref{sec:SS-setup} “describe[s] the evaluation fibration and its Serre spectral sequence,” and “recall[s] the relationship between the connecting map and the Samelson product.” Then Section~\\ref{sec:low} “contains the proof of Theorem~\\ref{thm:main}, including a detailed analysis of the bidegrees for $j\\le 10$ and the vanishing of $H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.”",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Next the authors describe the evaluation fibration and its Serre spectral sequence, and recall the relationship between the connecting map and the Samelson product. Then later they establish the main theorem, including a detailed analysis of the bidegrees for $j\\le 10$ and the vanishing of $H^j(\\Omega^3_0G_2)$ for $1\\le j\\le 4$.",
    "expanded_theorem": "[Low-degree structure of $H^*(B\\mathcal{G}_k)$]\\label{thm:main}\nWork $2$--locally and fix an integer $k$.  Consider the Serre spectral\nsequence\n\\begin{equation}\\label{eq:Serre-SS}\nE_2^{s,t}(k)\\;\\cong\\; H^s(\\BG;H^t(\\Omega^3_0\\G))\n\\;\\Longrightarrow\\; H^{s+t}(B\\mathcal{G}_k).\n\\end{equation}\nfor the fibration\n$\\Omega^3_0G_2\\to B\\mathcal{G}_k\\to B\\G$, and let\n$M=H^*(\\Omega^3_0G_2;\\mathbb{F}_2)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n  \\item In total degree $j=s+t\\le 10$ the $E_2$--term is naturally\n  isomorphic, as a bigraded $H^*(B\\G)$--module, to\n  \\[\n  E_2^{*,*}(k)^{\\le 10} \\;\\cong\\;\n  H^*(B\\G)\\otimes M^{\\le 10},\n  \\]\n  with $E_2^{s,t}=H^s(B\\G)\\otimes M^t$.\n\n  \\item Using the low-degree homotopy groups of $G_2$ and the Hurewicz\n  theorem, one has\n  \\[\n  M^j = H^j(\\Omega^3_0G_2;\\mathbb{F}_2)=0\\quad\\text{for }1\\le j\\le 4,\n  \\qquad\n  M^5\\neq 0.\n  \\]\n  In particular, there exists a nonzero class\n  \\[\n  u_5 \\in M^5\\cong E_2^{0,5}(k)\n  \\]\n  and $u_5$ is the first positive-degree generator of\n  $H^*(\\Omega^3_0G_2)$.\n\n  \\item In total degree $j\\le 10$ the differentials\n  \\[\n  d_r\\colon E_r^{0,t}(k)\\longrightarrow E_r^{r,t-r+1}(k)\n  \\]\n  with $2\\le r\\le 5$ vanish on all classes in bidegree $(0,t)$, and\n  the only page on which a differential with source in $E_r^{0,5}(k)$\n  can be nonzero is $r=6$.  After choosing a generator\n  $x_6\\in H^6(B\\G)\\cong E_2^{6,0}(k)$ and $u_5\\in M^5\\cong E_2^{0,5}(k)$\n  there is a uniquely determined scalar $\\varepsilon(k)\\in\\mathbb{F}_2$ such that\n  \\[\n  d_6(u_5) = \\varepsilon(k)\\, x_6\n  \\]\n  in $E_6^{6,0}(k)$.\n\n  \\item Let $U\\subset E_2^{*,*}(k)$ denote the $H^*(B\\G)$--submodule\n  generated by $u_5$.  In total degree $j\\le 10$ all differentials\n  $d_r$ with $2\\le r\\le 5$ vanish on $U$, and every differential\n  \\[\n  d_6\\colon E_6^{s,5}(k)\\longrightarrow E_6^{s+6,0}(k)\n  \\]\n  with $s+5\\le 10$ satisfies\n  \\[\n  d_6(x\\cdot u_5) = x\\cdot d_6(u_5)\n  \\qquad\\text{for }x\\in H^s(B\\G).\n  \\]\n  In particular, within $U$ the entire effect of the Serre\n  differentials in total degree $\\le 10$ is determined by the single\n  scalar $\\varepsilon(k)\\in\\mathbb{F}_2$.\n\n  \\item For each $j\\le 10$ the spectral sequence induces a finite\n  filtration of $H^j(B\\mathcal{G}_k)$ whose associated graded\n  contains, as a direct summand, the image of $U$ in\n  $E_\\infty^{*,*}(k)^j$.  This summand is explicitly computable from\n  $H^*(B\\G)\\otimes M^{\\le 10}$ together with $\\varepsilon(k)$, while\n  the remaining extension term depends only on $M^{\\le j}$ and the\n  multiplicative structure of the Serre spectral sequence.\n\\end{enumerate}\nIn particular, in total degree $j\\le 10$ the dependence of\n$H^j(B\\mathcal{G}_k;\\mathbb{F}_2)$ on the bundle class $k$ that is visible\nthrough the first nontrivial fibre class $u_5$ is completely encoded\nin the single Serre differential $d_6(u_5)=\\varepsilon(k)x_6$.",
    "theorem_type": [
      "Universal",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Working 2-locally and with mod-2 cohomology, let \\(k\\in\\mathbb Z\\), let \\(\\mathcal G_k=\\mathcal G_k(G_2)\\) denote the gauge group of the principal \\(G_2\\)-bundle over \\(S^4\\) classified by \\(k\\in \\pi_4(BG_2)\\cong \\mathbb Z\\), and consider the evaluation fibration \\(\\Omega_0^3G_2\\to B\\mathcal G_k\\to BG_2\\). Let\n\\[\nE_2^{s,t}(k)\\cong H^s\\!\\left(BG_2;H^t(\\Omega_0^3G_2;\\mathbb F_2)\\right)\\Longrightarrow H^{s+t}(B\\mathcal G_k;\\mathbb F_2)\n\\]\nbe its Serre spectral sequence, write \\(M=H^*(\\Omega_0^3G_2;\\mathbb F_2)\\), and for \\(N\\ge 0\\) write \\(M^{\\le N}=\\bigoplus_{j\\le N}M^j\\). Which statement holds for every such integer \\(k\\)?",
      "correct_choice": {
        "label": "A",
        "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon(k)\\). Finally, for each \\(j\\le 10\\), the spectral sequence induces a finite filtration of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) whose associated graded contains, as a direct summand, the image of \\(U\\) in the total-degree-\\(j\\) part of \\(E_\\infty\\); this summand is explicitly computable from \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\) together with \\(\\varepsilon(k)\\), while the remaining extension term depends only on \\(M^{\\le j}\\) and the multiplicative structure of the Serre spectral sequence. Equivalently, in total degree \\(j\\le 10\\), the dependence of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) on the bundle class \\(k\\) that is visible through the first nontrivial fibre class \\(u_5\\) is completely encoded by the single differential \\(d_6(u_5)=\\varepsilon(k)x_6\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 5\\) and \\(M^6\\neq 0\\), so there exists a nonzero class \\(u_6\\in M^6\\cong E_2^{0,6}(k)\\), and \\(u_6\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,6}(k)\\) can be nonzero is \\(r=7\\). After choosing generators \\(x_7\\in H^7(BG_2;\\mathbb F_2)\\cong E_2^{7,0}(k)\\) and \\(u_6\\in M^6\\cong E_2^{0,6}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_7(u_6)=\\varepsilon(k)x_7\\) in \\(E_7^{7,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_6\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 6\\) vanish on \\(U\\), and every differential \\(d_7:E_7^{s,6}(k)\\to E_7^{s+7,0}(k)\\) with \\(s+6\\le 10\\) satisfies \\(d_7(x\\cdot u_6)=x\\cdot d_7(u_6)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon(k)\\)."
        },
        {
          "label": "C",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon(k)\\in \\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\)."
        },
        {
          "label": "D",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a unique scalar \\(\\varepsilon\\in \\mathbb F_2\\), independent of \\(k\\), such that \\(d_6(u_5)=\\varepsilon x_6\\) in \\(E_6^{6,0}(k)\\) for every integer \\(k\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Hence, within \\(U\\), the entire effect of the Serre differentials in total degree \\(\\le 10\\) is determined by the single scalar \\(\\varepsilon\\), and therefore is independent of the bundle class \\(k\\)."
        },
        {
          "label": "E",
          "text": "In total degree \\(j=s+t\\le 10\\), the \\(E_2\\)-term is naturally isomorphic as a bigraded \\(H^*(BG_2;\\mathbb F_2)\\)-module to \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\), with \\(E_2^{s,t}(k)=H^s(BG_2;\\mathbb F_2)\\otimes M^t\\). Moreover, \\(M^j=0\\) for \\(1\\le j\\le 4\\) and \\(M^5\\neq 0\\), so there exists a nonzero class \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), and \\(u_5\\) is the first positive-degree generator of \\(H^*(\\Omega_0^3G_2;\\mathbb F_2)\\). In total degree \\(\\le 10\\), all differentials \\(d_r:E_r^{0,t}(k)\\to E_r^{r,t-r+1}(k)\\) with \\(2\\le r\\le 5\\) vanish on classes in bidegree \\((0,t)\\), and the only page on which a differential with source in \\(E_r^{0,5}(k)\\) can be nonzero is \\(r=6\\). After choosing generators \\(x_6\\in H^6(BG_2;\\mathbb F_2)\\cong E_2^{6,0}(k)\\) and \\(u_5\\in M^5\\cong E_2^{0,5}(k)\\), there is a uniquely determined scalar \\(\\varepsilon(k)\\in\\mathbb F_2\\) such that \\(d_6(u_5)=\\varepsilon(k)x_6\\) in \\(E_6^{6,0}(k)\\). If \\(U\\subset E_2^{*,*}(k)\\) is the \\(H^*(BG_2;\\mathbb F_2)\\)-submodule generated by \\(u_5\\), then in total degree \\(\\le 10\\) all \\(d_r\\) with \\(2\\le r\\le 5\\) vanish on \\(U\\), and every differential \\(d_6:E_6^{s,5}(k)\\to E_6^{s+6,0}(k)\\) with \\(s+5\\le 10\\) satisfies \\(d_6(x\\cdot u_5)=x\\cdot d_6(u_5)\\) for \\(x\\in H^s(BG_2;\\mathbb F_2)\\). Finally, for each \\(j\\le 10\\), the spectral sequence induces a finite filtration of \\(H^j(B\\mathcal G_k;\\mathbb F_2)\\) whose associated graded is completely and explicitly determined by \\(H^*(BG_2;\\mathbb F_2)\\otimes M^{\\le 10}\\) together with \\(\\varepsilon(k)\\); in particular, there are no additional extension terms depending on higher fibre classes or on the multiplicative structure of the Serre spectral sequence."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "bidegree-vanish",
          "tampered_component": "first_nonzero_fibre_degree",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "bidegree-vanish",
          "tampered_component": "drops the claims about the submodule U and the associated-graded direct-summand/extension-term conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence_of_epsilon_on_k",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "existence_of_remaining_extension_term",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem sets up notation and the spectral sequence but does not explicitly or implicitly reveal the correct choice. The key distinctions among the options are not leaked in the prompt."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not a direct restatement of the stem, but it mainly tests recognition of a precise theorem-level conclusion about the spectral sequence rather than deriving a new consequence from simpler premises."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the choices differ in subtle ways (first nonzero degree, page of the differential, dependence on k, extension claims). However, the pressure is weakened because choice C is a weaker true statement contained in A, while the stem asks only which statement 'holds,' not which strongest statement holds."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and target realistic errors: wrong degree/page shift (B), incorrect k-independence (D), and overclaiming collapse/extension control (E). But C is a poor distractor because it appears to be genuinely true, making the item ambiguous rather than cleanly single-answer."
      },
      "total_score": 5,
      "overall_assessment": "Technically sophisticated and free of answer leakage, but flawed as an MCQ because it appears to have more than one true option unless the prompt explicitly asks for the strongest conclusion. This reduces both discriminative power and generative reasoning value."
    }
  },
  {
    "id": "2512.12321v1",
    "paper_link": "http://arxiv.org/abs/2512.12321v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.",
      "start_pos": 2998,
      "end_pos": 3291,
      "label": "thm:Kitaev.vanishing"
    },
    "ref_dict": {
      "thm:Kitaev.vanishing": "\\begin{thm}\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.\n\\end{thm}",
      "prop:algebraic.step": "\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}",
      "eqn:Kitaev.class": "\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}",
      "eqn:HH.formula": "\\begin{equation}\\label{eqn:HH.formula}\n\\det(e^Ae^Be^{-A}e^{-B})=\\exp(\\mathrm{Tr}[A,B]).\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 436,
    "pre_theorem_intro_text": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.",
    "context": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.",
    "full_context": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.\n\n\\section{Introduction}\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\\det(UVU^{-1}V^{-1})$ can take \\emph{any} non-zero value \\cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \\cite{HH}, as well as $K$-theory \\cite{Brown}.\nNow, Kitaev's condition \\eqref{eqn:Kitaev.class} implies\n\\begin{equation}\\label{eqn:Kitaev.implies.det.class}\n\\begin{aligned}\nUVU^{-1}V^{-1}-\\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\\\\n&=\\big((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1})\\big)U^{-1}V^{-1}\\in\\mathcal{L}^1,\n\\end{aligned}\n\\end{equation}\nso that $\\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \\ref{sec:quantized.traces}), the veracity of Theorem \\ref{thm:Kitaev.vanishing} is of considerable general interest for applications.\n\n\\section{Proof of determinant trivialization theorem}\\label{sec:proof}\nFor each $n\\geq 1$, we write $M_n(-)$ for the $n\\times n$ matrix ring with entries in $(-)$, and $\\GL_n(\\mathcal{L})$ for the group of invertibles in $M_n(\\mathcal{L})$. We may identify $M_n(\\mathcal{L})$ with $\\mathcal{L}(\\mathcal{H}^{\\oplus n})$, and $M_n(\\mathcal{L}^1)$ with the ideal $\\mathcal{L}^1(\\mathcal{H}^{\\oplus n})$. Thus, there are Fredholm determinant homomorphisms\n\\[\n{\\det}^{(n)}:\\{U\\in \\GL_n(\\mathcal{L})\\,:\\,U-\\mathbf{1}^{\\oplus n}\\in M_n(\\mathcal{L}^1)\\}\\to\\mathbb{C}^\\times,\\qquad n\\geq 1,\n\\]\nand they are compatible with $\\det=\\det^{(1)}$ in the sense that\n\\begin{equation}\\label{eqn.det.stable}\n{\\det}^{(n)}\\begin{pmatrix} U & 0 \\\\ 0 & \\mathbf{1}^{\\oplus (n-1)}\\end{pmatrix} = {\\det}^{(1)}\\, U,\\qquad n\\geq 1.\n\\end{equation}\n\nBy combining the Steinberg relations with the Kitaev condition \\eqref{eqn:Kitaev.class}, and exploiting the redundancy Lemma \\ref{lem:redundant.elementary}, we can rewrite $\\det(UVU^{-1}V^{-1})$ as the determinant of a \\emph{single} commutator of \\emph{elementary matrices}:\n\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nAccording to the identities \\eqref{eqn:mult.commutator.stabalized}--\\eqref{eqn:Whitehead.big.UV},\n\\begin{equation}\\label{eqn:det.alg.calc.first.factorization}\n\\det(UVU^{-1}V^{-1})={\\det}^{(3)} \\big(d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\\big),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eqn:for.shuffling}\n\\begin{aligned}\nd_{12}(U) & = e_{21}(U^{-1})e_{12}(\\mathbf{1}-U)e_{21}(-\\mathbf{1})e_{12}(\\mathbf{1}-U^{-1})\\\\\nd_{13}(V) & = e_{31}(V^{-1})e_{13}(\\mathbf{1}-V)e_{31}(-\\mathbf{1})e_{13}(\\mathbf{1}-V^{-1}).\n\\end{aligned}\n\\end{equation}\nWe would like to swap the positions of $d_{12}(U)$ and $d_{13}(V)$ in \\eqref{eqn:det.alg.calc.first.factorization}, in order to cancel $d_{12}(U)^{-1}d_{13}(V)^{-1}$. This entails moving the elementary matrix factors of $d_{12}(U)$ past those of $d_{13}(V)$. Because of \\eqref{eqn:Steinberg.commutator.123}, we will pick up some commutator terms of the form $e_{ik}(ST)$ or $e_{ki}(-TS)$ along the way.\nIn general, a complicated collection of commutator terms will be acquired.\n\nFortunately, for the purposes of computing the determinant \\eqref{eqn:det.alg.calc.first.factorization}, Lemma \\ref{lem:redundant.elementary} tells us that we can discard any $e_{ik}(ST)$ or $e_{ki}(-TS)$ term with $ST\\in\\mathcal{L}^1$ or $TS\\in \\mathcal{L}^1$, since this kind of elementary matrix has trivial determinant. The Kitaev condition \\eqref{eqn:Kitaev.class} enters here: Because $(U-\\mathbf{1})(V-\\mathbf{1})$ is trace class, for $i\\neq k$,\n\\begin{align*}\n&{\\det}^{(3)}\\big(W e_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1}) W^\\prime\\big)\\\\\n&={\\det}^{(3)}\\big(W e_{ik}((U-\\mathbf{1})(V-\\mathbf{1}))e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}) W^\\prime\\big) & (\\mathrm{Eq.}\\,\\eqref{eqn:Steinberg.commutator.123})\\\\\n&={\\det}^{(3)}\\big(W e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}) W^\\prime\\big) & (\\mathrm{Lemma}\\;\\ref{lem:redundant.elementary})\n\\end{align*}\nholds whenever $W,W^\\prime$ are invertible operators such that the above determinants are defined. Similarly with $U$ and $V$ swapped. The above property is conveniently expressed as an extra ``commutativity'' relation,\n\\[\ne_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1})\\simeq e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}),\\qquad i\\neq k,\n\\]\n\\emph{where ``$\\simeq$'' indicates equality after taking a determinant}. \nKitaev's condition \\eqref{eqn:Kitaev.class} also implies\n\\[\n(U-\\mathbf{1})V=U-\\mathbf{1}=V(U-\\mathbf{1}),\\qquad\n(V-\\mathbf{1})U=V-\\mathbf{1}=U(V-\\mathbf{1})\\qquad \\mathrm{mod}\\;\\mathcal{L}^1,\n\\]\nThe extra relations made available by Kitaev's condition \\eqref{eqn:Kitaev.class} are summarized below:\n\\begin{equation}\\label{eqn:Steinberg.Kitaev.extra.identity.2}\n\\begin{aligned}\ne_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1}) &\\simeq e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}), & i\\neq k,\\\\\ne_{ij}(V-\\mathbf{1})e_{jk}(U-\\mathbf{1}) &\\simeq e_{jk}(U-\\mathbf{1})e_{ij}(V-\\mathbf{1}), & i\\neq k;\\\\\ne_{ij}((U-\\mathbf{1})V)&\\simeq e_{ij}(U-\\mathbf{1})\\simeq e_{ij}(V(U-\\mathbf{1})),\\\\\ne_{ij}((V-\\mathbf{1})U)&\\simeq e_{ij}(V-\\mathbf{1})\\simeq e_{ij}(U(V-\\mathbf{1})).\n\\end{aligned}\n\\end{equation}\nAs Kitaev's condition \\eqref{eqn:Kitaev.class} holds with $U$ replaced by $U^{-1}$ and/or $V$ replaced by $V^{-1}$, the same replacements can be made in \\eqref{eqn:Steinberg.Kitaev.extra.identity.2}.\n\nNow specialize to the case $R=\\mathcal{L}$ and $I=\\mathcal{L}^1$. For $U,V\\in \\mathcal{L}^\\times$ with determinant class $UVU^{-1}V^{-1}$, the above general definitions give\n\\begin{equation}\\label{eqn:Brown.observation}\n\\begin{aligned}\n\\check{\\det}\\circ\\partial\\{u,v\\}\n&=\\check{\\det}\\big[d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\\big]_{K_1(\\mathcal{L},\\mathcal{L}^1)}\\\\\n&=\\check{\\det}\\left[\\begin{pmatrix}UVU^{-1}V^{-1} & 0 & 0 \\\\ 0 & \\mathbf{1} & 0 \\\\ 0 & 0 & \\mathbf{1}\\end{pmatrix}\\right]_{K_1(\\mathcal{L},\\mathcal{L}^1)}\\\\\n&=\\det(UVU^{-1}V^{-1}).\n\\end{aligned}\n\\end{equation}\nThis $K$-theoretic nature of $\\det(UVU^{-1}V^{-1})$ was first observed by Brown \\cite{Brown}.\n\n\\subsection{Application to quantized traces}\\label{sec:quantized.traces}\\label{sec:application.quantized.traces}\nLet $A,B\\in\\mathcal{L}$, and suppose \n\\begin{equation}\\label{eqn:quantized.trace.hypothesis}\n[A,B]\\in \\mathcal{L}^1\\;\\;\\mathrm{and}\\;\\;(e^A-\\mathbf{1})(e^B-\\mathbf{1})\\in\\mathcal{L}^1.\n\\end{equation}\nThen, as in \\eqref{eqn:Kitaev.implies.det.class}, $(e^B-\\mathbf{1})(e^A-\\mathbf{1})$ is trace class as well. Thus, $e^A,e^B$ satisfy Kitaev's condition \\eqref{eqn:Kitaev.class}, so Theorem \\ref{thm:Kitaev.vanishing} guarantees that $\\det(e^Ae^Be^{-A}e^{-B})=1$. By the Pincus--Helton--Howe formula \\eqref{eqn:HH.formula}, we learn that the condition \\eqref{eqn:quantized.trace.hypothesis} implies the trace quantization,\n\\[\n\\frac{1}{2\\pi i}\\mathrm{Tr}[A,B]\\in\\mathbb{Z}.\n\\]",
    "post_theorem_intro_text_len": 4114,
    "post_theorem_intro_text": "This result was first suggested by A.~Kitaev in his influential paper \\cite[pp.~56]{Kitaev}, for the purpose of demonstrating the integrality of certain trace formulae appearing in modern quantum physics (see Section \\ref{sec:quantized.traces}). We shall therefore refer to \\eqref{eqn:Kitaev.class} as the \\emph{Kitaev condition} on an invertible pair $U,V\\in \\mathcal{L}^\\times$. \n\n\\medskip\nRecall that an operator $T\\in\\mathcal{L}$ is \\emph{trace class} if for an(y) orthonormal basis $\\mathcal{B}$, the sum $\\sum_{e\\in \\mathcal{B}}\\big\\langle e\\big||T|e\\big\\rangle$ is finite. In this case, its \\emph{trace}\n\\[\n\\mathrm{Tr}(T)=\\sum_{e\\in \\mathcal{B}}\\big\\langle e\\big|Te\\big\\rangle \\in\\mathbb{C}\n\\]\nis well-defined, independently of the choice of $\\mathcal{B}$. An operator $U\\in\\mathcal{L}$ is \\emph{determinant class} if $U-\\mathbf{1}\\in \\mathcal{L}^1$, in which case, its \\emph{(Fredholm) determinant} $\\det(U)\\in\\mathbb{C}$ is well-defined. See \\cite[Chapter 3]{Simon} for a detailed treatment. The identity\n\\[\n\\det(\\exp(T))=\\exp(\\mathrm{Tr}(T)),\\qquad T\\in\\mathcal{L}^1,\n\\]\nshows that a trivial determinant is intimately related to integrality of a corresponding trace (up to a $2\\pi i$ factor).\n\nThe Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\\det(UVU^{-1}V^{-1})$ can take \\emph{any} non-zero value \\cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \\cite{HH}, as well as $K$-theory \\cite{Brown}.\nNow, Kitaev's condition \\eqref{eqn:Kitaev.class} implies\n\\begin{equation}\\label{eqn:Kitaev.implies.det.class}\n\\begin{aligned}\nUVU^{-1}V^{-1}-\\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\\\\n&=\\big((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1})\\big)U^{-1}V^{-1}\\in\\mathcal{L}^1,\n\\end{aligned}\n\\end{equation}\nso that $\\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \\ref{sec:quantized.traces}), the veracity of Theorem \\ref{thm:Kitaev.vanishing} is of considerable general interest for applications. \n\nMore than fifteen years after Kitaev's paper \\cite{Kitaev} appeared, some progress was made in \\cite{EF}, where Theorem \\ref{thm:Kitaev.vanishing} was proved under the extra assumption that $(U^*-\\mathbf{1})(V-\\mathbf{1})$ and $(V-\\mathbf{1})(U^*-\\mathbf{1})$ are also trace class. Borel functional calculus played a key role in \\cite{EF}, and it is unclear whether the techniques there can be improved to address the full \\emph{adjoint-free} Theorem \\ref{thm:Kitaev.vanishing}. Nevertheless, the result of \\cite{EF} shows that there are no easy counterexamples to Theorem \\ref{thm:Kitaev.vanishing}.\n\nOur approach to Theorem \\ref{thm:Kitaev.vanishing} is completely different, and we believe that it gets to the heart of the problem. Our proof is presented in Section \\ref{sec:proof}, and comprises two main steps, both of which use the Kitaev condition \\eqref{eqn:Kitaev.class} in crucial ways.\n\\begin{enumerate}\n\\item An \\emph{algebraic} step, Prop.~\\ref{prop:algebraic.step}, simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices. This is the most novel part of the proof. Experts in algebraic $K$-theory may notice that the abstract concept of Steinberg symbols in $K_2$-theory is guiding our manipulations. We choose to keep the presentation concrete, and defer the $K$-theoretic interpretation to Section \\ref{sec:K.perspective}.\n\\item An \\emph{analytic} step computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\\eqref{eqn:HH.formula}), and Lidskii's trace theorem \\cite{Lidskii}.\n\\end{enumerate}\nFinally, the physical context for Theorem \\ref{thm:Kitaev.vanishing} is discussed in Section \\ref{sec:quantized.traces}.",
    "sketch": "The proof of Theorem~\\ref{thm:Kitaev.vanishing} (given in Section~\\ref{sec:proof}) has two main steps, both using the Kitaev condition \\eqref{eqn:Kitaev.class} in crucial ways:\n\\begin{enumerate}\n\\item An \\emph{algebraic} step (Prop.~\\ref{prop:algebraic.step}) that “simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.\n\\item An \\emph{analytic} step that “computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\\eqref{eqn:HH.formula}), and Lidskii's trace theorem.”\n\\end{enumerate}\nAdditionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \\eqref{eqn:Kitaev.implies.det.class}, so $\\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”",
    "expanded_sketch": "The proof of \\ref{thm:Kitaev.vanishing} (given later) has two main steps, both using the Kitaev condition\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nin crucial ways:\n\\begin{enumerate}\n\\item An \\emph{algebraic} step. We first prove the following proposition.\n\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}\nThis “simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.\n\\item An \\emph{analytic} step that “computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe,\n\\begin{equation}\\label{eqn:HH.formula}\n\\det(e^Ae^Be^{-A}e^{-B})=\\exp(\\mathrm{Tr}[A,B]).\n\\end{equation}\nand Lidskii's trace theorem.”\n\\end{enumerate}\nAdditionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \\eqref{eqn:Kitaev.implies.det.class}, so $\\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”",
    "expanded_theorem": "\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.",
    "theorem_type": [
      "Implication"
    ],
    "mcq": {
      "question": "Let mathcal{H} be a Hilbert space, let \\(\\mathcal{L}(\\mathcal{H})\\) be the algebra of bounded linear operators on \\(\\mathcal{H}\\), let \\(\\mathcal{L}^1(\\mathcal{H})\\) be the ideal of trace-class operators, let \\(\\mathcal{L}(\\mathcal{H})^\\times\\) denote the invertible bounded operators, and let \\(\\mathbf{1}\\) be the identity operator. Suppose \\(U,V\\in \\mathcal{L}(\\mathcal{H})^\\times\\) satisfy\n\\[\n(U-\\mathbf{1})(V-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\n\\quad\\text{and}\\quad\n(V-\\mathbf{1})(U-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H}).\n\\]\nFor an invertible operator of the form \\(\\mathbf{1}+K\\) with \\(K\\in \\mathcal{L}^1(\\mathcal{H})\\), write \\(\\det\\) for its Fredholm determinant. Under these hypotheses, which statement about the multiplicative commutator \\(UVU^{-1}V^{-1}\\) holds?",
      "correct_choice": {
        "label": "A",
        "text": "The Fredholm determinant of the commutator is trivial: \\(\\det(UVU^{-1}V^{-1})=1\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The multiplicative commutator is always determinant class under these hypotheses, and its Fredholm determinant is given by the trace formula\n\\[\n\\det(UVU^{-1}V^{-1})=\\exp\\!\\big(\\operatorname{Tr}((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1}))\\big).\n\\]"
        },
        {
          "label": "C",
          "text": "The multiplicative commutator is determinant class, i.e.\n\\[\nUVU^{-1}V^{-1}-\\mathbf{1}\\in \\mathcal{L}^1(\\mathcal{H}).\n\\]"
        },
        {
          "label": "D",
          "text": "If \\((U-\\mathbf{1})(V-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\\), then automatically\n\\[\n\\det(UVU^{-1}V^{-1})=1,\n\\]\nwithout any need to assume \\((V-\\mathbf{1})(U-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\\)."
        },
        {
          "label": "E",
          "text": "Under these hypotheses the commutator itself is trace class relative to the identity in the stronger sense that\n\\[\nUVU^{-1}V^{-1}=\\mathbf{1},\n\\]\nso in particular its Fredholm determinant equals \\(1\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "analytic-step trace formula applied directly to U,V rather than logarithmic elementary-matrix commutator",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "drop the final trivialization conclusion and keep only determinant-class/well-definedness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "two-sided Kitaev condition reduced to one-sided trace-class product hypothesis",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "algebraic.step",
          "tampered_component": "replace determinant triviality of the commutator by actual operator triviality",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states the hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that the determinant equals 1 or strongly telegraph choice A over the others."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a direct theorem-recall item: the hypotheses are presented in essentially theorem form and the correct option states the standard conclusion. However, the presence of nearby alternatives (weaker true statement, overstrong statement, and altered hypotheses) prevents it from being a pure tautological restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some reasoning to distinguish the exact conclusion from a merely weaker consequence (C), an unjustified stronger claim (E), and a one-sided hypothesis variant (D). Still, for a student who recognizes the theorem, the answer is fairly immediate rather than deeply generative."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing determinant triviality with trace-class membership (C), overextending to a one-sided condition (D), inserting an appealing but unjustified trace formula (B), or strengthening determinant triviality to operator equality (E)."
      },
      "total_score": 6,
      "overall_assessment": "A solid, high-quality theorem-based MCQ with strong distractors and no answer leakage, but it remains somewhat recall-oriented and only moderately non-tautological."
    }
  },
  {
    "id": "2512.12621v1",
    "paper_link": "http://arxiv.org/abs/2512.12621v1",
    "theorems_cnt": 10,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{Thm-Existence}\n\tFor all $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, there exists a positive constant $m_1$ depending on $n,\\alpha$ and $\\gamma$ such that for $0<m\\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem \\eqref{Prob-mini} with volume $m$ is given by a geodesic ball.",
      "start_pos": 16440,
      "end_pos": 16779,
      "label": "Thm-Existence"
    },
    "ref_dict": {
      "defn-Functional": "\\begin{equation}\\label{defn-Functional}\n\t\\mathcal{E}(F)=P(F)+\\gamma NL_{\\alpha}(F),\n\\end{equation}",
      "defn-doubleint": "\\begin{equation}\\label{defn-doubleint}\n \tNL_{\\alpha}(F)=\\int_{\\mathbb{H}^n}\\int_{\\mathbb{H}^n}{\\frac{\\chi_F(x)\\chi_F(y)}{{d_g(x,y)}^{\\alpha}}}\\,dV_g(x)dV_g(y),\n \\end{equation}",
      "Prob-mini": "\\begin{equation}\\label{Prob-mini}\n \tE(m)=\\inf_{|F|_g=m}\\mathcal{E}(F).\n \\end{equation}",
      "Thm-Existence": "\\begin{thm}\\label{Thm-Existence}\n\tFor all $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, there exists a positive constant $m_1$ depending on $n,\\alpha$ and $\\gamma$ such that for $0<m\\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem \\eqref{Prob-mini} with volume $m$ is given by a geodesic ball. \n\\end{thm}",
      "defn-Perimeter": "\\begin{equation}\\label{defn-Perimeter}\n P(F)=\\sup\\left\\{\\int_{F}\\mathrm{div}_g X\\,dV_g:X\\in C_c^1(\\mathbb{H}^n,\\mathrm{T}\\mathbb{H}^n),|X|_g\\leq 1\\right\\},\n \\end{equation}",
      "Con-Cri": "\\begin{con}[cf. \\cite{Choksi-Peletier11}]\\label{Con-Cri}\n\tGiven $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, for $m\\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\\mathcal{E}(\\cdot)$ among measurable sets $F\\in\\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.\n\t\\end{con}",
      "Thm-Nonexistence": "\\begin{thm}\\label{Thm-Nonexistence}\n\tFor all $n\\geq 2$, $0<\\alpha<2$ and $\\gamma>0$, there exists a positive constant $m_2$ depending on $n,\\alpha$ and $\\gamma$ such that for $m\\geq m_2$, the minimization problem \\eqref{Prob-mini} does not admit a minimizer with volume $m$.\n\\end{thm}",
      "Subsec-model": "\\label{Subsec-model} The $n$-dimensional hyperbolic space $\\mathbb{H}^n$ can be viewed as $\\mathbb{H}^n=(\\mathbb{R}^n_{+},g)$, where $\\mathbb{R}^n_{+}=\\{(x_1,x_2,\\dots,x_n)\\in\\mathbb{R}^n:x_n>0\\}$ is"
    },
    "pre_theorem_intro_text_len": 11253,
    "pre_theorem_intro_text": "Let $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$. For any measurable set $F\\subset\\mathbb{H}^n$, we consider the following functional:\n\\begin{equation}\\label{defn-Functional}\n\t\\mathcal{E}(F)=P(F)+\\gamma NL_{\\alpha}(F),\n\\end{equation}\nwhere\n\\begin{equation}\\label{defn-Perimeter}\n P(F)=\\sup\\left\\{\\int_{F}\\mathrm{div}_g X\\,dV_g:X\\in C_c^1(\\mathbb{H}^n,\\mathrm{T}\\mathbb{H}^n),|X|_g\\leq 1\\right\\},\n \\end{equation}\n denotes the perimeter of $F$ in the sense of  De Giorgi and the second term $NL_{\\alpha}(F)$ is defined as\n \\begin{equation}\\label{defn-doubleint}\n \tNL_{\\alpha}(F)=\\int_{\\mathbb{H}^n}\\int_{\\mathbb{H}^n}{\\frac{\\chi_F(x)\\chi_F(y)}{{d_g(x,y)}^{\\alpha}}}\\,dV_g(x)dV_g(y),\n \\end{equation}\n where $\\chi_F$ denotes the characteristic function of $F$. Furthermore,  in both \\eqref{defn-Perimeter} and \\eqref{defn-doubleint}, $g$ denotes the metric of $\\mathbb{H}^n$ and in this paper, all quantities with subscript $g$ indicate that they are considered in $\\mathbb{H}^n$. In the following, we investigate the minimization problem\n \\begin{equation}\\label{Prob-mini}\n \tE(m)=\\inf_{|F|_g=m}\\mathcal{E}(F).\n \\end{equation}\nThe minimization problem \\eqref{Prob-mini} of functional \\eqref{defn-Functional} has a physical background in Euclidean space $\\mathbb{R}^n$ and has been extensively studied. The case where $\\alpha=1$ in $\\mathbb{R}^3$ is particularly significant, recalling Gamow's liquid drop model for atomic nuclei. In this model, the nucleus is viewed as a region $\\Omega\\subset\\mathbb{R}^3$ densely packed with nucleons (protons and neutrons) at constant density, implying the nucleon count is proportional to the volume of $\\Omega$. The perimeter term in the energy functional corresponds to the surface tension binding the nucleus and the second term represents the Coulomb repulsion between protons (here all physical constants have been normalized to 1 for simplicity). Then the minimization problem is to ask that whether stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter.\n\nIn the case of $\\mathbb{R}^n$, the minimization problem of functional \\eqref{defn-Functional} is highly nontrivial since by the isoperimetric inequality, the perimeter  term attains its minimum on balls for fixed volume, while the second term attains its maximum on balls by the Riesz's rearrangement inequality (cf. \\cite[Theorem 3.7]{Leib-Loss01}). For set $F\\subset\\mathbb{R}^n$ measurable with volume $|F|=m$, we scale $F$ by $F\\to m^{\\frac{1}{n}}U$, then $|U|=1$ and using the scaling properties of $\\mathbb{R}^n$, we have\n\\begin{equation}\\label{Ex-scale}\n\t\\mathcal{E}(F)=m^{\\frac{n-1}{n}}\\left(P(U)+\\gamma m^{\\frac{n+1-\\alpha}{n}}NL_{\\alpha}(U)\\right),\n\\end{equation}\nwhich suggests that the perimeter term is dominant if volume $m$ is small and the nonlocal term $NL_{\\alpha}(\\cdot)$ is dominant if volume $m$ is large. Hence, we expect that there exist minimizers for small $m$ and there exist no minimizers for large $m$. Indeed, we define the critical volume $m_{*}$ to be the volume such that the value of the functional $\\mathcal{E}(\\cdot)$ of the ball with volume $m_{*}$ equals that of two balls with both volume $m_{*}/2$, spaced infinitely far apart, i.e.,\n\\begin{equation*}\n\t\\mathcal{E}\\left(\\left(\\frac{m_{*}}{b_n}\\right)^{\\frac{1}{n}}B\\right)=2\\mathcal{E}\\left(\\left(\\frac{m_{*}}{2 b_n}\\right)^{\\frac{1}{n}}B\\right),\n\\end{equation*}\nwhere $B$ denotes the unit ball in $\\mathbb{R}^n$ and $b_n=|B|$. In the follwing of paper, we also denote $\\omega_{n-1}:=|\\partial B|$, which is the area of the unit sphere in $\\mathbb{R}^n$. Note that by coarea formula, we have $\\omega_{n-1}=n b_n$.  A direct calculation shows\n\\begin{equation}\\label{Eq-Cri}\n\tm_{*}=\\left(\\frac{2^{\\frac{1}{n}}-1}{1-2^{\\frac{\\alpha-n}{n}}}\\frac{\\omega_{n-1}}{\\gamma NL_{\\alpha}(B)}\\right)^{\\frac{n}{n+1-\\alpha}}b_n.\n\\end{equation}\nThe significance of $m_{*}$ is that for $m>m_{*}$, the value of the functional $\\mathcal{E}(\\cdot)$ of two balls with both volume $m/2$, spaced infinitely far apart is less than that of a ball with volume $m$. Moreover, it is conjectured that\n\\begin{con}[cf. \\cite{Choksi-Peletier11}]\\label{Con-Cri}\n\tGiven $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, for $m\\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\\mathcal{E}(\\cdot)$ among measurable sets $F\\in\\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.\n\t\\end{con}\nSo far, Conjecture \\ref{Con-Cri} has not been completely solved. Kn$\\ddot{\\mathrm{u}}$pfer and Muratov \\cites{Knupfer-Muratov13,Knupfer-Muratov14} proved the following results:\n\\begin{itemize}\n\t\\item[(a)] For every $n\\geq 2$, $\\alpha\\in(0,n)$ and $\\gamma>0$, there exists a constant $m_{c_1}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has a minimizer for every $m\\leq m_{c_1}$.\n\t\\item[(b)] For every $n\\geq 2$, $\\alpha\\in(0,2)$ and $\\gamma>0$, there exists a constant $m_{c_2}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has no minimizer for every $m>m_{c_2}$.\n\t\\item[(c)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha>0$ is suffciently small, then $m_{c_1}=m_{c_2}=m_{*}$, and balls are unique minimizers for $E(m)$ when $m\\leq m_{*}$.\n\t\\item[(d)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha<2$, or if $3\\leq n\\leq 7$ and $\\alpha<n-1$, there exists $0<m_{c_0}\\leq m_{c_1}$, depending on $n,\\alpha$ and $\\gamma$ such that balls are unique minimizers for $E(m)$ with $m<m_{c_0}$.\n\\end{itemize}\n\nSince then, much progress has been made on the basis of these results. Julin \\cite{Julin14} extended the result of (d) to all $n\\geq 3$ and $\\alpha=n-2$. Bonacini and Cristoferi \\cite{Bonacini-Cristoferi14} extended the results of (c) and (d) to all $n\\geq 2$. Figalli, Fusco, Maggi, Millot and Mirini \\cite{Figalli15} extended the result of (d) to to all $n\\geq 2$ and $\\alpha\\in(0,n)$. Frank and Nam \\cite{Frank-Nam21} extended the result of (a) to $m_{c_1}=m_{*}$, they also showed that (b) holds for $0<\\alpha<n$ and $\\alpha\\leq 2$. There are also some quantitative estimates for $\\alpha$, $m_{c_0}$ and $m_{c_2}$. In the case of $n=2$, Muratov and Zaleski \\cite{Muratov15} showed that the result of (c) holds for $\\alpha\\leq 0.034$.  In the case of $n=3$ and $\\alpha=1$, Frank, Killip and Nam \\cite{Frank-Killp-Nam16} proved that $m_{c_2}\\leq 2.38m_{*}$, while Chodosh and Ruohoniemi \\cite{Chodosh-Ruohomiemi25} showed that $m_{c_0}\\geq 0.28m_{*}$. We also refer readers to \\cites{Alama21,Carazzato23,Carazzato25,Choksi17,Choksi22,Frank19,Frank-Lieb15,Goldman15,Goldman25,Muratov18,Novaga21,Novaga22,Pascale25} for some other interesting results on the nonlocal isoperimetric problem.\n\nNow we turn back to the case of $\\mathbb{H}^n$. In this situation, the perimeter term also attains its minimum on geodesic balls with fixed volume by the isoperimetric inequality, and the second term attains its maximum on geodesic balls with fixed volume by the Symmetrization Lemma (cf. \\cite[Page 225]{Beckner93}). Hence the minimization problem \\eqref{Prob-mini} in $\\mathbb{H}^n$ is also highly nontrivial and we expect that there are similar results in $\\mathbb{H}^n$ compared with $\\mathbb{R}^n$.\n\nBefore stating our main results, we make some measure-theoretic notations. We say that a function $u\\in L^1(\\mathbb{H}^n)$ has bounded variation, i.e. $u\\in BV(\\mathbb{H}^n)$, if there holds\n\\begin{equation*}\n\t\\int_{\\mathbb{H}^n}{|\\nabla u|_g}\\,dV_g:=\\sup\\left\\{\\int_{\\mathbb{H}^n}u\\,\\mathrm{div}_g X\\,dV_g: X\\in C_c^1(\\mathbb{H}^n,\\mathrm{T}\\mathbb{H}^n), |X|_g\\leq 1\\right\\}<\\infty.\n\\end{equation*}\nFor any measurable set $F\\subset\\mathbb{H}^n$, we say that $F$ has finite perimeter if its characteristic function $\\chi_F\\in BV(\\mathbb{H}^n)$ and then its perimeter can be denoted as $P(F):=\\int_{\\mathbb{H}^n}{|\\nabla\\chi_F|}\\,dV_g$. The $k$-dimensional Hausdorff measure with $k\\in[0,n]$ is denoted as $H^k(F)$.\n\nFor any $H^n$-measurable set $F$, its upper and lower densities at a point $x\\in\\mathbb{H}^n$ are defined respectively by \n\\begin{equation}\\label{Defn-uppden}\n\t\\overline{D}(F,x):=\\limsup_{r\\to 0}\\frac{|F\\cap B_r(x)|_g}{|B_r(x)|_g},\\quad  \\underline{D}(F,x):=\\liminf_{r\\to 0}\\frac{|F\\cap B_r(x)|_g}{|B_r(x)|_g},\n\\end{equation}\nwhere $B_r(x)\\subset\\mathbb{H}^n$ is the open geodesic ball with center $x\\in\\mathbb{H}^n$ and radius $r$. If the upper and lower densities are equal, their common value will be called the density of $F$ at $x$, and it will be denoted by $D(F,x)$. \n\nThe essential interior $\\mathring{F}^M$ of $F$ is then defined as the set of all $x\\in\\mathbb{H}^n$ for which $D(F,x)=1$, while the essential closure $\\bar{F}^{M}$ of $F$ is defined as the sets of all points where $\\overline{D}(F,x)>0$. The essential boundary $\\partial^M F$ of F is defined as the set of points where $\\overline{D}(F,x)>0$ and $\\overline{D}(\\mathbb{H}^n\\setminus F,x)>0$.  For any two sets $A, B\\subset\\mathbb{H}^n$, we denote $A\\Delta B:=(A\\setminus B)\\cup (B\\setminus A)$ as the symmetric difference of $A$ and $B$, then by the Lebesgue–Besicovitch Differentiation Theorem (cf. \\cite[Theorem 1.32]{Evans15}), we have\n\\begin{equation*}\n\t|F\\Delta\\mathring{F}^M|_g=|F\\Delta\\bar{F}^{M}|_g=0.\n\\end{equation*}\nWe also have that (cf. \\cite[Page 46]{Ambrosio01})\n\\begin{equation}\\label{Eq-Essenb}\n\t\\partial^M F=\\mathbb{H}^n\\setminus(\\mathring{F}^M\\cup(\\mathring{F}^c)^M).\n\\end{equation}\n\nThe reduced boundary $\\partial^{*}F$ of a set of finite perimeter F is defined as a set of points $x\\in\\partial^M F$ such that the measure-theoretic normal exists at $x$, i.e., if the following limit exists:\n\\begin{equation*}\n\t\\nu_F(x):=\\lim_{r\\to 0}\\frac{\\int_{B_r(x)}{\\nabla{\\chi_{F}}\\,dV_g}}{\\int_{B_r(x)}{|\\nabla{\\chi_{F}}|_g\\,dV_g}}\\quad \\text{and}\\quad |\\nu_F(x)|_g=1,\n\\end{equation*}\nwhere $\\nabla\\chi_F$ is the vector-valued Radon measure associated with the distributional derivative of $\\chi_F$ and $|\\nabla{\\chi_{F}}|_g$ coincides with the $H^{n-1}$ measure restricted to $\\partial^M F$. If $F$ is a set of finite perimeter, then by De Giorgi’s structure theorem (cf. \\cite[Theorem 3.59]{Ambrosio-Fusco-Pallara00}, \\cite[Theorem 15.9]{Maggi-12}), we have $P(F)=H^{n-1}(\\partial^{*}F)$. Also, by a result of Federer (cf. \\cite[Theorem 3.61]{Ambrosio-Fusco-Pallara00}), we have $\\partial^{*} F\\subset\\partial^M F$ and $H^{n-1}(\\partial^M F\\setminus\\partial^{*} F)$=0. \n\\begin{rem}\n\tFor any set $A\\subset\\mathbb{H}^n$, we can define the relative perimeter of $F$ in $A$ as $P(F;A):=|\\nabla{\\chi_{F}}|(A)=H^{n-1}(\\partial^{*}F\\cap A)$. We say that $F$ is a set of locally finite perimeter, if for every compact set $K\\subset\\mathbb{H}^n$, we have $P(F;K)<\\infty$.\n\\end{rem}\n\nWe say that a set $F$ of finite perimeter is essentially bounded, if its essential closure $\\bar{F}^{M}$ is bounded. Also, we say $F$ is decomposable, if there exists a partition $(A,B)$ of $F$ with both $|A|_g>0$ and $|B|_g>0$, such that $P(F)=P(A)+P(B)$. Otherwise, we say that $F$ is indecomposable.\n\\begin{rem}\n\tAll the measure-theoretic notations mentioned above suit the case of $\\mathbb{R}^n$ if the corresponding quantities are considered in $\\mathbb{R}^n$.\n\\end{rem}\n\nOur first result says that the functional $\\mathcal{E}(\\cdot)$ admits geodesic ball as the unique minimizer (up to hyperbolic isometries) when the volume $m$ is small.",
    "context": "In the case of $\\mathbb{R}^n$, the minimization problem of functional \\eqref{defn-Functional} is highly nontrivial since by the isoperimetric inequality, the perimeter term attains its minimum on balls for fixed volume, while the second term attains its maximum on balls by the Riesz's rearrangement inequality (cf. \\cite[Theorem 3.7]{Leib-Loss01}). For set $F\\subset\\mathbb{R}^n$ measurable with volume $|F|=m$, we scale $F$ by $F\\to m^{\\frac{1}{n}}U$, then $|U|=1$ and using the scaling properties of $\\mathbb{R}^n$, we have\n\\begin{equation}\\label{Ex-scale}\n \\mathcal{E}(F)=m^{\\frac{n-1}{n}}\\left(P(U)+\\gamma m^{\\frac{n+1-\\alpha}{n}}NL_{\\alpha}(U)\\right),\n\\end{equation}\nwhich suggests that the perimeter term is dominant if volume $m$ is small and the nonlocal term $NL_{\\alpha}(\\cdot)$ is dominant if volume $m$ is large. Hence, we expect that there exist minimizers for small $m$ and there exist no minimizers for large $m$. Indeed, we define the critical volume $m_{*}$ to be the volume such that the value of the functional $\\mathcal{E}(\\cdot)$ of the ball with volume $m_{*}$ equals that of two balls with both volume $m_{*}/2$, spaced infinitely far apart, i.e.,\n\\begin{equation*}\n \\mathcal{E}\\left(\\left(\\frac{m_{*}}{b_n}\\right)^{\\frac{1}{n}}B\\right)=2\\mathcal{E}\\left(\\left(\\frac{m_{*}}{2 b_n}\\right)^{\\frac{1}{n}}B\\right),\n\\end{equation*}\nwhere $B$ denotes the unit ball in $\\mathbb{R}^n$ and $b_n=|B|$. In the follwing of paper, we also denote $\\omega_{n-1}:=|\\partial B|$, which is the area of the unit sphere in $\\mathbb{R}^n$. Note that by coarea formula, we have $\\omega_{n-1}=n b_n$. A direct calculation shows\n\\begin{equation}\\label{Eq-Cri}\n m_{*}=\\left(\\frac{2^{\\frac{1}{n}}-1}{1-2^{\\frac{\\alpha-n}{n}}}\\frac{\\omega_{n-1}}{\\gamma NL_{\\alpha}(B)}\\right)^{\\frac{n}{n+1-\\alpha}}b_n.\n\\end{equation}\nThe significance of $m_{*}$ is that for $m>m_{*}$, the value of the functional $\\mathcal{E}(\\cdot)$ of two balls with both volume $m/2$, spaced infinitely far apart is less than that of a ball with volume $m$. Moreover, it is conjectured that\n\\begin{con}[cf. \\cite{Choksi-Peletier11}]\\label{Con-Cri}\n Given $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, for $m\\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\\mathcal{E}(\\cdot)$ among measurable sets $F\\in\\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.\n \\end{con}\nSo far, Conjecture \\ref{Con-Cri} has not been completely solved. Kn$\\ddot{\\mathrm{u}}$pfer and Muratov \\cites{Knupfer-Muratov13,Knupfer-Muratov14} proved the following results:\n\\begin{itemize}\n \\item[(a)] For every $n\\geq 2$, $\\alpha\\in(0,n)$ and $\\gamma>0$, there exists a constant $m_{c_1}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has a minimizer for every $m\\leq m_{c_1}$.\n \\item[(b)] For every $n\\geq 2$, $\\alpha\\in(0,2)$ and $\\gamma>0$, there exists a constant $m_{c_2}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has no minimizer for every $m>m_{c_2}$.\n \\item[(c)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha>0$ is suffciently small, then $m_{c_1}=m_{c_2}=m_{*}$, and balls are unique minimizers for $E(m)$ when $m\\leq m_{*}$.\n \\item[(d)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha<2$, or if $3\\leq n\\leq 7$ and $\\alpha<n-1$, there exists $0<m_{c_0}\\leq m_{c_1}$, depending on $n,\\alpha$ and $\\gamma$ such that balls are unique minimizers for $E(m)$ with $m<m_{c_0}$.\n\\end{itemize}\n\nSince then, much progress has been made on the basis of these results. Julin \\cite{Julin14} extended the result of (d) to all $n\\geq 3$ and $\\alpha=n-2$. Bonacini and Cristoferi \\cite{Bonacini-Cristoferi14} extended the results of (c) and (d) to all $n\\geq 2$. Figalli, Fusco, Maggi, Millot and Mirini \\cite{Figalli15} extended the result of (d) to to all $n\\geq 2$ and $\\alpha\\in(0,n)$. Frank and Nam \\cite{Frank-Nam21} extended the result of (a) to $m_{c_1}=m_{*}$, they also showed that (b) holds for $0<\\alpha<n$ and $\\alpha\\leq 2$. There are also some quantitative estimates for $\\alpha$, $m_{c_0}$ and $m_{c_2}$. In the case of $n=2$, Muratov and Zaleski \\cite{Muratov15} showed that the result of (c) holds for $\\alpha\\leq 0.034$. In the case of $n=3$ and $\\alpha=1$, Frank, Killip and Nam \\cite{Frank-Killp-Nam16} proved that $m_{c_2}\\leq 2.38m_{*}$, while Chodosh and Ruohoniemi \\cite{Chodosh-Ruohomiemi25} showed that $m_{c_0}\\geq 0.28m_{*}$. We also refer readers to \\cites{Alama21,Carazzato23,Carazzato25,Choksi17,Choksi22,Frank19,Frank-Lieb15,Goldman15,Goldman25,Muratov18,Novaga21,Novaga22,Pascale25} for some other interesting results on the nonlocal isoperimetric problem.\n\nNow we turn back to the case of $\\mathbb{H}^n$. In this situation, the perimeter term also attains its minimum on geodesic balls with fixed volume by the isoperimetric inequality, and the second term attains its maximum on geodesic balls with fixed volume by the Symmetrization Lemma (cf. \\cite[Page 225]{Beckner93}). Hence the minimization problem \\eqref{Prob-mini} in $\\mathbb{H}^n$ is also highly nontrivial and we expect that there are similar results in $\\mathbb{H}^n$ compared with $\\mathbb{R}^n$.\n\nWe say that a set $F$ of finite perimeter is essentially bounded, if its essential closure $\\bar{F}^{M}$ is bounded. Also, we say $F$ is decomposable, if there exists a partition $(A,B)$ of $F$ with both $|A|_g>0$ and $|B|_g>0$, such that $P(F)=P(A)+P(B)$. Otherwise, we say that $F$ is indecomposable.\n\\begin{rem}\n All the measure-theoretic notations mentioned above suit the case of $\\mathbb{R}^n$ if the corresponding quantities are considered in $\\mathbb{R}^n$.\n\\end{rem}\n\nOur first result says that the functional $\\mathcal{E}(\\cdot)$ admits geodesic ball as the unique minimizer (up to hyperbolic isometries) when the volume $m$ is small.",
    "full_context": "In the case of $\\mathbb{R}^n$, the minimization problem of functional \\eqref{defn-Functional} is highly nontrivial since by the isoperimetric inequality, the perimeter term attains its minimum on balls for fixed volume, while the second term attains its maximum on balls by the Riesz's rearrangement inequality (cf. \\cite[Theorem 3.7]{Leib-Loss01}). For set $F\\subset\\mathbb{R}^n$ measurable with volume $|F|=m$, we scale $F$ by $F\\to m^{\\frac{1}{n}}U$, then $|U|=1$ and using the scaling properties of $\\mathbb{R}^n$, we have\n\\begin{equation}\\label{Ex-scale}\n \\mathcal{E}(F)=m^{\\frac{n-1}{n}}\\left(P(U)+\\gamma m^{\\frac{n+1-\\alpha}{n}}NL_{\\alpha}(U)\\right),\n\\end{equation}\nwhich suggests that the perimeter term is dominant if volume $m$ is small and the nonlocal term $NL_{\\alpha}(\\cdot)$ is dominant if volume $m$ is large. Hence, we expect that there exist minimizers for small $m$ and there exist no minimizers for large $m$. Indeed, we define the critical volume $m_{*}$ to be the volume such that the value of the functional $\\mathcal{E}(\\cdot)$ of the ball with volume $m_{*}$ equals that of two balls with both volume $m_{*}/2$, spaced infinitely far apart, i.e.,\n\\begin{equation*}\n \\mathcal{E}\\left(\\left(\\frac{m_{*}}{b_n}\\right)^{\\frac{1}{n}}B\\right)=2\\mathcal{E}\\left(\\left(\\frac{m_{*}}{2 b_n}\\right)^{\\frac{1}{n}}B\\right),\n\\end{equation*}\nwhere $B$ denotes the unit ball in $\\mathbb{R}^n$ and $b_n=|B|$. In the follwing of paper, we also denote $\\omega_{n-1}:=|\\partial B|$, which is the area of the unit sphere in $\\mathbb{R}^n$. Note that by coarea formula, we have $\\omega_{n-1}=n b_n$. A direct calculation shows\n\\begin{equation}\\label{Eq-Cri}\n m_{*}=\\left(\\frac{2^{\\frac{1}{n}}-1}{1-2^{\\frac{\\alpha-n}{n}}}\\frac{\\omega_{n-1}}{\\gamma NL_{\\alpha}(B)}\\right)^{\\frac{n}{n+1-\\alpha}}b_n.\n\\end{equation}\nThe significance of $m_{*}$ is that for $m>m_{*}$, the value of the functional $\\mathcal{E}(\\cdot)$ of two balls with both volume $m/2$, spaced infinitely far apart is less than that of a ball with volume $m$. Moreover, it is conjectured that\n\\begin{con}[cf. \\cite{Choksi-Peletier11}]\\label{Con-Cri}\n Given $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, for $m\\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\\mathcal{E}(\\cdot)$ among measurable sets $F\\in\\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.\n \\end{con}\nSo far, Conjecture \\ref{Con-Cri} has not been completely solved. Kn$\\ddot{\\mathrm{u}}$pfer and Muratov \\cites{Knupfer-Muratov13,Knupfer-Muratov14} proved the following results:\n\\begin{itemize}\n \\item[(a)] For every $n\\geq 2$, $\\alpha\\in(0,n)$ and $\\gamma>0$, there exists a constant $m_{c_1}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has a minimizer for every $m\\leq m_{c_1}$.\n \\item[(b)] For every $n\\geq 2$, $\\alpha\\in(0,2)$ and $\\gamma>0$, there exists a constant $m_{c_2}>0$ depending on $n,\\alpha$ and $\\gamma$ such that $E(m)$ has no minimizer for every $m>m_{c_2}$.\n \\item[(c)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha>0$ is suffciently small, then $m_{c_1}=m_{c_2}=m_{*}$, and balls are unique minimizers for $E(m)$ when $m\\leq m_{*}$.\n \\item[(d)] For fixed $\\gamma>0$, if $n=2$ and $\\alpha<2$, or if $3\\leq n\\leq 7$ and $\\alpha<n-1$, there exists $0<m_{c_0}\\leq m_{c_1}$, depending on $n,\\alpha$ and $\\gamma$ such that balls are unique minimizers for $E(m)$ with $m<m_{c_0}$.\n\\end{itemize}\n\nSince then, much progress has been made on the basis of these results. Julin \\cite{Julin14} extended the result of (d) to all $n\\geq 3$ and $\\alpha=n-2$. Bonacini and Cristoferi \\cite{Bonacini-Cristoferi14} extended the results of (c) and (d) to all $n\\geq 2$. Figalli, Fusco, Maggi, Millot and Mirini \\cite{Figalli15} extended the result of (d) to to all $n\\geq 2$ and $\\alpha\\in(0,n)$. Frank and Nam \\cite{Frank-Nam21} extended the result of (a) to $m_{c_1}=m_{*}$, they also showed that (b) holds for $0<\\alpha<n$ and $\\alpha\\leq 2$. There are also some quantitative estimates for $\\alpha$, $m_{c_0}$ and $m_{c_2}$. In the case of $n=2$, Muratov and Zaleski \\cite{Muratov15} showed that the result of (c) holds for $\\alpha\\leq 0.034$. In the case of $n=3$ and $\\alpha=1$, Frank, Killip and Nam \\cite{Frank-Killp-Nam16} proved that $m_{c_2}\\leq 2.38m_{*}$, while Chodosh and Ruohoniemi \\cite{Chodosh-Ruohomiemi25} showed that $m_{c_0}\\geq 0.28m_{*}$. We also refer readers to \\cites{Alama21,Carazzato23,Carazzato25,Choksi17,Choksi22,Frank19,Frank-Lieb15,Goldman15,Goldman25,Muratov18,Novaga21,Novaga22,Pascale25} for some other interesting results on the nonlocal isoperimetric problem.\n\nNow we turn back to the case of $\\mathbb{H}^n$. In this situation, the perimeter term also attains its minimum on geodesic balls with fixed volume by the isoperimetric inequality, and the second term attains its maximum on geodesic balls with fixed volume by the Symmetrization Lemma (cf. \\cite[Page 225]{Beckner93}). Hence the minimization problem \\eqref{Prob-mini} in $\\mathbb{H}^n$ is also highly nontrivial and we expect that there are similar results in $\\mathbb{H}^n$ compared with $\\mathbb{R}^n$.\n\nWe say that a set $F$ of finite perimeter is essentially bounded, if its essential closure $\\bar{F}^{M}$ is bounded. Also, we say $F$ is decomposable, if there exists a partition $(A,B)$ of $F$ with both $|A|_g>0$ and $|B|_g>0$, such that $P(F)=P(A)+P(B)$. Otherwise, we say that $F$ is indecomposable.\n\\begin{rem}\n All the measure-theoretic notations mentioned above suit the case of $\\mathbb{R}^n$ if the corresponding quantities are considered in $\\mathbb{R}^n$.\n\\end{rem}\n\nOur first result says that the functional $\\mathcal{E}(\\cdot)$ admits geodesic ball as the unique minimizer (up to hyperbolic isometries) when the volume $m$ is small.\n\nHowever, such transformations are lacking in $\\mathbb{H}^n$. Fortunately, after comparing several common models of $\\mathbb{H}^n$, we have adopted the $\\Phi_{\\lambda}$-transformation (see \\S\\ref{Subsec-model} for details) in the upper half-space model of $\\mathbb{H}^n$, which has the advantage that by adjusting the value of parameter $\\lambda$, the volume of any set in $\\mathbb{H}^n$ can be scaled to a given multiple. By means of the $\\Phi_{\\lambda}$-transformation, we can establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability, and we can further deduce the existence of minimizers for all sufficiently small volumes $m\\leq m_0$ for some positive $m_0$ depending only on $n,\\alpha$ and $\\gamma$.\n\nOn the other hand, by adopting similar ideas from \\cite{Knupfer-Muratov14} and leveraging the $\\Phi_{\\lambda}$-transformation, we can prove that the functional $\\mathcal{E}(\\cdot)$ does not admit a minimizer for large volume $m$ with an extra assumption that $\\alpha<2$.\n\\begin{thm}\\label{Thm-Nonexistence}\n For all $n\\geq 2$, $0<\\alpha<2$ and $\\gamma>0$, there exists a positive constant $m_2$ depending on $n,\\alpha$ and $\\gamma$ such that for $m\\geq m_2$, the minimization problem \\eqref{Prob-mini} does not admit a minimizer with volume $m$.\n\\end{thm}\n\\begin{rem}\n We find it difficult to extend Theorem \\ref{Thm-Nonexistence} to the case of $\\alpha=2$, as in the work of Frank and Nam \\cite{Frank-Nam21}, the reason being that we are unsure how to obtain the estimate in \\cite[Lemma 7]{Frank-Nam21}.\n\\end{rem}\n\\begin{rem}\n Motivated by the classical results in $\\mathbb{R}^n$ (cf. \\cite[Theorem 2.9]{Bonacini-Cristoferi14} and \\cite[Theorem 1.5]{Figalli15}), we intend to calculate the first and second variations of the functional \\eqref{defn-Functional} in our subsequent work, so as to provide explicit bounds for the volume within which geodesic balls are local minimizers. As an application, we aim to derive similar quantitative results regarding parameter $m_1$ as those presented in \\cite{Chodosh-Ruohomiemi25}.\n\\end{rem}\nThe paper is organized as follows: In \\S\\ref{Sec-pre}, we collect some preliminaries which will be used later in this paper, including the upper-half space model of $\\mathbb{H}^n$, some isperimetric-type inequalities in $\\mathbb{H}^n$, two notions of quasiminimizer of the perimeter, and Fuglede’s type estimates of $P(E)$ and $NL_{\\alpha}(E)$ for a nearly spherical set $E\\subset\\mathbb{H}^n$. In \\S\\ref{Sec-Basic es}, we prove some basic properties for the minimizer of the functional $\\mathcal{E}(\\cdot)$, including the regularity result, the boundedness, the indecomposability and then prove a uniform lower density bound. During this process, we also derive a general criterion on a set of finite perimeter being not a minimizer. In \\S\\ref{Sec-Existence}, we prove that there exists a minimizer if the volume $m$ is small and it is contained in a geodesic ball with small radius. In \\S\\ref{Sec-Ball}, we prove that the functional $\\mathcal{E}(\\cdot)$ admits geodesic ball as unique minimizers (up to hyperbolic isometries) when volumes $m$ are small. In \\S\\ref{Sec-Nonex}, we firstly prove that both the the minimization value \\eqref{Prob-mini} and the diameter of the essential closure of a minimizer can be two-sided controlled by the volume $m$, and then complete the proof that there exist no minimizers for large volumes.\n\nThen we are left to prove that for $m\\leq m_0$, the minimization problem \\eqref{Prob-mini} admits a minimizer $E$ with volume $m$ which is contained in a geodesic ball of radius $(1+C_3 r^{\\frac{1}{2n}})r$. Firstly, we can choose a sequence of sets of finite perimeter $\\{F_k\\}$ with $|F_k|_g=m$ such that \n\\begin{equation*}\n \\lim_{k\\to\\infty}\\mathcal{E}(F_k)=\\inf_{|F|_g=m}{\\mathcal{E}(F)}.\n\\end{equation*}\nBy the argument above, we can further choose a minimizing sequence $\\{G_k\\}$, such that up to isometries, $G_k\\subset B_{(1+C_3 r^{\\frac{1}{2n}})r}(p)$ for some fixed $p\\in\\mathbb{H}^n$. Then by the lower semicontinuity of perimeter (cf. \\cite[Proposition 12.15]{Maggi-12}) and the compactness of perimeter (cf. \\cite[Theorem 12.26]{Maggi-12}), there exists a set of finite perimeter $E\\subset\\mathbb{H}^n$, a fixed point $p\\in\\mathbb{H}^n$ and a subsequence, which we also denote as $\\{G_k\\}$ such that\n\\begin{align}\n G_k\\to& E,\\quad E\\subset B_{(1+C_3 r^{\\frac{1}{2n}})r}(p),\\\\\n P(E)&\\leq\\liminf_{k\\to\\infty} P(G_k).\\label{Eq-peri}\n\\end{align}\nSince $G_k\\to E$, we have $m=|G_k|_g\\to|E|_g$ and hence $|E|_g=m$. Furthermore, by Lemma \\ref{Lem-NL}, we have\n\\begin{equation}\\label{Eq-NLa}\n \\lim_{k\\to\\infty}{NL_{\\alpha}(G_k)}=NL_{\\alpha}(E).\n\\end{equation}\nCombining \\eqref{Eq-peri} with \\eqref{Eq-NLa} gives\n\\begin{align*}\n \\inf_{|F|_g=m}{\\mathcal{E}(F)}\\leq\\mathcal{E}(E)&=P(E)+\\gamma NL_{\\alpha}(E)\\\\\n &\\leq \\liminf_{k\\to\\infty} P(G_k)+\\gamma\\lim_{k\\to\\infty}{NL_{\\alpha}(G_k)}\\\\\n &=\\liminf_{k\\to\\infty}\\mathcal{E}(G_k)\\\\\n &\\leq \\liminf_{k\\to\\infty}\\mathcal{E}(F_k)=\\inf_{|F|_g=m}{\\mathcal{E}(F)},\n\\end{align*}\nwhich implies that $E$ is a minimizer of the functional $\\mathcal{E}(\\cdot)$ with volume $m$. \n\\end{proof}\n\\section{Geodesic ball as the minimizer}\\label{Sec-Ball}\nIn this section, we give the proof of Theorem \\ref{Thm-Existence}. By Proposition \\ref{Prop-Quasi}, we know that for any given $r_0>0$, the minimizer to the minimization problem \\eqref{Prob-mini} is an $(\\omega,r_0)$-minimizer for the perimeter with some constant $\\omega>0$. Using Theorem \\ref{Lem-Min}, we can calculate which quantities $\\omega$ precisely depends on provided $m\\leq m_0$.\n\\begin{prop}\\label{Prop-Qp}\n Let $n\\geq 2$, $\\alpha\\in(0,n)$ and $\\gamma>0$. Assume that $E$ is a minimizer with $|E|_g=|B_r|_g=m$ for $m\\leq m_0$, then there exist two constants $\\Lambda_1$ and $\\Lambda_2$ depending only on $n$, $\\alpha$ and $\\gamma$ such that\n\\begin{equation}\\label{In-Qp}\n P(E)\\leq P(F)+\\left(\\frac{\\Lambda_1}{r}+\\Lambda_2\\right)|E\\Delta F|_g\n\\end{equation}\nfor every $F\\subset\\mathbb{H}^n$. Here $\\Lambda_1$ and $\\Lambda_2$ can be chosen as \n\\begin{align}\n \\Lambda_1&=\\frac{6C_4}{b_n},\\label{Defin-Lamb1}\\\\\n \\Lambda_2&=\\max\\left\\{\\frac{6C_4}{b_n}+14\\gamma c_3,\\gamma c_3\\left(2C_5^{\\frac{2(\\alpha+1-n)}{n-1}}+C_5+1\\right)\\right\\},\\label{Defin-Lamb2}\n\\end{align}\nwhere $C_4$ is the constant defined in \\eqref{Defn-C_4},\n\\begin{equation}\\label{Defn-C5}\n C_5=\\left(\\frac{2C_4}{nb_n}\\right)^{\\frac{n}{n-1}},\n\\end{equation}\nand $c_3$ is the constant defined in Lemma \\ref{Lem-Bound} depending on $n$, $\\alpha$ and $C_5 m_0$ (we choose $\\bar{m}=C_5 m_0$).\n\\end{prop}\n\\begin{proof}\n In the following of proof, we always assume that $P(F)\\leq P(E)$, since otherwise \\eqref{In-Qp} holds trivially. We claim that we can reduce to prove \\eqref{In-Qp} in the case that \n \\begin{equation}\\label{In-twos}\n \\frac{1}{2}\\leq\\frac{|F|_g}{m}\\leq C_5.\n \\end{equation}\n Indeed, if we compare $E$ with $B_r$, by \\eqref{Eq-Min0}, we have\n \\begin{equation}\\label{Qp1}\n \\mathcal{E}(E)=P(E)+\\gamma NL_{\\alpha}(E)\\leq C_4(r^{n-1}+r^n).\n \\end{equation}\n Then if $|F|_g<\\frac{1}{2}m$, we have\n \\begin{equation*}\n |E\\Delta F|_g\\geq |E|_g-|F|_g>\\frac{1}{2}m\\geq\\frac{b_n}{2} r^n.\n \\end{equation*}\n Combined with the definition of $\\Lambda_1$ and $\\Lambda_2$ gives \\eqref{In-Qp}. Also, if $|F|_g>C_5 m\\geq C_5 b_n r^n$, then by \\eqref{Eq-HEin} and $m\\leq m_0$ (which implies $r\\leq 1$ by \\eqref{Defn-m0}), we have\n \\begin{equation*}\n P(F)\\geq nb_n^{\\frac{1}{n}}|F|_g^{\\frac{n-1}{n}}\\geq nb_n C_5^{\\frac{n-1}{n}}r^{n-1}=2C_4 r^{n-1}\\geq P(E),\n \\end{equation*}\n which also gives \\eqref{In-Qp}.",
    "post_theorem_intro_text_len": 5717,
    "post_theorem_intro_text": "\\begin{rem}\n\tIt remains an open problem to provide  explicit lower bounds on $m_1$.\n\\end{rem}\n\\begin{rem}\n\tThrough a more refined analysis of the constants appearing in the estimates, we can further show that if $0<\\alpha\\leq\\bar{\\alpha}$ and $0<\\gamma\\leq\\bar{\\gamma}$ for two fixed $\\bar{\\alpha}\\in(0,n)$ and $\\bar{\\gamma}>0$, then the positive constant $m_1$ in Theorem \\ref{Thm-Existence} can be chosen to depend only on $n$, $\\bar{\\alpha}$ and $\\bar{\\gamma}$.\n\\end{rem}\nThe proof of this theorem in $\\mathbb{H}^n$ differs significantly from that in $\\mathbb{R}^n$. In $\\mathbb{R}^n$, scaling transformations possess particularly convenient geometric properties. In particular, they allow one to rescale the volume to an arbitrary prescribed value while preserving the shape of the underlying set, and geodesic balls are mapped to geodesic balls under such transformations. Consequently, the analysis of the shape of minimizers in the small-volume regime can be reduced to the study of the corresponding minimization problem for a fixed reference volume $b_n$, as presented in \\cites{Bonacini-Cristoferi14,Knupfer-Muratov13,Knupfer-Muratov14,Figalli15}. \n\nHowever, such transformations are lacking in $\\mathbb{H}^n$. Fortunately, after comparing several common models of $\\mathbb{H}^n$, we have adopted the $\\Phi_{\\lambda}$-transformation (see \\S\\ref{Subsec-model} for details) in the upper half-space model of $\\mathbb{H}^n$, which has the advantage that by adjusting the value of parameter $\\lambda$, the volume of any set in $\\mathbb{H}^n$ can be scaled to a given multiple. By means of the $\\Phi_{\\lambda}$-transformation, we can establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability, and we can further deduce the existence of minimizers for all sufficiently small volumes $m\\leq m_0$ for some positive $m_0$ depending only on $n,\\alpha$ and $\\gamma$. \n\nWhen proving that geodesic balls are the unique minimizers for small volumes, we encounter a new difficulty. Since our $\\Phi_{\\lambda}$-transformation cannot map geodesic balls to geodesic balls, the argument for a fixed volume of $b_n$ in $\\mathbb{R}^n$ is inapplicable. A key observation is that if the minimizer $E$ satisfies $|E|_g=|B_r|_g=m$ for $m\\leq m_0$ and some positive $r$, then $E$ is contained in a geodesic ball of radius $(1+C_3 r^{\\frac{1}{2n}})r$ for some uniform constant $C_3$. Combining this observation with a notion of quasiminimizer of the perimeter, we are able to show that $\\partial E$ can be viewed as a radial graph over $\\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough. Then by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term, we are able to prove that for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.\n\nOn the other hand, by adopting similar ideas from \\cite{Knupfer-Muratov14} and leveraging the $\\Phi_{\\lambda}$-transformation, we can prove that the functional $\\mathcal{E}(\\cdot)$ does not admit a minimizer for large volume $m$ with an extra assumption that $\\alpha<2$.\n\\begin{thm}\\label{Thm-Nonexistence}\n\tFor all $n\\geq 2$, $0<\\alpha<2$ and $\\gamma>0$, there exists a positive constant $m_2$ depending on $n,\\alpha$ and $\\gamma$ such that for $m\\geq m_2$, the minimization problem \\eqref{Prob-mini} does not admit a minimizer with volume $m$.\n\\end{thm}\n\\begin{rem}\n\tWe find it difficult to extend Theorem \\ref{Thm-Nonexistence} to the case of $\\alpha=2$, as in the work of Frank and Nam \\cite{Frank-Nam21}, the reason being that we are unsure how to obtain the estimate in \\cite[Lemma 7]{Frank-Nam21}.\n\\end{rem}\n\\begin{rem}\n\tMotivated by the classical results in $\\mathbb{R}^n$ (cf. \\cite[Theorem 2.9]{Bonacini-Cristoferi14} and \\cite[Theorem 1.5]{Figalli15}), we intend to calculate the first and second variations of the functional \\eqref{defn-Functional} in our subsequent work, so as to provide explicit bounds for the volume within which geodesic balls are local minimizers. As an application, we aim to derive similar quantitative results regarding parameter $m_1$ as those presented in \\cite{Chodosh-Ruohomiemi25}.\n\\end{rem}\nThe paper is organized as follows: In \\S\\ref{Sec-pre}, we collect some preliminaries which will be used later in this paper, including the upper-half space model of $\\mathbb{H}^n$, some isperimetric-type inequalities in $\\mathbb{H}^n$, two notions of quasiminimizer of the perimeter, and Fuglede’s type estimates of $P(E)$ and $NL_{\\alpha}(E)$ for a nearly spherical set $E\\subset\\mathbb{H}^n$. In \\S\\ref{Sec-Basic es}, we prove some basic properties for the minimizer of the functional $\\mathcal{E}(\\cdot)$, including the regularity result, the boundedness, the indecomposability and then prove a uniform lower density bound. During this process, we also derive a general criterion on a set of finite perimeter being not a minimizer. In \\S\\ref{Sec-Existence}, we prove that there exists a minimizer if the volume $m$ is small and it is contained in a geodesic ball with small radius. In \\S\\ref{Sec-Ball}, we prove that the functional $\\mathcal{E}(\\cdot)$ admits geodesic ball as unique minimizers (up to hyperbolic isometries) when volumes $m$ are small. In \\S\\ref{Sec-Nonex}, we firstly prove that both the the minimization value \\eqref{Prob-mini} and the diameter of the essential closure of a minimizer can be two-sided controlled by the volume $m$, and then complete the proof that there exist no minimizers for large volumes.\n\n\\begin{ack}\n\tThe research was supported by NSFC Grant No.12471047 and  China Postdoctoral Science Foundation No.2024M751605.\n\\end{ack}",
    "sketch": "Compared to $\\mathbb{R}^n$, the proof of Theorem~\\ref{Thm-Existence} in $\\mathbb{H}^n$ “differs significantly” because “such transformations are lacking in $\\mathbb{H}^n$.” The authors instead “adopted the $\\Phi_{\\lambda}$-transformation … in the upper half-space model,” which lets one “by adjusting the value of parameter $\\lambda$, [scale] the volume of any set in $\\mathbb{H}^n$ … to a given multiple.”\n\nUsing the $\\Phi_{\\lambda}$-transformation, they “establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability,” and “further deduce the existence of minimizers for all sufficiently small volumes $m\\leq m_0$.”\n\nFor uniqueness of geodesic balls at small volume, they note a key obstruction: “our $\\Phi_{\\lambda}$-transformation cannot map geodesic balls to geodesic balls,” so the Euclidean fixed-volume reduction is “inapplicable.” A “key observation” is that if a minimizer $E$ has $|E|_g=|B_r|_g=m\\leq m_0$, then “$E$ is contained in a geodesic ball of radius $(1+C_3 r^{\\frac{1}{2n}})r$.” Combining this with “a notion of quasiminimizer of the perimeter,” they show “$\\partial E$ can be viewed as a radial graph over $\\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough.” Finally, “by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term,” they conclude that “for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.”",
    "expanded_sketch": "Compared to $\\mathbb{R}^n$, the proof in $\\mathbb{H}^n$ “differs significantly” because “such transformations are lacking in $\\mathbb{H}^n$.” The authors instead “adopted the $\\Phi_{\\lambda}$-transformation … in the upper half-space model,” which lets one “by adjusting the value of parameter $\\lambda$, [scale] the volume of any set in $\\mathbb{H}^n$ … to a given multiple.”\n\nUsing the $\\Phi_{\\lambda}$-transformation, they “establish several fundamental properties of the minimizers, including their regularity, essential boundedness and indecomposability,” and “further deduce the existence of minimizers for all sufficiently small volumes $m\\leq m_0$.”\n\nFor the uniqueness of geodesic balls at small volume (which is the conclusion in establishing the main theorem), they note a key obstruction: “our $\\Phi_{\\lambda}$-transformation cannot map geodesic balls to geodesic balls,” so the Euclidean fixed-volume reduction is “inapplicable.” A “key observation” is that if a minimizer $E$ has $|E|_g=|B_r|_g=m\\leq m_0$, then “$E$ is contained in a geodesic ball of radius $(1+C_3 r^{\\frac{1}{2n}})r$.” Combining this with “a notion of quasiminimizer of the perimeter,” they show “$\\partial E$ can be viewed as a radial graph over $\\mathbb{S}^{n-1}$ with the radial function close to $r$ provided the volume $m$ is small enough.” Finally, “by means of proof by contradiction and Fuglede's type estiamtes for the perimeter and the nonlocal term,” they conclude that “for small volumes $m$, geodesic balls are unique minimizers up to hyperbolic isometries.”",
    "expanded_theorem": "\\label{Thm-Existence}\n\tFor all $n\\geq 2$, $0<\\alpha<n$ and $\\gamma>0$, there exists a positive constant $m_1$ depending on $n,\\alpha$ and $\\gamma$ such that for $0<m\\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem \n\\begin{equation}\\label{Prob-mini}\n \tE(m)=\\inf_{|F|_g=m}\\mathcal{E}(F).\n \\end{equation}\n with volume $m$ is given by a geodesic ball.,",
    "theorem_type": [
      "Existential–Universal",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(n\\ge 2\\), \\(0<\\alpha<n\\), and \\(\\gamma>0\\). For a measurable set \\(F\\subset \\mathbb H^n\\) with hyperbolic volume \\(|F|_g\\), consider the energy\n\\[\n\\mathcal E(F)=P(F)+\\gamma\\,NL_{\\alpha}(F),\n\\]\nwhere \\(P(F)\\) is the hyperbolic perimeter and \\(NL_{\\alpha}(F)\\) is the corresponding nonlocal interaction term. For each prescribed volume \\(m>0\\), define\n\\[\nE(m)=\\inf\\{\\mathcal E(F): |F|_g=m\\}.\n\\]\nWhich statement holds about the minimizers of this fixed-volume problem in \\(\\mathbb H^n\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant \\(m_1>0\\), depending only on \\(n\\), \\(\\alpha\\), and \\(\\gamma\\), such that for every \\(m\\) with \\(0<m\\le m_1\\), the problem \\(E(m)=\\inf_{|F|_g=m}\\mathcal E(F)\\) has a minimizer, and that minimizer is a geodesic ball of volume \\(m\\), unique up to hyperbolic isometries."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant \\(m_1>0\\), depending only on \\(n\\), \\(\\alpha\\), and \\(\\gamma\\), such that for every \\(m\\) with \\(0<m\\le m_1\\), the problem \\(E(m)=\\inf_{|F|_g=m}\\mathcal E(F)\\) has a minimizer, and every minimizer is a geodesic ball of volume \\(m\\), unique without passing to hyperbolic isometries."
        },
        {
          "label": "C",
          "text": "There exists a constant \\(m_1>0\\), depending only on \\(n\\), \\(\\alpha\\), and \\(\\gamma\\), such that for every \\(m\\) with \\(0<m\\le m_1\\), the problem \\(E(m)=\\inf_{|F|_g=m}\\mathcal E(F)\\) has a minimizer in \\(\\mathbb H^n\\)."
        },
        {
          "label": "D",
          "text": "There exists a constant \\(m_1>0\\), depending only on \\(n\\), \\(\\alpha\\), and \\(\\gamma\\), such that for every \\(m>0\\), the problem \\(E(m)=\\inf_{|F|_g=m}\\mathcal E(F)\\) has a minimizer, and that minimizer is a geodesic ball of volume \\(m\\), unique up to hyperbolic isometries."
        },
        {
          "label": "E",
          "text": "For every prescribed volume \\(m>0\\), there exists a constant \\(m_1>0\\), depending only on \\(n\\), \\(\\alpha\\), \\(\\gamma\\), and \\(m\\), such that whenever \\(0<m\\le m_1\\), the problem \\(E(m)=\\inf_{|F|_g=m}\\mathcal E(F)\\) has a minimizer, and that minimizer is a geodesic ball of volume \\(m\\), unique up to hyperbolic isometries."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "uniqueness only modulo hyperbolic isometries",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "geodesic-ball characterization and uniqueness up to isometries",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "small-volume restriction m\\le m_1",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "uniform dependence of threshold only on n,\\alpha,\\gamma",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only sets up the variational problem and asks which conclusion holds; it does not reveal the small-volume threshold, geodesic-ball characterization, or uniqueness up to isometries."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is essentially a direct theorem statement about small-volume minimizers being geodesic balls, so the item functions largely as theorem recall rather than requiring selection among substantively different mathematical conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing exact quantifiers and the role of uniqueness up to isometries versus absolute uniqueness, but overall the task mainly tests memory of the theorem's precise wording. Generative pressure is further weakened because option C is a weaker true statement."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and target common errors (dropping the small-volume restriction, mishandling uniqueness modulo isometries, changing parameter dependence), but the presence of C as a weaker true statement makes the distractor set flawed for a single-correct MCQ."
      },
      "total_score": 5,
      "overall_assessment": "No answer leakage, but the item is mostly a theorem-restatement question with only moderate reasoning demand. Distractors are conceptually relevant, yet the inclusion of a weaker true alternative undermines single-answer quality."
    }
  },
  {
    "id": "2512.12835v1",
    "paper_link": "http://arxiv.org/abs/2512.12835v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem: intro}\n\tConsistently from a measurable cardinal, there are two $\\sigma$-complete ultrafilters $\\mathcal{V}, \\mathcal{W}$ such that $\\mathcal{V}$ is Lipschitz below $\\mathcal{W}$, but $\\mathcal{V}$ and $\\mathcal{W}$ are Ketonen-incomparable. Furthermore, the same is also consistent with the assumption that the Weak Ultrapower Axiom holds.",
      "start_pos": 6822,
      "end_pos": 7206,
      "label": "theorem: intro"
    },
    "ref_dict": {
      "Def: Lipschitz order on ultrafilters": "\\begin{definition}[The Lipschitz order on ultrafilters]\\label{Def: Lipschitz order on ultrafilters}\nLet $U,W\\subseteq \\mathcal{P}(\\kappa)$ be ultrafilters on $\\kappa$. We say that $U$ is Lipschitz below $W$, and denote $U<_L W$, if one of the following equivalent conditions hold:\n\\begin{itemize}\n\t\\item player I has a winning strategy in $G_{\\kappa}(W,U)$.\n\t\\item there exists a super-Lipschitz function $f\\colon \\mathcal{P}(\\kappa)\\to \\mathcal{P}(\\kappa)$ that reduces $U$ to $W$, in the sense that $f^{-1}[W] = U$.\n\\end{itemize}\n\n\\end{definition}",
      "Theorem: Separating Lipschitz from Ketonen the Easy way": "\\begin{theorem}\\label{Theorem: Separating Lipschitz from Ketonen the Easy way}\n\tAssume $V= L[U]$, $\\kappa$ is the unique measurable cardinal, $\\po$ is the forcing notion described above and $G,G^*, U_0,U_1$ are as above. Denote $\\vc = U_0$ and $\\wc$ the measure derived from $j^{V[G]}_{(U_1)^2}$ using the ordinal $j_U(\\kappa)+\\kappa$ as a seed. Then $\\vc<_{L} \\wc$ but $\\vc, \\wc$ are Ketonen incomparable.\n\\end{theorem}",
      "Def: Ketonen order": "\\begin{definition}[The Ketonen order\\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \\cite{Ketonen1972strongcompactnesscardinalsins}.}]\\label{Def: Ketonen order}\n\tLet $U,W$ be $\\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:\n\t\\begin{enumerate}\n\t\t\\item There exists $I\\in W$ and a sequence $\\la U_\\xi \\colon \\xi\\in I \\ra$ of $\\sigma$-complete ultrafilters, such that each $U_\\xi$ concentrates on $\\xi$, and, for every $X\\subseteq \\kappa$ (where $\\kappa$ is the underlying ordinal of $U$), \n\t\t$$X\\in U \\iff \\{ \\xi\\in I \\colon X\\cap \\xi \\in U_\\xi \\}\\in W.$$\n\t\t\\item There exists a $\\sigma$-complete ultrafilter $U^*\\in M_W$ and an elementary embedding $k\\colon M_U\\to M^{M_W}_{U^*}$ such that $k\\circ j_U = j^{M_W}_{U^*}\\circ j_W$, and $k\\left( [Id]_U \\right)< j_{U^*}\\left( [Id]_W \\right)$.\n\t\\end{enumerate}\n\n\\end{definition}",
      "theorem: intro": "\\begin{theorem}\\label{theorem: intro}\n\tConsistently from a measurable cardinal, there are two $\\sigma$-complete ultrafilters $\\mathcal{V}, \\mathcal{W}$ such that $\\mathcal{V}$ is Lipschitz below $\\mathcal{W}$, but $\\mathcal{V}$ and $\\mathcal{W}$ are Ketonen-incomparable. Furthermore, the same is also consistent with the assumption that the Weak Ultrapower Axiom holds.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1804,
    "pre_theorem_intro_text": "The study and analysis of the various connections between $\\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts\nthat for every pair of $\\sigma$-complete ultrafilters, $U,W$, there are $\\sigma$-complete ultrafilters $W^*\\in M_U$ and $U^*\\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$.\\footnote{For a $\\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\\colon V\\to M_U$ denotes the corresponding elementary embedding. If $U\\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \\colon N\\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.} \nGoldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.\n\nIn his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \\cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:",
    "context": "The study and analysis of the various connections between $\\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts\nthat for every pair of $\\sigma$-complete ultrafilters, $U,W$, there are $\\sigma$-complete ultrafilters $W^*\\in M_U$ and $U^*\\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$.\\footnote{For a $\\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\\colon V\\to M_U$ denotes the corresponding elementary embedding. If $U\\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \\colon N\\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.} \nGoldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.\n\nIn his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \\cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:\n\n\\begin{definition}[The Ketonen order\\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \\cite{Ketonen1972strongcompactnesscardinalsins}.}]\\label{Def: Ketonen order}\n\tLet $U,W$ be $\\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:\n\t\\begin{enumerate}\n\t\t\\item There exists $I\\in W$ and a sequence $\\la U_\\xi \\colon \\xi\\in I \\ra$ of $\\sigma$-complete ultrafilters, such that each $U_\\xi$ concentrates on $\\xi$, and, for every $X\\subseteq \\kappa$ (where $\\kappa$ is the underlying ordinal of $U$), \n\t\t$$X\\in U \\iff \\{ \\xi\\in I \\colon X\\cap \\xi \\in U_\\xi \\}\\in W.$$\n\t\t\\item There exists a $\\sigma$-complete ultrafilter $U^*\\in M_W$ and an elementary embedding $k\\colon M_U\\to M^{M_W}_{U^*}$ such that $k\\circ j_U = j^{M_W}_{U^*}\\circ j_W$, and $k\\left( [Id]_U \\right)< j_{U^*}\\left( [Id]_W \\right)$.\n\t\\end{enumerate}\n\n\\end{definition}\n\n\\begin{definition}[The Lipschitz order on ultrafilters]\\label{Def: Lipschitz order on ultrafilters}\nLet $U,W\\subseteq \\mathcal{P}(\\kappa)$ be ultrafilters on $\\kappa$. We say that $U$ is Lipschitz below $W$, and denote $U<_L W$, if one of the following equivalent conditions hold:\n\\begin{itemize}\n\t\\item player I has a winning strategy in $G_{\\kappa}(W,U)$.\n\t\\item there exists a super-Lipschitz function $f\\colon \\mathcal{P}(\\kappa)\\to \\mathcal{P}(\\kappa)$ that reduces $U$ to $W$, in the sense that $f^{-1}[W] = U$.\n\\end{itemize}\n\n\\end{definition}",
    "full_context": "The study and analysis of the various connections between $\\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts\nthat for every pair of $\\sigma$-complete ultrafilters, $U,W$, there are $\\sigma$-complete ultrafilters $W^*\\in M_U$ and $U^*\\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$.\\footnote{For a $\\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\\colon V\\to M_U$ denotes the corresponding elementary embedding. If $U\\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \\colon N\\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.} \nGoldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.\n\nIn his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \\cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:\n\n\\begin{definition}[The Ketonen order\\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \\cite{Ketonen1972strongcompactnesscardinalsins}.}]\\label{Def: Ketonen order}\n\tLet $U,W$ be $\\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:\n\t\\begin{enumerate}\n\t\t\\item There exists $I\\in W$ and a sequence $\\la U_\\xi \\colon \\xi\\in I \\ra$ of $\\sigma$-complete ultrafilters, such that each $U_\\xi$ concentrates on $\\xi$, and, for every $X\\subseteq \\kappa$ (where $\\kappa$ is the underlying ordinal of $U$), \n\t\t$$X\\in U \\iff \\{ \\xi\\in I \\colon X\\cap \\xi \\in U_\\xi \\}\\in W.$$\n\t\t\\item There exists a $\\sigma$-complete ultrafilter $U^*\\in M_W$ and an elementary embedding $k\\colon M_U\\to M^{M_W}_{U^*}$ such that $k\\circ j_U = j^{M_W}_{U^*}\\circ j_W$, and $k\\left( [Id]_U \\right)< j_{U^*}\\left( [Id]_W \\right)$.\n\t\\end{enumerate}\n\n\\end{definition}\n\n\\begin{definition}[The Lipschitz order on ultrafilters]\\label{Def: Lipschitz order on ultrafilters}\nLet $U,W\\subseteq \\mathcal{P}(\\kappa)$ be ultrafilters on $\\kappa$. We say that $U$ is Lipschitz below $W$, and denote $U<_L W$, if one of the following equivalent conditions hold:\n\\begin{itemize}\n\t\\item player I has a winning strategy in $G_{\\kappa}(W,U)$.\n\t\\item there exists a super-Lipschitz function $f\\colon \\mathcal{P}(\\kappa)\\to \\mathcal{P}(\\kappa)$ that reduces $U$ to $W$, in the sense that $f^{-1}[W] = U$.\n\\end{itemize}\n\n\\end{definition}\n\n\\begin{abstract}\nIn his study of the Ultrapower Axiom (UA), Goldberg revealed a connection between UA and the determinacy of certain games that witness Lipschitz reducibility between ultrafilters. In particular, he analyzed the relationship between the Ketonen and Lipschitz orders—two natural extensions of the Mitchell order from normal measures to arbitrary $\\sigma$-complete ultrafilters—and proved that the Lipschitz order extends the Ketonen order. He further observed that under UA the two orders coincide. Goldberg asked if it's consistent that the orders differ from each other. We show that the answer is positive. In fact, even the Weak Ultrapower Axiom does not imply that the Ketonen and Lipschitz orders coincide.\n\\end{abstract}\n\nThe study and analysis of the various connections between $\\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts\nthat for every pair of $\\sigma$-complete ultrafilters, $U,W$, there are $\\sigma$-complete ultrafilters $W^*\\in M_U$ and $U^*\\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$.\\footnote{For a $\\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\\colon V\\to M_U$ denotes the corresponding elementary embedding. If $U\\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \\colon N\\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.} \nGoldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.\n\nIn his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \\cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:\n\nThe \\textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$. Models of the Weak UA+$\\neg$UA were constructed in \\cite{benhamouGoldberg2024applications} and more recently in \\cite{kaplan2025number}. We will rely on the construction from \\cite{kaplan2025number} in the proof of Theorem \\ref{theorem: intro}.\n\n\\begin{definition}[The Ketonen order\\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \\cite{Ketonen1972strongcompactnesscardinalsins}.}]\\label{Def: Ketonen order}\n Let $U,W$ be $\\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:\n \\begin{enumerate}\n \\item There exists $I\\in W$ and a sequence $\\la U_\\xi \\colon \\xi\\in I \\ra$ of $\\sigma$-complete ultrafilters, such that each $U_\\xi$ concentrates on $\\xi$, and, for every $X\\subseteq \\kappa$ (where $\\kappa$ is the underlying ordinal of $U$), \n $$X\\in U \\iff \\{ \\xi\\in I \\colon X\\cap \\xi \\in U_\\xi \\}\\in W.$$\n \\item There exists a $\\sigma$-complete ultrafilter $U^*\\in M_W$ and an elementary embedding $k\\colon M_U\\to M^{M_W}_{U^*}$ such that $k\\circ j_U = j^{M_W}_{U^*}\\circ j_W$, and $k\\left( [Id]_U \\right)< j_{U^*}\\left( [Id]_W \\right)$.\n \\end{enumerate}\n\nThe study of the Ketonen order is especially interesting in the context of Goldberg's Ultrapower Axiom. Recall that the \\textit{Ultrapower Axiom} (UA) is the assertion that for every pair of $\\sigma$-complete ultrafilters, $U,W$, there are $\\sigma$-complete ultrafilters $U^*\\in M_W$ and $W^*\\in M_U$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$, and $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$. The \\textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$.\n\nWe proceed and define the Lipschitz order between $\\sigma$-complete ultrafilters. For that, we describe the Lipschitz game of length $\\kappa$, $G_{\\kappa}(W,U)$, where $W,U$ are subsets of $\\mathcal{P}(\\kappa)$ for an ordinal $\\kappa$. The game is being held between two players, I and II, and consists of $\\kappa$ stages. On the $i$-th stage ($i<\\kappa$), Player I chooses a bit $a(i)\\in \\{0,1\\}$, and Player II responds with a bit $b(i)\\in \\{0,1\\}$. Since Player I moves first at any stage, they are aware of the sequences $\\la a(j) \\colon j<i \\ra, \\la b(j) \\colon j<i \\ra$ constructed so far; Player II moves second, being aware of the values of $\\la a(j) \\colon j\\leq i \\ra, \\la b(j) \\colon j<i \\ra$. After $\\kappa$ rounds, the players have constructed a pair of subsets of $\\kappa$,\n$$ a = \\{ i<\\kappa \\colon a(i) = 1 \\} $$\n$$ b = \\{ i<\\kappa \\colon b(i) = 1 \\}. $$\nPlayer II wins if $a\\in W\\leftrightarrow b\\in U$. Otherwise, Player I wins.\n\nThe Lipschitz order is a strict partial order on $\\sigma$-complete ultrafilters (see \\cite[Corollary 3.4.29]{GoldbergUABook}). To the best of our knowledge, it is still not known whether the Lipschitz order on $\\sigma$-complete ultrafilters must be well-founded (but it is well-founded under UA, see Corollary \\ref{Corollary: UA implies that Ketonen is Lipschitz} below). We remark that the Lipschitz order on arbitrary subsets of $\\mathcal{P}(\\kappa)$ is ill-founded in models of AC (see \\cite[Theorem 3.4.30]{GoldbergUABook}); however, in this paper, we focus only on the Lipschitz order restricted to $\\sigma$-complete ultrafilters. We freely refer to it as the “Lipschitz order,” without mentioning the restriction to pairs of ultrafilters.\nWe proceed and prove that the Lipschitz order extends the Ketonen order. We provide a proof directly from the definitions given above (for an alternative justification, see Proposition \\ref{Proposition: equivalent characterization of Lipschitz and Ketonen}).",
    "post_theorem_intro_text_len": 1236,
    "post_theorem_intro_text": "The \\textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\\circ j_U = j^{M_{W}}_{U^*}\\circ j_{W}$. Models of the Weak UA+$\\neg$UA were constructed in \\cite{benhamouGoldberg2024applications} and more recently in \\cite{kaplan2025number}. We will rely on the construction from \\cite{kaplan2025number} in the proof of Theorem \\ref{theorem: intro}.\n\nThe structure of this paper is as follows: \n\\begin{itemize}\n\t\\item In Section 2 we define the Ketonen and Lipschitz orders and outline their basic properties; all the results in this section are due to Goldberg. \n\t\\item In Section 3 we show that, consistently, the Ketonen and Lipschitz orders do not coincide (however, this is not done in a model of the Weak UA). See Theorem \\ref{Theorem: Separating Lipschitz from Ketonen the Easy way}. \n\t\\item In Section 4 we sketch the basic properties of forcing with  nonstationary support products.\n\t\\item In Section 5, we separate the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of Theorem \\ref{theorem: intro}.\n\\end{itemize}\n\n\\textbf{Acknowledgments:} The author deeply thanks Gabe Goldberg and Eilon Bilinsky for many conversations on the subject of this paper.",
    "sketch": "We will rely on the construction from \\cite{kaplan2025number} in the proof of Theorem \\ref{theorem: intro}. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties (Section 2); (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA) (Section 3, Theorem \\ref{Theorem: Separating Lipschitz from Ketonen the Easy way}); (3) sketching basic properties of forcing with nonstationary support products (Section 4); and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, \"thereby completing the proof of Theorem \\ref{theorem: intro}\" (Section 5).",
    "expanded_sketch": "We will rely on the construction from \\cite{kaplan2025number} in the proof of the main theorem. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties; (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA). We first prove the following theorem. \\begin{theorem}\\label{Theorem: Separating Lipschitz from Ketonen the Easy way}\n\tAssume $V= L[U]$, $\\kappa$ is the unique measurable cardinal, $\\po$ is the forcing notion described above and $G,G^*, U_0,U_1$ are as above. Denote $\\vc = U_0$ and $\\wc$ the measure derived from $j^{V[G]}_{(U_1)^2}$ using the ordinal $j_U(\\kappa)+\\kappa$ as a seed. Then $\\vc<_{L} \\wc$ but $\\vc, \\wc$ are Ketonen incomparable.\n\\end{theorem} (3) sketching basic properties of forcing with nonstationary support products; and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of the main theorem.",
    "expanded_theorem": "\\label{theorem: intro}\n\tConsistently from a measurable cardinal, there are two $\\sigma$-complete ultrafilters $\\mathcal{V}, \\mathcal{W}$ such that $\\mathcal{V}$ is Lipschitz below $\\mathcal{W}$, but $\\mathcal{V}$ and $\\mathcal{W}$ are Ketonen-incomparable. Furthermore, the same is also consistent with the assumption that the Weak Ultrapower Axiom holds.",
    "theorem_type": [
      "Independence or Consistency",
      "Existence"
    ],
    "mcq": {
      "question": "Assume the consistency of a measurable cardinal. For $\\sigma$-complete ultrafilters $U,W$ on ordinals, write $U<_L W$ if player I has a winning strategy in the length-$\\kappa$ game $G_{\\kappa}(W,U)$ (where at stage $i<\\kappa$ player I chooses $a(i)\\in\\{0,1\\}$, player II chooses $b(i)\\in\\{0,1\\}$, and player II wins iff $a=\\{i:a(i)=1\\}\\in W\\leftrightarrow b=\\{i:b(i)=1\\}\\in U$), and write $U<_k W$ if there is a $\\sigma$-complete ultrafilter $U^*\\in M_W$ and an elementary embedding $k\\colon M_U\\to M^{M_W}_{U^*}$ such that $k\\circ j_U=j^{M_W}_{U^*}\\circ j_W$ and $k([\\mathrm{Id}]_U)<j_{U^*}([\\mathrm{Id}]_W)$. The Weak Ultrapower Axiom says that for every pair of $\\sigma$-complete ultrafilters $U,W$, there are $\\sigma$-complete ultrafilters $U^*\\in M_W$ and $W^*\\in M_U$ such that $M^{M_U}_{W^*}=M^{M_W}_{U^*}$. Which statement is consistent under that large-cardinal assumption?",
      "correct_choice": {
        "label": "A",
        "text": "It is consistent that there exist two $\\sigma$-complete ultrafilters $\\mathcal V,\\mathcal W$ such that $\\mathcal V<_L\\mathcal W$, but $\\mathcal V$ and $\\mathcal W$ are Ketonen-incomparable (that is, neither $\\mathcal V<_k\\mathcal W$ nor $\\mathcal W<_k\\mathcal V$); moreover, it is also consistent that such a situation occurs in a model satisfying the Weak Ultrapower Axiom."
      },
      "choices": [
        {
          "label": "B",
          "text": "It is consistent that whenever $\\mathcal V,\\mathcal W$ are $\\sigma$-complete ultrafilters with $\\mathcal V<_L\\mathcal W$, one has $\\mathcal V<_k\\mathcal W$; in particular, in every model satisfying the Weak Ultrapower Axiom, the Lipschitz and Ketonen orders coincide on $\\sigma$-complete ultrafilters."
        },
        {
          "label": "C",
          "text": "It is consistent that there exist two $\\sigma$-complete ultrafilters $\\mathcal V,\\mathcal W$ such that $\\mathcal V<_L\\mathcal W$, but $\\mathcal V\\not<_k\\mathcal W$."
        },
        {
          "label": "D",
          "text": "It is consistent that there exist two $\\sigma$-complete ultrafilters $\\mathcal V,\\mathcal W$ such that $\\mathcal V<_L\\mathcal W$, but $\\mathcal W<_k\\mathcal V$; moreover, it is also consistent that such a situation occurs in a model satisfying the Weak Ultrapower Axiom."
        },
        {
          "label": "E",
          "text": "It is consistent that there exist two $\\sigma$-complete ultrafilters $\\mathcal V,\\mathcal W$ such that $\\mathcal V<_L\\mathcal W$, $\\mathcal V$ and $\\mathcal W$ are Ketonen-incomparable, and this already follows in every model satisfying the Weak Ultrapower Axiom."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "separation persists despite Weak UA",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped reverse non-comparability and Weak UA refinement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "incomparability replaced by reverse Ketonen comparability",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "consistency in some Weak-UA model strengthened to consequence of all Weak-UA models",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions of the Lipschitz order, Ketonen order, and Weak Ultrapower Axiom, but it does not explicitly or implicitly reveal which of the listed consistency statements is correct."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a mere restatement of a definition or theorem. The respondent must distinguish among several nearby consistency claims involving separation, incomparability, and the role of Weak UA."
      },
      "GPS": {
        "score": 1,
        "justification": "The item does require some reasoning about logical strength and the interaction between the two orders. However, the reasoning pressure is weakened because one distractor is a weaker statement entailed by the marked correct answer, so the task becomes ambiguous rather than cleanly generative."
      },
      "DQS": {
        "score": 0,
        "justification": "The distractors are not all functioning well. In particular, choice C is a weaker statement implied by A, so it is also consistent if A is consistent. That creates answer ambiguity. The other distractors are distinct, but the presence of an actually true weaker option is a major flaw."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically sophisticated and non-tautological item, but it is significantly weakened by poor option design: the marked correct answer entails another choice, so the MCQ does not have a uniquely correct answer as written."
    }
  },
  {
    "id": "2512.12931v1",
    "paper_link": "http://arxiv.org/abs/2512.12931v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "thmA",
      "content": "\\label{thmA}\nLet $M$ be a type $\\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\\theta \\in \\operatorname{Aut}(M)$ be a state preserving automorphism. Then the following conditions are equivalent:\n\\begin{enumerate}\n\t\\item $\\theta$ has infinite order in $\\mathrm{Out}(M)$;\n\t\\item there exists $u \\in \\mathcal U(M)$ such that $\\operatorname{Ad}(u)\\circ\\theta$ is state preserving and ergodic.\n\\end{enumerate}",
      "start_pos": 7213,
      "end_pos": 7648,
      "label": "thmA"
    },
    "ref_dict": {
      "thmA": "\\begin{thmA}\\label{thmA}\nLet $M$ be a type $\\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\\theta \\in \\Aut(M)$ be a state preserving automorphism. Then the following conditions are equivalent:\n\\begin{enumerate}\n\t\\item $\\theta$ has infinite order in $\\mathrm{Out}(M)$;\n\t\\item there exists $u \\in \\mathcal U(M)$ such that $\\Ad(u)\\circ\\theta$ is state preserving and ergodic.\n\\end{enumerate}\n\\end{thmA}",
      "thm-case2": "\\begin{thm}\\label{thm-case2}\nLet $M$ be a type ${\\rm III_1}$ factor with separable predual, $\\varphi\\in M_\\ast$ a faithful state, and $\\alpha\\colon \\mathbb Z \\actson (M,\\varphi)$ a state preserving outer action such that $\\Gamma_{\\mathrm{mod}} = p\\mathbb Z$ for some $p\\geq 1$.\nThen there exists a unitary $u\\in \\mathcal U(M)$ such that $\\alpha^u$ is an ergodic state preserving action.\n\\end{thm}",
      "Preliminaries": "\\label{Preliminaries}\n\n\tLet $M$ be a von Neumann algebra and $\\varphi\\in M_\\ast$ a faithful state. The \\textit{modular operator, conjugation}, and \\textit{action} are denoted by $\\Delta_\\varphi$, $J_\\",
      "thmC": "\\begin{thmA}\\label{thmC}\nLet $\\alpha \\colon \\Gamma \\actson M$ be a state preserving outer action of a countably infinite discrete group $\\Gamma$ on a type $\\mathrm{III}_1$ factor with separable predual. \nAssume that $\\Gamma$ is amenable and that $M$ has trivial bicentralizer. \nIf $\\Gamma_{\\mathrm{mod}}$ is trivial, then there exists an $\\alpha$-cocycle \n$u \\colon \\Gamma \\to \\mathcal U(M)$ such that the perturbed action \nis state preserving and weakly mixing.\n\\end{thmA}"
    },
    "pre_theorem_intro_text_len": 2080,
    "pre_theorem_intro_text": "Cocycle perturbations of group actions provide a useful method for modifying dynamical\nproperties while preserving the underlying von Neumann algebraic structure.\nIn the case of type $\\mathrm{II}_1$ factors, Marrakchi and Vaes \\cite{MV23} proved that for every\nouter action of a countable amenable group, one can find a 1-cocycle such that the \nperturbed action becomes ergodic, and moreover that such cocycles form a dense $G_\\delta$\nsubset of the cocycle space. \nWe refer to such a cocycle as an \\emph{ergodic cocycle}. \nThis settles the ergodicity problem for cocycle perturbations in the type $\\mathrm{II}_1$ setting, \nand in particular implies that for a single automorphism, the existence of an ergodic cocycle \nis equivalent to the outerness of all its nonzero powers.\n\nThe purpose of this paper is to extend the above framework to the \\emph{state preserving} \nsetting for type $\\mathrm{III}$ factors, where state preserving means that the action \nleaves a faithful normal state invariant.\nWhile many concrete examples of ergodic actions on type $\\mathrm{III}$ factors are known, \nit has remained open whether an outer action admits an ergodic cocycle.\nThe state preserving assumption ensures that the action extends to an isometry on the \nGNS Hilbert space, which underlies the topological arguments used in \\cite{MV23} and \nin the present work. \nWe also note that Marrakchi and Vaes \\cite[Open problem~4.6]{MV23} explicitly asked whether their \nresults could extend to type $\\mathrm{III}$ factors and suggested that the state preserving \ncase might be tractable.\n\nOur main theorem, stated below, gives an affirmative answer for type $\\mathrm{III}_1$ factors under the assumption of \\emph{trivial bicentralizer}, see Subsection~\\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\\mathrm{III}_1$ setting is settled in essentially all cases.",
    "context": "Cocycle perturbations of group actions provide a useful method for modifying dynamical\nproperties while preserving the underlying von Neumann algebraic structure.\nIn the case of type $\\mathrm{II}_1$ factors, Marrakchi and Vaes \\cite{MV23} proved that for every\nouter action of a countable amenable group, one can find a 1-cocycle such that the \nperturbed action becomes ergodic, and moreover that such cocycles form a dense $G_\\delta$\nsubset of the cocycle space. \nWe refer to such a cocycle as an \\emph{ergodic cocycle}. \nThis settles the ergodicity problem for cocycle perturbations in the type $\\mathrm{II}_1$ setting, \nand in particular implies that for a single automorphism, the existence of an ergodic cocycle \nis equivalent to the outerness of all its nonzero powers.\n\nThe purpose of this paper is to extend the above framework to the \\emph{state preserving} \nsetting for type $\\mathrm{III}$ factors, where state preserving means that the action \nleaves a faithful normal state invariant.\nWhile many concrete examples of ergodic actions on type $\\mathrm{III}$ factors are known, \nit has remained open whether an outer action admits an ergodic cocycle.\nThe state preserving assumption ensures that the action extends to an isometry on the \nGNS Hilbert space, which underlies the topological arguments used in \\cite{MV23} and \nin the present work. \nWe also note that Marrakchi and Vaes \\cite[Open problem~4.6]{MV23} explicitly asked whether their \nresults could extend to type $\\mathrm{III}$ factors and suggested that the state preserving \ncase might be tractable.\n\nOur main theorem, stated below, gives an affirmative answer for type $\\mathrm{III}_1$ factors under the assumption of \\emph{trivial bicentralizer}, see Subsection~\\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\\mathrm{III}_1$ setting is settled in essentially all cases.\n\n\\label{Preliminaries}\n\n\tLet $M$ be a von Neumann algebra and $\\varphi\\in M_\\ast$ a faithful state. The \\textit{modular operator, conjugation}, and \\textit{action} are denoted by $\\Delta_\\varphi$, $J_\\",
    "full_context": "Cocycle perturbations of group actions provide a useful method for modifying dynamical\nproperties while preserving the underlying von Neumann algebraic structure.\nIn the case of type $\\mathrm{II}_1$ factors, Marrakchi and Vaes \\cite{MV23} proved that for every\nouter action of a countable amenable group, one can find a 1-cocycle such that the \nperturbed action becomes ergodic, and moreover that such cocycles form a dense $G_\\delta$\nsubset of the cocycle space. \nWe refer to such a cocycle as an \\emph{ergodic cocycle}. \nThis settles the ergodicity problem for cocycle perturbations in the type $\\mathrm{II}_1$ setting, \nand in particular implies that for a single automorphism, the existence of an ergodic cocycle \nis equivalent to the outerness of all its nonzero powers.\n\nThe purpose of this paper is to extend the above framework to the \\emph{state preserving} \nsetting for type $\\mathrm{III}$ factors, where state preserving means that the action \nleaves a faithful normal state invariant.\nWhile many concrete examples of ergodic actions on type $\\mathrm{III}$ factors are known, \nit has remained open whether an outer action admits an ergodic cocycle.\nThe state preserving assumption ensures that the action extends to an isometry on the \nGNS Hilbert space, which underlies the topological arguments used in \\cite{MV23} and \nin the present work. \nWe also note that Marrakchi and Vaes \\cite[Open problem~4.6]{MV23} explicitly asked whether their \nresults could extend to type $\\mathrm{III}$ factors and suggested that the state preserving \ncase might be tractable.\n\nOur main theorem, stated below, gives an affirmative answer for type $\\mathrm{III}_1$ factors under the assumption of \\emph{trivial bicentralizer}, see Subsection~\\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\\mathrm{III}_1$ setting is settled in essentially all cases.\n\n\\label{Preliminaries}\n\n\tLet $M$ be a von Neumann algebra and $\\varphi\\in M_\\ast$ a faithful state. The \\textit{modular operator, conjugation}, and \\textit{action} are denoted by $\\Delta_\\varphi$, $J_\\\n\nOur main theorem, stated below, gives an affirmative answer for type $\\mathrm{III}_1$ factors under the assumption of \\emph{trivial bicentralizer}, see Subsection~\\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\\mathrm{III}_1$ setting is settled in essentially all cases.\n\nWhen $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\\mathrm{III}_1$ factor \\cite{KST89}. \nThis yields the following corollary.\n\n\\begin{thmA}\\label{thmC}\nLet $\\alpha \\colon \\Gamma \\actson M$ be a state preserving outer action of a countably infinite discrete group $\\Gamma$ on a type $\\mathrm{III}_1$ factor with separable predual. \nAssume that $\\Gamma$ is amenable and that $M$ has trivial bicentralizer. \nIf $\\Gamma_{\\mathrm{mod}}$ is trivial, then there exists an $\\alpha$-cocycle \n$u \\colon \\Gamma \\to \\mathcal U(M)$ such that the perturbed action \nis state preserving and weakly mixing.\n\\end{thmA}\n\n\\begin{lem}\\label{lem-relative-bicentralizer}\n Let $\\Gamma \\actson M$ be a state preserving outer action of a countable group on a von Neumann algebra. Assume that:\n\\begin{itemize}\n \\item $\\Gamma_{\\mathrm{mod}}$ is trivial;\n \\item $M$ is a type $\\mathrm{III}_1$ factor with separable predual and has trivial bicentralizer.\n\\end{itemize}\nThen we have $\\mathrm{BC}(M \\subset M\\rtimes_\\alpha \\Gamma, \\varphi)=\\mathbb C$ for all faithful state $\\varphi\\in M_\\ast$.\n\\end{lem}\n\n\\begin{lem}\\label{lem-cocycle3.6}\n Let $\\alpha\\colon \\Gamma \\actson (M,\\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual and with a faithful normal state. \nAssume that $\\Gamma$ is amenable and that $\\mathrm{BC}(M\\subset M\\rtimes \\Gamma,\\varphi)=\\C$. \nThen for every $\\varepsilon>0$ and every finite subset $F\\subset \\Gamma$, there exist $h\\in \\Gamma$ and $U\\in \\mathcal U(M^\\omega)$ such that:\n\\begin{itemize}\n \\item $\\varphi^\\omega(U)=0$, \\quad $\\|U\\varphi^\\omega-\\varphi^\\omega U\\|<\\varepsilon$,\\quad and \\quad $\\|\\alpha_g^\\omega(U)-U\\|_{\\varphi^\\omega}<\\varepsilon$ for all $g\\in F$;\n \\item $M$, $U$, and $\\alpha_h^\\omega(U)$ are $\\ast$-free with respect to $\\varphi^\\omega$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{lem}\\label{lem-cocycle3.5}\nLet $\\alpha\\colon \\Gamma \\actson (M,\\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual. Assume that $\\Gamma$ is amenable and that $\\mathrm{BC}(M\\subset M\\rtimes \\Gamma,\\varphi)=\\C$. For any $x\\in M\\setminus\\C$, any finite subset $E\\subset M\\setminus\\C$, and any $\\varepsilon>0$, the pair $(1,\\varphi)\\in\\mathcal C_{\\mathrm{state}}(\\alpha)$ can be approximated by elements of $\\mathcal U(x)\\cap \\mathcal V( E,\\varepsilon)$, where $1\\in\\mathcal C(\\alpha)$ is the trivial cocycle.\n\\end{lem}\n\\begin{proof}\n Fix $\\varepsilon>0$, $x\\in M\\setminus\\C$, finite subsets $F\\subset\\Gamma$ and $E\\subset M$. We will show that for every sufficiently small $\\delta>0$, there exists $u\\in\\mathcal U(M)$ satisfying the following conditions:\n\\begin{itemize}\n \\item[(a)] $\\|u-\\alpha_g(u)\\|_\\varphi<\\delta$ for all $g\\in F$, and $\\|u\\varphi-\\varphi u\\|<\\delta$;\n \\item[(b)] there exists $h\\in\\Gamma$ such that \n \\[\n \\|\\alpha_h(uxu^*)-uxu^*\\|_\\varphi>\\|uxu^*-\\varphi(uxu^*)\\|_\\varphi;\n \\]\n \\item[(c)] there exists $h\\in\\Gamma$ such that \n \\[\n \\sum_{a,b\\in E}\\bigl|\n \\varphi(\\alpha_h(uau^*)ubu^*)-\\varphi(uau^*)\\varphi(ubu^*)\n \\bigr|<\\varepsilon.\n \\]\n\\end{itemize}\nIf such a unitary $u$ exists, define an $\\alpha$-cocycle $v_g:=u^*\\alpha_g(u)$ for $g\\in\\Gamma$ and set $\\psi:=u^*\\varphi u$. Then $ \\alpha^v$ preserves $\\psi$, and by (a) the distance between $(1,\\varphi)$ and $(v,\\psi)$ is small. Moreover, (b) and (c) ensures that $(v,\\psi)\\in\\mathcal U(x) \\cap \\mathcal V(E,\\varepsilon)$. Thus to prove the lemma, it suffices to construct such a unitary $u$.\n\n\\begin{thm}\\label{thm-case1}\nLet $\\alpha \\colon \\Gamma \\actson M$ be a state preserving outer action satisfying the following assumptions:\n\\begin{itemize}\n \\item $\\Gamma$ is amenable and countably infinite;\n \\item $\\Gamma_{\\mathrm{mod}} = \\{e\\}$;\n \\item $M$ is a type ${\\rm III_1}$ factor with separable predual and has trivial bicentralizer.\n\\end{itemize}\nThen both $\\mathcal C_{\\mathrm{erg}}(\\alpha)$ and $\\mathcal C_{\\mathrm{wm}}(\\alpha)$ are dense $G_\\delta$ subsets of $\\mathcal C_{\\mathrm{state}}(\\alpha)$.\n\\end{thm}\n\n\\begin{cor}\nLet $M$ be a type $\\mathrm{III}_\\lambda$ factor $(0<\\lambda<1)$ with separable predual, and let \n$\\varphi\\in M_\\ast$ be a faithful state such that $\\sigma^\\varphi$ has period \n$T$, where $T=-2\\pi/\\log(\\lambda)$. \nLet $\\alpha \\colon \\Gamma \\actson (M,\\varphi)$ be a state preserving outer action \nsatisfying the following assumptions:\n\\begin{itemize}\n \\item $\\Gamma$ is amenable and countably infinite;\n \\item $\\Gamma_{\\mathrm{mod}} = \\{e\\}$.\n\\end{itemize}\nThen both $\\mathcal C_{\\mathrm{erg}}(\\alpha,\\varphi)$ and \n$\\mathcal C_{\\mathrm{wm}}(\\alpha,\\varphi)$ are dense $G_\\delta$ subsets \nof $\\mathcal C(\\alpha,\\varphi)$.\n\\end{cor}\n\n\\label{Preliminaries}\n\n\tLet $M$ be a von Neumann algebra and $\\varphi\\in M_\\ast$ a faithful state. The \\textit{modular operator, conjugation}, and \\textit{action} are denoted by $\\Delta_\\varphi$, $J_\\",
    "post_theorem_intro_text_len": 4505,
    "post_theorem_intro_text": "When $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\\mathrm{III}_1$ factor \\cite{KST89}. \nThis yields the following corollary.\n\n\\begin{corA}\\label{corB}\nEvery automorphism of the amenable type $\\mathrm{III}_1$ factor $R_\\infty$ that has infinite order in $\\mathrm{Out}(R_\\infty)$ admits an ergodic cocycle.\n\\end{corA}\n\nFor a von Neumann algebra $M$ with a faithful state $\\varphi \\in M_\\ast$, define\n\\[\n\\mathrm{Mod}(M)\n:= \\{\\, \\operatorname{Ad}(u)\\circ\\sigma_t^{\\varphi} \\in \\operatorname{Aut}(M) \\mid u \\in \\mathcal U(M),\\ t \\in \\mathbb R \\,\\}.\n\\]\nThis is a normal subgroup of $\\operatorname{Aut}(M)$ containing $\\mathrm{Int}(M)$. \nFor a discrete group action $\\alpha \\colon \\Gamma \\to \\operatorname{Aut}(M)$, we define the normal subgroup\n\\[\n\\Gamma_{\\mathrm{mod}} := \\alpha^{-1}(\\mathrm{Mod}(M)) \\le \\Gamma.\n\\]\nFor outer actions on type $\\mathrm{II}$ factors, one always has $\\Gamma_{\\mathrm{mod}}=\\{e\\}$. \nOne of the major differences in the type $\\mathrm{III}$ setting is that $\\Gamma_{\\mathrm{mod}}$\nmay be nontrivial.\nThe proof of Theorem~\\ref{thmA} will be divided according to whether \n$\\Gamma_{\\mathrm{mod}}$ is trivial or not.\n\n\\bigskip\n\\noindent\n{\\bf Case 1: $\\Gamma_{\\mathrm{mod}}$ is trivial.}\nIn the case where $\\Gamma_{\\mathrm{mod}}$ is trivial, we prove the following theorem, \nwhich is a direct generalization of \\cite{MV23} to the type III setting.\n\n\\begin{thmA}\\label{thmC}\nLet $\\alpha \\colon \\Gamma \\curvearrowright M$ be a state preserving outer action of a countably infinite discrete group $\\Gamma$ on a type $\\mathrm{III}_1$ factor with separable predual. \nAssume that $\\Gamma$ is amenable and that $M$ has trivial bicentralizer. \nIf $\\Gamma_{\\mathrm{mod}}$ is trivial, then there exists an $\\alpha$-cocycle \n$u \\colon \\Gamma \\to \\mathcal U(M)$ such that the perturbed action \nis state preserving and weakly mixing.\n\\end{thmA}\n\nUp to stabilizing by another type $\\mathrm{III}_1$ factor, we obtain the following consequence.\n\n\\begin{corA}\\label{corD}\nLet $\\alpha \\colon \\Gamma \\curvearrowright M$ be a state preserving outer action of a countably infinite discrete amenable group on a type $\\mathrm{III}_1$ factor with separable predual. Let $N$ be another type $\\mathrm{III}_1$ factor with separable predual. Then the action $\\alpha \\,\\overline\\otimes\\, \\id_N \\colon \\Gamma \\curvearrowright M \\,\\overline\\otimes\\, N$ admits a cocycle perturbation that is state preserving and weakly mixing.\n\\end{corA}\n\nThe original proof of \\cite{MV23} in the case of $\\mathrm{II}_1$ factors relies heavily on Popa's free independence result \\cite{Po95} (and its variant in \\cite{PSV18}). This free independence theorem has been generalized to the setting of type III factors under appropriate hypotheses in \\cite{HI14}. Therefore, if a state preserving action $\\alpha\\colon \\Gamma \\curvearrowright (M,\\varphi)$ in Theorem~\\ref{thmC} further satisfies $M_\\varphi' \\cap M=\\mathbb C$, then by making use of \\cite{HI14}, the proof becomes a rather direct adaptation of the argument in \\cite{MV23}.\n\nHowever, the theorem also covers the case where $M_\\varphi=\\mathbb C$, in which \\cite{HI14} cannot be applied directly. The key point of our proof is the Connes--St\\o rmer transitivity theorem for type $\\mathrm{III}_1$ factors, which ensures that all faithful states become unitarily conjugate at the level of ultrapowers. As a consequence, $(M^\\omega)_{\\varphi^\\omega}$ is sufficiently large for every $\\varphi$, and this fact is the key technical ingredient of our proof.\n\n\\bigskip\n\\noindent\n{\\bf Case 2: $\\Gamma_{\\mathrm{mod}}$ is nontrivial.}\nWhen $\\Gamma_{\\mathrm{mod}}$ is nontrivial, it is not clear how to deal with this situation in full generality. We therefore restrict ourselves to the case $\\Gamma=\\mathbb Z$, that is, to single automorphisms. In this case $\\Gamma_{\\mathrm{mod}}$ has the form $p\\mathbb Z \\leq \\mathbb Z$ for some $p>0$, and the action restricted to $p\\mathbb Z$ agrees with a modular action. By \\cite{MV23}, one obtains an ergodic cocycle for $p\\mathbb Z$. We then analyze how such a cocycle can be lifted appropriately to the entire group $\\mathbb Z$. This step does not appear in \\cite{MV23} and requires a new argument, which will be carried out in Theorem~\\ref{thm-case2}.\n\\bigskip\n\nIn the final section, we collect several related results, including the case of type $\\mathrm{III}_\\lambda$ factors and additional properties of actions and cocycles on ultrapowers.\n\n\\tableofcontents",
    "sketch": "The proof of Theorem~\\ref{thmA} is split depending on whether \\(\\Gamma_{\\mathrm{mod}}:=\\alpha^{-1}(\\mathrm{Mod}(M))\\) is trivial or not.\n\n\\noindent\\textbf{Case 1: \\(\\Gamma_{\\mathrm{mod}}\\) is trivial.} One proves Theorem~\\ref{thmC}, described as “a direct generalization of \\cite{MV23} to the type III setting,” producing an \\(\\alpha\\)-cocycle \\(u\\colon\\Gamma\\to\\mathcal U(M)\\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \\(M_\\varphi'\\cap M=\\mathbb C\\), then using the type III free independence result \\cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \\cite{MV23}.” In the remaining situation “where \\(M_\\varphi=\\mathbb C\\), in which \\cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\\o rmer transitivity theorem for type \\(\\mathrm{III}_1\\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\\((M^\\omega)_{\\varphi^\\omega}\\) is sufficiently large for every \\(\\varphi\\), and this fact is the key technical ingredient of our proof.”\n\n\\noindent\\textbf{Case 2: \\(\\Gamma_{\\mathrm{mod}}\\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \\(\\Gamma=\\mathbb Z\\) (single automorphisms). Then \\(\\Gamma_{\\mathrm{mod}}=p\\mathbb Z\\) for some \\(p>0\\) and the restriction to \\(p\\mathbb Z\\) “agrees with a modular action.” Using \\cite{MV23}, one “obtains an ergodic cocycle for \\(p\\mathbb Z\\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \\(\\mathbb Z\\).” This lifting step “does not appear in \\cite{MV23} and requires a new argument,” to be carried out later (Theorem~\\ref{thm-case2}).",
    "expanded_sketch": "The proof of the main theorem is split depending on whether \\(\\Gamma_{\\mathrm{mod}}:=\\alpha^{-1}(\\mathrm{Mod}(M))\\) is trivial or not.\n\n\\noindent\\textbf{Case 1: \\(\\Gamma_{\\mathrm{mod}}\\) is trivial.} We first prove the following theorem.\n\\begin{thmA}\\label{thmC}\nLet $\\alpha \\colon \\Gamma \\actson M$ be a state preserving outer action of a countably infinite discrete group $\\Gamma$ on a type $\\mathrm{III}_1$ factor with separable predual. \nAssume that $\\Gamma$ is amenable and that $M$ has trivial bicentralizer. \nIf $\\Gamma_{\\mathrm{mod}}$ is trivial, then there exists an $\\alpha$-cocycle \n$u \\colon \\Gamma \\to \\mathcal U(M)$ such that the perturbed action \nis state preserving and weakly mixing.\n\\end{thmA}\nThis theorem is described as “a direct generalization of \\cite{MV23} to the type III setting,” producing an \\(\\alpha\\)-cocycle \\(u\\colon\\Gamma\\to\\mathcal U(M)\\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \\(M_\\varphi'\\cap M=\\mathbb C\\), then using the type III free independence result \\cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \\cite{MV23}.” In the remaining situation “where \\(M_\\varphi=\\mathbb C\\), in which \\cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\\o rmer transitivity theorem for type \\(\\mathrm{III}_1\\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\\((M^\\omega)_{\\varphi^\\omega}\\) is sufficiently large for every \\(\\varphi\\), and this fact is the key technical ingredient of our proof.”\n\n\\noindent\\textbf{Case 2: \\(\\Gamma_{\\mathrm{mod}}\\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \\(\\Gamma=\\mathbb Z\\) (single automorphisms). Then \\(\\Gamma_{\\mathrm{mod}}=p\\mathbb Z\\) for some \\(p>0\\) and the restriction to \\(p\\mathbb Z\\) “agrees with a modular action.” Using \\cite{MV23}, one “obtains an ergodic cocycle for \\(p\\mathbb Z\\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \\(\\mathbb Z\\).” This lifting step “does not appear in \\cite{MV23} and requires a new argument,” to be carried out later. We next prove the following theorem.\n\\begin{thm}\\label{thm-case2}\nLet $M$ be a type ${\\rm III_1}$ factor with separable predual, $\\varphi\\in M_\\ast$ a faithful state, and $\\alpha\\colon \\mathbb Z \\actson (M,\\varphi)$ a state preserving outer action such that $\\Gamma_{\\mathrm{mod}} = p\\mathbb Z$ for some $p\\geq 1$.\nThen there exists a unitary $u\\in \\mathcal U(M)$ such that $\\alpha^u$ is an ergodic state preserving action.\n\\end{thm}",
    "expanded_theorem": "\\label{thmA}\nLet $M$ be a type $\\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\\theta \\in \\operatorname{Aut}(M)$ be a state preserving automorphism. Then the following conditions are equivalent:\n\\begin{enumerate}\n\t\\item $\\theta$ has infinite order in $\\mathrm{Out}(M)$;\n\t\\item there exists $u \\in \\mathcal U(M)$ such that $\\operatorname{Ad}(u)\\circ\\theta$ is state preserving and ergodic.\n\\end{enumerate}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(M\\) be a type \\(\\mathrm{III}_1\\) factor with separable predual and trivial bicentralizer, and let \\(\\theta\\in\\operatorname{Aut}(M)\\) be state preserving, meaning that \\(\\theta\\) leaves some faithful normal state on \\(M\\) invariant. Write \\(\\mathrm{Out}(M)=\\operatorname{Aut}(M)/\\operatorname{Inn}(M)\\), so that saying \\(\\theta\\) has infinite order in \\(\\mathrm{Out}(M)\\) means that \\([\\theta]^n\\neq e\\) for every nonzero integer \\(n\\). Also, \\(\\mathcal U(M)\\) denotes the unitary group of \\(M\\), \\(\\operatorname{Ad}(u)(x)=uxu^*\\), and an automorphism \\(\\beta\\) of \\(M\\) is ergodic if its fixed-point algebra \\(\\{x\\in M:\\beta(x)=x\\}\\) is \\(\\mathbb C1\\). Which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "The following are equivalent: (i) \\(\\theta\\) has infinite order in \\(\\mathrm{Out}(M)\\); (ii) there exists \\(u\\in\\mathcal U(M)\\) such that \\(\\operatorname{Ad}(u)\\circ\\theta\\) is state preserving and ergodic."
      },
      "choices": [
        {
          "label": "B",
          "text": "The following are equivalent: (i) \\(\\theta\\) has finite order in \\(\\mathrm{Out}(M)\\); (ii) there exists \\(u\\in\\mathcal U(M)\\) such that \\(\\operatorname{Ad}(u)\\circ\\theta\\) is state preserving and ergodic."
        },
        {
          "label": "C",
          "text": "If there exists \\(u\\in\\mathcal U(M)\\) such that \\(\\operatorname{Ad}(u)\\circ\\theta\\) is state preserving and ergodic, then \\(\\theta\\) has infinite order in \\(\\mathrm{Out}(M)\\)."
        },
        {
          "label": "D",
          "text": "The following are equivalent: (i) \\(\\theta\\) has infinite order in \\(\\mathrm{Out}(M)\\); (ii) there exists \\(u\\in\\mathcal U(M)\\) such that \\(\\operatorname{Ad}(u)\\circ\\theta\\) is ergodic."
        },
        {
          "label": "E",
          "text": "The following are equivalent: (i) every nonzero power \\(\\theta^n\\) is outer; (ii) for every faithful normal state \\(\\psi\\) on \\(M\\), there exists \\(u\\in\\mathcal U(M)\\) such that \\(\\operatorname{Ad}(u)\\circ\\theta\\) preserves \\(\\psi\\) and is ergodic."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "infinite-order-outerness condition",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped the converse implication from infinite order to existence of an ergodic state-preserving perturbation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "state-preserving requirement in the perturbation conclusion",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "existential state in the hypothesis replaced by uniform preservation for every faithful normal state",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or paraphrase it closely enough to make the answer trivial. It asks for an equivalent characterization, but the key property in the correct choice (existence of a unitary perturbation that is ergodic) is not given away in the question itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is very close to a direct theorem-recall item: it asks which statement is equivalent to a given property, and the correct answer is essentially the target equivalence itself. The task is mainly to identify the exact formulation rather than derive a new conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare nearby variants: weaker existence without ergodicity, stronger conditions like all powers ergodic, and quantifier changes over states. However, the item still primarily tests recognition of the precise equivalence rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful. They differ by common failure modes: dropping ergodicity, strengthening to weak mixing or all powers ergodic, and changing an existential statement to a universal one over preserved states."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall/comparison MCQ with strong distractors, but it is largely a direct restatement of a theorem and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.13085v1",
    "paper_link": "http://arxiv.org/abs/2512.13085v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of \\eqref{eq:a certain product} is additive.",
      "start_pos": 18160,
      "end_pos": 19153,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:a certain product": "\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}",
      "thm:main": "\\begin{theorem}\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of \\eqref{eq:a certain product} is additive.\n\t\\end{theorem}",
      "eq:normJprodukt": "\\begin{equation}\\label{eq:normJprodukt}\n\t\ta\\circ b :=\\frac12(ab+ba),\n\t\\end{equation}",
      "eq:the product": "\\begin{equation}\\label{eq:the product}\n\t\ta^{2} \\circ b = \\frac12(a^{2} b + b a^{2})\n\t\\end{equation}",
      "eq:glavnieq": "\\begin{equation}\\label{eq:glavnieq}\n\t\t\\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n\t\\end{equation}",
      "rem:k-potents not diagonalizable": "\\begin{remark}\\label{rem:k-potents not diagonalizable}\n\t\tSuppose that $n \\ge 2$. All matrices in $\\pot_k(M_n)$ are diagonalizable if and only if $\\chr(\\F)$ does not divide $k-1$. Indeed, suppose that $\\chr(\\F)$ does not divide $k-1$, and let $P \\in \\pot_k(M_n)$ be arbitrary. We have\n\t\t$$P^{k} = P \\, \\implies \\, P(P^{k-1}-I) = 0,$$\n\t\twhich implies that the minimal polynomial $m_P \\in \\F[x]$ of $P$ necessarily divides \n\t\t$$q(x):=x(x^{k-1}-1).$$ \n\t\tSince $\\F$ is algebraically closed, to show that $P$ is diagonalizable in $M_n$, it suffices to show that $q$ (and consequently $m_P$) is square-free, meaning that it decomposes into a product of distinct linear factors. We have\n\t\t$q'(x) = kx^{k-1}-1$. Assuming that $x_0 \\in \\F$ satisfies $q(x_0) = q'(x_0) = 0$ would lead to $k-1=0$, which is a contradiction.\n\n\t\tConversely, suppose that $\\chr(\\F) \\mid (k-1)$. For $n=2$, the matrix \n\t\t$$\n\t\tA := \\begin{bmatrix}\n\t\t\t1  & 1 \\\\ 0 & 1\n\t\t\\end{bmatrix} \\in M_2\n\t\t$$ \n\t\tsatisfies $$A^{k-1} = \\begin{bmatrix}\n\t\t\t1  & k-1 \\\\ 0 & 1\n\t\t\\end{bmatrix} = I \\implies A^k = A$$\n\t\tso $A$ is a $k$-potent in $M_2$. On the other hand, the minimal polynomial of $A$ is $m_A(x) = (x-1)^2 \\in \\F[x]$, which is not square-free in $\\F[x]$. Therefore, $A$ is not diagonalizable in $M_2$. When $n > 2$, a similar conclusion follows by considering the matrix $\\diag(A,I_{n-2}) \\in M_n$.\n\n\t\tAdditionally, the matrix $A$ demonstrates other interesting properties of the order $\\preceq$. It is not difficult to show directly that any matrix $B \\in M_2$ satisfies $AB=BA=B^2$ if and only if $B \\in \\{0,A\\}$. In particular, $A$ is a minimal element of the poset $(\\pot_k(M_2)\\setminus\\{0\\}, \\preceq)$, even though its rank is $2$. It is also a consequence that $A$ \\emph{cannot} be written in the form $A = P+Q$ for some nonzero mutually orthogonal $k$-potents $P,Q \\in \\pot_k(M_2)$. On the other hand, by Remark \\ref{re:maximal element}, $A$ is also a maximal element of the poset $(\\pot_k(M_2), \\preceq)$.\n\t\\end{remark}"
    },
    "pre_theorem_intro_text_len": 10794,
    "pre_theorem_intro_text": "The study of multiplicative maps between rings and algebras has a long and substantial history, centered on the question of when multiplicativity alone enforces stronger algebraic behavior, most notably additivity. Foundational contributions were made by Rickart~\\cite{Rickart} in 1948 and Johnson~\\cite{Johnson} in 1958, who established positive results under suitable structural assumptions and highlighted the connection with the rigidity of the additive structure of rings. A major advance was achieved by Martindale~\\cite{Martindale} in 1969, who proved that every bijective multiplicative map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is automatically additive. In the same year, Jodeit and Lam~\\cite{JodeitLam} classified all nondegenerate multiplicative self-maps of the matrix rings $M_n(\\mathcal{R})$ over a principal ideal domain~$\\mathcal{R}$, where nondegeneracy means that the map is not identically zero on the set of matrices with zero determinant. Specifically, for each such map $\\phi: M_n(\\ca{R}) \\to M_n(\\ca{R})$, either there exists a nonzero idempotent matrix $P \\in M_n(\\ca{R})$ such that $\\phi - P$ is multiplicative and degenerate, or there exists an invertible matrix $T \\in M_n(\\ca{R})$ and a ring endomorphism $\\omega$ of $\\ca{R}$ such that $\\phi$ takes one of the following two distinct forms:\n\t$$\n\t\\phi(X) = T \\, \\omega(X) \\, T^{-1} \\quad \\text{ or } \\quad \\phi(X) = T \\, \\omega(X)^* \\, T^{-1},\n\t$$\n\twhere $\\omega(X)$ denotes the matrix obtained by applying $\\omega$ entrywise, and $(\\cdot)^*$ is the corresponding cofactor matrix. In particular, all bijective multiplicative self-maps on $M_n(\\mathcal{R})$ are automatically additive and hence ring automorphisms. Taken together, these results expose a fundamental phenomenon: multiplicativity, when coupled with bijectivity, interacts strongly with the ambient ring structure, leaving little room for pathological nonadditive behavior. This rigidity principle later emerged as a central theme in preserver theory, especially in the context of matrix and operator algebras.\n\n\tThis viewpoint was later extended from ordinary ring homomorphisms to \\emph{Jordan homomorphisms}, which originate as homomorphisms in the category of \\emph{Jordan algebras}, introduced by Jordan, von Neumann, and Wigner \\cite{Jordan} in the 1930s as algebraic models for quantum observables.\n\tA \\emph{Jordan algebra} is a nonassociative algebra $\\ca{A}$ over a field equipped with a commutative multiplication $\\diamond$ satisfying the \\emph{Jordan identity}\n\t$$\n\ta^2 \\diamond (a \\diamond b) = a \\diamond (a^2 \\diamond b), \\quad \\text{ for all } a,b \\in \\ca{A}.\n\t$$\n\tJordan rings are similarly defined. Every associative algebra (or ring) $\\mathcal{A}$ naturally becomes a Jordan algebra (ring) when equipped with the \\emph{Jordan product}\n\t$$\n\ta \\diamond b := ab + ba.\n\t$$\n\tA Jordan algebra that is isomorphic to a Jordan subalgebra of an associative algebra is called a \\emph{special Jordan algebra} (see e.g. \\cite[Section~2.3]{HancheStormer}). Not all Jordan algebras are special; the Albert algebra~\\cite{Albert} provides the classical example of an \\emph{exceptional Jordan algebra}.\n\n\tIn an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n\t$$\n\t\\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n\t$$\n\tIf the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n\t$$\n\t\\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n\t$$\n\tMoreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n\t\\begin{equation}\\label{eq:normJprodukt}\n\t\ta\\circ b :=\\frac12(ab+ba),\n\t\\end{equation}\n\tand Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\n\tThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result  asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n\t\\begin{equation}\\label{eq:glavnieq}\n\t\t\\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n\t\\end{equation}\n\twhere $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\n\tIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all  $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}.   This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n\t$$\n\t\\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n\t$$\n\ttransforms $\\phi$ into a $\\circ$-preserving map $\\psi$.  This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n\twith the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\tA substantial body of literature also investigates multiplicative maps with respect to products other than the Jordan product, particularly various symmetrized or triple-product variants such as $(a,b) \\mapsto aba$, $(a,b,c) \\mapsto abc + cba$, and many others (see, e.g., \\cite{AnHou, Kuzma, LesnjakSze, Lu2, Molnar2} and references therein). More recently, preservers of the product\n\t\\begin{equation}\\label{eq:the product}\n\t\ta^{2} \\circ b = \\frac12(a^{2} b + b a^{2})\n\t\\end{equation}\n\twere studied by Ghorbanipour and Hejazian~\\cite{GhorbanipourHejazian}, who showed that any bijection $\\phi : \\mathcal{A} \\to \\mathcal{A}$ preserving \\eqref{eq:the product}, where $\\mathcal{A}$ is a real standard Jordan operator algebra acting on a Hilbert space of dimension at least $2$, is of the form $\\phi = \\pm \\psi$, where $\\psi$ is a Jordan automorphism of $\\mathcal{A}$.\n\n\tThe product $(a,b)\\mapsto a^2 \\circ b$ plays an important role in the theory of JB-algebras. Recall that a \\emph{JB-algebra} is a (typically real) Jordan algebra $(\\mathcal{A},\\circ)$ equipped with a complete submultiplicative norm $\\|\\cdot\\|$  (that is, a Jordan Banach algebra) which additionally satisfies\n\t$$\n\t\\|a\\|^2 \\le \\|a^2 + b^2\\|, \\quad \\text{ for all }  a,b \\in \\mathcal{A}.\n\t$$\n\tA central feature of JB-algebras is that many aspects of their structure, such as order, spectral theory, and the theory of ideals, are encoded in the associated quadratic operators (see, for example, \\cite{Battaglia}). In particular, the fundamental quadratic mapping $U_a : \\mathcal{A} \\to \\mathcal{A}$,\n\t$$\n\tU_{a}(b) := 2(a \\circ b) \\circ a - a^{2} \\circ b,\n\t$$\n\tplays a key role in the analysis of orthogonality, annihilators, and quadratic ideals. Moreover, the product $(a,b)\\mapsto a^2 \\circ b$ was used in \\cite{Battaglia} to define orthogonality in JB-algebras: elements $a,b \\in \\mathcal{A}$ are said to be orthogonal if $a^{2} \\circ b = 0$. In the setting of special JB-algebras, such as the self-adjoint part of a $C^{*}$-algebra, the Jordan product is given by \\eqref{eq:normJprodukt}. In this case, the expression $a^{2} \\circ b$ agrees exactly with the symmetrized product appearing in \\eqref{eq:the product}.\n\n\t\\smallskip\n\n\tThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n\t$$\n\ta^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n\t$$\n\tfor arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.",
    "context": "In an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n $$\n \\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n $$\n If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n $$\n \\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n $$\n Moreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n \\begin{equation}\\label{eq:normJprodukt}\n a\\circ b :=\\frac12(ab+ba),\n \\end{equation}\n and Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\nThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n \\begin{equation}\\label{eq:glavnieq}\n \\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n \\end{equation}\n where $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\nIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n $$\n \\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n $$\n transforms $\\phi$ into a $\\circ$-preserving map $\\psi$. This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\\smallskip\n\nThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n $$\n a^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n $$\n for arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.",
    "full_context": "In an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n $$\n \\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n $$\n If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n $$\n \\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n $$\n Moreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n \\begin{equation}\\label{eq:normJprodukt}\n a\\circ b :=\\frac12(ab+ba),\n \\end{equation}\n and Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\nThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n \\begin{equation}\\label{eq:glavnieq}\n \\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n \\end{equation}\n where $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\nIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n $$\n \\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n $$\n transforms $\\phi$ into a $\\circ$-preserving map $\\psi$. This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\\smallskip\n\nThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n $$\n a^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n $$\n for arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.\n\n\\smallskip\n\n\\section{Notation and Preliminaries}\\label{sec:prel}\n We begin by introducing notation that will be used throughout the paper. Let $\\F$ be a fixed algebraically closed field of characteristic different from $2$. We denote by $\\mathbb{F}^\\times$ the multiplicative group of nonzero elements of $\\mathbb{F}$, and by $\\mathbb{F}[x]$ the polynomial algebra in one variable over $\\mathbb{F}$.\n\n\\begin{lemma}\\label{le:h exists}\n Let $\\phi : M_n \\to M_n$ be a nonzero map which satisfies \\eqref{eq:a certain product} and $\\phi(0) = 0$. There exists a unique multiplicative map $\\omega : \\F \\to \\F$ such that \n \\begin{equation}\\label{eq:homogeneity}\n \\phi(\\lambda X) = \\omega(\\lambda) \\phi(X), \\quad \\text{ for all } \\lambda \\in \\F \\text{ and } X \\in M_n.\n \\end{equation}\n Moreover, there exists an invertible matrix $T \\in M_n$ and $\\varepsilon\\in\\Sph^1_k$ such that the map $$\\psi:= T^{-1}\\phi(\\cdot)T$$ satisfies $$\\psi(E_{jj}) = \\varepsilon E_{jj}, \\quad \\text{ for each } j \\in [n].$$\n Further, fix some distinct $i,j \\in [n]$. Then $\\psi(E_{ij})$ is a nonzero scalar multiple of $E_{ij}$ or $E_{ji}$. If $\\psi(E_{ij}) = \\beta_{ij} E_{ij}$ for some $\\beta_{ij} \\in \\F^\\times$, then for each $x,y \\in \\F$ we have\n \\begin{equation}\\label{eq: Eii+Eij 1}\n \\psi(E_{ii} + x E_{ij}) = \\varepsilon E_{ii} + \\omega(x) (\\beta_{ij} E_{ij}), \\qquad \\psi(E_{jj} + y E_{ij}) = \n \\varepsilon E_{jj} + \\omega(y) (\\beta_{ij} E_{ij}).\n \\end{equation}\n If, on the other hand, $\\psi(E_{ij}) = \\gamma_{ij} E_{ji}$ for some $\\gamma_{ij} \\in \\F^\\times$, then\n \\begin{equation}\\label{eq: Eii+Eij 2}\n \\psi(E_{ii} + x E_{ij}) = \\varepsilon E_{ii} + \\omega(x) (\\gamma_{ij} E_{ji}), \\qquad \\psi(E_{jj} + y E_{ij}) = \n \\varepsilon E_{jj} + \\omega(y) (\\gamma_{ij} E_{ji}).\n \\end{equation}\n Finally, if $n \\ge 2$, the map $\\omega : \\F \\to \\F$ is a ring monomorphism. In particular, $\\phi$ is $\\K$-homogeneous.\n \\end{lemma}\n \\begin{proof}\n In view of Lemma \\ref{le:basic properties II} (c) and (e), $\\phi(E_{11}), \\ldots ,\\phi(E_{nn})$ are mutually orthogonal rank-one $(k+1)$-potents and therefore can be simultaneously diagonalized (see e.g.\\ \\cite[Theorem~8 of \\S6.5]{HoffmanKunze}). Hence, by passing to map $T^{-1}\\phi(\\cdot)T$, for a suitable invertible matrix $T \\in M_n$, without loss of generality we can assume that \n \\begin{equation}\\label{eq:phi fiksira dijagonalne}\n \\phi(E_{jj}) = \\varepsilon_j E_{jj}, \\quad \\text{ for some $\\varepsilon_j \\in \\Sph^1_k$ for each } j \\in [n].\n \\end{equation}\n Note that by Lemma \\ref{le:basic properties II} (f), for each diagonal $(k+1)$-potent $D \\in \\ca{D}_n$ we have\n \\begin{equation}\\label{eq:diagonal tripotents}\n \\phi(D) = \n \\sum_{(j,j) \\in \\supp D} \\varepsilon_j E_{jj}.\n \\end{equation}\n By Lemma \\ref{le:zero map}, we have \n \\begin{equation}\\label{eq:lambda times ejj}\n \\phi(\\lambda E_{jj}) \\ne 0, \\quad\\text{ for all } j \\in [n] \\text{ and } \\lambda \\in \\F^\\times.\n \\end{equation} Further, for each $X \\in M_n$ and $S \\subseteq [n]$ we have\n \\begin{equation}\\label{eq:preserves support}\n \\supp X \\subseteq S \\times S \\implies \\supp \\phi(X) \\subseteq S \\times S.\n \\end{equation}\n Indeed, denote the diagonal idempotent \n $P := \\sum_{j \\in [n]\\setminus S} E_{jj}$ and note that a matrix $X\\in M_n$ is supported in $S \\times S$ if and only if $XP=PX=0$. In that case, obviously $X \\circ P^k = 0$, so \n $$0 = \\phi(X \\circ P^k) = \\phi(X) \\circ \\phi(P)^k \\stackrel{\\eqref{eq:diagonal tripotents}}= \\phi(X) \\circ P$$\n and hence Lemma \\ref{le:Jordan product calculations} (a) implies the claim.\n \\smallskip\n\nFix $(i,j), (j,p) \\in [n]^2 \\setminus \\Delta_n$. If $i \\ne p$, then by Lemma \\ref{le:h exists} we have\n \\begin{align*}\n \\frac12 g(i,p)E_{ip} &= \\phi\\left(\\frac12 E_{ip}\\right) =\\phi((E_{ii}+E_{ij})^k \\circ E_{jp}) = \\phi(E_{ii}+E_{ij})^k \\circ \\phi(E_{jp}) \\\\\n &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= (E_{ii}+g(i,j)E_{ij})^k \\circ (g(j,p)E_{jp})\\\\\n &= \\frac12 g(i,j)g(j,p)E_{ip},\n \\end{align*}\n which implies $g(i,p) = g(i,j)g(j,p)$.\n On the other hand, if $i = p$, then by Lemma \\ref{le:h exists} we have\n \\begin{align*}\n \\frac{1}{2}\\phi\\left(E_{ii} + E_{jj} + E_{ji}\\right) &= \\phi\\left(\\frac{1}{2}\\left(E_{ii} + E_{jj} + E_{ji}\\right)\\right) = \\phi((E_{ii}+E_{ij})^k \\circ E_{ji}) \\\\\n &= \\phi(E_{ii}+E_{ij})^k \\circ \\phi(E_{ji}) \\\\\n &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= (E_{ii}+ g(i,j)E_{ij})^k \\circ (g(j,i)E_{ji})\\\\\n &= \\frac12 g(j,i) \\Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\\Big).\n \\end{align*}\n Furthermore, we have\n \\begin{align*}\n E_{ii} +\\frac12g(j,i) E_{ji} \\,\\, &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= \\phi\\left(E_{ii} +\\frac12 E_{ji}\\right) =\\phi\\left((E_{ii} + E_{jj} + E_{ji}) \\circ E_{ii}^k\\right) \\\\\n &= \\phi\\left(E_{ii} + E_{jj} + E_{ji}\\right) \\circ \\phi(E_{ii})^k\\\\\n &= g(j,i) \\Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\\Big) \\circ E_{ii} \\\\\n &= g(j,i)\\left(g(i,j) E_{ii} +\\frac12 E_{ji}\\right)\n \\end{align*}\n We obtain \n $$\n g(i,j)g(j,i) = 1,\n $$\n as desired. Following \\cite{Coelho}, denote by $$g^* : M_n \\to M_n, \\qquad g^*(E_{ij}) := g(i,j)E_{ij}$$\n the induced algebra automorphism of $M_n$. Note that $g^*$ is implemented via conjugation by the invertible diagonal matrix\n $$\n D := \\operatorname{diag}\\big(g(1,1), g(2,1), \\ldots, g(n,1)\\big) \\in \\ca{D}_n,\n $$\n i.e.\\ $g^*(\\cdot)=D(\\cdot)D^{-1}$. Hence, by passing to the map $(g^*)^{-1}(\\phi(\\cdot)) = D^{-1}\\phi(\\cdot)D$, without loss of generality we can assume that \n $$\\phi(E_{ij}) = E_{ij}, \\quad \\text{ for all } (i,j) \\in [n]^2.$$\n In view of Lemma \\ref{le:h exists}, denote by $\\omega :\\F \\to \\F$ the induced ring monomorphism that satisfies \\eqref{eq:homogeneity}. We claim that \n \\begin{equation}\\label{eq:phi = omega}\n \\phi(X) = \\omega(X), \\quad \\text{ for all } X \\in M_n.\n \\end{equation}\n First assume that $X=[x_{ij}] \\in M_n$ is of the form $X = Y^k$ for some $Y \\in M_n$. Fix $(i,j) \\in [n]^2$ and denote $\\phi(X)=[\\phi(X)_{ij}]$. If $i = j$, then by Lemma \\ref{le:preserves triple product} we have \n \\begin{align*}\n \\omega(x_{ii})E_{ii} &= \\phi(x_{ii}E_{ii}) = \\phi(E_{ii}Y^k E_{ii}) = \\phi(E_{ii})\\phi(Y)^k \\phi(E_{ii}) = E_{ii}\\phi(X) E_{ii} \\\\\n &= \\phi(X)_{ii}E_{ii},\n \\end{align*}\n which shows that\n \\begin{equation}\\label{eq:Yii}\n \\phi(X)_{ii} = \\omega(x_{ii}).\n \\end{equation} Now assume $i \\ne j$. Denote $P:= E_{ii}+E_{ji}$ and note that \n $$\n PAP = (a_{ii}+a_{ij})P, \\quad \\text{ for all matrices $A=[a_{ij}] \\in M_n$}.\n $$ \n By \\eqref{eq: Eii+Eij 1} we have $\\phi(P) = P$ and hence\n \\begin{align*}\n \\omega(x_{ii}+x_{ij})P &=\\phi((x_{ii}+x_{ij})P) = \\phi(PXP) \\stackrel{\\text{Lemma }\\ref{le:preserves triple product}}= \\phi(P)\\phi(X)\\phi(P) \\\\\n &= P\\phi(X)P = (\\phi(X)_{ii}+\\phi(X)_{ij})P.\n \\end{align*}\n Therefore, by the additivity of $\\omega$ and \\eqref{eq:Yii} we obtain\n $$\\omega(x_{ii})+\\omega(x_{ij}) = \\phi(X)_{ii}+\\phi(X)_{ij} = \\omega(x_{ii})+\\phi(X)_{ij}.$$\n This implies $\\phi(X)_{ij} = \\omega(x_{ij})$, which shows \\eqref{eq:phi = omega} for all matrices $X \\in M_n(\\F)$ which are a $k$-th power of some matrix. Using the notation of Lemma \\ref{le:iterated R}, this means \n $$\\phi(X) = \\omega(X), \\quad \\text{ for all } X \\in \\ca{R}_0^n.$$\n We now extend this conclusion to all matrices in $M_n$. Note that if $A,B \\in M_n$ satisfy $\\phi(A) = \\omega(A)$ and $\\phi(B) = \\omega(B)$, then clearly\n $$\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B) = \\omega(A)^k \\circ \\omega(B) = \\omega(A^k \\circ B).$$\n Again returning to the notation of Lemma \\ref{le:iterated R}, this inductively implies that $$\\phi(X) = \\omega(X), \\quad \\text{ for all } X\\in \\ca{R}^n.$$\n By the same lemma we have $\\ca{R}^n = M_n$, which completes the proof of the theorem.\n \\end{proof}\n We conclude the paper with the following generalization of \\cite[Corollary~3.5]{GogicTomasevic-AM}, which follows immediately from the first part of the proof of Theorem~\\ref{thm:main}.\n \\begin{corollary} \n Let $m < n$. A map $\\phi : M_n \\to M_m$ satisfies \\eqref{eq:a certain product} if and only if it is constant and equal to a fixed $(k+1)$-potent. \n \\end{corollary}",
    "post_theorem_intro_text_len": 1130,
    "post_theorem_intro_text": "This result extends all previously known Jordan-type additivity results on $M_n(\\F)$, at least for algebraically closed fields with $\\operatorname{char}(\\F) \\neq 2$. The proof of Theorem~\\ref{thm:main}, presented in Section~\\S\\ref{sec:main}, follows the general strategy of \\cite[Theorem~1.1]{GogicTomasevic-AM}. The classification relies on the fact that maps satisfying \\eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements. These notions will be introduced and discussed in the next section. A notable difference from the case $k=1$, which corresponds to idempotents, is that when $\\operatorname{char}(\\F)$ is positive and divides $k$, $(k+1)$-potents may fail to be diagonalizable (see Remark~\\ref{rem:k-potents not diagonalizable}). This phenomenon requires more refined arguments in order to handle the mixed Jordan--power preservers effectively. We also note that linear and closely related variants of $k$-potent preservers on matrix and operator algebras have been studied in papers such as \\cite{HouHou,SongCao,YouWang}.",
    "sketch": "The proof of Theorem~\\ref{thm:main} (given in Section~\\S\\ref{sec:main}) “follows the general strategy of \\cite[Theorem~1.1]{GogicTomasevic-AM}.” The classification “relies on the fact that maps satisfying \\eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\\operatorname{char}(\\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”",
    "expanded_sketch": "The proof of the main theorem (given later) “follows the general strategy of Gogic--Tomasevic, Theorem~1.1 (GogicTomasevic-AM).” The classification “relies on the fact that maps satisfying\n\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\npreserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\\operatorname{char}(\\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”",
    "expanded_theorem": "\t\t\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tis additive.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb F\\) be an algebraically closed field with \\(\\operatorname{char}(\\mathbb F)\\neq 2\\), let \\(n\\ge 2\\) and \\(k\\in\\mathbb N\\), and write \\(A\\circ B:=\\tfrac12(AB+BA)\\) for the Jordan product on \\(M_n(\\mathbb F)\\). Which maps \\(\\phi:M_n(\\mathbb F)\\to M_n(\\mathbb F)\\) satisfy\n\\[\n\\phi(A^k\\circ B)=\\phi(A)^k\\circ \\phi(B)\\qquad\\text{for all }A,B\\in M_n(\\mathbb F)?\n\\]",
      "correct_choice": {
        "label": "A",
        "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\) (that is, \\(P^{k+1}=P\\)); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1},\n\\]\nwhere \\(\\omega(X)\\) means applying \\(\\omega\\) entrywise. In particular, every nonconstant such \\(\\phi\\) is additive."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\) and a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1}.\n\\]\nIn particular, every nonconstant such \\(\\phi\\) is additive."
        },
        {
          "label": "C",
          "text": "Every map \\(\\phi:M_n(\\mathbb F)\\to M_n(\\mathbb F)\\) of one of the following forms satisfies\n\\[\n\\phi(A^k\\circ B)=\\phi(A)^k\\circ\\phi(B)\\qquad\\text{for all }A,B\\in M_n(\\mathbb F):\n\\]\neither \\(\\phi\\) is constant with value a fixed \\((k+1)\\)-potent matrix \\(P\\), or there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1}.\n\\]"
        },
        {
          "label": "D",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed idempotent matrix \\(P\\) (that is, \\(P^2=P\\)); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a scalar \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1},\n\\]\nwhere \\(\\omega(X)\\) is applied entrywise. In particular, every nonconstant such \\(\\phi\\) is additive."
        },
        {
          "label": "E",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a field automorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}.\n\\]\nIn particular, every nonconstant such \\(\\phi\\) is additive."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "k-th-root-of-unity scalar factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "replaced classification iff by one-way sufficient condition",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "constant-value type and unrestricted scalar factor",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "transpose branch and monomorphism generality",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the exact classification in the correct option. It only gives the functional identity and asks for an equivalent characterization."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct option is the classification theorem itself, stated in near-final form. It does not meaningfully reframe the result into a new setting or application."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle structural details (monomorphism vs automorphism, k-th vs (k+1)-th conditions, global vs pointwise scalar factor). However, the task mainly tests precise recall/discrimination rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong potent class, overly strong automorphism assumption, loss of the scalar factor, and incorrect pointwise dependence. They are distinct and nontrivial."
      },
      "total_score": 5,
      "overall_assessment": "A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and has excellent distractors, but it is largely a direct restatement test rather than a genuinely generative reasoning problem."
    }
  },
  {
    "id": "2512.13085v1",
    "paper_link": "http://arxiv.org/abs/2512.13085v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of \\eqref{eq:a certain product} is additive.",
      "start_pos": 18160,
      "end_pos": 19153,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:a certain product": "\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}",
      "thm:main": "\\begin{theorem}\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of \\eqref{eq:a certain product} is additive.\n\t\\end{theorem}",
      "eq:normJprodukt": "\\begin{equation}\\label{eq:normJprodukt}\n\t\ta\\circ b :=\\frac12(ab+ba),\n\t\\end{equation}",
      "eq:the product": "\\begin{equation}\\label{eq:the product}\n\t\ta^{2} \\circ b = \\frac12(a^{2} b + b a^{2})\n\t\\end{equation}",
      "eq:glavnieq": "\\begin{equation}\\label{eq:glavnieq}\n\t\t\\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n\t\\end{equation}",
      "rem:k-potents not diagonalizable": "\\begin{remark}\\label{rem:k-potents not diagonalizable}\n\t\tSuppose that $n \\ge 2$. All matrices in $\\pot_k(M_n)$ are diagonalizable if and only if $\\chr(\\F)$ does not divide $k-1$. Indeed, suppose that $\\chr(\\F)$ does not divide $k-1$, and let $P \\in \\pot_k(M_n)$ be arbitrary. We have\n\t\t$$P^{k} = P \\, \\implies \\, P(P^{k-1}-I) = 0,$$\n\t\twhich implies that the minimal polynomial $m_P \\in \\F[x]$ of $P$ necessarily divides \n\t\t$$q(x):=x(x^{k-1}-1).$$ \n\t\tSince $\\F$ is algebraically closed, to show that $P$ is diagonalizable in $M_n$, it suffices to show that $q$ (and consequently $m_P$) is square-free, meaning that it decomposes into a product of distinct linear factors. We have\n\t\t$q'(x) = kx^{k-1}-1$. Assuming that $x_0 \\in \\F$ satisfies $q(x_0) = q'(x_0) = 0$ would lead to $k-1=0$, which is a contradiction.\n\n\t\tConversely, suppose that $\\chr(\\F) \\mid (k-1)$. For $n=2$, the matrix \n\t\t$$\n\t\tA := \\begin{bmatrix}\n\t\t\t1  & 1 \\\\ 0 & 1\n\t\t\\end{bmatrix} \\in M_2\n\t\t$$ \n\t\tsatisfies $$A^{k-1} = \\begin{bmatrix}\n\t\t\t1  & k-1 \\\\ 0 & 1\n\t\t\\end{bmatrix} = I \\implies A^k = A$$\n\t\tso $A$ is a $k$-potent in $M_2$. On the other hand, the minimal polynomial of $A$ is $m_A(x) = (x-1)^2 \\in \\F[x]$, which is not square-free in $\\F[x]$. Therefore, $A$ is not diagonalizable in $M_2$. When $n > 2$, a similar conclusion follows by considering the matrix $\\diag(A,I_{n-2}) \\in M_n$.\n\n\t\tAdditionally, the matrix $A$ demonstrates other interesting properties of the order $\\preceq$. It is not difficult to show directly that any matrix $B \\in M_2$ satisfies $AB=BA=B^2$ if and only if $B \\in \\{0,A\\}$. In particular, $A$ is a minimal element of the poset $(\\pot_k(M_2)\\setminus\\{0\\}, \\preceq)$, even though its rank is $2$. It is also a consequence that $A$ \\emph{cannot} be written in the form $A = P+Q$ for some nonzero mutually orthogonal $k$-potents $P,Q \\in \\pot_k(M_2)$. On the other hand, by Remark \\ref{re:maximal element}, $A$ is also a maximal element of the poset $(\\pot_k(M_2), \\preceq)$.\n\t\\end{remark}"
    },
    "pre_theorem_intro_text_len": 10794,
    "pre_theorem_intro_text": "The study of multiplicative maps between rings and algebras has a long and substantial history, centered on the question of when multiplicativity alone enforces stronger algebraic behavior, most notably additivity. Foundational contributions were made by Rickart~\\cite{Rickart} in 1948 and Johnson~\\cite{Johnson} in 1958, who established positive results under suitable structural assumptions and highlighted the connection with the rigidity of the additive structure of rings. A major advance was achieved by Martindale~\\cite{Martindale} in 1969, who proved that every bijective multiplicative map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is automatically additive. In the same year, Jodeit and Lam~\\cite{JodeitLam} classified all nondegenerate multiplicative self-maps of the matrix rings $M_n(\\mathcal{R})$ over a principal ideal domain~$\\mathcal{R}$, where nondegeneracy means that the map is not identically zero on the set of matrices with zero determinant. Specifically, for each such map $\\phi: M_n(\\ca{R}) \\to M_n(\\ca{R})$, either there exists a nonzero idempotent matrix $P \\in M_n(\\ca{R})$ such that $\\phi - P$ is multiplicative and degenerate, or there exists an invertible matrix $T \\in M_n(\\ca{R})$ and a ring endomorphism $\\omega$ of $\\ca{R}$ such that $\\phi$ takes one of the following two distinct forms:\n\t$$\n\t\\phi(X) = T \\, \\omega(X) \\, T^{-1} \\quad \\text{ or } \\quad \\phi(X) = T \\, \\omega(X)^* \\, T^{-1},\n\t$$\n\twhere $\\omega(X)$ denotes the matrix obtained by applying $\\omega$ entrywise, and $(\\cdot)^*$ is the corresponding cofactor matrix. In particular, all bijective multiplicative self-maps on $M_n(\\mathcal{R})$ are automatically additive and hence ring automorphisms. Taken together, these results expose a fundamental phenomenon: multiplicativity, when coupled with bijectivity, interacts strongly with the ambient ring structure, leaving little room for pathological nonadditive behavior. This rigidity principle later emerged as a central theme in preserver theory, especially in the context of matrix and operator algebras.\n\n\tThis viewpoint was later extended from ordinary ring homomorphisms to \\emph{Jordan homomorphisms}, which originate as homomorphisms in the category of \\emph{Jordan algebras}, introduced by Jordan, von Neumann, and Wigner \\cite{Jordan} in the 1930s as algebraic models for quantum observables.\n\tA \\emph{Jordan algebra} is a nonassociative algebra $\\ca{A}$ over a field equipped with a commutative multiplication $\\diamond$ satisfying the \\emph{Jordan identity}\n\t$$\n\ta^2 \\diamond (a \\diamond b) = a \\diamond (a^2 \\diamond b), \\quad \\text{ for all } a,b \\in \\ca{A}.\n\t$$\n\tJordan rings are similarly defined. Every associative algebra (or ring) $\\mathcal{A}$ naturally becomes a Jordan algebra (ring) when equipped with the \\emph{Jordan product}\n\t$$\n\ta \\diamond b := ab + ba.\n\t$$\n\tA Jordan algebra that is isomorphic to a Jordan subalgebra of an associative algebra is called a \\emph{special Jordan algebra} (see e.g. \\cite[Section~2.3]{HancheStormer}). Not all Jordan algebras are special; the Albert algebra~\\cite{Albert} provides the classical example of an \\emph{exceptional Jordan algebra}.\n\n\tIn an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n\t$$\n\t\\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n\t$$\n\tIf the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n\t$$\n\t\\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n\t$$\n\tMoreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n\t\\begin{equation}\\label{eq:normJprodukt}\n\t\ta\\circ b :=\\frac12(ab+ba),\n\t\\end{equation}\n\tand Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\n\tThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result  asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n\t\\begin{equation}\\label{eq:glavnieq}\n\t\t\\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n\t\\end{equation}\n\twhere $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\n\tIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all  $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}.   This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n\t$$\n\t\\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n\t$$\n\ttransforms $\\phi$ into a $\\circ$-preserving map $\\psi$.  This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n\twith the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\tA substantial body of literature also investigates multiplicative maps with respect to products other than the Jordan product, particularly various symmetrized or triple-product variants such as $(a,b) \\mapsto aba$, $(a,b,c) \\mapsto abc + cba$, and many others (see, e.g., \\cite{AnHou, Kuzma, LesnjakSze, Lu2, Molnar2} and references therein). More recently, preservers of the product\n\t\\begin{equation}\\label{eq:the product}\n\t\ta^{2} \\circ b = \\frac12(a^{2} b + b a^{2})\n\t\\end{equation}\n\twere studied by Ghorbanipour and Hejazian~\\cite{GhorbanipourHejazian}, who showed that any bijection $\\phi : \\mathcal{A} \\to \\mathcal{A}$ preserving \\eqref{eq:the product}, where $\\mathcal{A}$ is a real standard Jordan operator algebra acting on a Hilbert space of dimension at least $2$, is of the form $\\phi = \\pm \\psi$, where $\\psi$ is a Jordan automorphism of $\\mathcal{A}$.\n\n\tThe product $(a,b)\\mapsto a^2 \\circ b$ plays an important role in the theory of JB-algebras. Recall that a \\emph{JB-algebra} is a (typically real) Jordan algebra $(\\mathcal{A},\\circ)$ equipped with a complete submultiplicative norm $\\|\\cdot\\|$  (that is, a Jordan Banach algebra) which additionally satisfies\n\t$$\n\t\\|a\\|^2 \\le \\|a^2 + b^2\\|, \\quad \\text{ for all }  a,b \\in \\mathcal{A}.\n\t$$\n\tA central feature of JB-algebras is that many aspects of their structure, such as order, spectral theory, and the theory of ideals, are encoded in the associated quadratic operators (see, for example, \\cite{Battaglia}). In particular, the fundamental quadratic mapping $U_a : \\mathcal{A} \\to \\mathcal{A}$,\n\t$$\n\tU_{a}(b) := 2(a \\circ b) \\circ a - a^{2} \\circ b,\n\t$$\n\tplays a key role in the analysis of orthogonality, annihilators, and quadratic ideals. Moreover, the product $(a,b)\\mapsto a^2 \\circ b$ was used in \\cite{Battaglia} to define orthogonality in JB-algebras: elements $a,b \\in \\mathcal{A}$ are said to be orthogonal if $a^{2} \\circ b = 0$. In the setting of special JB-algebras, such as the self-adjoint part of a $C^{*}$-algebra, the Jordan product is given by \\eqref{eq:normJprodukt}. In this case, the expression $a^{2} \\circ b$ agrees exactly with the symmetrized product appearing in \\eqref{eq:the product}.\n\n\t\\smallskip\n\n\tThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n\t$$\n\ta^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n\t$$\n\tfor arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.",
    "context": "In an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n $$\n \\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n $$\n If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n $$\n \\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n $$\n Moreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n \\begin{equation}\\label{eq:normJprodukt}\n a\\circ b :=\\frac12(ab+ba),\n \\end{equation}\n and Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\nThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n \\begin{equation}\\label{eq:glavnieq}\n \\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n \\end{equation}\n where $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\nIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n $$\n \\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n $$\n transforms $\\phi$ into a $\\circ$-preserving map $\\psi$. This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\\smallskip\n\nThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n $$\n a^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n $$\n for arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.",
    "full_context": "In an associative setting, a linear (or additive) map $\\phi: \\ca{A} \\to \\ca{B}$ between associative algebras (or rings) $\\ca{A}$ and $\\ca{B}$ is called a \\emph{Jordan homomorphism} if it satisfies\n $$\n \\phi(a \\diamond b) = \\phi(a) \\diamond \\phi(b), \\quad \\text{ for all } a, b \\in \\ca{A}.\n $$\n If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\\phi$ preserves squares with respect to the underlying product of $\\mathcal{A}$, that is,\n $$\n \\phi(a^2) = \\phi(a)^2, \\quad \\text{ for all } a \\in \\ca{A}.\n $$\n Moreover, if $\\ca{A}$ is an algebra over a field $\\F$ with $\\chr(\\F) \\ne 2$, the Jordan structure is commonly (esp.\\ in quantum mechanics contexts) induced by the \\emph{normalized Jordan product}\n \\begin{equation}\\label{eq:normJprodukt}\n a\\circ b :=\\frac12(ab+ba),\n \\end{equation}\n and Jordan homomorphisms are equivalently defined as $\\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.\n\nThe problem of automatic additivity for \\emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\\circ$ or $\\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\\'ar \\cite{Molnar}, who characterized $\\circ$-preserving bijections $\\phi:\\mathcal{A} \\to \\mathcal{B}$ between standard operator algebras $\\mathcal{A}$ and $\\mathcal{B}$ with $\\dim(\\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\\'ar's result asserts that any $\\circ$-preserving bijection $\\phi : M_n(\\C)\\to M_n(\\C)$, $n \\ge 2$, takes the form\n \\begin{equation}\\label{eq:glavnieq}\n \\phi(X)=T\\,\\omega(X)\\,T^{-1} \\quad \\text{ or } \\quad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1},\n \\end{equation}\n where $T\\in M_n(\\C)$ is invertible, $\\omega$ is a ring automorphism of $\\C$ applied entrywise, and $(\\cdot)^t$ denotes matrix transposition. Lu \\cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \\cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\\diamond$- or $\\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \\cite{Ji} showed that if $\\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\\phi:\\mathcal{A}\\to \\mathcal{B}$ to another Jordan algebra $\\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.\n\nIn the setting of full matrix algebras, our recent work \\cite{GogicTomasevic-AM} provides a complete classification of all $\\circ$-preserving maps $\\phi : M_n(\\mathbb{F}) \\to M_n(\\mathbb{F})$, where $\\chr(\\mathbb{F}) \\neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\\eqref{eq:glavnieq}, where $T \\in M_n(\\mathbb{F})$ is invertible and $\\omega : \\mathbb{F} \\to \\mathbb{F}$ is a ring monomorphism \\cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\\phi : \\mathcal{A} \\to \\mathcal{B}$ is any $\\diamond$-preserving map between $\\mathbb{F}$-algebras (with $\\chr(\\mathbb{F}) \\neq 2$), then the simple normalization\n $$\n \\psi(x) := 2\\,\\phi\\left(\\frac{x}{2}\\right), \\quad x \\in \\mathcal{A},\n $$\n transforms $\\phi$ into a $\\circ$-preserving map $\\psi$. This reduction allows us to obtain a corresponding classification of $\\diamond$-preserving maps on $M_n(\\mathbb{F})$, \n with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \\in M_n(\\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\\mathbb{R}$ and $\\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \\cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \\emph{structural matrix algebras}, i.e.\\ subalgebras of $M_n(\\mathbb{F})$ containing all diagonal matrices \\cite{vanWyk} (see also \\cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \\cite{Coelho, GogicTomasevic-LAA}.\n\n\\smallskip\n\nThis motivates our study of \\emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product\n $$\n a^k \\circ b= \\frac{1}{2}(a^k b + b a^k),\n $$\n for arbitrary $k \\in \\N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.\n\n\\smallskip\n\n\\section{Notation and Preliminaries}\\label{sec:prel}\n We begin by introducing notation that will be used throughout the paper. Let $\\F$ be a fixed algebraically closed field of characteristic different from $2$. We denote by $\\mathbb{F}^\\times$ the multiplicative group of nonzero elements of $\\mathbb{F}$, and by $\\mathbb{F}[x]$ the polynomial algebra in one variable over $\\mathbb{F}$.\n\n\\begin{lemma}\\label{le:h exists}\n Let $\\phi : M_n \\to M_n$ be a nonzero map which satisfies \\eqref{eq:a certain product} and $\\phi(0) = 0$. There exists a unique multiplicative map $\\omega : \\F \\to \\F$ such that \n \\begin{equation}\\label{eq:homogeneity}\n \\phi(\\lambda X) = \\omega(\\lambda) \\phi(X), \\quad \\text{ for all } \\lambda \\in \\F \\text{ and } X \\in M_n.\n \\end{equation}\n Moreover, there exists an invertible matrix $T \\in M_n$ and $\\varepsilon\\in\\Sph^1_k$ such that the map $$\\psi:= T^{-1}\\phi(\\cdot)T$$ satisfies $$\\psi(E_{jj}) = \\varepsilon E_{jj}, \\quad \\text{ for each } j \\in [n].$$\n Further, fix some distinct $i,j \\in [n]$. Then $\\psi(E_{ij})$ is a nonzero scalar multiple of $E_{ij}$ or $E_{ji}$. If $\\psi(E_{ij}) = \\beta_{ij} E_{ij}$ for some $\\beta_{ij} \\in \\F^\\times$, then for each $x,y \\in \\F$ we have\n \\begin{equation}\\label{eq: Eii+Eij 1}\n \\psi(E_{ii} + x E_{ij}) = \\varepsilon E_{ii} + \\omega(x) (\\beta_{ij} E_{ij}), \\qquad \\psi(E_{jj} + y E_{ij}) = \n \\varepsilon E_{jj} + \\omega(y) (\\beta_{ij} E_{ij}).\n \\end{equation}\n If, on the other hand, $\\psi(E_{ij}) = \\gamma_{ij} E_{ji}$ for some $\\gamma_{ij} \\in \\F^\\times$, then\n \\begin{equation}\\label{eq: Eii+Eij 2}\n \\psi(E_{ii} + x E_{ij}) = \\varepsilon E_{ii} + \\omega(x) (\\gamma_{ij} E_{ji}), \\qquad \\psi(E_{jj} + y E_{ij}) = \n \\varepsilon E_{jj} + \\omega(y) (\\gamma_{ij} E_{ji}).\n \\end{equation}\n Finally, if $n \\ge 2$, the map $\\omega : \\F \\to \\F$ is a ring monomorphism. In particular, $\\phi$ is $\\K$-homogeneous.\n \\end{lemma}\n \\begin{proof}\n In view of Lemma \\ref{le:basic properties II} (c) and (e), $\\phi(E_{11}), \\ldots ,\\phi(E_{nn})$ are mutually orthogonal rank-one $(k+1)$-potents and therefore can be simultaneously diagonalized (see e.g.\\ \\cite[Theorem~8 of \\S6.5]{HoffmanKunze}). Hence, by passing to map $T^{-1}\\phi(\\cdot)T$, for a suitable invertible matrix $T \\in M_n$, without loss of generality we can assume that \n \\begin{equation}\\label{eq:phi fiksira dijagonalne}\n \\phi(E_{jj}) = \\varepsilon_j E_{jj}, \\quad \\text{ for some $\\varepsilon_j \\in \\Sph^1_k$ for each } j \\in [n].\n \\end{equation}\n Note that by Lemma \\ref{le:basic properties II} (f), for each diagonal $(k+1)$-potent $D \\in \\ca{D}_n$ we have\n \\begin{equation}\\label{eq:diagonal tripotents}\n \\phi(D) = \n \\sum_{(j,j) \\in \\supp D} \\varepsilon_j E_{jj}.\n \\end{equation}\n By Lemma \\ref{le:zero map}, we have \n \\begin{equation}\\label{eq:lambda times ejj}\n \\phi(\\lambda E_{jj}) \\ne 0, \\quad\\text{ for all } j \\in [n] \\text{ and } \\lambda \\in \\F^\\times.\n \\end{equation} Further, for each $X \\in M_n$ and $S \\subseteq [n]$ we have\n \\begin{equation}\\label{eq:preserves support}\n \\supp X \\subseteq S \\times S \\implies \\supp \\phi(X) \\subseteq S \\times S.\n \\end{equation}\n Indeed, denote the diagonal idempotent \n $P := \\sum_{j \\in [n]\\setminus S} E_{jj}$ and note that a matrix $X\\in M_n$ is supported in $S \\times S$ if and only if $XP=PX=0$. In that case, obviously $X \\circ P^k = 0$, so \n $$0 = \\phi(X \\circ P^k) = \\phi(X) \\circ \\phi(P)^k \\stackrel{\\eqref{eq:diagonal tripotents}}= \\phi(X) \\circ P$$\n and hence Lemma \\ref{le:Jordan product calculations} (a) implies the claim.\n \\smallskip\n\nFix $(i,j), (j,p) \\in [n]^2 \\setminus \\Delta_n$. If $i \\ne p$, then by Lemma \\ref{le:h exists} we have\n \\begin{align*}\n \\frac12 g(i,p)E_{ip} &= \\phi\\left(\\frac12 E_{ip}\\right) =\\phi((E_{ii}+E_{ij})^k \\circ E_{jp}) = \\phi(E_{ii}+E_{ij})^k \\circ \\phi(E_{jp}) \\\\\n &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= (E_{ii}+g(i,j)E_{ij})^k \\circ (g(j,p)E_{jp})\\\\\n &= \\frac12 g(i,j)g(j,p)E_{ip},\n \\end{align*}\n which implies $g(i,p) = g(i,j)g(j,p)$.\n On the other hand, if $i = p$, then by Lemma \\ref{le:h exists} we have\n \\begin{align*}\n \\frac{1}{2}\\phi\\left(E_{ii} + E_{jj} + E_{ji}\\right) &= \\phi\\left(\\frac{1}{2}\\left(E_{ii} + E_{jj} + E_{ji}\\right)\\right) = \\phi((E_{ii}+E_{ij})^k \\circ E_{ji}) \\\\\n &= \\phi(E_{ii}+E_{ij})^k \\circ \\phi(E_{ji}) \\\\\n &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= (E_{ii}+ g(i,j)E_{ij})^k \\circ (g(j,i)E_{ji})\\\\\n &= \\frac12 g(j,i) \\Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\\Big).\n \\end{align*}\n Furthermore, we have\n \\begin{align*}\n E_{ii} +\\frac12g(j,i) E_{ji} \\,\\, &\\leftstackrel{\\eqref{eq: Eii+Eij 1}}= \\phi\\left(E_{ii} +\\frac12 E_{ji}\\right) =\\phi\\left((E_{ii} + E_{jj} + E_{ji}) \\circ E_{ii}^k\\right) \\\\\n &= \\phi\\left(E_{ii} + E_{jj} + E_{ji}\\right) \\circ \\phi(E_{ii})^k\\\\\n &= g(j,i) \\Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\\Big) \\circ E_{ii} \\\\\n &= g(j,i)\\left(g(i,j) E_{ii} +\\frac12 E_{ji}\\right)\n \\end{align*}\n We obtain \n $$\n g(i,j)g(j,i) = 1,\n $$\n as desired. Following \\cite{Coelho}, denote by $$g^* : M_n \\to M_n, \\qquad g^*(E_{ij}) := g(i,j)E_{ij}$$\n the induced algebra automorphism of $M_n$. Note that $g^*$ is implemented via conjugation by the invertible diagonal matrix\n $$\n D := \\operatorname{diag}\\big(g(1,1), g(2,1), \\ldots, g(n,1)\\big) \\in \\ca{D}_n,\n $$\n i.e.\\ $g^*(\\cdot)=D(\\cdot)D^{-1}$. Hence, by passing to the map $(g^*)^{-1}(\\phi(\\cdot)) = D^{-1}\\phi(\\cdot)D$, without loss of generality we can assume that \n $$\\phi(E_{ij}) = E_{ij}, \\quad \\text{ for all } (i,j) \\in [n]^2.$$\n In view of Lemma \\ref{le:h exists}, denote by $\\omega :\\F \\to \\F$ the induced ring monomorphism that satisfies \\eqref{eq:homogeneity}. We claim that \n \\begin{equation}\\label{eq:phi = omega}\n \\phi(X) = \\omega(X), \\quad \\text{ for all } X \\in M_n.\n \\end{equation}\n First assume that $X=[x_{ij}] \\in M_n$ is of the form $X = Y^k$ for some $Y \\in M_n$. Fix $(i,j) \\in [n]^2$ and denote $\\phi(X)=[\\phi(X)_{ij}]$. If $i = j$, then by Lemma \\ref{le:preserves triple product} we have \n \\begin{align*}\n \\omega(x_{ii})E_{ii} &= \\phi(x_{ii}E_{ii}) = \\phi(E_{ii}Y^k E_{ii}) = \\phi(E_{ii})\\phi(Y)^k \\phi(E_{ii}) = E_{ii}\\phi(X) E_{ii} \\\\\n &= \\phi(X)_{ii}E_{ii},\n \\end{align*}\n which shows that\n \\begin{equation}\\label{eq:Yii}\n \\phi(X)_{ii} = \\omega(x_{ii}).\n \\end{equation} Now assume $i \\ne j$. Denote $P:= E_{ii}+E_{ji}$ and note that \n $$\n PAP = (a_{ii}+a_{ij})P, \\quad \\text{ for all matrices $A=[a_{ij}] \\in M_n$}.\n $$ \n By \\eqref{eq: Eii+Eij 1} we have $\\phi(P) = P$ and hence\n \\begin{align*}\n \\omega(x_{ii}+x_{ij})P &=\\phi((x_{ii}+x_{ij})P) = \\phi(PXP) \\stackrel{\\text{Lemma }\\ref{le:preserves triple product}}= \\phi(P)\\phi(X)\\phi(P) \\\\\n &= P\\phi(X)P = (\\phi(X)_{ii}+\\phi(X)_{ij})P.\n \\end{align*}\n Therefore, by the additivity of $\\omega$ and \\eqref{eq:Yii} we obtain\n $$\\omega(x_{ii})+\\omega(x_{ij}) = \\phi(X)_{ii}+\\phi(X)_{ij} = \\omega(x_{ii})+\\phi(X)_{ij}.$$\n This implies $\\phi(X)_{ij} = \\omega(x_{ij})$, which shows \\eqref{eq:phi = omega} for all matrices $X \\in M_n(\\F)$ which are a $k$-th power of some matrix. Using the notation of Lemma \\ref{le:iterated R}, this means \n $$\\phi(X) = \\omega(X), \\quad \\text{ for all } X \\in \\ca{R}_0^n.$$\n We now extend this conclusion to all matrices in $M_n$. Note that if $A,B \\in M_n$ satisfy $\\phi(A) = \\omega(A)$ and $\\phi(B) = \\omega(B)$, then clearly\n $$\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B) = \\omega(A)^k \\circ \\omega(B) = \\omega(A^k \\circ B).$$\n Again returning to the notation of Lemma \\ref{le:iterated R}, this inductively implies that $$\\phi(X) = \\omega(X), \\quad \\text{ for all } X\\in \\ca{R}^n.$$\n By the same lemma we have $\\ca{R}^n = M_n$, which completes the proof of the theorem.\n \\end{proof}\n We conclude the paper with the following generalization of \\cite[Corollary~3.5]{GogicTomasevic-AM}, which follows immediately from the first part of the proof of Theorem~\\ref{thm:main}.\n \\begin{corollary} \n Let $m < n$. A map $\\phi : M_n \\to M_m$ satisfies \\eqref{eq:a certain product} if and only if it is constant and equal to a fixed $(k+1)$-potent. \n \\end{corollary}",
    "post_theorem_intro_text_len": 1130,
    "post_theorem_intro_text": "This result extends all previously known Jordan-type additivity results on $M_n(\\F)$, at least for algebraically closed fields with $\\operatorname{char}(\\F) \\neq 2$. The proof of Theorem~\\ref{thm:main}, presented in Section~\\S\\ref{sec:main}, follows the general strategy of \\cite[Theorem~1.1]{GogicTomasevic-AM}. The classification relies on the fact that maps satisfying \\eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements. These notions will be introduced and discussed in the next section. A notable difference from the case $k=1$, which corresponds to idempotents, is that when $\\operatorname{char}(\\F)$ is positive and divides $k$, $(k+1)$-potents may fail to be diagonalizable (see Remark~\\ref{rem:k-potents not diagonalizable}). This phenomenon requires more refined arguments in order to handle the mixed Jordan--power preservers effectively. We also note that linear and closely related variants of $k$-potent preservers on matrix and operator algebras have been studied in papers such as \\cite{HouHou,SongCao,YouWang}.",
    "sketch": "The proof of Theorem~\\ref{thm:main} (given in Section~\\S\\ref{sec:main}) “follows the general strategy of \\cite[Theorem~1.1]{GogicTomasevic-AM}.” The classification “relies on the fact that maps satisfying \\eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\\operatorname{char}(\\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”",
    "expanded_sketch": "The proof of the main theorem (given later) “follows the general strategy of Gogic--Tomasevic, Theorem~1.1 (GogicTomasevic-AM).” The classification “relies on the fact that maps satisfying\n\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\npreserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\\operatorname{char}(\\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”",
    "expanded_theorem": "\t\t\\label{thm:main}\n\t\tLet $\\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \\in \\N$ with $n \\ge 2$. A map $\\phi : M_n(\\F) \\to M_n(\\F)$\n\t\tsatisfies the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tif and only if one of the following holds:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] $\\phi$ is constant, taking a fixed $(k+1)$-potent value;\n\t\t\t\\item[(b)] there exist an invertible matrix $T \\in M_n(\\F)$, a ring monomorphism $\\omega : \\F \\to \\F$, and a $k$-th root of unity $\\varepsilon \\in \\F$ such that\n\t\t\t\\begin{equation}\\label{eq:form of phi}\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)\\, T^{-1}\n\t\t\t\t\\quad \\text{ or } \\quad\n\t\t\t\t\\phi(X) = \\varepsilon\\, T\\, \\omega(X)^t\\, T^{-1}.\n\t\t\t\\end{equation}\n\t\t\\end{itemize}\n\t\tIn particular, every nonconstant solution of the mixed Jordan--power identity\n\t\t\\begin{equation}\\label{eq:a certain product}\n\t\t\t\\phi(A^k \\circ B) = \\phi(A)^k \\circ \\phi(B), \\qquad \\text{for all } A,B \\in M_n(\\F),\n\t\t\\end{equation}\n\t\tis additive.,",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\(\\mathbb F\\) be an algebraically closed field with \\(\\operatorname{char}(\\mathbb F)\\neq 2\\), let \\(n\\ge 2\\) and \\(k\\in\\mathbb N\\), and write \\(A\\circ B:=\\tfrac12(AB+BA)\\) for the Jordan product on \\(M_n(\\mathbb F)\\). Which maps \\(\\phi:M_n(\\mathbb F)\\to M_n(\\mathbb F)\\) satisfy\n\\[\n\\phi(A^k\\circ B)=\\phi(A)^k\\circ \\phi(B)\\qquad\\text{for all }A,B\\in M_n(\\mathbb F)?\n\\]",
      "correct_choice": {
        "label": "A",
        "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\) (that is, \\(P^{k+1}=P\\)); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1},\n\\]\nwhere \\(\\omega(X)\\) means applying \\(\\omega\\) entrywise. In particular, every nonconstant such \\(\\phi\\) is additive."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\) and a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=T\\,\\omega(X)^t\\,T^{-1}.\n\\]\nIn particular, every nonconstant such \\(\\phi\\) is additive."
        },
        {
          "label": "C",
          "text": "Every map \\(\\phi:M_n(\\mathbb F)\\to M_n(\\mathbb F)\\) of one of the following forms satisfies\n\\[\n\\phi(A^k\\circ B)=\\phi(A)^k\\circ\\phi(B)\\qquad\\text{for all }A,B\\in M_n(\\mathbb F):\n\\]\neither \\(\\phi\\) is constant with value a fixed \\((k+1)\\)-potent matrix \\(P\\), or there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1}.\n\\]"
        },
        {
          "label": "D",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed idempotent matrix \\(P\\) (that is, \\(P^2=P\\)); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a ring monomorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a scalar \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}\\qquad\\text{or}\\qquad \\phi(X)=\\varepsilon T\\,\\omega(X)^t\\,T^{-1},\n\\]\nwhere \\(\\omega(X)\\) is applied entrywise. In particular, every nonconstant such \\(\\phi\\) is additive."
        },
        {
          "label": "E",
          "text": "Exactly those maps \\(\\phi\\) for which either: (i) \\(\\phi\\) is constant, with value a fixed \\((k+1)\\)-potent matrix \\(P\\); or (ii) there exist an invertible matrix \\(T\\in M_n(\\mathbb F)\\), a field automorphism \\(\\omega:\\mathbb F\\to\\mathbb F\\), and a \\(k\\)-th root of unity \\(\\varepsilon\\in\\mathbb F\\) such that for every \\(X\\in M_n(\\mathbb F)\\),\n\\[\n\\phi(X)=\\varepsilon T\\,\\omega(X)\\,T^{-1}.\n\\]\nIn particular, every nonconstant such \\(\\phi\\) is additive."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "k-th-root-of-unity scalar factor",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "replaced classification iff by one-way sufficient condition",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "constant-value type and unrestricted scalar factor",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "transpose branch and monomorphism generality",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not disclose the classification or directly hint at the correct structural form of the maps. The correct answer is not leaked by wording in the prompt."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially an exact-classification recall question: the correct option states the theorem itself. It does not ask the student to derive a new consequence from the hypothesis, but to recognize the theorem statement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but important ways (e.g. iff vs one-way implication, presence of transpose, monomorphism vs automorphism, k-th root of unity factor). However, the item mainly tests theorem recognition rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic confusions: omitting the root-of-unity factor, weakening an iff classification to a sufficient condition, replacing (k+1)-potent by idempotent and allowing arbitrary scalars, or excluding the transpose/monomorphism cases. They are distinct and well aligned with common failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of the classification result rather than a genuine generative-reasoning task."
    }
  },
  {
    "id": "2512.13369v1",
    "paper_link": "http://arxiv.org/abs/2512.13369v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{EucTrees}\n In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\\mathrm{MST}} = \\Theta(\\sqrt n)$ w.h.p.\\ and $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt{n})$ w.h.p.",
      "start_pos": 12024,
      "end_pos": 12235,
      "label": "EucTrees"
    },
    "ref_dict": {
      "th2": "\\begin{theorem}\\label{th2} \nThe minimum cost solution of the following problems have value $O(1)$ w.h.p. The proofs are non-constructive.\n\\begin{enumerate}[(a)]\n\\item Traveling Salesperson Problem in $K_n$ with at least $n$ colors for the edges. \n\\item Minimum cost perfect matching in complete $k$-uniform hypergraphs, $k\\geq 2$ with at least $n/k$ colors for the edges.\n\\item Minimum cost classes of degree bounded spanning trees in $K_n$ with at least $n-1$ colors for the edges e.g. spanning binary trees.\n\\end{enumerate}\n\\end{theorem}",
      "EucTours": "\\begin{theorem}\\label{EucTours}\nIn the Euclidean setting, with edges labeled randomly by $q=(1+\\ep)n$ colors, we have (a) $Z_{\\TSP}=O(n^{1/2}\\log n)$, and (b) $Z_{\\TSP}-Z_{\\TSP}^*=\\Omega(\\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.\n\\end{theorem}",
      "EucTrees": "\\begin{theorem}\\label{EucTrees}\n In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\\MST} = \\Theta(\\sqrt n)$ w.h.p.\\ and $Z_{\\MST}-Z^*_{\\MST}=\\Theta(\\sqrt{n})$ w.h.p.\\end{theorem}",
      "rweight": "\\begin{theorem}\\label{rweight}\nLet $\\cH$ be a $\\k$-spread, $r$-uniform hypergraph on vertex set $X$. Suppose that (i) $\\k=\\Omega(r)$ and (ii) $|X|\\leq \\k^2r/\\log^5r$. If we randomly color the elements of $\\cH$ with $k\\geq r$ colors, then $\\xi_\\cH\\leq 3C_0r/\\k$ w.h.p. (assuming $r\\to\\infty$) for some absolute constant $C_0>0$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3741,
    "pre_theorem_intro_text": "The minimum length of structures like spanning trees and Hamilton cycles among random points in Euclidean space has been a topic of interest for the better part of a century.  Beardwood, Halton and Hammersley \\cite{BHH} showed that the minimum cost of a tour through $n$ random points in the unit square is asymptotic to $\\b_{TSP} n^{1/2}$ with high probability (we say an event $\\cE_n$ occurs {\\em with high probability} or w.h.p.\\ if $\\mathbb{P}(\\cE_n) \\rightarrow 1$ as $n \\rightarrow \\infty$). Here $\\b_{TSP}$ is a positive constant, still unknown after more than 60 years. Steele \\cite{Steele1} generalised this result to {\\em Euclidean Functionals} and showed that the minimum cost of a spanning tree on $n$ random points is asymptotic to $\\b_{MST} n^{1/2}$, w.h.p. In both cases, the cost of edges is determined by the Euclidean distance between points. Steele's work shows that there are many instances of optimisation problems where the expected minimum cost grows like $\\beta n^{1/2}$ for unknown constants $\\beta$. Frieze and Pegden \\cite{FP1} showed the constants for lengths of lower bound structures for the tour (like the minimum spanning tree, or twice a matching) really are distinct from the constant for the TSP, even though the best known explicit bounds on the constants generally overlap.\n\nWhen the costs are generated independently, the situation becomes much clearer. In the case of spanning trees where the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ random costs, Frieze \\cite{F1} showed that $\\mathbb{E}(L_n)\\sim\\zeta(3)=\\sum_{k=1}^\\infty\\frac{1}{k^3}$, where $L_n$ denotes the (random) minimum cost of a spanning tree. \n\nThe expected cost of the minimum cost Hamilton cycle in the complete graph $K_{n}$ is asymptotic to $\\tau=\\frac12\\int_{x=0}^\\infty y(x)dx$ where $y$ is the positive solution to $\\brac{1+\\frac{x}{2}}e^{-x}+\\brac{1+\\frac{y}{2}}e^{-y}=1$. This was proved by Wastl\\\"und \\cite{Wast1}. Frieze \\cite{F2} showed that the expected minimum cost of a Hamilton cycle is asymptotically equal to  the expected minimum cost of a 2-factor. \n\nIf we orient the edges then it follows from the work of Karp \\cite{K1} and Aldous \\cite{A1} that the expected minimum cost of a directed Hamilton cycle is asymptotic to $\\zeta(2)=\\sum_{k=1}^\\infty\\frac{1}{k^2}=\\frac{\\pi^2}{6}$.\n\nIn this work we add a spot of color to the problems. We assume that the edges of the graphs in question have been given uniformly random colors and ask for minimum cost rainbow Hamilton cycles or spanning trees. In this context, a set $S$ of edges is {\\em rainbow} colored if all the colors in $S$ are distinct. We do this within two contexts: either we have $n$ random points in $[0,1]^2$ and the cost of an edge is the Euclidean distance between its endpoints, or else we have independent uniform $[0,1]$ random costs.\n\nIn the first context we let ${\\mathcal X}=\\left\\{\\bx_1,\\bx_2,\\ldots,\\bx_n\\right\\}$ where the $\\bx_i$ are independently chosen uniformly from the unit square $[0,1]^2$. The cost of edge $\\left\\{\\bx_i,\\bx_j\\right\\}$ is the Euclidean length $|\\bx_i-\\bx_j|$. Our first results concern the length $Z_{\\mathrm{MST}}$ of the minimum rainbow spanning tree in this model,  when we randomly color labeled from a set of $n-1$ colors.  (Note that with fewer colors, no rainbow spanning tree would exist. Also, even if we have enough colors there is always some small probability that there is no rainbow spanning tree at all, in which case $Z_{\\mathrm{MST}}$ is undefined.)  We let $Z^*_{\\mathrm{MST}}$ denote the length of the minimum spanning tree (without the rainbow constraint); recall that $Z^*_{\\mathrm{MST}}\\sim \\beta_{\\mathrm{MST}} \\sqrt n$ w.h.p., for some constant $\\beta_{\\mathrm{MST}}$.",
    "context": "The minimum length of structures like spanning trees and Hamilton cycles among random points in Euclidean space has been a topic of interest for the better part of a century. Beardwood, Halton and Hammersley \\cite{BHH} showed that the minimum cost of a tour through $n$ random points in the unit square is asymptotic to $\\b_{TSP} n^{1/2}$ with high probability (we say an event $\\cE_n$ occurs {\\em with high probability} or w.h.p.\\ if $\\mathbb{P}(\\cE_n) \\rightarrow 1$ as $n \\rightarrow \\infty$). Here $\\b_{TSP}$ is a positive constant, still unknown after more than 60 years. Steele \\cite{Steele1} generalised this result to {\\em Euclidean Functionals} and showed that the minimum cost of a spanning tree on $n$ random points is asymptotic to $\\b_{MST} n^{1/2}$, w.h.p. In both cases, the cost of edges is determined by the Euclidean distance between points. Steele's work shows that there are many instances of optimisation problems where the expected minimum cost grows like $\\beta n^{1/2}$ for unknown constants $\\beta$. Frieze and Pegden \\cite{FP1} showed the constants for lengths of lower bound structures for the tour (like the minimum spanning tree, or twice a matching) really are distinct from the constant for the TSP, even though the best known explicit bounds on the constants generally overlap.\n\nWhen the costs are generated independently, the situation becomes much clearer. In the case of spanning trees where the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ random costs, Frieze \\cite{F1} showed that $\\mathbb{E}(L_n)\\sim\\zeta(3)=\\sum_{k=1}^\\infty\\frac{1}{k^3}$, where $L_n$ denotes the (random) minimum cost of a spanning tree.\n\nThe expected cost of the minimum cost Hamilton cycle in the complete graph $K_{n}$ is asymptotic to $\\tau=\\frac12\\int_{x=0}^\\infty y(x)dx$ where $y$ is the positive solution to $\\brac{1+\\frac{x}{2}}e^{-x}+\\brac{1+\\frac{y}{2}}e^{-y}=1$. This was proved by Wastl\\\"und \\cite{Wast1}. Frieze \\cite{F2} showed that the expected minimum cost of a Hamilton cycle is asymptotically equal to the expected minimum cost of a 2-factor.\n\nIf we orient the edges then it follows from the work of Karp \\cite{K1} and Aldous \\cite{A1} that the expected minimum cost of a directed Hamilton cycle is asymptotic to $\\zeta(2)=\\sum_{k=1}^\\infty\\frac{1}{k^2}=\\frac{\\pi^2}{6}$.\n\nIn this work we add a spot of color to the problems. We assume that the edges of the graphs in question have been given uniformly random colors and ask for minimum cost rainbow Hamilton cycles or spanning trees. In this context, a set $S$ of edges is {\\em rainbow} colored if all the colors in $S$ are distinct. We do this within two contexts: either we have $n$ random points in $[0,1]^2$ and the cost of an edge is the Euclidean distance between its endpoints, or else we have independent uniform $[0,1]$ random costs.\n\nIn the first context we let ${\\mathcal X}=\\left\\{\\bx_1,\\bx_2,\\ldots,\\bx_n\\right\\}$ where the $\\bx_i$ are independently chosen uniformly from the unit square $[0,1]^2$. The cost of edge $\\left\\{\\bx_i,\\bx_j\\right\\}$ is the Euclidean length $|\\bx_i-\\bx_j|$. Our first results concern the length $Z_{\\mathrm{MST}}$ of the minimum rainbow spanning tree in this model, when we randomly color labeled from a set of $n-1$ colors. (Note that with fewer colors, no rainbow spanning tree would exist. Also, even if we have enough colors there is always some small probability that there is no rainbow spanning tree at all, in which case $Z_{\\mathrm{MST}}$ is undefined.) We let $Z^*_{\\mathrm{MST}}$ denote the length of the minimum spanning tree (without the rainbow constraint); recall that $Z^*_{\\mathrm{MST}}\\sim \\beta_{\\mathrm{MST}} \\sqrt n$ w.h.p., for some constant $\\beta_{\\mathrm{MST}}$.",
    "full_context": "The minimum length of structures like spanning trees and Hamilton cycles among random points in Euclidean space has been a topic of interest for the better part of a century. Beardwood, Halton and Hammersley \\cite{BHH} showed that the minimum cost of a tour through $n$ random points in the unit square is asymptotic to $\\b_{TSP} n^{1/2}$ with high probability (we say an event $\\cE_n$ occurs {\\em with high probability} or w.h.p.\\ if $\\mathbb{P}(\\cE_n) \\rightarrow 1$ as $n \\rightarrow \\infty$). Here $\\b_{TSP}$ is a positive constant, still unknown after more than 60 years. Steele \\cite{Steele1} generalised this result to {\\em Euclidean Functionals} and showed that the minimum cost of a spanning tree on $n$ random points is asymptotic to $\\b_{MST} n^{1/2}$, w.h.p. In both cases, the cost of edges is determined by the Euclidean distance between points. Steele's work shows that there are many instances of optimisation problems where the expected minimum cost grows like $\\beta n^{1/2}$ for unknown constants $\\beta$. Frieze and Pegden \\cite{FP1} showed the constants for lengths of lower bound structures for the tour (like the minimum spanning tree, or twice a matching) really are distinct from the constant for the TSP, even though the best known explicit bounds on the constants generally overlap.\n\nWhen the costs are generated independently, the situation becomes much clearer. In the case of spanning trees where the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ random costs, Frieze \\cite{F1} showed that $\\mathbb{E}(L_n)\\sim\\zeta(3)=\\sum_{k=1}^\\infty\\frac{1}{k^3}$, where $L_n$ denotes the (random) minimum cost of a spanning tree.\n\nThe expected cost of the minimum cost Hamilton cycle in the complete graph $K_{n}$ is asymptotic to $\\tau=\\frac12\\int_{x=0}^\\infty y(x)dx$ where $y$ is the positive solution to $\\brac{1+\\frac{x}{2}}e^{-x}+\\brac{1+\\frac{y}{2}}e^{-y}=1$. This was proved by Wastl\\\"und \\cite{Wast1}. Frieze \\cite{F2} showed that the expected minimum cost of a Hamilton cycle is asymptotically equal to the expected minimum cost of a 2-factor.\n\nIf we orient the edges then it follows from the work of Karp \\cite{K1} and Aldous \\cite{A1} that the expected minimum cost of a directed Hamilton cycle is asymptotic to $\\zeta(2)=\\sum_{k=1}^\\infty\\frac{1}{k^2}=\\frac{\\pi^2}{6}$.\n\nIn this work we add a spot of color to the problems. We assume that the edges of the graphs in question have been given uniformly random colors and ask for minimum cost rainbow Hamilton cycles or spanning trees. In this context, a set $S$ of edges is {\\em rainbow} colored if all the colors in $S$ are distinct. We do this within two contexts: either we have $n$ random points in $[0,1]^2$ and the cost of an edge is the Euclidean distance between its endpoints, or else we have independent uniform $[0,1]$ random costs.\n\nIn the first context we let ${\\mathcal X}=\\left\\{\\bx_1,\\bx_2,\\ldots,\\bx_n\\right\\}$ where the $\\bx_i$ are independently chosen uniformly from the unit square $[0,1]^2$. The cost of edge $\\left\\{\\bx_i,\\bx_j\\right\\}$ is the Euclidean length $|\\bx_i-\\bx_j|$. Our first results concern the length $Z_{\\mathrm{MST}}$ of the minimum rainbow spanning tree in this model, when we randomly color labeled from a set of $n-1$ colors. (Note that with fewer colors, no rainbow spanning tree would exist. Also, even if we have enough colors there is always some small probability that there is no rainbow spanning tree at all, in which case $Z_{\\mathrm{MST}}$ is undefined.) We let $Z^*_{\\mathrm{MST}}$ denote the length of the minimum spanning tree (without the rainbow constraint); recall that $Z^*_{\\mathrm{MST}}\\sim \\beta_{\\mathrm{MST}} \\sqrt n$ w.h.p., for some constant $\\beta_{\\mathrm{MST}}$.\n\n\\begin{theorem}\\label{EucTours}\nIn the Euclidean setting, with edges labeled randomly by $q=(1+\\ep)n$ colors, we have (a) $Z_{\\TSP}=O(n^{1/2}\\log n)$, and (b) $Z_{\\TSP}-Z_{\\TSP}^*=\\Omega(\\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.\n\\end{theorem}\nHere like before the starred version $Z^*_{\\TSP}$ represents the minimum length in the absence of the rainbow condition.\n\nWe define a weighted bipartite (multi-)graph $\\G$ with bipartition $\\cX_1 \\cup C$ where $C=[n-1]$ is our set of colors. The edge set will be the union $E_1\\cup E_2$ of sets which we define next; for both sets $E_1$ and $E_2$, an edge $(\\bx,c)$ will correspond to some edge $(\\bx,\\bx')$ with $\\xi_2(\\bx) < \\xi_2(\\bx')$ in the embedded graph that has been assigned color $c$. This last property guarantees that a perfect matching in $\\G$ corresponds to a rainbow spanning tree in the embedded graph. So to prove the first assertion in Theorem~\\ref{EucTrees}, it suffices to show that w.h.p.\n\\begin{equation}\\label{match}\n\\text{$\\G$ contains a perfect matching,}\n\\end{equation}\nand\n\\begin{equation}\\label{wt}\n\\text{the total weight of all edges in $\\G$ is $O(n^{1/2})$.}\n\\end{equation}\n\n\\begin{claim}\\label{clm:3}\n If the event $\\cG$ holds (where $\\cG$ is defined above \\eqref{negG}), then for any fixed $S \\subseteq A$ of size $s$, (under the probability of choosing random colors for each edge)\n\\[\n \\Pr(\\cE_c(S)\\mid A) \\leq \\bfrac{s}{|A|}^{\\sqrt{K}} + O(n^{-\\sqrt{K} + 2}\\log^{8\\sqrt{K}} n).\n\\]\n \\end{claim}\n \\begin{proof}\n Fix $\\bx\\in A$ and recall the definition of $\\nu_\\bx$ above \\eqref{negG}. Let $\\cA$ be the event that for each $\\bx \\in A$, each color appears fewer than $\\sqrt{K}$ times on edges of length at most $2Bn^{-1/2}\\log^4 n$. The probability of this event failing for fixed $\\bx \\in A$ is at most\n \\[|C| {\\nu_\\bx \\choose \\sqrt{K}}|C|^{-\\sqrt{K}},\\]\n and taking a union bound over $A$ and using \\eqref{negG} gives the second bound in the claim.\n\nWe begin with a simple lemma.\n\\begin{lemma}\\label{repeat}\nLet $S$ be a set of size $\\a=\\Theta(n)$ that is randomly colored using $\\b=\\Theta(n)$ colors. Then w.h.p. there are $\\Theta(n)$ colors that are used more than once.\n\\end{lemma}\n\\begin{proof}\nLet $Z$ denote the number of colors that appear more than once. Then \n\\[\n\\E(Z)=\\b\\brac{1-\\brac{1-\\frac{1}{\\b}}^{\\a}-\\a \\cdot \\frac{1}{\\b}\\brac{1-\\frac{1}{\\b}}^{\\a-1}}=\\Theta(n).\n\\]\nNow changing the color of one edge changes $Z$ by at most one. So, applying McDiarmid's inequality \\cite{McD} we have \n\\[\n\\Pr\\brac{Z\\leq \\E(Z)/2}\\leq \\exp\\set{-\\frac{\\E(Z)^2}{2\\a}}=e^{-\\Omega(n)}.\n\\]\n\\end{proof}\nNow back to the main argument. For notational convenience, in this section we scale $\\cX$ so that we are instead working with a set $\\cY_n$ of $n$ random points in a square of side length $\\sqrt{n}$; note that there is thus one point on average, per unit of area.\n\nLet $X$ be a finite set and let $\\cH$ be an $r$-bounded hypergraph on $X$. For $k \\geq r$, let $X^* = X \\times [k]$. Let $\\cH^*$ denote the family consisting of all possible rainbow copies of $\\cH$ using the color set $[k]$. Let $\\xi_x\\,(x\\in X)$ be independent random variables, each uniform from $[0,1]$, and set\n\\[\n\\xi_{\\cH}=\\min_{S\\in \\cH^*}\\sum_{x\\in S}\\xi_x.\n\\]\nWe prove\n\\begin{theorem}\\label{rweight}\nLet $\\cH$ be a $\\k$-spread, $r$-uniform hypergraph on vertex set $X$. Suppose that (i) $\\k=\\Omega(r)$ and (ii) $|X|\\leq \\k^2r/\\log^5r$. If we randomly color the elements of $\\cH$ with $k\\geq r$ colors, then $\\xi_\\cH\\leq 3C_0r/\\k$ w.h.p. (assuming $r\\to\\infty$) for some absolute constant $C_0>0$.\n\\end{theorem}\n(The paper \\cite{HY} removes conditions (i) and (ii), but the proof doesn't adapt well to deal with the costs, as far as we can see.)\n\\begin{proof}\nWe let $\\ell_0$ be the least $i$ such that $r/2^i<\\log r$, and let $\\ell$ be the least $i>\\ell_0$ such that $r/2^i<1$. We sort the elements of $X=\\set{x_1,x_2,\\ldots,x_N}$ so that $\\xi_{x_1}<\\xi_{x_2}<\\cdots<\\xi_{x_N}$. We then let $p_i=C_i/\\k$ where \n\\[\nC_i=\\begin{cases}C_0&i\\leq \\ell_0.\\\\\\frac{\\log r}{\\log\\log r}&\\ell_0<i\\leq \\ell.\\end{cases}\n\\]\nHere $C_0$ is a large constant. Let $L_i=Np_i$ and define $n_0$ to be the smallest integer such that $2^{-n_0}\\leq \\log r$ and let $n$ be the least $i>n_0$ such that $r/2^i<1$. Then for $i\\in[n]$ we let \n\\[\na_i=\\sum_{j=1}^iL_i\\text{ and }\\e_i=\\frac{2a_i}{N}.\n\\]\nThen let $W^*_1=\\set{x_1,x_2,\\ldots,x_{a_1}},\\,W^*_2=\\set{x_{a_1+1},x_{a_1+2},\\ldots,x_{a_2}}$ etc. Note that $W^*_1,W^*_2,\\ldots,W^*_n$ are randomly colored random sets of size $L_i$ . The Chernoff bounds imply that\n\\begin{align}\n\\Pr(\\exists i:\\max\\set{\\xi_x:x\\in W^*_i}\\geq \\e_i)&=\\sum_{i=1}^n\\Pr(Bin\\brac{N,\\e_i}\\leq\\e_i N)\\nn\\\\\n& \\leq \\sum_{i=1}^n e^{-\\e_iN/8}\\leq 2e^{-a_1/4}=2e^{-C_0N/\\k}.\\label{max}\n\\end{align}\nIn \\cite{BFM} (as we will explain shortly), it is proved that with probability $1-\\frac{\\log\\log r}{\\log^{1/4}r}$, there is a rainbow colored edge $e$ such that $|e\\setminus \\bigcup_{j\\leq i}W^*_j|\\leq r/2^{i-1}$. We will use the main theorem (and its proof which we discuss below) from \\cite{BFM}:\n\\begin{theorem}[Theorem 2 in \\cite{BFM}]\\label{thrainbow}\nLet $\\cH$ be an $r$-bounded, $\\k$-spread hypergraph and let $X=V(\\cH)$ be randomly colored from $Q=[q]$ where $q\\ge r$. Suppose also that (i) $\\k=\\Omega(r)$, that is, there exists a constant $L>0$ such that $\\kappa\\ge Lr$ for all valid $r$, and that (ii) $N\\leq \\k^2r/\\log^5r$. Then there is a constant $C>0$ such that if \n\\beq{mbounds}{\nm\\geq\\frac{(C\\log_2r)|X|}{\\k}\n}\nthen $X_m$ contains a rainbow colored edge of $\\cH$ w.h.p. \n\\end{theorem}\n The proof of Theorem \\ref{thrainbow} in \\cite{BFM} proceeds by iteratively choosing random sets of colored elements $W^*_i$ (just as we have in our current proof). Theorem \\ref{thrainbow} is proved by iteratively applying the lemma we state next. Here $\\cH^*$ is the set of possible rainbow colored edges of $\\cH$. For $H^* \\in \\cH^*$, $T^*(W^*,H^*)$ is $G^* \\setminus W^*$ for some $G^* \\in \\cH^*$.\n \\begin{lemma}[Lemma 5 in \\cite{BFM}] \\label{ML}\nLet $H^*\\in\\cH^*$ be good with respect to $W^*$ if $H^*\\sim W^*$ and $|T^*(W^*,H^*)|<r/2$. Let \n\\[\n\\text{$\\cS$ be the event $|\\set{H^*\\in \\cH^*:H^*\\text{ is good}}|<(1-\\e)|\\cH^*|(1-p)^r$.}\n\\]\nThen\n\\[\n\\Pr(\\cS)\\leq \\exp\\set{-\\frac{\\e^2\\k^2q(1-p)}{16e^3N}} +\\frac{2K}{C_0^{r/3}\\e\\brac{1-p}^r}\n\\]\n\\end{lemma}\nIn other words, so long as the unlikely event $\\cS$ does not occur, we have that for each rainbow colored edge $H^*$, there is some rainbow colored edge $G^*$ that is mostly contained in $W^*$. The proof of Theorem \\ref{thrainbow} proceeds by iteratively applying Lemma \\ref{ML}, each time replacing $\\cH^*$ with the hypergraph whose edges are the $T^*(W^*, H^*)$ (i.e. portions of edges of $\\cH^*$ which are not contained in $W^*$). In \\cite{BFM} they show that during all applications of the Lemma \\ref{ML}, \n the unlikely event $\\cS$ never happens w.h.p. This means that the rainbow edge $H^*$ we obtain in the end was good for every application of Lemma \\ref{ML}. In particular, we have\n \\beq{forcost}{\n|H^*\\setminus \\bigcup_{j\\leq i}W^*_j|\\leq \\frac{r}{2^{i-1}}, \\qquad \\text{i=1, \\ldots, $\\ell$.}\n}\nIn our setting, since $W^*_i$ was the set of vertices of cost at most $\\e_i$, the rainbow edge $H^*$ we obtain from Theorem \\ref{thrainbow} costs at most\n\\mults{\n\\sum_{i=1}^n\\frac{r\\e_i}{2^{i-1}}=\\frac{r}{N}\\sum_{i=1}^n\\frac{1}{2^{i-1}}\\sum_{j=1}^iNp_i\\\\\n=r\\brac{\\sum_{i=1}^{\\ell_0}\\frac{C_i}{2^{i-1}\\k}+\\sum_{i=\\ell_0+1}^\\ell\\frac{C_{\\ell_0}+(i-\\ell_0)\\log r}{2^{i-1}\\k}}\\leq \\frac{3C_0r}{\\k}=O(1).\n}",
    "post_theorem_intro_text_len": 3994,
    "post_theorem_intro_text": "In particular, Theorem \\ref{EucTrees} says that the minimum length doesn't change by more than a constant factor under a rainbow constraint, but that a constant-factor change does necessarily occur.  To put Theorem \\ref{EucTrees} in context, there is an important sense in which it is much less obvious that the minimum rainbow spanning tree should have length $\\Theta(\\sqrt{n})$ than it is for the various optimization problems (matchings, 2-factors, linear programming relaxations of the TSP, etc) to which the general methods derived from Beardwood, Halton, and Hammersley and from Steele have been applied.  In particular, the rainbow condition seems to preclude a straightforward application of subadditivity, which explains, for example, why we do not yet know that $Z_{\\mathrm{MST}}/\\sqrt{n}$ has a limit.  In some sense, the success of subadditivity approaches to structure lengths in random point-sets reflects the sense in which the asymptotic structure lengths really depend only on the local geometry of the ambient space.  (For example, in the Beardwood-Halton-Hammersley theorem, the unit square can be replaced by any bounded open set of $\\mathbb{R}^2$ of measure 1, and the asymptotic length of a tour through $n$ points remains the same, with the same unknown constant $\\beta_{\\mathrm{TSP}}$. But the rainbow condition we impose is a global constraint on the spanning tree that cannot be easily decomposed into noninteracting parts.\n\nFor the minimum length $Z_{\\mathrm{TSP}}$ of a rainbow Hamilton cycle in the Euclidean model, we do not achieve an optimal result up to constant factors unless we allow a slight excess in the number of available colors.  \n\n\\begin{theorem}\\label{EucTours}\nIn the Euclidean setting, with edges labeled randomly by $q=(1+\\varepsilon)n$ colors, we have (a) $Z_{\\mathrm{TSP}}=O(n^{1/2}\\log n)$, and (b) $Z_{\\mathrm{TSP}}-Z_{\\mathrm{TSP}}^*=\\Omega(\\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.\n\\end{theorem}\nHere like before the starred version $Z^*_{\\mathrm{TSP}}$ represents the minimum length in the absence of the rainbow condition.\n\nIn the second scenario, $[0,1]^2$ is replaced by the complete graph $K_n$ and the costs $C(i,j)$ are independent uniform $[0,1]$ random variables.  We prove a general theorem (see Theorem \\ref{rweight}) for cheap rainbow substructures in randomly colored structures with random costs. The proof is an adaptation of some of the recent work on thresholds in spread hypergraphs (see \\cite{BFM}) based on Frankston, Kahn, Narayanan, and Park's recent proof of the fractional Kahn--Kalai conjecture \\cite{FKNP}. Our Theorem \\ref{rweight} implies the following:\n\\begin{theorem}\\label{th2} \nThe minimum cost solution of the following problems have value $O(1)$ w.h.p. The proofs are non-constructive.\n\\begin{enumerate}[(a)]\n\\item Traveling Salesperson Problem in $K_n$ with at least $n$ colors for the edges. \n\\item Minimum cost perfect matching in complete $k$-uniform hypergraphs, $k\\geq 2$ with at least $n/k$ colors for the edges.\n\\item Minimum cost classes of degree bounded spanning trees in $K_n$ with at least $n-1$ colors for the edges e.g. spanning binary trees.\n\\end{enumerate}\n\\end{theorem}\nThe rest of the paper is organized as follows. In Section \\ref{sec:treeUB} we prove the first assertion of Theorem \\ref{EucTrees}, i.e.\\ that $Z_{\\mathrm{MST}} = \\Theta(\\sqrt{n})$ w.h.p. In Section \\ref{sec:tours} we prove the first assertion of Theorem \\ref{EucTours}, i.e.\\ that $Z_{\\mathrm{TSP}}=O(\\sqrt {n}\\log n)$ for $n$ colors. In Section \\ref{sec:separate} we prove the remaining assertions in Theorems \\ref{EucTrees} and \\ref{EucTours}, i.e.\\ that $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt{n})$ for $n$ colors and that $Z_{\\mathrm{TSP}}-Z_{\\mathrm{TSP}}^*=\\Theta(\\sqrt{n})$ for $(1+\\varepsilon)n$ colors. In Section \\ref{sec:spread} we prove Theorem \\ref{th2}. Finally, we make some observations and state open questions in Section \\ref{sec:conclusion}.",
    "sketch": "The post-theorem discussion does not give a proof outline for Theorem~\\ref{EucTrees} beyond noting why it is \"much less obvious\" (the rainbow condition is a \"global constraint\" that \"seems to preclude a straightforward application of subadditivity\"). The only structural indication is the paper roadmap: in Section~\\ref{sec:treeUB} they \"prove the first assertion of Theorem~\\ref{EucTrees}, i.e.\\ that $Z_{\\mathrm{MST}}=\\Theta(\\sqrt n)$ w.h.p.,\" and in Section~\\ref{sec:separate} they \"prove the remaining assertions in Theorem~\\ref{EucTrees}\\ldots i.e.\\ that $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt n)$ for $n$ colors.\"",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{EucTrees}\n In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\\mathrm{MST}} = \\Theta(\\sqrt n)$ w.h.p.\\ and $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt{n})$ w.h.p.,",
    "theorem_type": [
      "Asymptotic or Limit",
      "Universal"
    ],
    "mcq": {
      "question": "Let ${\\mathcal X}=\\{\\mathbf x_1,\\dots,\\mathbf x_n\\}$ be points chosen independently and uniformly from the unit square $[0,1]^2$. Form the complete graph on ${\\mathcal X}$, give each edge its Euclidean length, and label the edges randomly using exactly $n-1$ colors. A spanning tree is called rainbow if its $n-1$ edges all have distinct colors. Let $Z_{\\mathrm{MST}}$ be the total Euclidean length of a minimum-length rainbow spanning tree, and let $Z^*_{\\mathrm{MST}}$ be the total Euclidean length of the ordinary minimum spanning tree without the rainbow requirement. Here “with high probability” means with probability tending to $1$ as $n\\to\\infty$. Which statement holds with high probability in this model?",
      "correct_choice": {
        "label": "A",
        "text": "Both $Z_{\\mathrm{MST}}=\\Theta(\\sqrt n)$ and $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt n)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "Both $Z_{\\mathrm{MST}}=\\Theta(\\sqrt n)$ and $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=o(\\sqrt n)$."
        },
        {
          "label": "C",
          "text": "$Z_{\\mathrm{MST}}=\\Theta(\\sqrt n)$."
        },
        {
          "label": "D",
          "text": "Both $Z_{\\mathrm{MST}}=\\Theta(\\sqrt n\\log n)$ and $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=\\Theta(\\sqrt n)$."
        },
        {
          "label": "E",
          "text": "There exists a constant $C>0$ such that $Z_{\\mathrm{MST}}\\le C\\sqrt n$ w.h.p., but $Z_{\\mathrm{MST}}-Z^*_{\\mathrm{MST}}=O(1)$ w.h.p."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "order_of_rainbow_penalty",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped_gap_asymptotic",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "extra_log_in_total_length",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "collapsed_gap_to_constant",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the model and quantities but does not explicitly reveal the asymptotic behavior. There are no clear verbal cues that point directly to choice A over the others."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: one option appears to restate a specific asymptotic result almost verbatim. It is not a pure tautology because the student must distinguish among nearby asymptotic alternatives, but it still mainly tests recognition of a known conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare orders such as Θ(√n), o(√n), Θ(√n log n), and O(1). However, the question mostly rewards prior knowledge of the result rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 1,
        "justification": "Choices B, D, and E are plausible asymptotic confusions, but choice C is a weaker statement that would also be true if A is true. That makes the option set less clean for a single-correct MCQ and lowers distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "Moderately good but theorem-recall-heavy. It avoids direct answer leakage, yet it is somewhat close to restating a known result, and the weaker-true option creates ambiguity in the distractor set."
    }
  },
  {
    "id": "2512.13445v1",
    "paper_link": "http://arxiv.org/abs/2512.13445v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "[{\\cite[Theorem~5.14]{Guterman2025}}]\\label{MainTheoremEvenIntro}Assume that $k \\ge 4, n \\ge k + 2, n + k$ is even and $|{\\mathbb F}| > k$. Let $T\\colon \\M_{n\\,k} ({\\mathbb F}) \\to \\M_{n\\,k} ({\\mathbb F})$ be a linear map. Then $\\det_{n\\,k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} ({\\mathbb F})$ if and only if there exist $A \\in \\M_{n\\,n}({\\mathbb F})$ and $B \\in \\M_{k\\,k}({\\mathbb F})$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and\n\\[\nT(X) = AXB\n\\]\nfor all $X \\in \\M_{n\\,k} ({\\mathbb F}).$",
      "start_pos": 32043,
      "end_pos": 32658,
      "label": "MainTheoremEvenIntro"
    },
    "ref_dict": {
      "cor:AdditionLinearMapNPLusKOdd": "\\begin{corollary}\\label{cor:AdditionLinearMapNPLusKOdd}Suppose that $n + k$ is odd. Let \n$\\phi \\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$\nbe a linear map. Denote by $S$ a linear map $\\M_{n}(\\F) \\to \\M_{n\\,k}(\\F)$ defined by\n\\begin{equation}\\label{eq:AddingKernel}\nS(X) = X + \\phi (X)\n\\end{equation}\nfor all $X \\in \\M_{n\\,k}(\\F).$ Then\n\\[\n\\det_{n\\, k} (S(X)) = \\det_{n\\,k}(X)\n\\]\nfor all $X \\in \\M_{n\\,k} (\\F).$\n\\end{corollary}",
      "lem:FromNKOddToNKEvenIfKerW": "\\begin{lemma}\\label{lem:FromNKOddToNKEvenIfKerW}Assume that  $|\\F| > k$, $n > k$ and $n + k$ is odd. Let $T\\colon \\M_{n\\, k} (\\F) \\to \\M_{n\\, k} (\\F)$ be a linear map such that $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ and $\\Img(T) = \\M^0_{n\\,k}(\\F)$. Then the following statements hold.\n\\begin{enumerate}[label=(\\alph*), ref=(\\alph*)]\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part4} $\\Ker(T) = \\rad(\\det_{n\\,k})$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part5} $L^{+}\\circ L^{-} \\circ T = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part6} $T \\circ L^{+} \\circ L^{-} = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part2} If a linear map $S$ is defined by \n\\begin{equation}\\label{lem:FromNKOddToNKEvenIfKerW:eq}\nS = L^{-} \\circ T \\circ L^{+},\n\\end{equation}\nthen $T = L^{+} \\circ S \\circ L^{-}$.\n\\end{enumerate}\n\\end{lemma}",
      "thm:MainTheoremCullisNKOdd": "\\begin{theorem}\\label{thm:MainTheoremCullisNKOdd}Assume that $|\\F| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is odd. Let $T\\colon \\M_{n\\,k} (\\F) \\to \\M_{n\\,k} (\\F)$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} (\\F)$ if and only if  there exist $A \\in \\M_{n\\, n}(\\F)$ and $B \\in \\M_{k\\, k}(\\F)$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eqq}\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\end{equation}\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and a linear map\\linebreak $\\phi\\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eq}\nT(X) = AXB + \\phi(X)\n\\end{equation}\nfor all $X \\in \\M_{n\\,k} (\\F).$\n\nHere $W_{n\\,k} \\subseteq \\M_{n\\,k}(\\F)$ is the space of matrices, all rows of which are equal, which is defined in Definition~\\ref{def:WNK} and  is a radical of $\\det_{n\\,k}$ by Lemma~\\ref{lem:NKOddRadDetNK}.\n\\end{theorem}",
      "lem:FromNKOddToNKEvenIfKerW:part6": "\\begin{enumerate}[label=(\\alph*), ref=(\\alph*)]\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part4} $\\Ker(T) = \\rad(\\det_{n\\,k})$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part5} $L^{+}\\circ L^{-} \\circ T = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part6} $T \\circ L^{+} \\circ L^{-} = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part2} If a linear map $S$ is defined by \n\\begin{equation}\\label{lem:FromNKOddToNKEvenIfKerW:eq}\nS = L^{-} \\circ T \\circ L^{+},\n\\end{equation}\nthen $T = L^{+} \\circ S \\circ L^{-}$.\n\\end{enumerate}",
      "def:radical": "\\begin{definition}[Cf.~{\\cite[text at the beginning of Section~1]{Waterhouse1983}}]\\label{def:radical}Let $\\F$ be a field, $V$ be a finite-dimensional vector space over $\\F$, and let $f$ be a function from $V$ to $\\F$. The \\emph{radical} of $f$, denoted by $\\rad (f)$ is a subset of $V$ defined by\n\\[\n\\rad (f) = \\{\\mathbf w \\mid f(\\mathbf v + \\lambda \\mathbf w) = f(\\mathbf v)\\;\\;\\mbox{for all}\\;\\; \\mathbf v \\in V,\\;\\; \\lambda \\in \\F\\}.\n\\]\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 1380,
    "pre_theorem_intro_text": "The theory of linear maps preserving matrix invariants is well-known direction of research in Linear Algebra connected with other fields of mathematics. It has a long history which dates back to the beginning of 20th century in the works of Frobenius~\\cite{GF} and continues at present time (see~\\cite{LAMA199233} for the survey of the results until the end of 20th century and~\\cite{hogben_handbook_2014} for a brief introduction).\n\nThe Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\\cite{cullis1913} and studied later by Radi\\'{c} in~\\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\\l{}kowski in~\\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\\cite{amiri2010}, and Nakagami and Yanay in~\\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\\cite{Guterman2025}. \nIn particular, \\cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\\det_{n\\,k}$ for the case when $ k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\\ref{sec:prelim}).",
    "context": "The theory of linear maps preserving matrix invariants is well-known direction of research in Linear Algebra connected with other fields of mathematics. It has a long history which dates back to the beginning of 20th century in the works of Frobenius~\\cite{GF} and continues at present time (see~\\cite{LAMA199233} for the survey of the results until the end of 20th century and~\\cite{hogben_handbook_2014} for a brief introduction).\n\nThe Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\\cite{cullis1913} and studied later by Radi\\'{c} in~\\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\\l{}kowski in~\\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\\cite{amiri2010}, and Nakagami and Yanay in~\\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\\cite{Guterman2025}. \nIn particular, \\cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\\det_{n\\,k}$ for the case when $ k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\\ref{sec:prelim}).",
    "full_context": "The theory of linear maps preserving matrix invariants is well-known direction of research in Linear Algebra connected with other fields of mathematics. It has a long history which dates back to the beginning of 20th century in the works of Frobenius~\\cite{GF} and continues at present time (see~\\cite{LAMA199233} for the survey of the results until the end of 20th century and~\\cite{hogben_handbook_2014} for a brief introduction).\n\nThe Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\\cite{cullis1913} and studied later by Radi\\'{c} in~\\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\\l{}kowski in~\\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\\cite{amiri2010}, and Nakagami and Yanay in~\\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\\cite{Guterman2025}. \nIn particular, \\cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\\det_{n\\,k}$ for the case when $ k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\\ref{sec:prelim}).\n\nThe Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\\cite{cullis1913} and studied later by Radi\\'{c} in~\\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\\l{}kowski in~\\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\\cite{amiri2010}, and Nakagami and Yanay in~\\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\\cite{Guterman2025}. \nIn particular, \\cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\\det_{n\\,k}$ for the case when $ k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\\ref{sec:prelim}).\n\nThe parity of $n + k$ is significant because if $n + k$ is odd, then there exist linear maps preserving the Cullis' determinant which do not have such form (Corollary~\\ref{cor:AdditionLinearMapNPLusKOdd}).\n\n\\begin{theorem}[Theorem~\\ref{thm:MainTheoremCullisNKOdd}]Assume that $|\\F| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is odd. Let $T\\colon \\M_{n\\,k} (\\F) \\to \\M_{n\\,k} (\\F)$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} (\\F)$ if and only if there exist $A \\in \\M_{n\\, n}(\\F)$ and $B \\in \\M_{k\\, k}(\\F)$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and a linear map\\linebreak $\\phi\\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and\n\\begin{equation*}\nT(X) = AXB + \\phi(X)\n\\end{equation*}\nfor all $X \\in \\M_{n\\,k} (\\F).$ Here $W_{n\\,k} \\subseteq \\M_{n\\,k}(\\F)$ denotes the space of matrices, all rows of which are equal, being a radical of $\\det_{n\\,k}$.\n\\end{theorem}\n\n\\begin{lemma}[{\\cite[Lemma~5.13]{Guterman2025}}]\\label{lem:CullisDeg1Rank1}Assume that $|\\F| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even. Let $T\\colon \\M_{n\\, k} (\\F) \\to \\M_{n\\, k} (\\F)$ be a linear map such that $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\, k} (\\F).$ Then $\\rk (X) = 1$ implies $\\rk (T(X)) = 1.$\n\\end{lemma}\n\nWe omit the subscripts for $\\sgn_{[n]}$ and $\\sgn_{n\\,k}$ if this cannot lead to a misunderstanding.\\begin{definition}[\\cite{NAKAGAMI2007422}, Theorem 13]\\label{def:CullisDet} Let $n \\ge k$, $X \\in \\M_{n\\,k} (\\mathbb F)$. Then Cullis' determinant $\\det_{n\\, k}(X)$ of $X$ is defined to be the function:\n$$\n\\det_{n\\, k} (X) = \\sum_{\\sigma \\in \\mathcal C_{k}^{[n]}} \\sgn_{n\\,k} (\\sigma) x_{\\sigma(1)\\, 1}x_{\\sigma(2)\\, 2}\\ldots x_{\\sigma(k)\\, k}.\n$$\nWe also denote $\\det_{n\\, k} (X)$ as follows\n\\[\n\\det_{n\\,k}(X)=\\begin{vmatrix}x_{1\\,1} & \\cdots & x_{1\\,k}\\\\\n\\vdots & \\cdots & \\vdots\\\\\nx_{n\\,1} & \\cdots & x_{n\\,k}\n\\end{vmatrix}_{n\\,k}.\n\\]\nIf $n = k$, then we also write $\\det_{k}$ or $\\det$ instead of $\\det_{n\\,k}$ because in this case $\\det_{n\\,k}$ is clearly equal to an ordinary determinant of a square matrix.\n\\end{definition}\n\n\\begin{lemma}[{\\cite[Lemma~3.1]{Guterman2025}}]\\label{lem:TwoSidedMulPreservesDet}Let $n \\ge k$, $A \\in \\M_{n\\,n}(\\F)$, $B \\in \\M_{k\\,k}(\\F)$, and let $T\\colon \\M_{n\\,k}(\\F) \\to \\M_{n\\,k}(\\F)$ be a linear map defined by\n\\begin{equation}\\label{eq:TwoSideMul}\nT(X) = AXB\n\\end{equation}\nfor all $X \\in \\M_{n\\,k}(\\F).$ Then\n\\[\n\\det_{n\\,k} (T(X)) = \\det_{n\\,k}(X)\n\\]\nfor all $X \\in \\M_{n\\,k} (\\F)$ if and only if\n\\begin{equation}\\label{eq:TwoSideMul2}\n\\det_{n\\,k} \\Bigl(A(|d]\\Bigr)\\cdot \\det_k \\Bigl(B\\Bigr) = \\sgn (d)\n\\end{equation}\nfor all $d \\in \\binom{[n]}{k}.$\n\\end{lemma}\n\n\\begin{lemma}[{\\cite[Theorem~5.14]{Guterman2025}}]\\label{lem:MainTheoremEven}Assume that $|\\F| > k \\ge 4, n \\ge k + 2$ and $n + k$ is even. Let $T\\colon \\M_{n\\, k} (\\F) \\to \\M_{n\\, k} (\\F)$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\, k} (\\F)$ if and only if there exist $A \\in \\M_{n\\, n}(\\F)$ and $B \\in \\M_{k\\, k}(\\F)$ such that\n\\begin{equation}\\label{thm:MainTheoremEvenKGe4:eq}\n\\det_{n\\, k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\cdot \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\end{equation}\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and\n\\[\nT(X) = AXB\n\\]\nfor all $X \\in \\M_{n\\, k} (\\F).$\n\\end{lemma}\n\n\\begin{theorem}\\label{thm:MainTheoremCullisNKOdd}Assume that $|\\F| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is odd. Let $T\\colon \\M_{n\\,k} (\\F) \\to \\M_{n\\,k} (\\F)$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} (\\F)$ if and only if there exist $A \\in \\M_{n\\, n}(\\F)$ and $B \\in \\M_{k\\, k}(\\F)$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eqq}\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\end{equation}\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and a linear map\\linebreak $\\phi\\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eq}\nT(X) = AXB + \\phi(X)\n\\end{equation}\nfor all $X \\in \\M_{n\\,k} (\\F).$\n\n\\begin{corollary}\\label{cor:AdditionLinearMapNPLusKOdd}Suppose that $n + k$ is odd. Let \n$\\phi \\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$\nbe a linear map. Denote by $S$ a linear map $\\M_{n}(\\F) \\to \\M_{n\\,k}(\\F)$ defined by\n\\begin{equation}\\label{eq:AddingKernel}\nS(X) = X + \\phi (X)\n\\end{equation}\nfor all $X \\in \\M_{n\\,k}(\\F).$ Then\n\\[\n\\det_{n\\, k} (S(X)) = \\det_{n\\,k}(X)\n\\]\nfor all $X \\in \\M_{n\\,k} (\\F).$\n\\end{corollary}\n\n\\begin{theorem}\\label{thm:MainTheoremCullisNKOdd}Assume that $|\\F| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is odd. Let $T\\colon \\M_{n\\,k} (\\F) \\to \\M_{n\\,k} (\\F)$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} (\\F)$ if and only if  there exist $A \\in \\M_{n\\, n}(\\F)$ and $B \\in \\M_{k\\, k}(\\F)$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eqq}\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\end{equation}\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and a linear map\\linebreak $\\phi\\colon \\M_{n\\,k}(\\F) \\to W_{n\\,k}$ such that\n\\begin{equation}\\label{thm:MainTheoremCullisNKOdd:eq}\nT(X) = AXB + \\phi(X)\n\\end{equation}\nfor all $X \\in \\M_{n\\,k} (\\F).$\n\nHere $W_{n\\,k} \\subseteq \\M_{n\\,k}(\\F)$ is the space of matrices, all rows of which are equal, which is defined in Definition~\\ref{def:WNK} and  is a radical of $\\det_{n\\,k}$ by Lemma~\\ref{lem:NKOddRadDetNK}.\n\\end{theorem}",
    "post_theorem_intro_text_len": 4421,
    "post_theorem_intro_text": "The parity of $n + k$ is significant because if $n + k$ is odd, then there exist linear maps preserving the Cullis' determinant which do not have such form (Corollary~\\ref{cor:AdditionLinearMapNPLusKOdd}). \n\nHowever, in this paper we show that the case $(n,k)$ with $n \\ge k$ and $n + k$ odd reduces to the case $(n-1,k)$ (Lemma~\\ref{lem:FromNKOddToNKEvenIfKerW}\\ref{lem:FromNKOddToNKEvenIfKerW:part6}). Using this reduction we prove the main theorem of this paper providing the description of linear maps preserving $\\det_{n\\,k}$ if $n + k$ is odd.\n\n\\begin{theorem}[Theorem~\\ref{thm:MainTheoremCullisNKOdd}]Assume that $|{\\mathbb F}| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is odd. Let $T\\colon \\M_{n\\,k} ({\\mathbb F}) \\to \\M_{n\\,k} ({\\mathbb F})$ be a linear map. Then $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} ({\\mathbb F})$ if and only if  there exist $A \\in \\M_{n\\, n}({\\mathbb F})$ and $B \\in \\M_{k\\, k}({\\mathbb F})$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and a linear map\\linebreak $\\phi\\colon \\M_{n\\,k}({\\mathbb F}) \\to W_{n\\,k}$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and\n\\begin{equation*}\nT(X) = AXB + \\phi(X)\n\\end{equation*}\nfor all $X \\in \\M_{n\\,k} ({\\mathbb F}).$ Here $W_{n\\,k} \\subseteq \\M_{n\\,k}({\\mathbb F})$ denotes the space of matrices, all rows of which are equal, being a radical of $\\det_{n\\,k}$.\n\\end{theorem}\n\nOur proof of the main theorem relies on the notion of the radical of a function which was introduced and studied by Waterhouse~\\cite{Waterhouse1983} (Definition~\\ref{def:radical} below). The reduction of the case $(n,k)$ with $k \\ge 4, n \\ge k + 2$ and $n + k$ odd to the case $(n-1,k)$ with $k \\ge 4, n \\ge k + 2$ and $(n-1) + k$ even is done by finding the radical of $\\det_{n\\,k}$ explicitly and using its properties.\n\nThus, the results of the current paper and its predecessor~\\cite{Guterman2025} provide the solution to linear preserver problem for the Cullis' determinant if $k \\ge 4, n \\ge k + 2$ and the ground field is large enough. The cases $k = 1$ and $k = 2$ are considered in~\\cite{Guterman2025} as well. The case $n = k + 1$ is considered in~\\cite{Guterman2024}, where the first attempt to find the description of linear maps preserving the Cullis' determinant is made. The remaining case $k = 3$ need a special approach because the proof for the case $k \\ge 4$ relies on the following lemma and will be discussed separately. We remark that in this paper we apply the results obtained in~\\cite{Guterman2025} and we do not use any results from~\\cite{Guterman2024}.\n\n\\begin{lemma}[{\\cite[Lemma~5.13]{Guterman2025}}]\\label{lem:CullisDeg1Rank1}Assume that $|{\\mathbb F}| > k \\ge 4$, $n \\ge k + 2$ and $n + k$ is even. Let  $T\\colon \\M_{n\\, k} ({\\mathbb F}) \\to \\M_{n\\, k} ({\\mathbb F})$ be a linear map such that  $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\, k} ({\\mathbb F}).$ Then $\\rk (X) = 1$ implies $\\rk (T(X)) = 1.$\n\\end{lemma}\n\nThe example below shows that this lemma does not hold for $k = 3$.\n\n\\begin{example}[{\\cite[Example~5.11]{Guterman2025}}]\\label{cex:K3NKEvenInt}Let $B = \\begin{psmallmatrix}\n                                 1 & 0 & 0 & 0 & -1 & 0 & \\cdots & 0\\\\\n                                 0 & 1 & 0 & -1 & 0 & 0 & \\cdots & 0\\\\\n                                 0 & 0 & 0 & 0 & 0 & 0 & \\cdots & 0\n                                        \\end{psmallmatrix}^t \\in \\M_{n\\,3}({\\mathbb F}).$ Then $\\rk (B) = 2$ and $\\deg_{\\lambda} (\\det_{n\\,k}(A + \\lambda B)) \\le 1$ for all $A \\in \\M_{n\\,3}({\\mathbb F})$.\n\\end{example}\n\nNevertheless, if $k = 3$, then linear maps preserving the Cullis determinant admit the same description as in the case when $k \\ge 4$ and will be studied separately.\\bigskip\n\nThe paper is organized as follows: in Section~\\ref{sec:prelim} we provide the basic facts regarding $\\det_{n\\,k}$; in Section~\\ref{sec:kernels} we find the radical of $\\det_{n\\,k}$ if $n + k$ is odd; in Section~\\ref{sec:maintheorem} we reduce the linear preserver problem for $\\det_{n\\,k}$ and $n + k$ is odd to the linear preserver problem for $\\det_{(n-1)\\,k}$ and prove the main characterization theorem for $n + k$ odd.",
    "sketch": "The introduction explains that “the parity of $n+k$ is significant” since when $n+k$ is odd “there exist linear maps preserving the Cullis' determinant which do not have such form.” It then outlines a reduction argument: “the case $(n,k)$ with $n\\ge k$ and $n+k$ odd reduces to the case $(n-1,k)$” (via Lemma~\\ref{lem:FromNKOddToNKEvenIfKerW}…), and “using this reduction we prove the main theorem … for $n+k$ … odd.”\n\nIt further specifies what drives this reduction: “Our proof of the main theorem relies on the notion of the radical of a function … (Definition~\\ref{def:radical} below). The reduction … is done by finding the radical of $\\det_{n\\,k}$ explicitly and using its properties.” In particular, for $n+k$ odd one identifies the radical space $W_{n\\,k}$ (“the space of matrices, all rows of which are equal, being a radical of $\\det_{n\\,k}$”), which leads to the possibility of an additive radical term $\\phi(X)$ in the odd-parity characterization.",
    "expanded_sketch": "The introduction explains that “the parity of $n+k$ is significant” since when $n+k$ is odd “there exist linear maps preserving the Cullis' determinant which do not have such form.” It then outlines a reduction argument: “the case $(n,k)$ with $n\\ge k$ and $n+k$ odd reduces to the case $(n-1,k)$” (via the following lemma).\n\n\\begin{lemma}\\label{lem:FromNKOddToNKEvenIfKerW}Assume that  $|\\F| > k$, $n > k$ and $n + k$ is odd. Let $T\\colon \\M_{n\\, k} (\\F) \\to \\M_{n\\, k} (\\F)$ be a linear map such that $\\det_{n\\, k} (T(X)) = \\det_{n\\,k}(X)$ and $\\Img(T) = \\M^0_{n\\,k}(\\F)$. Then the following statements hold.\n\\begin{enumerate}[label=(\\alph*), ref=(\\alph*)]\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part4} $\\Ker(T) = \\rad(\\det_{n\\,k})$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part5} $L^{+}\\circ L^{-} \\circ T = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part6} $T \\circ L^{+} \\circ L^{-} = T$.\n\\item\\label{lem:FromNKOddToNKEvenIfKerW:part2} If a linear map $S$ is defined by \n\\begin{equation}\\label{lem:FromNKOddToNKEvenIfKerW:eq}\nS = L^{-} \\circ T \\circ L^{+},\n\\end{equation}\nthen $T = L^{+} \\circ S \\circ L^{-}$.\n\\end{enumerate}\n\\end{lemma}\n\nUsing this reduction we prove the main theorem for $n+k$ odd.\n\nIt further specifies what drives this reduction: “Our proof of the main theorem relies on the notion of the radical of a function …” where the radical is defined as follows.\n\n\\begin{definition}[Cf.~{\\cite[text at the beginning of Section~1]{Waterhouse1983}}]\\label{def:radical}Let $\\F$ be a field, $V$ be a finite-dimensional vector space over $\\F$, and let $f$ be a function from $V$ to $\\F$. The \\emph{radical} of $f$, denoted by $\\rad (f)$ is a subset of $V$ defined by\n\\[\n\\rad (f) = \\{\\mathbf w \\mid f(\\mathbf v + \\lambda \\mathbf w) = f(\\mathbf v)\\;\\;\\mbox{for all}\\;\\; \\mathbf v \\in V,\\;\\; \\lambda \\in \\F\\}.\n\\]\n\\end{definition}\n\nThe reduction is done by finding the radical of $\\det_{n\\,k}$ explicitly and using its properties. In particular, for $n+k$ odd one identifies the radical space $W_{n\\,k}$ (“the space of matrices, all rows of which are equal, being a radical of $\\det_{n\\,k}$”), which leads to the possibility of an additive radical term $\\phi(X)$ in the odd-parity characterization.",
    "expanded_theorem": "[{Guterman2025, Theorem~5.14} ]\\label{MainTheoremEvenIntro}Assume that $k \\ge 4, n \\ge k + 2, n + k$ is even and $|{\\mathbb F}| > k$. Let $T\\colon \\M_{n\\,k} ({\\mathbb F}) \\to \\M_{n\\,k} ({\\mathbb F})$ be a linear map. Then $\\det_{n\\,k} (T(X)) = \\det_{n\\,k}(X)$ for all $X \\in \\M_{n\\,k} ({\\mathbb F})$ if and only if there exist $A \\in \\M_{n\\,n}({\\mathbb F})$ and $B \\in \\M_{k\\,k}({\\mathbb F})$ such that\n\\[\n\\det_{n\\,k} \\Bigl(A(|i_1,\\ldots, i_k]\\Bigr) \\det_k \\Bigl(B\\Bigr) = (-1)^{i_1 + \\ldots + i_k - 1 - \\ldots - k}\n\\]\nfor all increasing sequences $1 \\le i_1 < \\ldots < i_k \\le n $ and\n\\[\nT(X) = AXB\n\\]\nfor all $X \\in \\M_{n\\,k} ({\\mathbb F}).$",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Let \\(\\M_{n,k}(\\mathbb F)\\) be the space of \\(n\\times k\\) matrices over a field \\(\\mathbb F\\), and let \\(\\det_{n,k}\\) denote the Cullis determinant on \\(\\M_{n,k}(\\mathbb F)\\), a determinant-like invariant for rectangular matrices that agrees with the usual determinant when \\(n=k\\). Assume \\(k\\ge 4\\), \\(n\\ge k+2\\), \\(n+k\\) is even, and \\(|\\mathbb F|>k\\). Let \\(T\\colon \\M_{n,k}(\\mathbb F)\\to \\M_{n,k}(\\mathbb F)\\) be linear. Which statement characterizes exactly when \\(T\\) preserves the Cullis determinant, that is, when \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for every \\(X\\in \\M_{n,k}(\\mathbb F)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The equality \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for all \\(X\\in \\M_{n,k}(\\mathbb F)\\) holds if and only if there exist \\(A\\in \\M_{n,n}(\\mathbb F)\\) and \\(B\\in \\M_{k,k}(\\mathbb F)\\) such that, for every increasing sequence \\(1\\le i_1<\\cdots<i_k\\le n\\),\n\\[\n\\det_{n,k}\\bigl(A(|i_1,\\ldots,i_k]\\bigr)\\,\\det_k(B)=(-1)^{i_1+\\cdots+i_k-(1+\\cdots+k)},\n\\]\nwhere \\(A(|i_1,\\ldots,i_k]\\) is the \\(n\\times k\\) submatrix of \\(A\\) formed by columns \\(i_1,\\ldots,i_k\\), and\n\\[\nT(X)=AXB\n\\]\nfor all \\(X\\in \\M_{n,k}(\\mathbb F)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The equality \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for all \\(X\\in \\M_{n,k}(\\mathbb F)\\) holds if and only if there exist \\(A\\in \\M_{n,n}(\\mathbb F)\\), \\(B\\in \\M_{k,k}(\\mathbb F)\\), and a linear map \\(\\phi\\colon \\M_{n,k}(\\mathbb F)\\to W_{n,k}\\), where \\(W_{n,k}\\subseteq \\M_{n,k}(\\mathbb F)\\) is the subspace of matrices whose rows are all equal, such that for every increasing sequence \\(1\\le i_1<\\cdots<i_k\\le n\\),\n\\[\n\\det_{n,k}\\bigl(A(|i_1,\\ldots,i_k]\\bigr)\\,\\det_k(B)=(-1)^{i_1+\\cdots+i_k-(1+\\cdots+k)},\n\\]\nand\n\\[\nT(X)=AXB+\\phi(X)\n\\]\nfor all \\(X\\in \\M_{n,k}(\\mathbb F)\\)."
        },
        {
          "label": "C",
          "text": "If there exist \\(A\\in \\M_{n,n}(\\mathbb F)\\) and \\(B\\in \\M_{k,k}(\\mathbb F)\\) such that, for every increasing sequence \\(1\\le i_1<\\cdots<i_k\\le n\\),\n\\[\n\\det_{n,k}\\bigl(A(|i_1,\\ldots,i_k]\\bigr)\\,\\det_k(B)=(-1)^{i_1+\\cdots+i_k-(1+\\cdots+k)},\n\\]\nand\n\\[\nT(X)=AXB\n\\]\nfor all \\(X\\in \\M_{n,k}(\\mathbb F)\\), then \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for all \\(X\\in \\M_{n,k}(\\mathbb F)\\)."
        },
        {
          "label": "D",
          "text": "The equality \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for all \\(X\\in \\M_{n,k}(\\mathbb F)\\) holds if and only if there exist \\(A\\in \\M_{n,n}(\\mathbb F)\\) and \\(B\\in \\M_{k,k}(\\mathbb F)\\) such that, for some increasing sequence \\(1\\le i_1<\\cdots<i_k\\le n\\),\n\\[\n\\det_{n,k}\\bigl(A(|i_1,\\ldots,i_k]\\bigr)\\,\\det_k(B)=(-1)^{i_1+\\cdots+i_k-(1+\\cdots+k)},\n\\]\nand\n\\[\nT(X)=AXB\n\\]\nfor all \\(X\\in \\M_{n,k}(\\mathbb F)\\)."
        },
        {
          "label": "E",
          "text": "The equality \\(\\det_{n,k}(T(X))=\\det_{n,k}(X)\\) for all \\(X\\in \\M_{n,k}(\\mathbb F)\\) holds if and only if there exist invertible matrices \\(A\\in \\M_{n,n}(\\mathbb F)\\) and \\(B\\in \\M_{k,k}(\\mathbb F)\\) such that\n\\[\n\\det(A)\\det(B)=1\n\\]\nand\n\\[\nT(X)=AXB\n\\]\nfor all \\(X\\in \\M_{n,k}(\\mathbb F)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "parity-dependent radical term",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the converse direction of the characterization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "universality over all increasing index sequences",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "specific minor-sign constraint replaced by ordinary determinant condition",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct characterization; it only asks for the exact preservation criterion. Although the wording signals that an 'if and only if' theorem is sought, it does not itself disclose the content of the answer."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the correct option is the full preservation characterization itself. The task mainly asks the student to recognize the exact statement of the theorem rather than derive or apply it."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact characterization from a weaker one-way implication, a quantifier-weakened variant, and a superficially similar determinant condition. However, the item still primarily tests recognition/recall of the theorem rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and target plausible failure modes: adding an extra radical-like term, dropping the converse direction, weakening a universal quantifier to an existential one, and confusing the rectangular invariant with the ordinary determinant-preserver condition."
      },
      "total_score": 5,
      "overall_assessment": "A technically solid but theorem-recall-heavy MCQ. It avoids direct answer leakage and has strong distractors, but it is largely tautological and only mildly pressures genuine reasoning."
    }
  },
  {
    "id": "2512.13450v1",
    "paper_link": "http://arxiv.org/abs/2512.13450v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "Theoremstar",
      "content": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$",
      "start_pos": 45297,
      "end_pos": 45623,
      "label": null
    },
    "ref_dict": {
      "fig:sigma2": "\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}",
      "sigma2": "\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}",
      "defsigma2": "\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}",
      "genre2": "\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\n\\begin{proof}\nBy a finite Fourier transform over $\\Z/2p\\Z$, we can write for $m\\in\\Z$ :\n\\begin{equation}\\label{FFT}\n  (-1)^{\\lfloor m/p\\rfloor}=\n  \\frac 1 p\n  \\sum_{n=1,\\text{ odd}}^{2p-1}\\frac{\\sin(\\pi n(2m+1)/2p)}{\\sin(\\pi n/2p)}\n  =\\frac 1 p  \\sum_{n=1,\\text{ odd}}^{2p-1} \\frac {t^{2m+1}-t^{-2m-1}}{t-t^{-1}}\n    \\Big|_{t=e^{\\frac {i\\pi n}{2p}}~.\n      }\n\\end{equation}\nIn the sequel, we set $\\O_p=\\{1,3,\\ldots,2p-1\\}$ and $\\O_p^+=\\{1,3,\\ldots,p\\}$. Using Formula \\eqref{FFT} for each sign $\\epsilon_{j+1}=(-1)^{\\lfloor q(j+1)/p\\rfloor}$ and plugging it into the formula of Lemma \\ref{triplepsilon}, we get \n\\begin{equation}\\nonumber\\sigma_2({\\textstyle{\\frac q p}})\\,\\,\n  =\\sum_{(j,k,\\ell)\\in T_p} \\epsilon_{j+1}\\epsilon_{k+1}\\epsilon_{\\ell+1}\n  =\\frac1 {p^3} \\sum_{(a,b,c)\\in \\O_p^3}\\psi(a,b,c)\\end{equation} \nwhere\n\\begin{equation}\n\\psi(a,b,c)=\\!\\!\\!\\!\\sum\\limits_{(j,k,l)\\in T_p}\\!\\!\\!\\!\n\\frac{(x^{2q(j+1)+1}-x^{-2q(j+1)-1})(y^{2q(k+1)+1}-y^{-2q(k+1)-1})(z^{2q(\\ell+1)+1}-z^{-2q(\\ell+1)-1})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\nonumber\n\\end{equation}\nwith $x=e^{\\frac{i\\pi a}{2p}}$,  $y=e^{\\frac{i\\pi b}{2p}}$ and $z=e^{\\frac{i\\pi c}{2p}}$.\n\nThe function $\\psi$ is clearly symmetric in its three variables and satisfies also $\\psi(2p-a,b,c)=\\psi(a,b,c)$. We can hence rewrite $\\sigma_2(\\frac q p)$  as \n\\begin{equation}\\label{eq4n}\\sigma_2({\\textstyle{\\frac q p}})=\n  \\frac1 {p^3}\n  \\sum_{a,b,c\\in \\O_p^+} 2^{m_a+m_b+m_c}\\psi(a,b,c)\\end{equation}\nwhere $m_a=1$ if $a\\ne p$ and $m_p=0$.\n\nIn order to compute $\\psi(a,b,c)$, we rewrite it as follows:\n\\begin{equation} \\psi(a,b,c)= \\sum\\limits_{\\epsilon,\\mu,\\nu=\\pm 1}\\epsilon\\mu\\nu \\frac{ x^{\\epsilon}y^{\\mu}z^{\\nu}F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\label{eq5n}\n\\end{equation} where $F$ is the polynomial defined by $F(X,Y,Z)=\\sum_{(j,k,l)\\in T_p}X^{j+1}Y^{k+1}Z^{\\ell +1}$.\n\nThe computation of $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ is simplified by the following observation:  we have $$F(X,Y,Z)=X^pY^p F(X^{-1}, Y^{-1},Z)~,$$ since the involution $(j,k,\\ell ) \\mapsto (p-2-j,p-2-k,\\ell)$ preserves the set $T_p$. Therefore, since $F$ is symmetric in its three variables,  whenever $F(X,Y,Z)$ is evaluated at numbers $X=x^{2q}$, $Y=y^{2q}$, and $Z=z^{2q}$\nsatisfying $X^p=Y^p=Z^p=-1$, then for  any $\\epsilon,\\mu,\\nu=\\pm 1$, we have that $F(X^\\epsilon, Y^\\mu,Z^\\nu)$ is equal to either $F(X,Y,Z)$ (when $\\epsilon\\mu\\nu =1$) or to  $F(X^{-1}, Y^{-1},Z^{-1})= \\overline {F(X,Y,Z)}$ (when $\\epsilon\\mu\\nu =-1$).\n\\begin{Remark}\\label{rk33}{\\em For later use, we note the following special case: when $F(x^{2q},y^{2q},z^{2q})$ happens to be real, then $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ \n    does not depend at all on $\\epsilon,\\mu$, or $\\nu$, and formula \\eqref{eq5n} for $\\psi(a,b,c)$ simply becomes $\\psi(a,b,c)=F(x^{2q},y^{2q},z^{2q})$.}\n    \\end{Remark}"
    },
    "pre_theorem_intro_text_len": 2862,
    "pre_theorem_intro_text": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups  and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by  \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$. \n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$.  The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten  \\cite{Witten} to show that for $g>1$,  \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for  $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following",
    "context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following",
    "full_context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following\n\nOur main result is the following\n\nFigure \\ref{fig:sigma2} suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$.\n\nWe briefly review the properties of the convergents $q_k/p_k$ of $\\theta$ that we will need. See {\\em e.g.} Perron \\cite{P} or Khinchin \\cite{Kh1} for more details\\footnote{Unfortunately, \\cite{Kh1} writes $p_k$ for our $q_k$ and {\\em vice versa.}}. Each irrational $\\theta\\in {\\mathbb R}$ has a unique regular continued fraction expansion $$\\theta = [a_0; a_1, a_2, \\ldots ] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{a_2+\\dots}} $$ where regular means that $a_0\\in \\Z$ and $a_i\\in \\Z_{\\geq 1}$ for $i\\geq 1$. Note that in our case $a_0=0$ since\n$\\theta \\in [0,1]\\setminus \\Q$.\nThe convergents\n$$\\frac{q_k}{p_k} = [a_0; a_1, a_2, \\ldots, a_k] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{\\dots+ \\frac{1}{a_k}}} $$ can also be defined recursively by\n\\begin{eqnarray} q_k&=a_k q_{k-1} + q_{k-2}, \\ \\ \\ \\ q_0=a_0, \\ \\ \\ \\ q_{-1}=1 \\label{rec5}\\\\\n p_k&=a_k p_{k-1} + p_{k-2}, \\ \\ \\ \\ p_0=1, \\ \\ \\ \\ p_{-1}=0 \\label{rec6}\n\\end{eqnarray}\nThe following classical facts are well-known. \n\\begin{Proposition}\\label{qpfacts}\n For all irrational $\\theta$, the convergents $q_k/p_k$ of $\\theta$ satisfy the following properties:\n \\begin{enumerate} \n \\item[(i)] One has $1=p_0\\leq p_1<p_2<\\ldots$, {\\em i.e.} the denominators $p_k$ are positive integers and moreover strictly increasing from $p_1$ onwards.\n \\item[(ii)] For all $k\\geq 1$, $q_k$ and $p_k$ are coprime.\n \\item[(iii)] $\\lim_{k\\rightarrow\\infty} q_k/p_k = \\theta$.\n \\item[(iv)]\n For all $i\\geq 1$, we have\n \\begin{equation}\\nonumber \n \\frac 1 {p_i(p_i+p_{i+1})} <\n \\Big|\\theta - \\frac {q_i}{p_i} \\Big| < \\frac 1 {p_ip_{i+1}} < \\frac 1 {p_i^2}~.\n \\end{equation}\n \\item[(v)] {\\em [Lagrange]}\\footnote{For a proof, see Perron \\cite[\\S 15, Satz 2.17]{P}.} For all $1\\leq i<k$, and any $n,\\ell \\in\\Z$ with $1\\leq n<p_{i+1}$, we have\n \\begin{equation}\\nonumber \n \\Big| n\\frac {q_k}{p_k} -\\ell \\Big|\n \\geq \\Big|p_i\\frac {q_k}{p_k} - q_i\\Big| \\geq \\frac{1}{p_i + p_{i+1}} ~.\n \\end{equation}\n \\end{enumerate}\n \\end{Proposition}\n\n\\begin{proof}[\\bf{Proof of Theorem~\\ref{asympto}}] We start from the trigonometric formula for $\\sigma_2(\\frac{q}{p})$ given in Theorem~\\ref{sigma2}. The first observation is that the term $(1-p^2)/(6p^2)$ in this formula stays bounded as $p\\rightarrow \\infty$ and so does not contribute to the limit of $\\sigma_2(\\frac{q_k}{p_k})/ p_k^2$. Thus it suffices to show\n \\begin{equation}\\label{eq4}\n \\lim_{k\\rightarrow \\infty} \\frac{1}{4p_k^4} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{f(n,p_k,q_k)}{\\sin^3({n\\pi}/{2p_k})\\sin^2({q_kn\\pi}/{p_k})} =\\frac{16}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}\n \\end{equation} with $f(n,p,q)$ given as in Theorem~\\ref{sigma2}.\n\nLet $S^{(1)}(q_k,p_k,\\theta)$ denote the sum $S^{(1)}(q_k,p_k)$ but with $\\sin(n\\pi\\theta)$ in the denominator in place of $\\sin({q_kn\\pi}/{p_k})$. Using (\\ref{sin-ineq}), we see that the $n$-th summand of $S^{(1)}(q_k,p_k,\\theta)$ is bounded from above by $1/(n^3\\sin(n\\pi\\theta))$. So we can apply the Lebesgue dominated convergence theorem to this sum. We get\n $$\\lim_{k\\rightarrow \\infty}S^{(1)}(q_k,p_k,\\theta) =\\frac{8}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}~.$$\n Thus to prove Lemma~\\ref{3.3} it suffices to show that the difference $d_k=S^{(1)}(q_k,p_k) -S^{(1)}(q_k,p_k,\\theta)$ goes to zero as $k\\rightarrow \\infty$. We have \n $$d_k= \\frac{1}{p_k^3} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{\\cos (n\\pi/2p_k)}{\\sin^3({n\\pi}/{2p_k}) }\\Biggl (\\frac {1} {\\sin({q_kn\\pi}/{p_k})} - \\frac {1} {\\sin(n\\pi\\theta)}\\Biggr)~.$$ We can bound the denominators from below using (\\ref{sin-ineq}), (\\ref{sinn-inequ}), and the trivial lower bound $|\\sin({q_kn\\pi}/{p_k})| \\geq \\sin (\\pi/p_k) \\geq 2/p_k$. Thus for all $\\epsilon >0$ we have\n\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.4}}]\n We assume that $\\mu(\\theta)=2$ and that $\\theta$ satisfies the conditions\n in Theorem~\\ref{ThKL}.\n Recall $S^{(2)}(q_k,p_k)$, defined in (\\ref{sum2}), is a sum from $n=1$ to $n=p_k-2$. We cut\nit into two parts, as follows. Put\n\\begin{equation}\\label{floor} N(k)= \\lfloor 3k/4 \\rfloor\n\\end{equation} and write $$S^{(2)}(q_k,p_k) = S^{(2.1)}(q_k,p_k)\n+S^{(2.2)}(q_k,p_k)~,$$ where $S^{(2.1)}(q_k,p_k)$ is the sum from\n$n=1$ to $n=p_{N(k)+1}-1$ and $S^{(2.2)}(q_k,p_k)$ is the sum from\n$n=p_{N(k)+1}$ to $p_k-2$. The following two lemmas provide upper bounds for these two sums.\n\\begin{Lemma}\\label{3.9} We have $$|S^{(2.1)}(q_k,p_k) |\\leq \\frac\n{p_{N(k)}} {p_k} F_7(k)$$ where $F_7(k) ={\\mathcal O}(k^7)$.\n\\end{Lemma}\n\\begin{Lemma}\\label{3.10} We have $$|S^{(2.2)}(q_k,p_k)| \\leq \\frac{1}{2} \\zeta(2)\n\\frac {p_{k}} {p_{N(k)+1}^2}~.$$\n\\end{Lemma}\nAssuming these two lemmas for the moment, since $\\lim_{m\\rightarrow\\infty} p_m^{1/m}= L>1$ by the Khinchin-L{\\'e}vy Theorem~\\ref{ThKL}(ii), $p_k$ grows like $L^k$ as $k\\rightarrow \\infty$. It is not hard to deduce from this that both $ S^{(2.1)}(q_k,p_k)$ and $ S^{(2.2)}(q_k,p_k)$ converge to zero.~\\footnote{Here is how to see this for $ S^{(2.1)}(q_k,p_k)$. Theorem~\\ref{ThKL}(ii) gives that for all $\\epsilon>0$ there exist $m_0(\\epsilon)$ so that $L^{1-\\epsilon} \\leq p_m^{1/m} \\leq L^{1+\\epsilon}$ holds for for all $m\\geq m_0(\\epsilon)$. Thus also $L^{m-m\\epsilon} \\leq p_m\\leq L^{m+m\\epsilon}$ for all $m\\geq m_0(\\epsilon)$. Applying now the first inequality with $m=k$ and the second one with $m=N(k)=\\lfloor 3k/4\\rfloor$ and taking $\\epsilon$ small enough, it follows that $p_{N(k)}/p_k \\leq L^{-\\delta k}$ for some $\\delta>0$ as soon as $k\\geq 2 m_0(\\epsilon)$. Thus $p_{N(k)} F_7(k)/ {p_k} \\leq k^7 L^{-\\delta k}$ for $k$ big enough, proving that the limit as $k\\rightarrow\\infty$ is zero, as asserted. }\nThis proves Lemma~\\ref{3.4}. \n\\end{proof}\nIt remains to prove Lemmas~\\ref{3.9} and~\\ref{3.10}. Let us first deal\nwith Lemma~\\ref{3.10} whose proof is easy.\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.10}}] Since $n\\geq p_{N(k)+1}$\n in this sum,\n we can bound the\n$\\sin^2(n\\pi/2p_k)$ term in the denominator from below by\n$p_{N(k)+1}^2/p_k^2$. Thus\n\\begin{equation}\\label{sum3}|S^{(2.2)}(q_k,p_k)| \\leq \\frac 1 {p_k\np_{N(k)+1}^2}\\sum_{n=1,\\textrm{ odd}}^{p_k-2} \\frac 1\n{\\sin^2(q_kn\\pi/p_k)}~.\\end{equation}\n\n\\section{The higher genus case}\\label{sec5}\nIn this section, we briefly discuss how the asymptotic behaviour of the genus $2$ signature might generalize to higher genus. Numerical experiments (see below) indicate that\n $\\sigma_{g,n}(\\frac{q}{p};\\lambda)$ might grow like $p^{\\max\\{g,2g-2\\}}$ when $\\frac q p$ goes to an irrational $\\theta\\in[0,1]$. So we ask the following question.\n\\begin{Question}\\label{51}\n Given $g,n\\in \\Z_{\\geq 0}$ and $n$-tuples of integers $\\lambda=(\\lambda_1,\\ldots,\\lambda_n)\\in \\Z^n_{\\geq 0}$,\n is there a function $F_{g;\\lambda}:\n [0,1]\n \\to \\R$ defined almost everywhere satisfying the following: \n For almost all irrational $\\theta\\in[0,1]$,\n if we write $\\frac{q_k}{p_k}$ for the convergents of $\\theta$, one has \n$$ \\lim_{k\\to\\infty}\\frac{\\sigma_{g,n}(\\frac{q_k}{p_k};\\lambda)}{p_k^{\\max\\{g,2g-2\\}}}=F_{g;\\lambda}(\\theta)$$\n\\end{Question}\nTheorem \\ref{asympto} shows that the answer is yes for $g=2$ and $n=0$. Let us\ncompare this to the asymptotics of the dimension\nof the TQFT vector spaces\nwhich \nis well-known: it has a semi-classical interpretation for which we refer to \\cite[Section 3]{Witten}. Precisely:\n$$ \\lim_{p\\to\\infty}\\frac{\\dim \\boV_p(S_g)}{p^{\\max\\{g,3g-3\\}}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2)$$\nwhere the last equality holds only for $g>0$ and $\\mathcal{M}_g$ is the character variety of representations of $\\pi_1(S_g)$ into SU$_2$, endowed with its Liouville measure. We have no conceptual explanation why the order of growth $3g-3$\nfor the dimension\nshould be replaced by $2g-2$\nfor the signature.",
    "post_theorem_intro_text_len": 4698,
    "post_theorem_intro_text": "Figure \\ref{fig:sigma2}  suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}\n\\end{figure}\n\nThe proof of this theorem involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}. We first apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$. We then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement): this step already presents miraculous cancellations for which we do not have any conceptual explanation. \n\nThen in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper. In order to control its denominators, we need to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly. We use for that  some classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.\n\nIn higher genus one can also write down a formula for\n  $\\sigma_g(q/p)$ as the sum of signs indexed by lattice points inside some polytope (see \\cite[Remark 4.12]{BHMV}~\\footnote{Our $\\boV_p$ corresponds to the $V_{2p}$ of \\cite{BHMV}.}.) But the signs are much more complicated than in Formula \\eqref{defsigma2} which prevents us from generalizing our theorem. We briefly discuss in Section~\\ref{sec5} how the asymptotics might look like for general surfaces including marked points, and illustrate this by some numerical experiments.\n\nIn the last section of the paper, we discuss some modular properties of the signature. We observe that the function \n$$\\Lambda(\\theta)=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}$$\nis the boundary value of an Eichler integral of a modular form of weight $4$ for the level $2$ group $\\Gamma(2)$. In addition to the obvious property $\\Lambda(\\theta+1)=-\\Lambda(\\theta)$, it satisfies the following transformation law:\n\\begin{equation}\\label{Lambdamod}\n\\Lambda\\Big(\\frac{\\theta}{2\\theta+1}\\Big)(2\\theta+1)^2-\\Lambda(\\theta)=2\\theta^2+2\\theta+1.\n\\end{equation}\nThis functional equation suggests that a similar integral version holds for signatures as stated in the following conjecture, checked by computer for all $0<q<p<100$. \n\\begin{Conjecturestar}\nFor any odd coprime integers $0<q<p$ one has\n$$\\sigma_2\\Big(\\frac{q}{2q+p}\\Big)-\\sigma_2\\Big(\\frac{q}{p}\\Big)=2q^2+2qp+p^2-1.$$\n\\end{Conjecturestar}\n\nThis formula is reminiscent of the reciprocity formula of Dedekind sums: in fact we will make this analogy more precise in Section~\\ref{SectionS}. We prove that the signature at $\\zeta=e^{i\\pi q/p}$ of  the TQFT-vector space associated to the 4 times punctured sphere (with  colors $(p-1)/2$  at the marked points)  is actually equal to a 2-smoothed version, $S(q/p)$,  of the Dedekind sum $s(q,p)$. It is then classical that  this function $S(q/p)$ is the boundary value of a modular form (this time of weight $1/2$) and that it satisfies the following transformation law\n\n$$S\\Big(\\frac{q}{2q+p}\\Big)-S\\Big(\\frac{q}{p}\\Big)=1.$$\nWe provide an elementary proof of these facts for the benefit of the non-specialist reader. \n\nThis very satisfactory state of things would hopefully generalize to higher genus as well. However, even in genus 2, a precise (non asymptotic) relation between the signature $\\sigma_g(q/p)$\nand the Eichler integral is still lacking. Such a relation would presumably lead to a proof  of the conjectured transformation law for $\\sigma_2$. \n\nTo end this introduction, we would like to mention the parallel paper \\cite{MY} in which the first author and S. Yoon compute the asymptotics of the function\n$\\sigma_g(q/p)$, but for sequences of the form $\\frac{q_n}{p_n}=\\frac{a+bn}{c+dn}$ where $\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)\\in \\Gamma(2)$. This regime is somewhat orthogonal to the one described in the present paper. The techniques used and the results are also very different. A synthesis of the two papers has yet to be made.\n\\vskip 8pt\n\n{\\bf Acknowledgements:} It is our pleasure to thank Pierre Charollois for early discussions on many aspects of this paper, Gaëtan Chenevier for his expertise on modular forms, \nand Don Zagier for very helpful explanations about Eichler integrals and quantum modular forms.",
    "sketch": "The proof “involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}.” First, “we apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.” Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement),” a step that “already presents miraculous cancellations.” Then, “in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”",
    "expanded_sketch": "The proof “involves several steps and occupies” the part starting at\n\n\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\nand the later section \\ref{sec2}.\n\nFirst, “we apply” a finite Fourier transform in the defining expression\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\nto transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.\n\nNext, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$,” namely the formula stated in\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\na step that “already presents miraculous cancellations.” Then, “we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”",
    "expanded_theorem": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$",
    "theorem_type": [
      "Asymptotic or Limit",
      "Universal"
    ],
    "mcq": {
      "question": "For a reduced rational number \\(q/p\\in(0,1)\\), define\n\\[\n\\sigma_2\\!\\left(\\frac qp\\right)=\\sum_{(a,b,c)\\in \\Delta_p}(-1)^{\\lfloor aq/p\\rfloor+\\lfloor bq/p\\rfloor+\\lfloor cq/p\\rfloor},\n\\]\nwhere \\(\\Delta_p\\) is the set of triples \\((a,b,c)\\) of integers such that\n\\[\n0<a,b,c<p,\\qquad a<b+c,\\quad b<a+c,\\quad c<a+b,\n\\]\n\\[\na+b+c<2p,\\qquad a+b+c\\text{ is odd}.\n\\]\nNow let \\(\\theta\\in[0,1]\\) be irrational, and let \\(q_k/p_k\\) be the convergents of the regular continued-fraction expansion of \\(\\theta\\). Which statement holds for Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
        },
        {
          "label": "C",
          "text": "For Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\), the limit\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n\\]\nexists."
        },
        {
          "label": "D",
          "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\,|\\sin(n\\pi\\theta)|}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every irrational \\(\\theta\\in[0,1]\\), if \\(q_k/p_k\\) are the convergents of its regular continued-fraction expansion, then\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "odd-index restriction in the trigonometric expansion",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit identification of the limiting value by the odd-series formula",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "signed denominator \\(\\sin(n\\pi\\theta)\\) replaced by absolute value, destroying the exact Fourier-cancellation structure",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "almost-everywhere metric Diophantine hypothesis strengthened to all irrationals",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct limit formula. It defines the quantity and asks for the asymptotic statement; the correct answer is not directly embedded in the wording."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: one option is essentially the exact stated limiting theorem, with distractors formed by small perturbations. It is not a pure tautology, but it is only a mild reformulation of a known result."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare quantifiers, normalization, and the odd-index restriction, but the question mostly tests recognition/recall of the theorem rather than genuine derivation or generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: removing the odd restriction, weakening to mere existence, overstrengthening 'almost all' to 'every,' and changing the normalization to p_k^3 are all distinct and aligned with common failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A reasonably well-constructed theorem-recognition MCQ with strong distractors and little direct answer leakage, but it only moderately avoids tautology and does not strongly test generative reasoning."
    }
  },
  {
    "id": "2512.13450v1",
    "paper_link": "http://arxiv.org/abs/2512.13450v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "Theoremstar",
      "content": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$",
      "start_pos": 45297,
      "end_pos": 45623,
      "label": null
    },
    "ref_dict": {
      "fig:sigma2": "\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}",
      "sigma2": "\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}",
      "defsigma2": "\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}",
      "genre2": "\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\n\\begin{proof}\nBy a finite Fourier transform over $\\Z/2p\\Z$, we can write for $m\\in\\Z$ :\n\\begin{equation}\\label{FFT}\n  (-1)^{\\lfloor m/p\\rfloor}=\n  \\frac 1 p\n  \\sum_{n=1,\\text{ odd}}^{2p-1}\\frac{\\sin(\\pi n(2m+1)/2p)}{\\sin(\\pi n/2p)}\n  =\\frac 1 p  \\sum_{n=1,\\text{ odd}}^{2p-1} \\frac {t^{2m+1}-t^{-2m-1}}{t-t^{-1}}\n    \\Big|_{t=e^{\\frac {i\\pi n}{2p}}~.\n      }\n\\end{equation}\nIn the sequel, we set $\\O_p=\\{1,3,\\ldots,2p-1\\}$ and $\\O_p^+=\\{1,3,\\ldots,p\\}$. Using Formula \\eqref{FFT} for each sign $\\epsilon_{j+1}=(-1)^{\\lfloor q(j+1)/p\\rfloor}$ and plugging it into the formula of Lemma \\ref{triplepsilon}, we get \n\\begin{equation}\\nonumber\\sigma_2({\\textstyle{\\frac q p}})\\,\\,\n  =\\sum_{(j,k,\\ell)\\in T_p} \\epsilon_{j+1}\\epsilon_{k+1}\\epsilon_{\\ell+1}\n  =\\frac1 {p^3} \\sum_{(a,b,c)\\in \\O_p^3}\\psi(a,b,c)\\end{equation} \nwhere\n\\begin{equation}\n\\psi(a,b,c)=\\!\\!\\!\\!\\sum\\limits_{(j,k,l)\\in T_p}\\!\\!\\!\\!\n\\frac{(x^{2q(j+1)+1}-x^{-2q(j+1)-1})(y^{2q(k+1)+1}-y^{-2q(k+1)-1})(z^{2q(\\ell+1)+1}-z^{-2q(\\ell+1)-1})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\nonumber\n\\end{equation}\nwith $x=e^{\\frac{i\\pi a}{2p}}$,  $y=e^{\\frac{i\\pi b}{2p}}$ and $z=e^{\\frac{i\\pi c}{2p}}$.\n\nThe function $\\psi$ is clearly symmetric in its three variables and satisfies also $\\psi(2p-a,b,c)=\\psi(a,b,c)$. We can hence rewrite $\\sigma_2(\\frac q p)$  as \n\\begin{equation}\\label{eq4n}\\sigma_2({\\textstyle{\\frac q p}})=\n  \\frac1 {p^3}\n  \\sum_{a,b,c\\in \\O_p^+} 2^{m_a+m_b+m_c}\\psi(a,b,c)\\end{equation}\nwhere $m_a=1$ if $a\\ne p$ and $m_p=0$.\n\nIn order to compute $\\psi(a,b,c)$, we rewrite it as follows:\n\\begin{equation} \\psi(a,b,c)= \\sum\\limits_{\\epsilon,\\mu,\\nu=\\pm 1}\\epsilon\\mu\\nu \\frac{ x^{\\epsilon}y^{\\mu}z^{\\nu}F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\label{eq5n}\n\\end{equation} where $F$ is the polynomial defined by $F(X,Y,Z)=\\sum_{(j,k,l)\\in T_p}X^{j+1}Y^{k+1}Z^{\\ell +1}$.\n\nThe computation of $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ is simplified by the following observation:  we have $$F(X,Y,Z)=X^pY^p F(X^{-1}, Y^{-1},Z)~,$$ since the involution $(j,k,\\ell ) \\mapsto (p-2-j,p-2-k,\\ell)$ preserves the set $T_p$. Therefore, since $F$ is symmetric in its three variables,  whenever $F(X,Y,Z)$ is evaluated at numbers $X=x^{2q}$, $Y=y^{2q}$, and $Z=z^{2q}$\nsatisfying $X^p=Y^p=Z^p=-1$, then for  any $\\epsilon,\\mu,\\nu=\\pm 1$, we have that $F(X^\\epsilon, Y^\\mu,Z^\\nu)$ is equal to either $F(X,Y,Z)$ (when $\\epsilon\\mu\\nu =1$) or to  $F(X^{-1}, Y^{-1},Z^{-1})= \\overline {F(X,Y,Z)}$ (when $\\epsilon\\mu\\nu =-1$).\n\\begin{Remark}\\label{rk33}{\\em For later use, we note the following special case: when $F(x^{2q},y^{2q},z^{2q})$ happens to be real, then $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ \n    does not depend at all on $\\epsilon,\\mu$, or $\\nu$, and formula \\eqref{eq5n} for $\\psi(a,b,c)$ simply becomes $\\psi(a,b,c)=F(x^{2q},y^{2q},z^{2q})$.}\n    \\end{Remark}"
    },
    "pre_theorem_intro_text_len": 2862,
    "pre_theorem_intro_text": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups  and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by  \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$. \n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$.  The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten  \\cite{Witten} to show that for $g>1$,  \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for  $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following",
    "context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following",
    "full_context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following\n\nOur main result is the following\n\nFigure \\ref{fig:sigma2} suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$.\n\nWe briefly review the properties of the convergents $q_k/p_k$ of $\\theta$ that we will need. See {\\em e.g.} Perron \\cite{P} or Khinchin \\cite{Kh1} for more details\\footnote{Unfortunately, \\cite{Kh1} writes $p_k$ for our $q_k$ and {\\em vice versa.}}. Each irrational $\\theta\\in {\\mathbb R}$ has a unique regular continued fraction expansion $$\\theta = [a_0; a_1, a_2, \\ldots ] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{a_2+\\dots}} $$ where regular means that $a_0\\in \\Z$ and $a_i\\in \\Z_{\\geq 1}$ for $i\\geq 1$. Note that in our case $a_0=0$ since\n$\\theta \\in [0,1]\\setminus \\Q$.\nThe convergents\n$$\\frac{q_k}{p_k} = [a_0; a_1, a_2, \\ldots, a_k] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{\\dots+ \\frac{1}{a_k}}} $$ can also be defined recursively by\n\\begin{eqnarray} q_k&=a_k q_{k-1} + q_{k-2}, \\ \\ \\ \\ q_0=a_0, \\ \\ \\ \\ q_{-1}=1 \\label{rec5}\\\\\n p_k&=a_k p_{k-1} + p_{k-2}, \\ \\ \\ \\ p_0=1, \\ \\ \\ \\ p_{-1}=0 \\label{rec6}\n\\end{eqnarray}\nThe following classical facts are well-known. \n\\begin{Proposition}\\label{qpfacts}\n For all irrational $\\theta$, the convergents $q_k/p_k$ of $\\theta$ satisfy the following properties:\n \\begin{enumerate} \n \\item[(i)] One has $1=p_0\\leq p_1<p_2<\\ldots$, {\\em i.e.} the denominators $p_k$ are positive integers and moreover strictly increasing from $p_1$ onwards.\n \\item[(ii)] For all $k\\geq 1$, $q_k$ and $p_k$ are coprime.\n \\item[(iii)] $\\lim_{k\\rightarrow\\infty} q_k/p_k = \\theta$.\n \\item[(iv)]\n For all $i\\geq 1$, we have\n \\begin{equation}\\nonumber \n \\frac 1 {p_i(p_i+p_{i+1})} <\n \\Big|\\theta - \\frac {q_i}{p_i} \\Big| < \\frac 1 {p_ip_{i+1}} < \\frac 1 {p_i^2}~.\n \\end{equation}\n \\item[(v)] {\\em [Lagrange]}\\footnote{For a proof, see Perron \\cite[\\S 15, Satz 2.17]{P}.} For all $1\\leq i<k$, and any $n,\\ell \\in\\Z$ with $1\\leq n<p_{i+1}$, we have\n \\begin{equation}\\nonumber \n \\Big| n\\frac {q_k}{p_k} -\\ell \\Big|\n \\geq \\Big|p_i\\frac {q_k}{p_k} - q_i\\Big| \\geq \\frac{1}{p_i + p_{i+1}} ~.\n \\end{equation}\n \\end{enumerate}\n \\end{Proposition}\n\n\\begin{proof}[\\bf{Proof of Theorem~\\ref{asympto}}] We start from the trigonometric formula for $\\sigma_2(\\frac{q}{p})$ given in Theorem~\\ref{sigma2}. The first observation is that the term $(1-p^2)/(6p^2)$ in this formula stays bounded as $p\\rightarrow \\infty$ and so does not contribute to the limit of $\\sigma_2(\\frac{q_k}{p_k})/ p_k^2$. Thus it suffices to show\n \\begin{equation}\\label{eq4}\n \\lim_{k\\rightarrow \\infty} \\frac{1}{4p_k^4} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{f(n,p_k,q_k)}{\\sin^3({n\\pi}/{2p_k})\\sin^2({q_kn\\pi}/{p_k})} =\\frac{16}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}\n \\end{equation} with $f(n,p,q)$ given as in Theorem~\\ref{sigma2}.\n\nLet $S^{(1)}(q_k,p_k,\\theta)$ denote the sum $S^{(1)}(q_k,p_k)$ but with $\\sin(n\\pi\\theta)$ in the denominator in place of $\\sin({q_kn\\pi}/{p_k})$. Using (\\ref{sin-ineq}), we see that the $n$-th summand of $S^{(1)}(q_k,p_k,\\theta)$ is bounded from above by $1/(n^3\\sin(n\\pi\\theta))$. So we can apply the Lebesgue dominated convergence theorem to this sum. We get\n $$\\lim_{k\\rightarrow \\infty}S^{(1)}(q_k,p_k,\\theta) =\\frac{8}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}~.$$\n Thus to prove Lemma~\\ref{3.3} it suffices to show that the difference $d_k=S^{(1)}(q_k,p_k) -S^{(1)}(q_k,p_k,\\theta)$ goes to zero as $k\\rightarrow \\infty$. We have \n $$d_k= \\frac{1}{p_k^3} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{\\cos (n\\pi/2p_k)}{\\sin^3({n\\pi}/{2p_k}) }\\Biggl (\\frac {1} {\\sin({q_kn\\pi}/{p_k})} - \\frac {1} {\\sin(n\\pi\\theta)}\\Biggr)~.$$ We can bound the denominators from below using (\\ref{sin-ineq}), (\\ref{sinn-inequ}), and the trivial lower bound $|\\sin({q_kn\\pi}/{p_k})| \\geq \\sin (\\pi/p_k) \\geq 2/p_k$. Thus for all $\\epsilon >0$ we have\n\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.4}}]\n We assume that $\\mu(\\theta)=2$ and that $\\theta$ satisfies the conditions\n in Theorem~\\ref{ThKL}.\n Recall $S^{(2)}(q_k,p_k)$, defined in (\\ref{sum2}), is a sum from $n=1$ to $n=p_k-2$. We cut\nit into two parts, as follows. Put\n\\begin{equation}\\label{floor} N(k)= \\lfloor 3k/4 \\rfloor\n\\end{equation} and write $$S^{(2)}(q_k,p_k) = S^{(2.1)}(q_k,p_k)\n+S^{(2.2)}(q_k,p_k)~,$$ where $S^{(2.1)}(q_k,p_k)$ is the sum from\n$n=1$ to $n=p_{N(k)+1}-1$ and $S^{(2.2)}(q_k,p_k)$ is the sum from\n$n=p_{N(k)+1}$ to $p_k-2$. The following two lemmas provide upper bounds for these two sums.\n\\begin{Lemma}\\label{3.9} We have $$|S^{(2.1)}(q_k,p_k) |\\leq \\frac\n{p_{N(k)}} {p_k} F_7(k)$$ where $F_7(k) ={\\mathcal O}(k^7)$.\n\\end{Lemma}\n\\begin{Lemma}\\label{3.10} We have $$|S^{(2.2)}(q_k,p_k)| \\leq \\frac{1}{2} \\zeta(2)\n\\frac {p_{k}} {p_{N(k)+1}^2}~.$$\n\\end{Lemma}\nAssuming these two lemmas for the moment, since $\\lim_{m\\rightarrow\\infty} p_m^{1/m}= L>1$ by the Khinchin-L{\\'e}vy Theorem~\\ref{ThKL}(ii), $p_k$ grows like $L^k$ as $k\\rightarrow \\infty$. It is not hard to deduce from this that both $ S^{(2.1)}(q_k,p_k)$ and $ S^{(2.2)}(q_k,p_k)$ converge to zero.~\\footnote{Here is how to see this for $ S^{(2.1)}(q_k,p_k)$. Theorem~\\ref{ThKL}(ii) gives that for all $\\epsilon>0$ there exist $m_0(\\epsilon)$ so that $L^{1-\\epsilon} \\leq p_m^{1/m} \\leq L^{1+\\epsilon}$ holds for for all $m\\geq m_0(\\epsilon)$. Thus also $L^{m-m\\epsilon} \\leq p_m\\leq L^{m+m\\epsilon}$ for all $m\\geq m_0(\\epsilon)$. Applying now the first inequality with $m=k$ and the second one with $m=N(k)=\\lfloor 3k/4\\rfloor$ and taking $\\epsilon$ small enough, it follows that $p_{N(k)}/p_k \\leq L^{-\\delta k}$ for some $\\delta>0$ as soon as $k\\geq 2 m_0(\\epsilon)$. Thus $p_{N(k)} F_7(k)/ {p_k} \\leq k^7 L^{-\\delta k}$ for $k$ big enough, proving that the limit as $k\\rightarrow\\infty$ is zero, as asserted. }\nThis proves Lemma~\\ref{3.4}. \n\\end{proof}\nIt remains to prove Lemmas~\\ref{3.9} and~\\ref{3.10}. Let us first deal\nwith Lemma~\\ref{3.10} whose proof is easy.\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.10}}] Since $n\\geq p_{N(k)+1}$\n in this sum,\n we can bound the\n$\\sin^2(n\\pi/2p_k)$ term in the denominator from below by\n$p_{N(k)+1}^2/p_k^2$. Thus\n\\begin{equation}\\label{sum3}|S^{(2.2)}(q_k,p_k)| \\leq \\frac 1 {p_k\np_{N(k)+1}^2}\\sum_{n=1,\\textrm{ odd}}^{p_k-2} \\frac 1\n{\\sin^2(q_kn\\pi/p_k)}~.\\end{equation}\n\n\\section{The higher genus case}\\label{sec5}\nIn this section, we briefly discuss how the asymptotic behaviour of the genus $2$ signature might generalize to higher genus. Numerical experiments (see below) indicate that\n $\\sigma_{g,n}(\\frac{q}{p};\\lambda)$ might grow like $p^{\\max\\{g,2g-2\\}}$ when $\\frac q p$ goes to an irrational $\\theta\\in[0,1]$. So we ask the following question.\n\\begin{Question}\\label{51}\n Given $g,n\\in \\Z_{\\geq 0}$ and $n$-tuples of integers $\\lambda=(\\lambda_1,\\ldots,\\lambda_n)\\in \\Z^n_{\\geq 0}$,\n is there a function $F_{g;\\lambda}:\n [0,1]\n \\to \\R$ defined almost everywhere satisfying the following: \n For almost all irrational $\\theta\\in[0,1]$,\n if we write $\\frac{q_k}{p_k}$ for the convergents of $\\theta$, one has \n$$ \\lim_{k\\to\\infty}\\frac{\\sigma_{g,n}(\\frac{q_k}{p_k};\\lambda)}{p_k^{\\max\\{g,2g-2\\}}}=F_{g;\\lambda}(\\theta)$$\n\\end{Question}\nTheorem \\ref{asympto} shows that the answer is yes for $g=2$ and $n=0$. Let us\ncompare this to the asymptotics of the dimension\nof the TQFT vector spaces\nwhich \nis well-known: it has a semi-classical interpretation for which we refer to \\cite[Section 3]{Witten}. Precisely:\n$$ \\lim_{p\\to\\infty}\\frac{\\dim \\boV_p(S_g)}{p^{\\max\\{g,3g-3\\}}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2)$$\nwhere the last equality holds only for $g>0$ and $\\mathcal{M}_g$ is the character variety of representations of $\\pi_1(S_g)$ into SU$_2$, endowed with its Liouville measure. We have no conceptual explanation why the order of growth $3g-3$\nfor the dimension\nshould be replaced by $2g-2$\nfor the signature.",
    "post_theorem_intro_text_len": 4698,
    "post_theorem_intro_text": "Figure \\ref{fig:sigma2}  suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}\n\\end{figure}\n\nThe proof of this theorem involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}. We first apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$. We then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement): this step already presents miraculous cancellations for which we do not have any conceptual explanation. \n\nThen in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper. In order to control its denominators, we need to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly. We use for that  some classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.\n\nIn higher genus one can also write down a formula for\n  $\\sigma_g(q/p)$ as the sum of signs indexed by lattice points inside some polytope (see \\cite[Remark 4.12]{BHMV}~\\footnote{Our $\\boV_p$ corresponds to the $V_{2p}$ of \\cite{BHMV}.}.) But the signs are much more complicated than in Formula \\eqref{defsigma2} which prevents us from generalizing our theorem. We briefly discuss in Section~\\ref{sec5} how the asymptotics might look like for general surfaces including marked points, and illustrate this by some numerical experiments.\n\nIn the last section of the paper, we discuss some modular properties of the signature. We observe that the function \n$$\\Lambda(\\theta)=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}$$\nis the boundary value of an Eichler integral of a modular form of weight $4$ for the level $2$ group $\\Gamma(2)$. In addition to the obvious property $\\Lambda(\\theta+1)=-\\Lambda(\\theta)$, it satisfies the following transformation law:\n\\begin{equation}\\label{Lambdamod}\n\\Lambda\\Big(\\frac{\\theta}{2\\theta+1}\\Big)(2\\theta+1)^2-\\Lambda(\\theta)=2\\theta^2+2\\theta+1.\n\\end{equation}\nThis functional equation suggests that a similar integral version holds for signatures as stated in the following conjecture, checked by computer for all $0<q<p<100$. \n\\begin{Conjecturestar}\nFor any odd coprime integers $0<q<p$ one has\n$$\\sigma_2\\Big(\\frac{q}{2q+p}\\Big)-\\sigma_2\\Big(\\frac{q}{p}\\Big)=2q^2+2qp+p^2-1.$$\n\\end{Conjecturestar}\n\nThis formula is reminiscent of the reciprocity formula of Dedekind sums: in fact we will make this analogy more precise in Section~\\ref{SectionS}. We prove that the signature at $\\zeta=e^{i\\pi q/p}$ of  the TQFT-vector space associated to the 4 times punctured sphere (with  colors $(p-1)/2$  at the marked points)  is actually equal to a 2-smoothed version, $S(q/p)$,  of the Dedekind sum $s(q,p)$. It is then classical that  this function $S(q/p)$ is the boundary value of a modular form (this time of weight $1/2$) and that it satisfies the following transformation law\n\n$$S\\Big(\\frac{q}{2q+p}\\Big)-S\\Big(\\frac{q}{p}\\Big)=1.$$\nWe provide an elementary proof of these facts for the benefit of the non-specialist reader. \n\nThis very satisfactory state of things would hopefully generalize to higher genus as well. However, even in genus 2, a precise (non asymptotic) relation between the signature $\\sigma_g(q/p)$\nand the Eichler integral is still lacking. Such a relation would presumably lead to a proof  of the conjectured transformation law for $\\sigma_2$. \n\nTo end this introduction, we would like to mention the parallel paper \\cite{MY} in which the first author and S. Yoon compute the asymptotics of the function\n$\\sigma_g(q/p)$, but for sequences of the form $\\frac{q_n}{p_n}=\\frac{a+bn}{c+dn}$ where $\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)\\in \\Gamma(2)$. This regime is somewhat orthogonal to the one described in the present paper. The techniques used and the results are also very different. A synthesis of the two papers has yet to be made.\n\\vskip 8pt\n\n{\\bf Acknowledgements:} It is our pleasure to thank Pierre Charollois for early discussions on many aspects of this paper, Gaëtan Chenevier for his expertise on modular forms, \nand Don Zagier for very helpful explanations about Eichler integrals and quantum modular forms.",
    "sketch": "The proof “involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}.” First, “we apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.” Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement),” a step that “already presents miraculous cancellations.” Then, “in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”",
    "expanded_sketch": "The proof “involves several steps and occupies” the part starting at\n\n\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\nand the later section \\ref{sec2}.\n\nFirst, “we apply” a finite Fourier transform in the defining expression\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\nto transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.\n\nNext, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$,” namely the formula stated in\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\na step that “already presents miraculous cancellations.” Then, “we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”",
    "expanded_theorem": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$",
    "theorem_type": [
      "Asymptotic or Limit",
      "Universal"
    ],
    "mcq": {
      "question": "For a reduced rational number \\(q/p\\in(0,1)\\), define\n\\[\n\\sigma_2\\!\\left(\\frac qp\\right)=\\sum_{(a,b,c)\\in \\Delta_p}(-1)^{\\lfloor aq/p\\rfloor+\\lfloor bq/p\\rfloor+\\lfloor cq/p\\rfloor},\n\\]\nwhere \\(\\Delta_p\\) is the set of triples \\((a,b,c)\\) of integers such that\n\\[\n0<a,b,c<p,\\qquad a<b+c,\\quad b<a+c,\\quad c<a+b,\n\\]\n\\[\na+b+c<2p,\\qquad a+b+c\\text{ is odd}.\n\\]\nNow let \\(\\theta\\in[0,1]\\) be irrational, and let \\(q_k/p_k\\) be the convergents of the regular continued-fraction expansion of \\(\\theta\\). Which statement holds for Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
        },
        {
          "label": "C",
          "text": "For Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\), the limit\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n\\]\nexists."
        },
        {
          "label": "D",
          "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\,|\\sin(n\\pi\\theta)|}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every irrational \\(\\theta\\in[0,1]\\), if \\(q_k/p_k\\) are the convergents of its regular continued-fraction expansion, then\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "odd-index restriction in the trigonometric expansion",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit identification of the limiting value by the odd-series formula",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "signed denominator \\(\\sin(n\\pi\\theta)\\) replaced by absolute value, destroying the exact Fourier-cancellation structure",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "almost-everywhere metric Diophantine hypothesis strengthened to all irrationals",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct formula or uniquely point to choice A. It sets up the objects and asks for the valid almost-everywhere statement, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a recognition item for a specific theorem: the correct option is the theorem’s full conclusion, while the others are small perturbations of it. It mainly restates known content rather than asking for an independently derived conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants such as odd-only summation, signed versus absolute value, and 'almost every' versus 'every' irrational. However, the item primarily tests exact recall of the theorem statement rather than substantial generative reasoning from the definitions."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: dropping the odd-index restriction, weakening to mere existence, inserting an absolute value, and overstrengthening an almost-everywhere claim to a universal one."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests genuine reasoning."
    }
  },
  {
    "id": "2512.13528v1",
    "paper_link": "http://arxiv.org/abs/2512.13528v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theoremA",
      "content": "\\label{A}\nLet \\( (M^4, g_H) \\) be a closed hyperbolic \\(4\\)-manifold.\nSuppose \\( g_1 \\) is a Riemannian metric on \\( M^4 \\) with \n\\( \\mathrm{Sc}_{g_1} \\ge -12 \\).\nAssume in addition that one of the following holds:\n\\begin{enumerate}\n    \\item[(i)] \\( g_1 \\) is self–dual or anti–self–dual;\n    \\item[(ii)] the Yamabe metric \\( g \\in [g_1] \\) with \\( \\mathrm{Sc}_g = -12 \\)\n    satisfies one of the conditions:\n    \\begin{enumerate}\n        \\item[(a)] \\( g \\) is Einstein;\n        \\item[(b)] for every point \\( x \\in (M^4, g) \\), there exists \n        \\( r(x) > 0 \\) such that for all \\( 0 < r \\le r(x) \\), the geodesic \\( r \\)-balls satisfy $  \\mathrm{Vol}_g(B^g_r(x)) \\ge  \\mathrm{Vol}_{g_H}(B^{g_H}_r(x)). $\n    \\end{enumerate}\n\\end{enumerate}\n\nThen $\\mathrm{Vol}_{g_1}(M^4) \\geq \\mathrm{Vol}_{g_H}(M^4)$. Moreover, if equality holds, i.e., \\( \\mathrm{Vol}_{g_1}(M^4) = \\mathrm{Vol}_{g_H}(M^4) \\), then  \\( g_1 \\) is isometric to \\( g_H \\).",
      "start_pos": 77164,
      "end_pos": 78137,
      "label": "A"
    },
    "ref_dict": {
      "Liouville theorem": "\\begin{theorem}[Liouville theorem]\\label{Liouville theorem}\nLet \\( (M^n, g) \\), \\( n \\ge 3 \\), be a complete, locally conformally flat manifold with nonnegative scalar curvature.  \nIf \\( \\Phi : M^n \\to S^n \\) is a conformal map, then \\( \\Phi \\) is injective and the boundary \\( \\partial \\Phi(M) \\) has zero Newtonian capacity (i.e.,  $2$-capacity).\n\\end{theorem}",
      "the universal cover": "\\begin{theorem} \\label{the universal cover}\nLet \\( (M^4, g) \\) be a closed, locally conformally flat, and scalar–flat \\(4\\)-manifold with \\( \\pi_2(M^4) \\neq 0 \\).  Then its Riemannian universal cover \\( (\\widetilde{M}, \\tilde{g}) \\) is isometric to $(\\mathbb{H}^2  \\times S^2 ,  g_{H} \\oplus g_{st})$ up to homothety.\n\\end{theorem}",
      "3": "\\begin{proof}[Proof of Theorem \\ref{A}]\nWhen the Yamabe metric is Einstein, Proposition \\ref{ncsc} applies to complete the proof.\nThe remaining cases are handled by Theorem \\ref{LCF} and Theorem \\ref{local}.\n\\end{proof}\n\n\\section{The topology of  \\(4\\)-manifolds with PSC}\\label{3}\n\nWithin the known classifications of closed, oriented \\(3\\)-manifolds admitting Riemannian metrics of positive scalar curvature and of locally conformally flat \\(4\\)-manifolds with positive scalar curvature (up to diffeomorphism type), one may further ask to classify closed, oriented Riemannian \\(4\\)-manifolds with positive scalar curvature up to homeomorphism type.  \n\nHowever, Carr \\cite{MR936805} shows that for any nontrivial finitely presented group \\(\\pi\\), there exists a closed Riemannian \\(4\\)-manifold \\(M\\) with positive scalar curvature such that \\(\\pi_1(M)=\\pi\\). By Markov’s theorem, it follows that closed \\(4\\)-manifolds with positive scalar curvature cannot be classified up to homeomorphism. In fact, Carr’s result extends to all dimensions \\(n>4\\): for every finitely presented group \\(\\pi\\) and \\(n\\ge4\\), there exists a closed smooth stably parallelizable Riemannian \\(n\\)-manifold with positive scalar curvature; moreover, if \\(\\pi\\) contains a subgroup of index two, there also exists a closed smooth non-orientable Riemannian \\(n\\)-manifold with positive scalar curvature \\cite[Theorems 6 and 7]{zbMATH07432160}.  \n\nTo circumvent this group-theoretic obstruction, one may instead aim to classify those Riemannian \\(n\\)-manifolds with a fixed fundamental group. The simplest case is the simply connected one. For \\(n\\ge5\\), it is known that a simply connected smooth manifold \\(M^n\\) admits a metric of positive scalar curvature if and only if \\(M^n\\) is non-spin, or it is spin and its \\(\\alpha\\)-invariant vanishes in \\(\\mathrm{KO}_n\\). In particular, since \\(\\mathrm{KO}_n = 0\\) for \\(n \\equiv 3,5,6,7 \\pmod{8}\\) by Bott periodicity for the $\\mathrm{KO}_n$-theory of a point, every simply connected smooth \\(n\\)-manifold in these dimensions admits a metric of positive scalar curvature. Consequently, in these cases, the geometric classification problem reduces entirely to a topological one.\n\nThe classical results of Freedman and Donaldson in  dimension 4 imply that if \n\\( M^4 \\) is a smoothable, closed, simply connected, topological \\(4\\)-manifold, \nthen \\( M^4 \\) is homeomorphic to one of the following:\n$S^4$, $\\#^m \\mathbb{C}P^2 \\,\\#^n \\overline{\\mathbb{C}P^2}$, or $  \\#^{\\pm m} M_{E_8} \\,\\#^n (S^2 \\times S^2),$\nfor some integers \\( m,n \\geq 0 \\), where \\(M_{E_8}\\) denotes the topological $4$–manifold whose intersection form is given by the $E_8$ lattice.\n\n For a simply connected closed \\(4\\)-manifold \\(M^4\\), it admits a spin structure if and only if its intersection form is even. The intersection form of $\\#^{\\pm m} M_{E_8} \\,\\#^n (S^2 \\times S^2)$\nhas rank \\(8m+n\\), signature \\(\\pm 8m\\), and is even. Furthermore, by the Hirzebruch signature theorem, the \\(\\hat{A}\\)-genus of a \\(4\\)-manifold equals \\(-\\tfrac{1}{8}\\) times its signature. Hence, \n\\(\\#^{\\pm m} M_{E_8} \\,\\#^n (S^2 \\times S^2)\\) does not admit a metric of positive scalar curvature.  \n\nTherefore, a closed simply connected \\(4\\)-manifold admitting a metric of positive scalar curvature is homeomorphic to one of the following:\n\\[\nS^4, \\quad \n\\#^m \\mathbb{C}P^2 \\,\\#^n \\overline{\\mathbb{C}P^2}, \\quad \n\\#^n (S^2 \\times S^2),\n\\]\nor a connected sum of these manifolds (see \\cite[Theorem 7.8]{MR3450199}).  This classification cannot, in general, be improved to the diffeomorphism type.  Indeed, for some \\(k\\), Teicher \\cite[Theorem~5.8]{MR1720873} constructs examples of simply connected complex surfaces of general type which are spin and have vanishing signature.  These manifolds are homeomorphic to \\( \\#^k (S^2 \\times S^2) \\), but, being of general type, they do not admit metrics of positive scalar curvature.  \n\nThe next step is to analyze \\(4\\)-manifolds which are not simply connected. If its fundamental group is finite, we have the following result.\n\n\\begin{proposition}\\label{finite}\nLet $M^4$ be a connected, closed, non-simply connected Riemannian $4$-manifold with positive scalar curvature and finite fundamental group. Then:  \n\\begin{enumerate}\n\\item[(i)] If $M$ is spin, then its universal cover $\\tilde{M}$ is homeomorphic to \n$\\#^{k} (S^2 \\times S^2)$ for some integer $k \\ge 1$, where $k = \\tfrac{1}{2} b_2(\\tilde{M})$.\n\n\\item[(ii)] If $M$ is non-spin, then $\\tilde{M}$ is homeomorphic to one of the following:\n\\begin{enumerate}\n\\item[(a)] $\\#^{k} \\mathbb{CP}^2 \\,\\#^{l} \\overline{\\mathbb{CP}}^{\\,2}$, \n for some non-negative integers $k+l \\ge 1$, if $\\tilde{M}$ is non-spin;\n\\item[(b)] $\\#^{m} (S^2 \\times S^2)$, for some integer $m \\ge 0$, if $\\tilde{M}$ is spin.\n\\end{enumerate}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nIf $M$ is spin, then its universal cover $\\tilde{M}$ is also spin  and the intersection form of $\\tilde{M}$ is even. The lifted positive scalar curvature metric implies that the signature of $\\tilde{M}$ vanishes. Hence, $\\tilde{M}$ is homeomorphic to $\\#^{k} (S^2 \\times S^2)$.  \nIf $k = 0$, then $M$ would be homeomorphic to $\\mathbb{RP}^4$. However, $\\mathbb{RP}^4$ is non-spin, which leads to a contradiction. Therefore, $k \\ge 1$.\n\nNow suppose $M$ is non-spin.  \nIf $\\tilde{M}$ is spin, then as above $\\tilde{M}$ is homeomorphic to $\\#^{m} (S^2 \\times S^2)$ for some $m \\ge 0$.  \nIf $\\tilde{M}$ is non-spin, then the intersection form of $\\tilde{M}$ is odd. By the classification of odd unimodular forms and Freedman's theorem, $\\tilde{M}$ must be homeomorphic to a connected sum of copies of the complex projective plane $\\mathbb{CP}^2$ and its orientation-reversed version $\\overline{\\mathbb{CP}}^{\\,2}$ for some non-negative integers $k+l \\ge 1$.\n\\end{proof}",
      "2": "\\label{2}\n\nThe Bishop–Gromov volume comparison theorem for Ricci curvature bounded below has many applications.  \nIt is natural to seek analogous volume comparison results for scalar curvature bounded",
      "5": "\\begin{remark}\nFor \\( \\alpha = 2 \\) and \\( \\beta = \\tfrac{m+1}{m} \\) with \\( m > 0 \\), Case defines the weighted Yamabe constant \\( \\Lambda(g, e^{-f} \\, d\\mathrm{Vol}_g, m) \\) and solves the corresponding weighted Yamabe problem in \\cite{zbMATH06537658}. However, the weighted Yamabe invariant for general parameters \\( \\alpha \\neq 0 \\) and \\( \\beta \\neq 0 \\) has not yet been defined or systematically studied. It is also natural to ask whether weighted analogues of the results in this section can be formulated and established. In particular, if the manifold admits a hyperbolic metric, one may ask whether the inequality\n\\[\n\\Lambda(g, e^{-f} \\, d\\mathrm{Vol}_g, m) \\leq \\Lambda(g_H, e^{-f} \\, d\\mathrm{Vol}_{g_H}, m)\n\\]\nholds for a fixed function \\( f \\in C^{\\infty}(M^n) \\) and \\( m > 0 \\).\n\n\\end{remark}\n\n\\section{LCF  $4$-manifolds with NSC}\\label{5}\n\nLet \\( (M^n, g) \\) be a closed, oriented, smooth, locally conformally flat Riemannian manifold with \\( n \\geq 4 \\) and positive scalar curvature. Schoen and Yau~\\cite[Theorem~4.5]{zbMATH04075988} prove that the  developing (conformal) map of \\( (\\widetilde{M^n}, \\tilde{g}) \\) is injective, and that \\( \\pi_1(M^n) \\) admits a faithful holonomy representation into a discrete subgroup of \\( \\operatorname{Conf}(S^n) \\). Let \\( \\Gamma \\) denote the image of \\( \\pi_1(M^n) \\) under the representation induced by the developing map. The image of the developing map is an open subset \\( \\Omega \\subset S^n \\) on which \\( \\Gamma \\) acts properly discontinuously, and \\( M^n \\) is diffeomorphic to the quotient \\( \\Omega / \\Gamma \\).\n\nMoreover, \\( \\Omega \\) coincides with the domain of discontinuity of \\( \\Gamma \\), and can be written as \\( \\Omega = S^n \\setminus \\Lambda \\), where \\( \\Lambda \\subset S^n \\) is the limit set of \\( \\Gamma \\). In particular, \\( \\Lambda \\) is the minimal closed \\( \\Gamma \\)-invariant subset of \\( S^n \\), and \\( \\Omega \\) serves both as the image of the developing map and as the maximal open set on which the action of \\( \\Gamma \\) is properly discontinuous.\n\nThe above results of Schoen and Yau~\\cite{zbMATH04075988}   hold in dimensions not less than four under the assumption \\( \\mathrm{Sc}_g \\geq c > 0 \\).\n  In that paper, they also remarked that their result \\cite[Proposition 4.4$^{\\prime}$]{zbMATH04075988}  would remain valid under the assumption of nonnegative scalar curvature, provided the positive energy theorem can be extended to the case of complete manifolds—namely, when \\( M \\) has one asymptotically flat end and other ends that are merely complete.  \nRecently, Lesourd, Unger, and Yau~\\cite[Theorem~1.2]{MR4836036} used minimal hypersurfaces to prove that there does not exist a complete Riemannian metric with positive scalar curvature on \\( T^3 \\# X \\), where \\( X \\) is an arbitrary (possibly noncompact) manifold.  \nFurthermore, they show that the nonexistence of a complete smooth metric with positive scalar curvature on \\( T^n \\# X \\) $(n\\geq 3)$ implies the following Liouville theorem~\\cite[Theorem~1.7]{MR4836036}, see also~\\cite[Corollary~4]{zbMATH07817078}:\n\n\\begin{theorem}[Liouville theorem]\\label{Liouville theorem}\nLet \\( (M^n, g) \\), \\( n \\ge 3 \\), be a complete, locally conformally flat manifold with nonnegative scalar curvature.  \nIf \\( \\Phi : M^n \\to S^n \\) is a conformal map, then \\( \\Phi \\) is injective and the boundary \\( \\partial \\Phi(M) \\) has zero Newtonian capacity (i.e.,  $2$-capacity).\n\\end{theorem}\n\nIn another paper, Lesourd, Unger, and Yau~\\cite[Theorem~1.2]{MR4773185} use $\\nu$-buddle to prove the above Schoen–Yau conjecture and obtain a new proof of Liouville theorem, which is completely in the spirit of the approach outlined in~\\cite{zbMATH04075988}.\n\n A Kleinian group $\\Gamma$ is called \\textit{elementary} if its limit set $\\Lambda$ is empty or consists of at most two points; otherwise, it is called \\textit{non-elementary}. If $\\Lambda$ is empty, then $\\Gamma$ is finite. If $\\Lambda$ consists of a single point, then $\\Gamma$ contains an abelian subgroup of finite index of rank $k$ with $k \\leq n$. If $\\Lambda$ consists of two points, then $\\Gamma$ contains an infinite cyclic subgroup of finite index. If the limit set $\\Lambda(\\Gamma)$ does not consist of exactly two points, $\\Lambda$ can be characterized as the minimal closed \\( \\Gamma \\)-invariant subset of \\( S^n \\). \n\n Let \\( B^{n+1} := \\{ x \\in \\mathbb{R}^{n+1} \\mid |x| < 1 \\} \\) be the Poincaré ball model endowed with the hyperbolic metric  $\ng_{H} = 4(1 - |x|^2)^{-2} \\sum_{i=1}^{n+1} (dx^i)^2.\n$ \nEvery element of the conformal group \\( \\mathrm{Conf}(S^n) \\) extends to a diffeomorphism of the closed ball \\( \\overline{B^{n+1}} := B^{n+1} \\cup S^n \\), and restricts to an isometry of \\( B^{n+1} \\) with respect to the hyperbolic metric. Conversely, every isometry in \\( \\mathrm{Isom}(B^{n+1}) \\) extends continuously to the boundary \\( S^n \\), acting as a conformal transformation of the standard sphere \\( (S^n, g_{st}) \\). Therefore, there is a natural group isomorphism: $\\mathrm{Isom}(B^{n+1}) \\cong \\mathrm{Conf}(S^n).$\n\nFor an infinite Kleinian group \\( \\Gamma \\), the \\emph{critical exponent} \\( \\delta(\\Gamma) \\) is defined by  \n\\[\n\\delta(\\Gamma) := \\inf\\left\\{ s > 0 \\,\\middle|\\, \\sum_{\\gamma \\in \\Gamma} \\exp\\left( -s \\, \\mathrm{dist}(x, \\gamma y) \\right) < \\infty \\right\\},\n\\]\nwhere  $x, y \\in B^{n+1}$ and \\( \\mathrm{dist}(\\cdot, \\cdot) \\) denotes the hyperbolic distance function on the Poincaré ball model \\( B^{n+1} \\). The value of \\( \\delta(\\Gamma) \\) is independent of the particular choice of \\( x \\) and \\( y \\). If \\( \\Gamma \\) is non-elementary, then \\( 0 < \\delta(\\Gamma) \\leq n \\). If \\( \\Gamma'  \\leq \\Gamma \\) is a subgroup, then   $\\delta(\\Gamma') \\leq \\delta(\\Gamma).$\n\nA Kleinian group \\( \\Gamma \\) is said to be \\emph{convex cocompact} if the quotient\n$(\\Omega(\\Gamma) \\cup B^{n+1}) / \\Gamma$ is compact. Equivalently, \\( \\Gamma \\) is convex cocompact if the hyperbolic convex hull \\( \\mathrm{CH}(\\Gamma) \\subset B^{n+1} \\) of the limit set \\( \\Lambda(\\Gamma) \\) satisfies that \\( \\mathrm{CH}(\\Gamma)/\\Gamma \\) is nonempty and compact. Here, \\( \\mathrm{CH}(\\Gamma) \\) denotes the minimal convex subset of \\( B^{n+1} \\) whose closure in the compactified ball \\( \\overline{B^{n+1}} := B^{n+1} \\cup S^n \\) contains \\( \\Lambda(\\Gamma) \\).\n\nA Kleinian group \\( \\Gamma \\) is said to be \\emph{geometrically finite} if there exists a uniform bound on the orders of its finite subgroups and the \\( \\epsilon \\)-neighborhood of \\( \\mathrm{CH}(\\Gamma)/\\Gamma \\) in \\( B^{n+1}/\\Gamma \\) has finite volume for some \\( \\epsilon > 0 \\). Equivalently, \\( \\Gamma \\) is geometrically finite if it admits a fundamental polyhedron in \\( B^{n+1} \\) with finitely many faces.\n\nConvex cocompact Kleinian groups are geometrically finite. Conversely, geometrically finite Kleinian groups without parabolic elements are precisely the convex cocompact ones.\n\nIf \\( \\Gamma \\) is a non-elementary geometrically finite Kleinian group, then the Patterson–Sullivan theorem~\\cite[Theorem~1]{zbMATH03903608} states that\n\\begin{equation*}\\label{Sullivan}\n\\delta(\\Gamma) = \\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)),\n\\end{equation*}\nwhere \\( \\dim_{\\mathcal{H}} \\) denotes the Hausdorff dimension of the limit set \\( \\Lambda(\\Gamma) \\).\n\nIn the following, we  study the topological types of locally conformally flat \\(4\\)-manifolds with scalar-flat metrics. One reason this is interesting is that the Riemannian connection associated with a locally conformally flat scalar-flat metric achieves an absolute minimum of the Yang–Mills functional on a closed oriented \\(4\\)-manifold. Therefore, it is natural to study the topological types of closed \\(4\\)-manifolds \\(M^4\\) admitting locally conformally flat scalar-flat metrics.\nBased on the topological classification of orientable, simply connected, closed \\(4\\)-manifolds, LeBrun and Maskit~\\cite[Corollary~1.2]{zbMATH05319680} characterized the manifolds that admit scalar-flat anti-self-dual metrics as follows: A compact, simply connected topological \\(4\\)-manifold \\(M\\) admits a smooth structure supporting a scalar-flat anti-self-dual metric \\(g\\) if and only if \\(M\\) is homeomorphic to one of the following:\n$K3$,  $\\mathbb{C}P^2 \\# k\\, \\overline{\\mathbb{C}P}^2 \\ (k \\ge 10)$, or $ k\\, \\overline{\\mathbb{C}P}^2 \\ (k \\ge 5).$\n\nWe now consider the case where the manifold \\( M^4 \\) is not necessarily simply connected.  By the Gauss–Bonnet–Chern formula, one has \\(\\chi(M^4) \\le 0\\), with equality if and only if \\( M^4 \\) admits a flat metric. In 1978, Brown, Bülow, Neubüser, Wondratschek, and Zassenhaus complete a computer-assisted classification of all isomorphism classes of \\(4\\)-dimensional crystallographic groups, thereby classifying closed flat \\(4\\)-manifolds \\cite{MR484179}. They obtained \\(74\\) homeomorphism equivalence classes of closed flat \\(4\\)-manifolds, consisting of \\(27\\) orientable classes and \\(47\\) non-orientable classes. \n\nFrom now on, suppose that \\(\\chi(M^4) < 0\\), and let \\(g\\) be a locally conformally flat scalar-flat metric on \\(M^4\\). In this case, the Ricci tensor of \\(g\\) must be nonzero, since otherwise \\(\\chi(M^4) = 0\\). \nThus, the metric \\(g\\) can be deformed to a metric \\(g'\\) with positive scalar curvature. \nHowever, the metric \\(g'\\) may not have vanishing Weyl tensor, \nbecause the existence of a locally conformally flat metric with positive scalar curvature implies that the second Betti number \\(b_2(M)\\) vanishes. \n\nMoreover, even when \\(b_2(M) = 0\\), the metric \\(g\\) cannot, in general, be deformed to a locally conformally flat metric with positive scalar curvature, as the following example shows.\n\n\\begin{example}\\label{example scalar flat}\nLet \\((\\Sigma_g, g_H)\\) be a hyperbolic surface that admits an orientation-reversing isometry \\(r: \\Sigma_g \\to \\Sigma_g\\). Consider \\((X, g) = (S^2 \\times \\Sigma_g, g_{st} \\oplus g_H)\\), and let \\(A: (S^2, g_{st}) \\to (S^2, g_{st})\\) be the antipodal map, \ni.e., the orientation-reversing fixed-point-free isometry. \nThen \\(F := A \\times r\\) acts freely and isometrically on \n\\((X, g) = (S^2 \\times \\Sigma_g, g_{st} \\oplus g_H)\\). \nSince \\(\\mathrm{deg}(F)=\\mathrm{deg}(A)\\mathrm{deg}(r) = (-1)\\cdot(-1) = +1\\), the map \\(F\\) is orientation-preserving. \n\nLet \\(M := X / \\langle F \\rangle\\). \nBecause \\(F\\) is an isometry, the metric \\(g\\) descends to a smooth metric \\(\\bar{g}\\) on \\(M\\); and since both the locally conformally flat and scalar-flat properties are local, \\((M, \\bar{g})\\) is still locally conformally flat and scalar-flat. For a free finite group action one has $H^2(M; \\mathbb{Q}) \\cong H^2(X; \\mathbb{Q})^{\\langle F \\rangle}$ (the $G$-invariants of the $H^2(X; \\mathbb{Q})$).\nNow, $H^2(X; \\mathbb{Q}) \\cong H^2(S^2; \\mathbb{Q}) \\oplus H^2(\\Sigma_g; \\mathbb{Q}),$\nwhich is generated by the area classes of the two factors. \nSince both \\(A\\) and \\(r\\) reverse orientation on their respective factors, \n\\(F_*\\) acts by \\(-1\\) on each summand. \nHence, the \\(\\langle F \\rangle\\)-invariant subspace of \\(H^2(X; \\mathbb{Q})\\) is trivial, and therefore $b_2(M) = \\dim H^2(M; \\mathbb{Q}) = 0.$ Thus, \\((M, \\bar{g})\\) is a closed, nonflat, locally conformally flat, scalar-flat \\(4\\)-manifold with \\(b_2(M) = 0\\).\n\nThe manifold \\( M \\) does not admit a locally conformally flat metric with positive scalar curvature.  \nOtherwise, a finite cover of \\( M^4 \\) would be diffeomorphic to \\( k\\#(S^1 \\times S^3) \\) for some \\( k \\ge 2 \\).  \nIts universal cover \\( \\tilde{M} \\) would then be homeomorphic to the complement of a Cantor set in \\( S^4 \\), \nso its second homotopy group would vanish.  \nHowever, \\( \\tilde{M} \\) is homeomorphic to \\( S^2 \\times \\mathbb{H}^2 \\), whose second homotopy group does not vanish — a contradiction.\n\\end{example}\n\nFor a closed, orientable, locally conformally flat \\(2n\\)-manifold $N^{2n}$ \\((n \\ge 2)\\) with nonnegative scalar curvature, \nNoronha~\\cite[Theorem~2]{MR1235219} analyzes the possible holonomy groups case by case and shows that either \\(b_n(N^{2n}) = 0\\), or \\(N^{2n}\\) is covered by \\(\\mathbb{E}^{2n}\\) or by \\(S^n \\times \\mathbb{H}^n\\).  \n\nFurthermore, by imposing the additional condition that, on a closed locally conformally flat scalar-flat \\(4\\)-manifold, the largest eigenvalue of the Ricci operator is not greater than the absolute value of its smallest eigenvalue, Noronha applies the Weitzenböck formula to \\(2\\)-forms to show that \\(\\nabla \\mathrm{Rm} \\equiv 0\\) in that paper; that is, the manifold is locally symmetric, and is covered by either \\(\\mathbb{E}^4\\) or \\(S^2 \\times \\mathbb{H}^2\\).\n\nNoronha's results were motivated by the following question posed in her paper concerning compact \\(4\\)-manifolds:\n\\begin{quote}\nIf \\( \\pi_2(M^4) \\neq 0 \\) and \\( M^4 \\) admits a conformally flat metric with zero scalar curvature, is \\( M^4 \\) covered by \\( S^2 \\times \\mathbb{H}^2 \\)?\n\\end{quote}\n\nWe will give a positive answer to her question in the following theorem.\n\n\\begin{theorem} \\label{the universal cover}\nLet \\( (M^4, g) \\) be a closed, locally conformally flat, and scalar–flat \\(4\\)-manifold with \\( \\pi_2(M^4) \\neq 0 \\).  Then its Riemannian universal cover \\( (\\widetilde{M}, \\tilde{g}) \\) is isometric to $(\\mathbb{H}^2  \\times S^2 ,  g_{H} \\oplus g_{st})$ up to homothety.\n\\end{theorem}\n\n\\begin{proof}\nThe manifold \\( M^4 \\) admits neither a flat metric nor a locally conformally flat metric with positive scalar curvature, since \\( \\pi_2(M^4) \\neq 0 \\). Identifying \\( \\pi_1(M) \\) with its holonomy image \\( \\Gamma \\), We have the diffeomorphisms  \n$M \\cong \\Omega / \\Gamma$, $\\widetilde{M} \\cong \\Omega = S^4 \\setminus \\Lambda(\\Gamma).$\n\nThe group \\( \\Gamma \\) is non-elementary.  Otherwise, \\(\\Lambda(\\Gamma)\\) would have at most two points, which would imply \\(\\pi_2(S^4 \\setminus \\Lambda(\\Gamma)) = 0\\).  \nHowever, by assumption, \\(\\pi_2(\\widetilde{M}) \\cong \\pi_2(\\Omega) = \\pi_2(S^4 \\setminus \\Lambda(\\Gamma)) \\neq 0\\), leading to a contradiction.\n\nThe group \\( \\Gamma \\) is geometrically finite. \nSince \\( M \\cong \\Omega / \\Gamma \\) is compact and admits a locally conformally flat, scalar–flat metric, we have \n\\[\n\\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) \\le 1 < 4.\n\\]\nChang, Qing, and Yang~\\cite[Theorem 0.1]{MR2070141} show that if \\( \\Gamma \\) is a nonelementary, finitely generated, conformally finite  subgroup of \\( \\mathrm{Conf}(S^n) \\), then \\( \\Gamma \\) is geometrically finite if and only if \\( \\dim_\\mathcal{H}(\\Lambda(\\Gamma)) < n \\). Here,  conformally finite means that \n$\\Omega(\\Gamma)/\\Gamma$ is the disjoint union of a compact set and finitely many standard conformal cusp ends.  Since \\( M \\cong \\Omega / \\Gamma \\) is the closed manifold, the group \\( \\Gamma \\) is finitely generated and conformally finite. Combining this with \\( \\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) \\le 1 < 4 \\), we conclude that \\( \\Gamma \\) is geometrically finite.\n\nThen, by the Patterson–Sullivan theorem, we have \n$\\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) = \\delta(\\Lambda),$ where \\( \\delta(\\Lambda) \\) is the critical exponent of \\( \\Gamma \\).  \nThe existence of the metric \\( g \\) implies \\( \\delta(\\Lambda) = 1 \\) by Nayatani’s theorem~\\cite[Corollary~3.4]{zbMATH01028179}. \n\nThe Čech–Alexander duality theorem states that for any nonempty compact subset \n\\( K \\subset S^n \\),\n\\[\n\\widetilde{H}_k(S^n \\setminus K; G) \n\\cong \n\\check{\\widetilde{H}}^{\\,n-k-1}(K; G),\n\\]\nthat is, the reduced singular homology of the complement is isomorphic to the reduced Čech cohomology of the set,\nfor every abelian coefficient group \\( G \\) and every \\( k \\ge 0 \\).\n\nSince \\( \\widetilde{M} \\) is simply connected and \\( \\pi_2(M) \\neq 0 \\), \nby Hurewicz theorem, one has\n\\[\n0\\neq \\pi_2(\\widetilde{M}) \\cong H_2(\\widetilde{M}; \\mathbb{Z}) \\cong \\widetilde{H}_2(S^4 \\setminus \\Lambda(\\Gamma); \\mathbb{Z})  \\cong \\check{\\widetilde{H}}^{\\,1}(\\Lambda; \\mathbb{Z})\\cong \\check{H}^{\\,1}(\\Lambda; \\mathbb{Z}).\n\\]\nHence \\( H_2(\\widetilde{M}; \\mathbb{Z}) \\neq 0 \\) implies $\\check{H}^{\\,1}(\\Lambda; \\mathbb{Z}) \\neq 0.$\n\nRecall the cohomological dimension \\( \\dim_G(X) \\) of a topological space \\( X \\) with respect to an abelian group \\( G \\) as the largest integer \\( n \\) such that there exists a closed subset \\( A \\subset X \\) with $\\check{H}^{\\,n}(X, A; G) \\neq 0.$ That is,\n\\[\n\\dim_G X := \\sup \\{\\, n : \\exists\\, A \\subset X \\text{ such that } \\check{H}^{\\,n}(X, A; G) \\neq 0 \\,\\}.\n\\]\n\nFor any abelian group \\( G \\) and any compact space \\( X \\), one has\n\\[\n\\dim_G X \\leq \\dim_{\\mathbb{Z}} X \\leq \\dim X,\n\\]\nwhere \\( \\dim X \\) denotes the topological (Lebesgue covering) dimension of \\( X \\).\n\nIn fact, by the Alexandroff theorem, if \\( X \\) is a compact metric space of finite topological dimension, then $\\dim_{\\mathbb{Z}} X = \\dim X.$\n\nThus, \\( \\check{H}^{\\,1}(\\Lambda; \\mathbb{Z}) \\neq 0 \\) implies that \n\\( \\dim_{\\mathbb{Z}}(\\Lambda(\\Gamma)) \\geq 1 \\), \nand hence the topological dimension of the limit set satisfies \n\\( \\dim(\\Lambda(\\Gamma)) \\geq 1 \\). (It also follows that \\( \\Gamma \\) is non-elementary.)\nTherefore,\n\\[\n1 \n\\le \\dim(\\Lambda(\\Gamma)) \n\\le \\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) \n\\le 1\n\\;\\implies\\;\n\\dim(\\Lambda(\\Gamma)) \n= \\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) = 1.\n\\]\n\nFinally, we need the following theorem of Kapovich~\\cite[Theorem~1.3]{MR2491697}:  \n\nSuppose that \\( \\Gamma \\subset \\mathrm{Isom}(\\mathbb{H}^n) \\) is a nonelementary, geometrically finite group such that the Hausdorff dimension of its limit set equals its topological dimension \\(d\\).  \nThen the limit set of \\( \\Gamma \\) is a round \\(d\\)-sphere.  \n\nThus, in our case, \\( \\Lambda(\\Gamma) \\) is a round circle.  Consequently, \\( \\widetilde{M},\\tilde{g}$ is conformal to $  (\\Omega = S^4 \\setminus S^1, g_{st}|_{S^4 \\setminus S^1}) \\).  By the following Lemma~\\ref{Stereographic}, it follows that \\( \\widetilde{M} \\) is conformal to $(\\mathbb{H}^2  \\times S^2 ,  g_{H} \\oplus g_{st}).$ Finally, Lemma~\\ref{iso} implies that \\( \\widetilde{M} \\) is isometric to  \n\\( (\\mathbb{H}^{2} \\times S^{2},\\, g_{H} \\oplus g_{st}) \\) up to homothety.\n\\end{proof}\n\n\\begin{lemma}\\label{Stereographic}\nLet $m\\ge 2$ and $1\\le k\\le m-2$. Then\n$$(S^{m}\\setminus S^{k}, g_{st}|_{S^{m}\\setminus S^{k}})\\ \\cong_{\\mathrm{conf}}\\ (\\mathbb H^{k+1}\\times S^{\\,m-k-1}, g_{H}\\oplus g_{st}).$$\nIn particular, for $m=2n$ and $k=n-1$,\n$S^{2n}\\setminus S^{n-1}\\ \\cong_{\\mathrm{conf}}\\ \\mathbb H^{n}\\times S^{n}.$\n\\end{lemma}\n\n\\begin{proof}\n\nFix $p\\in S^{k}\\subset S^{m}$ and let $\\sigma_p:\\ S^{m}\\setminus\\{p\\}\\longrightarrow \\mathbb R^{m}$ be stereographic projection from $p$. If a round subsphere passes through $p$,\nits image under $\\sigma_p$ is an affine linear subspace. Hence\n$\\Pi:=\\sigma_p(S^{k}\\setminus\\{p\\})$ is an affine $k$--plane in $\\mathbb R^{m}$ and\n\\[\n\\sigma_p:\\ S^{m}\\setminus S^{k}\\xrightarrow{\\ \\cong\\ }\\mathbb R^{m}\\setminus \\Pi.\n\\]\nChoose orthogonal coordinates $\\mathbb R^{m}=\\mathbb R^{k}\\oplus\\mathbb R^{m-k}$ so that\n$\\Pi=\\mathbb R^{k}\\times\\{0\\}$ and write $x=(u,v)$ with $u\\in\\mathbb R^{k}$, $v\\in\\mathbb R^{m-k}$.\nThen $\\mathbb R^{m}\\setminus\\Pi\\ \\cong\\ \\mathbb R^{k}\\times\\big(\\mathbb R^{m-k}\\setminus\\{0\\}\\big),$ and since $m-k\\ge 2$, the complement is connected.\n\nFor convenience in this proof, denote by \\( g_H = g_{\\mathbb{H}^{k+1}} \\) the hyperbolic metric on \\( \\mathbb{H}^{k+1} \\), and by \\( g_{\\mathrm{st}} = g_{S^{m-k-1}} \\) the standard round metric on \\( S^{m-k-1} \\). Write $v=\\rho\\,\\omega$ with $\\rho:=|v|>0$ and $\\omega\\in S^{m-k-1}$. The Euclidean metric splits as\n\\[\nds_{\\mathbb E^{m}}^{2}=|du|^{2}+d\\rho^{2}+\\rho^{2}\\,d\\omega^{2}.\n\\]\nRescaling by $\\rho^{-2}$ yields\n\\begin{equation*}\\label{eq:key}\n\\rho^{-2}\\,ds_{\\mathbb E^{m}}^{2}\n=\\frac{|du|^{2}+d\\rho^{2}}{\\rho^{2}}+d\\omega^{2}.\n\\end{equation*}\nIdentify $\\big(\\mathbb R^{k}\\times(0,\\infty), (|du|^{2}+d\\rho^{2})/\\rho^{2}\\big)$ with the upper half--space\nmodel of $\\mathbb H^{k+1}$ and $(S^{m-k-1},d\\omega^{2})$ with the round sphere. Thus\n\\[\n\\rho^{-2}\\,ds_{\\mathbb E^{m}}^{2} \\;=\\; g_{\\mathbb H^{k+1}}\\oplus g_{S^{m-k-1}}.\n\\]\nDefine\n\\[\nJ:\\ \\mathbb R^{m}\\setminus\\Pi\\longrightarrow \\mathbb H^{k+1}\\times S^{m-k-1},\\qquad\nJ(u,\\rho,\\omega)=\\big((u,\\rho),\\,\\omega\\big),\n\\]\nso that\n\\begin{equation}\\label{eq:pull}\nJ^{*}\\big(g_{\\mathbb H^{k+1}}\\oplus g_{S^{m-k-1}}\\big)=\\rho^{-2}\\,ds_{\\mathbb E^{m}}^{2}.\n\\end{equation}\n\nStereographic projection is conformal with\n\\begin{equation}\\label{eq:stereo}\n\\sigma_p^{*}\\,ds_{\\mathbb E^{m}}^{2}\n=\\Big(\\frac{1+|x|^{2}}{2}\\Big)^{2} g_{S^{m}},\\qquad x=(u,\\rho\\omega)=\\sigma_p(y).\n\\end{equation}\nLet $F:=J\\circ\\sigma_p:\\ S^{m}\\setminus S^{k}\\to \\mathbb H^{k+1}\\times S^{m-k-1}$. Combining\n\\eqref{eq:pull} and \\eqref{eq:stereo},\n\\[\n\\begin{aligned}\nF^{*}\\big(g_{\\mathbb H^{k+1}}\\oplus g_{S^{m-k-1}}\\big)\n&=(\\sigma_p)^{*}\\big(\\rho^{-2}\\,ds_{\\mathbb E^{m}}^{2}\\big)\n=\\rho^{-2}\\,\\sigma_p^{*} ds_{\\mathbb E^{m}}^{2}\\\\\n&=\\Big(\\frac{1+|x|^{2}}{2\\rho}\\Big)^{2} g_{S^{m}},\n\\end{aligned}\n\\]\na smooth positive multiple of $g_{S^{m}}$. Hence $F$ is a conformal diffeomorphism, proving\n\\[\nS^{m}\\setminus S^{k}\\ \\cong_{\\mathrm{conf}}\\ \\mathbb H^{k+1}\\times S^{m-k-1}.\n\\]\n\\end{proof}\n\n\\begin{lemma}\\label{iso}\nSuppose the Riemannian universal cover \\( (\\widetilde{M}, \\tilde{g}) \\) of a closed manifold \\( (M^{2n}, g) \\) ($n\\geq 2$) with \\( \\mathrm{Sc}_{\\tilde{g}} = 0 \\)  \nis conformal to\n$\n\\bigl( S^{n} \\times \\mathbb{H}^{n},\\;\ng_{0} := g_{st} \\oplus g_{H} \\bigr).$\nThen there exists a constant \\( c > 0 \\) such that\n$(\\widetilde{M}, \\tilde{g})$  is isometric to \n$\\bigl( S^{n} \\times \\mathbb{H}^{n},\\, c^{2} g_{0} \\bigr).\n$\n\\end{lemma}\n\n\\begin{proof}\nLet $F : (\\widetilde M, \\tilde g) \\longrightarrow S^{n} \\times \\mathbb{H}^{n}$\nbe a conformal diffeomorphism.  \nThen there exists a smooth function \n$\\varphi : \\widetilde M \\longrightarrow \\mathbb{R}$\nsuch that\n\\begin{equation*}\nF^{*}(g_{0}) = e^{-2\\varphi}\\,\\tilde g .\n\\end{equation*}\nEquivalently,\n$\\tilde g = e^{2\\varphi}\\, F^{*}(g_{0}).$\n\nSince $\\mathrm{Scal}(g_{0}) = 0$ and $\\mathrm{Scal}(\\tilde g)=0$, the standard conformal--change formula for the scalar curvature implies that\n\\[\n\\Delta_{\\tilde g}\\varphi \n= \\frac{m-2}{2}\\,|\\nabla\\varphi|^{2}_{\\tilde g} \\;\\geq\\; 0\n\\quad\\text{on } \\widetilde M.\n\\]\n\nLet $\\Gamma = \\pi_{1}(M)$ denote the group of deck transformations of the universal covering \n$\\widetilde M \\to M$. \nEach $\\gamma \\in \\Gamma$ acts on $\\widetilde M$ by an isometry of $(\\widetilde M,\\tilde g)$.\nDefine\n\\[\nf_{\\gamma} := F \\circ \\gamma \\circ F^{-1} : \nS^{n} \\times \\mathbb H^{n} \\longrightarrow S^{n} \\times \\mathbb H^{n}.\n\\]\nWe now compute the conformal factor of $f_{\\gamma}$ with respect to $g_{0}$.\n\nLet $\\psi := \\varphi \\circ F^{-1} : S^{n} \\times \\mathbb H^{n} \\to \\mathbb{R}$.\nThen we can rewrite\n\\[\n\\tilde g = e^{2\\varphi}\\,F^{*}(g_{0})\n\\qquad\\Longrightarrow\\qquad\n(F^{-1})^{*}\\tilde g = e^{2\\psi}\\,g_{0}.\n\\]\nSince $\\gamma$ is an isometry of $(\\widetilde M,\\tilde g)$, we have $\\gamma^{*}\\tilde g = \\tilde g$, and hence\n\\[\n\\gamma^{*}F^{*}g_{0}\n= \\gamma^{*}\\bigl(e^{-2\\varphi}\\tilde g\\bigr)\n= e^{-2\\varphi\\circ\\gamma}\\,\\tilde g.\n\\]\nUsing $f_{\\gamma} = F\\circ\\gamma\\circ F^{-1}$ and pulling back by $F^{-1}$, we obtain\n\\begin{align*}\nf_{\\gamma}^{*}g_{0}\n&= (F\\gamma F^{-1})^{*}g_{0}\n= (F^{-1})^{*}\\bigl(\\gamma^{*}F^{*}g_{0}\\bigr) \\\\\n&= (F^{-1})^{*}\\bigl(e^{-2\\varphi\\circ\\gamma}\\tilde g\\bigr) \\\\\n&= e^{-2(\\varphi\\circ\\gamma)\\circ F^{-1}} \\,(F^{-1})^{*}\\tilde g \\\\\n&= e^{-2\\psi\\circ f_{\\gamma}} \\, e^{2\\psi} g_{0}\n= e^{2(\\psi - \\psi\\circ f_{\\gamma})}\\,g_{0}.\n\\end{align*}\nThus each $f_{\\gamma}$ is a conformal diffeomorphism of $(S^{n} \\times \\mathbb H^{n},g_{0})$ with conformal factor $e^{2(\\psi - \\psi\\circ f_{\\gamma})}$.\n\nBy a result of Jimenez and Tojeiro \\cite[Cor.~2]{MR4374774}, every conformal diffeomorphism of a product \n$\\mathbb H^{k} \\times S^{\\ell}$ is in fact an isometry. \nApplying this to $f_{\\gamma}$, we conclude that $f_{\\gamma}$ is an isometry of $(S^{n} \\times \\mathbb H^{n},g_{0})$ for every $\\gamma \\in \\Gamma$. \nHence its conformal factor must be identically $1$, so that\n\\[\n\\psi \\circ f_{\\gamma} = \\psi\n\\qquad \\text{for all } \\gamma\\in\\Gamma.\n\\]\n\nReturning to $\\varphi$ on $\\widetilde M$, we note that\n\\[\n\\psi \\circ f_{\\gamma} \n= \\psi \\quad\\Longleftrightarrow\\quad \n(\\varphi \\circ F^{-1}) \\circ (F\\circ\\gamma\\circ F^{-1}) = \\varphi \\circ F^{-1}\n\\quad\\Longleftrightarrow\\quad \n\\varphi \\circ \\gamma = \\varphi.\n\\]\nThus\n\\begin{equation*}\n\\varphi \\circ \\gamma = \\varphi\n\\qquad\\text{for all } \\gamma \\in \\Gamma,\n\\end{equation*}\nand $\\varphi$ is $\\Gamma$--invariant. \nTherefore $\\varphi$ descends to a smooth function on the closed manifold $M$.\nSince $\\varphi$ is $\\Gamma$--invariant, there exists a smooth function \n$\\tilde\\varphi : M \\to \\mathbb R$ such that\n$\\varphi = \\tilde\\varphi \\circ \\pi.$\nBecause $\\pi : (\\widetilde M,\\tilde g) \\to (M,g)$ is a Riemannian covering, it is a local isometry and \n$\\tilde g = \\pi^{*}g$. In particular, for any smooth function $\\tilde\\varphi$ on $M$ one has\n\\[\n\\nabla_{\\tilde g}(\\tilde\\varphi \\circ \\pi) = (\\nabla_{g}\\tilde\\varphi)\\circ \\pi,\n\\qquad\n\\Delta_{\\tilde g}(\\tilde\\varphi \\circ \\pi) = (\\Delta_{g}\\tilde\\varphi)\\circ \\pi,\n\\]\nand therefore\n\\[\n|\\nabla\\varphi|^{2}_{\\tilde g}\n= |\\nabla(\\tilde\\varphi\\circ\\pi)|^{2}_{\\tilde g}\n= \\bigl(|\\nabla\\tilde\\varphi|^{2}_{g}\\bigr)\\circ\\pi.\n\\]\nSubstituting $\\varphi = \\tilde\\varphi\\circ\\pi$ into the identity\n\\[\n\\Delta_{\\tilde g}\\varphi = \\frac{m-2}{2}\\,|\\nabla\\varphi|^{2}_{\\tilde g}\n\\quad\\text{on } \\widetilde M\n\\]\nwe obtain\n\\[\n\\bigl(\\Delta_{g}\\tilde\\varphi\\bigr)\\circ\\pi\n= \\frac{m-2}{2}\\,\\bigl(|\\nabla\\tilde\\varphi|^{2}_{g}\\bigr)\\circ\\pi.\n\\]\nSince $\\pi$ is surjective, it follows that\n\\begin{equation*}\n\\Delta_{g}\\tilde\\varphi = \\frac{m-2}{2}\\,|\\nabla\\tilde\\varphi|^{2}_{g}\n\\qquad\\text{on } M.\n\\end{equation*}\n\nSince $\\tilde\\varphi$ is now globally defined on $M$, we may integrate the identity\n\\[\n\\Delta_{g}\\tilde\\varphi = \\frac{m-2}{2}\\,|\\nabla\\tilde\\varphi|^{2}_{ g}\n\\]\nover $M$:\n\\[\n0 = \\int_{M} \\Delta_{ g}\\tilde\\varphi\\, d\\mathrm{vol}_{ g}\n= \\frac{m-2}{2} \\int_{M} |\\nabla\\tilde\\varphi|^{2}_{g}\\, d\\mathrm{vol}_{ g}.\n\\]\nIt follows that $|\\nabla\\tilde\\varphi|_{ g} \\equiv 0$ on $M$, and hence $\\varphi$ is constant on $\\widetilde M$ as well. \nLet $c := e^{\\varphi} > 0$ denote this constant. \nThus, we obtain\n\\[\n\\tilde g = e^{2\\varphi} F^{*} g_{0} = c^{2} F^{*}g_{0},\n\\]\nand, upon identifying $\\widetilde M$ with $S^{n} \\times \\mathbb H^{n}$ via $F$, we conclude\n$\\tilde g = c^{2} g_{0}.$\nThus $(\\widetilde M,\\tilde g)$ is isometric to $\\bigl(S^{n} \\times \\mathbb H^{n}, c^{2} g_{0}\\bigr)$, \nas claimed.\n\\end{proof}\n\n\\begin{remark}\nThe condition \\(\\pi_2(M^4) \\neq 0\\) in Theorem~\\ref{the universal cover} cannot be replaced by the nonexistence of a flat metric.  \nFor example, LeBrun and Maskit~\\cite[Proposition~3.3]{zbMATH05319680} show that there exists a smooth family of metrics \\(h_t\\) on \\((S^1 \\times S^3) \\# (S^1 \\times S^3)\\), \\(t \\in [-1,1]\\), such that for each \\(t\\), the metric \\(h_t\\) is locally conformally flat, with $Y(M, [h_1]) > 0$ and  $Y(M, [h_{-1}]) < 0.$ On the other hand, Große and Nardmann~\\cite[Theorem~1]{MR3192307} show that if \\(N\\) is compact, then the Yamabe constant \\(Y(N,[g])\\) is continuous at \\(g\\) with respect to the compact-open \\(C^2\\)-topology.  Therefore, there exists \\(s \\in (-1,1)\\) such that \\(Y(M, [h_s]) = 0\\).  \nThus, \\(h_s\\) is conformal to a locally conformally flat scalar-flat metric.  However, the universal cover of \\((S^1 \\times S^3) \\# (S^1 \\times S^3)\\) is homeomorphic to the complement of a Cantor set in \\(S^4\\). Its second fundamental group is trivial, so it is not of the form \\(S^2 \\times \\mathbb{H}^2\\).  \n\n LeBrun and Maskit’s examples also show that the Hausdorff dimension of a Kleinian group depends not only on the group’s algebraic structure but also on its specific representation in the isometry group of hyperbolic space. For instance, for \n\\[\n(M, h_1) = \\bigl((S^1 \\times S^3) \\# (S^1 \\times S^3),\\, h_1\\bigr),\n\\quad \\Gamma_1 = \\rho_1(\\pi_1(M)),\n\\]\none has \\( \\dim_{\\mathcal{H}}(\\Gamma_1) < 1 \\), \nwhereas for \n\\[\n(M, h_s) = \\bigl((S^1 \\times S^3) \\# (S^1 \\times S^3),\\, h_s\\bigr),\n\\quad \\Gamma_s = \\rho_s(\\pi_1(M)),\n\\]\none obtains \\( \\dim_{\\mathcal{H}}(\\Gamma_s) = 1 \\).\n\n\\end{remark}",
      "4": "\\begin{proof}[Proof of Theorem \\ref{B}]\nIf the bundle is trivial, Proposition \\ref{split4} implies the conclusion.  \nIf the fiber or the base is $S^2$, Proposition \\ref{S^2} implies the conclusion.  \nIf the fiber is $S^1$, Proposition \\ref{S^1-fiber} implies the conclusion.  \n\nAny smooth fiber bundle $\\pi: M \\to S^1$ with fiber $N$ is isomorphic to the mapping torus $M_\\varphi$ of a diffeomorphism $\\varphi: N \\to N$.  \nThus, when the base is $S^1$, $M^4$ is the mapping torus of $N^3$, and Propositions \\ref{base manifold} and \\ref{mapping torus} complete the proof.\n\\end{proof}\n\nIt remains an open problem to classify the topological types of closed $4$–manifolds that admit scalar-flat metrics but no metrics of positive scalar curvature. In other words, the topological classification of Ricci-flat $4$–manifolds is still not fully understood. In particular, no examples are known of closed simply connected Ricci-flat manifolds with generic holonomy, and the existence of a Ricci-flat metric on $S^n$ for $n \\ge 4$ remains a major open problem.\n\n\\section{Rigidity theorems about scalar curvature}\\label{4}\n\nObata \\cite[Proposition 6.1.]{zbMATH03374588}  shows that if $g \\in [g_{st}]$ with $Sc_g\\equiv n(n-1)$ on $S^n$, then the metric $g$ is isometric to  $g_{st}$. In fact, if $(M^n, g)$ is Einstein with positive scalar curvature, then $g$ is the unique constant scalar curvature metric in its conformal class (up to  a scaling factor). However, on $S^n$, if the conformal class of a metric does not contain  an Einstein metric, then the number of constant positive scalar curvature metrics within  the conformal class may  not be unique up to  a scaling factor. \n\nThe above result of Obata is a clue for Obata's solution of the conjecture on conformal transformation of Riemannian manifold and the proof of the sharp Sobolev inequality on ($S^n,g_{st}$) (see, \\cite[P. 121, Theorem 5.1]{MR1688256}), which means that every smooth function $f \\in C^{\\infty}(S^n)$ satisfies  the following inequality:\n\\begin{equation*}\n(\\int_{S^n}|f|^{\\frac{2n}{n-2}}dv_{g_{st}})^\\frac{n-2}{n}\\leq \\frac{4}{n(n-2)\\mathrm{Vol}_{g_{st}}(S^n)^{\\frac{2}{n}}}\\int_{S^n}|\\nabla f|^2_{g_{st}}dv_{g_{st}}  + \\frac{1}{\\mathrm{Vol}_{g_{st}}(S^n)^{\\frac{2}{n}}} \\int_{S^n}f^2dv_{g_{st}}.\n\\end{equation*}\nEquality holds if and only if $f$ is a constant. Conversely, the sharp Sobolev inequality can be used to establish the following weaker version of Obata's result.\n\n\\begin{theorem}\\label{LCF n}\nLet ($M^n,g$) ($n\\geq 3$) be a oriented, closed, simply connected, locally conformally flat manifold  with $\\mathrm{Sc}_g = n(n-1)$, then \\( \\mathrm{Vol}_g(M) \\geq \\mathrm{Vol}_{g_{st}}(S^n) \\). If the equality holds, then ($M^n,g$) is isometric to ($S^n,g_{st}$).\n\\end{theorem}\n\n\\begin{proof}\nSince a closed, simply connected, locally conformally flat manifold is conformally diffeomorphic to the round sphere ($S^n, g_{st}$), the above assumptions guarantee the existence of a positive smooth function \\( u \\) on \\( S^n \\) and a diffeomorphism \\( \\phi: (S^n, g_{st}) \\to (M,g) \\) such that    $\\phi^*g=u^{4/(n-2)}g_{st}$.  We continue to denote \\( \\phi^*g \\) by \\( g \\).   Moreover, the assumption \\( \\mathrm{Sc}_g \\equiv n(n-1) \\), together with the formula for scalar curvature under a conformal transformation, implies that the function \\( u \\) must satisfy the following equation:\n\\begin{equation}\\label{n(n-1)}\n\\frac{4(n-1)}{(n-2)}\\Delta_{g_{st}}u  +  n(n-1)u =n(n-1)u^{\\frac{n+2}{n-2}}.\n\\end{equation}\n\nOn the other hand, the function  $u$ also satisfies the sharp Sobolev inequality,\n\\begin{equation*}\n(\\int_{S^n}u^{\\frac{2n}{n-2}}dv_{g_{st}})^\\frac{n-2}{n}\\leq \\frac{1}{n(n-1)\\mathrm{Vol}_{g_{st}}(S^n)^{\\frac{2}{n}}}\\int_{S^n}(\\frac{4(n-1)}{n-2}|\\nabla u|^2_{g_{st}}+n(n-1)u^2)dv_{g_{st}}.\n\\end{equation*}\nEquality holds if and only if $u$ is a constant.  Multiplying equation (\\ref{n(n-1)}) by \\( u \\) and integrating, we obtain:\n\\begin{equation*}\n\\int_{S^n}(\\frac{4(n-1)}{n-2}|\\nabla u|^2_{g_{st}}+n(n-1)u^2)dv_{g_{st}}=n(n-1)\\int_{S^n}u^{\\frac{2n}{n-2}}dv_{g_{st}}.\n\\end{equation*}\nHence, we have \n\\begin{equation}\\label{Sobolev}\n(\\int_{S^n}u^{\\frac{2n}{n-2}}dv_{g_{st}})^\\frac{n-2}{n}\\leq \\frac{1}{\\mathrm{Vol}_{g_{st}}(S^n)^{\\frac{2}{n}}}\\int_{S^n}u^{\\frac{2n}{n-2}}dv_{g_{st}}.\n\\end{equation}\n\nSince \n\\begin{equation*}\n  \\mathrm{Vol}_g(M)=\\int_{S^n}u^{\\frac{2n}{n-2}}dv_{g_{st}},\n\\end{equation*}\nthus, inequality  (\\ref{Sobolev}) means\n\\begin{equation*}\n \\mathrm{Vol}_g(M)^\\frac{n-2}{n} \\leq \\frac{\\mathrm{Vol}_g(M)}{\\mathrm{Vol}_{g_{st}}(S^n)^{\\frac{2}{n}}}\n\\end{equation*}\nThat means \\( \\mathrm{Vol}_g(M) \\geq \\mathrm{Vol}_{g_{st}}(S^n) \\). If \\( \\mathrm{Vol}_g(M) = \\mathrm{Vol}_{g_{st}}(S^n) \\), then the sharp Sobolev inequality must achieve equality, which implies \\( u \\equiv 1 \\), i.e., \\( (M^n, g) \\) is isometric to \\( (S^n, g_{st}) \\).\n\n\\end{proof}"
    },
    "pre_theorem_intro_text_len": 741,
    "pre_theorem_intro_text": "For metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally.  Globally, however, the behavior is more subtle.  \nIn the study of the Yamabe problem, Schoen \\cite{MR994021} conjectures that if \\( g \\) is any Riemannian metric on a closed hyperbolic \\( n \\)-manifold \\((M^n, g_H)\\) with \\( n \\ge 3 \\) and its scalar curvature satisfies $\\mathrm{Sc}_g \\ge -n(n-1),$ then the total volume of \\( M \\) with respect to \\( g \\) is at least as large as the hyperbolic volume of \\( M \\). In dimension 3, Schoen's conjecture has been studied by Anderson \\cite[page~132]{MR2213687}. In this paper, we  give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.",
    "context": "For metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally. Globally, however, the behavior is more subtle. \nIn the study of the Yamabe problem, Schoen \\cite{MR994021} conjectures that if \\( g \\) is any Riemannian metric on a closed hyperbolic \\( n \\)-manifold \\((M^n, g_H)\\) with \\( n \\ge 3 \\) and its scalar curvature satisfies $\\mathrm{Sc}_g \\ge -n(n-1),$ then the total volume of \\( M \\) with respect to \\( g \\) is at least as large as the hyperbolic volume of \\( M \\). In dimension 3, Schoen's conjecture has been studied by Anderson \\cite[page~132]{MR2213687}. In this paper, we give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.",
    "full_context": "For metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally. Globally, however, the behavior is more subtle. \nIn the study of the Yamabe problem, Schoen \\cite{MR994021} conjectures that if \\( g \\) is any Riemannian metric on a closed hyperbolic \\( n \\)-manifold \\((M^n, g_H)\\) with \\( n \\ge 3 \\) and its scalar curvature satisfies $\\mathrm{Sc}_g \\ge -n(n-1),$ then the total volume of \\( M \\) with respect to \\( g \\) is at least as large as the hyperbolic volume of \\( M \\). In dimension 3, Schoen's conjecture has been studied by Anderson \\cite[page~132]{MR2213687}. In this paper, we give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.\n\n\\tableofcontents\n\\section{Introduction}\nFor metrics with negative scalar curvature, making the scalar curvature more negative tends to increase the volume locally. Globally, however, the behavior is more subtle. \nIn the study of the Yamabe problem, Schoen \\cite{MR994021} conjectures that if \\( g \\) is any Riemannian metric on a closed hyperbolic \\( n \\)-manifold \\((M^n, g_H)\\) with \\( n \\ge 3 \\) and its scalar curvature satisfies $\\mathrm{Sc}_g \\ge -n(n-1),$ then the total volume of \\( M \\) with respect to \\( g \\) is at least as large as the hyperbolic volume of \\( M \\). In dimension 3, Schoen's conjecture has been studied by Anderson \\cite[page~132]{MR2213687}. In this paper, we give a partial confirmation of Schoen's conjecture in dimension 4 in the following cases.\n\nThe strategy of the proof is to reduce the problem to the Yamabe metric.\nFirst, we establish a Schoen–conjecture–type volume inequality for the Yamabe metric of negative scalar curvature in the given conformal class.\nSecond, we apply the Gauss–Bonnet–Chern formula to the Yamabe metric to complete the argument.\n\n\\begin{theorem} \\label{local}\nLet \\( g \\) be a Riemannian metric with constant scalar curvature \\( \\mathrm{Sc}_g \\equiv -12 \\) on a closed, oriented hyperbolic 4-manifold \\( (M^4, g_H) \\). Assume that for every point \\( x \\in M^4 \\), there exists a sufficiently small radius \\( r(x) > 0 \\) such that for all \\( 0 < r \\leq r(x) \\), the volumes of the geodesic \\( r \\)-balls \\( B_r(x) \\) satisfy\n\\[\n\\mathrm{Vol}_g(B_r(x)) \\geq \\mathrm{Vol}_{g_H}(B_r(x)).\n\\]\nThen it follows that\n$ \\mathrm{Vol}_g(M^4) \\geq \\mathrm{Vol}_{g_H}(M^4)$. Moreover, if equality holds, i.e., \\( \\mathrm{Vol}_g(M^4) = \\mathrm{Vol}_{g_H}(M^4) \\), then \\( g \\equiv g_H \\).\n\n\\begin{corollary}\nLet \\( g \\) be a Riemannian metric with constant scalar curvature \\( \\mathrm{Sc}_g \\equiv 12 \\) on a $(S^4,g_{st})$. Assume that for every point \\( x \\in S^4 \\), there exists a sufficiently small radius \\( r(x) > 0 \\) such that for all \\( 0 < r \\leq r(x) \\), the volumes of the geodesic \\( r \\)-balls \\( B_r(x) \\) satisfy\n\\[\n\\mathrm{Vol}_g(B_r(x)) \\geq \\mathrm{Vol}_{g_{st}}(B_r(x)).\n\\]\nThen it follows that\n$ \\mathrm{Vol}_g(S^4) \\geq \\mathrm{Vol}_{g_{st}}(S^4)$. Moreover, if equality holds, i.e., \\( \\mathrm{Vol}_g(S^4) = \\mathrm{Vol}_{g_{st}}(S^4) \\), then \\( g \\) is isometric to $g_{st}$.\n\nMotivated by Conjecture \\ref{4D CSC}, we propose the following volume rigidity conjecture in higher dimensions:\n\\begin{conjecture}[Volume Rigidity Conjecture] \\label{HYC}\n Assume a Riemannian metric $g$ on a closed hyperbolic $n$-manifold $(M^n,g_H)$ ($n\\geq 5$) satisfies $\\mathrm{Sc}_g=-n(n-1)$ and $\\mathrm{Vol}_g(M^n)=\\mathrm{Vol}_{g_H}(M^n)$, then the metric $g$ is isometric to the hyperbolic metric $g_H$. \n\\end{conjecture}\n\n\\begin{theorem}\\label{Einstein}\nLet $(N,g_N)$ be an orientable closed (Riemannian) Einstein $n$-manifold with scalar curvature $R^*<0$ and $(M,g_M)$ be an orientable closed Riemannian $n$-manifold with $\\mathrm{Sc}_g\\geq R^*$. Suppose there exists a smooth non-zero degree $1$-expansive map $f: (M,g_M) \\to (N, g_N)$, i.e., $|f_*v|_{g_N}\\geq |v|_{g_M}$ for $v\\in TM$, and the map $f$ is harmonic with condition $C\\leq 0$, then $f$ is a locally isometric map.\n\nInspired by Theorem \\ref{Einstein}, we propose the following Llarull-type rigidity conjecture in the hyperbolic setting.\n\\begin{conjecture}[Hyperbolic Rigidity Conjecture]\nAssume $(N^n,g)$ is an orientable closed $n$-manifold with $\\mathrm{Sc}_g\\geq -n(n-1)$ and $(M^n,g_H)$ be an orientable closed hyperbolic $n$-manifold. Suppose there exists a smooth non-zero degree $1$-expansive map $f: (N^n,g) \\to (M^n, g_H)$, i.e., $|f_*v|_{g_H}\\geq |v|_{g_N}$ for $v\\in TN$, then $f$ is a isometric map. \n\\end{conjecture}\n\n\\begin{lemma}\\label{iso}\nSuppose the Riemannian universal cover \\( (\\widetilde{M}, \\tilde{g}) \\) of a closed manifold \\( (M^{2n}, g) \\) ($n\\geq 2$) with \\( \\mathrm{Sc}_{\\tilde{g}} = 0 \\) \nis conformal to\n$\n\\bigl( S^{n} \\times \\mathbb{H}^{n},\\;\ng_{0} := g_{st} \\oplus g_{H} \\bigr).$\nThen there exists a constant \\( c > 0 \\) such that\n$(\\widetilde{M}, \\tilde{g})$ is isometric to \n$\\bigl( S^{n} \\times \\mathbb{H}^{n},\\, c^{2} g_{0} \\bigr).\n$\n\\end{lemma}",
    "post_theorem_intro_text_len": 5960,
    "post_theorem_intro_text": "The strategy of the proof is to reduce the problem to the Yamabe metric.\nFirst, we establish a Schoen–conjecture–type volume inequality for the Yamabe metric of negative scalar curvature in the given conformal class.\nSecond, we apply the Gauss–Bonnet–Chern formula to the Yamabe metric to complete the argument.\n\nSince any finitely presented group $\\pi$ can be realized as the fundamental group of a closed Riemannian $4$--manifold with positive scalar curvature, additional assumptions are necessary in order to classify Riemannian $4$--manifolds admitting positive scalar curvature. For example, when the fundamental group is trivial, the combined results of Freedman and Donaldson lead to a classification, up to homeomorphism, of closed Riemannian $4$--manifolds that admit metrics of positive scalar curvature. Independently, by imposing the assumption that the manifold is the total space of a fiber bundle with fiber of positive dimension, one obtains the following classification.\n\n\\begin{theoremB}\\label{B}\nLet $M^4$ be a closed, oriented, smooth $4$–manifold which is the total space of a fiber bundle. Then:\n\\begin{enumerate}\n\\item[(1)] If either the fiber or the base is $S^2$, then $M^4$ admits a Riemannian metric of positive scalar curvature.\n\\item[(2)] If either the fiber or the base is a closed, oriented, connected $3$–manifold $N^3$, then $M^4$ admits a Riemannian metric of positive scalar curvature if and only if $N^3$ does.\n\\end{enumerate}\n\\end{theoremB}\n\nWhen the fiber or the base is \\( S^2 \\), the proof relies on results concerning the group of orientation-preserving diffeomorphisms of oriented closed surfaces; \nWhen the fiber is  \\( N^3 \\), the argument depends on the classification of \\(3\\)-manifolds admitting metrics of positive scalar curvature;  \n When the base is $N^3$ and $N^3$ admits a metric of positive scalar curvature, we use the fact that the space of positive scalar curvature metrics on $N^3$ is path-connected in order to glue concordance metrics and thereby construct a metric of positive scalar curvature on $M^4$. The main result of Bamler and Kleiner \\cite{2019arXiv190908710B} shows that the space of positive scalar curvature metrics on $N^3$ is contractible.\n Relying on this result, one can complete the proof.\n\nOn the other hand, closed locally conformally flat Riemannian $4$--manifolds with positive scalar curvature were classified by Hamilton, Chen, Tang, and Zhu \\cite{zbMATH06081388} using the Ricci flow. A natural next step is to classify locally conformally flat $4$--manifolds admitting scalar-flat metrics. One motivation for this study is that the Riemannian connection associated with a locally conformally flat, scalar-flat metric attains an absolute minimum of the Yang--Mills functional on a closed oriented $4$--manifold.\n Note that for a locally conformally flat $4$-manifold $M^4$ with scalar-flat metric, its universal cover may not be the Euclidean space $\\mathbb{E}^4$ or the product $(\\mathbb{H}^2 \\times S^2, g_{H} \\oplus g_{st})$.  \n\nIn 1993, Noronha \\cite{MR1235219} asks whether, under the additional assumption that $\\pi_2(M^4) \\neq 0$, the universal cover of $M^4$ is $(\\mathbb{H}^2 \\times S^2, g_{H} \\oplus g_{st})$.  She shows that if $b_2(M^4) > 0$, then $M^4$ is covered by $(\\mathbb{H}^2 \\times S^2, g_{H} \\oplus g_{st})$ using Bochner-type formulas.  We provide a positive answer to Noronha's question in the following.\n\n\\begin{theoremC}[Theorem \\ref{the universal cover}]\\label{D}\nLet \\( (M^4, g) \\) be a closed, locally conformally flat, and scalar–flat \\(4\\)-manifold with \\( \\pi_2(M^4) \\neq 0 \\).  Then its Riemannian universal cover \\( (\\widetilde{M}, \\tilde{g}) \\) is isometric to $(\\mathbb{H}^2  \\times S^2 ,  g_{H} \\oplus g_{st})$ up to homothety.\n\\end{theoremC}\n\nUsing the Liouville theorem (Theorem \\ref{Liouville theorem}) of Schoen and Yau \\cite{MR4836036} together with the theorem of Chang–Qing–Yang \\cite{MR2070141}, we show that $\\pi_1(M^{4})$, viewed as a Kleinian group $\\Gamma$, is geometrically finite.  \nIt then follows from the Patterson–Sullivan theorem ~\\cite{zbMATH03903608} and Nayatani's theorem \\cite{zbMATH01028179} that the critical exponent satisfies \n$\\delta(\\Gamma) = \\dim_{\\mathcal{H}}(\\Lambda(\\Gamma)) = 2.$  \nThe condition \\( \\pi_2(M^4) \\neq 0 \\)  further implies, via Alexander duality, that the topological dimension satisfies $\\dim(\\Lambda(\\Gamma)) \\geq 2$.  \nFinally, by Kapovich's theorem \\cite{MR2491697}, $\\Lambda(\\Gamma)$ is embedded as a round $2$-sphere, which completes the proof. The proof also holds for manifolds $M^{2n}$ ($n \\geq 2$) if one replaces the condition $\\pi_2(M^4) \\neq 0$ by $\\widetilde{H}_n(\\widetilde{M}; \\mathbb{Z}) \\neq 0$.\n\n\\paragraph*{Organization of the Paper.}\nIn Section~\\ref{2}, we partially verify Schoen's conjecture in dimension~4. In Section~\\ref{3}, we  characterize Riemannian $4$–manifolds with positive scalar curvature that arise as the total space of a fiber bundle. In Section~\\ref{4}, results and questions related to the rigidity of negative scalar curvature are  discussed. In Section~\\ref{5},  we  characterize the universal cover of locally conformally flat scalar-flat $2n$–manifolds under an additional condition.\n\n\\paragraph*{Acknowledgment.}          \nThe author acknowledges support from the Oberwolfach Leibniz Fellows programme (MFO), the YMSC Overseas Shuimu Scholarship, the Simons Center for Geometry and Physics, and ICMS Edinburgh (workshops on Geometric Measure Theory on Metric Spaces with Applications to Physics and Geometry and Geometric Moduli Spaces, respectively). I  thank Gerhard Huisken for discussions on the Ricci flow. This work originates from a broader project initiated during my postdoctoral stay at the Yau Center. During that time, this work was also supported by NSFC  12401063 and partially by  NSFC  12271284. I am deeply grateful to Shing-Tung Yau and Akito Futaki for their trust and support, which allowed me to pursue independent research.",
    "sketch": "To prove Theorem~\\ref{A}, the proof strategy is to \\emph{reduce the problem to the Yamabe metric}. First, the authors \\emph{establish a Schoen--conjecture--type volume inequality} for the Yamabe metric of negative scalar curvature in the given conformal class. Second, they \\emph{apply the Gauss--Bonnet--Chern formula} to the Yamabe metric to complete the argument.",
    "expanded_sketch": "To prove the main theorem, the proof strategy is to \\emph{reduce the problem to the Yamabe metric}. First, the authors \\emph{establish a Schoen--conjecture--type volume inequality} for the Yamabe metric of negative scalar curvature in the given conformal class. Second, they \\emph{apply the Gauss--Bonnet--Chern formula} to the Yamabe metric to complete the argument.",
    "expanded_theorem": "\\label{A}\nLet \\( (M^4, g_H) \\) be a closed hyperbolic \\(4\\)-manifold.\nSuppose \\( g_1 \\) is a Riemannian metric on \\( M^4 \\) with \n\\( \\mathrm{Sc}_{g_1} \\ge -12 \\).\nAssume in addition that one of the following holds:\n\\begin{enumerate}\n    \\item[(i)] \\( g_1 \\) is self–dual or anti–self–dual;\n    \\item[(ii)] the Yamabe metric \\( g \\in [g_1] \\) with \\( \\mathrm{Sc}_g = -12 \\)\n    satisfies one of the conditions:\n    \\begin{enumerate}\n        \\item[(a)] \\( g \\) is Einstein;\n        \\item[(b)] for every point \\( x \\in (M^4, g) \\), there exists \n        \\( r(x) > 0 \\) such that for all \\( 0 < r \\le r(x) \\), the geodesic \\( r \\)-balls satisfy $  \\mathrm{Vol}_g(B^g_r(x)) \\ge  \\mathrm{Vol}_{g_H}(B^{g_H}_r(x)). $\n    \\end{enumerate}\n\\end{enumerate}\n\nThen $\\mathrm{Vol}_{g_1}(M^4) \\geq \\mathrm{Vol}_{g_H}(M^4)$. Moreover, if equality holds, i.e., \\( \\mathrm{Vol}_{g_1}(M^4) = \\mathrm{Vol}_{g_H}(M^4) \\), then  \\( g_1 \\) is isometric to \\( g_H \\).,",
    "theorem_type": [
      "Inequality or Bound",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\((M^4,g_H)\\) be a closed hyperbolic 4-manifold, where \\(g_H\\) is the hyperbolic metric. Let \\(g_1\\) be a Riemannian metric on \\(M^4\\) with scalar curvature \\(\\mathrm{Sc}_{g_1}\\ge -12\\). Assume in addition that either\n(i) \\(g_1\\) is self-dual or anti-self-dual, or\n(ii) if \\(g\\in[g_1]\\) denotes the Yamabe metric in the conformal class of \\(g_1\\) with constant scalar curvature \\(\\mathrm{Sc}_g=-12\\), then either\n(a) \\(g\\) is Einstein, or\n(b) for every point \\(x\\in M^4\\), there exists \\(r(x)>0\\) such that for every \\(0<r\\le r(x)\\), the geodesic balls satisfy\n\\[\n\\mathrm{Vol}_g(B_r^g(x))\\ge \\mathrm{Vol}_{g_H}(B_r^{g_H}(x)),\n\\]\nwhere \\(B_r^g(x)\\) and \\(B_r^{g_H}(x)\\) are the radius-\\(r\\) geodesic balls centered at \\(x\\) in the metrics \\(g\\) and \\(g_H\\), respectively.\n\nWhich quantitative global volume statement holds under these assumptions?",
      "correct_choice": {
        "label": "A",
        "text": "\\(\\mathrm{Vol}_{g_1}(M^4)\\ge \\mathrm{Vol}_{g_H}(M^4)\\). Moreover, if \\(\\mathrm{Vol}_{g_1}(M^4)=\\mathrm{Vol}_{g_H}(M^4)\\), then \\(g_1\\) is isometric to \\(g_H\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\(\\mathrm{Vol}_{g_1}(M^4)\\le \\mathrm{Vol}_{g_H}(M^4)\\). Moreover, if \\(\\mathrm{Vol}_{g_1}(M^4)=\\mathrm{Vol}_{g_H}(M^4)\\), then \\(g_1\\) is isometric to \\(g_H\\)."
        },
        {
          "label": "C",
          "text": "\\(\\mathrm{Vol}_{g_1}(M^4)\\ge \\mathrm{Vol}_{g_H}(M^4)\\)."
        },
        {
          "label": "D",
          "text": "\\(\\mathrm{Vol}_{g_1}(M^4)\\ge \\mathrm{Vol}_{g_H}(M^4)\\) provided that in case \\((ii)(b)\\) there exists a single radius \\(r_0>0\\) such that for every point \\(x\\in M^4\\) and every \\(0<r\\le r_0\\), the geodesic balls satisfy \\(\\mathrm{Vol}_g(B_r^g(x))\\ge \\mathrm{Vol}_{g_H}(B_r^{g_H}(x))\\). Moreover, if equality holds, then \\(g_1\\) is isometric to \\(g_H\\)."
        },
        {
          "label": "E",
          "text": "\\(\\mathrm{Vol}_{g}(M^4)\\ge \\mathrm{Vol}_{g_H}(M^4)\\). Moreover, if \\(\\mathrm{Vol}_{g}(M^4)=\\mathrm{Vol}_{g_H}(M^4)\\), then the Yamabe metric \\(g\\) is isometric to \\(g_H\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "direction of global volume inequality after Yamabe reduction",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped rigidity/equality conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "pointwise-dependent small-radius condition replaced by a uniform radius",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "global conclusion for \\(g_1\\) replaced by conclusion only for the Yamabe metric \\(g\\)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It sets up a theorem-like hypothesis package, but the exact direction, sharpness, and rigidity conclusion still have to be chosen from the options."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-completion item: the hypotheses are stated in full and the student is asked to identify the conclusion. That makes it very close to a direct restatement rather than an independently posed problem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish the exact sharp conclusion from nearby variants (wrong inequality direction, weaker true statement, altered quantifier, wrong metric in the conclusion). However, it mostly tests precise theorem recall rather than generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: B reverses the inequality, C gives a weaker true statement, D subtly changes the quantifier structure in the local ball condition, and E shifts the conclusion from g1 to the Yamabe metric g. These reflect plausible failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with high-quality distractors and little direct answer leakage, but it is strongly tautological and only moderately tests genuine reasoning."
    }
  },
  {
    "id": "2512.13556v1",
    "paper_link": "http://arxiv.org/abs/2512.13556v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "conjecture",
      "content": "[{\\cite[Conjecture 5]{motivatedintro}}]\\label{conj}\n    Let $G$ be a unipotent group over $\\overline{\\F}_q$. $G$ is easy if and only if it has trivial $\\mathbb{L}$-packets.",
      "start_pos": 9482,
      "end_pos": 9670,
      "label": "conj"
    },
    "ref_dict": {
      "thm": "\\begin{theorem}\\label{thm}\n    If $G$ has trivial $\\L$-packets, then $G$ is easy.\n\\end{theorem}",
      "asai prop": "\\begin{theorem}\\label{asai prop}\n    Let $G$ be a connected algebraic group. The Asai twisting operator is trivial on $C(G^{F^m}/\\sim)$ for all $m \\in \\Z_{>0}$ if and only if $G$ is easy.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 534,
    "pre_theorem_intro_text": "The theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \\cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \\cite{foundations}, \\cite{characters}, and \\cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\\cite{motivatedintro}). Five of the six were proved in \\cite{foundations} and \\cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:",
    "context": "The theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \\cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \\cite{foundations}, \\cite{characters}, and \\cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\\cite{motivatedintro}). Five of the six were proved in \\cite{foundations} and \\cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:",
    "full_context": "The theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \\cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \\cite{foundations}, \\cite{characters}, and \\cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\\cite{motivatedintro}). Five of the six were proved in \\cite{foundations} and \\cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:\n\n\\begin{abstract}\n In 2006, Boyarchenko and Drinfeld conjectured that for a unipotent algebraic group over $\\fqb$, every geometric point is contained in the neutral connected component of its centralizer if and only if its $\\L$-packets of character sheaves are singletons. In 2013, Boyarchenko proved the ``only if\" direction of this conjecture. In this paper, we complete the proof. Along the way, we explore the relationship between general algebraic groups satisfying this property and their Asai twisting operator.\n\\end{abstract}\n\\section*{Introduction}\nThe theory of character sheaves for unipotent groups was first conjectured by George Lusztig in \\cite{lusztig2006}. This theory was later developed by Mitya Boyarchenko and Vladimir Drinfeld in \\cite{foundations}, \\cite{characters}, and \\cite{chsheaves}. In 2006, Boyarchenko and Drinfeld proposed six initial conjectures in the development of this theory (\\cite{motivatedintro}). Five of the six were proved in \\cite{foundations} and \\cite{chsheaves}. The goal of this paper is to complete the proof of the last remaining conjecture:\n\nIn \\cite{chsheaves}, Boyarchenko proved the ``only if\" direction. The following theorem completes the proof of the conjecture:\n\n\\begin{theorem}[{\\cite[Theorem 1.15]{foundations}}]\\label{edgg cat thm}\n Let $G$ be a unipotent algebraic group over $k$, and let $e\\in \\dgg$ be a minimal closed idempotent.\n \\begin{itemize}\n \\item[(a)]$\\mathscr{M}_e^{perv}$ is a semisimple abelian category with finitely many simple objects. In particular, $\\L$-packets of character sheaves on $G$ are finite.\n \\item[(b)]There exists a (necessarily unique) integer $n_e$ such that $e[-n_e] \\in \\mathscr{M}_e^{perv}$. One has $0\\le n_e \\le \\dim G$. The subcategory $\\mathscr{M}_e := \\mathscr{M}_e^{perv}[n_e]$ of the monoidal category $e\\dgg$ is monoidal.\n \\item[(c)]The perverse t-structure on $\\dg$ induces a t-structure on $e\\dgg$, and the canonical functor $D^b(\\mathscr{M}_e^{perv}) \\rightarrow e\\dgg$ is an equivalence.\n \\end{itemize}\n\\end{theorem}\n\n\\begin{prop}[{\\cite[Proposition 4.4]{foundations}}]\\label{ad pair prop}\n Let $G$ be a unipotent group over $k$ and $M \\in \\dg$, $M \\ne 0$. Suppose that $\\hl \\in \\mathscr{P}_{norm}(G)$ is maximal among all pairs $(A,\\NN) \\in \\mathscr{P}_{norm}(G)$ that are compatible with $M$. If $\\LL$ is invariant under the conjugation action of $G$, then the pair $(H,\\LL)$ is admissible for $G$.\n\\end{prop}\n\n\\begin{lemma}[{\\cite[Lemma 4.16]{characters}}]\\label{easy twists}\n Let $G$ be an easy unipotent group over a field $k$ of characteristic $p>0$. Then, for every object $M \\in \\dgg$, its twist automorphism $\\theta_M$ is trivial. \n\\end{lemma}\n\n\\begin{theorem}[{\\cite[Theorem 1.8(b)]{chsheaves}}]\\label{basis}\n Let $G$ be a connected unipotent group. The functions\n $$\\{t_M : G^F \\to \\qlb \\;| \\;M\\in CS(G)^F\\} $$\n form a basis for the space $C(G^F/ \\sim)$ which is orthonormal with respect to the inner product \n $$\\langle f_1 \\; | \\;f_2\\rangle = \\sum_{g \\in G^F}f_1(g) \\overline{f_2(g)}.$$\n\\end{theorem}\n\n\\begin{theorem}\\label{asai prop}\n Let $G$ be a connected algebraic group. The Asai twisting operator is trivial on $C(G^{F^m}/\\sim)$ for all $m \\in \\Z_{>0}$ if and only if $G$ is easy.\n\\end{theorem}\n\nFrom now on, let $G$ be a unipotent group over $\\fqb$. The proof of \\Cref{thm} relies on the structure of twists in the category $\\dgg$ and their relationship to the Asai twisting operator. \n\\begin{lemma}\\label{triv twists}\n If the $\\L$-packet of character sheaves corresponding to $e$ is a singleton, then the twists in the Hecke subcategory $e\\dgg$ are trivial.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm}\n    If $G$ has trivial $\\L$-packets, then $G$ is easy.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1154,
    "post_theorem_intro_text": "In \\cite{chsheaves}, Boyarchenko proved the ``only if\" direction. The following theorem completes the proof of the conjecture:\n\n\\begin{mainthm}[\\ref{thm}]\n    If $G$ is a unipotent group with trivial $\\mathbb{L}$-packets, then $G$ is easy.\n\\end{mainthm}\n\nThe proof relies on the relationship between character sheaves and Shintani descent described by Deshpande in \\cite{unipshint}.  In the process, we prove a statement about general algebraic groups:\n\n\\begin{mainthm} [\\ref{asai prop}]  Let $G$ be a connected algebraic group over $\\overline{\\F}_q$. The Asai twisting operator is trivial on $C(G^{F^m}/\\sim)$ for all $m \\in \\Z_{>0}$ if and only if $G$ is easy.\n\\end{mainthm}\n\n\\subsection*{Acknowledgments} I thank Tanmay Deshpande for generously sharing his expertise and ideas. I am grateful to my advisor, Charlotte Chan, for introducing me to the subject area, many helpful meetings, and comments on a previous draft. I also thank Sydney Mathematical Research Institute for excellent working conditions. This work was partially supported by NSF grants DMS-2507946 and DMS-1840234 (RTG) as well as a Simons Dissertation Fellowship (MPS-SDF-00014744).",
    "sketch": "The proof is said to rely on “the relationship between character sheaves and Shintani descent described by Deshpande in \\cite{unipshint}.” In the process, the authors “prove a statement about general algebraic groups,” namely: for a connected algebraic group $G$ over $\\overline{\\F}_q$, “the Asai twisting operator is trivial on $C(G^{F^m}/\\sim)$ for all $m\\in\\Z_{>0}$ if and only if $G$ is easy.”",
    "expanded_sketch": "The proof is said to rely on “the relationship between character sheaves and Shintani descent described by Deshpande in \\cite{unipshint}.” In the process, the authors “prove a statement about general algebraic groups,” namely: for a connected algebraic group $G$ over $\\overline{\\F}_q$, “the Asai twisting operator is trivial on $C(G^{F^m}/\\sim)$ for all $m\\in\\Z_{>0}$ if and only if $G$ is easy.”,",
    "expanded_theorem": "[{{\\cite[Conjecture 5]{motivatedintro}}}]\\label{conj}\n    Let $G$ be a unipotent group over $\\overline{\\F}_q$. $G$ is easy if and only if it has trivial $\\mathbb{L}$-packets.",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Let $G$ be a unipotent algebraic group over $\\overline{\\mathbb F}_q$. Call $G$ easy if for every geometric point $g$ of $G$, one has $g \\in Z_G(g)^\\circ$, where $Z_G(g)^\\circ$ is the neutral connected component of the centralizer of $g$. Say that $G$ has trivial $\\mathbb L$-packets if every $\\mathbb L$-packet of character sheaves on $G$ is a singleton. Which statement holds for such a group $G$?",
      "correct_choice": {
        "label": "A",
        "text": "$G$ is easy if and only if it has trivial $\\mathbb L$-packets."
      },
      "choices": [
        {
          "label": "B",
          "text": "$G$ is easy if and only if at least one $\\mathbb L$-packet of character sheaves on $G$ is a singleton."
        },
        {
          "label": "C",
          "text": "If $G$ is easy, then it has trivial $\\mathbb L$-packets."
        },
        {
          "label": "D",
          "text": "If $G$ has trivial $\\mathbb L$-packets, then the Asai twisting operator on $C(G^{F^m}/\\sim)$ is trivial for some $m\\in \\mathbb Z_{>0}$, and hence $G$ is easy."
        },
        {
          "label": "E",
          "text": "$G$ is easy if and only if, for every geometric point $g$ of $G$, the full centralizer $Z_G(g)$ is connected."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "global singleton condition replaced by existence of one singleton packet",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the converse implication",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity in the Asai criterion over all positive integers m",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "membership in the neutral connected component of the centralizer replaced by connectedness of the entire centralizer",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the two properties but does not explicitly reveal the equivalence. There is no direct verbal cue that singles out choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially asking for the exact theorem statement: whether 'easy' is equivalent to having trivial ℒ-packets. This is close to a direct restatement rather than an application or consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some logical comparison among choices (equivalence vs one-way implication vs altered conditions), but the problem mainly tests recall/recognition of a theorem rather than generating a conclusion from mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically distinct: a weakened implication, a quantifier error, a technical but incorrect criterion involving the Asai operator, and a nearby but stronger connectedness condition. These reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall-style MCQ with strong distractors and no answer leakage, but it is largely theorem restatement and therefore only moderately effective at testing genuine generative reasoning."
    }
  },
  {
    "id": "2512.13952v1",
    "paper_link": "http://arxiv.org/abs/2512.13952v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).",
      "start_pos": 7841,
      "end_pos": 8549,
      "label": "dimension bound theorem"
    },
    "ref_dict": {
      "definition of polynomially bounded growth in introduction": "\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}",
      "dimension bound theorem": "\\begin{theorem}\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 4953,
    "pre_theorem_intro_text": "The relationship between the geometry of manifolds and the analytic properties of functions on manifolds is a defining theme of geometric analysis. Our direction starts with the Liouville theorems for harmonic functions on \\(\\mathbb{R}^n\\) and Yau's generalization.\n\nYau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}. \n\nA natural generalization is to try to show this result for solutions of the heat equation. However, the heat equation is very flexible compared to the Laplace equation, and since there are bounded solutions to the heat equation, no Liouville theorem is possible in general.\n\nDespite this, if we restrict attention to specifically \\textit{ancient} solutions of the heat equation, that is, solutions which are defined for all time going back to \\(-\\infty\\), then Liouville theorems actually do become possible. Indeed, in \\cite{colding2021optimal}, Colding and Minicozzi generalize \\cite{colding1997harmonic} to show that the space \\(\\mathcal{P}_d(M)\\) of ancient solutions of the heat equation with polynomially bounded growth also has finite dimension. In (\\cite{colding2020complexity}, \\cite{colding2019search}, \\cite{colding2019liouville}), Colding and Minicozzi show how the spaces \\(\\mathcal{P}_d(M)\\) are relevant to geometric flows.\n\nContinuing to more types of equations, Wang and Zhu recently generalized the result of \\cite{colding1997harmonic} to biharmonic functions \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, i.e., functions \\(u: M \\rightarrow \\mathbb{R}\\) solving\n\\begin{equation*}\n    \\Delta\\Delta u = 0.\n\\end{equation*}\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n    \\begin{equation*}\n        \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n    \\end{equation*}\n    such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n    \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n    \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as",
    "context": "Yau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}.\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n \\begin{equation*}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n \\end{equation*}\n such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as\n\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}",
    "full_context": "Yau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}.\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n \\begin{equation*}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n \\end{equation*}\n such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as\n\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nCombining this with Wang and Zhu's result \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:\n\n\\begin{corollary}\n Let \\(M\\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \\(k, \\ell \\ge 0\\) the spaces \\(\\mathcal{P}_{4k, 4\\ell}(M)\\) are finite dimensional.\n\\end{corollary}\n\nSince there are \\(d + 1\\) vectors \\((1, t_i, \\ldots, t_i^{d + 1})\\), they span \\(\\mathbb{R}^{d + 1}\\), and so there are constants \\(b_i^j\\) such that\n \\begin{equation}\\label{bdefinitions}\n e_j = b_i^j(1, t_i, \\ldots, t_i^{d}).\n \\end{equation}\n It now follows that\n \\begin{equation}\\label{expression of p in b}\n p_j(x) = b_i^ju(x, t_i), \\quad \\nabla p_j(x) = b_i^j\\nabla u(x, t_i),\n \\end{equation}\n and we conclude that \\(p_j\\) can grow at most polynomially of degree \\(4k\\) and \\(\\nabla p_j\\) can grow at most polynomially of degree \\(4\\ell\\). Because \\(p_j\\) vanishes when \\(j > k\\) and \\(\\nabla p_j\\) vanishes when \\(j > \\ell + 1\\), we have\n \\begin{equation}\n u = p_0 + t p_1 + \\ldots + t^{k}p_{k} \\quad \\text{and} \\quad \\nabla u = \\nabla p_0 + t\\nabla p_1 + \\ldots + t^{\\ell}\\nabla p_{\\ell},\n \\end{equation}\n it follows that\n \\begin{equation}\n \\begin{split}\n |u(x, t)| \\le C(1 + |t|^k + |x|^{4k}) \\quad \\text{and} \\quad |\\nabla u(x, t)| \\le C(1 + |t|^{\\ell} + |x|^{4\\ell})\n \\end{split}\n \\end{equation}\n From equation (\\ref{bdefinitions}) we have\n \\begin{equation}\n \\begin{split}\n \\sum_i b_i^ju(x, R^4t_i) & = \\sum_i\\sum_{m}b_i^jp_{m}(x)R^{4j}t_i^{m} = R^{4j}\\sum_mp_m(x)\\left(\\sum_ib_{i}^jt_i^m\\right) \\\\ & = \\sum R^{4j}\\sum_m p_m(x)\\delta_{mj} \\\\ & = R^{4j}p_j(x).\n \\end{split}\n \\end{equation}\n Similarly,\n \\begin{equation}\n \\sum_i b_i^j\\nabla u(x, R^4t_i) = R^{4j}\\nabla p_j(x)\n \\end{equation}\n Thus\n \\begin{equation}\n \\begin{split}\n |R^{4j}p_j(x)| & = \\left|\\sum_i b_i^ju(x, R^4t_i)\\right| \\le A\\sum_i\\left|u(x, R^4t_i)\\right| \\\\ & \\le A(1 + |x|^{4k} + \\sum_i|Rt_i|^{4d}) \\le AR^{4k}\n \\end{split}\n \\end{equation}\n and similarly\n \\begin{equation}\n \\begin{split}\n |R^{4j}\\nabla p_j(x)| & \\le A'(1 + |x|^{4\\ell} + \\sum_i|Rt_i|^{4\\ell}) \\le A'R^{4\\ell},\n \\end{split}\n \\end{equation}\n so that \\(|p_j(x)| \\le A_jR^{4(k - j)}\\) and \\(|p_j(x)| \\le A_j'R^{4(\\ell - j)}\\).\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{dimension bound theorem}]\n Choose some \\(u \\in {P}_{4k, 4\\ell}(M)\\) and suppose \\(u = p_0(x) + tp_1(x) + \\cdots + t^dp_d(x)\\), as in lemma \\ref{coefficientbound}. Then \\(\\Delta\\Delta p_{d} = 0\\) and for \\(j < d\\), \\(\\Delta\\Delta p_j = -(j+1)p_{j + 1}\\). Thus, there a linear map \\(\\Psi_0: \\mathcal{P}_{4k, 4\\ell} \\rightarrow \\mathcal{H}_{4k, 4\\ell}\\) defined by \\(\\Psi_0u = p_{d}\\) (here we use the coefficient estimate in (\\ref{coefficients in polynomial expression bounds})). If we let \\(\\mathcal{K}_0 = \\ker \\Psi_0\\) we find\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell} \\le \\dim \\mathcal{K}_0 + \\dim \\mathcal{H}_{4k, 4\\ell}\n \\end{equation}\n If \\(u \\in \\mathcal{K}_0\\), then \\(p_{d} = 0\\) and \\(\\Delta\\Delta p_{d - 1} = -dp_{d} = 0\\), and so we have a map \\(\\Psi_1: \\mathcal{K}_0 \\rightarrow \\mathcal{H}_{4(k - 1), 4(\\ell - 1)}\\) defined by setting \\(\\Psi_1u = p_{d - 1}\\). Letting \\(\\ker \\Psi_1 = \\mathcal{K}_1\\) then\n \\begin{equation}\n \\dim \\mathcal{K}_0 \\le \\dim \\mathcal{K}_1 + \\dim \\mathcal{H}_{4k, 4\\ell}\n \\end{equation}\n When \\(k \\le \\ell + 1\\), we can repeat this \\(k\\) times to get\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M)\n \\end{equation}\n When \\(k > \\ell + 1\\), we have from lemma \\ref{coefficientbound} that \\(\\nabla p_{\\ell + 1} = 0\\), and so \\(p_{\\ell + 1} = p_d\\) is a constant. Thus in this case \\(p_d\\) lies in a one dimensional subspace of \\(H_{4k, 4\\ell}(M)\\), so that\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell} = \\dim \\mathcal{K}_0 + 1.\n \\end{equation}\n We then iterate the same argument as before, \\(\\ell\\) times, to get the second inequality in (\\ref{equation in the first statement of the main theorem}).\n\n\\end{proof}\n\\subsection{Bicaloric Polynomials in \\(\\mathbb{R}^n\\)}\nNow we consider polynomially bounded solutions to the biharmonic heat equation in \\(\\mathbb{R}^n\\).\n\\begin{proposition}\\label{bicaloric functions in Rn are polynomials}\n Let \\(u \\in \\mathcal{P}_{k, \\ell}(\\mathbb{R}^n)\\). Then \\(u\\) is a polynomial in \\(x_i\\) and \\(t\\).\n\\end{proposition}\n\\begin{proof}\n As before, this follows from the reverse Poincaré estimate and the fact that the operators \\(\\partial_{x_i}\\), \\(\\partial_t\\), and \\(\\Delta\\) commute in \\(\\mathbb{R}^n\\).\n\\end{proof}\n\\begin{corollary}\n If \\(u \\in \\mathcal{P}_{k, \\ell}(\\mathbb{R}^n)\\), then there is some \\(d\\) such that \\(u \\in \\mathcal{P}_{d, d-1}(\\mathbb{R}^n)\\).\n\\end{corollary}\\label{fewer caloric spaces in R^n}\n\\begin{proof}\n This follows the same way as before.\n\\end{proof}\nGiven a monomial in \\(x_i\\) and \\(t\\), we define its biparabolic degree as follows: for \\(t^{n_0}\\prod x_i^{n_i}\\), the biparabolic degree is \\(4n_0 + \\sum n_i\\). The degree of a polynomial is then the maximal degree of the monomials summing to it. Let \\(\\mathcal{A}_j^n\\) be the set of homogeneous polynomials in \\(\\mathbb{R}^n\\) of biparabolic degree \\(j\\). We have\n\\begin{equation}\n \\mathcal{A}_d^n = A_d^n \\oplus t A_{d - 4}^n \\oplus t^2A_{d - 8}^n \\oplus \\cdots\n\\end{equation}\n\\begin{lemma}\\label{dim for bicaloric polynomials}\n For \\(d > 0\\) we have \\(\\dim(\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n) = \\dim A_d^n\\) and\n \\begin{equation}\n \\dim \\mathcal{P}_{d, d-1}(\\mathbb{R}^n) = \\sum_{j = 0}^d\\dim A_j^n.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Both \\(\\partial_t\\) and \\(\\Delta\\Delta\\) map \\(\\mathcal{A}_d^n\\) to \\(\\mathcal{A}_{d-4}^n\\). We note that for \\(u \\in \\mathcal{A}_{d-4}^n\\) we have\n \\begin{equation}\n (\\partial_t + \\Delta\\Delta)\\left(tu - \\frac{1}{2}t^2(\\partial_t + \\Delta\\Delta)u + \\frac{1}{6}t^3(\\partial_t + \\Delta\\Delta)^2 u - \\cdots\\right) = u,\n \\end{equation}\n so that \\(\\partial_t + \\Delta\\Delta\\) is surjective. Thus,\n \\begin{equation}\n \\dim (\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n) = \\dim \\mathcal{A}_d^n - \\dim\\mathcal{A}_{d-4}^n = \\dim A_d^n\n \\end{equation}\n since \\(\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n\\) is the kernel of \\(\\partial_t + \\Delta\\Delta\\) restricted to \\(\\mathcal{A}_d^n\\). Summing gives the second claim.\n\n\\end{proof}\nNow finally we show that the estimate in Theorem \\ref{dimension bound theorem} is sharp in \\(\\mathbb{R}^n\\).\n\\begin{corollary}\\label{sharp in Rn}\n For positive integers \\(d > 0\\)\n \\begin{equation}\n \\dim \\mathcal{P}_{4d, 4d - 1}(\\mathbb{R}^n) = \\sum_{i = 0}^{d} \\dim \\mathcal{H}_{4(d - i), 4(d - i) - 1}(\\mathbb{R}^n).\n \\end{equation}\n\\end{corollary}\n\\begin{proof}\n From lemmas \\ref{dim for bicaloric polynomials} and \\ref{dim for biharmonic polynomials} we have\n \\begin{equation}\n \\begin{split}\n \\dim \\mathcal{P}_{4d, 4d-1}(\\mathbb{R}^n) & = \\sum_{j = 0}^{4d}\\dim A_j^n = \\sum_{j = 0}^d(\\dim A_{4j}^n + \\dim A_{4j - 1}^n + \\dim A_{4j - 2}^n + \\dim A_{4j - 3}^n) \\\\ & = \\sum_{j = 0}^d\\dim \\mathcal{H}_{4d, 4d - 1}(\\mathbb{R}^n).\n \\end{split}\n \\end{equation}\n Setting \\(k = \\ell = d\\) in Theorem \\ref{dimension bound theorem} and noting \\(\\mathcal{P}_{4d, 4d}(\\mathbb{R}^n) = \\mathcal{P}_{4d, 4d-1}(\\mathbb{R}^n)\\) and \\(\\mathcal{H}_{4d, 4d}(\\mathbb{R}^n) = \\mathcal{H}_{4d, 4d-1}(\\mathbb{R}^n)\\) shows that the estimate in Theorem \\ref{dimension bound theorem} is sharp.",
    "post_theorem_intro_text_len": 4214,
    "post_theorem_intro_text": "Combining this with Wang and Zhu's result \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:\n\n\\begin{corollary}\n    Let \\(M\\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \\(k, \\ell \\ge 0\\) the spaces \\(\\mathcal{P}_{4k, 4\\ell}(M)\\) are finite dimensional.\n\\end{corollary}\n\n\\subsection{Harmonic and biharmonic functions}\n\nBiharmonic functions arise in several variational problems. Just as minimizing \\(\\int|\\nabla u|^2\\) leads one to the Laplace and heat equations, minimizing \\(\\int |\\Delta u|^2\\) leads to the biharmonic and biharmonic heat equations.\n\nIn general, fourth order elliptic operators arise naturally when taking variations involving second order objects, one major example being variations of metrics in conformal geometry (see \\cite{chang1995extremal}, \\cite{Lin1998classification}). They also arise in the study of the Willmore energy. For an immersed surface \\(\\phi: M^2 \\rightarrow \\mathbb{R}^3\\), the Willmore energy is defined as\n\\begin{equation}\\label{Willmore functional}\n    \\mathcal{W}(\\phi) = \\int_{M}H^2\\,dA\n\\end{equation}\nwhere \\(dA\\) is the induced volume element and \\(H\\) is the mean curvature \\cite{willmore2000surfaces}. In studying critical points of this functional one arrives at the Euler-Lagrange equation\n\\begin{equation}\\label{euler-lagrange equation}\n    \\Delta H + 2H(H^2 - K) = 0,\n\\end{equation}\na fourth order elliptic operator. The biharmonic heat equation similarly arises when studying the gradient flow of the Willmore energy (\\cite{kuwert2002gradient}, \\cite{lamm2005biharmonic}). Ancient solutions to heat equations often appear when doing blowup analysis of general solutions to a variational problem. See \\cite{kuwert2004removable} for blowup analysis of singularities of Willmore flows. We also again reference (\\cite{colding2020complexity}, \\cite{colding2019search}, \\cite{colding2019liouville}) for more on how ancient solutions to heat equations with polynomially bounded growth are relevant to geometric flows.\n\nAlthough both arise from variational problems, biharmonic functions in general differ significantly from harmonic functions, because no maximum principle holds for biharmonic functions. This limits the kinds of estimates we can find for biharmonic functions. In particular, the usual pointwise derivative estimates one can find for harmonic functions on a ball cannot be found for a biharmonic function.\n\nOn the bright side, energy methods for harmonic and caloric functions seem to have analogs for biharmonic and bicaloric functions, which we will see as we prove Theorem \\ref{dimension bound theorem}. We are still limited to some extent, however, because when performing integrations by parts we are forced to use the Bochner formula\n\\begin{equation*}\n    \\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\end{equation*}\nto control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\). It is the appearance of the Ricci term here that makes the decay on Ricci curvature crucial for our result.\n\nOur methodology is inspired by Colding and Minicozzi's in \\cite{colding2021optimal}. We will show a reverse Poincaré inequality for bicaloric functions on ``heat balls'' \\(B_R(p) \\times [-R^4, 0]\\). Because we are considering ancient bicaloric functions, we will be able to apply the inequality as \\(R \\rightarrow \\infty\\) to get strong, global control of their behavior. In particular we will see that high order time derivatives \\(\\partial_t^ku\\) must vanish identically, allowing us to write for some finite \\(d\\):\n\\[u(x, t) = p_d(x)t^d + \\cdots + p_1(x)t + p_0(x)\\]\nwith \\(\\Delta \\Delta p_d = 0\\) and \\(\\Delta \\Delta p_j = -(j + 1)p_{j + 1}\\). This will allow us to directly compare the spaces \\(\\mathcal{H}_{4k, 4\\ell}(M)\\) with \\(\\mathcal{P}_{4k, 4\\ell}(M)\\).\n\nTo show the dimension estimates are sharp in \\(\\mathbb{R}^n\\), we will consider biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions of the spaces \\(\\mathcal{H}_{4k, 4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k, 4\\ell}(\\mathbb{R}^n)\\).",
    "sketch": "The introduction indicates the proof strategy for Theorem~\\ref{dimension bound theorem} as follows.\n\n1. **Use energy methods and Bochner to handle fourth order terms.** It says that while working \"as we prove Theorem \\ref{dimension bound theorem}\" one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is \"forced to use the Bochner formula\" \n\\[\n\\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\]\n\"to control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\),\" and \"the appearance of the Ricci term here\" is why the quadratic lower bound/decay assumption on Ricci is crucial.\n\n2. **Establish a reverse Poincar\\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will \"show a reverse Poincar\\'e inequality for bicaloric functions on `heat balls' \\(B_R(p)\\times[-R^4,0]\\).\"\n\n3. **Send \\(R\\to\\infty\\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, \"we will be able to apply the inequality as \\(R\\to\\infty\\) to get strong, global control of their behavior.\"\n\n4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, \"high order time derivatives \\(\\partial_t^k u\\) must vanish identically,\" so for some finite \\(d\\)\n\\[\nu(x,t)=p_d(x)t^d+\\cdots+p_1(x)t+p_0(x),\n\\]\nwith relations \"\\(\\Delta\\Delta p_d=0\\)\" and \"\\(\\Delta\\Delta p_j=-(j+1)p_{j+1}\\).\"\n\n5. **Compare \\(\\mathcal{H}_{4k,4\\ell}(M)\\) and \\(\\mathcal{P}_{4k,4\\ell}(M)\\) via the coefficient relations.** The stated expansion and recursions \"will allow us to directly compare the spaces \\(\\mathcal{H}_{4k,4\\ell}(M)\\) with \\(\\mathcal{P}_{4k,4\\ell}(M)\\),\" yielding the dimension bound.\n\n6. **Sharpness in \\(\\mathbb{R}^n\\) by explicit polynomial models.** To show sharpness, it says they will consider \"biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions\" of \\(\\mathcal{H}_{4k,4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k,4\\ell}(\\mathbb{R}^n)\\).",
    "expanded_sketch": "The introduction indicates the proof strategy for the following theorem.\n\n\\begin{theorem}\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).\n\\end{theorem}\n\n1. **Use energy methods and Bochner to handle fourth order terms.** In establishing the main theorem, one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is forced to use the Bochner formula\n\\[\n\\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\]\nso as to control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\); the appearance of the Ricci term here is why the quadratic lower bound/decay assumption on Ricci is crucial.\n\n2. **Establish a reverse Poincar\\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will show a reverse Poincar\\'e inequality for bicaloric functions on “heat balls” \\(B_R(p)\\times[-R^4,0]\\).\n\n3. **Send \\(R\\to\\infty\\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, one will be able to apply the inequality as \\(R\\to\\infty\\) to get strong, global control of their behavior.\n\n4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, high order time derivatives \\(\\partial_t^k u\\) must vanish identically, so for some finite \\(d\\)\n\\[\nu(x,t)=p_d(x)t^d+\\cdots+p_1(x)t+p_0(x),\n\\]\nwith relations “\\(\\Delta\\Delta p_d=0\\)” and “\\(\\Delta\\Delta p_j=-(j+1)p_{j+1}\\).”\n\n5. **Compare \\(\\mathcal{H}_{4k,4\\ell}(M)\\) and \\(\\mathcal{P}_{4k,4\\ell}(M)\\) via the coefficient relations.** The stated expansion and recursions allow one to directly compare the spaces \\(\\mathcal{H}_{4k,4\\ell}(M)\\) with \\(\\mathcal{P}_{4k,4\\ell}(M)\\), yielding the dimension bound \\eqref{equation in the first statement of the main theorem}.\n\n6. **Sharpness in \\(\\mathbb{R}^n\\) by explicit polynomial models.** To show sharpness, one considers biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling one to explicitly compute the dimensions of \\(\\mathcal{H}_{4k,4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k,4\\ell}(\\mathbb{R}^n)\\).",
    "expanded_theorem": "\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(M\\) be a complete Riemannian manifold. Say that \\(M\\) has polynomial volume growth if there exist \\(p\\in M\\) and constants \\(C,d_V>0\\) such that \\(\\operatorname{Vol}(B_R(p))\\le C(1+R)^{d_V}\\) for all \\(R>0\\). Say that the Ricci curvature of \\(M\\) is bounded below quadratically if there exist \\(p\\in M\\) and \\(K>0\\) such that \\(\\sup_{v\\in TB_R(p)}\\frac{\\operatorname{Ric}(v,v)}{|v|^2}\\ge -\\frac{K}{R^2}\\) for all \\(R>0\\). For \\(d,d'\\), let \\(\\mathcal{P}_{d,d'}(M)\\) denote the space of ancient functions \\(u:M\\times(-\\infty,0]\\to\\mathbb{R}\\) satisfying the biharmonic heat equation \\(\\partial_t u+\\Delta\\Delta u=0\\) and such that for some constants \\(C,C'>0\\), \\(\\sup_{B_R(q)\\times[-R^4,0]}|u(x,t)|\\le C(1+R)^d\\) and \\(\\sup_{B_R(q)\\times[-R^4,0]}|\\nabla u(x,t)|\\le C'(1+R)^{d'}\\) for every \\(q\\in M\\) and \\(R>0\\). Let \\(\\mathcal{H}_{d,d'}(M)\\) denote the space of functions \\(u:M\\to\\mathbb{R}\\) satisfying \\(\\Delta\\Delta u=0\\) and the analogous bounds over \\(B_R(q)\\). For nonnegative integers \\(k\\) and \\(\\ell\\), which statement holds for every such manifold \\(M\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\\dim \\mathcal{P}_{4k,4\\ell}(M)\\le \\begin{cases}\\displaystyle \\sum_{i=0}^{k}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k\\le \\ell+1,\\\\[4pt]\\displaystyle 1+\\sum_{i=0}^{\\ell}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k>\\ell+1,\\end{cases}\\] and moreover these inequalities are sharp in \\(\\mathbb{R}^n\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\\dim \\mathcal{P}_{4k,4\\ell}(M)\\le \\sum_{i=0}^{\\min\\{k,\\ell\\}}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M),\\] and moreover this inequality is sharp in \\(\\mathbb{R}^n\\)."
        },
        {
          "label": "C",
          "text": "\\[\\dim \\mathcal{P}_{4k,4\\ell}(M)<\\infty\\] for all nonnegative integers \\(k\\) and \\(\\ell\\)."
        },
        {
          "label": "D",
          "text": "\\[\\dim \\mathcal{P}_{4k,4\\ell}(M)\\le \\begin{cases}\\displaystyle \\sum_{i=0}^{k}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k\\le \\ell,\\\\[4pt]\\displaystyle 1+\\sum_{i=0}^{\\ell}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k>\\ell,\\end{cases}\\] and moreover these inequalities are sharp in \\(\\mathbb{R}^n\\)."
        },
        {
          "label": "E",
          "text": "\\[\\dim \\mathcal{P}_{4k,4\\ell}(M)= \\begin{cases}\\displaystyle \\sum_{i=0}^{k}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k\\le \\ell+1,\\\\[4pt]\\displaystyle 1+\\sum_{i=0}^{\\ell}\\dim \\mathcal{H}_{4(k-i),4(\\ell-i)}(M), & k>\\ell+1,\\end{cases}\\] and moreover these equalities are sharp in \\(\\mathbb{R}^n\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "extra_constant_and_upper_limit_in_second_case",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit_piecewise_dimension_bound_and_sharpness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "threshold_k_leq_ell_plus_1",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "inequality_vs_equality_for_general_manifolds",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and hypotheses but does not reveal the dimension bound itself. There is no explicit or obvious hint that singles out choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially asking for the exact theorem statement under the stated hypotheses. The multiple close alternatives prevent it from being a pure verbatim restatement, but it still mainly tests theorem recall rather than an independently derived conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants: off-by-one thresholding, inequality versus equality, and the extra constant term in one regime. However, the task is mostly recognition of the correct theorem statement rather than substantial mathematical generation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one is a weaker true statement, others are plausible near-miss versions with boundary errors or overstrong equality claims. They are distinct and reflect realistic failure modes in reading or recalling the result."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it mainly tests precise recall/discrimination among near-identical statements rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.14016v1",
    "paper_link": "http://arxiv.org/abs/2512.14016v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.",
      "start_pos": 8009,
      "end_pos": 8522,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{theorem}\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.\n\\end{theorem}",
      "eq:Ein-assumptions": "\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}",
      "thm:Ein-fill": "\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2754,
    "pre_theorem_intro_text": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds.  \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n    \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$.  More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$.  In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant.  This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.",
    "context": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds. \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$. In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}",
    "full_context": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds. \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$. In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}\n\n\\begin{abstract}\nWe study the smallest area $A_{\\min}(M,g)$ of a $2$-dimensional stationary integral varifold in a closed Einstein $4$-manifold $(M^4,g)$ with\n\\[\n\\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\]\nBuilding on the previous work \\cite{LW_HF1} on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class there is an upper bound\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D),\n\\]\nwhere $F_{\\Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\\varepsilon$–regularity constants for Einstein metrics. \n\\end{abstract}\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\nAs in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} proceeds via a\nhomological filling inequality expressed in terms of the first homological\nfilling function $\\HF_1$ of $(M,g)$.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property. If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:main} is then an immediate consequence of\nTheorem~\\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman\n\\cite{nabutovsky2006curvature}, which relates $A_{\\min}(M,g)$ to $\\HF_1$.\nIn the Einstein setting we can track all constants arising from the Sobolev\ninequality, the $L^2$–curvature bound and the bubble–tree decomposition,\nthus making $F_{\\Ein}(v,D)$ completely explicit once the analytic data are fixed.\n\n\\begin{theorem}[Einstein $\\varepsilon$-regularity via Sobolev]\\label{thm:eps_reg_Einstein}\nThere exist dimensional constants $\\varepsilon_{\\mathrm{Ein}}>0$ and $c_{\\mathrm{Ein}}>0$,\ndepending only on $C_S(v,D)$, such that the following holds.\n\n\\begin{theorem}[Nabutovsky--Rotman~{\\cite{nabutovsky2006curvature}}]\\label{thm:NR}\nLet $M$ be a closed $n$-dimensional Riemannian manifold with $H_1(M;\\Z)=0$\nand diameter $\\diam(M)=D$. Let $A_{\\min}(M)$ denote the area of a smallest\n$2$-dimensional stationary integral varifold in $M$. Then\n\\[\n A_{\\min}(M)\\;\\le\\;\\frac{(n+1)!}{2}\\,\\HF_1(2D).\n\\]\nIn particular, for $n=4$,\n\\[\n A_{\\min}(M)\\;\\le\\;60\\,\\HF_1(2D).\n\\]\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1183,
    "post_theorem_intro_text": "As in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} proceeds via a\nhomological filling inequality expressed in terms of the first homological\nfilling function $\\HF_1$ of $(M,g)$.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\mathbb{Z})$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\mathbb{Z})$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:main} is then an immediate consequence of\nTheorem~\\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman\n\\cite{nabutovsky2006curvature}, which relates $A_{\\min}(M,g)$ to $\\HF_1$.\nIn the Einstein setting we can track all constants arising from the Sobolev\ninequality, the $L^2$–curvature bound and the bubble–tree decomposition,\nthus making $F_{\\Ein}(v,D)$ completely explicit once the analytic data are fixed.",
    "sketch": "As in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} \"proceeds via a homological filling inequality expressed in terms of the first homological filling function $\\HF_1$ of $(M,g)$.\" One first proves Theorem~\\ref{thm:Ein-fill}, i.e. a linear filling estimate for every singular Lipschitz $1$-cycle $C$ by a singular $2$-chain $E$ with \\(\\mass_2(E)\\le f_1^{\\Ein}(v,D)\\,\\mass_1(C)+f_2^{\\Ein}(v,D)\\), equivalently \\(\\HF_1(l)\\le f_1^{\\Ein}(v,D)\\,l+f_2^{\\Ein}(v,D)\\). Then Theorem~\\ref{thm:main} is \"an immediate consequence\" of Theorem~\\ref{thm:Ein-fill} together with the min--max theorem of Nabutovsky--Rotman \\cite{nabutovsky2006curvature}, \"which relates $A_{\\min}(M,g)$ to $\\HF_1$.\" In the Einstein setting, the constants are tracked using \"the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,\" making $F_{\\Ein}(v,D)$ explicit once the analytic data are fixed.",
    "expanded_sketch": "As in Larry Guth and David W. Rotman, \\emph{The first homology of a Riemannian manifold and the first homological filling function}, the proof of the main theorem “proceeds via a homological filling inequality expressed in terms of the first homological filling function $\\HF_1$ of $(M,g)$.” We first prove the following theorem.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTo prove the main theorem, we then combine this filling estimate with the min--max theorem of Nabutovsky--Rotman (Alexander Nabutovsky and Regina Rotman, \\emph{Curvature-Free Bounds for the Least Area of a Surface}, 2006), which “relates $A_{\\min}(M,g)$ to $\\HF_1$.” In the Einstein setting, the constants are tracked using “the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,” making $F_{\\Ein}(v,D)$ explicit once the analytic data are fixed.",
    "expanded_theorem": "\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.,",
    "theorem_type": [
      "Existential–Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let $v,D>0$. Suppose $(M^4,g)$ is a closed Einstein $4$-manifold in the class $\\Ee_1(4,v,D)$, meaning that\n\\[\n\\Ric_g=\\lambda g,\\qquad |\\lambda|\\le 3,\\qquad \\Vol(M,g)\\ge v,\\qquad \\diam(M,g)\\le D,\\qquad H_1(M;\\mathbb Z)=0.\n\\]\nLet $A_{\\min}(M,g)$ denote the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$. Which conclusion about $A_{\\min}(M,g)$ is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant $F_{\\Ein}(v,D)>0$, depending only on $v$ and $D$, such that\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D).\n\\]\nMoreover, $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\\varepsilon$-regularity constants."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a universal constant $F_{\\Ein}>0$, independent of $v$ and $D$, such that for every $(M^4,g)\\in \\Ee_1(4,v,D)$ one has\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}.\n\\]\nMoreover, this bound can be written explicitly in terms of dimensional Einstein $\\varepsilon$-regularity constants alone."
        },
        {
          "label": "C",
          "text": "There exists a constant $F_{\\Ein}(v,D)>0$ such that\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D)\n\\]\nfor every $(M^4,g)\\in \\Ee_1(4,v,D)$."
        },
        {
          "label": "D",
          "text": "For every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following property: if $(M^4,g)\\in \\Ee_1(4,v,D)$, then every $2$-dimensional stationary integral varifold in $(M,g)$ has area at most $F_{\\Ein}(v,D)$. Moreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\\varepsilon$-regularity constants."
        },
        {
          "label": "E",
          "text": "For every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following property. If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D),\n\\]\nand one may choose $F_{\\Ein}(v,D)$ to depend only on $v$, $D$, and the diameter term appearing in the Nabutovsky--Rotman estimate, without any use of the Sobolev constant, the global $L^2$-curvature bound, or Einstein $\\varepsilon$-regularity constants."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the bound on v and D and analytic data",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "explicit description of how F_{\\Ein}(v,D) is written in terms of Sobolev/L^2-curvature/\\varepsilon-regularity data",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "smallest varifold versus arbitrary stationary varifold",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "necessity of Sobolev, L^2-curvature, and bubble-tree/\\varepsilon-regularity input for explicit constants",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or uniquely identify choice A. It presents the hypothesis class and asks which uniform existence statement is valid, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall question: the correct option is essentially the target theorem stated under the given hypotheses. However, the presence of nearby variants means it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the strongest valid conclusion from weakened, overgeneralized, or tampered versions. Still, the task is mostly recognition/comparison of theorem formulations rather than genuine mathematical generation."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and reflect realistic failure modes (omitting the H1 hypothesis, overstating independence of analytic constants, inserting an unjustified factor). But choice C is a weaker true statement, which creates ambiguity in a single-best-answer format and weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably constructed theorem-discrimination MCQ with no major answer leakage, but it is closer to recall than to generative reasoning, and the weaker-true distractor makes the single-correct-answer structure somewhat ambiguous."
    }
  },
  {
    "id": "2512.14016v1",
    "paper_link": "http://arxiv.org/abs/2512.14016v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.",
      "start_pos": 8009,
      "end_pos": 8522,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{theorem}\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.\n\\end{theorem}",
      "eq:Ein-assumptions": "\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}",
      "thm:Ein-fill": "\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2754,
    "pre_theorem_intro_text": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds.  \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n    \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$.  More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$.  In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant.  This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.",
    "context": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds. \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$. In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}",
    "full_context": "Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\\min}(M,g)$ \nthe area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.\nIn general, the existence of such a minimal object follows from the\nAlmgren--Pitts min--max theory \\cite{pitts2014existence}, and in low\ncodimension and low dimension one can often upgrade the varifold to a smooth\nembedded minimal hypersurface using the regularity theory of\nSchoen--Simon--Yau \\cite{schoen1975,schoen1981regularity}.\n\nIn this article we are interested in bounding $A_{\\min}(M,g)$ in terms of\nglobal geometric data in the special case when $(M,g)$ is an Einstein\n$4$-manifold subject to natural non-collapsing and diameter bounds. \nThroughout we assume that\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\mathbb{Z})=0.\n\\end{equation}\nWe write $\\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds\nsatisfying \\eqref{eq:Ein-assumptions}.\n\nThis paper is a sequel to our previous work \\cite{LW_HF1}, where we considered\nthe larger class with $ |{\\rm Ric}_g|\\le 3,\\ \\Vol(M,g)\\ge v>0,\\\n \\diam(M,g)\\le D,\\ H_1(M;\\mathbb{Z})=0, $\nand proved a linear upper bound for the first homological filling function\n$\\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\\in\\Mm_1(4,v,D)$ we\nshowed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that\n\\[\n \\HF_1(l)\\ \\le\\ f_1(v,D)\\,l + f_2(v,D)\\qquad\\text{for all }l\\ge 0.\n\\]\nCombined with an effective version of the Almgren--Pitts min--max theory due to\nNabutovsky--Rotman \\cite{nabutovsky2006curvature}, this yields\n\\[\n A_{\\min}(M,g)\\ \\le\\ F(v,D)\n\\]\nfor some function $F$ depending only on $v$ and $D$. In that setting, however,\nthe dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made\nexplicit, since the constants coming from the Cheeger--Naber bubble--tree\ndecomposition \\cite{cheeger2014regularity} were only known to exist abstractly.\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\n\\begin{equation}\\label{eq:Ein-assumptions}\n \\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad\n \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\end{equation}\n\n\\begin{abstract}\nWe study the smallest area $A_{\\min}(M,g)$ of a $2$-dimensional stationary integral varifold in a closed Einstein $4$-manifold $(M^4,g)$ with\n\\[\n\\Ric_g = \\lambda g,\\quad |\\lambda|\\le 3,\\quad \\Vol(M,g)\\ge v>0,\\quad \\diam(M,g)\\le D,\\quad H_1(M;\\Z)=0.\n\\]\nBuilding on the previous work \\cite{LW_HF1} on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class there is an upper bound\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D),\n\\]\nwhere $F_{\\Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\\varepsilon$–regularity constants for Einstein metrics. \n\\end{abstract}\n\nIn the present work we revisit this question in the more rigid Einstein setting.\nFor $(M^4,g)\\in\\Ee_1(4,v,D)$ the curvature solves an elliptic system, and\nglobal Sobolev inequalities can be used, following Anderson\n\\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature\nbounds and $\\varepsilon$–regularity with constants depending only on a Sobolev\nconstant. This analytic input, combined with the Einstein specialization of the\nCheeger--Naber bubble--tree decomposition and an improved combinatorial filling\nestimate, allows us to obtain an explicit upper bound for $A_{\\min}(M,g)$ in\nterms of $(v,D)$ and the Sobolev constants.\n\nOur first main result is the following quantitative bound for minimal\n$2$-varifolds in Einstein $4$-manifolds.\n\nAs in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} proceeds via a\nhomological filling inequality expressed in terms of the first homological\nfilling function $\\HF_1$ of $(M,g)$.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property. If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:main} is then an immediate consequence of\nTheorem~\\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman\n\\cite{nabutovsky2006curvature}, which relates $A_{\\min}(M,g)$ to $\\HF_1$.\nIn the Einstein setting we can track all constants arising from the Sobolev\ninequality, the $L^2$–curvature bound and the bubble–tree decomposition,\nthus making $F_{\\Ein}(v,D)$ completely explicit once the analytic data are fixed.\n\n\\begin{theorem}[Einstein $\\varepsilon$-regularity via Sobolev]\\label{thm:eps_reg_Einstein}\nThere exist dimensional constants $\\varepsilon_{\\mathrm{Ein}}>0$ and $c_{\\mathrm{Ein}}>0$,\ndepending only on $C_S(v,D)$, such that the following holds.\n\n\\begin{theorem}[Nabutovsky--Rotman~{\\cite{nabutovsky2006curvature}}]\\label{thm:NR}\nLet $M$ be a closed $n$-dimensional Riemannian manifold with $H_1(M;\\Z)=0$\nand diameter $\\diam(M)=D$. Let $A_{\\min}(M)$ denote the area of a smallest\n$2$-dimensional stationary integral varifold in $M$. Then\n\\[\n A_{\\min}(M)\\;\\le\\;\\frac{(n+1)!}{2}\\,\\HF_1(2D).\n\\]\nIn particular, for $n=4$,\n\\[\n A_{\\min}(M)\\;\\le\\;60\\,\\HF_1(2D).\n\\]\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1183,
    "post_theorem_intro_text": "As in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} proceeds via a\nhomological filling inequality expressed in terms of the first homological\nfilling function $\\HF_1$ of $(M,g)$.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\mathbb{Z})$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\mathbb{Z})$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:main} is then an immediate consequence of\nTheorem~\\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman\n\\cite{nabutovsky2006curvature}, which relates $A_{\\min}(M,g)$ to $\\HF_1$.\nIn the Einstein setting we can track all constants arising from the Sobolev\ninequality, the $L^2$–curvature bound and the bubble–tree decomposition,\nthus making $F_{\\Ein}(v,D)$ completely explicit once the analytic data are fixed.",
    "sketch": "As in \\cite{LW_HF1}, the proof of Theorem~\\ref{thm:main} \"proceeds via a homological filling inequality expressed in terms of the first homological filling function $\\HF_1$ of $(M,g)$.\" One first proves Theorem~\\ref{thm:Ein-fill}, i.e. a linear filling estimate for every singular Lipschitz $1$-cycle $C$ by a singular $2$-chain $E$ with \\(\\mass_2(E)\\le f_1^{\\Ein}(v,D)\\,\\mass_1(C)+f_2^{\\Ein}(v,D)\\), equivalently \\(\\HF_1(l)\\le f_1^{\\Ein}(v,D)\\,l+f_2^{\\Ein}(v,D)\\). Then Theorem~\\ref{thm:main} is \"an immediate consequence\" of Theorem~\\ref{thm:Ein-fill} together with the min--max theorem of Nabutovsky--Rotman \\cite{nabutovsky2006curvature}, \"which relates $A_{\\min}(M,g)$ to $\\HF_1$.\" In the Einstein setting, the constants are tracked using \"the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,\" making $F_{\\Ein}(v,D)$ explicit once the analytic data are fixed.",
    "expanded_sketch": "As in Larry Guth and David W. Rotman, \\emph{The first homology of a Riemannian manifold and the first homological filling function}, the proof of the main theorem “proceeds via a homological filling inequality expressed in terms of the first homological filling function $\\HF_1$ of $(M,g)$.” We first prove the following theorem.\n\n\\begin{theorem}\\label{thm:Ein-fill}\nFor every $v,D>0$ there exist functions $$f_1^{\\Ein}(v,D),f_2^{\\Ein}(v,D)>0, $$\nwith the following property.  If $(M^4,g)\\in \\Ee_1(4,v,D)$ and\n$C\\in\\Zz_1(M;\\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a\nsingular $2$-chain $E\\in\\Cc_2(M;\\Z)$ such that\n\\[\n \\mass_2(E)\\ \\le\\ f_1^{\\Ein}(v,D)\\,\\mass_1(C) + f_2^{\\Ein}(v,D).\n\\]\nEquivalently, the first homological filling function of $(M,g)$ satisfies\n\\[\n \\HF_1(l)\\ \\le\\ f_1^{\\Ein}(v,D)\\,l + f_2^{\\Ein}(v,D)\n \\qquad\\text{for all }l\\ge 0.\n\\]\n\\end{theorem}\n\nTo prove the main theorem, we then combine this filling estimate with the min--max theorem of Nabutovsky--Rotman (Alexander Nabutovsky and Regina Rotman, \\emph{Curvature-Free Bounds for the Least Area of a Surface}, 2006), which “relates $A_{\\min}(M,g)$ to $\\HF_1$.” In the Einstein setting, the constants are tracked using “the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,” making $F_{\\Ein}(v,D)$ explicit once the analytic data are fixed.",
    "expanded_theorem": "\\label{thm:main}\nFor every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following\nproperty.  If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a\nsmallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\n A_{\\min}(M,g)\\ \\le\\ F_{\\Ein}(v,D).\n\\]\nMoreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev\nconstant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative\nEinstein $\\varepsilon$–regularity constants.,",
    "theorem_type": [
      "Existential–Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let $v,D>0$. Suppose $(M^4,g)$ is a closed Einstein $4$-manifold in the class $\\Ee_1(4,v,D)$, meaning that\n\\[\n\\Ric_g=\\lambda g,\\qquad |\\lambda|\\le 3,\\qquad \\Vol(M,g)\\ge v,\\qquad \\diam(M,g)\\le D,\\qquad H_1(M;\\mathbb Z)=0.\n\\]\nLet $A_{\\min}(M,g)$ denote the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$. Which conclusion about $A_{\\min}(M,g)$ is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a constant $F_{\\Ein}(v,D)>0$, depending only on $v$ and $D$, such that\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D).\n\\]\nMoreover, $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\\varepsilon$-regularity constants."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a universal constant $F_{\\Ein}>0$, independent of $v$ and $D$, such that for every $(M^4,g)\\in \\Ee_1(4,v,D)$ one has\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}.\n\\]\nMoreover, this bound can be written explicitly in terms of dimensional Einstein $\\varepsilon$-regularity constants alone."
        },
        {
          "label": "C",
          "text": "There exists a constant $F_{\\Ein}(v,D)>0$ such that\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D)\n\\]\nfor every $(M^4,g)\\in \\Ee_1(4,v,D)$."
        },
        {
          "label": "D",
          "text": "For every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following property: if $(M^4,g)\\in \\Ee_1(4,v,D)$, then every $2$-dimensional stationary integral varifold in $(M,g)$ has area at most $F_{\\Ein}(v,D)$. Moreover $F_{\\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\\varepsilon$-regularity constants."
        },
        {
          "label": "E",
          "text": "For every $v,D>0$ there exists a constant $F_{\\Ein}(v,D)>0$ with the following property. If $(M^4,g)\\in \\Ee_1(4,v,D)$, then the area $A_{\\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies\n\\[\nA_{\\min}(M,g)\\le F_{\\Ein}(v,D),\n\\]\nand one may choose $F_{\\Ein}(v,D)$ to depend only on $v$, $D$, and the diameter term appearing in the Nabutovsky--Rotman estimate, without any use of the Sobolev constant, the global $L^2$-curvature bound, or Einstein $\\varepsilon$-regularity constants."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the bound on v and D and analytic data",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "explicit description of how F_{\\Ein}(v,D) is written in terms of Sobolev/L^2-curvature/\\varepsilon-regularity data",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "smallest varifold versus arbitrary stationary varifold",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "necessity of Sobolev, L^2-curvature, and bubble-tree/\\varepsilon-regularity input for explicit constants",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option explicitly or by an obvious cue. It only states the geometric hypotheses and asks for the valid conclusion."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to a theorem-recall question: under the stated hypotheses, select the conclusion. It is not a pure verbatim restatement because the options vary in strength and in the dependence of the constant, but it remains largely theorem matching."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing between a uniform bound, a bound for the minimum varifold versus all varifolds, and the role of explicit analytic inputs. However, the presence of a weaker true statement among the choices reduces the need for genuine generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and target natural mistakes: over-uniformity, over-strong quantification, and incorrect dependence of constants. But choice C appears to be a weaker true consequence of A, so the distractor set is not cleanly single-correct."
      },
      "total_score": 5,
      "overall_assessment": "Mathematically substantive and mostly free of answer leakage, but the item is weakened by theorem-like phrasing and, more seriously, by including a weaker statement that also appears valid, creating ambiguity in a single-answer MCQ."
    }
  },
  {
    "id": "2512.14262v2",
    "paper_link": "http://arxiv.org/abs/2512.14262v2",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nIf \\(n \\neq 6\\), then\n    $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$",
      "start_pos": 26104,
      "end_pos": 26321,
      "label": "thm:main"
    },
    "ref_dict": {
      "que:hk_torsion": "\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\  manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}",
      "cor:nice_statement_even": "\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}"
    },
    "pre_theorem_intro_text_len": 2102,
    "pre_theorem_intro_text": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\  manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however,  Kapfer and Menet give a positive answer in the case of Kummer fourfolds. \n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:",
    "context": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:",
    "full_context": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe strategy of the proof is inspired by and generalizes the approach of \\cite{kapfermenet}:\nwe first consider a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology, which forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion. By considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\), we then show that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \\(n\\) is odd, we obtain Corollary \\ref{cor:nice_statement_even}.\nWe are then left with asking ourselves the following\n\n\\end{lemma}\n\\begin{proof}\nSince \\(H^3(-,\\Z)_{\\tors}\\) is a birational invariant, by passing to a resolution of indeterminacies we may assume that \\(f\\) is everywhere defined. The result then follows from the projection formula.\n\\end{proof}\n\\begin{corollary}\\label{cor:n-tors_of_Kum}\n The abelian group $H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors}$ is $n$-torsion.\n\\end{corollary}\n\\begin{proof}\n The map \\(K_{n-1}(A) \\dashrightarrow \\Kum_{n-1}(A)\\) factors through the quotient by \\(\\mathfrak{S}_{n-1}\\) acting on the first \\(n-1\\) entries of \\(A^n\\), yielding a map of degree \\(n\\). \n The quotient \\(K_{n-1}(A)/\\mathfrak{S}_{n-1}\\) is birational to \\(A^{[n-1]}\\), hence we have a generically finite rational map of degree \\(n\\)\n \\[A^{[n-1]} \\dashrightarrow \\Kum_{n-1}(A).\\] \nBy \\cite{Markman2007_integral} the group $H^3(A^{[n-1]},\\Z)$ is torsion-free. The result then follows from Lemma \\ref{lem:dn_tors} with \\(N=1\\).\n\\end{proof}\n\\section{Quotient by the alternating group}\nIdeally, one would like to apply the argument of the previous section to the rational double cover $$A^{n-1} / \\mathfrak A_n \\dashrightarrow \\Kum_{n-1}(A).$$ However, there is no smooth projective model \\(Z\\) of $A^{n-1} / \\alt_n$ for which \\(H^3(Z, \\Z)_{\\tors}\\) is known.\n\nLet $U \\coloneqq K_{n-1}(A) \\cap (A^n\\setminus \\Delta_2) \\subset A^n$ and $V \\coloneqq \\Kum_{n-1}(A) \\cap (A^{[n]} \\setminus D_2) \\subset A^{[n]}$.\nFor codimension reasons, avoiding the smaller diagonals does not change $H^3(-, \\mathbb Z)_{\\tors}$:\n\\begin{lemma}\\label{lem:torsvkum}\n We have\n $H^3(V, \\mathbb Z)_{\\tors} \\simeq H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors}.$\n\\end{lemma}\n\\begin{proof}\nWe consider the stratification \n\\[D_{2}= Z_0 \\supset Z_1\\supset \\cdots \\]\nwhere \\(Z_{i+1}\\) is the singular locus of \\(Z_i\\). By the long exact sequence of relative cohomology and the Thom isomorphism, we have\n\\[H^3(\\Kum_{n-1}\\setminus Z_{1},\\Z)\\simeq H^3(\\Kum_{n-1}\\setminus Z_{2},\\Z) \\simeq \\dots \\simeq H^3(\\Kum_{n-1}(A), \\mathbb Z),\\]\nsee \\cite[Lemma 11.13]{Voisin_book_I}. By the same argument we also have the exact sequence \n$$ 0\\to H^3(\\Kum_{n-1} \\setminus Z_1,\\mathbb Z) \\to H^3( \\Kum_{n-1}(A) \\setminus Z_0, \\mathbb Z) \\to \\mathbb Z^{\\pi_0(Z_0 \\setminus Z_1)}.$$\nThus \n\\[H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\simeq H^3(\\Kum_{n-1}(A)\\setminus Z_0,\\Z)_{\\tors}.\\]\nSince $V = \\Kum_{n-1}(A) \\setminus Z_0$, we conclude.\n\\end{proof}\nSince the alternating group acts freely on $U$, the relation between the quotient $U / \\alt_n$ and the open subset $V \\subset \\Kum_{n-1}(A)$ can be made precise as follows:\n\\begin{lemma}\\label{lem:doublecover_V}\nThere exists a ramified double cover \n $$\\pi \\colon \\Bl_{\\overline{\\Delta}} (U/\\alt_n) \\to V,$$\n where \\(\\overline{\\Delta}\\) is the image in \\(U/\\alt_n\\) of \\(\\Delta\\subset U\\).\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Sec.\\ 8, p.\\ 770]{beauville}, we have\n\\[(\\Bl_{\\Delta} U )/ \\mathfrak S_n \\simeq V\\]\nMoreover, since $\\mathfrak A_n$ acts freely on $U$ and stabilizes $\\Delta$, we have\n$$\\Bl_{\\Delta}(U) / \\mathfrak A_n \\simeq \\Bl_{\\overline{\\Delta}}(U / \\mathfrak A_n).$$\nThe action of \\(\\Z/2\\Z\\simeq\\mathfrak{S}_n/\\alt_n\\) induced onto the quotient \\(\\Bl_{\\Delta}(U) / \\mathfrak A_n \\) produces a double cover\n $$\\Bl_{\\overline\\Delta}(U / \\mathfrak A_n) \\simeq (\\Bl_{\\Delta} U ) / \\mathfrak A_n \\to (\\Bl_{\\Delta} U) / \\mathfrak S_n \\simeq V,$$\n which is what we wanted.\n \\end{proof}\n\n\\begin{proposition}\\label{prop:spectral_sequence_argument}\n If there are integers $N_p$ for $1 \\leq p \\leq 3$ such that $$N_p \\cdot H^p(\\mathfrak A_n, H^{3-p}(K_{n-1}(A), \\mathbb Z)) = 0,$$ then we have\n $$2N_1N_2N_3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\n\\end{proposition}\n\\begin{proof}\nSince the action of $\\mathfrak A_n$ on $U$ is free, there is the Cartan--Leray spectral sequence\n$$H^p(\\mathfrak A_n, H^q(U, \\mathbb Z)) \\Rightarrow H^{p+q}(U / \\mathfrak A_n, \\mathbb Z).$$\nNote that for $p \\leq 6$, by the same argument as in Lemma \\ref{lem:torsvkum} we have $H^p(U, \\mathbb Z) \\simeq H^p(K_{n-1}(A), \\mathbb Z)$, as the complement of $U \\subset K_{n-1}(A)$ is of codimension four. In particular, $H^0(\\mathfrak A_n, H^3(U, \\mathbb Z)) \\simeq H^3(K_{n-1}(A))^{\\mathfrak A_n}$ is torsion-free. \nIt then follows that $H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors}$ is an extension of subquotients of $H^{p}(\\mathfrak A_n, H^{3-p}(K_{n-1}(A, \\mathbb Z)))$ for $1 \\leq p \\leq 3$, which implies that \\(N_1 N_2 N_3 \\cdot H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors} = 0\\) .\nAs blowing-up smooth subvarieties leaves $H^3(-, \\mathbb Z)_{\\tors}$ invariant, we have\n$$H^3(\\Bl_{\\overline{\\Delta}}(U / \\mathfrak A_n), \\mathbb Z)_{\\tors} \\simeq H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors}.$$\nBy applying \\cite[Thm.\\ 5.4]{aguilarprieto} to the double cover of Lemma \\ref{lem:doublecover_V} $$\\pi \\colon \\Bl_{\\overline\\Delta}(U / \\mathfrak A_n) \\to V,$$\nwe have that \\(\\pi_* \\pi^* \\alpha =2 \\alpha\\) for any \\(\\alpha \\in H^*(V,\\Z)\\), which allows us to conclude by Lemma \\ref{lem:torsvkum}.\n\\end{proof}\n\n\\begin{theorem*}[Theorem \\ref{thm:main}]\nIf \\(n \\neq 6\\), then\n $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$\n\\end{theorem*}\n\n\\begin{proof}\nOn the one hand, we have $n \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ by Corollary \\ref{cor:n-tors_of_Kum}. Note that this already implies the result for $n \\in \\{4, 6, 8\\}$. On the other hand, by combining Proposition \\ref{prop:spectral_sequence_argument} and Proposition \\ref{prop:spectralval}, we have $2^3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ for $n = 3$ or $n \\geq 8$ and $2 \\cdot 12^3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ for $n=5,7$. The claim then follows by taking the greatest common divisor of $n$ and $2^3$ (respectively \\ $2 \\cdot 12^3$).\n\\end{proof}\n\n\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}",
    "post_theorem_intro_text_len": 2336,
    "post_theorem_intro_text": "For odd values of \\(n\\), this amounts to the following:\n\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}\nAs already mentioned, the case \\(n=2\\) follows from \\cite{kapfermenet}. This result confirms folklore expectations, as stated for example in \\cite[Rem.\\ 2.1.(ii)]{huyb2024}\n\nThe strategy of the proof is inspired by and generalizes the approach of \\cite{kapfermenet}:\nwe first consider a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology, which forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion. By considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\), we then show that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \\(n\\) is odd, we obtain Corollary \\ref{cor:nice_statement_even}.\nWe are then left with asking ourselves the following\n\n\\begin{question}\n    Does \\(H^3(\\Kum_{2n+1}(A),\\Z)_{\\tors}\\) vanish for \\(n \\geq 1\\) as well?\n\\end{question}\n\n\\subsection{Acknowledgements}\nThe authors would like to thank Daniel Huybrechts and Emanuele Macrì for their interest in the project, as well as Nick Addington for sharing his computations and suggesting that earlier bounds could be improved. The first author would like to thank Claire Voisin for the invitation to Paris, where this project was initiated. \nMany thanks to Daniel Huybrechts and Paolo Stellari, respectively, for bringing up Question \\ref{que:hk_torsion} at the ``K3 surfaces and friends\"\\footnote{Lorentz Center, Leiden, June 2025  (\\href{https://www.lorentzcenter.nl/k3-surfaces-en-friends-brauer-groups-and-moduli.html}{\\underline{link}}) }\nand PRAGMATIC\\footnote{University of Catania, Italy, September 2025 (\\href{https://www.dmi.unict.it/pragmatic/docs/Pragmatic2025.html}{\\underline{link}})} summer schools. We thank the organizers of said events for the wonderful research environment.\nBoth authors were supported by the ERC Synergy Grant 854361 HyperK.\nThe first author is grateful for the support provided by the International Max Planck Research School on Moduli Spaces at the Max Planck Institute for Mathematics in Bonn.",
    "sketch": "The strategy of the proof (inspired by and generalizing \\,\\cite{kapfermenet}) is as follows. To prove Theorem~\\ref{thm:main}, one first considers “a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology,” which “forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion.” Then, “by considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\),” one shows that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) “is a \\(2\\)-group,” using “an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence.” When \\(n\\) is odd, combining these conclusions yields Corollary~\\ref{cor:nice_statement_even}.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:main}\nIf \\(n \\neq 6\\), then\n    $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$,",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(A\\) be an abelian surface, and for an integer \\(n\\ge 2\\) let \\(\\Kum_{n-1}(A)\\) denote the corresponding generalized Kummer variety. Write \\(H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}\\) for the torsion subgroup of its third integral cohomology. Which annihilation statement holds for this torsion subgroup?",
      "correct_choice": {
        "label": "A",
        "text": "If \\(n\\neq 6\\), then \\[\\gcd(2^3,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\] If \\(n=6\\), then \\[6\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every integer \\(n\\ge 2\\), one has\n\\[2^3\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]"
        },
        {
          "label": "C",
          "text": "If \\(n\\) is odd, then\n\\[H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]"
        },
        {
          "label": "D",
          "text": "If \\(n\\neq 6\\), then\n\\[\\operatorname{lcm}(2^3,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]\nIf \\(n=6\\), then\n\\[6\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"
        },
        {
          "label": "E",
          "text": "If \\(n\\neq 6\\), then\n\\[\\gcd(2^2,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]\nIf \\(n=6\\), then\n\\[2\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "exceptional_case_n_eq_6_and_gcd_dependence",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "full_annihilator_replaced_by_odd_n_consequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "gcd_replaced_by_lcm",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "sharp_2_primary_exponent_and_special_n_6_constant",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the setup and asks which conclusion holds; it does not state or strongly hint at the specific gcd(2^3,n) bound or the exceptional n=6 case."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-selection item: the correct option is the precise statement of a known conclusion, though the presence of nearby variants prevents it from being a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires distinguishing between closely related bounds (gcd vs lcm, sharp exponent, exceptional case), but success is driven more by recall/recognition of the exact theorem statement than by substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrong uniform bounds, weaker true-but-not-best statements, replacing gcd by lcm, and tampering with the exceptional constant."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than deep generative mathematical reasoning."
    }
  },
  {
    "id": "2512.14403v1",
    "paper_link": "http://arxiv.org/abs/2512.14403v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:Cd-main}\nFor every $d \\geq 1$ and every finite set $A\\subset \\mathbb{R}^m$ with exponential Freiman dimension $d$, we have \n\\begin{equation} \\label{eq:main-tight}\n  |A+A|\\geq C_d(|A|-2^d)+3^d,\n\\end{equation}\nwhere\n\\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]",
      "start_pos": 11292,
      "end_pos": 11744,
      "label": "thm:Cd-main"
    },
    "ref_dict": {
      "eq:example-0.2": "\\begin{align}\n  A &= \\{0,1\\}^{d-1} \\times [0,N], \\label{eq:example-0.1}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{d/2} \\times [0,N]^{d/2}\\bigr)\n      &&\\text{for \\(d\\) even}, \\label{eq:example-0.2}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{(d-1)/2} \\times \\{0,1\\}\\times [0,N]^{(d-1)/2}\\bigr)\n      &&\\text{for \\(d\\) odd}. \\label{eq:example-0.3}\n\\end{align}",
      "thm:Cd-main": "\\begin{theorem}\\label{thm:Cd-main}\nFor every $d \\geq 1$ and every finite set $A\\subset \\R^m$ with exponential Freiman dimension $d$, we have \n\\begin{equation} \\label{eq:main-tight}\n  |A+A|\\geq C_d(|A|-2^d)+3^d,\n\\end{equation}\nwhere\n\\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]\n\\end{theorem}",
      "eq:example-0.1": "\\begin{align}\n  A &= \\{0,1\\}^{d-1} \\times [0,N], \\label{eq:example-0.1}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{d/2} \\times [0,N]^{d/2}\\bigr)\n      &&\\text{for \\(d\\) even}, \\label{eq:example-0.2}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{(d-1)/2} \\times \\{0,1\\}\\times [0,N]^{(d-1)/2}\\bigr)\n      &&\\text{for \\(d\\) odd}. \\label{eq:example-0.3}\n\\end{align}",
      "eq:example-0.3": "\\begin{align}\n  A &= \\{0,1\\}^{d-1} \\times [0,N], \\label{eq:example-0.1}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{d/2} \\times [0,N]^{d/2}\\bigr)\n      &&\\text{for \\(d\\) even}, \\label{eq:example-0.2}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{(d-1)/2} \\times \\{0,1\\}\\times [0,N]^{(d-1)/2}\\bigr)\n      &&\\text{for \\(d\\) odd}. \\label{eq:example-0.3}\n\\end{align}",
      "eq:main-tight": "\\begin{equation} \\label{eq:main-tight}\n  |A+A|\\geq C_d(|A|-2^d)+3^d,\n\\end{equation}",
      "lem:mainineq": "\\begin{theorem} \\label{lem:mainineq}\n    Let $0\\leq k<d$ be integers. Let $\\{x_u: u\\in \\{0,1\\}^k\\}$ and $\\{y_u: u\\in \\{0,1,2\\}^k\\}$ be sets of non-negative real numbers such that $y_{u+v}\\geq x_u+x_v$ for each $u,v\\in \\{0,1\\}^k$. Assume further that $x_u\\geq x_v$ if $v\\succeq u$. Then we have\n    \\begin{equation} \\label{anineq}\n        \\sum_{u\\in \\{0,1,2\\}^k} y_u^{d-k}\\geq C_d \\sum_{u\\in \\{0,1\\}^k} x_u^{d-k},\n    \\end{equation}\n    where $C_d$ is as in Theorem \\ref{thm:Cd-main}, namely\n    \\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]\n\\end{theorem}",
      "sqrt2": "\\begin{equation} \\label{sqrt2}\n  |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n  \\end{equation}"
    },
    "pre_theorem_intro_text_len": 3746,
    "pre_theorem_intro_text": "A central theme in additive combinatorics suggests that in any abelian group, finite sets $A$ with  small doubling\n\\[\n  \\sigma(A) \\;:=\\; \\frac{|A+A|}{|A|}\n\\]\nmust exhibit strong additive structure.  Here, $A+ A = \\{ a+ a' : a,a' \\in A\\}$ denotes the sumset of $A$. Classical results of Freiman and their later refinements show that such sets are efficiently contained in\ngeneralized arithmetic progressions or subspaces with bounded dimension; see, for example, Freiman's monograph\n\\cite{FreimanBook} and the book of Tao and Vu~\\cite{TaoVu}.\n\nWhen the abelian group above has no torsion, these questions are often related to studying the following problem: given a finite set $A \\subset \\mathbb{R}^m$, what type of geometric conditions must $A$ satisfy so as to ensure that the doubling $\\sigma(A)$ grows rapidly, see \\cite{Bilu, Chang}.\nA basic instance of this is \\emph{Freiman's lemma}, which states that if a finite set \\(A\\subset\\mathbb{R}^m\\) contains the vertices of a non-degenerate \\(d\\)-simplex, then\n\\[\n  |A+A| \\geq (d+1)|A| - \\tfrac12 d(d+1) \\geq d|A|/2.\n\\]\nA non-degenerate \\(d\\)-simplex is a set $P \\;=\\; \\{v_0,\\,v_0+v_1,\\dots,v_0+v_d\\}$\nwith \\(v_1,\\dots,v_d\\) linearly independent. So the mere presence of a simplex of dimension \\(d\\) forces the additive doubling $\\sigma(A)$ to grow linearly in $d$.\n\nThis can be compared with the continuous setting where given a non-empty, compact set $\\mathcal{A} \\subset \\mathbb{R}^d$,  one can apply the Brunn--Minkowski inequality to deduce that $\\mu(\\mathcal{A} + \\mathcal{A}) \\geq 2^d \\mu(\\mathcal{A})$, with $\\mu$ denoting the Lebesgue measure in $\\mathbb{R}^d$.  Thus it is natural to ask what local conditions can be prescribed for a finite set $A \\subseteq \\mathbb{R}^d$ to ensure that $\\sigma(A)$ grows exponentially in $d$. \n\nIn this direction, Green and Tao \\cite{GTcompressions} introduced the notion of \\emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \\subset \\mathbb{R}^m$, we define its exponential Freiman dimension  to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped\n\\[\n  P \\;=\\; v_0 + \\{0,1\\} \\cdot v_1 + \\cdots + \\{0,1\\} \\cdot v_d\n\\]\nwith \\(v_1,\\dots,v_d\\) linearly independent. Green and Tao \\cite{GTcompressions} proved that if $A \\subset \\mathbb{R}^m$ has exponential Freiman dimension $d$, then \n\\begin{equation} \\label{sqrt2}\n  |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n  \\end{equation}\nSuch inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\\'{a}lv\\\"{o}lgyi--Zhelezov ~ \\cite{PZ2020}. This circle of ideas has been significantly generalized in \\cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \\cite{PFR} on the  polynomial Freiman--Ruzsa conjecture over $\\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\\mathbb{R}$.\n\nIn \\cite{GTcompressions}, Green and Tao noted that the inequality \\eqref{sqrt2} is somewhat close to sharp: for example, if $A = \\{0,1\\}^d\\subset\\mathbb{R}^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that\n\\begin{equation}\\label{eq:Cd-def}\n  |A+A| \\;\\ge\\; C_d\\,|A| - O_d(1)\n\\end{equation}\nholds for all finite sets $A \\subset \\mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that \n$$(\\sqrt{2})^{d} \\leq C_{d} \\leq 2(3/2)^{d-1},$$\nwith the upper bound following from considering the set $A = \\{0,1\\}^{d-1} \\times \\{0,1,\\dots, N\\}$. \n\nOur main result closes the above exponential gap by determining the exact value of \\(C_d\\) for every $d \\geq 1$.",
    "context": "A central theme in additive combinatorics suggests that in any abelian group, finite sets $A$ with small doubling\n\\[\n \\sigma(A) \\;:=\\; \\frac{|A+A|}{|A|}\n\\]\nmust exhibit strong additive structure. Here, $A+ A = \\{ a+ a' : a,a' \\in A\\}$ denotes the sumset of $A$. Classical results of Freiman and their later refinements show that such sets are efficiently contained in\ngeneralized arithmetic progressions or subspaces with bounded dimension; see, for example, Freiman's monograph\n\\cite{FreimanBook} and the book of Tao and Vu~\\cite{TaoVu}.\n\nWhen the abelian group above has no torsion, these questions are often related to studying the following problem: given a finite set $A \\subset \\mathbb{R}^m$, what type of geometric conditions must $A$ satisfy so as to ensure that the doubling $\\sigma(A)$ grows rapidly, see \\cite{Bilu, Chang}.\nA basic instance of this is \\emph{Freiman's lemma}, which states that if a finite set \\(A\\subset\\mathbb{R}^m\\) contains the vertices of a non-degenerate \\(d\\)-simplex, then\n\\[\n |A+A| \\geq (d+1)|A| - \\tfrac12 d(d+1) \\geq d|A|/2.\n\\]\nA non-degenerate \\(d\\)-simplex is a set $P \\;=\\; \\{v_0,\\,v_0+v_1,\\dots,v_0+v_d\\}$\nwith \\(v_1,\\dots,v_d\\) linearly independent. So the mere presence of a simplex of dimension \\(d\\) forces the additive doubling $\\sigma(A)$ to grow linearly in $d$.\n\nThis can be compared with the continuous setting where given a non-empty, compact set $\\mathcal{A} \\subset \\mathbb{R}^d$, one can apply the Brunn--Minkowski inequality to deduce that $\\mu(\\mathcal{A} + \\mathcal{A}) \\geq 2^d \\mu(\\mathcal{A})$, with $\\mu$ denoting the Lebesgue measure in $\\mathbb{R}^d$. Thus it is natural to ask what local conditions can be prescribed for a finite set $A \\subseteq \\mathbb{R}^d$ to ensure that $\\sigma(A)$ grows exponentially in $d$.\n\nIn this direction, Green and Tao \\cite{GTcompressions} introduced the notion of \\emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \\subset \\mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped\n\\[\n P \\;=\\; v_0 + \\{0,1\\} \\cdot v_1 + \\cdots + \\{0,1\\} \\cdot v_d\n\\]\nwith \\(v_1,\\dots,v_d\\) linearly independent. Green and Tao \\cite{GTcompressions} proved that if $A \\subset \\mathbb{R}^m$ has exponential Freiman dimension $d$, then \n\\begin{equation} \\label{sqrt2}\n |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n \\end{equation}\nSuch inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\\'{a}lv\\\"{o}lgyi--Zhelezov ~ \\cite{PZ2020}. This circle of ideas has been significantly generalized in \\cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \\cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\\mathbb{R}$.\n\nIn \\cite{GTcompressions}, Green and Tao noted that the inequality \\eqref{sqrt2} is somewhat close to sharp: for example, if $A = \\{0,1\\}^d\\subset\\mathbb{R}^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that\n\\begin{equation}\\label{eq:Cd-def}\n |A+A| \\;\\ge\\; C_d\\,|A| - O_d(1)\n\\end{equation}\nholds for all finite sets $A \\subset \\mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that \n$$(\\sqrt{2})^{d} \\leq C_{d} \\leq 2(3/2)^{d-1},$$\nwith the upper bound following from considering the set $A = \\{0,1\\}^{d-1} \\times \\{0,1,\\dots, N\\}$.\n\nOur main result closes the above exponential gap by determining the exact value of \\(C_d\\) for every $d \\geq 1$.\n\n\\begin{equation} \\label{sqrt2}\n  |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n  \\end{equation}",
    "full_context": "A central theme in additive combinatorics suggests that in any abelian group, finite sets $A$ with small doubling\n\\[\n \\sigma(A) \\;:=\\; \\frac{|A+A|}{|A|}\n\\]\nmust exhibit strong additive structure. Here, $A+ A = \\{ a+ a' : a,a' \\in A\\}$ denotes the sumset of $A$. Classical results of Freiman and their later refinements show that such sets are efficiently contained in\ngeneralized arithmetic progressions or subspaces with bounded dimension; see, for example, Freiman's monograph\n\\cite{FreimanBook} and the book of Tao and Vu~\\cite{TaoVu}.\n\nWhen the abelian group above has no torsion, these questions are often related to studying the following problem: given a finite set $A \\subset \\mathbb{R}^m$, what type of geometric conditions must $A$ satisfy so as to ensure that the doubling $\\sigma(A)$ grows rapidly, see \\cite{Bilu, Chang}.\nA basic instance of this is \\emph{Freiman's lemma}, which states that if a finite set \\(A\\subset\\mathbb{R}^m\\) contains the vertices of a non-degenerate \\(d\\)-simplex, then\n\\[\n |A+A| \\geq (d+1)|A| - \\tfrac12 d(d+1) \\geq d|A|/2.\n\\]\nA non-degenerate \\(d\\)-simplex is a set $P \\;=\\; \\{v_0,\\,v_0+v_1,\\dots,v_0+v_d\\}$\nwith \\(v_1,\\dots,v_d\\) linearly independent. So the mere presence of a simplex of dimension \\(d\\) forces the additive doubling $\\sigma(A)$ to grow linearly in $d$.\n\nThis can be compared with the continuous setting where given a non-empty, compact set $\\mathcal{A} \\subset \\mathbb{R}^d$, one can apply the Brunn--Minkowski inequality to deduce that $\\mu(\\mathcal{A} + \\mathcal{A}) \\geq 2^d \\mu(\\mathcal{A})$, with $\\mu$ denoting the Lebesgue measure in $\\mathbb{R}^d$. Thus it is natural to ask what local conditions can be prescribed for a finite set $A \\subseteq \\mathbb{R}^d$ to ensure that $\\sigma(A)$ grows exponentially in $d$.\n\nIn this direction, Green and Tao \\cite{GTcompressions} introduced the notion of \\emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \\subset \\mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped\n\\[\n P \\;=\\; v_0 + \\{0,1\\} \\cdot v_1 + \\cdots + \\{0,1\\} \\cdot v_d\n\\]\nwith \\(v_1,\\dots,v_d\\) linearly independent. Green and Tao \\cite{GTcompressions} proved that if $A \\subset \\mathbb{R}^m$ has exponential Freiman dimension $d$, then \n\\begin{equation} \\label{sqrt2}\n |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n \\end{equation}\nSuch inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\\'{a}lv\\\"{o}lgyi--Zhelezov ~ \\cite{PZ2020}. This circle of ideas has been significantly generalized in \\cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \\cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\\mathbb{R}$.\n\nIn \\cite{GTcompressions}, Green and Tao noted that the inequality \\eqref{sqrt2} is somewhat close to sharp: for example, if $A = \\{0,1\\}^d\\subset\\mathbb{R}^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that\n\\begin{equation}\\label{eq:Cd-def}\n |A+A| \\;\\ge\\; C_d\\,|A| - O_d(1)\n\\end{equation}\nholds for all finite sets $A \\subset \\mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that \n$$(\\sqrt{2})^{d} \\leq C_{d} \\leq 2(3/2)^{d-1},$$\nwith the upper bound following from considering the set $A = \\{0,1\\}^{d-1} \\times \\{0,1,\\dots, N\\}$.\n\nOur main result closes the above exponential gap by determining the exact value of \\(C_d\\) for every $d \\geq 1$.\n\n\\begin{equation} \\label{sqrt2}\n  |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n  \\end{equation}\n\nIn this direction, Green and Tao \\cite{GTcompressions} introduced the notion of \\emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \\subset \\mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped\n\\[\n P \\;=\\; v_0 + \\{0,1\\} \\cdot v_1 + \\cdots + \\{0,1\\} \\cdot v_d\n\\]\nwith \\(v_1,\\dots,v_d\\) linearly independent. Green and Tao \\cite{GTcompressions} proved that if $A \\subset \\mathbb{R}^m$ has exponential Freiman dimension $d$, then \n\\begin{equation} \\label{sqrt2}\n |A+A| \\geq (\\sqrt{2})^{d} \\,|A|.\n \\end{equation}\nSuch inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\\'{a}lv\\\"{o}lgyi--Zhelezov ~ \\cite{PZ2020}. This circle of ideas has been significantly generalized in \\cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \\cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\\mathbb{R}$.\n\nIn \\cite{GTcompressions}, Green and Tao noted that the inequality \\eqref{sqrt2} is somewhat close to sharp: for example, if $A = \\{0,1\\}^d\\subset\\R^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that\n\\begin{equation}\\label{eq:Cd-def}\n |A+A| \\;\\ge\\; C_d\\,|A| - O_d(1)\n\\end{equation}\nholds for all finite sets $A \\subset \\mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that \n$$(\\sqrt{2})^{d} \\leq C_{d} \\leq 2(3/2)^{d-1},$$\nwith the upper bound following from considering the set $A = \\{0,1\\}^{d-1} \\times \\{0,1,\\dots, N\\}$.\n\nA perhaps surprising aspect of Theorem~\\ref{thm:Cd-main} is that the natural upper bound example $A = \\{0,1\\}^{d-1} \\times \\{0,1,2,\\dots, N\\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \\leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\\sigma(A)$.\nWriting \\([0,N] := \\{0,1,\\dots,N\\}\\)\nfor any \\(N\\in\\N\\), we describe these examples below.\n\\begin{align}\n A &= \\{0,1\\}^{d-1} \\times [0,N], \\label{eq:example-0.1}\\\\\n A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{d/2} \\times [0,N]^{d/2}\\bigr)\n &&\\text{for \\(d\\) even}, \\label{eq:example-0.2}\\\\\n A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{(d-1)/2} \\times \\{0,1\\}\\times [0,N]^{(d-1)/2}\\bigr)\n &&\\text{for \\(d\\) odd}. \\label{eq:example-0.3}\n\\end{align}\nExample~\\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \\{0,1,2\\}^{\\,d-1}\\times[0,2N]$, so \n$$|A+A| = 3^{\\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$\nExample~\\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and \n\\[ A+ A = \\{0,1,2\\}^d \\cup (\\{0,1\\}^{d/2} \\times [0,N+1]^{d/2}) \\cup (\\{0\\}^{d/2} \\times [0, 2N]^{d/2}), \\]\nwhence, \n\\[ \\frac{|A+A|}{|A|} = \\frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \\rightarrow 2 \\cdot 2^{d/2}-1 \\]\nas $N\\rightarrow\\infty$.\nFinally, Example~\\eqref{eq:example-0.3} satisfies $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and\n\\begin{align*}\n A+A &= \\{0,1,2\\}^d \\cup (\\{0,1\\}^{(d-1)/2}\\times \\{0,1,2\\}\\times [0,N+1]^{(d-1)/2}) \\\\\n & \\quad \\cup (\\{0\\}^{(d-1)/2}\\times \\{0,1,2\\}\\times [0,2N]^{(d-1)/2}),\n\\end{align*}\nand so, we get that \n\\begin{align*}\n \\frac{|A+A|}{|A|} &= \\frac{3\\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\\cdot (2N+1)^{(d-1)/2} - 3\\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\\\\n &\\rightarrow 3\\cdot 2^{(d-1)/2} - \\tfrac32\n\\end{align*}\nas $N\\rightarrow\\infty$.\n\n\\begin{theorem} \\label{lem:mainineq}\n Let $0\\leq k<d$ be integers. Let $\\{x_u: u\\in \\{0,1\\}^k\\}$ and $\\{y_u: u\\in \\{0,1,2\\}^k\\}$ be sets of non-negative real numbers such that $y_{u+v}\\geq x_u+x_v$ for each $u,v\\in \\{0,1\\}^k$. Assume further that $x_u\\geq x_v$ if $v\\succeq u$. Then we have\n \\begin{equation} \\label{anineq}\n \\sum_{u\\in \\{0,1,2\\}^k} y_u^{d-k}\\geq C_d \\sum_{u\\in \\{0,1\\}^k} x_u^{d-k},\n \\end{equation}\n where $C_d$ is as in Theorem \\ref{thm:Cd-main}, namely\n \\[\n C_d \\;=\\;\n \\begin{cases}\n 2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n 2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n 3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n \\end{cases}\n\\]\n\\end{theorem}\n\n\\begin{definition}[$i$-compression]\nLet $A \\subset \\R^{d}$ be finite and fix $1 \\le i \\le d$. For each $\\mathbf{u}\\in\\R^{d}$, set\n\\[\nm(\\mathbf{u}) := |A \\cap H_{\\mathbf{u}}|.\n\\]\nWe define the \\emph{$i$-compression} $C_i(A)$ of $A$ by\n\\[\nC_i(A)\n := \\Bigl\\{\n(u_1,\\dots,u_{i-1},t,u_{i+1},\\dots,u_d) : \nt\\in \\N_0, 0 \\le t < m(\\mathbf{u}) \\text{ for some } \\mathbf{u}\n\\Bigr\\},\n\\]\nthat is, in each fiber $H_{\\mathbf{u}}$, we replace $A \\cap H_{\\mathbf{u}}$ by the initial segment\n$\\{0,1,\\dots,m(\\mathbf{u})-1\\}$.\n\\end{definition}\n\nWe begin by recording some useful notations. First, for vectors $u = (u_1, \\dots, u_d)$ and $v = (v_1, \\dots, v_d)$ in $\\{0,1\\}^d$, we define the Hamming distance $d(u,v)$ to be\n\\[ d(u,v) = \\sum_{1 \\leq i \\leq d: \\ u_i \\neq v_i} 1 = \\sum_{1 \\leq i \\leq d} |u_i - v_i|. \\]\nMoreover, we define\n\\[ |u| = d(u,0) = u_1 + \\dots + u_d. \\]\nNext, recall that\n\\[C_d \\;=\\;\n \\begin{cases}\n 2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n 2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n 3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n \\end{cases}\n\\]\n\nJust as before, we will redistribute some of the weight from $D_{k-m-2}$ to $D_{k-m-1}$ and $D_{k-m}$. By \\nameref{lem:redistribution} twice, we have \n\\[X_{k-m-2} \\geq \\frac{k-m-1}{m+2} X_{k-m-1}\\geq \\frac{(k-m-1)(k-m)}{(m+2)(m+1)} X_{k-m}.\\]\nThus we will be done if we can show that\n\\[\\frac{(m+2)(m+1)}{(k-m-1)(k-m)}(C_d-D_{k-m})+\\frac{m+1}{k-m}(C_d-D_{k-m-1})\\leq D_{k-m-2} - C_d,\\]\nor equivalently,\n\\begin{equation} \\label{eq:doddweight}\n\\frac{(m+2)(m+1)}{(k-m-1)(k-m)}\\left(2^{k-m}-\\frac{3}{2}\\right)+\\frac{m+1}{k-m}\\left(2^{k-m-1}-\\frac{3}{2}\\right)\\leq 3\\cdot 2^{m-1}-2^{k-m-2} + \\frac{3}{2}.\n\\end{equation}\nWriting\n\\begin{align*}\n f_1(k,m) &= (m+2)(m+1)\\left(2^{k-m}-\\frac{3}{2}\\right)+(m+1)(k-m-1)\\left(2^{k-m-1}-\\frac{3}{2}\\right)\\\\\n &\\quad - (k-m-1)(k-m)\\left(3\\cdot 2^{m-1}-2^{k-m-2}+\\frac{3}{2}\\right)\n\\end{align*}\nfor all $k,m \\in \\mathbb{Z}$, we see that \\eqref{eq:doddweight} follows for $k,m$ if $f_1(k,m)\\leq 0$.\n\nThe permutohedron $P_{d}$ lives inside the hyperplane $\\left\\{x \\in \\mathbb{R}^{d}:\\ x_1+\\ldots+x_d = \\binom{d+1}{2}\\right\\}$, and has a number of remarkable properties (see for example \\cite{Postnikov} or \\cite{Stanley}). \nFor instance, when $d$ is even, the vertex set of $P_d$ contains $d/2$-dimensional affine cubes. Indeed, consider the $d/2$ disjoint adjacent transpositions\n\\[\nt_j = (2j-1\\;\\;2j) \\in S_{d}, \\qquad j = 1,\\dots,d/2.\n\\]\nThese transpositions commute and each of them has order two. Thereby, for each \n$\\varepsilon = (\\varepsilon_1,\\dots,\\varepsilon_{d/2})\\in\\{0,1\\}^{d/2}$, we can define the permutation\n\\[\n\\sigma_\\varepsilon := t_1^{\\varepsilon_1} \\circ t_2^{\\varepsilon_2} \\cdots \\circ t_{d/2}^{\\varepsilon_{d/2}} \\in S_{d}.\n\\]\nThen the $2^{d/2}$ vertices of $P_{d}$ defined by\n\\[\n\\Bigl\\{\\,\\bigl(\\sigma_\\varepsilon(1),\\dots,\\sigma_\\varepsilon(d)\\bigr)\n : \\varepsilon\\in\\{0,1\\}^{d/2} \\Bigr\\}\n\\]\nform an affine copy of the ${d/2}$-dimensional cube $\\{0,1\\}^{d/2}$. Consequently, if a finite set \\(A \\subset \\mathbb{Z}^d\\) contains the vertex set of the permutohedron, then \\(A\\) automatically contains \nan affine copy of \\(\\{0,1\\}^{\\lfloor d/2 \\rfloor}\\). \nOur Theorem \\ref{thm:Cd-main} therefore implies\n\\begin{equation} \\label{cubeinP}\n|A+A|\\geq C_{d/2}(|A|-2^{d/2})+3^{d/2}.\n\\end{equation}",
    "post_theorem_intro_text_len": 6928,
    "post_theorem_intro_text": "A perhaps surprising aspect of Theorem~\\ref{thm:Cd-main} is  that the natural upper bound example $A = \\{0,1\\}^{d-1} \\times \\{0,1,2,\\dots, N\\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \\leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\\sigma(A)$.\nWriting \\([0,N] := \\{0,1,\\dots,N\\}\\)\nfor any \\(N\\in\\mathbb{N}\\), we describe these examples below.\n\\begin{align}\n  A &= \\{0,1\\}^{d-1} \\times [0,N], \\label{eq:example-0.1}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{d/2} \\times [0,N]^{d/2}\\bigr)\n      &&\\text{for \\(d\\) even}, \\label{eq:example-0.2}\\\\\n  A &= \\{0,1\\}^d \\,\\cup\\, \\bigl(\\{0\\}^{(d-1)/2} \\times \\{0,1\\}\\times [0,N]^{(d-1)/2}\\bigr)\n      &&\\text{for \\(d\\) odd}. \\label{eq:example-0.3}\n\\end{align}\nExample~\\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \\{0,1,2\\}^{\\,d-1}\\times[0,2N]$, so \n$$|A+A| = 3^{\\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$\nExample~\\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and \n\\[ A+ A = \\{0,1,2\\}^d \\cup (\\{0,1\\}^{d/2} \\times [0,N+1]^{d/2}) \\cup (\\{0\\}^{d/2} \\times [0, 2N]^{d/2}),  \\]\nwhence, \n\\[ \\frac{|A+A|}{|A|} = \\frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \\rightarrow 2 \\cdot 2^{d/2}-1 \\]\nas $N\\rightarrow\\infty$.\nFinally, Example~\\eqref{eq:example-0.3} satisfies  $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and\n\\begin{align*}\n    A+A &= \\{0,1,2\\}^d \\cup (\\{0,1\\}^{(d-1)/2}\\times \\{0,1,2\\}\\times [0,N+1]^{(d-1)/2}) \\\\\n    & \\quad \\cup (\\{0\\}^{(d-1)/2}\\times \\{0,1,2\\}\\times [0,2N]^{(d-1)/2}),\n\\end{align*}\nand so, we get that \n\\begin{align*}\n    \\frac{|A+A|}{|A|} &= \\frac{3\\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\\cdot (2N+1)^{(d-1)/2} - 3\\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\\\\n    &\\rightarrow 3\\cdot 2^{(d-1)/2} - \\tfrac32\n\\end{align*}\nas $N\\rightarrow\\infty$.\n\nTaking the minimum of these three upper bounds for each \\(d\\) recovers exactly\nthe values in Theorem~\\ref{thm:Cd-main}. Furthermore, \\eqref{eq:main-tight} is exactly tight for $d\\leq 5$. Interestingly, for sufficiently large $d$, since $C_d<(3/2)^d$, Theorem \\ref{thm:Cd-main} yields the bound\n\\[ |A+A|\\geq C_d|A|+(3^d-2^dC_d), \\]\nwherein the error term is actually positive.\n\nThe main difficulty will be to show that $C_{d}$ is at least the indicated values above. Before we move on to the proof of Theorem \\ref{thm:Cd-main} in \\S2, let us start by discussing some of the high-level ideas behind it. \n\n\\medskip\n\n{\\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume\nthat $P=\\left\\{0,1\\right\\}^{d} \\subseteq A \\subset \\mathbb{R}^{d}$. By further applying translations\nand coordinate-wise compressions, we may also assume that $A$ is a \\emph{down-set} in $\\N_0^d$, that is, for any \\(u\\in A\\) and any $v \\in \\mathbb{Z}^d$ satisfying the inequality \\(0\\le v\\le u\\) coordinate-wise, we have \\(v\\in A\\).  In particular, the\nassumption \\(\\{0,1\\}^d \\subseteq A\\) says that the ``bottom layer'' of \\(A\\) is a\ncomplete discrete hypercube.  We then partition \\(A\\) according to which\ncoordinates lie below or above a fixed threshold, and obtain a parallel\npartition of the sumset \\(B := A+A\\).  We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \\cite{GTcompressions}, to get lower bounds on the sizes of the blocks of \\(B\\)\nin terms of the blocks of \\(A\\).\n\nAfter a suitable renormalisation, this reduces Theorem~\\ref{thm:Cd-main} to a\npurely analytic inequality for families of non-negative real numbers\n\\(\\{x_u\\}_{u\\in\\{0,1\\}^k}\\) and \\(\\{y_w\\}_{w\\in\\{0,1,2\\}^k}\\), indexed by slices\nof the discrete (Hamming) hypercube and hypergrid, subject to two constraints:\na collection of inequalities of the form \\(y_{u+v}\\ge x_u+x_v\\) arising from the sumset structure, and a monotonicity condition reflecting the downset\nstructure. The monotonicity condition is not strictly necessary, but it is convenient to have. In order to state this, we will require one further piece of notation, and so, given vectors $u=(u_1, \\dots, u_d)$ and $v= (v_1, \\dots, v_d)$ in $\\mathbb{R}^d$, we say that $v\\succeq u$ if $v_i \\geq u_i$ for every $1 \\leq i \\leq d$. \n\n\\begin{theorem} \\label{lem:mainineq}\n    Let $0\\leq k<d$ be integers. Let $\\{x_u: u\\in \\{0,1\\}^k\\}$ and $\\{y_u: u\\in \\{0,1,2\\}^k\\}$ be sets of non-negative real numbers such that $y_{u+v}\\geq x_u+x_v$ for each $u,v\\in \\{0,1\\}^k$. Assume further that $x_u\\geq x_v$ if $v\\succeq u$. Then we have\n    \\begin{equation} \\label{anineq}\n        \\sum_{u\\in \\{0,1,2\\}^k} y_u^{d-k}\\geq C_d \\sum_{u\\in \\{0,1\\}^k} x_u^{d-k},\n    \\end{equation}\n    where $C_d$ is as in Theorem \\ref{thm:Cd-main}, namely\n    \\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]\n\\end{theorem}\n\nWe remark that the constant $C_d$ in Theorem~\\ref{thm:Cd-main} is tight for at least one value of $k$ for each $d$. In particular, for $d\\leq 5$, tightness occurs at $k=d-1$, with $x_u=1$ for all $u$. For even $d\\geq 6$, tightness occurs at $k=d/2$ with $x_0=1$ and $x_u=0$ for all $u\\neq 0$. Finally, for odd $d\\geq 7$, tightness occurs at $k=(d+1)/2$ with $x_0=x_{e_1}=1$ and $x_u=0$ for all other $u$. These equality cases naturally result in the constructions given in \\eqref{eq:example-0.1}, \\eqref{eq:example-0.2} and \\eqref{eq:example-0.3}.\n\nTheorem \\ref{lem:mainineq} is proved by a\ncombination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight.  The\ndimension-dependent constants that arise along the way match precisely the\nvalues of \\(C_d\\) asserted in Theorem~\\ref{thm:Cd-main}.\n\n\\medskip\n\n{\\bf{Organization of the paper}}. In \\S2 we discuss various prerequisites concerning compressions and prove how  Theorem~\\ref{thm:Cd-main} follows from Theorem \\ref{lem:mainineq}. We lay some preliminary brickwork for proving our main analytic inequality in \\S3, which already turns out to be sufficient to deal with the small-dimensional cases \\(d \\in \\{3,4,5\\}\\). The case when $d \\geq 6$ is even is resolved in \\S4, while the case when $d \\geq 7$ is odd is analysed in \\S5.  The odd case is slightly more involved compared to the even case, since the corresponding doubling constant $C_d$ is larger and the construction \\eqref{eq:example-0.3} attaining ``equality'' is more complicated. In \\S6, we will discuss a few natural future directions.\n\n\\medskip\n\n{\\bf{Acknowledgments.}} The authors would like to thank Ben Green for useful discussions. J.L. was supported by ERC Advanced Grant 883810. A.M. was supported by Leverhulme early career fellowship ECF-2025-148. C.P. was supported by NSF grant DMS-2246659. X.S. was supported by NSF grant DMS-2452462.\n\n\\medskip",
    "sketch": "{\\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\\left\\{0,1\\right\\}^{d} \\subseteq A \\subset \\mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \\emph{down-set} in $\\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \\cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.\n\nAfter a suitable renormalisation, this reduces Theorem~\\ref{thm:Cd-main} to a purely analytic inequality for families of non-negative real numbers $\\{x_u\\}_{u\\in\\{0,1\\}^k}$ and $\\{y_w\\}_{w\\in\\{0,1,2\\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. This analytic inequality is stated as Theorem~\\ref{lem:mainineq}.\n\nFinally, Theorem~\\ref{lem:mainineq} is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in Theorem~\\ref{thm:Cd-main}.",
    "expanded_sketch": "{\\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\\left\\{0,1\\right\\}^{d} \\subseteq A \\subset \\mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \\emph{down-set} in $\\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \\cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.\n\nAfter a suitable renormalisation, to prove the main theorem we reduce to a purely analytic inequality for families of non-negative real numbers $\\{x_u\\}_{u\\in\\{0,1\\}^k}$ and $\\{y_w\\}_{w\\in\\{0,1,2\\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. We first prove the following theorem.\n\n\\begin{theorem} \\label{lem:mainineq}\n    Let $0\\leq k<d$ be integers. Let $\\{x_u: u\\in \\{0,1\\}^k\\}$ and $\\{y_u: u\\in \\{0,1,2\\}^k\\}$ be sets of non-negative real numbers such that $y_{u+v}\\geq x_u+x_v$ for each $u,v\\in \\{0,1\\}^k$. Assume further that $x_u\\geq x_v$ if $v\\succeq u$. Then we have\n    \\begin{equation} \\label{anineq}\n        \\sum_{u\\in \\{0,1,2\\}^k} y_u^{d-k}\\geq C_d \\sum_{u\\in \\{0,1\\}^k} x_u^{d-k},\n    \\end{equation}\n    where $C_d$ is as in Theorem \\ref{thm:Cd-main}, namely\n    \\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]\n\\end{theorem}\n\nFinally, the theorem above is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in the main theorem.",
    "expanded_theorem": "\\label{thm:Cd-main}\nFor every $d \\geq 1$ and every finite set $A\\subset \\mathbb{R}^m$ with exponential Freiman dimension $d$, we have \n\\begin{equation} \\label{eq:main-tight}\n  |A+A|\\geq C_d(|A|-2^d)+3^d,\n\\end{equation}\nwhere\n\\[\n  C_d \\;=\\;\n  \\begin{cases}\n    2\\bigl(\\tfrac32\\bigr)^{d-1}, & d \\le 5,\\\\[4pt]\n    2^{d/2+1} - 1, & d \\ge 6 \\text{ even},\\\\[4pt]\n    3\\cdot 2^{(d-1)/2} - \\tfrac32, & d \\ge 7 \\text{ odd}.\n  \\end{cases}\n\\]",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let $d\\ge 1$ be an integer, and let $A\\subset \\mathbb{R}^m$ be a finite set for some $m$. Write $A+A=\\{a+a':a,a'\\in A\\}$. Say that $A$ has exponential Freiman dimension $d$ if $d$ is the largest positive integer for which $A$ contains the vertex set of a non-degenerate $d$-dimensional parallelepiped\n\\[\nP=v_0+\\{0,1\\}\\cdot v_1+\\cdots+\\{0,1\\}\\cdot v_d,\n\\]\nwhere $v_1,\\dots,v_d$ are linearly independent. Which statement holds for every such $d$ and every such finite set $A$?",
      "correct_choice": {
        "label": "A",
        "text": "One always has\n\\[\n|A+A|\\ge C_d\\bigl(|A|-2^d\\bigr)+3^d,\n\\]\nwhere\n\\[\nC_d=\n\\begin{cases}\n2\\bigl(\\tfrac32\\bigr)^{d-1}, & d\\le 5,\\\\[4pt]\n2^{d/2+1}-1, & d\\ge 6 \\text{ even},\\\\[4pt]\n3\\cdot 2^{(d-1)/2}-\\tfrac32, & d\\ge 7 \\text{ odd}.\n\\end{cases}\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "One always has\n\\[\n|A+A|\\ge C_d\\bigl(|A|-2^d\\bigr)+3^d,\n\\]\nwhere\n\\[\nC_d=\n\\begin{cases}\n2\\bigl(\\tfrac32\\bigr)^{d-1}, & d\\le 5,\\\\[4pt]\n2^{d/2+1}-1, & d\\ge 6,\\\\[4pt]\n3\\cdot 2^{(d-1)/2}-\\tfrac32, & d\\ge 7 \\text{ odd}.\n\\end{cases}\n\\]"
        },
        {
          "label": "C",
          "text": "One always has\n\\[\n|A+A|\\ge C_d\\bigl(|A|-2^d\\bigr),\n\\]\nwhere\n\\[\nC_d=\n\\begin{cases}\n2\\bigl(\\tfrac32\\bigr)^{d-1}, & d\\le 5,\\\\[4pt]\n2^{d/2+1}-1, & d\\ge 6 \\text{ even},\\\\[4pt]\n3\\cdot 2^{(d-1)/2}-\\tfrac32, & d\\ge 7 \\text{ odd}.\n\\end{cases}\n\\]"
        },
        {
          "label": "D",
          "text": "One always has\n\\[\n|A+A|\\ge C_d\\,|A|-O_d(1),\n\\]\nwhere\n\\[\nC_d=\n\\begin{cases}\n2\\bigl(\\tfrac32\\bigr)^{d-1}, & d\\le 5,\\\\[4pt]\n2^{d/2+1}-1, & d\\ge 6 \\text{ even},\\\\[4pt]\n3\\cdot 2^{(d-1)/2}-\\tfrac32, & d\\ge 7 \\text{ odd},\n\\end{cases}\n\\]\nand the implicit constant in \\(O_d(1)\\) may be chosen independently of \\(d\\) and \\(m\\)."
        },
        {
          "label": "E",
          "text": "If \\(A\\subset \\mathbb{R}^m\\) contains the vertex set of some non-degenerate \\(d\\)-dimensional parallelepiped, then one always has\n\\[\n|A+A|\\ge C_d\\bigl(|A|-2^d\\bigr)+3^d,\n\\]\nwhere\n\\[\nC_d=\n\\begin{cases}\n2\\bigl(\\tfrac32\\bigr)^{d-1}, & d\\le 5,\\\\[4pt]\n2^{d/2+1}-1, & d\\ge 6 \\text{ even},\\\\[4pt]\n3\\cdot 2^{(d-1)/2}-\\tfrac32, & d\\ge 7 \\text{ odd}.\n\\end{cases}\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "parity-dependent piecewise definition of C_d",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the additive terminal term +3^d",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "dependence of the error term on d",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "largest-dimension hypothesis replaced by mere existence of a d-parallelepiped",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the setting and asks for a universal valid statement, but it does not reveal the bound, the constants, or the crucial quantifier structure of the correct answer."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-identification item: the correct option is the exact theorem statement, while the others are nearby variants. It tests recall/recognition of the statement more than independent conclusion-drawing."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing subtle changes in parity conditions, additive terms, quantifiers, and uniformity of constants, but the task is still mainly to recognize the exact theorem rather than generate it from mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are close to the true statement and reflect realistic failure modes such as dropping the +3^d term, weakening the hypothesis, mishandling parity cases, or overstating uniformity."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is fairly tautological because it mostly asks for the exact statement rather than eliciting genuine generative reasoning."
    }
  },
  {
    "id": "2512.14581v1",
    "paper_link": "http://arxiv.org/abs/2512.14581v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "ex",
      "content": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}",
      "start_pos": 75967,
      "end_pos": 77444,
      "label": null
    },
    "ref_dict": {
      "sct:main2": "\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}\n\n\\section{Power counting when allowing eigenvalues with vanishing derivative}\\label{sct:main2}\nIn our first main theorem, Theorem \\ref{thm:main}, we assumed that $f'[\\lambda_{k_1},\\ldots,\\lambda_{k_n}]\\neq0$. With $\\{\\lambda_k\\}_{k\\geq1}$ in general position, this assumption is typically satisfied. However, there will be at least one~${x\\in\\R}$ such that $f'(x)=0$, which means that the assumption of nonvanishing divided differences is sensitive to infinitesimal changes in $\\{\\lambda_k\\}_{k\\geq1}$. In fact, we shall show that the actual order of divergence of many graphs is sensitive to the question of whether or not $f'$ vanishes on $\\{\\lambda_k\\}_{k\\geq1}$.\n\nWe will refer to faces with an index $i_0$ such that $f'(\\lambda_{i_0})=0$ as \\emph{0-faces} or \\emph{singular faces}, and similarly for indices and eigenvalues.\nAs in Section~\\ref{sect: lower}, given a subset $\\bfr\\subseteq\\mathcal U$ of the set of unbroken faces of a Feynman ribbon graph~$G$, we let~$G_{\\bfr}$ be the graph obtained from $G$ by artificially declaring the faces in $\\mathfrak{b}$ to be broken. Elements of $\\mathfrak{b}$ are called 0-faces.\\begin{thm}\\label{thm:main2}\nLet $f\\in C^{\\infty}(\\mathbb{R})_\\R$ be an even function with $f'$ of precise order $-p-1$ for some $p \\in\\mathbb{R}_{\\ge0}$.\nLet~$\\{\\lambda_k\\}_{k=1}^\\infty$ be a sequence of real numbers with constants $K,c_1,c_2>0$ such that $c_1k^{1/d}\\leq |\\lambda_k|\\leq c_2k^{1/d}$ for all $k\\geq K$. Assume that for all $k,l\\in\\N_{\\geq1}$ we have $f'[\\lambda_{k},\\lambda_{l}]\\neq0$. Then, for any Feynman ribbon graph $G=(G^0,n,G^1)$ whose vertices have valence $\\geq3$, for all external indices $i_1,\\ldots,i_n\\in\\N_{\\geq1}$, there exist $M,c_4>0$ such that for all~$N\\geq M$,\n$$|\\Ampl_{N,i_1,\\ldots,i_n}(G)|\\leq c_4 N^{\\tilde\\omega(G)},$$\n$$\\tilde \\omega(G):=\\max_{\\bfr\\subseteq\\mathcal U}\\omega_{\\bfr}(G_{\\bfr}):=\\max_{\\bfr\\subseteq\\mathcal U}(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),$$\nwhere $U^{\\bfr}$ is the number of unbroken faces of $G_{\\bfr}$, $E^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal edges of $G_{\\bfr}$ (propagators bordered on both sides by unbroken faces) and $V^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal vertices of $G_{\\bfr}$ (vertices bordered on all sides by unbroken faces). Respectively, $E^{\\bfr}_{10}$ and $V^{\\bfr}_{10}$ are the number of edges and vertices of~$G_{\\bfr}$ that border exactly one 0-face and for the rest unbroken faces.\n\\end{thm}\n\\begin{proof}\nUsing the precise order of $f'$, there exists $R>0$ such that, for all $x\\in\\mathbb{R}\\setminus[-R,R]$ and all~$0\\le k\\le \\max_{v\\in G^{0}}\\;\\deg(v)$, equation \\eqref{eqn: precise_order} is satisfied and the conclusion of Lemma \\ref{lem:zero-index bounds} holds. Let~$i_R$ be the lowest index such that $|\\lambda_i| > R$ for $i\\ge i_R$. For brevity we assume -- without loss of generality -- that $f'(\\lambda_0)=0$ and $f'(\\lambda_k)\\neq0$ for~${k\\geq 1}$. As in the proof of Lemma~\\ref{lem: restricted_lower_sufficient}, a sum splitting argument yields\n\\begin{equation*}\n    \\Ampl_{N,i_1,\\ldots,i_n}(G) = \\sum_{\\bfr\\subseteq \\mathcal{U}}\\sum_{\\gamma:\\bfr\\to\\{0,\\dotsc,i_R-1\\}}\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr}).\n\\end{equation*}\nWe claim that it suffices to show that \\begin{align}\\label{eqn: ampl_Gbgamma}\n|\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr})|&\\lesssim N^{\\omega_{\\bfr,\\gamma}(G_{\\bfr})},\\\\\n\\omega_{\\bfr,\\gamma}(G_{\\bfr})&:=(U^{\\bfr}+\\frac{p}{d}(E_{\\textnormal{fi}}^b-V_{\\textnormal{fi}}^b)+\\frac{p+1}{d}(E_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma})),\\nonumber\n\\end{align}\nas the following graph-theoretical argument shows that $\\omega_{\\bfr,\\gamma}(G_{\\bfr})\\le \\omega_{\\bfr}(G_{\\bfr})$ for all such $\\gamma$. In particular, if we let $\\gamma_0$ denote the map that sends all artificially broken faces to the zero index, then $\\omega_{\\bfr,\\gamma_0}(G_{\\bfr})= \\omega_{\\bfr}(G_{\\bfr})$, hence this bound is sharp.\n\nMore generally, we have $\\omega_{\\bfr}(G_{\\bfr,\\gamma})\\leq\\omega_{\\bfr}(G_{\\bfr,\\gamma'})$ whenever $\\gamma'$ is obtained from $\\gamma$ by setting \\begin{equation*}\n    \\gamma'(b)=\\begin{cases}\n        0,&b=b', \\\\\n        \\gamma(b),&\\text{else},\n    \\end{cases}\n\\end{equation*}\nfor some $b'$ with $\\gamma(b')\\neq0$. Indeed, as the values $U^{\\bfr}$, $\\Efi^b$, and $\\Vfi^b$ do not depend on $\\gamma$, it suffices to show \\begin{equation*}\nE_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma'}.\n\\end{equation*}\nTowards this, first note that $E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma}$ equals the number of edges that border the face $b'$ on the one side and border an unbroken face (i.e.\\ an element of $\\mathcal{U}\\setminus\\bfr$) on the other. Moreover, $V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}$ equals the number of vertices that border $b'$ and for the rest only unbroken faces. For each distinct vertex of the latter kind there is at least one distinct edge of the former kind, implying that $$ V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma},$$which concludes the argument. \n\nIt remains to prove~\\eqref{eqn: ampl_Gbgamma}. The strategy is similar to the proof of Theorem \\ref{thm:upper}, but with different bounds for the edges and vertices, in which we distinguish three different types of indices and their corresponding eigenvalues.\nWe distinguish between (i) eigenvalues $\\lambda_0$ such that $f'(\\lambda_0)=0$, (ii) eigenvalues $\\lambda_i$ for which $f'(\\lambda_i)\\neq0$ and the index $i<R$ is fixed, and (iii) eigenvalues $\\lambda_k$ for which $f'(\\lambda_k)\\neq0$ and the index $k\\geq R$ is running. Analogously to the proof of Theorem \\ref{thm:upper}, noting Remark \\ref{rem:apparent shortcut} and its resolution, the above estimate follows from the bounds on edges and vertices that are summarized and proved in Lemma \\ref{lem:final bounds} below, where we stress that all constants that arise are independent of the running indices.\n\\end{proof}\n\n\\begin{lem}\\label{lem:final bounds}\nLet $n\\geq3$. Let $f,\\{\\lambda_k\\}_{k=1}^\\infty$ be as in Theorem \\ref{thm:main2}. Let $\\lambda_0\\in\\R$ be such that $f'(\\lambda_0)=0$. There exist $c_1,c_2>0$ such that we have\n\\begin{multicols}{2}\n\\begin{enumerate}\n\\item $|f'\\{\\lambda_{k_1},\\lambda_{k_2}\\}|>c_1|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_{i},\\lambda_{k}\\}|>c_1$,\n\\item $|f'\\{\\lambda_{i_1},\\lambda_{i_2}\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_k\\}|> c_1|\\lambda_k|^{-p-1}$,\n\\item $|f'\\{\\lambda_0,\\lambda_i\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_0\\}|>c_1$,\n\\item $|f'\\{\\lambda_{k_1},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_0,\\lambda_{k_2},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_2}|^{-p-1}$,\n\\item $|f'\\{\\lambda_{j_1},\\ldots,\\lambda_{j_n}\\}|\\leq c_2$,\nll}] \nll}] \nll}]\n\\end{enumerate}\n\\end{multicols}\n\\noindent\nfor all $i,i_1,i_2\\leq i_R$, $k,k_1,\\ldots,k_n\\geq i_R$ such that $f'(\\lambda_j)\\neq0$ for $j\\in\\{i,i_1,i_2,k,k_1,\\ldots,k_n\\}$ and such that $|\\lambda_{k_1}|\\leq\\cdots\\leq|\\lambda_{k_n}|$, and for all $j_1,\\ldots,j_n\\in\\N$.\n\\end{lem}\n\n\\begin{proof}\nThe estimates 1.\\ and 7.\\ follow from Theorem~\\ref{thm:weighted_divdiff_bound}, 4.\\ and 8.\\ follow from Lemma~\\ref{lem:zero-index bounds}, and 9.\\ was shown in Corollary~\\ref{cor:wdivdifs are bounded}. Estimates 3., 5., and 6.\\ follow from the fact that there is a finite amount of indices $i<i_R$, and for each $i_1,i_2<i_R$ the (weighted) divided difference is nonzero. {Estimate 2.\\ follows from Theorem~\\ref{thm:no_vanishing_divdifs}.}\n\\end{proof}\nWe conclude this section with some remarks and examples illustrating the influence of 0-faces on the behaviour of amplitudes. \n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}",
      "sct:Definitions": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\bfr$ of unbroken faces, $U^{\\bfr},\\Efi^b,\\Vfi^b$ are as before when designating the elements of $\\bfr$ as broken, and $V^{\\bfr}_{10}/E^{\\bfr}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\bfr$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $\\Efi-\\Vfi\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $\\Efi-\\Vfi$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\R^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\N$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.\n\n\\section{Definitions}\n\\label{sct:Definitions}\n\nThroughout we fix a positive real number $d\\in\\R_{>0}$, and a sequence $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\R$ with the property that there exist numbers $K\\in\\N$, $c_1,c_2\\in\\R_{>0}$ such that\n\\begin{align}\\label{eq:eigenvalues asymptotics assumption}\n    c_1 k^{1/d}\\leq|\\lambda_k|\\leq c_2 k^{1/d}\n\\end{align}",
      "eq:power counting formula": "\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\f}-V_{\\f}).\n\\end{align}",
      "rem:2-point 2-loop": "\\begin{rem}\\label{rem:2-point 2-loop}\nFrom Theorem \\ref{thm:main} we may derive, at each loop order, the set of relevant diagrams. For example, the set of 2-point 2-loop diagrams (with vertices of valence $\\geq3$) with maximal order of divergence is\n\\begin{align*}\n~&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=45,in=90] (0,0);\n\\draw (-.8,0) to[out=-45,in=-90] (0,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (-0.3,0);\n\\draw (-.3,0) to[out=45,in=90] (.5,0);\n\\draw (-.3,0) to[out=-45,in=-90] (.5,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{-.3,0};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to[out=-160,in=180] (0,-.1);\n\\draw (0,.5) to[out=-20,in=0] (0,-.1);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to (0,.05);\n\\draw (0,.05) arc (90:450:.2cm);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\vertex{0,.05};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=0,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to[out=180,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (-.4,.4) arc (180:360:.4cm);\n\\vertex{-.4,.4};\n\\vertex{.4,.4};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\quad\"1\\leftrightarrow2\"\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (0,.5) to (0,-.5);\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}.\n\\end{align*}\nEach has an order of divergence of $2+\\frac{p}{d}$ (and no less).\nOne notices these diagrams are automatically planar. They are moreover connected and, though not necessarily 1PI, satisfy a similar connectivity property. Namely, their dual graphs stay connected after removing the vertices that correspond to the broken faces of the original graph. For instance, the diagram\n$$\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}",
      "rem:UV/IR": "\\begin{rem}\\label{rem:UV/IR}\nSmooth even functions $f$ satisfy $f'(0)=0$, and if $\\{\\lambda_k\\}$ is the spectrum of a typical Dirac operator, these eigenvalues correspond to modes of zero momentum. The fact that UV-divergences can become higher as external momenta vanish is reminiscent of noncommutative quantum field theory, cf.\\ \\cite{MRS2000}. For instance, in the naive version of noncommutative $\\phi^4_4$, such behavior was shown to lead to UV/IR mixing and, consequently, nonrenormalizability \\cite{CR2000,CR2001,GW2005a}. This prompted the Grosse--Wulkenhaar model which solved the UV/IR problem \\cite{GW2005b} and proved incredibly successful \\cite{DGMR2007,GW2014}.\n\nIn light of this, it seems instructive to determine whether UV/IR-mixing is present in the spectral action matrix model and its relatives. This is a difficult question to answer,\nrequiring a careful renormalization analysis and the passage to continuous spectrum.\n\\end{rem}",
      "rem:conjectures": "\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}",
      "eq:power counting formula 2": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}",
      "eq:path integral": "\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 5228,
    "pre_theorem_intro_text": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0,0);\n\t\\draw (-0.5,0.5) to (0,0);\n\t\\draw (0.5,0.5) to (0,0);\n\t\\draw (0.5,-0.5) to (0,0);\n\t\\node at (-0.5,0) {$i$};\n\t\\node at (0,0.5) {$j$};\n\t\\node at (0.5,0) {$k$};\n\t\\node at (0,-0.5) {$l$};\n\tlldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0.5,0.5);\n\t\\node at (-0.25,0.25) {$i$};\n\t\\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}",
    "context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}",
    "full_context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}\n\nFor renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.\n\n\\begin{rem}\\label{rem:apparent shortcut}\nWe explain here a problem with an apparent shortcut to the above proof. Indeed, given positive~${n_1,\\ldots,n_U}$ it is not hard to show (using $\\lambda_j\\sim j^{1/d}$) that\n\\begin{align}\\label{eq:sums like integrals}\n \\sum_{j_1=1}^N\\cdots\\sum_{j_U=1}^N \\lambda_{j_1}^{n_1}\\cdots\\lambda_{j_U}^{n_U}=\\O(N^{U+(n_1+\\ldots+n_U)/d}),\n\\end{align} \nexactly as in the case with integrals instead of sums. If one skips the graph-theoretical Lemma \\ref{lem:injection fi vertices to fi edges} and applies the estimates of Theorem \\ref{thm:weighted_divdiff_bound} for arbitrary bordering indices of the fully internal vertices and edges, one can estimate the amplitude by the sum on the left-hand side of \\eqref{eq:sums like integrals} where indeed the powers automatically add up to $n_1+\\ldots+n_U=p(\\Efi-\\Vfi)$ as required! However, the $n_1,\\ldots,n_U$ may not be positive, which invalidates \\eqref{eq:sums like integrals}. The following example shows that such an estimate of the amplitude by \\eqref{eq:sums like integrals} really is too coarse in general. We surely have\n\\begin{align}\\label{eq:example diagram}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n \\draw (0.45,-.7) to (1,0);\n \\draw (1.55,-.7) to (1,0);\n \\draw (1,0.054) arc (-90:270:0.3cm);\n \\draw (1,-0.035) arc (-90:270:.7cm);\n \\node at (1,-.55) {$i_1$};\n \\node at (0,1) {$i_2$};\n \\vertex{1,0};\n\\end{tikzpicture}\n}\n=\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\sum_{k=1}^N\\sum_{l=1}^N\\frac{f'\\{i_1,k,l,k,i_1,i_2\\}}{f'\\{i_1,k\\}f'\\{k,l\\}}\\lesssim\n\\sum_{k=1}^N\\sum_{l=1}^N\\frac{l^{-p/d}}{k^{-p/d}k^{-p/d}}=\\sum_{k=1}^N k^{2p/d}\\sum_{l=1}^N l^{-p/d}.\n\\end{align}\nNaively adding up the orders gives the correct result, $\\O(N^{2+\\frac{p}{d}})$. But because $-p/d$ is negative, we cannot add up the powers: Assuming $p/d>1$, the sequence $(\\sum_{l=1}^N l^{-p/d})_{N\\in\\N}$ is convergent with nonzero limit. Hence, $\\sum_{l=1}^N l^{-p/d}$ is of order $N^0$, not of order $N^{1-p/d}$. The right-hand side of \\eqref{eq:example diagram} is therefore $\\O(N^{1+2p/d})$, and as we now know we have a better bound for the left-hand side. The trick is simply to choose the indices so that each vertex contribution is canceled by an edge contribution.\n\\end{rem}\n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}\n}\n=-\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\lambda_{i_3}^{-1}\\sum_{k=1}^N\\frac{f'\\{i_1,i_2,i_3,k\\}f'\\{i_1,i_3,k\\}}{f'\\{i_1,k\\}f'\\{i_3,k\\}}.\n\\end{align*}\nContrary to the situation where $f'[i_1,i_2,i_3]\\neq0$, the factor $|f'\\{i_1,i_2,i_3,k\\}|$ obtained from the 4-vertex is proportional to $|\\lambda_k|^{-1}$ as $k\\to\\infty$.\nThe other factors are proportional to $1$ as usual. We obtain\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}}\n\\sim\\sum_{k=1}^N|\\lambda_k|^{-1}\\sim\\sum_{k=1}^Nk^{-\\frac{1}{d}}<\\O(N).\n\\end{align*}\nThe order depends on $d$ but (since $d\\geq0$) at least it is smaller than $\\O(N)$ in the sense that there exists no $c$ such that the lower bound is $\\geq cN$. For $d<1$ the graph is in fact finite.\n\\end{rem}\n\n\\begin{rem}\nEven though there exist only finitely many eigenmodes $\\lambda_k$ with $f'(\\lambda_k)=0$, these singular modes can boost the order of divergence not only when occurring as external indices. For instance, assuming non-singular external indices, we have the divergences\n\\begin{align*}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\end{tikzpicture}}\n=\\O(N^{2+\\frac{p}{d}})\n\\quad\\text{and}\\quad\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (0,.5) {$0$};\n\\end{tikzpicture}}\n=\\O(N^{1+\\frac{p+1}{d}}).\n\\end{align*}\nThe latter is larger than the former precisely if $d<1$. More generally, at second loop order, the boost of UV-divergence by internal singular indices is not apparent for $d\\geq1$. Indeed, if $d\\geq1$, then the diagrams of Remark \\ref{rem:2-point 2-loop} remain precisely those of maximal order, also in the more general setting of Theorem \\ref{thm:main2}. \nHowever, for any $d\\in\\N$, at loop order $L=d+2$ the maximal diagrams become those where one of the faces is artificially broken by a singular index, such as\n\\begin{align}\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n~~\n=\\O(N^{4+3p/d})\\quad\\text{and}\\quad\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\node at (0,0) {$0$};\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n=\\O(N^{3+3(p+1)/d}).\\label{eq:3-point 4-loop}\n\\end{align}\nThe latter is larger than the former precisely if $d<3$. Graphs similar to the examples above yield divergences increased by breaking faces to singular modes for any $d<L-1$.\n\\end{rem}\n\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}",
    "post_theorem_intro_text_len": 5794,
    "post_theorem_intro_text": "For renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior. \n\n\\paragraph{Main results}\nSuppose that $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\mathbb{R}$ is a sequence of asymptotic behavior $\\lambda_k\\asymp k^{1/d}$ for $d\\in\\R_{>0}$. Suppose that $f$ is smooth, even, and satisfies, for $x\\to\\infty$, $f^{(n)}(x)\\asymp (-1)^nx^{-p-n}$ ($n\\in\\mathbb{N}$) for some~$p>0$. Suppose moreover that the divided differences of $f$ do not vanish on $\\{\\lambda_k\\}_{k=1}^\\infty$. For a graph $G$ with $U$ unbroken faces, $E_{\\textnormal{fi}}$ edges which do not border a broken face, and $V_{\\textnormal{fi}}$ vertices which do not border a broken face, the amplitude of $G$ is bounded from above and below by a constant times $N^{\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\textnormal{fi}}-V_{\\textnormal{fi}}).\n\\end{align}\n\nHowever, when the assumption of vanishing divided differences is not satisfied, the divergences become (for certain graphs \\textit{strictly}) larger. In this case, the amplitude is bounded from above by $N^{\\tilde\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\mathfrak{b}}+\\frac{p}{d}(E^{\\mathfrak{b}}_{\\textnormal{fi}}-V^{\\mathfrak{b}}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\mathfrak{b}}_{10}-V^{\\mathfrak{b}}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\mathfrak{b}$ of unbroken faces, $U^{\\mathfrak{b}},E_\\textnormal{fi}^b,V_\\textnormal{fi}^b$ are as before when designating the elements of $\\mathfrak{b}$ as broken, and $V^{\\mathfrak{b}}_{10}/E^{\\mathfrak{b}}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\mathfrak{b}$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $E_\\textnormal{fi}-V_\\textnormal{fi}\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $E_\\textnormal{fi}-V_\\textnormal{fi}$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\mathbb{N}$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.",
    "sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”",
    "expanded_sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”,",
    "expanded_theorem": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}",
    "theorem_type": [
      "Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Consider the spectral action matrix model determined by a self-adjoint diagonal operator $D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ and a sufficiently regular function $f:\\mathbb R\\to\\mathbb R$. For ribbon-graph amplitudes, use the Feynman rules from the spectral action model: a vertex bordered by faces with labels $i_1,\\dots,i_n$ contributes $f'[\\lambda_{i_1},\\dots,\\lambda_{i_n}]$, an internal edge bordered by face labels $a,b$ contributes $1/f'[\\lambda_a,\\lambda_b]$, and each unbroken internal face is summed over its label from $1$ to $N$. For the two one-vertex 2-point ribbon graphs with external face labels $j$ and $i$—the first having one internal loop face, and the second having two nested internal loop faces— which statement gives their amplitudes for all choices of the external labels?",
      "correct_choice": {
        "label": "A",
        "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
      },
      "choices": [
        {
          "label": "B",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        },
        {
          "label": "C",
          "text": "Their amplitudes are obtained by summing over the internal face labels and taking the product of the vertex factor with the reciprocal edge factors. In particular, the first graph is a single sum over one internal face label and the second graph is a double sum over two internal face labels, with denominators \\(f'[\\lambda_j,\\lambda_k]\\) and \\(f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]\\), respectively."
        },
        {
          "label": "D",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        },
        {
          "label": "E",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_j]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "cyclic face-label repetition at the vertex",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "explicit vertex numerators were dropped while retaining the summation/edge-factor structure",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "edge denominator depends on adjacent face labels, not the external pair",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "ordered incidence data of face labels around the vertex",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the general Feynman-style rules, but the student still has to apply them to the two specific ribbon graphs and track the face labels correctly."
      },
      "TAS": {
        "score": 1,
        "justification": "This is not a pure theorem restatement, but it is very close to a direct instantiation of the stated rules. The task is essentially to translate the graph data into the prescribed amplitude formula."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to determine the cyclic order in the divided differences and the correct edge denominators, especially for the nested-loop graph. However, the problem is largely procedural rather than strongly generative."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors reflect plausible mistakes: using the wrong adjacent labels in edge factors, omitting repeated labels in the vertex term, or inserting incorrect denominator powers. But choice C is a weaker true statement, which reduces answer exclusivity and weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably solid application-style MCQ with no direct answer leakage and some mathematical checking required, but it is fairly close to rule-following and is weakened by the presence of a partially correct distractor."
    }
  },
  {
    "id": "2512.14581v1",
    "paper_link": "http://arxiv.org/abs/2512.14581v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "ex",
      "content": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}",
      "start_pos": 75967,
      "end_pos": 77444,
      "label": null
    },
    "ref_dict": {
      "sct:main2": "\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}\n\n\\section{Power counting when allowing eigenvalues with vanishing derivative}\\label{sct:main2}\nIn our first main theorem, Theorem \\ref{thm:main}, we assumed that $f'[\\lambda_{k_1},\\ldots,\\lambda_{k_n}]\\neq0$. With $\\{\\lambda_k\\}_{k\\geq1}$ in general position, this assumption is typically satisfied. However, there will be at least one~${x\\in\\R}$ such that $f'(x)=0$, which means that the assumption of nonvanishing divided differences is sensitive to infinitesimal changes in $\\{\\lambda_k\\}_{k\\geq1}$. In fact, we shall show that the actual order of divergence of many graphs is sensitive to the question of whether or not $f'$ vanishes on $\\{\\lambda_k\\}_{k\\geq1}$.\n\nWe will refer to faces with an index $i_0$ such that $f'(\\lambda_{i_0})=0$ as \\emph{0-faces} or \\emph{singular faces}, and similarly for indices and eigenvalues.\nAs in Section~\\ref{sect: lower}, given a subset $\\bfr\\subseteq\\mathcal U$ of the set of unbroken faces of a Feynman ribbon graph~$G$, we let~$G_{\\bfr}$ be the graph obtained from $G$ by artificially declaring the faces in $\\mathfrak{b}$ to be broken. Elements of $\\mathfrak{b}$ are called 0-faces.\\begin{thm}\\label{thm:main2}\nLet $f\\in C^{\\infty}(\\mathbb{R})_\\R$ be an even function with $f'$ of precise order $-p-1$ for some $p \\in\\mathbb{R}_{\\ge0}$.\nLet~$\\{\\lambda_k\\}_{k=1}^\\infty$ be a sequence of real numbers with constants $K,c_1,c_2>0$ such that $c_1k^{1/d}\\leq |\\lambda_k|\\leq c_2k^{1/d}$ for all $k\\geq K$. Assume that for all $k,l\\in\\N_{\\geq1}$ we have $f'[\\lambda_{k},\\lambda_{l}]\\neq0$. Then, for any Feynman ribbon graph $G=(G^0,n,G^1)$ whose vertices have valence $\\geq3$, for all external indices $i_1,\\ldots,i_n\\in\\N_{\\geq1}$, there exist $M,c_4>0$ such that for all~$N\\geq M$,\n$$|\\Ampl_{N,i_1,\\ldots,i_n}(G)|\\leq c_4 N^{\\tilde\\omega(G)},$$\n$$\\tilde \\omega(G):=\\max_{\\bfr\\subseteq\\mathcal U}\\omega_{\\bfr}(G_{\\bfr}):=\\max_{\\bfr\\subseteq\\mathcal U}(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),$$\nwhere $U^{\\bfr}$ is the number of unbroken faces of $G_{\\bfr}$, $E^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal edges of $G_{\\bfr}$ (propagators bordered on both sides by unbroken faces) and $V^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal vertices of $G_{\\bfr}$ (vertices bordered on all sides by unbroken faces). Respectively, $E^{\\bfr}_{10}$ and $V^{\\bfr}_{10}$ are the number of edges and vertices of~$G_{\\bfr}$ that border exactly one 0-face and for the rest unbroken faces.\n\\end{thm}\n\\begin{proof}\nUsing the precise order of $f'$, there exists $R>0$ such that, for all $x\\in\\mathbb{R}\\setminus[-R,R]$ and all~$0\\le k\\le \\max_{v\\in G^{0}}\\;\\deg(v)$, equation \\eqref{eqn: precise_order} is satisfied and the conclusion of Lemma \\ref{lem:zero-index bounds} holds. Let~$i_R$ be the lowest index such that $|\\lambda_i| > R$ for $i\\ge i_R$. For brevity we assume -- without loss of generality -- that $f'(\\lambda_0)=0$ and $f'(\\lambda_k)\\neq0$ for~${k\\geq 1}$. As in the proof of Lemma~\\ref{lem: restricted_lower_sufficient}, a sum splitting argument yields\n\\begin{equation*}\n    \\Ampl_{N,i_1,\\ldots,i_n}(G) = \\sum_{\\bfr\\subseteq \\mathcal{U}}\\sum_{\\gamma:\\bfr\\to\\{0,\\dotsc,i_R-1\\}}\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr}).\n\\end{equation*}\nWe claim that it suffices to show that \\begin{align}\\label{eqn: ampl_Gbgamma}\n|\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr})|&\\lesssim N^{\\omega_{\\bfr,\\gamma}(G_{\\bfr})},\\\\\n\\omega_{\\bfr,\\gamma}(G_{\\bfr})&:=(U^{\\bfr}+\\frac{p}{d}(E_{\\textnormal{fi}}^b-V_{\\textnormal{fi}}^b)+\\frac{p+1}{d}(E_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma})),\\nonumber\n\\end{align}\nas the following graph-theoretical argument shows that $\\omega_{\\bfr,\\gamma}(G_{\\bfr})\\le \\omega_{\\bfr}(G_{\\bfr})$ for all such $\\gamma$. In particular, if we let $\\gamma_0$ denote the map that sends all artificially broken faces to the zero index, then $\\omega_{\\bfr,\\gamma_0}(G_{\\bfr})= \\omega_{\\bfr}(G_{\\bfr})$, hence this bound is sharp.\n\nMore generally, we have $\\omega_{\\bfr}(G_{\\bfr,\\gamma})\\leq\\omega_{\\bfr}(G_{\\bfr,\\gamma'})$ whenever $\\gamma'$ is obtained from $\\gamma$ by setting \\begin{equation*}\n    \\gamma'(b)=\\begin{cases}\n        0,&b=b', \\\\\n        \\gamma(b),&\\text{else},\n    \\end{cases}\n\\end{equation*}\nfor some $b'$ with $\\gamma(b')\\neq0$. Indeed, as the values $U^{\\bfr}$, $\\Efi^b$, and $\\Vfi^b$ do not depend on $\\gamma$, it suffices to show \\begin{equation*}\nE_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma'}.\n\\end{equation*}\nTowards this, first note that $E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma}$ equals the number of edges that border the face $b'$ on the one side and border an unbroken face (i.e.\\ an element of $\\mathcal{U}\\setminus\\bfr$) on the other. Moreover, $V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}$ equals the number of vertices that border $b'$ and for the rest only unbroken faces. For each distinct vertex of the latter kind there is at least one distinct edge of the former kind, implying that $$ V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma},$$which concludes the argument. \n\nIt remains to prove~\\eqref{eqn: ampl_Gbgamma}. The strategy is similar to the proof of Theorem \\ref{thm:upper}, but with different bounds for the edges and vertices, in which we distinguish three different types of indices and their corresponding eigenvalues.\nWe distinguish between (i) eigenvalues $\\lambda_0$ such that $f'(\\lambda_0)=0$, (ii) eigenvalues $\\lambda_i$ for which $f'(\\lambda_i)\\neq0$ and the index $i<R$ is fixed, and (iii) eigenvalues $\\lambda_k$ for which $f'(\\lambda_k)\\neq0$ and the index $k\\geq R$ is running. Analogously to the proof of Theorem \\ref{thm:upper}, noting Remark \\ref{rem:apparent shortcut} and its resolution, the above estimate follows from the bounds on edges and vertices that are summarized and proved in Lemma \\ref{lem:final bounds} below, where we stress that all constants that arise are independent of the running indices.\n\\end{proof}\n\n\\begin{lem}\\label{lem:final bounds}\nLet $n\\geq3$. Let $f,\\{\\lambda_k\\}_{k=1}^\\infty$ be as in Theorem \\ref{thm:main2}. Let $\\lambda_0\\in\\R$ be such that $f'(\\lambda_0)=0$. There exist $c_1,c_2>0$ such that we have\n\\begin{multicols}{2}\n\\begin{enumerate}\n\\item $|f'\\{\\lambda_{k_1},\\lambda_{k_2}\\}|>c_1|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_{i},\\lambda_{k}\\}|>c_1$,\n\\item $|f'\\{\\lambda_{i_1},\\lambda_{i_2}\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_k\\}|> c_1|\\lambda_k|^{-p-1}$,\n\\item $|f'\\{\\lambda_0,\\lambda_i\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_0\\}|>c_1$,\n\\item $|f'\\{\\lambda_{k_1},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_0,\\lambda_{k_2},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_2}|^{-p-1}$,\n\\item $|f'\\{\\lambda_{j_1},\\ldots,\\lambda_{j_n}\\}|\\leq c_2$,\nll}] \nll}] \nll}]\n\\end{enumerate}\n\\end{multicols}\n\\noindent\nfor all $i,i_1,i_2\\leq i_R$, $k,k_1,\\ldots,k_n\\geq i_R$ such that $f'(\\lambda_j)\\neq0$ for $j\\in\\{i,i_1,i_2,k,k_1,\\ldots,k_n\\}$ and such that $|\\lambda_{k_1}|\\leq\\cdots\\leq|\\lambda_{k_n}|$, and for all $j_1,\\ldots,j_n\\in\\N$.\n\\end{lem}\n\n\\begin{proof}\nThe estimates 1.\\ and 7.\\ follow from Theorem~\\ref{thm:weighted_divdiff_bound}, 4.\\ and 8.\\ follow from Lemma~\\ref{lem:zero-index bounds}, and 9.\\ was shown in Corollary~\\ref{cor:wdivdifs are bounded}. Estimates 3., 5., and 6.\\ follow from the fact that there is a finite amount of indices $i<i_R$, and for each $i_1,i_2<i_R$ the (weighted) divided difference is nonzero. {Estimate 2.\\ follows from Theorem~\\ref{thm:no_vanishing_divdifs}.}\n\\end{proof}\nWe conclude this section with some remarks and examples illustrating the influence of 0-faces on the behaviour of amplitudes. \n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}",
      "sct:Definitions": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\bfr$ of unbroken faces, $U^{\\bfr},\\Efi^b,\\Vfi^b$ are as before when designating the elements of $\\bfr$ as broken, and $V^{\\bfr}_{10}/E^{\\bfr}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\bfr$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $\\Efi-\\Vfi\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $\\Efi-\\Vfi$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\R^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\N$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.\n\n\\section{Definitions}\n\\label{sct:Definitions}\n\nThroughout we fix a positive real number $d\\in\\R_{>0}$, and a sequence $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\R$ with the property that there exist numbers $K\\in\\N$, $c_1,c_2\\in\\R_{>0}$ such that\n\\begin{align}\\label{eq:eigenvalues asymptotics assumption}\n    c_1 k^{1/d}\\leq|\\lambda_k|\\leq c_2 k^{1/d}\n\\end{align}",
      "eq:power counting formula": "\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\f}-V_{\\f}).\n\\end{align}",
      "rem:2-point 2-loop": "\\begin{rem}\\label{rem:2-point 2-loop}\nFrom Theorem \\ref{thm:main} we may derive, at each loop order, the set of relevant diagrams. For example, the set of 2-point 2-loop diagrams (with vertices of valence $\\geq3$) with maximal order of divergence is\n\\begin{align*}\n~&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=45,in=90] (0,0);\n\\draw (-.8,0) to[out=-45,in=-90] (0,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (-0.3,0);\n\\draw (-.3,0) to[out=45,in=90] (.5,0);\n\\draw (-.3,0) to[out=-45,in=-90] (.5,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{-.3,0};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to[out=-160,in=180] (0,-.1);\n\\draw (0,.5) to[out=-20,in=0] (0,-.1);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to (0,.05);\n\\draw (0,.05) arc (90:450:.2cm);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\vertex{0,.05};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=0,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to[out=180,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (-.4,.4) arc (180:360:.4cm);\n\\vertex{-.4,.4};\n\\vertex{.4,.4};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\quad\"1\\leftrightarrow2\"\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (0,.5) to (0,-.5);\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}.\n\\end{align*}\nEach has an order of divergence of $2+\\frac{p}{d}$ (and no less).\nOne notices these diagrams are automatically planar. They are moreover connected and, though not necessarily 1PI, satisfy a similar connectivity property. Namely, their dual graphs stay connected after removing the vertices that correspond to the broken faces of the original graph. For instance, the diagram\n$$\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}",
      "rem:UV/IR": "\\begin{rem}\\label{rem:UV/IR}\nSmooth even functions $f$ satisfy $f'(0)=0$, and if $\\{\\lambda_k\\}$ is the spectrum of a typical Dirac operator, these eigenvalues correspond to modes of zero momentum. The fact that UV-divergences can become higher as external momenta vanish is reminiscent of noncommutative quantum field theory, cf.\\ \\cite{MRS2000}. For instance, in the naive version of noncommutative $\\phi^4_4$, such behavior was shown to lead to UV/IR mixing and, consequently, nonrenormalizability \\cite{CR2000,CR2001,GW2005a}. This prompted the Grosse--Wulkenhaar model which solved the UV/IR problem \\cite{GW2005b} and proved incredibly successful \\cite{DGMR2007,GW2014}.\n\nIn light of this, it seems instructive to determine whether UV/IR-mixing is present in the spectral action matrix model and its relatives. This is a difficult question to answer,\nrequiring a careful renormalization analysis and the passage to continuous spectrum.\n\\end{rem}",
      "rem:conjectures": "\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}",
      "eq:power counting formula 2": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}",
      "eq:path integral": "\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 5228,
    "pre_theorem_intro_text": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0,0);\n\t\\draw (-0.5,0.5) to (0,0);\n\t\\draw (0.5,0.5) to (0,0);\n\t\\draw (0.5,-0.5) to (0,0);\n\t\\node at (-0.5,0) {$i$};\n\t\\node at (0,0.5) {$j$};\n\t\\node at (0.5,0) {$k$};\n\t\\node at (0,-0.5) {$l$};\n\tlldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0.5,0.5);\n\t\\node at (-0.25,0.25) {$i$};\n\t\\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}",
    "context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}",
    "full_context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}\n\nFor renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.\n\n\\begin{rem}\\label{rem:apparent shortcut}\nWe explain here a problem with an apparent shortcut to the above proof. Indeed, given positive~${n_1,\\ldots,n_U}$ it is not hard to show (using $\\lambda_j\\sim j^{1/d}$) that\n\\begin{align}\\label{eq:sums like integrals}\n \\sum_{j_1=1}^N\\cdots\\sum_{j_U=1}^N \\lambda_{j_1}^{n_1}\\cdots\\lambda_{j_U}^{n_U}=\\O(N^{U+(n_1+\\ldots+n_U)/d}),\n\\end{align} \nexactly as in the case with integrals instead of sums. If one skips the graph-theoretical Lemma \\ref{lem:injection fi vertices to fi edges} and applies the estimates of Theorem \\ref{thm:weighted_divdiff_bound} for arbitrary bordering indices of the fully internal vertices and edges, one can estimate the amplitude by the sum on the left-hand side of \\eqref{eq:sums like integrals} where indeed the powers automatically add up to $n_1+\\ldots+n_U=p(\\Efi-\\Vfi)$ as required! However, the $n_1,\\ldots,n_U$ may not be positive, which invalidates \\eqref{eq:sums like integrals}. The following example shows that such an estimate of the amplitude by \\eqref{eq:sums like integrals} really is too coarse in general. We surely have\n\\begin{align}\\label{eq:example diagram}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n \\draw (0.45,-.7) to (1,0);\n \\draw (1.55,-.7) to (1,0);\n \\draw (1,0.054) arc (-90:270:0.3cm);\n \\draw (1,-0.035) arc (-90:270:.7cm);\n \\node at (1,-.55) {$i_1$};\n \\node at (0,1) {$i_2$};\n \\vertex{1,0};\n\\end{tikzpicture}\n}\n=\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\sum_{k=1}^N\\sum_{l=1}^N\\frac{f'\\{i_1,k,l,k,i_1,i_2\\}}{f'\\{i_1,k\\}f'\\{k,l\\}}\\lesssim\n\\sum_{k=1}^N\\sum_{l=1}^N\\frac{l^{-p/d}}{k^{-p/d}k^{-p/d}}=\\sum_{k=1}^N k^{2p/d}\\sum_{l=1}^N l^{-p/d}.\n\\end{align}\nNaively adding up the orders gives the correct result, $\\O(N^{2+\\frac{p}{d}})$. But because $-p/d$ is negative, we cannot add up the powers: Assuming $p/d>1$, the sequence $(\\sum_{l=1}^N l^{-p/d})_{N\\in\\N}$ is convergent with nonzero limit. Hence, $\\sum_{l=1}^N l^{-p/d}$ is of order $N^0$, not of order $N^{1-p/d}$. The right-hand side of \\eqref{eq:example diagram} is therefore $\\O(N^{1+2p/d})$, and as we now know we have a better bound for the left-hand side. The trick is simply to choose the indices so that each vertex contribution is canceled by an edge contribution.\n\\end{rem}\n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}\n}\n=-\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\lambda_{i_3}^{-1}\\sum_{k=1}^N\\frac{f'\\{i_1,i_2,i_3,k\\}f'\\{i_1,i_3,k\\}}{f'\\{i_1,k\\}f'\\{i_3,k\\}}.\n\\end{align*}\nContrary to the situation where $f'[i_1,i_2,i_3]\\neq0$, the factor $|f'\\{i_1,i_2,i_3,k\\}|$ obtained from the 4-vertex is proportional to $|\\lambda_k|^{-1}$ as $k\\to\\infty$.\nThe other factors are proportional to $1$ as usual. We obtain\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}}\n\\sim\\sum_{k=1}^N|\\lambda_k|^{-1}\\sim\\sum_{k=1}^Nk^{-\\frac{1}{d}}<\\O(N).\n\\end{align*}\nThe order depends on $d$ but (since $d\\geq0$) at least it is smaller than $\\O(N)$ in the sense that there exists no $c$ such that the lower bound is $\\geq cN$. For $d<1$ the graph is in fact finite.\n\\end{rem}\n\n\\begin{rem}\nEven though there exist only finitely many eigenmodes $\\lambda_k$ with $f'(\\lambda_k)=0$, these singular modes can boost the order of divergence not only when occurring as external indices. For instance, assuming non-singular external indices, we have the divergences\n\\begin{align*}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\end{tikzpicture}}\n=\\O(N^{2+\\frac{p}{d}})\n\\quad\\text{and}\\quad\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (0,.5) {$0$};\n\\end{tikzpicture}}\n=\\O(N^{1+\\frac{p+1}{d}}).\n\\end{align*}\nThe latter is larger than the former precisely if $d<1$. More generally, at second loop order, the boost of UV-divergence by internal singular indices is not apparent for $d\\geq1$. Indeed, if $d\\geq1$, then the diagrams of Remark \\ref{rem:2-point 2-loop} remain precisely those of maximal order, also in the more general setting of Theorem \\ref{thm:main2}. \nHowever, for any $d\\in\\N$, at loop order $L=d+2$ the maximal diagrams become those where one of the faces is artificially broken by a singular index, such as\n\\begin{align}\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n~~\n=\\O(N^{4+3p/d})\\quad\\text{and}\\quad\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\node at (0,0) {$0$};\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n=\\O(N^{3+3(p+1)/d}).\\label{eq:3-point 4-loop}\n\\end{align}\nThe latter is larger than the former precisely if $d<3$. Graphs similar to the examples above yield divergences increased by breaking faces to singular modes for any $d<L-1$.\n\\end{rem}\n\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}",
    "post_theorem_intro_text_len": 5794,
    "post_theorem_intro_text": "For renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior. \n\n\\paragraph{Main results}\nSuppose that $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\mathbb{R}$ is a sequence of asymptotic behavior $\\lambda_k\\asymp k^{1/d}$ for $d\\in\\R_{>0}$. Suppose that $f$ is smooth, even, and satisfies, for $x\\to\\infty$, $f^{(n)}(x)\\asymp (-1)^nx^{-p-n}$ ($n\\in\\mathbb{N}$) for some~$p>0$. Suppose moreover that the divided differences of $f$ do not vanish on $\\{\\lambda_k\\}_{k=1}^\\infty$. For a graph $G$ with $U$ unbroken faces, $E_{\\textnormal{fi}}$ edges which do not border a broken face, and $V_{\\textnormal{fi}}$ vertices which do not border a broken face, the amplitude of $G$ is bounded from above and below by a constant times $N^{\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\textnormal{fi}}-V_{\\textnormal{fi}}).\n\\end{align}\n\nHowever, when the assumption of vanishing divided differences is not satisfied, the divergences become (for certain graphs \\textit{strictly}) larger. In this case, the amplitude is bounded from above by $N^{\\tilde\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\mathfrak{b}}+\\frac{p}{d}(E^{\\mathfrak{b}}_{\\textnormal{fi}}-V^{\\mathfrak{b}}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\mathfrak{b}}_{10}-V^{\\mathfrak{b}}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\mathfrak{b}$ of unbroken faces, $U^{\\mathfrak{b}},E_\\textnormal{fi}^b,V_\\textnormal{fi}^b$ are as before when designating the elements of $\\mathfrak{b}$ as broken, and $V^{\\mathfrak{b}}_{10}/E^{\\mathfrak{b}}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\mathfrak{b}$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $E_\\textnormal{fi}-V_\\textnormal{fi}\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $E_\\textnormal{fi}-V_\\textnormal{fi}$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\mathbb{N}$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.",
    "sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”",
    "expanded_sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”,",
    "expanded_theorem": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}",
    "theorem_type": [
      "Universal",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Consider the spectral action matrix model determined by a self-adjoint diagonal operator $D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ and a sufficiently regular function $f:\\mathbb R\\to\\mathbb R$. For ribbon-graph amplitudes, use the Feynman rules from the spectral action model: a vertex bordered by faces with labels $i_1,\\dots,i_n$ contributes $f'[\\lambda_{i_1},\\dots,\\lambda_{i_n}]$, an internal edge bordered by face labels $a,b$ contributes $1/f'[\\lambda_a,\\lambda_b]$, and each unbroken internal face is summed over its label from $1$ to $N$. For the two one-vertex 2-point ribbon graphs with external face labels $j$ and $i$—the first having one internal loop face, and the second having two nested internal loop faces— which statement gives their amplitudes for all choices of the external labels?",
      "correct_choice": {
        "label": "A",
        "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
      },
      "choices": [
        {
          "label": "B",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        },
        {
          "label": "C",
          "text": "Their amplitudes are obtained by summing over the internal face labels and taking the product of the vertex factor with the reciprocal edge factors. In particular, the first graph is a single sum over one internal face label and the second graph is a double sum over two internal face labels, with denominators \\(f'[\\lambda_j,\\lambda_k]\\) and \\(f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]\\), respectively."
        },
        {
          "label": "D",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        },
        {
          "label": "E",
          "text": "Their amplitudes are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_j]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the graph with one internal loop face, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested internal loop faces."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "cyclic face-label repetition at the vertex",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "explicit vertex numerators were dropped while retaining the summation/edge-factor structure",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "edge denominator depends on adjacent face labels, not the external pair",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "ordered incidence data of face labels around the vertex",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the general Feynman rules but does not reveal the final amplitudes for the two specific graphs. The correct choice must still be derived by applying those rules to the stated ribbon-graph configurations."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a mere restatement of the rules/theorem. The test-taker must instantiate the vertex and edge rules on concrete graph topologies and track the face-label sequence around the vertex."
      },
      "GPS": {
        "score": 1,
        "justification": "Some genuine reasoning is required: one must determine the correct summation indices, adjacent-face edge denominators, and cyclic label order at the vertex. However, once the rules are known, the task is largely procedural rather than deeply generative, and the missing diagrams may make it more about reconstruction than conceptual choice."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors target plausible errors: missing repeated face labels at the vertex (B), wrong edge-label dependence (D), and altered cyclic ordering (E). But choice C is a weaker true statement rather than a clearly false alternative, which undermines single-best-answer quality and makes the distractor set mixed rather than strong."
      },
      "total_score": 6,
      "overall_assessment": "A reasonably solid application-based MCQ with no answer leakage and little tautology, but its distractor set is weakened by an arguably true-but-incomplete option and by reliance on verbally described graph structure rather than explicit diagrams."
    }
  },
  {
    "id": "2512.14627v1",
    "paper_link": "http://arxiv.org/abs/2512.14627v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.",
      "start_pos": 12114,
      "end_pos": 12850,
      "label": "main1"
    },
    "ref_dict": {
      "main1-eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}",
      "change-var": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}",
      "main2-eq": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}",
      "main1": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}",
      "scalar.eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\R^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\pd\\Om$ has a neighborhood $U_x$ such that $\\pd \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\div  G, \\quad u|_{\\pd \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\R^3)$, $\\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}",
      "drifteq": "\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G",
      "1.2": "\\label{1.2}\n\\De u = \\div (\\nb u),\\quad\nb\\cdot\\nabla u = \\div (b\\otimes u), \\quad \\int_\\Om  \\div \\GG \\cdot \\zeta = -\n\\int_\\Om  \\GG:\\nb \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we",
      "energy_2": "\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}",
      "pressure_bound_est": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}",
      "soln-pair": "\\label{soln-pair}\n\\int_{\\Om} (\\nb u - b \\otimes u): \\nb \\zeta - \\pi \\div \\zeta  = -\\int_\\Om \\GG : \\nb \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts",
      "th1.2": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3896,
    "pre_theorem_intro_text": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$  in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and  force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$  we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om  \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om  \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\  \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies  \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta  = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta   = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$.  The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for   higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.",
    "context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.",
    "full_context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\R^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nb u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\R^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\nUsing the same proof scheme we can also prove the following theorem for the scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$. Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\n\\section{Existence of weak solutions}\\label{sec3}\nIn this section we will prove existence of weak solutions to the perturbed Stokes system \\eqref{drifteq} for $b\\in L^2$ or $L^{2,\\infty}$, $\\div b=0$,\n\\EQ{ \\label{eq3.1}\n-\\Delta u+ b\\cdot\\nabla u + \\nb \\pi =f,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nin $\\Om \\subset \\R^2$. When $b\\in L^2$, we will further show uniqueness.\nRecall that a weak solution $u$ belongs to $ W^{1,2}_{0,\\sigma}(\\Omega)$ and satisfies a weak form of \\eqref{drifteq} similar to \\eqref{weak-soln}.\n\\begin{theorem}[Weak solutions for weak $L^2$ drift]\\label{energy}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Moreover, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\n\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\nApproximating $u$ in $W^{1,2}_{0,\\si}(\\Om)$ by smooth $u_j$\nand using $\\div b=0$ conclude that\n\\EQ{\\label{eq3.4}\n\\int_\\Om (b\\cdot \\nb) u\\cdot u = \\lim_{j \\to \\infty} \\int_\\Om (b\\cdot \\nb) u_j\\cdot u_j = \\int_\\Om b\\cdot \\nb(\\frac12|u_j|^2) = 0.\n}\n Thus we get the coercivity of this bilinear form, indeed,\n\\EQ{\n\\mathcal{B}(u,u)=\\|\\nabla u\\|_{L^2(\\Om)}^2, \\quad \\forall u\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nTherefore, by Lax-Milgram theorem, for any $f \\in W^{-1,2}(\\Omega)^n\\subset (W_{0,\\sigma}^{1,2}(\\Om)^n)^*$, there exists a unique $u\\in W_{0,\\sigma}^{1,2}(\\Om)$ such that\n\\EQ{\n\\mathcal{B}(u,v)=\\langle f,v\\rangle,\\quad \\forall v\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nThis is the weak form of \\eqref{drifteq} and hence\n$u$ is the unique weak solution. By taking $v=u$, we get $\\norm{u}_{W^{1,2}_0(\\Om)} \\le C\\norm{f}_{W^{-1,2}(\\Om)}$.\nLastly, we apply Lemma \\ref{pressure} with $q=2$ (and using that $\\Om$ is bounded and Lipschitz) to get an associated pressure $\\pi\\in L^2_0(\\Om)$ that satisfies the bound.\n\\end{proof}\n\n\\begin{theorem}\\label{theoremL2wolf}\nAssume $p>1$ and $V\\subset \\R^n$, $n\\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.\nThen the problem \\eqref{Stokes} has a unique weak solution pair $(v,\\pi)$ with $v \\in W^{1,p}_{0,\\si}(V)$ and $\\pi \\in L^p_0(V)$ for given $f=\\div \\FF$, $ \\FF \\in L^p(\\Om)^{n\\times n}$, and the local pressure projection in $V$,\n\\[\n\\mathcal{W}_{p,V}(\\div \\FF)=\\nabla\\pi,\n\\]\nsatisfies the following inequality\n\\EQ{\\label{pWolfbound}\n\\|\\pi\\|_{L^p(V)}\\le c_1\\|\\FF\\|_{L^p(V)},\n}\nwhere $c_1=c_1(n,V,p)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{Capriori}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L \\le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nAssume further $b \\in L^{2+\\delta}$ for some $\\delta>0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $2<p<p_0$,\nany weak solution $u\\in W^{1,2}_{0,\\si}(\\Omega)$ of \\eqref{drifteq} with force $\\GG\\in L^p(\\Om)^{2\\times 2}$\nis in $ W^{1,p}(\\Omega)$ and\n\\EQ{\\label{eq5.1}\n\\|u\\|_{W^{1,p}(\\Omega)}\\le c\\|u\\|_{W^{1,2}(\\Omega)}+ c\\|\\GG\\|_{L^p(\\Omega)}\n}\nfor some constant $c(\\Omega,\\|b\\|_{L^{2,\\infty}})>0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 7623,
    "post_theorem_intro_text": "\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\mathbb{R}^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\partial\\Om$ has a neighborhood $U_x$ such that $\\partial \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\mathop{\\rm div}\\nolimits  G, \\quad u|_{\\partial \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\mathbb{R}^3)$, $\\mathop{\\rm div}\\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\mathbb{R}^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\nThis theorem improves $\\alpha=0$ case of \\cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \\cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \\in (2,q)$ depending on $q$, $\\left  \\| b  \\right \\|_{L^{2,\\infty}}$ and $\\Om$ such that for any $G \\in L^q(\\Om)$, there is a unique $p$-weak solution $u$ of \\eqref{scalar.eq} with force $G$. Moreover,\n\\[\n\\left  \\| \\nabla u  \\right \\|_{L^p(\\Om)} \\le C \\left  \\| G  \\right \\|_{L^q(\\Om)}.\n\\]\nTheorem \\ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \\in (p_0',2]$ by duality, which yields existence and uniqueness\n for $q \\in (p_0',2]$.\n\nWe skip the proof of Theoem \\ref{th1.2} since it is similar to Theoem \\ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\\partial \\Om$ is allowed to be arbitrarily large.\n\n\\medskip\n\nWe now explain the new difficulties and ideas.\nTheorem \\ref{main1}\nextends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the 2D case, and\nwe will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.\n\nThe first new difficulty is that we cannot use $\\zeta = u \\eta^4$ for some cut-off function $\\eta$ as the test function in the weak form \\eqref{soln-pair}, because $b \\otimes u$ is no longer in $L^2$ as in the 3D case, and\n the term\n\\EQ{\\label{drift-integral}\n\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)\n}\nis not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\\infty}(\\Omega)$ to $L^{2,\\infty}(\\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.\n\nThe second new difficulty comes from the lower integrability of the pressure.\nWe will\nbound the pressure using Wolf's local pressure projection with a uniform constant as in \\cite{CT1}.\nDenote $\\Omega_R=\\Omega\\cap B_R(x_0)$ for some $x_0\\in \\overline \\Om$. In \\cite{CT1}, it suffices to bound $\\pi -(\\pi)_{\\Omega_R}$ in $L^2(\\Omega_R)$. However,  for dimension 2  we need to bound $\\pi -(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)\nWe have to choose $q<2$ because\n\\[\n\\pi {\\ \\lesssim \\ } \\nabla u + b \\otimes u -\\mathbb{G}\n\\]\nand $b \\otimes u$ is at best in $L^{2-}$ for $u \\in W^{1,2}_\\mathrm{loc}$ even if we assume $b \\in L^2$. This would not be an issue if $b \\in L^{r}$, $r>2$, and we can only assume $u \\in W^{1,2}_\\mathrm{loc}$ which is the assumption of the reversed H\\\"older inequality.\n\nTo obtain a uniform constant of Wolf's local pressure projection in $L^q(\\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.\n\nThe third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\\in L^{2,\\infty}(\\Om)$, $\\mathop{\\rm div}\\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\\|b_n\\|_{L^{2,\\infty}(\\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \\cite{Verchota_1,Verchota_2} and later Kwon \\cite{Kwon}, Chernobai-Shilkin \\cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \\ref{sec6} for details.\n\nThe rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \\ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.",
    "sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to proving Theorem~\\ref{main1} in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\\ref{main1}.”",
    "expanded_sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to establishing the main theorem in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”",
    "expanded_theorem": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G such that\n\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming the preceding estimate.",
    "theorem_type": [
      "Existential–Universal",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\\pi ∈ L^q(Ω) : \\int_Ω \\pi\\,dx = 0}. A weak solution pair (u,\\pi) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nis one with u ∈ W^{1,q}_{0,σ}(Ω), \\pi ∈ L^q_0(Ω), and\n\\int_Ω (∇u - b\\otimes u):∇\\zeta - \\pi\\,div\\,\\zeta = -\\int_Ω \\mathbb{G}:∇\\zeta \\quad \\text{for all } \\zeta \\in C_c^\\infty(Ω)^2.\nIf p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?",
      "correct_choice": {
        "label": "A",
        "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} ∈ L^q(Ω)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) ∈ W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(Ω)} + \\|\\pi\\|_{L^q(Ω)} \\le C(Ω,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(Ω)}.\nEquivalently, for each such q, the solution pair is unique in W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) even without assuming the estimate a priori."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with 1 < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        },
        {
          "label": "C",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nsuch that\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        },
        {
          "label": "D",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)},\nwith a constant independent of b."
        },
        {
          "label": "E",
          "text": "If \\Omega \\subset \\mathbb{R}^2 is a bounded Lipschitz domain and b \\in L^{2,\\infty}(\\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "lower q-threshold p0'<q",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniqueness clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of constants on \\|b\\|_{L^{2,\\infty}}",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "small Lipschitz constant requirement for Wolf projection/boundary stretching",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the PDE setting and definition of weak solution, but the key conclusion—exact q-range, uniqueness strength, and parameter dependence—must be inferred from the choices."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement. Still, the alternatives vary in subtle ways (endpoint inclusion, uniqueness, dependence on b, q-range), so it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to identify the strongest valid statement among nearby variants, especially regarding open vs. closed ranges and dependence of p0. However, the task mainly tests recognition of the exact theorem rather than substantial derivation or generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: they alter lower endpoints, endpoint inclusion, uniqueness, and norm/parameter dependence in ways that match common failure modes when recalling regularity theorems."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans heavily toward precise theorem recall rather than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.14627v1",
    "paper_link": "http://arxiv.org/abs/2512.14627v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.",
      "start_pos": 12114,
      "end_pos": 12850,
      "label": "main1"
    },
    "ref_dict": {
      "main1-eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}",
      "change-var": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}",
      "main2-eq": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}",
      "main1": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}",
      "scalar.eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\R^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\pd\\Om$ has a neighborhood $U_x$ such that $\\pd \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\div  G, \\quad u|_{\\pd \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\R^3)$, $\\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}",
      "drifteq": "\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G",
      "1.2": "\\label{1.2}\n\\De u = \\div (\\nb u),\\quad\nb\\cdot\\nabla u = \\div (b\\otimes u), \\quad \\int_\\Om  \\div \\GG \\cdot \\zeta = -\n\\int_\\Om  \\GG:\\nb \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we",
      "energy_2": "\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}",
      "pressure_bound_est": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}",
      "soln-pair": "\\label{soln-pair}\n\\int_{\\Om} (\\nb u - b \\otimes u): \\nb \\zeta - \\pi \\div \\zeta  = -\\int_\\Om \\GG : \\nb \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts",
      "th1.2": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3896,
    "pre_theorem_intro_text": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$  in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and  force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$  we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om  \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om  \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\  \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies  \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta  = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta   = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$.  The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for   higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.",
    "context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.",
    "full_context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\R^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nb u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\R^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\nUsing the same proof scheme we can also prove the following theorem for the scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$. Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\n\\section{Existence of weak solutions}\\label{sec3}\nIn this section we will prove existence of weak solutions to the perturbed Stokes system \\eqref{drifteq} for $b\\in L^2$ or $L^{2,\\infty}$, $\\div b=0$,\n\\EQ{ \\label{eq3.1}\n-\\Delta u+ b\\cdot\\nabla u + \\nb \\pi =f,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nin $\\Om \\subset \\R^2$. When $b\\in L^2$, we will further show uniqueness.\nRecall that a weak solution $u$ belongs to $ W^{1,2}_{0,\\sigma}(\\Omega)$ and satisfies a weak form of \\eqref{drifteq} similar to \\eqref{weak-soln}.\n\\begin{theorem}[Weak solutions for weak $L^2$ drift]\\label{energy}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Moreover, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\n\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\nApproximating $u$ in $W^{1,2}_{0,\\si}(\\Om)$ by smooth $u_j$\nand using $\\div b=0$ conclude that\n\\EQ{\\label{eq3.4}\n\\int_\\Om (b\\cdot \\nb) u\\cdot u = \\lim_{j \\to \\infty} \\int_\\Om (b\\cdot \\nb) u_j\\cdot u_j = \\int_\\Om b\\cdot \\nb(\\frac12|u_j|^2) = 0.\n}\n Thus we get the coercivity of this bilinear form, indeed,\n\\EQ{\n\\mathcal{B}(u,u)=\\|\\nabla u\\|_{L^2(\\Om)}^2, \\quad \\forall u\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nTherefore, by Lax-Milgram theorem, for any $f \\in W^{-1,2}(\\Omega)^n\\subset (W_{0,\\sigma}^{1,2}(\\Om)^n)^*$, there exists a unique $u\\in W_{0,\\sigma}^{1,2}(\\Om)$ such that\n\\EQ{\n\\mathcal{B}(u,v)=\\langle f,v\\rangle,\\quad \\forall v\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nThis is the weak form of \\eqref{drifteq} and hence\n$u$ is the unique weak solution. By taking $v=u$, we get $\\norm{u}_{W^{1,2}_0(\\Om)} \\le C\\norm{f}_{W^{-1,2}(\\Om)}$.\nLastly, we apply Lemma \\ref{pressure} with $q=2$ (and using that $\\Om$ is bounded and Lipschitz) to get an associated pressure $\\pi\\in L^2_0(\\Om)$ that satisfies the bound.\n\\end{proof}\n\n\\begin{theorem}\\label{theoremL2wolf}\nAssume $p>1$ and $V\\subset \\R^n$, $n\\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.\nThen the problem \\eqref{Stokes} has a unique weak solution pair $(v,\\pi)$ with $v \\in W^{1,p}_{0,\\si}(V)$ and $\\pi \\in L^p_0(V)$ for given $f=\\div \\FF$, $ \\FF \\in L^p(\\Om)^{n\\times n}$, and the local pressure projection in $V$,\n\\[\n\\mathcal{W}_{p,V}(\\div \\FF)=\\nabla\\pi,\n\\]\nsatisfies the following inequality\n\\EQ{\\label{pWolfbound}\n\\|\\pi\\|_{L^p(V)}\\le c_1\\|\\FF\\|_{L^p(V)},\n}\nwhere $c_1=c_1(n,V,p)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{Capriori}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L \\le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nAssume further $b \\in L^{2+\\delta}$ for some $\\delta>0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $2<p<p_0$,\nany weak solution $u\\in W^{1,2}_{0,\\si}(\\Omega)$ of \\eqref{drifteq} with force $\\GG\\in L^p(\\Om)^{2\\times 2}$\nis in $ W^{1,p}(\\Omega)$ and\n\\EQ{\\label{eq5.1}\n\\|u\\|_{W^{1,p}(\\Omega)}\\le c\\|u\\|_{W^{1,2}(\\Omega)}+ c\\|\\GG\\|_{L^p(\\Omega)}\n}\nfor some constant $c(\\Omega,\\|b\\|_{L^{2,\\infty}})>0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 7623,
    "post_theorem_intro_text": "\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\mathbb{R}^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\partial\\Om$ has a neighborhood $U_x$ such that $\\partial \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\mathop{\\rm div}\\nolimits  G, \\quad u|_{\\partial \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\mathbb{R}^3)$, $\\mathop{\\rm div}\\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\mathbb{R}^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\nThis theorem improves $\\alpha=0$ case of \\cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \\cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \\in (2,q)$ depending on $q$, $\\left  \\| b  \\right \\|_{L^{2,\\infty}}$ and $\\Om$ such that for any $G \\in L^q(\\Om)$, there is a unique $p$-weak solution $u$ of \\eqref{scalar.eq} with force $G$. Moreover,\n\\[\n\\left  \\| \\nabla u  \\right \\|_{L^p(\\Om)} \\le C \\left  \\| G  \\right \\|_{L^q(\\Om)}.\n\\]\nTheorem \\ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \\in (p_0',2]$ by duality, which yields existence and uniqueness\n for $q \\in (p_0',2]$.\n\nWe skip the proof of Theoem \\ref{th1.2} since it is similar to Theoem \\ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\\partial \\Om$ is allowed to be arbitrarily large.\n\n\\medskip\n\nWe now explain the new difficulties and ideas.\nTheorem \\ref{main1}\nextends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the 2D case, and\nwe will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.\n\nThe first new difficulty is that we cannot use $\\zeta = u \\eta^4$ for some cut-off function $\\eta$ as the test function in the weak form \\eqref{soln-pair}, because $b \\otimes u$ is no longer in $L^2$ as in the 3D case, and\n the term\n\\EQ{\\label{drift-integral}\n\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)\n}\nis not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\\infty}(\\Omega)$ to $L^{2,\\infty}(\\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.\n\nThe second new difficulty comes from the lower integrability of the pressure.\nWe will\nbound the pressure using Wolf's local pressure projection with a uniform constant as in \\cite{CT1}.\nDenote $\\Omega_R=\\Omega\\cap B_R(x_0)$ for some $x_0\\in \\overline \\Om$. In \\cite{CT1}, it suffices to bound $\\pi -(\\pi)_{\\Omega_R}$ in $L^2(\\Omega_R)$. However,  for dimension 2  we need to bound $\\pi -(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)\nWe have to choose $q<2$ because\n\\[\n\\pi {\\ \\lesssim \\ } \\nabla u + b \\otimes u -\\mathbb{G}\n\\]\nand $b \\otimes u$ is at best in $L^{2-}$ for $u \\in W^{1,2}_\\mathrm{loc}$ even if we assume $b \\in L^2$. This would not be an issue if $b \\in L^{r}$, $r>2$, and we can only assume $u \\in W^{1,2}_\\mathrm{loc}$ which is the assumption of the reversed H\\\"older inequality.\n\nTo obtain a uniform constant of Wolf's local pressure projection in $L^q(\\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.\n\nThe third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\\in L^{2,\\infty}(\\Om)$, $\\mathop{\\rm div}\\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\\|b_n\\|_{L^{2,\\infty}(\\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \\cite{Verchota_1,Verchota_2} and later Kwon \\cite{Kwon}, Chernobai-Shilkin \\cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \\ref{sec6} for details.\n\nThe rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \\ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.",
    "sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to proving Theorem~\\ref{main1} in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\\ref{main1}.”",
    "expanded_sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to establishing the main theorem in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”",
    "expanded_theorem": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G such that\n\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming the preceding estimate.",
    "theorem_type": [
      "Existential–Universal",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\\pi ∈ L^q(Ω) : \\int_Ω \\pi\\,dx = 0}. A weak solution pair (u,\\pi) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nis one with u ∈ W^{1,q}_{0,σ}(Ω), \\pi ∈ L^q_0(Ω), and\n\\int_Ω (∇u - b\\otimes u):∇\\zeta - \\pi\\,div\\,\\zeta = -\\int_Ω \\mathbb{G}:∇\\zeta \\quad \\text{for all } \\zeta \\in C_c^\\infty(Ω)^2.\nIf p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?",
      "correct_choice": {
        "label": "A",
        "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} ∈ L^q(Ω)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) ∈ W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(Ω)} + \\|\\pi\\|_{L^q(Ω)} \\le C(Ω,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(Ω)}.\nEquivalently, for each such q, the solution pair is unique in W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) even without assuming the estimate a priori."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with 1 < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        },
        {
          "label": "C",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nsuch that\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        },
        {
          "label": "D",
          "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)},\nwith a constant independent of b."
        },
        {
          "label": "E",
          "text": "If \\Omega \\subset \\mathbb{R}^2 is a bounded Lipschitz domain and b \\in L^{2,\\infty}(\\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "lower q-threshold p0'<q",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniqueness clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "dependence of constants on \\|b\\|_{L^{2,\\infty}}",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "small Lipschitz constant requirement for Wolf projection/boundary stretching",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the PDE setup and definitions, but the decisive theorem-level conclusion (exact q-range, uniqueness, and estimate dependence) is left to the choices."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially a theorem-identification question: it asks which theorem statement is correct for the given problem. While the options differ in meaningful ways, the task is still close to selecting the precise restatement of a known result rather than deriving a new conclusion from the setup."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants: the sharp range p0' < q < p0, the uniqueness clause, dependence on ||b||_{L^{2,\\infty}}, and the small-Lipschitz assumption. However, the problem mainly tests recall/recognition of the exact theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted. They reflect common failure modes: using the wrong q-range, dropping uniqueness, claiming b-independent constants, or omitting the small Lipschitz requirement. They are distinct and close enough to the correct theorem to be effective."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it leans more toward precise recall than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.14939v1",
    "paper_link": "http://arxiv.org/abs/2512.14939v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.",
      "start_pos": 5302,
      "end_pos": 5622,
      "label": "thm: main"
    },
    "ref_dict": {
      "prop: rank 3 excluded minor": "\\begin{proposition}\\label{prop: rank 3 excluded minor}\n    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors}.\n\\end{proposition}",
      "prob: Kung problem": "\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}",
      "thm: main": "\\begin{theorem} \\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}",
      "thm: exlude a line": "\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}",
      "thm: complex exclude a line": "\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}",
      "prop: rank 4 excluded minor": "\\begin{proposition}\\label{prop: rank 4 excluded minor}\n    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor}.\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 2450,
    "pre_theorem_intro_text": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors. \n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.",
    "context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.",
    "full_context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.\n\nThis result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm: main result with proof}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor} and no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}\n\\begin{proof}\nFirst suppose that $\\ell = 1$.\nIf $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \\ge 2$ and $M$ has a circuit with at least three elements.\nBut every circuit with at least three elements has a $U_{2,3}$-minor \\cite[Example 3.1.9]{Oxley-2011}, a contradiction.\nSo $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.\nSo we may assume that $\\ell \\ge 2$.\nWe will proceed by induction on $r$.\nThe statement is clearly true when $r \\le 2$ so we may assume that $r \\ge 3$.\nIf $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have\n\\begin{align*}\n |M| = 1 + |M/e| \\le 1 + \\ell(r - 2) + 1 = \\ell(r - 1) + 2 - \\ell < \\ell(r - 1) + 1,\n\\end{align*}\nas desired.\nSo we may assume that every element of $M$ is on a long line.\nLet $L$ be a long line of $M$.\nWe will now consider the connected components of $M/L$.\nLet $\\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.\nLet $X \\in \\cX$, and let $N = M|(L \\cup X)$.\nWe will first bound the number of elements of $N \\del L$.\n\n\\begin{claim} \\label{claim: bound the size of each component}\n At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \\del L| \\le \\ell(r(N) - 2)$.\n\\end{claim}\n\\begin{proof}\nSuppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.\nLet $K = \\si(N/L)$.\nBy deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.\nNote that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.\nFor $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\\cl_N(L \\cup L_i)$ of $N$.\nSince $K$ is connected, it follows from \\cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \\del D/C$ on ground set $\\{p_1, p_2, p_3\\}$.\nWe may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.\nSince $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.\nLet $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\\cl_N(L \\cup d) - L$, and then contracting $C$.\nThen $K_1 = \\si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.\nWe will show that $N_1$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or \\ref{fig: rank-4 excluded minor}.\n\n\\begin{claim}\n Some $x \\in L$ is on more than one long line of $M$.\n\\end{claim}\n\\begin{proof}\nLet $e \\in L$ and let $M_1 = M/e$.\nLet $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.\nAlso note that $\\elem(M_1) = |M| - \\ell = \\ell(r(M_1) - 1) + 1$,\nso by induction, $\\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\\ell+1}$.\nThis implies that $P$ is on a long line $L_1$ of $M_1$.\nLet $F$ be the plane $L \\cup L_1$ of $M$.\nNote that $|F| \\ge |L| + |L_1| - 1 \\ge 2(\\ell + 1) - 1 = 2\\ell + 1$.\nIf $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \\ell+2}$-minor, a contradiction.\nSo $F - L$ has rank $2$.\nIf $\\cl(F - L)$ does not intersect $L$, then for every $f \\in F - L$ we have $(M|F)/f \\cong U_{2,\\ell+2}$, a contradiction. \nSo $\\cl(F - L)$ intersects $L$ in some element $x$.\nSince $|F - L| \\ge \\ell \\ge 2$, it follows that $\\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.\n\\end{proof}\n\nLet $r, \\ell$ be integers with $r \\ge 2$ and $\\ell \\ge 3$.\nLet $W(r,\\ell)$ be the matroid with a basis $B = \\{b_1, b_2, \\dots, b_r\\}$ and $\\lfloor \\frac{\\ell - 1}{2} \\rfloor$ elements freely placed (see \\cite[page 270]{Oxley-2011}) in the span of $\\{b_i, b_{i+1}\\}$ for all $i \\in [r]$, reading indices modulo $r$.\nLet $W(r, \\ell)^+$ be obtained from $W(r, \\ell)$ by freely placing one point in $\\cl(\\{b_1, b_2\\})$.\nThen $W(2, \\ell) \\cong U_{2,\\lfloor \\frac{\\ell - 1}{2} \\rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\\mathcal W^r$ when $r \\ge 3$.\nSince the whirl $\\mathcal W^r$ is $3$-connected (see \\cite[Example 8.3.4]{Oxley-2011}) and $W(r,\\ell)$ has $\\mathcal W^r$ as a restriction, it follows from \\cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\\ell)$ and $W(r, \\ell)^+$ are $3$-connected.\nAnd it is straightforward to check that the natural cyclic ordering of $E(W(r, \\ell))$ (first $b_1$, then all elements in $\\cl(\\{b_1, b_2\\}) - b_2$, then $b_2$, and so on) or $W(r,\\ell)^+$ satisfies the condition of Theorem \\ref{thm:Bonin's condition}, so $W(r, \\ell)$ and $W(r, \\ell)^+$ are positroids.\nFinally, one can check that if $r \\ge 4$ and $e \\notin \\{b_1, \\dots, b_r\\}$, then $\\si(W(r, \\ell)/e) \\cong W(r, \\ell)$, and that $\\si(W(r, \\ell)/b_i) \\cong W(r,\\ell) \\del (\\cl(\\{b_{i-1},b_i, b_{i+1}) - \\{b_{i-1}, b_{i+1}\\})$, and from these observations one can prove that $W(r,\\ell)$ is $U_{2,\\ell+2}$-minor-free, and when $\\ell$ is even, $W(r, \\ell)^+$ is $U_{2, \\ell+2}$-minor-free.\nNoting that $|W(r, \\ell)| = r + r\\lfloor \\frac{\\ell-1}{2} \\rfloor$, we make the following conjecture.",
    "post_theorem_intro_text_len": 5248,
    "post_theorem_intro_text": "This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\nProjective geometries are not representable over fields of characteristic zero (see \\cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\\bF$-representable matroid with no $U_{2, \\ell+2}$-minor?\nThis was answered by Geelen, Nelson, and Walsh \\cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\\bF = \\bC$.\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\nWhen $\\bF \\in \\{\\bR, \\mathbb Q\\}$, there is a lower bound of $2\\binom{r}{2} + r$ and an upper bound of $2 \\binom{r}{2} + f(\\ell)\\cdot r$ with $f$ a function of $\\ell$, due to Geelen, Nelson, and Walsh \\cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\\ref{prob: Kung problem} is known for all $\\ell$. \nTheorem \\ref{thm: main} can be seen as a further specialization of Theorems \\ref{thm: exlude a line} and \\ref{thm: complex exclude a line} to the subclass of $\\bR$-representable matroids consisting of positroids.\n\nTheorem \\ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.\nFor example, Bonin \\cite{Bonin-2024} studied matroid amalagams of positroids, Quail \\cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \\cite{Quail-Rombach-2025} studied graphic positroids, and Park \\cite{Park-2023} studied excluded minors for positroids.\nOur proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids.\n\nFinally, \\Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$. \nFor example, Sims \\cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \\cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \\cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.\nMore recently, work of Pendavingh and van der Pol \\cite{Pendavingh-vanderPol-2015} and van der Pol \\cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.\nMost notably for this paper, Kung studied Problem \\ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \\ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\\ell^{\\ell - 1} + 8\\ell)\\cdot r$ elements \\cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \\ell+2}$-minor. \nTo the best of our knowledge, this was in fact the previous best upper bound for this quantity.\nThere are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.\nWe will discuss these further in Section \\ref{sec: future work}.\n\nThe rest of this paper is structured as follows.\nIn Section \\ref{sec: preliminaries}, we will give more background on matroids and positroids.\nIn Section \\ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.\nIn Section \\ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \\ref{thm: main} and prove that they are in fact positroids and are $U_{2, \\ell+2}$-minor-free.\nWe will prove Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}, and conclude in Section \\ref{sec: future work} by discussing several directions for future work.",
    "sketch": "Our proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \\ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \\ref{thm: main} and proves that they are positroids and are $U_{2, \\ell+2}$-minor-free (Section \\ref{sec: the extremal examples}); and (iii) proves Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.",
    "theorem_type": [
      "Inequality or Bound",
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Let $r,2 ell\\ge 1$ be integers. Recall that a positroid is a matroid representable over $\\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\\ge 2$ when does equality occur?",
      "correct_choice": {
        "label": "A",
        "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The bound $|M|\\le \\ell r+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M\\cong U_{2,\\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\\ell+2}$."
        },
        {
          "label": "C",
          "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds."
        },
        {
          "label": "D",
          "text": "For each fixed $\\ell\\ge 1$, there exists an integer $r_0(\\ell)$ such that whenever $r\\ge r_0(\\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, one has $|M|\\le \\ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."
        },
        {
          "label": "E",
          "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\\ell+1$ points."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "excluded-line size in extremal family",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the equality characterization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "all-ranks statement replaced by sufficiently-large-r",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "global parallel-connection structure replaced by local long-line condition",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the equality characterization, and it does not mention parallel connections or the specific building block $U_{2,\\ell+1}$. It states the bound and asks for the equivalent equality case without giving away the correct option."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to theorem recall: the stem presents the extremal bound and asks for the equality characterization. However, it is not a verbatim restatement because the student must still distinguish among several structurally similar conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to reject plausible alternatives such as a direct sum, a weaker minor statement, or the wrong line size. Still, the item mainly tests recognition of the exact characterization rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and well-targeted: B confuses parallel connection with direct sum, C is a weaker true-type statement, D overgeneralizes the line sizes, and E incorrectly uses the excluded threshold. These reflect realistic mathematical failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based."
    }
  },
  {
    "id": "2512.14939v1",
    "paper_link": "http://arxiv.org/abs/2512.14939v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.",
      "start_pos": 5302,
      "end_pos": 5622,
      "label": "thm: main"
    },
    "ref_dict": {
      "prop: rank 3 excluded minor": "\\begin{proposition}\\label{prop: rank 3 excluded minor}\n    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors}.\n\\end{proposition}",
      "prob: Kung problem": "\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}",
      "thm: main": "\\begin{theorem} \\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}",
      "thm: exlude a line": "\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}",
      "thm: complex exclude a line": "\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}",
      "prop: rank 4 excluded minor": "\\begin{proposition}\\label{prop: rank 4 excluded minor}\n    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor}.\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 2450,
    "pre_theorem_intro_text": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors. \n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.",
    "context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.",
    "full_context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.\n\nThis result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm: main result with proof}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor} and no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}\n\\begin{proof}\nFirst suppose that $\\ell = 1$.\nIf $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \\ge 2$ and $M$ has a circuit with at least three elements.\nBut every circuit with at least three elements has a $U_{2,3}$-minor \\cite[Example 3.1.9]{Oxley-2011}, a contradiction.\nSo $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.\nSo we may assume that $\\ell \\ge 2$.\nWe will proceed by induction on $r$.\nThe statement is clearly true when $r \\le 2$ so we may assume that $r \\ge 3$.\nIf $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have\n\\begin{align*}\n |M| = 1 + |M/e| \\le 1 + \\ell(r - 2) + 1 = \\ell(r - 1) + 2 - \\ell < \\ell(r - 1) + 1,\n\\end{align*}\nas desired.\nSo we may assume that every element of $M$ is on a long line.\nLet $L$ be a long line of $M$.\nWe will now consider the connected components of $M/L$.\nLet $\\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.\nLet $X \\in \\cX$, and let $N = M|(L \\cup X)$.\nWe will first bound the number of elements of $N \\del L$.\n\n\\begin{claim} \\label{claim: bound the size of each component}\n At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \\del L| \\le \\ell(r(N) - 2)$.\n\\end{claim}\n\\begin{proof}\nSuppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.\nLet $K = \\si(N/L)$.\nBy deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.\nNote that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.\nFor $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\\cl_N(L \\cup L_i)$ of $N$.\nSince $K$ is connected, it follows from \\cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \\del D/C$ on ground set $\\{p_1, p_2, p_3\\}$.\nWe may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.\nSince $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.\nLet $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\\cl_N(L \\cup d) - L$, and then contracting $C$.\nThen $K_1 = \\si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.\nWe will show that $N_1$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or \\ref{fig: rank-4 excluded minor}.\n\n\\begin{claim}\n Some $x \\in L$ is on more than one long line of $M$.\n\\end{claim}\n\\begin{proof}\nLet $e \\in L$ and let $M_1 = M/e$.\nLet $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.\nAlso note that $\\elem(M_1) = |M| - \\ell = \\ell(r(M_1) - 1) + 1$,\nso by induction, $\\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\\ell+1}$.\nThis implies that $P$ is on a long line $L_1$ of $M_1$.\nLet $F$ be the plane $L \\cup L_1$ of $M$.\nNote that $|F| \\ge |L| + |L_1| - 1 \\ge 2(\\ell + 1) - 1 = 2\\ell + 1$.\nIf $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \\ell+2}$-minor, a contradiction.\nSo $F - L$ has rank $2$.\nIf $\\cl(F - L)$ does not intersect $L$, then for every $f \\in F - L$ we have $(M|F)/f \\cong U_{2,\\ell+2}$, a contradiction. \nSo $\\cl(F - L)$ intersects $L$ in some element $x$.\nSince $|F - L| \\ge \\ell \\ge 2$, it follows that $\\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.\n\\end{proof}\n\nLet $r, \\ell$ be integers with $r \\ge 2$ and $\\ell \\ge 3$.\nLet $W(r,\\ell)$ be the matroid with a basis $B = \\{b_1, b_2, \\dots, b_r\\}$ and $\\lfloor \\frac{\\ell - 1}{2} \\rfloor$ elements freely placed (see \\cite[page 270]{Oxley-2011}) in the span of $\\{b_i, b_{i+1}\\}$ for all $i \\in [r]$, reading indices modulo $r$.\nLet $W(r, \\ell)^+$ be obtained from $W(r, \\ell)$ by freely placing one point in $\\cl(\\{b_1, b_2\\})$.\nThen $W(2, \\ell) \\cong U_{2,\\lfloor \\frac{\\ell - 1}{2} \\rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\\mathcal W^r$ when $r \\ge 3$.\nSince the whirl $\\mathcal W^r$ is $3$-connected (see \\cite[Example 8.3.4]{Oxley-2011}) and $W(r,\\ell)$ has $\\mathcal W^r$ as a restriction, it follows from \\cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\\ell)$ and $W(r, \\ell)^+$ are $3$-connected.\nAnd it is straightforward to check that the natural cyclic ordering of $E(W(r, \\ell))$ (first $b_1$, then all elements in $\\cl(\\{b_1, b_2\\}) - b_2$, then $b_2$, and so on) or $W(r,\\ell)^+$ satisfies the condition of Theorem \\ref{thm:Bonin's condition}, so $W(r, \\ell)$ and $W(r, \\ell)^+$ are positroids.\nFinally, one can check that if $r \\ge 4$ and $e \\notin \\{b_1, \\dots, b_r\\}$, then $\\si(W(r, \\ell)/e) \\cong W(r, \\ell)$, and that $\\si(W(r, \\ell)/b_i) \\cong W(r,\\ell) \\del (\\cl(\\{b_{i-1},b_i, b_{i+1}) - \\{b_{i-1}, b_{i+1}\\})$, and from these observations one can prove that $W(r,\\ell)$ is $U_{2,\\ell+2}$-minor-free, and when $\\ell$ is even, $W(r, \\ell)^+$ is $U_{2, \\ell+2}$-minor-free.\nNoting that $|W(r, \\ell)| = r + r\\lfloor \\frac{\\ell-1}{2} \\rfloor$, we make the following conjecture.",
    "post_theorem_intro_text_len": 5248,
    "post_theorem_intro_text": "This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\nProjective geometries are not representable over fields of characteristic zero (see \\cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\\bF$-representable matroid with no $U_{2, \\ell+2}$-minor?\nThis was answered by Geelen, Nelson, and Walsh \\cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\\bF = \\bC$.\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\nWhen $\\bF \\in \\{\\bR, \\mathbb Q\\}$, there is a lower bound of $2\\binom{r}{2} + r$ and an upper bound of $2 \\binom{r}{2} + f(\\ell)\\cdot r$ with $f$ a function of $\\ell$, due to Geelen, Nelson, and Walsh \\cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\\ref{prob: Kung problem} is known for all $\\ell$. \nTheorem \\ref{thm: main} can be seen as a further specialization of Theorems \\ref{thm: exlude a line} and \\ref{thm: complex exclude a line} to the subclass of $\\bR$-representable matroids consisting of positroids.\n\nTheorem \\ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.\nFor example, Bonin \\cite{Bonin-2024} studied matroid amalagams of positroids, Quail \\cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \\cite{Quail-Rombach-2025} studied graphic positroids, and Park \\cite{Park-2023} studied excluded minors for positroids.\nOur proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids.\n\nFinally, \\Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$. \nFor example, Sims \\cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \\cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \\cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.\nMore recently, work of Pendavingh and van der Pol \\cite{Pendavingh-vanderPol-2015} and van der Pol \\cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.\nMost notably for this paper, Kung studied Problem \\ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \\ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\\ell^{\\ell - 1} + 8\\ell)\\cdot r$ elements \\cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \\ell+2}$-minor. \nTo the best of our knowledge, this was in fact the previous best upper bound for this quantity.\nThere are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.\nWe will discuss these further in Section \\ref{sec: future work}.\n\nThe rest of this paper is structured as follows.\nIn Section \\ref{sec: preliminaries}, we will give more background on matroids and positroids.\nIn Section \\ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.\nIn Section \\ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \\ref{thm: main} and prove that they are in fact positroids and are $U_{2, \\ell+2}$-minor-free.\nWe will prove Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}, and conclude in Section \\ref{sec: future work} by discussing several directions for future work.",
    "sketch": "Our proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \\ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \\ref{thm: main} and proves that they are positroids and are $U_{2, \\ell+2}$-minor-free (Section \\ref{sec: the extremal examples}); and (iii) proves Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.",
    "theorem_type": [
      "Inequality or Bound",
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Let $r,2 ell\\ge 1$ be integers. Recall that a positroid is a matroid representable over $\\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\\ge 2$ when does equality occur?",
      "correct_choice": {
        "label": "A",
        "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The bound $|M|\\le \\ell r+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M\\cong U_{2,\\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\\ell+2}$."
        },
        {
          "label": "C",
          "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds."
        },
        {
          "label": "D",
          "text": "For each fixed $\\ell\\ge 1$, there exists an integer $r_0(\\ell)$ such that whenever $r\\ge r_0(\\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, one has $|M|\\le \\ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."
        },
        {
          "label": "E",
          "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\\ell+1$ points."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "excluded-line size in extremal family",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the equality characterization",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "all-ranks statement replaced by sufficiently-large-r",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "global parallel-connection structure replaced by local long-line condition",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the bound or the equality characterization; it only states the hypotheses and asks for the correct conclusion. There is no explicit answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem asks exactly for the quantitative bound and equality case under the stated hypotheses. The correct choice is basically the theorem statement itself."
      },
      "GPS": {
        "score": 1,
        "justification": "Some discrimination is required because several choices are close variants (weaker true statement, asymptotic weakening, local-vs-global condition, wrong excluded-minor size). However, solving it mainly depends on remembering the exact theorem rather than generating a conclusion from mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and distinct: B uses a natural off-by-one/extremal-family error, C is a weaker true statement, D weakens the quantifier to sufficiently large rank, and E replaces the structural characterization by an insufficient local condition. These align well with realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it scores lower on tautology avoidance and generative reasoning because it mostly asks for the exact recalled statement of a result."
    }
  },
  {
    "id": "2512.14949v1",
    "paper_link": "http://arxiv.org/abs/2512.14949v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "proposition",
      "content": "\\label{prop:Buo=order} For a sequence $(x_n)$ in a vector lattice $E$ and $x\\in E$ the following are equivalent: \\begin{enumerate}[label=(\\roman*)] \\item $x_n\\xrightarrow{\\Buo} x$; \\item $x_n\\xrightarrow{\\mathrm{o}} x$ (order convergence). \\end{enumerate}",
      "start_pos": 6829,
      "end_pos": 7121,
      "label": "prop:Buo=order"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 1812,
    "pre_theorem_intro_text": "In his 1981 paper \\cite{Bourgain81}, J.~Bourgain investigated a strengthened\nmode of convergence on $\\ell_1$: coordinatewise convergence combined with\ndomination by a single element of $\\ell_1$.  Bourgain showed, via an elegant\nfinite-block construction, that this convergence is not sequentially complete,\nanswering a question posed by Jan Mikusiński.\n\nA natural abstract framework for Bourgain’s phenomenon is provided by vector\nlattices.  In this setting, coordinatewise convergence is replaced by\n\\emph{unbounded order} ($\\uo$) convergence, while domination is interpreted as\norder boundedness by a single positive element.  Recall that a sequence\n$(x_n)$ in a vector lattice $E$ is said to \\emph{\\uo-converge} to $x\\in E$,\nwritten $x_n\\xrightarrow{\\uo}x$, if\n\\[\n|x_n-x|\\wedge u \\xrightarrow{\\mathrm{o}} 0\n\\qquad\\text{for every }u\\in E_+,\n\\]\nthat is, all truncations of $|x_n-x|$ converge to zero in order.\n\nIn many familiar settings, $\\uo$-convergence coincides with classical pointwise\nor almost-everywhere convergence.  For example, in $c_0$ and $\\ell_p$ it is\nexactly coordinatewise convergence, while in standard function spaces it agrees\nwith almost-everywhere convergence.  This leads naturally to the following\ndefinition.\n\n\\begin{definition}[Bourgain--uo convergence]\nA sequence $(x_n)\\subset E$ \\emph{\\Buo-converges} to $x\\in E$ if\n$x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that\n$|x_n|\\leqslant y$ for all $n$.\n\\end{definition}\n\nAlthough this definition may appear stronger than order convergence, we show\nthat for sequences it introduces no genuinely new convergence notion: Bourgain--uo\nconvergence coincides with ordinary order convergence; only the Cauchy concept we use later is non-standard. We are indebted to Vladimir Troitsky and Mitchell Taylor for pointing this out to us.",
    "context": "In his 1981 paper \\cite{Bourgain81}, J.~Bourgain investigated a strengthened\nmode of convergence on $\\ell_1$: coordinatewise convergence combined with\ndomination by a single element of $\\ell_1$. Bourgain showed, via an elegant\nfinite-block construction, that this convergence is not sequentially complete,\nanswering a question posed by Jan Mikusiński.\n\nA natural abstract framework for Bourgain’s phenomenon is provided by vector\nlattices. In this setting, coordinatewise convergence is replaced by\n\\emph{unbounded order} ($\\uo$) convergence, while domination is interpreted as\norder boundedness by a single positive element. Recall that a sequence\n$(x_n)$ in a vector lattice $E$ is said to \\emph{\\uo-converge} to $x\\in E$,\nwritten $x_n\\xrightarrow{\\uo}x$, if\n\\[\n|x_n-x|\\wedge u \\xrightarrow{\\mathrm{o}} 0\n\\qquad\\text{for every }u\\in E_+,\n\\]\nthat is, all truncations of $|x_n-x|$ converge to zero in order.\n\nIn many familiar settings, $\\uo$-convergence coincides with classical pointwise\nor almost-everywhere convergence. For example, in $c_0$ and $\\ell_p$ it is\nexactly coordinatewise convergence, while in standard function spaces it agrees\nwith almost-everywhere convergence. This leads naturally to the following\ndefinition.\n\n\\begin{definition}[Bourgain--uo convergence]\nA sequence $(x_n)\\subset E$ \\emph{\\Buo-converges} to $x\\in E$ if\n$x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that\n$|x_n|\\leqslant y$ for all $n$.\n\\end{definition}\n\nAlthough this definition may appear stronger than order convergence, we show\nthat for sequences it introduces no genuinely new convergence notion: Bourgain--uo\nconvergence coincides with ordinary order convergence; only the Cauchy concept we use later is non-standard. We are indebted to Vladimir Troitsky and Mitchell Taylor for pointing this out to us.",
    "full_context": "In his 1981 paper \\cite{Bourgain81}, J.~Bourgain investigated a strengthened\nmode of convergence on $\\ell_1$: coordinatewise convergence combined with\ndomination by a single element of $\\ell_1$. Bourgain showed, via an elegant\nfinite-block construction, that this convergence is not sequentially complete,\nanswering a question posed by Jan Mikusiński.\n\nA natural abstract framework for Bourgain’s phenomenon is provided by vector\nlattices. In this setting, coordinatewise convergence is replaced by\n\\emph{unbounded order} ($\\uo$) convergence, while domination is interpreted as\norder boundedness by a single positive element. Recall that a sequence\n$(x_n)$ in a vector lattice $E$ is said to \\emph{\\uo-converge} to $x\\in E$,\nwritten $x_n\\xrightarrow{\\uo}x$, if\n\\[\n|x_n-x|\\wedge u \\xrightarrow{\\mathrm{o}} 0\n\\qquad\\text{for every }u\\in E_+,\n\\]\nthat is, all truncations of $|x_n-x|$ converge to zero in order.\n\nIn many familiar settings, $\\uo$-convergence coincides with classical pointwise\nor almost-everywhere convergence. For example, in $c_0$ and $\\ell_p$ it is\nexactly coordinatewise convergence, while in standard function spaces it agrees\nwith almost-everywhere convergence. This leads naturally to the following\ndefinition.\n\n\\begin{definition}[Bourgain--uo convergence]\nA sequence $(x_n)\\subset E$ \\emph{\\Buo-converges} to $x\\in E$ if\n$x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that\n$|x_n|\\leqslant y$ for all $n$.\n\\end{definition}\n\nAlthough this definition may appear stronger than order convergence, we show\nthat for sequences it introduces no genuinely new convergence notion: Bourgain--uo\nconvergence coincides with ordinary order convergence; only the Cauchy concept we use later is non-standard. We are indebted to Vladimir Troitsky and Mitchell Taylor for pointing this out to us.\n\n\\begin{definition}[Bourgain--uo convergence]\nA sequence $(x_n)\\subset E$ \\emph{\\Buo-converges} to $x\\in E$ if\n$x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that\n$|x_n|\\le y$ for all $n$.\n\\end{definition}\n\nThe novelty lies instead\nin the associated Cauchy condition.\n\n\\begin{proposition}\\label{prop:CX-metric}\nLet $(X,d)$ be a metric space with a non-isolated point $x_0\\in X$. Then:\n\\begin{enumerate}[\\upshape(i)]\n\\item if $X$ is compact, the Banach lattice $C(X)$ is not sequentially \\Buo-complete;\n\\item the Banach lattice $C_b(X)$ of bounded continuous functions on $X$ is not sequentially \\Buo-complete;\n\\item if $X$ is locally compact and there exists a decreasing sequence of relatively compact open neighbourhoods $(U_n)$ of $x_0$ with $\\bigcap_{n}U_n=\\{x_0\\}$, then $C_0(X)$ is not sequentially \\Buo-complete.\n\\end{enumerate}\n\\end{proposition}\n\nWe verify directly that $(f_n)$ is \\Buo-Cauchy in $C(X)$ (respectively $C_b(X)$).\nLet $(n_k)$ be strictly increasing and set $d_k:=f_{n_{k+1}}-f_{n_k}$. For each\n$x\\in X$ the scalar sequence $(f_n(x))_n$ is decreasing and convergent, hence\n$d_k(x)\\to0$. Thus $d_k\\xrightarrow{\\uo}0$ (uo-convergence is pointwise in vector\nlattices of real-valued functions). Moreover, $|d_k|\\le 1$ for all $k$, and the\nconstant function $1$ belongs to $C(X)$ and to $C_b(X)$, so the differences are\norder bounded by a single element. Hence $d_k\\xrightarrow{\\Buo}0$, and by\nProposition~\\ref{prop:Buo=order} we have $d_k\\xrightarrow{\\mathrm{o}}0$.\nSince $(n_k)$ was arbitrary, $(f_n)$ is \\Buo-Cauchy.\n\n\\begin{proposition}\\label{prop:Lipb-uniformly-discrete}\nLet $(X,d)$ be a metric space and let $\\mathrm{Lip}_b(X)$ be the vector lattice of bounded\nLipschitz functions $X\\to\\R$, ordered pointwise. Then $\\mathrm{Lip}_b(X)$ is sequentially\n\\Buo-complete if and only if $\\delta(X)>0$.\n\\end{proposition}\n\n\\smallskip\\noindent\n\\emph{Step 3: order convergence.}\nFor $m\\in\\N$ define\n\\[\nz_m(x):=\\sup_{n\\ge m}|f_n(x)-f(x)|,\\qquad x\\in X.\n\\]\nThen $0\\le z_{m+1}\\le z_m$ pointwise, and $z_m(x)\\downarrow0$ for each $x$ since $f_n(x)\\to f(x)$.\nAlso $z_m(x)\\le 2M$ for all $x$, hence $z_m\\in\\ell_\\infty(X)$.\nFinally, for every $n\\ge m$ we have $|f_n-f|\\le z_m$, so $f_n\\xrightarrow{\\mathrm{o}}f$ in $\\ell_\\infty(X)$.\nBy Proposition~\\ref{prop:Buo=order}, this is equivalent to $f_n\\xrightarrow{\\Buo}f$.\nTherefore $\\ell_\\infty(X)$ (and hence $\\mathrm{Lip}_b(X)$) is sequentially \\Buo-complete.\n\\end{proof}\n\n\\begin{remark}\\label{rem:Lipb-characterisation}\nProposition~\\ref{prop:Lipb-uniformly-discrete} shows that sequential \\Buo-completeness of $\\mathrm{Lip}_b(X)$\ndepends only on the metric geometry of $X$: it holds exactly when $X$ is uniformly discrete.\nIn particular:\n\\begin{enumerate}[\\upshape(i)]\n\\item if $X$ has a non-isolated point, then $\\delta(X)=0$ and $\\mathrm{Lip}_b(X)$ is not sequentially \\Buo-complete;\n\\item if $X$ is discrete but not uniformly discrete (so $d(x)>0$ for all $x$ but $\\delta(X)=0$), then $\\mathrm{Lip}_b(X)$ is not sequentially \\Buo-complete;\n\\item if $\\delta(X)>0$, then $\\mathrm{Lip}_b(X)=\\ell_\\infty(X)$ and $\\mathrm{Lip}_b(X)$ is sequentially \\Buo-complete.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{question}\\label{q:FBL-lp-sequential}\nLet $1<p<\\infty$ and $\\Gamma$ infinite.\nFor every \\Buo-Cauchy sequence $(f_n)$ in $\\mathrm{FBL}[\\ell_p(\\Gamma)]$,\ndoes the function\n\\[\nz_m(x^\\ast)=\\sup_{n\\ge m}|f_n(x^\\ast)-f(x^\\ast)|\n\\]\nlie in $\\mathrm{FBL}[\\ell_p(\\Gamma)]$ for each $m$?\nEquivalently, must $(f_n)$ converge in order (and hence in the \\Buo\\ sense)?\n\\end{question}",
    "post_theorem_intro_text_len": 3090,
    "post_theorem_intro_text": "\\begin{proof} (i)$\\Rightarrow$(ii): Suppose $x_n\\xrightarrow{\\uo}x$ and $|x_n|\\leqslant y$ for some $y\\in E_+$. Set $w:=(|x|-y)^+\\in E_+$. For every $n$ we have $|x_n|\\leqslant y$, hence on $\\{w>0\\}$ we have $|x_n|\\leqslant y<|x|$, so \\[ |x_n-x|\\wedge w \\ \\geqslant\\ (\\,|x|-|x_n|\\,)\\wedge w \\ \\geqslant\\ (|x|-y)\\wedge w \\ =\\ w. \\] If $w>0$, this contradicts $|x_n-x|\\wedge w\\xrightarrow{\\mathrm{o}}0$ (take $u=w$ in the definition of $\\uo$). Hence $w=0$ and so $|x|\\leqslant y$, i.e.\\ the sequence $(x_n)$ and its limit $x$ all lie in the order interval $[-y,y]$. On $[-y,y]$ we have \\[ |x_n-x| \\ =\\ |x_n-x|\\wedge (2y) \\ \\leqslant\\ 2\\bigl(|x_n-x|\\wedge y\\bigr), \\] and since $\\uo$-convergence gives $|x_n-x|\\wedge y\\xrightarrow{\\mathrm{o}}0$, we obtain $|x_n-x|\\xrightarrow{\\mathrm{o}}0$, i.e.\\ $x_n\\xrightarrow{\\mathrm{o}}x$. (ii)$\\Rightarrow$(i): If $x_n\\xrightarrow{\\mathrm{o}}x$, there exists a decreasing sequence $(z_k)\\downarrow 0$ with $|x_n-x|\\leqslant z_k$ for all $n\\geqslant N_k$. In particular the set $\\{x_n:n\\in\\mathbb N\\}\\cup\\{x\\}$ is order bounded, so there is $y\\in E_+$ with $|x_n|\\leqslant y$ for all $n$ (for instance $y:=|x|+\\sup_k z_k$). For every $u\\in E_+$ and $n\\geqslant N_k$, \\[ |x_n-x|\\wedge u \\ \\leqslant\\ z_k\\wedge u\\ \\downarrow 0, \\] so $x_n\\xrightarrow{\\uo}x$, and thus $x_n\\xrightarrow{\\Buo}x$ by definition. \\end{proof}\n\nThe novelty lies instead\nin the associated Cauchy condition.\n\n\\begin{definition}\nA sequence $(x_n)$ in a vector lattice $E$ is called \\Buo-\\emph{Cauchy} if for\nevery strictly increasing sequence $(n_k)$ the differences\n$x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$.\n\\end{definition}\n\nThis Cauchy condition is strictly stronger than the usual one induced by order\nconvergence and captures the oscillatory behaviour underlying Bourgain’s\nconstruction.  The central theme of this paper is to determine which classical\nBanach lattices are sequentially complete with respect to this notion.\n\nOur first main result concerns sequence spaces.  We show that both $c_0$ and\n$\\ell_\\infty$ are sequentially \\Buo-complete: every \\Buo-Cauchy sequence admits\na pointwise limit in the same space and converges to it in order.\n\nWe then turn to function lattices.  For bounded Lipschitz functions, the answer\nis governed entirely by the metric geometry of the underlying space.  Writing\n$\\mathrm{Lip}_b(X)$ for the vector lattice of bounded Lipschitz functions on a\nmetric space $(X,d)$, we prove that $\\mathrm{Lip}_b(X)$ is sequentially\n\\Buo-complete if and only if $X$ is uniformly discrete.  In particular,\n$\\mathrm{Lip}_b(X)$ fails \\Buo-completeness whenever $X$ has a non-isolated point\nor, more generally, whenever points of $X$ can accumulate at arbitrarily small\nscales.\n\nAs further applications, we show that the Banach lattices $C(X)$, $C_b(X)$ and\n$C_0(X)$ are not sequentially \\Buo-complete under mild topological assumptions\non $X$.  We conclude with several open problems, including the\n\\Buo-sequential completeness of $\\ell_p$ spaces in the reflexive range\n$1<p<\\infty$ and of free Banach lattices $\\mathrm{FBL}[\\ell_p(\\Gamma)]$.",
    "sketch": "To prove Proposition~\\ref{prop:Buo=order}:\n\n(i)$\\Rightarrow$(ii): Assume $x_n\\xrightarrow{\\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\\le y$ for some $y\\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\\le y$, on $\\{w>0\\}$ one has $|x_n|\\le y<|x|$, hence\n\\[\n|x_n-x|\\wedge w\\ \\ge\\ (|x|-|x_n|)\\wedge w\\ \\ge\\ (|x|-y)\\wedge w\\ =\\ w.\n\\]\nIf $w>0$ this contradicts $|x_n-x|\\wedge w\\xrightarrow{\\mathrm{o}}0$ (take $u=w$ in the definition of $\\uo$). Thus $w=0$ and $|x|\\le y$, so $x_n,x\\in[-y,y]$. On $[-y,y]$,\n\\[\n|x_n-x|=|x_n-x|\\wedge (2y)\\le 2\\bigl(|x_n-x|\\wedge y\\bigr),\n\\]\nand since $\\uo$-convergence gives $|x_n-x|\\wedge y\\xrightarrow{\\mathrm{o}}0$, it follows that $|x_n-x|\\xrightarrow{\\mathrm{o}}0$, i.e. $x_n\\xrightarrow{\\mathrm{o}}x$.\n\n(ii)$\\Rightarrow$(i): If $x_n\\xrightarrow{\\mathrm{o}}x$, there exists $(z_k)\\downarrow 0$ with $|x_n-x|\\le z_k$ for all $n\\ge N_k$. Then $\\{x_n\\}\\cup\\{x\\}$ is order bounded, so choose $y\\in E_+$ with $|x_n|\\le y$ for all $n$ (e.g. $y:=|x|+\\sup_k z_k$). For every $u\\in E_+$ and $n\\ge N_k$,\n\\[\n|x_n-x|\\wedge u\\le z_k\\wedge u\\downarrow 0,\n\\]\nso $x_n\\xrightarrow{\\uo}x$, and hence $x_n\\xrightarrow{\\Buo}x$ “by definition.”",
    "expanded_sketch": "To prove the main theorem:\n\n(i)$\\Rightarrow$(ii): Assume $x_n\\xrightarrow{\\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\\le y$ for some $y\\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\\le y$, on $\\{w>0\\}$ one has $|x_n|\\le y<|x|$, hence\n\\[\n|x_n-x|\\wedge w\\ \\ge\\ (|x|-|x_n|)\\wedge w\\ \\ge\\ (|x|-y)\\wedge w\\ =\\ w.\n\\]\nIf $w>0$ this contradicts $|x_n-x|\\wedge w\\xrightarrow{\\mathrm{o}}0$ (take $u=w$ in the definition of $\\uo$). Thus $w=0$ and $|x|\\le y$, so $x_n,x\\in[-y,y]$. On $[-y,y]$,\n\\[\n|x_n-x|=|x_n-x|\\wedge (2y)\\le 2\\bigl(|x_n-x|\\wedge y\\bigr),\n\\]\nand since $\\uo$-convergence gives $|x_n-x|\\wedge y\\xrightarrow{\\mathrm{o}}0$, it follows that $|x_n-x|\\xrightarrow{\\mathrm{o}}0$, i.e. $x_n\\xrightarrow{\\mathrm{o}}x$.\n\n(ii)$\\Rightarrow$(i): If $x_n\\xrightarrow{\\mathrm{o}}x$, there exists $(z_k)\\downarrow 0$ with $|x_n-x|\\le z_k$ for all $n\\ge N_k$. Then $\\{x_n\\}\\cup\\{x\\}$ is order bounded, so choose $y\\in E_+$ with $|x_n|\\le y$ for all $n$ (e.g. $y:=|x|+\\sup_k z_k$). For every $u\\in E_+$ and $n\\ge N_k$,\n\\[\n|x_n-x|\\wedge u\\le z_k\\wedge u\\downarrow 0,\n\\]\nso $x_n\\xrightarrow{\\uo}x$, and hence $x_n\\xrightarrow{\\Buo}x$ “by definition.”,",
    "expanded_theorem": "\\label{prop:Buo=order} For a sequence $(x_n)$ in a vector lattice $E$ and $x\\in E$ the following are equivalent: \\begin{enumerate}[label=(\\roman*)] \\item $x_n\\xrightarrow{\\Buo} x$; \\item $x_n\\xrightarrow{\\mathrm{o}} x$ (order convergence). \\end{enumerate}",
    "theorem_type": [
      "Biconditional or Equivalence"
    ],
    "mcq": {
      "question": "Let $E$ be a vector lattice, let $(x_n)$ be a sequence in $E$, and let $x\\in E$. Recall that $(x_n)$ is said to \\emph{Bourgain--uo converge} to $x$, written $x_n\\xrightarrow{\\Buo}x$, if $x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that $|x_n|\\le y$ for all $n$, where $x_n\\xrightarrow{\\uo}x$ means $|x_n-x|\\wedge u\\xrightarrow{\\mathrm{o}}0$ for every $u\\in E_+$. Which of the following statements is equivalent to $x_n\\xrightarrow{\\Buo}x$?",
      "correct_choice": {
        "label": "A",
        "text": "$x_n\\xrightarrow{\\mathrm{o}}x$; that is, $(x_n)$ converges to $x$ in order in $E$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$x_n\\xrightarrow{\\uo}x$ and there exists, for every $u\\in E_+$, an element $y_u\\in E_+$ such that $|x_n|\\wedge u\\le y_u$ for all $n$; equivalently, each truncation $(|x_n|\\wedge u)$ is order bounded."
        },
        {
          "label": "C",
          "text": "$x_n\\xrightarrow{\\uo}x$ and the set $\\{x_n:n\\in\\mathbb N\\}\\cup\\{x\\}$ is order bounded in $E$."
        },
        {
          "label": "D",
          "text": "$x_n\\xrightarrow{\\uo}x$ and there exists $y\\in E_+$ such that $|x_n-x|\\le y$ for all $n$."
        },
        {
          "label": "E",
          "text": "$x_n\\xrightarrow{\\uo}x$ and there exists a sequence $(y_m)\\subset E_+$ with $y_m\\downarrow 0$ such that for every $m$ one has $|x_n|\\le y_m$ for all sufficiently large $n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "single global dominator",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "replace domination of the sequence alone by order boundedness of the sequence together with the limit",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "bound on terms instead of bound used to force the limit into the same order interval",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "eventual vanishing domination replacing fixed order bound",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines Bourgain--uo convergence but does not directly state that it is equivalent to order convergence. The correct option is not leaked explicitly; one must know or derive the relevant theorem."
      },
      "TAS": {
        "score": 1,
        "justification": "This is not a verbatim restatement of the definition, since the intended answer invokes a different convergence notion. However, it is still essentially an equivalence-checking theorem question rather than a substantially new inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required: the solver must connect 'uo + order bounded' with order convergence and distinguish this from nearby variants. But the pressure is weakened because the question is mainly recall of a standard equivalence, and choice C appears so close to the intended theorem that uniqueness is doubtful."
      },
      "DQS": {
        "score": 1,
        "justification": "B, D, and E are plausible distractors built from common quantifier and domination confusions. However, C is not a clean distractor: it looks mathematically equivalent or at least extremely close to the intended correct statement, making the item potentially ambiguous."
      },
      "total_score": 5,
      "overall_assessment": "Moderately good concept-check question with no direct answer leakage, but it is mostly an equivalence-recall item and is weakened by a distractor that appears genuinely true or too close to the correct answer."
    }
  },
  {
    "id": "2512.15253v1",
    "paper_link": "http://arxiv.org/abs/2512.15253v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "rmk",
      "content": "The results of the Main Theorem still hold for $ C^{1+\\alpha} $ mappings. To cite some conclusions in \\cite{Qian} conveniently, we therefore assume $ C^2 $ regularity here.",
      "start_pos": 14004,
      "end_pos": 14201,
      "label": null
    },
    "ref_dict": {
      "separation": "\\begin{lem}$\\mathrm{(Lemma \\textbf{ }2.6 \\text{ of }\\cite{Zhu})}$\\label{separation}\n\t\tIf $ f:M\\to M$ is a non-degenerate endomorphism, or, more generally, if $ f:M\\to M $ is a local homeomorphism, then $ f $ has uniform separation of preimages.\n\t\\end{lem}"
    },
    "pre_theorem_intro_text_len": 9022,
    "pre_theorem_intro_text": "In modern dynamical systems and ergodic theory, both entropy and pressure are the fundamental quantities that measure the complexity of a system. These concepts are closely related to physics and thermodynamic mechanisms; see the monographs by Ruelle \\cite{Ruelle} and Bowen \\cite{Bowen book}. Roughly speaking, the topological pressure can be interpreted as the maximal free energy that a system can attain. Then, the Gibbs distribution, which can maximize free energy, corresponds to the equilibrium state in ergodic theory. If one neglects the potential energy, topological pressure reduces to topological entropy.\n\n\tInvariant measures play central roles in ergodic theory. Let $f$ be a continuous map on a compact metric space $X$. Given an $f$-invariant measure $\\mu$, one can define the measure-theoretic entropy $h_{\\mu}(f)$. The variational principle establishes a relationship between the measure-theoretic entropy and the topological entropy  $h(f)$ as\n\t$$\n\th(f)=\\sup\\{h_{\\mu}(f) \\;|\\; \\mu\\in \\mathcal{M}_f\\},\n\t$$\n\twhere $ \\mathcal{M}_f $ is the set of $ f $-invariant Borel probability measures.\tThe element in $ \\mathcal{M}_f $ that makes the equality hold is called the \\emph{measure of maximal entropy}.\n\tGiven a continuous function $\\phi$ on $X$, referred to as a potential, the variational principle for pressure then yields\n\t$$\n\tP(f,\\phi)=\\sup\\{h_{\\mu}(f)+\\int \\phi d\\mu \\;|\\; \\mu\\in \\mathcal{M}_f\\}.\n\t$$\n\tThe element in $ \\mathcal{M}_f $ that reaches the supremum is called the \\emph{equilibrium state} for $\\phi$. It follows that when the potential function vanishes, the topological pressure reduces to the topological entropy, and the corresponding equilibrium state coincides with a measure of maximal entropy.\n\n\tThe investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic,  there are two classical ones established in the  1970s. The first one is using the transfer operator theory, see in \\cite{Bowen book} and \\cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\\Lambda$ and then obtain a symbolic extension of $f_\\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \\cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\\Lambda$ is a topologically mixing compact locally maximal hyperbolic set\n\tfor a diffeomorphism $f$, then $f_\\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.\n\n\tIn recent years, the study of dynamics of the differentiable systems beyond uniform hyperbolicity, such as non-uniformly hyperbolic systems and partially hyperbolic systems,  has attracted much attention and made a series of important advances. For partially hyperbolic diffeomorphisms, a center direction is allowed in addition to the hyperbolic directions. The presence of the center direction permits a very rich type of structure. Investigating the thermodynamic mechanisms, especially the existence and uniqueness of the equilibrium state, for partially hyperbolic systems require more exquisite techniques.\n\n\tRecently, Climenhaga and Thompson \\cite{CT21} ingeniously developed the criterion of Bowen \\cite{Bowen74} in settings beyond uniform hyperbolicity\n\tusing weakened versions of specification and expansivity. They first considered the case of symbolic systems.\n\tRather than assuming specification on the entire symbolic system, they identified a collection of ``good\" words on which the specification property holds, while showing that the set of ``bad\" words has topological entropy strictly smaller than that of the full system. Using this decomposition, they established the uniqueness of the measure of maximal entropy and the equilibrium state in \\cite{CT12} and \\cite{CT13}, respectively. Further, Climenhaga and Thompson weakened expansivity by introducing a weaker, non-uniform notion applicable to general compact metric spaces; they also formulate correspondingly weakened, non-uniform versions of specification, expansivity, and the Bowen property in \\cite{CT14} and \\cite{CT16}. For applications of the method mentioned above, one can see \\cite{CFT18} and \\cite{CFT19}. The general strategy is to first identify obstructions to expansivity and specification, and then to control their entropy or pressure. Pacifico, F. Yang, and J. Yang \\cite{Yang} further refined the conditions in the Climenhaga-Thompson approach, allowing the framework to be applied in settings with fixed parameters.\n\n\tFrom \\cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \\cite{Mongez} substituted the condition on the central Lyapunov exponent used in \\cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \\cite{Yan21, HHW}.  Liu and Liao  \\cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\\\"{o}lder continuous potentials with not very large oscillation.\n\n\tPartially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \\cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \\cite{Wang, Zhu}.   Álvarez and Cantarino \\cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and  Cabezas  \\cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.\n\n\tOur main purpose in this paper is to generalize the results in \\cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \\ref{separation}), then we can deal with the preimages on a small domain. Denote  $U_a(\\phi):=\\{\\psi\\in C(X)\\;|\\; \\lVert \\phi-\\psi \\rVert <a\\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \\lVert \\phi \\rVert=\\sup_{x\\in X}\\lvert \\phi(x)\\rvert $.\n\n\t\\begin{mainthm}\\label{Main}\n\t\tLet $ f:M\\to M $ be a non-degenerate $ C^2 $  partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \\phi:M\\to \\mathbb{R} $ be a H\\\"{o}lder continuous potential. Assume that  $ P^u(f,\\phi)>P^s(f,\\phi)$ and  the unstable foliation $\\tilde{ \\mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\\frac{P^u(f,\\phi)-P^s(f,\\phi)}{2} $\n\t\tthere exists a $ C^1 $ neighborhood $ \\mathcal{U}_f $ of $ f $ such that for any H\\\"{o}lder continuous potential $\\psi\\in U_{a}(\\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\\in \\mathcal{U}_f $ there   has a unique equilibrium state for $(g,\\psi)$.\n\t\\end{mainthm}",
    "context": "The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic, there are two classical ones established in the 1970s. The first one is using the transfer operator theory, see in \\cite{Bowen book} and \\cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\\Lambda$ and then obtain a symbolic extension of $f_\\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \\cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\\Lambda$ is a topologically mixing compact locally maximal hyperbolic set\n for a diffeomorphism $f$, then $f_\\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.\n\nFrom \\cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \\cite{Mongez} substituted the condition on the central Lyapunov exponent used in \\cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \\cite{Yan21, HHW}. Liu and Liao \\cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\\\"{o}lder continuous potentials with not very large oscillation.\n\nPartially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \\cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \\cite{Wang, Zhu}. Álvarez and Cantarino \\cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and Cabezas \\cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.\n\nOur main purpose in this paper is to generalize the results in \\cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \\ref{separation}), then we can deal with the preimages on a small domain. Denote $U_a(\\phi):=\\{\\psi\\in C(X)\\;|\\; \\lVert \\phi-\\psi \\rVert <a\\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \\lVert \\phi \\rVert=\\sup_{x\\in X}\\lvert \\phi(x)\\rvert $.\n\n\\begin{mainthm}\\label{Main}\n Let $ f:M\\to M $ be a non-degenerate $ C^2 $ partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \\phi:M\\to \\mathbb{R} $ be a H\\\"{o}lder continuous potential. Assume that $ P^u(f,\\phi)>P^s(f,\\phi)$ and the unstable foliation $\\tilde{ \\mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\\frac{P^u(f,\\phi)-P^s(f,\\phi)}{2} $\n there exists a $ C^1 $ neighborhood $ \\mathcal{U}_f $ of $ f $ such that for any H\\\"{o}lder continuous potential $\\psi\\in U_{a}(\\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\\in \\mathcal{U}_f $ there has a unique equilibrium state for $(g,\\psi)$.\n \\end{mainthm}",
    "full_context": "The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic, there are two classical ones established in the 1970s. The first one is using the transfer operator theory, see in \\cite{Bowen book} and \\cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\\Lambda$ and then obtain a symbolic extension of $f_\\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \\cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\\Lambda$ is a topologically mixing compact locally maximal hyperbolic set\n for a diffeomorphism $f$, then $f_\\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.\n\nFrom \\cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \\cite{Mongez} substituted the condition on the central Lyapunov exponent used in \\cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \\cite{Yan21, HHW}. Liu and Liao \\cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\\\"{o}lder continuous potentials with not very large oscillation.\n\nPartially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \\cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \\cite{Wang, Zhu}. Álvarez and Cantarino \\cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and Cabezas \\cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.\n\nOur main purpose in this paper is to generalize the results in \\cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \\ref{separation}), then we can deal with the preimages on a small domain. Denote $U_a(\\phi):=\\{\\psi\\in C(X)\\;|\\; \\lVert \\phi-\\psi \\rVert <a\\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \\lVert \\phi \\rVert=\\sup_{x\\in X}\\lvert \\phi(x)\\rvert $.\n\n\\begin{mainthm}\\label{Main}\n Let $ f:M\\to M $ be a non-degenerate $ C^2 $ partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \\phi:M\\to \\mathbb{R} $ be a H\\\"{o}lder continuous potential. Assume that $ P^u(f,\\phi)>P^s(f,\\phi)$ and the unstable foliation $\\tilde{ \\mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\\frac{P^u(f,\\phi)-P^s(f,\\phi)}{2} $\n there exists a $ C^1 $ neighborhood $ \\mathcal{U}_f $ of $ f $ such that for any H\\\"{o}lder continuous potential $\\psi\\in U_{a}(\\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\\in \\mathcal{U}_f $ there has a unique equilibrium state for $(g,\\psi)$.\n \\end{mainthm}\n\nThe following results in \\cite{CT16} and \\cite{Yang} respectively proved the uniqueness of equilibrium states for certain systems.\n \\begin{thm} $(\\mathrm{Theorem \\text{ }5.6 \\text{ of } \\cite{CT16}})$\\label{CT}\n Let $ X $ be a compact metric space, $ f:X\\to X $ a homeomorphism and $ \\phi:X\\to \\mathbb{R} $ a continuous potential function. Suppose that there are $ \\varepsilon>0$, $\\delta>0 $ with $ \\varepsilon>40\\delta $ such that $P^\\perp_{\\mathrm{exp}}(f,\\phi,\\varepsilon)<P(f,\\phi) $ and there exists $ \\mathcal{D}\\subset X\\times \\mathbb{N} $ admits a decomposition $ (\\mathcal{P},\\mathcal{G},\\mathcal{S}) $ with the following properties:\n \\begin{flushleft}\n (1) For every $ N\\in\\mathbb{N} $, $ \\mathcal{G}^N $ has tail $ (W) $-specification at scale $ \\delta $;\\\\\n (2) $ \\phi $ has the Bowen property at scale $ \\varepsilon $ on $ \\mathcal{G} $;\\\\\n (3) $P(\\mathcal{P}\\cup\\mathcal{S}\\cup\\mathcal{D}^c,\\phi,\\delta,\\varepsilon)<P(f,\\phi) $.\\\\\n Then there exists a unique equilibrium state for $ (f,\\phi) $.\n \\end{flushleft}\n \\end{thm}\n \\begin{thm} $(\\mathrm{Theorem \\text{ }A} \\text{ of }\\cite{Yang})$\\label{2.2}\n Let $ X $ be a compact metric space, $ f:X\\to X $ a homeomorphism and $ \\phi:X\\to \\mathbb{R} $ a continuous potential function. Suppose that there exist $ \\varepsilon>0$, $\\delta>0 $ with $ \\varepsilon\\geq 2000\\delta $ such that $ P^\\perp_{\\mathrm{exp}}(f,\\phi,\\varepsilon)<P(f,\\phi) $ and $ X\\times \\mathbb{N} $ admits a decomposition $ (\\mathcal{P},\\mathcal{G},\\mathcal{S})$ with the following properties:\n \\begin{flushleft}\n (1) $ \\mathcal{G} $ has $ (W) $-specification at scale $ \\delta $;\\\\\n (2) $ \\phi $ has the Bowen property at scale $ \\varepsilon $ on $ \\mathcal{G} $;\\\\\n (3) $ P(\\mathcal{P}\\cup\\mathcal{S},\\phi,\\delta,\\varepsilon)<P(f,\\phi) $.\\\\\n Then there exists a unique equilibrium state for $ (f,\\phi) $.\n \\end{flushleft}\n \\end{thm}\n Theorem \\ref{CT} is sometimes referred to as the Climenhaga-Thompson Criterion (abbrev. CT Criterion). As an application of the above criteria, Mongez and Pacifico \\cite{Mongez} gave the following result for certain partially hyperbolic diffeomorphisms.\n \\begin{thm}$(\\mathrm{Theorem \\text{ }A} \\text{ of }\\cite{Mongez})$ \\label{Mon}\n Let $f: M\\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact manifold $M$ with $TM=E^u\\oplus E^c\\oplus E^s$ and $\\phi:M\\to \\mathbb{R}$ a H\\\"{o}lder continuous potential. Assume that dim($E^c$)=1 and the unstable foliation $\\mathcal{F}^u(f)$ is minimal. If $h^u(f)-h^s(f)>\\sup\\phi-\\inf\\phi\\geq 0$, then there exists a $C^1$ neighborhood $\\mathcal{U}$ of $f$ so that $(g,\\phi)$ has a unique equilibrium state for every $C^{1+}$ diffeomorphism $g\\in \\mathcal{U}$.\n \\end{thm}\n The main task of this paper is to obtain the counterpart of Theorem \\ref{Mon} in the setting of partially hyperbolic endomorphisms under weaker conditions.\n\nNext, we will use the results in \\cite{Yun, Chung, Gelfert} that for any $ C^{1+\\alpha} $ mapping, there exists a horseshoe whose topological pressure can approximate the sum of the metric entropy and the integral of the potential whenever the metric entropy is positive for an ergodic hyperbolic measure.\n\nAs we know that the horseshoe for $ C^{1+\\alpha} $ mapping is structurally stable under small $ C^1 $ perturbation in \\cite{Chung}, i.e. there is a $ C^1 $ neighborhood $ \\mathcal{U}_1 $ of $ f $ such that there exists a hyperbolic horseshoe $ H_g $ and $\n \\tau|_{(H_g)^g}: (H_g)^g \\to (H_g)^g $ is topologically conjugate to $\\tau|_{(H_\\varepsilon)^f}: (H_\\varepsilon)^f \\to (H_\\varepsilon)^f $ by $ h_g:(H_g)^g \\to (H_\\varepsilon)^f $ with $ \\lVert h_g-id \\rVert<\\delta $\n for any $ g\\in \\mathcal{U}_1 $ where $(H_g)^g$, $(H_\\varepsilon)^f$ are the inverse limit spaces corresponding to the horseshoes respectively and $ \\lVert \\phi^*\\circ h_g-\\phi^*\\rVert<\\varepsilon $ if $ \\lVert h_g-id \\rVert<\\delta $. Then we have\n \\begin{align*}\n P(\\tau|_{(H_{\\varepsilon})^f},\\phi^*|_{(H_{\\varepsilon})^f})=P(\\tau|_{(H_g)^g}, \\phi^*\\circ h_g).\n \\end{align*}\n and\n \\begin{align*}\n P^u(f,\\phi)-2\\varepsilon<P(g|_{H_g}, \\phi|_{H_g}).\n \\end{align*}\n In order to relate the above inequality to the unstable pressure of $g$, we need to establish the existence of the equilibrium state for $g|_{H_g}$ as follows.\n \\begin{claim}\n If $ H_g $ is a hyperbolic set for the endomorphism $ g $, then $ \\tau|_{(H_g)^g} $ is expansive.\n \\end{claim}\n In other words, we claim that there is a constant $ \\eta>0 $ small enough such that for any $ \\tilde{x},\\tilde{y} \\in (H_g)^g$ with $ \\tilde{x}\\neq \\tilde{y}$ then $ \\tilde{d}(\\tau^k\\tilde{x}, \\tau^k\\tilde{y})>\\eta $ for some $ k\\in \\mathbb{Z} $. Otherwise $ \\tilde{y}\\in \\tilde{W}^u_{\\eta}(\\tilde{x})\\cap \\tilde{W}^s_{\\eta}(\\tilde{x})$.\n Since $ \\tilde{x}\\in \\tilde{W}^u_{\\eta}(\\tilde{x})\\cap \\tilde{W}^s_{\\eta}(\\tilde{x})$ and $ \\Pi_{\\tilde{W}^u_{loc}(\\tilde{x})}: \\tilde{W}^u_{loc}(\\tilde{x}) \\to W^u_{loc}(\\tilde{x}) $ is bijective, then by the local product structure in Theorem IV.2.3 of \\cite{Qian}, we have $\\tilde{x}=\\tilde{y} $. This completes the proof of this claim.\n\nNow we assume that $f $ is a $ C^2 $ non-degenerate partially hyperbolic endomorphism on the compact Riemannian manifold with dim$(E^c)=1 $ and $ \\phi $ is a H\\\"{o}lder continuous potential function on $ M $. Recall that let $\\phi^*=\\phi\\circ \\Pi$. Given an ergodic measure $ \\mu\\in \\mathcal{M}^e_f $, we can consider the central Lyapunov exponents by Theorem I.2.1 and Proposition I.3.5 of \\cite{Qian}. We use $\\tilde{\\mu} $ to denote the ergodic $ \\tau $-invariant Borel probability measure on $ M^f $ such that $ \\Pi \\tilde{\\mu}=\\mu $. We denote $ \\lambda^c(\\tilde{\\mu},\\tau) $ (resp. $\\lambda^c(\\mu,f)$) be the central Lyapunov exponent of $ \\tilde{\\mu} $ (resp. $ \\mu$) whenever $ \\tilde{\\mu}$ (resp. $\\mu $) is ergodic, i.e. for $\\tilde{\\mu}$-$a.e.$ $ \\tilde{x}=(x_n)_{n\\in\\mathbb{Z}}\\in M^f$ (resp. $ \\mu$-$a.e.$ $x_0\\in M $), we have\n \\begin{align*}\n \\lambda^c(\\mu,f)=\\lambda^c(\\tilde{\\mu},\\tau)\n &=\\lim\\limits_{m\\to\\infty}\\frac{1}{m}\\log\\lvert D_{x_0}f^m \\xi\\rvert\\\\\n &=\\lim\\limits_{m\\to\\infty}\\frac{1}{m}\\log\\lVert D_{x_0}f^m |_{E^c(\\tilde{x})}\\rVert\\\\\n &=\\lim\\limits_{m\\to\\infty}\\frac{1}{m}\\log\\lVert D_{x_{m-1}}f\\circ \\cdots \\circ D_{x_1}f \\circ D_{x_0}f |_{E^c(\\tilde{x})}\\rVert\\\\\n &=\\lim\\limits_{m\\to\\infty}\\frac{1}{m}\\sum_{k=0}^{m-1}\\log\\lVert D_{x_k}f |_{E^c(\\tau^k\\tilde{x})}\\rVert\\\\\n &=\\int_{M^f} \\log\\lVert Df |_{E^c(\\tilde{x})}\\rVert d\\tilde{\\mu}, \n \\end{align*}\n where $ \\tilde{x}\\in M^f $ with $ \\Pi(\\tilde{x})=x_0 $, $ 0\\neq\\xi\\in E^c(\\tilde{x}) $, and define $\\tilde{\\varphi}^c(\\tilde{x}):=\\log\\lVert Df |_{E^c(\\tilde{x})}\\rVert$.\n\n\\begin{lem}\n There exists $ r>0 $ such that\n $ \\sup\\{h_{\\tilde{\\mu}}(\\tau)+\\int\\phi^* d\\tilde{\\mu}: \\tilde{\\mu}\\in \\mathcal{M}_\\tau, \\lambda^c(\\tilde{\\mu},\\tau)\\geq -r\\}<P(\\tau,\\phi^*) $.\n \\end{lem}\n \\begin{pf}\n First, we claim that\n $ \\sup\\{h_{\\tilde{\\mu}}(\\tau)+\\int\\phi^* d\\tilde{\\mu}: \\tilde{\\mu}\\in \\mathcal{M}_\\tau, \\lambda^c(\\tilde{\\mu},\\tau)\\geq 0\\}<P(\\tau,\\phi^*) $. If not, we can assume there exists an invariant probability measure $ \\tilde{\\mu} $ with $ \\lambda^c(\\tilde{\\mu},\\tau)\\geq 0 $, but $ \\tilde{\\mu} $ is an equilibrium state for $ (\\tau,\\phi^*) $. By the ergodic decomposition theorem, we have\n $ \\lambda^c(\\tilde{\\mu},\\tau)=\\int \\tilde{\\varphi}^c d\\tilde{\\mu}=\\int_{\\mathcal{M}_\\tau^e} (\\int \\tilde{\\varphi}^c dm) d\\tau(m) \\geq 0$,\n and by Theorem 8.4 in \\cite{Walters} we get\n $ h_{\\tilde{\\mu}}(\\tau)=\\int_{\\mathcal{M}_\\tau^e} h_m(\\tau)d\\tau(m) $.\n Then\n $$\n P(\\tau,\\phi^*)=h_{\\tilde{\\mu}}(\\tau)+\\int \\phi^* d\\tilde{\\mu}\n =\\int_{\\mathcal{M}_\\tau^e} (h_m(\\tau)+\\int \\phi^* dm)d\\tau(m),\n $$\n so each $ m\\in \\mathcal{M}_\\tau^e $ must be the equilibrium state for $ (\\tau,\\phi^*) $. Therefore, there exists a measure $ m_0\\in\\mathcal{M}_\\tau^e $ such that\n $ \\lambda^c(m_0,\\tau)=\\int \\tilde{\\varphi}^c dm_0\\geq 0 $\n and\n $ P(\\tau,\\phi^*)=h_{m_0}(\\tau)+\\int \\phi^* dm_0 $,\n which contradicts with $ P^+(\\tau,\\phi^*)<P^-(\\tau,\\phi^*) $.",
    "post_theorem_intro_text_len": 3651,
    "post_theorem_intro_text": "The proof of the main theorem in this paper is inspired by arguments established in \\cite{CT21} and \\cite{Mongez}. Compared with the case of diffeomorphisms, the main difficulty here comes from non-invertibility. In order to overcome this deficiency, we use the technique of inverse limit. For certain non-degenerate partially hyperbolic endomorphisms with one-dimensional center and any  H\\\"{o}lder continuous potential function satisfy specific conditions, we aim to construct a decomposition of orbit segments in the inverse limit spaces. An important condition in the main theorem is the gap between the unstable pressure and the stable pressure. For partially hyperbolic diffeomorphisms, the unstable entropy and unstable pressure for a continuous function were introduced in \\cite{HHW, HWZ}, respectively. As a generalization, the concepts of unstable entropy and unstable pressure were given by Wang, Wu and Zhu in \\cite{Wang} for partially hyperbolic endomorphisms. In \\cite{Zhu}, Wu and Zhu introduced the stable entropy for non-degenerate $C^1$  partially hyperbolic endomorphisms and their stable pressure is given in \\cite{Li}. Different from the condition given by Mongez and Pacifico in \\cite{Mongez}, we change the condition ``$h^u(f)-h^s(f)>\\sup\\phi-\\inf\\phi\\geq 0$'' to ``$P^u(f,\\phi)>P^s(f,\\phi)$'', where\n\t$P^u(f,\\phi)=\\sup\\{h^u_{\\mu}(f)+\\int \\phi d\\mu\\;|\\; \\mu\\in \\mathcal{M}_f^e\\}$ and $P^s(f,\\phi)=\\sup\\{h^s_{\\mu}(f)+\\int \\phi d\\mu\\;|\\; \\mu\\in \\mathcal{M}_f^e\\}$.\n\tSince for each ergodic $f$-invariant measure $\\mu$, there is\n\t\\begin{align*}\n\t\th^u_\\mu(f)+\\min\\phi \\leq h^u_\\mu(f)+\\int \\phi d\\mu,\n\t\\end{align*}\n\twe have $P^u(f,\\phi)\\geq h^u(f)+\\min\\phi$.\n\tMeanwhile, $h^s(f)+\\max\\phi \\geq P^s(f,\\phi)$. Therefore, the condition ``$P^u(f,\\phi)>P^s(f,\\phi)$'' is weaker than ``$h^u(f)-h^s(f)>\\sup\\phi-\\inf\\phi\\geq 0$'' since the space we are considering is compact so $\\sup\\phi = \\max\\phi$ and $\\inf\\phi=\\min\\phi$.\n\tUnder the condition $ P^u(f,\\phi)>P^s(f,\\phi) $ and assuming the existence of unstable foliation with some sense of minimality, we show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction. Moreover, the potential $ \\phi $ satisfies the Bowen property on these good orbit segments. Additionally, the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system. As a consequence, there exists a unique equilibrium state on the inverse limit space, and its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism. Furthermore, we also demonstrate that the conditions of the theorem are inherently robust, which further implies the robustness of the equilibrium state.\n\tAt the same time, H\\\"{o}lder continuous potential functions can also vary within a certain range,  thus allowing the uniqueness of the equilibrium state of the system under small perturbation can be obtained.\n\n\tThis paper is organized as follows. In Section 2, we give some preliminaries, including the concepts of the decomposition of the orbit segments, the pressure of the obstructions to expansivity, partially hyperbolic endomorphisms, inverse limit space, (un)stable en        tropy and (un)stable pressure, and other necessary definitions. In Section 3, we demonstrate the robustness of the conditions in Main Theorem. In Section 4, we prove  Main Theorem using the improved Climenhaga-Thompson criterion. In Section 5, we give an example satisfying the conditions in our main theorem.",
    "sketch": "The proof is said to be “inspired by arguments established in \\cite{CT21} and \\cite{Mongez},” with the “main difficulty” (vs. diffeomorphisms) coming from “non-invertibility,” which is handled by “the technique of inverse limit.” Under the condition $P^u(f,\\phi)>P^s(f,\\phi)$ (introduced as a weakening of Mongez–Pacifico’s condition), the strategy is to “construct a decomposition of orbit segments in the inverse limit spaces.” Assuming “the existence of unstable foliation with some sense of minimality,” they “show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction,” and that the potential $\\phi$ “satisfies the Bowen property on these good orbit segments.” They also show that “the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system.” Then, “as a consequence, there exists a unique equilibrium state on the inverse limit space,” and “its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism.” Finally, they “demonstrate that the conditions of the theorem are inherently robust,” implying “robustness of the equilibrium state,” and note that H\\\"older potentials “can also vary within a certain range,” giving uniqueness “under small perturbation.”",
    "expanded_sketch": "The proof is said to be “inspired by arguments established in \\cite{CT21} and \\cite{Mongez},” with the “main difficulty” (vs. diffeomorphisms) coming from “non-invertibility,” which is handled by “the technique of inverse limit.” Under the condition $P^u(f,\\phi)>P^s(f,\\phi)$ (introduced as a weakening of Mongez–Pacifico’s condition), the strategy is to “construct a decomposition of orbit segments in the inverse limit spaces.” Assuming “the existence of unstable foliation with some sense of minimality,” they “show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction,” and that the potential $\\phi$ “satisfies the Bowen property on these good orbit segments.” They also show that “the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system.” Then, “as a consequence, there exists a unique equilibrium state on the inverse limit space,” and “its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism.” Finally, they “demonstrate that the conditions of the theorem are inherently robust,” implying “robustness of the equilibrium state,” and note that H\\\"older potentials “can also vary within a certain range,” giving uniqueness “under small perturbation.”",
    "expanded_theorem": "The results of the Main Theorem still hold for $ C^{1+\\alpha} $ mappings. To cite some conclusions in \\cite{Qian} conveniently, we therefore assume $ C^2 $ regularity here.,",
    "theorem_type": [
      "Universal"
    ],
    "mcq": {
      "question": "Let $M$ be a compact Riemannian manifold, let $f:M\\to M$ be a non-degenerate $C^{1+\\alpha}$ partially hyperbolic endomorphism for some $\\alpha>0$ with one-dimensional center bundle, and let $\\phi:M\\to\\mathbb{R}$ be a H\\\"older continuous potential. Assume that the unstable pressure and stable pressure satisfy $P^u(f,\\phi)>P^s(f,\\phi)$, and that the unstable foliation $\\tilde{\\mathcal W}^u$ (defined on the inverse-limit space associated to $f$) is minimal, meaning every unstable leaf is dense. For $a>0$, define\n$$U_a(\\phi):=\\{\\psi\\in C(M):\\|\\phi-\\psi\\|_\\infty<a\\}.$$ \nFor any\n$$0<a<\\frac{P^u(f,\\phi)-P^s(f,\\phi)}{2},$$\nwhich statement holds for every such $f$, $\\phi$, and $a$?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a $C^1$ neighborhood $\\mathcal U$ of $f$ such that for every $g\\in\\mathcal U$ and every $\\psi\\in U_a(\\phi)$, the system $g$ admits a unique equilibrium state for the potential $\\psi$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a $C^1$ neighborhood $\\mathcal U$ of $f$ such that for every $g\\in\\mathcal U$ and every $\\psi\\in U_a(\\phi)$, the system $g$ admits at least one equilibrium state for the potential $\\psi$, and this equilibrium state is unique whenever $P^u(g,\\psi)>P^s(g,\\psi)$."
        },
        {
          "label": "C",
          "text": "There exists a $C^1$ neighborhood $\\mathcal U$ of $f$ such that the system $g$ admits a unique equilibrium state for the potential $\\phi$ for every $g\\in\\mathcal U$."
        },
        {
          "label": "D",
          "text": "There exists a $C^1$ neighborhood $\\mathcal U$ of $f$ such that for every $g\\in\\mathcal U$ and every $\\psi\\in U_a(\\phi)$, the system $g$ admits a unique equilibrium state for the potential $\\psi$ whenever $0<a<P^u(f,\\phi)-P^s(f,\\phi)$."
        },
        {
          "label": "E",
          "text": "For every $\\psi\\in U_a(\\phi)$, there exists a $C^1$ neighborhood $\\mathcal U_\\psi$ of $f$ such that for every $g\\in\\mathcal U_\\psi$, the system $g$ admits a unique equilibrium state for the potential $\\psi$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "robust uniqueness tied to fixed pressure-gap/minimality hypotheses rather than rechecking perturbed pressures",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the uniformity over all nearby potentials $\\psi\\in U_a(\\phi)$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "half-gap bound on allowable oscillation radius",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniform neighborhood independent of the perturbed potential",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or identify choice A. Although the hypotheses are tailored to the target theorem, the correct answer is not directly leaked from the wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem gives the exact hypotheses and asks for the conclusion that holds. Choice A is basically the theorem statement, with distractors formed by small perturbations of it."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to distinguish the exact quantifiers and the sharp half-gap condition from nearby false variants, but the main task is recognizing the precise theorem conclusion rather than generating a new argument."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: they vary uniformity in the neighborhood, dependence on the perturbed potential, and the sharpness of the pressure-gap bound. These align with common failure modes in reading technical statements."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors, but it is close to a direct restatement of the source theorem, so it tests recall/statement parsing more than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.15255v1",
    "paper_link": "http://arxiv.org/abs/2512.15255v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{HI_theorem}\nSuppose $p,q$ satisfy (\\ref{eq_range_q}). Then there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to (\\ref{11}) with integrability deficit $\\delta\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\nabla u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\nabla u|)^\\delta+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\mathit{data},\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\nabla u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\mathit{data}$, $\\delta$, $\\| H(z,|\\nabla u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.",
      "start_pos": 14161,
      "end_pos": 15015,
      "label": "HI_theorem"
    },
    "ref_dict": {
      "def_E": "\\begin{align}\\label{def_E}\n\tE(\\La)=\\left\\{z\\in \\RR^{n+1}: \\Upsilon(z)\\le\\La\\right\\}.\n\\end{align}",
      "def_weak": "\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\RR^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\na u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to \\eqref{11} with integrability deficit $\\de \\in (0,1)$, if\n\\[\n    \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mA(z,\\na u)\\cdot \\na\\varphi\\right)\\,dz\n    =\n    \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\na \\varphi+a(z)|F|^{q-2}F\\cdot \\na\\varphi  \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\RR^N)$.\n\\end{definition}",
      "11": "\\begin{align}\\label{11}\n\tu_t-\\dv\\mA(z,\\na u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}",
      "lem_theta_2": "\\begin{lemma}\\label{lem_theta_2}\n\tSuppose $i\\in \\Theta_2$ and $50U_i\\cap \\Q_2\\ne\\emptyset$. Then there exists $c=c(\\data)$ such that \n    \\[\n    \\rho_2-\\rho_1 \\leq c\\theta^{\\frac\n    {2-q}{p}}r_i.\n    \\]\n\\end{lemma}",
      "eq_range_q": "\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}",
      "HI_theorem": "\\begin{theorem}\\label{HI_theorem}\nSuppose $p,q$ satisfy \\eqref{eq_range_q}. Then there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to \\eqref{11} with integrability deficit $\\de\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\na u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\na u|)^\\de+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\de)}{2-q(1-\\de)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\data,\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\na u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\data$, $\\delta$, $\\| H(z,|\\na u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6649,
    "pre_theorem_intro_text": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nThis approach with intrinsic geometry has also been applied to other classes of nonlinear parabolic PDEs.\nIn the setting of porous medium equations, DiBenedetto, Gianazza and Vespri proved Hölder continuity and Harnack inequalities in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while Gianazza and Schwarzacher obtained gradient higher integrability in \\cite{MR3928041,MR4019094}. \nThe same methodology has been adapted to equations that are nonlinear in both time and spatial derivative parts. \nFor doubly nonlinear equations, higher integrability, Hölder continuity and gradient Hölder continuity have been developed in a sequence of works \\cite{bögelein2025schauderestimatesparabolicplaplace, MR4163123,  MR4287785, MR4603642, MR4500278,MR4373238, MR4797292} by Bögelein, Duzaar, Liao, Moring, Schätzler, Scheven and their collaborators.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n\tu_t-\\dv\\mathcal{A}(z,\\nabla u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n\t\t\\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n      \\quad\\text{and}\\quad \n\t\t|\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n\t\t|a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n\tH(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n    \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n    =\n    \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi  \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.",
    "context": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=\n -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad \n |\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n |a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n H(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n =\n \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.",
    "full_context": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=\n -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad \n |\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n |a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n H(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n =\n \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to \\eqref{11} is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\de = 1$. The constants depend on \n\\[\n\\data = (n,N,p,q,L_1,L_2,[a]_\\al)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\data)$ for a positive constant depending at most on the constants above.\n\n\\item[(W10)]\\label{viii} If $U_i=G_{i}$, then \n $\\tfrac{a(z_i)}{2}\\le a(z)\\le 2a(z_i)$ for every $z\\in 200 Q_{r_i}(z_i)$.\n \\item[(W11)]\\label{ix} For any $i\\in\\mathbb{N}$, the cardinality of $\\mathcal{I}_i$, denoted by $|\\mathcal{I}_i|$, is finite. Moreover, there exists a constant $c= c(n)$, such that $|\\mathcal{I}_i|\\le c$.\n \\end{enumerate}\n Additionally, there exists a partition of unity $\\{\\om_i\\}_{i\\in\\mathbb{N}}$ subordinate to the Whitney decomposition $\\{2U_i\\}_{i\\in \\mathbb{N}}$ with the following properties:\n \\begin{enumerate}[resume*=theoremconditions]\n \\item[(W12)]\\label{x} $0\\le \\om_i\\le 1$, $\\om_i\\in C_0^\\infty(2U_i)$ for every $i\\in\\mathbb{N}$ and $\\sum_{i\\in\\mathbb{N}}\\om_i=1$ on $E(\\La)^c$.\n \\item[(W13)]\\label{xi}There exists a constant $c=c(n)$ such that $\\lVert\\na \\om_j\\rVert_{\\infty}\\le cr_i^{-1}$,\n for every $i\\in\\mathbb{N}$ and $j\\in \\mathcal{I}_i$.\n \\item[(W14)]\\label{xii} There exists a constant $c=c(n)$ such that \n \\[\n \\lVert \\pa_t\\om_j\\rVert_{\\infty}\\le cs_i^{-1}\n \\]\n for every $i\\in\\mathbb{N}$ and $j\\in \\mathcal{I}_i$.\n \\end{enumerate}\n\\end{proposition}\nThe proof of Proposition~\\ref{prop_whitney} is otherwise the same as in \\cite[Proposition~3.2]{KKS}, but there is difference in the proofs of \\ref{UUi} and \\ref{viii}. Thus, we provide only the proofs of \\ref{UUi} and \\ref{viii}.\n\\begin{proof}[Proof of \\ref{UUi}]\n Since we have $Q_i\\subset E(\\La)^c$, there holds\n \\[\nint_{Q_i} \\Upsilon^\\mu\\,dz=\\frac{\\la_i^{p-2}}{2|B_1|r_i^{n+2}} \\iint_{\\mathbb{R}^{n+1}} \\Upsilon^{\\mu}\\,dz.\n \\]\n Multiplying both sides by $\\la_i^{-p\\mu}r_i^{n+2}$ and then taking exponent $\\tfrac{\\alpha}{n+2}$, we get\n \\[\n r_i^\\alpha < K\\la_i^{-\\frac{2\\alpha}{n+2}+\\frac{\\alpha p}{n+2} (1-\\mu)}.\n \\] \n Note that by the choice of $\\mu$ in \\eqref{def_mu} and $\\mfs$ in \\eqref{def_mfs}, it follows that\n \\[\n -\\frac{2\\alpha}{n+2}+\\frac{\\alpha p}{n+2} (1-\\mu)\n<-\\frac{2\\alpha}{n+2}+\\frac{\\alpha q}{n+2} (1-\\mu)<p-q-\\mfs\n \\]\n and therefore,\n \\[\n r_i^\\alpha < K \\la_i^{-\\mfs}\\la_i^{p-q}.\n \\]\n Moreover, since we have $\\theta \\mfc<\\theta\\Pi<\\La<2\\la_i^p$ and \\eqref{def_mfc}, it follows that\n \\[\n r_i^\\alpha < K \\la_i^{-\\mfs}\\la_i^{p-q}\\le K \\left( \\frac{\\theta \\mfc}{2} \\right)^{-\\frac{\\mfs}{p}}\\la_i^{p-q}\\le \\frac{1}{800(1+[a]_\\alpha)}\\la_i^{p-q}.\n \\]\n\\end{proof}\n\\begin{proof}[Proof of \\ref{viii}]\n We claim that\n \\[\n 2[a]_\\alpha (200r_i)^\\alpha < a(z_i).\n \\]\n Once the claim holds true, then we have\n \\[\n 2[a]_\\alpha (200r_i)^\\alpha < a(z_i)< \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\\alpha (200 r_i)^\\alpha.\n \\]\n and therefore,\n \\[\n [a]_\\alpha (200 r_i)^\\alpha< \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)\n \\]\n Moreover, we get\n \\begin{align*}\n \\begin{split}\n \\sup_{z\\in 200 Q_{r_i}(z_i)} a(z)\\le \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\\alpha (200 r_i)^\\alpha< 2\\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)\n \\end{split}\n \\end{align*}\n and thus, the proof is completed.\n To prove the claim, we argue as in the proof of \\ref{UUi}. We have\n \\[\n (a(z_i)\\la_i^{q})^\\mu < \\frac{\\La\\la_i^{-2}}{2|B_1|r_i^{n+2}}\\iint_{\\mathbb{R}^{n+1}}\\Upsilon^\\mu\\,dz<\\frac{a(z_i)\\la_i^{q-2}}{|B_1|r_i^{n+2}}\\iint_{\\mathbb{R}^{n+1}}\\Upsilon^\\mu\\,dz,\n \\]\n where to obtain the last inequality, we used the fact that $\\la_i^p<a(z_i)\\la_i^q$. Multiplying both sides by $(a(z_i)\\la_i^{q})^{-\\mu} r_i^{n+2}$ and then taking exponent $\\tfrac{\\alpha}{n+2}$, we have\n \\[\n r_i^\\alpha < \\la_i^{-\\frac{2\\alpha}{n+2}+\\frac{\\alpha q}{n+2}(1-\\mu)} \\left(\\frac{a(z_i)^{1-\\mu}}{|B_1|} \\iint_{\\mathbb{R}^{n+1}} \\Upsilon^\\mu \\,dz \\right)^\\frac{\\alpha}{n+2}\n <K \\la_i^{-\\mfs} \\la_i^{p-q}.\n \\]\n Again, since $\\theta\\mfc<\\theta\\Pi<\\La\\le 2a(z_i)\\lai^q \\le 2\\|a\\|_\\infty \\lai^q $ and \\eqref{def_mfc} holds, we get\n \\[\n r_i^\\alpha < K\\left( \\frac{\\theta \\mfc}{2\\|a\\|_\\infty} \\right)^{-\\frac{\\mfs}{q}}\\lai^{p-q}<\\frac{1}{800[a]_\\alpha}\\lai^{p-q}.\n \\]\n\n\\subsection{The $p$-intrinsic case}\nWe begin by proving a parabolic Poincar\\'e inequality in the $p$-intrinsic case.\nFirst we estimate the last term in Lemma~\\ref{lem_parabolic_poincare}.\n\\begin{lemma}\\label{sec4:lem:1}\n Let $u$ be a very weak solution to \\eqref{11}. Then, for $\\theta\\in((q-1)/p,\\de]$ and $s\\in[2\\rho,4\\rho]$, \n there exists a constant $c=c(\\data)$ such that\n \\begin{align*}\n \\begin{split}\nint_{Q_{s}^\\pi(z_0)}(|\\na u|^{p-1}+a(z)|\\na u|^{q-1}+|F|^{p-1} + a(z) |F|^{q-1} )\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} (|\\na u| + |F| )^{p-1}\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)}a(z)^\\frac{q-1}{q} ( |\\na u|^{q-1} + |F|^{q-1} )\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} ( |\\na u| + |F| )^{\\theta p}\\,dz\\right)^{\\frac{p-1}{\\theta p}},\n \\end{split}\n \\end{align*}\n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2}.\n\\end{lemma}\n\nWe remark that we have calculated the following in the proof of the previous lemma.\n\\begin{lemma}\\label{coro_p_phase}\n Let $u$ be a very weak solution to \\eqref{11}. Then for $\\theta\\in(\\mu,\\de]$ and $s\\in[2\\rho,4\\rho]$, there exist constants $c=c(\\data)$ and $R_0=R_0(\\data, \\delta ,\\| u\\|_{C_{\\loc}(0,T;L_{\\loc}^2(\\Om,\\mathbb{R}^N))} )\\in (0,1)$ such that\n \\begin{equation*}\n \\begin{split}\nint_{Q_{s}^\\pi(z_0)} H \\left( z, \\frac{|u-u_{Q_{s}^\\pi(z_0)}| }{s} \\right)^\\theta \\,dz\nint_{Q_{s}^\\pi(z_0)}H(z,|\\na u|)^\\theta\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz \\right)^\\frac{\\theta}{\\delta}\n \\end{split}\n \\end{equation*}\n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2} with $\\rho\\in(0,R_0)$.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem_reverse_Hölder}\nLet $u$ be a very weak solution to \\eqref{11}.\nThere exist constants $c=c(\\data)$, $\\delta_2=\\delta_2(\\data)\\in(\\delta_1,1)$ with $2-q(1-\\delta_2)>0$, $\\theta_0=\\theta_0(\\data)\\in(0,\\delta_2) $, $R_0=R_0(\\data, \\delta ,\\| u\\|_{C_{\\loc}(0,T;L_{\\loc}^2(\\Om,\\mathbb{R}^N))} )\\in (0,1)$ such that\n \\begin{align*}\n \\begin{split}\n \\pi^{\\de p}\nint_{Q_{2\\rho}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz,\n \\end{split}\n \\end{align*} \n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2} with $\\rho\\in(0,R_0)$ and $\\delta\\in (\\delta_2,1)$.\n Moreover, we have \n \\begin{align}\\label{eq_revH_Psi}\n \\begin{split}\n \\iint_{Q_{16\\rho}^\\pi(z_0)}H(z,|\\na u|)^\\delta\\,dz\n &\\le c\\Pi^{\\delta-\\theta_0}\\iint_{Q_{2\\rho}^\\pi(z_0)\\cap \\Psi(c^{-1}\\Pi)}H(z,|\\na u|)^{\\theta_0}\\,dz\\\\\n &\\qquad +c \\iint_{Q_{2\\rho}^\\pi(z_0)\\cap \\Phi(c^{-1}\\Pi)}H(z,|F|)^\\delta \\,dz,\n \\end{split}\n \\end{align}\n where $\\Psi(\\cdot)$ and $\\Phi(\\cdot)$ are defined in \\eqref{sec2:2}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem_KKM_4_3_pq}\n Let $u$ be a very weak solution to \\eqref{11}. Then there exists a constant $c=c(n,N,p,q,L_1,L_2)$ such that \n \\begin{align*}\nnt_{B_{2\\rho}(x_0)}\\frac{|u-u_{G_{2\\rho}^\\pi(z_0)}|^2}{(2\\rho)^2\\mh^{1-\\de}}\\,dx\\le c\\pi^2\\Pi^{\\de-1},\n \\end{align*} \n whenever $G_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pqcase}-\\eqref{eq_pq4}.\n\\end{lemma}\n\\begin{proof}\n By Lemma~\\ref{lem_caccioppoli}, there exists a constant $c=c(n,p,q,L_1,L_2)$ such that\n \\begin{equation}\\label{sec5:61}\n \\begin{split}\nnt_{B_{2\\rho}(x_0)}\\frac{|u-u_{G_{2\\rho}^\\pi(z_0)}|^2}{(2\\rho)^2\\mh^{1-\\de}}\\,dx\\\\\nint_{G_{4\\rho}^\\pi(z_0)}\\left(\\frac{|u-u_{G_{4\\rho}^\\pi(z_0)}|^{\\de p}}{(4\\rho)^{\\de p}}+a(z)^\\delta\\frac{|u-u_{G_{4\\rho}^\\pi(z_0)}|^{\\de q}}{(4\\rho)^{\\de q}}\\right)\\,dz\\\\\nint_{G_{4\\rho}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz.\n \\end{split}\n \\end{equation}",
    "post_theorem_intro_text_len": 5230,
    "post_theorem_intro_text": "For weak solutions to elliptic $p$-Laplace systems, gradient higher integrability results go back to Elcrat and Meyers in \\cite{Meyers}. The core principle of the proof is already the same as in our paper. First a Caccioppoli inequality and a Sobolev-Gagliardo-Nirenberg type inequality are used to obtain a reverse Hölder inequality, after which higher integrability follows from an application of Gehring's lemma. Higher integrability result similar to Theorem~\\ref{HI_theorem} for elliptic $p$-Laplace systems was proven for very weak solutions by Iwaniec and Sbordone in \\cite{Iwaniec_Sbordone} and independently by Lewis in \\cite{Lewis_elliptic}. We mention that a conjecture in \\cite{Iwaniec_Sbordone} on minimal integrability deficit for elliptic $p$-Laplace problems was  recently answered by Colombo and Tione in \\cite{colombo_jems}. There is a difficulty in showing the Caccioppoli inequality for very weak solutions, as a very weak solution is not an admissible test function. This problem was overcome in \\cite{Lewis_elliptic} by using a type of Lipschitz truncation based on Whitney decomposition. \n\nTheorem~\\ref{HI_theorem} was proven for parabolic $p$-Laplace systems in \\cite{KL_veryweak}. In the parabolic case, intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse Hölder inequality. Based on \\cite{KL_veryweak}, gradient higher integrability of very weak solutions was extended to variable exponent problem and higher order problem of $p$-Laplace type in \\cite{MR4074602,verena_thesis,bogelein_li,Li}. Furthermore, corresponding arguments have been implemented in the context of Calder\\'on-Zygmund type estimates in \\cite{MR3928653,MR3483172} and applied to establish existence of very weak solutions for elliptic $p$-Laplace problems \\cite{MR3531368,MR3500290} and linear parabolic problems \\cite{MR3985550}. \nThe parabolic Lipschitz truncation method has been considered also in relation to caloric approximations in \\cite{MR3672391,MR4525732} and existence of solutions to fluid models in \\cite{MR2668872}. \n\n     For parabolic double-phase problems higher integrability for weak solutions was shown in \\cite{KKM,KS}. In there, phase analysis is used to distinguish whether the energy estimates of double-phase systems are perturbed into that of $p$-Laplace systems or $(p,q)$-Laplace systems depending on the pointwise value of $a$. Based on the phase division, we construct a corresponding intrinsic geometry.\n Similar phase analysis was used in \\cite{KKS} to obtain standard energy estimates for weak solutions. Although a Lipschitz truncation method was applied in \\cite{KKS}, the argument is different for very weak solutions.\nOther recent result for parabolic double-phase problems include partial regularity in the nondegenerate case in \\cite{MR4774133,kim2025boundedsolutionsinterpolativegap,MR4926910,oh2025gradient,ok2025partialregularityparabolicsystems,sen2025lipschitzregularityparabolicdouble}.\n\nCorresponding result to Theorem~\\ref{HI_theorem} for very weak solutions to the elliptic case was proved in \\cite{BBK}. Obtaining the higher integrability result for very weak solutions to parabolic double-phase problems presents several new challenges. In order to show that there exists $\\de_0$ satisfying Theorem~\\ref{HI_theorem}, the higher integrability gained in the argument has to be quantified and independent from $\\delta$. In \\cite{KKM} the phase analysis in the reverse Hölder inequality argument for weak solutions is done with respect to a constant $K$ depending on the integral of the gradient. Utilizing the same argument for very weak solutions would lead to estimates depending on $\\delta$. To avoid with this problem, we consider cylinders with radii bounded by a constant $R_0$ appearing in Theorem~\\ref{HI_theorem} and use a new phase analysis argument.\n\nAlso in the proof of the Caccioppoli inequality there are novel challenges. In \\cite{KKS} the Lipschitz truncation method is used to prove a Caccioppoli inequality for weak solutions to parabolic double-phase problems. In the argument in \\cite{KKS}, the term $\\Lambda|E(\\Lambda)^c|$ in (\\ref{def_E}) vanishes as $\\Lambda \\to \\infty$. For very weak solutions this does not happen and finer estimates are required. In particular, we have to compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders defined in Section~\\ref{sec_setup}. In our context these cylinders have different geometries and to compare them we provide a new argument in Lemma~\\ref{lem_theta_2}. Our proof of the Caccioppoli inequality is self-contained. To simplify some of the arguments of previous papers, we have included the term $\\tfrac{|u-u_0|}{\\rho}$ in the definition of $E(\\Lambda)$.              \n\nFinally, we note that the range of $q$ in the parabolic double-phase system is not completely understood, however, the strict inequality for the upper bound of $q$ in (\\ref{eq_range_q}) seems to be natural. For the elliptic case, the range of $q$ is well understood and its optimality is known. When we consider very weak solutions in the elliptic setting, the natural integrability of the gradient does not allow the borderline case, see \\cite{BBK} for the details.",
    "sketch": "To prove Theorem~\\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \\emph{Caccioppoli inequality} and combines it with a \\emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \\emph{reverse H\\\"older inequality}; then the higher integrability conclusion follows by applying \\emph{Gehring's lemma}.\n\nFor \\emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \\emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\\\"older inequality.”\n\nIn the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \\(\\delta_0\\) as in Theorem~\\ref{HI_theorem}, “the higher integrability gained in the argument has to be quantified and independent from \\(\\delta\\).” Since using the phase analysis from \\cite{KKM} for very weak solutions “would lead to estimates depending on \\(\\delta\\),” the approach is to “consider cylinders with radii bounded by a constant \\(R_0\\) appearing in Theorem~\\ref{HI_theorem} and use a new phase analysis argument.”\n\nThere are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \\cite{KKS}, the term “\\(\\Lambda|E(\\Lambda)^c|\\) … vanishes as \\(\\Lambda\\to\\infty\\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries,” and this comparison is handled via “a new argument in Lemma~\\ref{lem_theta_2}.” The authors also note a simplification: they “included the term \\(\\tfrac{|u-u_0|}{\\rho}\\) in the definition of \\(E(\\Lambda)\\).”",
    "expanded_sketch": "To prove Theorem~\\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \\emph{Caccioppoli inequality} and combines it with a \\emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \\emph{reverse H\\\"older inequality}; then the higher integrability conclusion follows by applying \\emph{Gehring's lemma}.\n\nFor \\emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \\emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\\\"older inequality.”\n\nIn the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \\(\\delta_0\\) as in the main theorem, “the higher integrability gained in the argument has to be quantified and independent from \\(\\delta\\).” Since using the phase analysis from \\cite{KKM} for very weak solutions “would lead to estimates depending on \\(\\delta\\),” the approach is to “consider cylinders with radii bounded by a constant \\(R_0\\) appearing in the main theorem and use a new phase analysis argument.”\n\nThere are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \\cite{KKS}, the term “\\(\\Lambda|E(\\Lambda)^c|\\) … vanishes as \\(\\Lambda\\to\\infty\\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries.” This comparison is handled via the following lemma.\n\n\\begin{lemma}\\label{lem_theta_2}\n\tSuppose $i\\in \\Theta_2$ and $50U_i\\cap \\Q_2\\ne\\emptyset$. Then there exists $c=c(\\data)$ such that \n    \\[\n    \\rho_2-\\rho_1 \\leq c\\theta^{\\frac\n    {2-q}{p}}r_i.\n    \\]\n\\end{lemma}\n\nThe authors also note a simplification: they “included the term \\(\\tfrac{|u-u_0|}{\\rho}\\) in the definition of \\(E(\\Lambda)\\).”",
    "expanded_theorem": "\\label{HI_theorem}\nSuppose $p,q$ satisfy \n\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\nThen there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to\n\\begin{align}\\label{11}\n\\tu_t-\\dv\\mA(z,\\na u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwith integrability deficit $\\delta\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\nabla u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\nabla u|)^\\delta+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\mathit{data},\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\nabla u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\mathit{data}$, $\\delta$, $\\| H(z,|\\nabla u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.,",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega_T=\\Omega\\times(0,T)\\), and define \\(H(z,s)=s^p+a(z)s^q\\). Assume \\(a\\ge 0\\) belongs to \\(C^{\\alpha,\\alpha/2}(\\Omega_T)\\), \\(\\alpha\\in(0,1]\\), and that \\(\\mathcal A:\\Omega_T\\times\\mathbb R^{Nn}\\to\\mathbb R^{Nn}\\) is a Carath\\'eodory vector field satisfying the double-phase structure conditions\n\\[\n\\mathcal A(z,\\xi)\\cdot \\xi\\ge L_1\\big(|\\xi|^p+a(z)|\\xi|^q\\big),\n\\qquad\n|\\mathcal A(z,\\xi)|\\le L_2\\big(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}\\big)\n\\]\nfor a.e. \\(z\\in\\Omega_T\\) and all \\(\\xi\\in\\mathbb R^{Nn}\\). Suppose also that\n\\[\nq<p+\\frac{2\\alpha}{n+2},\\qquad p>2.\n\\]\nIf \\(u\\) is a very weak solution of\n\\[\nu_t-\\operatorname{div}\\mathcal A(z,\\nabla u)=-\\operatorname{div}\\big(|F|^{p-2}F+a(z)|F|^{q-2}F\\big)\n\\quad\\text{in }\\Omega_T,\n\\]\nwith integrability deficit \\(\\delta\\in(\\delta_0,1)\\) and \\(u\\in C(0,T;L^2(\\Omega,\\mathbb R^N))\\), and where \\(Q_r(z_0)\\) denotes a standard parabolic cylinder, which local higher-integrability estimate holds for every \\(Q_{2R}(z_0)\\subset\\Omega_T\\) with \\(R\\in(0,R_0)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends only on the structural data and \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and one may take \\(R_0=1\\)."
        },
        {
          "label": "C",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\) with \\(R\\in(0,R_0)\\), for some constants \\(c\\ge 1\\) and \\(R_0\\in(0,1)\\)."
        },
        {
          "label": "D",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta)>0\\) for every \\(\\delta\\in(\\delta_0,1)\\), and for every such \\(\\delta\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)^\\delta\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
        },
        {
          "label": "E",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "small-radius restriction and dependence of constants on global \\(L^\\delta\\) data",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit dependence of \\(c\\) and \\(R_0\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "source term appears as \\(H(z,|F|)\\), not \\(H(z,|F|)^\\delta\\)",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "precise Gehring exponent \\(1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}\\)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or uniquely signal A. It states the assumptions and asks for the valid conclusion, without giving away the dependence structure or exact estimate."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially a theorem-statement recognition task: under the listed hypotheses, the student must pick the exact existence/local estimate conclusion. This is very close to a direct restatement rather than an application to a new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants (radius restriction, norm dependence of constants, whether H(z,|F|) or H(z,|F|)^delta appears, and the Gehring-type exponent). However, the task is still mainly identification of the canonical theorem statement rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: over-uniformity in R, omission of dependency data, incorrect source-term integrability, and loss of the higher exponent. They are distinct and well aligned with theorem-misstatement errors."
      },
      "total_score": 5,
      "overall_assessment": "A technically solid MCQ with strong, plausible distractors and no obvious answer leakage, but it mainly tests recall/recognition of a theorem statement rather than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.15255v1",
    "paper_link": "http://arxiv.org/abs/2512.15255v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{HI_theorem}\nSuppose $p,q$ satisfy (\\ref{eq_range_q}). Then there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to (\\ref{11}) with integrability deficit $\\delta\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\nabla u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\nabla u|)^\\delta+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\mathit{data},\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\nabla u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\mathit{data}$, $\\delta$, $\\| H(z,|\\nabla u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.",
      "start_pos": 14161,
      "end_pos": 15015,
      "label": "HI_theorem"
    },
    "ref_dict": {
      "def_E": "\\begin{align}\\label{def_E}\n\tE(\\La)=\\left\\{z\\in \\RR^{n+1}: \\Upsilon(z)\\le\\La\\right\\}.\n\\end{align}",
      "def_weak": "\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\RR^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\na u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to \\eqref{11} with integrability deficit $\\de \\in (0,1)$, if\n\\[\n    \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mA(z,\\na u)\\cdot \\na\\varphi\\right)\\,dz\n    =\n    \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\na \\varphi+a(z)|F|^{q-2}F\\cdot \\na\\varphi  \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\RR^N)$.\n\\end{definition}",
      "11": "\\begin{align}\\label{11}\n\tu_t-\\dv\\mA(z,\\na u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}",
      "lem_theta_2": "\\begin{lemma}\\label{lem_theta_2}\n\tSuppose $i\\in \\Theta_2$ and $50U_i\\cap \\Q_2\\ne\\emptyset$. Then there exists $c=c(\\data)$ such that \n    \\[\n    \\rho_2-\\rho_1 \\leq c\\theta^{\\frac\n    {2-q}{p}}r_i.\n    \\]\n\\end{lemma}",
      "eq_range_q": "\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}",
      "HI_theorem": "\\begin{theorem}\\label{HI_theorem}\nSuppose $p,q$ satisfy \\eqref{eq_range_q}. Then there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to \\eqref{11} with integrability deficit $\\de\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\na u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\na u|)^\\de+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\de)}{2-q(1-\\de)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\data,\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\na u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\data$, $\\delta$, $\\| H(z,|\\na u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6649,
    "pre_theorem_intro_text": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nThis approach with intrinsic geometry has also been applied to other classes of nonlinear parabolic PDEs.\nIn the setting of porous medium equations, DiBenedetto, Gianazza and Vespri proved Hölder continuity and Harnack inequalities in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while Gianazza and Schwarzacher obtained gradient higher integrability in \\cite{MR3928041,MR4019094}. \nThe same methodology has been adapted to equations that are nonlinear in both time and spatial derivative parts. \nFor doubly nonlinear equations, higher integrability, Hölder continuity and gradient Hölder continuity have been developed in a sequence of works \\cite{bögelein2025schauderestimatesparabolicplaplace, MR4163123,  MR4287785, MR4603642, MR4500278,MR4373238, MR4797292} by Bögelein, Duzaar, Liao, Moring, Schätzler, Scheven and their collaborators.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n\tu_t-\\dv\\mathcal{A}(z,\\nabla u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n\t\t\\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n      \\quad\\text{and}\\quad \n\t\t|\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n\t\t|a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n\tH(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n    \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n    =\n    \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi  \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.",
    "context": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=\n -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad \n |\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n |a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n H(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n =\n \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.",
    "full_context": "In this paper, we study parabolic double-phase systems with the prototype equation\n\\[\nu_t-\\dv(|\\nabla u|^{p-2}\\nabla u + a(z)|\\nabla u|^{q-2}\\nabla u)=-\\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),\n\\]\nin $\\Omega_T=\\Omega\\times(0,T)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, and $T>0$. Here $1<p<q<\\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.\n\nIn the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\\equiv 0$) in \\cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed. \nWe refer to DiBenedetto’s book \\cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \\cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \\cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \\cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \\cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \\cite{MR1939688}.\n\nRegarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \\cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \\cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \\cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \\cite{MR3360738,MR3294408}. \nMoreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \\cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \\cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \\cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.\n\nThe exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \\begin{align}\\label{11}\n u_t-\\dv\\mathcal{A}(z,\\nabla u)=\n -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwhere $\\mathcal{A}(z,\\nabla u):\\Omega_T\\times \\mathbb{R}^{Nn}\\longrightarrow \\mathbb{R}^{Nn}$ with $N\\ge1$ is a Carath\\'eodory vector field satisfying the structure assumptions\n\\begin{align}\\label{12}\n \\mathcal{A}(z,\\xi)\\cdot \\xi\\ge L_1(|\\xi|^p+a(z)|\\xi|^q)\n \\quad\\text{and}\\quad \n |\\mathcal{A}(z,\\xi)|\\le L_2(|\\xi|^{p-1}+a(z)|\\xi|^{p-1})\n\\end{align}\nfor a.e. $z\\in \\Omega_T$ and every $\\xi\\in \\mathbb{R}^{Nn}$ with constants $0<L_1\\le L_2<\\infty$. We assume that the coefficient function $a:\\Omega_T\\longrightarrow\\mathbb{R}^+$ is non-negative and satisfies $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ for some $\\alpha \\in (0,1]$. Here $a\\in C^{\\alpha,\\alpha/2}(\\Omega_T)$ means that $a\\in L^{\\infty}(\\Omega_T)$ and there exists a constant $[a]_{\\alpha,\\alpha/2;\\Omega_T}=[a]_\\alpha>0$, such that\n\\begin{align}\\label{eq_holder_reg}\n |a(x,t)-a(y,s)|\\le [a]_{\\alpha,\\alpha/2;\\Omega_T}\\max\\{ |x-y|^\\alpha,|t-s|^\\frac{\\alpha}{2} \\}\n\\end{align}\nfor every $(x,y)\\in\\Omega$ and $(t,s)\\in (0,T)$. \nFor the source term $F$, we assume that $F:\\Om_T\\longrightarrow\\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\\in L^1(\\Om_T)$. Throughout the paper, we denote $H(z,s):\\Omega_T\\times \\mathbb{R}^{+ }\\longrightarrow \\mathbb{R}^+$ as\n\\begin{align*}\n H(z,s)=s^p+a(z)s^q.\n\\end{align*}\nRegarding the exponents, we assume that\n\\begin{align}\\label{eq_range_q}\n q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\n\n\\begin{definition}\\label{def_weak}\nA function $u\\in L^1(0,T;W^{1,1}(\\Omega,\\mathbb{R}^N))$ satisfying\n\\[\n\\iint_{\\Omega_T} H(z,|\\nabla u|)^\\delta\\,dz <\\infty\n\\]\nis a very weak solution to (\\ref{11}) with integrability deficit $\\delta \\in (0,1)$, if\n\\[\n \\iint_{\\Omega_T}\\left(-u\\cdot \\varphi_t+\\mathcal{A}(z,\\nabla u)\\cdot \\nabla\\varphi\\right)\\,dz\n =\n \\iint_{\\Om_T} \\left( |F|^{p-2}F\\cdot \\nabla \\varphi+a(z)|F|^{q-2}F\\cdot \\nabla\\varphi \\right)\\,dz\n\\]\nfor every $\\varphi\\in C_0^\\infty(\\Omega_T,\\mathbb{R}^N)$.\n\\end{definition}\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\\ref{11}) is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\delta = 1$. The constants depend on \n\\[\n\\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\\alpha)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\mathit{data})$ for a positive constant depending at most on the constants above.\n\nOur main theorem states that if the integrability deficit is close enough to 1, a very weak solution to \\eqref{11} is actually a weak solution, i.e., satisfies Definition~\\ref{def_weak} with $\\de = 1$. The constants depend on \n\\[\n\\data = (n,N,p,q,L_1,L_2,[a]_\\al)\n\\]\nand here, as in the rest of the paper, we denote $c=c(\\data)$ for a positive constant depending at most on the constants above.\n\n\\item[(W10)]\\label{viii} If $U_i=G_{i}$, then \n $\\tfrac{a(z_i)}{2}\\le a(z)\\le 2a(z_i)$ for every $z\\in 200 Q_{r_i}(z_i)$.\n \\item[(W11)]\\label{ix} For any $i\\in\\mathbb{N}$, the cardinality of $\\mathcal{I}_i$, denoted by $|\\mathcal{I}_i|$, is finite. Moreover, there exists a constant $c= c(n)$, such that $|\\mathcal{I}_i|\\le c$.\n \\end{enumerate}\n Additionally, there exists a partition of unity $\\{\\om_i\\}_{i\\in\\mathbb{N}}$ subordinate to the Whitney decomposition $\\{2U_i\\}_{i\\in \\mathbb{N}}$ with the following properties:\n \\begin{enumerate}[resume*=theoremconditions]\n \\item[(W12)]\\label{x} $0\\le \\om_i\\le 1$, $\\om_i\\in C_0^\\infty(2U_i)$ for every $i\\in\\mathbb{N}$ and $\\sum_{i\\in\\mathbb{N}}\\om_i=1$ on $E(\\La)^c$.\n \\item[(W13)]\\label{xi}There exists a constant $c=c(n)$ such that $\\lVert\\na \\om_j\\rVert_{\\infty}\\le cr_i^{-1}$,\n for every $i\\in\\mathbb{N}$ and $j\\in \\mathcal{I}_i$.\n \\item[(W14)]\\label{xii} There exists a constant $c=c(n)$ such that \n \\[\n \\lVert \\pa_t\\om_j\\rVert_{\\infty}\\le cs_i^{-1}\n \\]\n for every $i\\in\\mathbb{N}$ and $j\\in \\mathcal{I}_i$.\n \\end{enumerate}\n\\end{proposition}\nThe proof of Proposition~\\ref{prop_whitney} is otherwise the same as in \\cite[Proposition~3.2]{KKS}, but there is difference in the proofs of \\ref{UUi} and \\ref{viii}. Thus, we provide only the proofs of \\ref{UUi} and \\ref{viii}.\n\\begin{proof}[Proof of \\ref{UUi}]\n Since we have $Q_i\\subset E(\\La)^c$, there holds\n \\[\nint_{Q_i} \\Upsilon^\\mu\\,dz=\\frac{\\la_i^{p-2}}{2|B_1|r_i^{n+2}} \\iint_{\\mathbb{R}^{n+1}} \\Upsilon^{\\mu}\\,dz.\n \\]\n Multiplying both sides by $\\la_i^{-p\\mu}r_i^{n+2}$ and then taking exponent $\\tfrac{\\alpha}{n+2}$, we get\n \\[\n r_i^\\alpha < K\\la_i^{-\\frac{2\\alpha}{n+2}+\\frac{\\alpha p}{n+2} (1-\\mu)}.\n \\] \n Note that by the choice of $\\mu$ in \\eqref{def_mu} and $\\mfs$ in \\eqref{def_mfs}, it follows that\n \\[\n -\\frac{2\\alpha}{n+2}+\\frac{\\alpha p}{n+2} (1-\\mu)\n<-\\frac{2\\alpha}{n+2}+\\frac{\\alpha q}{n+2} (1-\\mu)<p-q-\\mfs\n \\]\n and therefore,\n \\[\n r_i^\\alpha < K \\la_i^{-\\mfs}\\la_i^{p-q}.\n \\]\n Moreover, since we have $\\theta \\mfc<\\theta\\Pi<\\La<2\\la_i^p$ and \\eqref{def_mfc}, it follows that\n \\[\n r_i^\\alpha < K \\la_i^{-\\mfs}\\la_i^{p-q}\\le K \\left( \\frac{\\theta \\mfc}{2} \\right)^{-\\frac{\\mfs}{p}}\\la_i^{p-q}\\le \\frac{1}{800(1+[a]_\\alpha)}\\la_i^{p-q}.\n \\]\n\\end{proof}\n\\begin{proof}[Proof of \\ref{viii}]\n We claim that\n \\[\n 2[a]_\\alpha (200r_i)^\\alpha < a(z_i).\n \\]\n Once the claim holds true, then we have\n \\[\n 2[a]_\\alpha (200r_i)^\\alpha < a(z_i)< \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\\alpha (200 r_i)^\\alpha.\n \\]\n and therefore,\n \\[\n [a]_\\alpha (200 r_i)^\\alpha< \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)\n \\]\n Moreover, we get\n \\begin{align*}\n \\begin{split}\n \\sup_{z\\in 200 Q_{r_i}(z_i)} a(z)\\le \\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\\alpha (200 r_i)^\\alpha< 2\\inf_{z\\in 200 Q_{r_i}(z_i)} a(z)\n \\end{split}\n \\end{align*}\n and thus, the proof is completed.\n To prove the claim, we argue as in the proof of \\ref{UUi}. We have\n \\[\n (a(z_i)\\la_i^{q})^\\mu < \\frac{\\La\\la_i^{-2}}{2|B_1|r_i^{n+2}}\\iint_{\\mathbb{R}^{n+1}}\\Upsilon^\\mu\\,dz<\\frac{a(z_i)\\la_i^{q-2}}{|B_1|r_i^{n+2}}\\iint_{\\mathbb{R}^{n+1}}\\Upsilon^\\mu\\,dz,\n \\]\n where to obtain the last inequality, we used the fact that $\\la_i^p<a(z_i)\\la_i^q$. Multiplying both sides by $(a(z_i)\\la_i^{q})^{-\\mu} r_i^{n+2}$ and then taking exponent $\\tfrac{\\alpha}{n+2}$, we have\n \\[\n r_i^\\alpha < \\la_i^{-\\frac{2\\alpha}{n+2}+\\frac{\\alpha q}{n+2}(1-\\mu)} \\left(\\frac{a(z_i)^{1-\\mu}}{|B_1|} \\iint_{\\mathbb{R}^{n+1}} \\Upsilon^\\mu \\,dz \\right)^\\frac{\\alpha}{n+2}\n <K \\la_i^{-\\mfs} \\la_i^{p-q}.\n \\]\n Again, since $\\theta\\mfc<\\theta\\Pi<\\La\\le 2a(z_i)\\lai^q \\le 2\\|a\\|_\\infty \\lai^q $ and \\eqref{def_mfc} holds, we get\n \\[\n r_i^\\alpha < K\\left( \\frac{\\theta \\mfc}{2\\|a\\|_\\infty} \\right)^{-\\frac{\\mfs}{q}}\\lai^{p-q}<\\frac{1}{800[a]_\\alpha}\\lai^{p-q}.\n \\]\n\n\\subsection{The $p$-intrinsic case}\nWe begin by proving a parabolic Poincar\\'e inequality in the $p$-intrinsic case.\nFirst we estimate the last term in Lemma~\\ref{lem_parabolic_poincare}.\n\\begin{lemma}\\label{sec4:lem:1}\n Let $u$ be a very weak solution to \\eqref{11}. Then, for $\\theta\\in((q-1)/p,\\de]$ and $s\\in[2\\rho,4\\rho]$, \n there exists a constant $c=c(\\data)$ such that\n \\begin{align*}\n \\begin{split}\nint_{Q_{s}^\\pi(z_0)}(|\\na u|^{p-1}+a(z)|\\na u|^{q-1}+|F|^{p-1} + a(z) |F|^{q-1} )\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} (|\\na u| + |F| )^{p-1}\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)}a(z)^\\frac{q-1}{q} ( |\\na u|^{q-1} + |F|^{q-1} )\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} ( |\\na u| + |F| )^{\\theta p}\\,dz\\right)^{\\frac{p-1}{\\theta p}},\n \\end{split}\n \\end{align*}\n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2}.\n\\end{lemma}\n\nWe remark that we have calculated the following in the proof of the previous lemma.\n\\begin{lemma}\\label{coro_p_phase}\n Let $u$ be a very weak solution to \\eqref{11}. Then for $\\theta\\in(\\mu,\\de]$ and $s\\in[2\\rho,4\\rho]$, there exist constants $c=c(\\data)$ and $R_0=R_0(\\data, \\delta ,\\| u\\|_{C_{\\loc}(0,T;L_{\\loc}^2(\\Om,\\mathbb{R}^N))} )\\in (0,1)$ such that\n \\begin{equation*}\n \\begin{split}\nint_{Q_{s}^\\pi(z_0)} H \\left( z, \\frac{|u-u_{Q_{s}^\\pi(z_0)}| }{s} \\right)^\\theta \\,dz\nint_{Q_{s}^\\pi(z_0)}H(z,|\\na u|)^\\theta\\,dz\\\\\nint_{Q_{s}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz \\right)^\\frac{\\theta}{\\delta}\n \\end{split}\n \\end{equation*}\n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2} with $\\rho\\in(0,R_0)$.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem_reverse_Hölder}\nLet $u$ be a very weak solution to \\eqref{11}.\nThere exist constants $c=c(\\data)$, $\\delta_2=\\delta_2(\\data)\\in(\\delta_1,1)$ with $2-q(1-\\delta_2)>0$, $\\theta_0=\\theta_0(\\data)\\in(0,\\delta_2) $, $R_0=R_0(\\data, \\delta ,\\| u\\|_{C_{\\loc}(0,T;L_{\\loc}^2(\\Om,\\mathbb{R}^N))} )\\in (0,1)$ such that\n \\begin{align*}\n \\begin{split}\n \\pi^{\\de p}\nint_{Q_{2\\rho}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz,\n \\end{split}\n \\end{align*} \n whenever $Q_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pcase}-\\eqref{eq_p2} with $\\rho\\in(0,R_0)$ and $\\delta\\in (\\delta_2,1)$.\n Moreover, we have \n \\begin{align}\\label{eq_revH_Psi}\n \\begin{split}\n \\iint_{Q_{16\\rho}^\\pi(z_0)}H(z,|\\na u|)^\\delta\\,dz\n &\\le c\\Pi^{\\delta-\\theta_0}\\iint_{Q_{2\\rho}^\\pi(z_0)\\cap \\Psi(c^{-1}\\Pi)}H(z,|\\na u|)^{\\theta_0}\\,dz\\\\\n &\\qquad +c \\iint_{Q_{2\\rho}^\\pi(z_0)\\cap \\Phi(c^{-1}\\Pi)}H(z,|F|)^\\delta \\,dz,\n \\end{split}\n \\end{align}\n where $\\Psi(\\cdot)$ and $\\Phi(\\cdot)$ are defined in \\eqref{sec2:2}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem_KKM_4_3_pq}\n Let $u$ be a very weak solution to \\eqref{11}. Then there exists a constant $c=c(n,N,p,q,L_1,L_2)$ such that \n \\begin{align*}\nnt_{B_{2\\rho}(x_0)}\\frac{|u-u_{G_{2\\rho}^\\pi(z_0)}|^2}{(2\\rho)^2\\mh^{1-\\de}}\\,dx\\le c\\pi^2\\Pi^{\\de-1},\n \\end{align*} \n whenever $G_{16\\rho}^{\\pi}(z_0)\\subset\\Omega_T$ satisfies \\eqref{eq_pqcase}-\\eqref{eq_pq4}.\n\\end{lemma}\n\\begin{proof}\n By Lemma~\\ref{lem_caccioppoli}, there exists a constant $c=c(n,p,q,L_1,L_2)$ such that\n \\begin{equation}\\label{sec5:61}\n \\begin{split}\nnt_{B_{2\\rho}(x_0)}\\frac{|u-u_{G_{2\\rho}^\\pi(z_0)}|^2}{(2\\rho)^2\\mh^{1-\\de}}\\,dx\\\\\nint_{G_{4\\rho}^\\pi(z_0)}\\left(\\frac{|u-u_{G_{4\\rho}^\\pi(z_0)}|^{\\de p}}{(4\\rho)^{\\de p}}+a(z)^\\delta\\frac{|u-u_{G_{4\\rho}^\\pi(z_0)}|^{\\de q}}{(4\\rho)^{\\de q}}\\right)\\,dz\\\\\nint_{G_{4\\rho}^\\pi(z_0)} H(z,|F|)^\\delta \\,dz.\n \\end{split}\n \\end{equation}",
    "post_theorem_intro_text_len": 5230,
    "post_theorem_intro_text": "For weak solutions to elliptic $p$-Laplace systems, gradient higher integrability results go back to Elcrat and Meyers in \\cite{Meyers}. The core principle of the proof is already the same as in our paper. First a Caccioppoli inequality and a Sobolev-Gagliardo-Nirenberg type inequality are used to obtain a reverse Hölder inequality, after which higher integrability follows from an application of Gehring's lemma. Higher integrability result similar to Theorem~\\ref{HI_theorem} for elliptic $p$-Laplace systems was proven for very weak solutions by Iwaniec and Sbordone in \\cite{Iwaniec_Sbordone} and independently by Lewis in \\cite{Lewis_elliptic}. We mention that a conjecture in \\cite{Iwaniec_Sbordone} on minimal integrability deficit for elliptic $p$-Laplace problems was  recently answered by Colombo and Tione in \\cite{colombo_jems}. There is a difficulty in showing the Caccioppoli inequality for very weak solutions, as a very weak solution is not an admissible test function. This problem was overcome in \\cite{Lewis_elliptic} by using a type of Lipschitz truncation based on Whitney decomposition. \n\nTheorem~\\ref{HI_theorem} was proven for parabolic $p$-Laplace systems in \\cite{KL_veryweak}. In the parabolic case, intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse Hölder inequality. Based on \\cite{KL_veryweak}, gradient higher integrability of very weak solutions was extended to variable exponent problem and higher order problem of $p$-Laplace type in \\cite{MR4074602,verena_thesis,bogelein_li,Li}. Furthermore, corresponding arguments have been implemented in the context of Calder\\'on-Zygmund type estimates in \\cite{MR3928653,MR3483172} and applied to establish existence of very weak solutions for elliptic $p$-Laplace problems \\cite{MR3531368,MR3500290} and linear parabolic problems \\cite{MR3985550}. \nThe parabolic Lipschitz truncation method has been considered also in relation to caloric approximations in \\cite{MR3672391,MR4525732} and existence of solutions to fluid models in \\cite{MR2668872}. \n\n     For parabolic double-phase problems higher integrability for weak solutions was shown in \\cite{KKM,KS}. In there, phase analysis is used to distinguish whether the energy estimates of double-phase systems are perturbed into that of $p$-Laplace systems or $(p,q)$-Laplace systems depending on the pointwise value of $a$. Based on the phase division, we construct a corresponding intrinsic geometry.\n Similar phase analysis was used in \\cite{KKS} to obtain standard energy estimates for weak solutions. Although a Lipschitz truncation method was applied in \\cite{KKS}, the argument is different for very weak solutions.\nOther recent result for parabolic double-phase problems include partial regularity in the nondegenerate case in \\cite{MR4774133,kim2025boundedsolutionsinterpolativegap,MR4926910,oh2025gradient,ok2025partialregularityparabolicsystems,sen2025lipschitzregularityparabolicdouble}.\n\nCorresponding result to Theorem~\\ref{HI_theorem} for very weak solutions to the elliptic case was proved in \\cite{BBK}. Obtaining the higher integrability result for very weak solutions to parabolic double-phase problems presents several new challenges. In order to show that there exists $\\de_0$ satisfying Theorem~\\ref{HI_theorem}, the higher integrability gained in the argument has to be quantified and independent from $\\delta$. In \\cite{KKM} the phase analysis in the reverse Hölder inequality argument for weak solutions is done with respect to a constant $K$ depending on the integral of the gradient. Utilizing the same argument for very weak solutions would lead to estimates depending on $\\delta$. To avoid with this problem, we consider cylinders with radii bounded by a constant $R_0$ appearing in Theorem~\\ref{HI_theorem} and use a new phase analysis argument.\n\nAlso in the proof of the Caccioppoli inequality there are novel challenges. In \\cite{KKS} the Lipschitz truncation method is used to prove a Caccioppoli inequality for weak solutions to parabolic double-phase problems. In the argument in \\cite{KKS}, the term $\\Lambda|E(\\Lambda)^c|$ in (\\ref{def_E}) vanishes as $\\Lambda \\to \\infty$. For very weak solutions this does not happen and finer estimates are required. In particular, we have to compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders defined in Section~\\ref{sec_setup}. In our context these cylinders have different geometries and to compare them we provide a new argument in Lemma~\\ref{lem_theta_2}. Our proof of the Caccioppoli inequality is self-contained. To simplify some of the arguments of previous papers, we have included the term $\\tfrac{|u-u_0|}{\\rho}$ in the definition of $E(\\Lambda)$.              \n\nFinally, we note that the range of $q$ in the parabolic double-phase system is not completely understood, however, the strict inequality for the upper bound of $q$ in (\\ref{eq_range_q}) seems to be natural. For the elliptic case, the range of $q$ is well understood and its optimality is known. When we consider very weak solutions in the elliptic setting, the natural integrability of the gradient does not allow the borderline case, see \\cite{BBK} for the details.",
    "sketch": "To prove Theorem~\\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \\emph{Caccioppoli inequality} and combines it with a \\emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \\emph{reverse H\\\"older inequality}; then the higher integrability conclusion follows by applying \\emph{Gehring's lemma}.\n\nFor \\emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \\emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\\\"older inequality.”\n\nIn the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \\(\\delta_0\\) as in Theorem~\\ref{HI_theorem}, “the higher integrability gained in the argument has to be quantified and independent from \\(\\delta\\).” Since using the phase analysis from \\cite{KKM} for very weak solutions “would lead to estimates depending on \\(\\delta\\),” the approach is to “consider cylinders with radii bounded by a constant \\(R_0\\) appearing in Theorem~\\ref{HI_theorem} and use a new phase analysis argument.”\n\nThere are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \\cite{KKS}, the term “\\(\\Lambda|E(\\Lambda)^c|\\) … vanishes as \\(\\Lambda\\to\\infty\\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries,” and this comparison is handled via “a new argument in Lemma~\\ref{lem_theta_2}.” The authors also note a simplification: they “included the term \\(\\tfrac{|u-u_0|}{\\rho}\\) in the definition of \\(E(\\Lambda)\\).”",
    "expanded_sketch": "To prove Theorem~\\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \\emph{Caccioppoli inequality} and combines it with a \\emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \\emph{reverse H\\\"older inequality}; then the higher integrability conclusion follows by applying \\emph{Gehring's lemma}.\n\nFor \\emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \\emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\\\"older inequality.”\n\nIn the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \\(\\delta_0\\) as in the main theorem, “the higher integrability gained in the argument has to be quantified and independent from \\(\\delta\\).” Since using the phase analysis from \\cite{KKM} for very weak solutions “would lead to estimates depending on \\(\\delta\\),” the approach is to “consider cylinders with radii bounded by a constant \\(R_0\\) appearing in the main theorem and use a new phase analysis argument.”\n\nThere are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \\cite{KKS}, the term “\\(\\Lambda|E(\\Lambda)^c|\\) … vanishes as \\(\\Lambda\\to\\infty\\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries.” This comparison is handled via the following lemma.\n\n\\begin{lemma}\\label{lem_theta_2}\n\tSuppose $i\\in \\Theta_2$ and $50U_i\\cap \\Q_2\\ne\\emptyset$. Then there exists $c=c(\\data)$ such that \n    \\[\n    \\rho_2-\\rho_1 \\leq c\\theta^{\\frac\n    {2-q}{p}}r_i.\n    \\]\n\\end{lemma}\n\nThe authors also note a simplification: they “included the term \\(\\tfrac{|u-u_0|}{\\rho}\\) in the definition of \\(E(\\Lambda)\\).”",
    "expanded_theorem": "\\label{HI_theorem}\nSuppose $p,q$ satisfy \n\\begin{align}\\label{eq_range_q}\n    q < p+\\frac{2\\alpha}{n+2} \\quad \\text{and} \\quad p>2. \n\\end{align}\nThen there exists $\\de_0=\\de_0(\\mathit{data})\\in(0,1)$ with $2-q(1-\\delta_0)>0$\nsuch that if $u$ is a very weak solution to\n\\begin{align}\\label{11}\n\\tu_t-\\dv\\mA(z,\\na u)=\n    -\\dv (|F|^{p-2}F+a(z)|F|^{q-2}F) \n    \\quad\\text{in}\\ \\Omega_T,\n\\end{align}\nwith integrability deficit $\\delta\\in(\\de_0,1)$ and $u\\in C(0,T;L^2(\\Omega_T,\\mathbb{R}^N))$, we have \n\t\\begin{align*}\n\t\t\\begin{split}\nint_{Q_{R} (z_0) }H(z,|\\nabla u|)\\,dz \nint_{Q_{2R} (z_0) } (  H(z,|\\nabla u|)^\\delta+ H(z,|F|) ) \\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\t\t\\end{split}\n\t\\end{align*}\n\t for every $Q_{2R}\\subset\\Om_T$ where $c\\ge\n     1$ depending on $\\mathit{data},\\|a\\|_{L^\\infty(\\Om_T)},\\| (H(z,|\\nabla u|)+H(z,|F|)+1 )\\|_{L^\\delta(\\Om_T)}$, and $R\\in(0,R_0)$ for $R_0\\in(0,1)$ depending on $\\mathit{data}$, $\\delta$, $\\| H(z,|\\nabla u|) \\|_{L^\\delta(\\Om_T)}$, $\\| H(z,|F|) \\|_{L^\\delta(\\Om_T)}$, $\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb{R}^N))}$.,",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(\\Omega_T=\\Omega\\times(0,T)\\), and define \\(H(z,s)=s^p+a(z)s^q\\). Assume \\(a\\ge 0\\) belongs to \\(C^{\\alpha,\\alpha/2}(\\Omega_T)\\), \\(\\alpha\\in(0,1]\\), and that \\(\\mathcal A:\\Omega_T\\times\\mathbb R^{Nn}\\to\\mathbb R^{Nn}\\) is a Carath\\'eodory vector field satisfying the double-phase structure conditions\n\\[\n\\mathcal A(z,\\xi)\\cdot \\xi\\ge L_1\\big(|\\xi|^p+a(z)|\\xi|^q\\big),\n\\qquad\n|\\mathcal A(z,\\xi)|\\le L_2\\big(|\\xi|^{p-1}+a(z)|\\xi|^{q-1}\\big)\n\\]\nfor a.e. \\(z\\in\\Omega_T\\) and all \\(\\xi\\in\\mathbb R^{Nn}\\). Suppose also that\n\\[\nq<p+\\frac{2\\alpha}{n+2},\\qquad p>2.\n\\]\nIf \\(u\\) is a very weak solution of\n\\[\nu_t-\\operatorname{div}\\mathcal A(z,\\nabla u)=-\\operatorname{div}\\big(|F|^{p-2}F+a(z)|F|^{q-2}F\\big)\n\\quad\\text{in }\\Omega_T,\n\\]\nwith integrability deficit \\(\\delta\\in(\\delta_0,1)\\) and \\(u\\in C(0,T;L^2(\\Omega,\\mathbb R^N))\\), and where \\(Q_r(z_0)\\) denotes a standard parabolic cylinder, which local higher-integrability estimate holds for every \\(Q_{2R}(z_0)\\subset\\Omega_T\\) with \\(R\\in(0,R_0)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends only on the structural data and \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and one may take \\(R_0=1\\)."
        },
        {
          "label": "C",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\) with \\(R\\in(0,R_0)\\), for some constants \\(c\\ge 1\\) and \\(R_0\\in(0,1)\\)."
        },
        {
          "label": "D",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta)>0\\) for every \\(\\delta\\in(\\delta_0,1)\\), and for every such \\(\\delta\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)^\\delta\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
        },
        {
          "label": "E",
          "text": "There exists \\(\\delta_0=\\delta_0(n,N,p,q,L_1,L_2,[a]_{\\alpha,\\alpha/2})\\in(0,1)\\) such that \\(2-q(1-\\delta_0)>0\\), and for every \\(\\delta\\in(\\delta_0,1)\\), if \\(u\\) is as above, then\n\\[\n\\fint_{Q_R(z_0)} H(z,|\\nabla u|)\\,dz\n\\le c\\left(\\fint_{Q_{2R}(z_0)}\\big(H(z,|\\nabla u|)^\\delta+H(z,|F|)\\big)\\,dz+1\\right)^{1+\\frac{q(1-\\delta)}{2}}\n\\]\nfor every \\(Q_{2R}(z_0)\\subset\\Omega_T\\), where \\(c\\ge 1\\) depends on the structural data, \\(\\|a\\|_{L^\\infty(\\Omega_T)}\\), and \\(\\|H(z,|\\nabla u|)+H(z,|F|)+1\\|_{L^\\delta(\\Omega_T)}\\), and \\(R_0\\in(0,1)\\) depends on the structural data, \\(\\delta\\), \\(\\|H(z,|\\nabla u|)\\|_{L^\\delta(\\Omega_T)}\\), \\(\\|H(z,|F|)\\|_{L^\\delta(\\Omega_T)}\\), and \\(\\|u\\|_{C(0,T;L^2(\\Omega,\\mathbb R^N))}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "small-radius restriction and dependence of constants on global \\(L^\\delta\\) data",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit dependence of \\(c\\) and \\(R_0\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "source term appears as \\(H(z,|F|)\\), not \\(H(z,|F|)^\\delta\\)",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "precise Gehring exponent \\(1+\\frac{q(1-\\delta)}{2-q(1-\\delta)}\\)",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives assumptions and asks for the resulting local estimate, but it does not explicitly state the conclusion or uniquely reveal the correct option. The answer must be identified from the choices."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the assumptions are listed and the task is to select the exact theorem conclusion. It is very close to a direct restatement rather than an independent problem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but important ways (dependence of constants, radius restriction, source-term exponent, Gehring exponent, weaker-vs-sharper statement). However, the item mainly tests recognition of the precise theorem statement rather than generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are mathematically nearby variants involving common failure modes such as omitting small-scale restrictions, weakening dependence statements, altering the source-term integrability, or changing the precise exponent. They are distinct and plausible."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-identification MCQ with strong distractors and little answer leakage, but it is largely tautological and tests precise recall/comparison more than genuine generative reasoning."
    }
  },
  {
    "id": "2512.15307v1",
    "paper_link": "http://arxiv.org/abs/2512.15307v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP \\eqref{kdV} admits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}",
      "start_pos": 16368,
      "end_pos": 17220,
      "label": "main"
    },
    "ref_dict": {
      "kdV": "\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}",
      "fig:network": "\\label{fig:network}\n\t\t\t\\caption{Network with $N$ edges}\t\n\t\\end{figure}\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n\t \\begin{itemize}\n\t \\item[i.] The",
      "main": "\\begin{theorem}\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP \\eqref{kdV} admits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 11439,
    "pre_theorem_intro_text": "In mathematics and physics, a quantum graph is a network composed of vertices connected by edges, where each edge carries a differential or pseudo-differential equation. When every edge is endowed with a natural metric, the structure is referred to as a metric graph. A common illustration is a power network: the transmission lines act as edges connecting transformer stations (vertices). In such a case, the governing differential equations may describe the voltage along each line, with boundary conditions at the vertices ensuring that the total current flowing into and out of each vertex sums to zero.\n\nThe concept of quantum graphs traces back to the work of Linus Pauling in the 1930s, who used them to model free electrons in organic molecules. Since then, they have emerged in a wide range of contexts, including quantum chaos, waveguide analysis, photonic crystals, and Anderson localization—the suppression of wave propagation in disordered media. Quantum graphs also arise as limit models of systems composed of thin wires and serve as important theoretical tools in mesoscopic physics and nanotechnology. A simplified formulation of quantum graphs was later proposed by Freedman \\textit{et al.} in \\cite{Freedman}.\n\nBeyond solving the differential equations in specific applications, the study of quantum graphs encompasses fundamental questions such as well-posedness, controllability, and identifiability. For example, controllability concerns determining the inputs required to steer the system toward a desired configuration, such as ensuring stable power distribution across a grid. Identifiability, on the other hand, focuses on optimal measurement placement—for instance, choosing pressure gauges in a water network to detect and locate potential leaks.\n\n\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nOther nonlinear dispersive systems on graphs have also been the subject of extensive study. For example, in the context of well-posedness theory, Cavalcante \\cite{Cav} established local well-posedness for the Cauchy problem associated with the Korteweg–de Vries (KdV) equation on a metric star graph composed of one negative and two positive half-lines joined at a common vertex $\\nu = 0$ (the so-called $\\mathcal{Y}$-junction).\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nSignificant progress has also been achieved in the fields of control theory and inverse problems on networks. Ignat \\textit{et al.} \\cite{Ignat2011} examined inverse problems for the heat and Schrödinger equations on tree-like structures, while Baudouin and Yamamoto \\cite{Baudouin} proposed a unified and simplified framework for coefficient identification problems. Furthermore, Ammari and Crépeau \\cite{Ammari and Crepeau 2018}, Cerpa \\textit{et al.} \\cite{Cerpa, Cerpa1}, and Parada \\textit{et al.} \\cite{Parada2022a, Parada2022b} established stabilization and boundary controllability results for the KdV equation on star-shaped graphs, contributing substantially to the control-theoretic understanding of dispersive models on networks. Recently, in \\cite{CapistranoParadaDaSilva25}, we studied the analytical structure of the entire functions associated with the KdV operator, addressing control problems involving mixed Dirichlet–Neumann boundary controls.\n\n\\subsection{Setting of problem and functional framework} In the context of graph structure, let us consider the following: given $N\\geq 2$, consider a star-shaped network $\\mathcal{T}$ with $N$ edges described by intervals $I_j=(0,l_j)$, $l_j>0$ for $j=1,...,N$. Denoting by $e_1,...,e_N$ the edges of $\\mathcal{T}$ one has $\\mathcal{T}=\\bigcup_{j=1}^Ne_j$. Here, we study the local  well-posedness for a system composed of $N$ nonlinear KdV equations posed on $\\mathcal{T}$ and coupled by the boundary, namely\n\t\t\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\nwhere $u=(u_1,...,u_N)$ stands the state of the system, $p_j$ and $g_j$ are the controls inputs and $\\alpha>\\frac{N}{2}$. We represent this situation in the figure \\eqref{fig:network}. \n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw(0, 0) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.02, 0.09) -- (0.5, 1.95);\n\t\t\t\t\\draw[fill=black](0.53, 2.04) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.07, 0.07) -- (1.33, 1.5);\n\t\t\t\t\\draw[fill=black](1.39, 1.57) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.1, 0.04) -- (1.85, 0.78);\n\t\t\t\t\\draw[fill=black](1.93, 0.815) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.08, -0.04) -- (1.75, -1);\n\t\t\t\t\\draw[fill=black](1.825, -1.045) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.05, -0.09) -- (1.02, -1.73);\n\t\t\t\t\\draw[fill=black](1.07, -1.8) circle (1mm);\n\t\t\t\t\\draw[fill=black](1.45, 0.2) circle (0.3mm);\n\t\t\t\t\\draw[fill=black](1.45, 0.05) circle (0.3mm);\n\t\t\t\t\\draw[fill=black](1.45, -0.1) circle (0.3mm);\n\t\t\t\t\\draw node (no1) at (-0.3,0) {$0$};\n\t\t\t\t\\draw node (no1) at (0.19, 1.7) {$e_1$};\n\t\t\t\t\\draw node (no1) at (0.86, 1.35) {$e_2$};\n\t\t\t\t\\draw node (no1) at (1.35, 0.8) {$e_3$};\n\t\t\t\t\\draw node (no1) at (1.7, -0.6) {$e_{_{N-1}}$};\n\t\t\t\t\\draw node (no1) at (1.06, -1.25) {$e_{_N}$};\n\t\t\t\t\\draw(3.95,-0.5) circle (1mm);\n\t\t\t\t\\draw node (no1) at (6.4,-0.5) {{\\footnotesize \\hspace{10mm} internal controlled node}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ };\n\t\t\t\t\\draw[fill=black](3.95,-1) circle (1mm);\n\t\t\t\t\\draw node (no1) at (6.4,-1) {{\\footnotesize \\hspace{13mm} external controlled node}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ };\n\t\t\t\\end{tikzpicture}\n\t\t\t\\label{fig:network}\n\t\t\t\\caption{Network with $N$ edges}\t\n\t\\end{figure}\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n\t \\begin{itemize}\n\t \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n\tand initial data is $u^0=(u^0_1,...,u^0_N)$. \n\t \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by  $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n\tMoreover,  $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n\t \\end{itemize}\n\t\\begin{itemize}\n\t\t\\item[iii.] Define also the following spaces:\n\t\t$$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n\t\tand\n\t\t$$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n\t\twhere $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n\t\t$$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n\t\t\\item[iv.] Consider the following characterization: \n\t\t$$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n\t\tas the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n\tC^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n\tL^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in \tC^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.",
    "context": "\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n \\begin{itemize}\n \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n and initial data is $u^0=(u^0_1,...,u^0_N)$. \n \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n Moreover, $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n \\end{itemize}\n \\begin{itemize}\n \\item[iii.] Define also the following spaces:\n $$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n and\n $$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n where $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n $$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n \\item[iv.] Consider the following characterization: \n $$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.",
    "full_context": "\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n \\begin{itemize}\n \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n and initial data is $u^0=(u^0_1,...,u^0_N)$. \n \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n Moreover, $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n \\end{itemize}\n \\begin{itemize}\n \\item[iii.] Define also the following spaces:\n $$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n and\n $$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n where $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n $$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n \\item[iv.] Consider the following characterization: \n $$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.\n\n\\begin{lemma}\\label{lemma for v}\n Let $(u^0,g_0, g,p)\\in \\mathbb{X}_{3,T}$, $f\\in W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))$ and $v$ the solution of \\eqref{v_j}. Then for $\\beta=0,1,2,3$ we have:\n \\begin{enumerate}\n \\item[i.] The function\n \\begin{align*}\n \\begin{array}{rcl}\n [0,T]&\\rightarrow&\\mathcal{D}'(0,l_j)\\\\\n s&\\mapsto&\\partial_x^\\beta v_j(s,\\cdot)\n \\end{array}\n \\end{align*}\n is integrable in $[0,T]$.\n \\item[ii.]For any $t \\in [0,T]$ we have $\\displaystyle\\int_0^tv_j(s,\\cdot)ds \\in H^4(0,l_j)$ and for $\\beta=0,1,2,3$ we have\n \\begin{align*}\n \\partial_x^\\beta\\int_0^tv_j(s,\\cdot)ds=\\int_0^t\\partial_x^\\beta v_j(s,\\cdot)ds.\n \\end{align*}\n \\end{enumerate}\n\\end{lemma}\nThe previous lemma can give us the regularity for $u$.\n\\begin{proposition}\\label{W.P._s=3}\nLet $(u^0,g_0,g,p) \\in \\mathbb{H}^3(\\mathcal{T})\\times H^{\\frac{3-1}{3}}(0,T)\\times [H^\\frac{3}{3}(0,T)]^N\\times [H^{\\frac{3+1}{3}}(0,T)]^N$ satisfying the linear compatibility conditions for $s=3$ and $f\\in W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))$ be. For $j=1,...,N$ the function $t\\mapsto u_j(t,\\cdot)$ given by \\eqref{u_j} satisfies the following items:\n\\begin{enumerate}\n \\item[i.] $u_j \\in C^1([0,T],L^2(0,l_j))\\cap C([0,T];H^3(0,l_j))$;\n \\item[ii.] $u\\in \\mathbb{B}_{3,T}=C([0,T];\\mathbb{H}^3(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^4(\\mathcal{T}))$ is the unique classical solution to the problem \\eqref{LkdV}; and\n \\begin{equation*}\n \\begin{split}\n \\|u\\|_{\\mathbb{B}_{3,T}}\\leq &C\\left( \\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\\n &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right)\n\\end{split}\n \\end{equation*}\n for some constant $C>0$.\n \\item[iii.] $u \\in H^1(0,T;\\mathbb{H}^1(\\mathcal{T}))$ and\n \\begin{equation*}\n \\begin{split}\n \\|u\\|_{H^1(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\ &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right),\n \\end{split}\n \\end{equation*}\n for some constant $C>0$.\n \\item [iv.] $\\partial_x^ku_j\\in L_x^\\infty\\left([0,l_j];H^\\frac{3+1-k}{3}(0,T)\\right)$ and \n \\begin{align*}\n \\|\\partial_x^ku_j\\|_{L_x^\\infty\\left([0,l_j];H^\\frac{3+1-k}{3}(0,T)\\right)}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\ &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right).\n \\end{align*}\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nConsider $j \\in\\{1,...,N\\}$. \n\\vspace{0.2 cm}\n\\noindent\\textbf{i.} Since $v_j\\in C([0,T],L^2(0,l_j))$ it is immediate that by \\eqref{u_j} that $u_j \\in C^{1}([0,T],L^2(0,l_j))$.\n\n\\section{Local well-posedness of the nonlinear problem}\\label{sec4}\nNow, we are in a position to prove the first main result of the article, namely, the local well-posedness of the following problem\n\\begin{align}\\label{NLKdV}\n \\left\\{\n \\begin{array}{ll}\n \\partial_t u_j+\\partial_x [(a_j+1)u_j]+\\partial_x^3 u_j+u_j\\partial_x u_j=f_j,&t \\in (0,T),\\ x \\in (0,l_j),\\\\\n u_j(t,0)=u_1(t,0),&t\\in (0,T),\\\\\n \\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \\in (0,T)\\\\ \n u_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\\\\n u_j(0,x)=u_j^0(x),&x \\in (0,l_j),\n \\end{array}\n \\right.\n\\end{align}\nwhere $j=1,...,N$ and $a$ is a given function. Remember that: for $s\\geq 0$ and $T>0$ the space\n\\begin{align*}\n\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}_{s,T}^*;\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\}\n\\end{align*}\nis a Banach space with the norm\n\\begin{align*}\n\\|v\\|_{\\mathcal{X}_{s,T}}=\\|v\\|_{\\mathbb{B}_{s,T}^*}+\\sum_{k=0}^2\\|\\partial_x^kv\\|_{\\prod_{j=1}^NL_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T))}.\n\\end{align*}\nNote that \\eqref{kdV} is a particular case of \\eqref{NLKdV} with $a=0$.\n\\begin{proof}[Proof of Theorem \\ref{main}] Fix $\\theta \\in (0,T)$ and define $\\Gamma_a:\\mathcal{X}_{s,\\theta}\\rightarrow \\mathcal{X}_{s,\\theta}$ putting, for each $u \\in \\mathcal{X}_{s,\\theta}$, $\\Gamma_au$ as being the solution $v$ of\n\\begin{align*}\n \\left\\{\n \\begin{array}{ll}\n \\partial_t v_j+\\partial_x [(a_j+1)v_j]+\\partial_x^3 v_j=-u_j\\partial_x u_j+f_j,&t \\in (0,\\theta),\\ x \\in (0,l_j),\\ j=1,...,N,\\\\\n v_j(t,0)=v_1(t,0),&t\\in (0,\\theta),\\ j=1,...,N,\\\\\n \\displaystyle\\sum_{j=1}^N\\partial_x^2 v_j(t,0)=-\\alpha v_1(t,0)-\\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \\in (0,\\theta,)\\\\ \n v_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x v_j(t,l_j)=g_j(t),&t\\in (0,\\theta),\\ j=1,...,N,\\\\\n v_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N.\n \\end{array}\n \\right.\n\\end{align*}\nFrom Proposition \\ref{W.P.LKdV-a} we have\n\\begin{align*}\n& \\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0-\\frac{N}{3}(u_1+a_1)^2(\\cdot,0)\\|_{H^\\frac{s-1}{3}(0,\\theta)}+\\|g\\|_{[H^\\frac{s}{3}(0,\\theta)]^N}\\right.\\\\\n&\\left.+\\|p\\|_{[H^\\frac{s+1}{3}(0,\\theta)]^N}+\\|u\\partial_xu\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}+\\|f\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))} \\right).\n\\end{align*}\nUsing Lemma \\ref{lemma 2.4} we obtain\n\\begin{align*}\n \\|\\partial_xu v\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\leq C(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})\\|u\\|_{\\mathbb{B}_{s,\\theta}^*}\\|u\\|_{\\mathbb{B}_{s,\\theta}^*}.\n\\end{align*}\nFurthermore, Lemma \\ref{lemma 2.1} gives us\n\\begin{align*}\n\\left\\|\\frac{N}{3}(u_1+a_1)^2(\\cdot,0)\\right\\|_{H^\\frac{s-1}{3}(0,\\theta)}&\\leq C\\theta^\\alpha\\frac{N}{3}\\left\\|(u_1+a_1)(\\cdot,0)\\right\\|_{H^\\frac{s+1}{3}(0,\\theta)}^2\\\\\n&\\leq C\\theta^\\alpha\\frac{N}{3}\\|u+a\\|_{\\mathcal{X}_{s,\\theta}}^2\\\\\n&\\leq \\frac{4}{3}CN\\theta^\\alpha \\left(\\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\|a\\|_{\\mathcal{X}_{s,\\theta}}^2\\right),\n\\end{align*}\nwhere $\\alpha>0$ is a constant. Thus,\n\\begin{align*}\n \\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}\\right.\\\\\n&+\\|p\\|_{[H^\\frac{s+1}{3}(0,T)]^N}+\\|f\\|_{W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\\\\n&\\left.+C(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})\\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\frac{4}{3}CN\\theta^\\alpha \\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\frac{4}{3}CN\\theta^\\alpha\\|a\\|_{\\mathcal{X}_{s,\\theta}}^2 \\right).\n\\end{align*}\nDefining\n\\begin{align*}\nr=&4C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}\\right.\\\\\n&\\left.+\\|p\\|_{[H^\\frac{s+1}{3}(0,T)]^N}+\\|f\\|_{W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\right),\n\\end{align*}\nit follows that, for $u \\in B=\\{v \\in \\mathcal{X}_{s,\\theta};\\ \\|v\\|_{\\mathcal{X}_{s,\\theta}}\\leq r\\}$,\n\\begin{align}\\label{eq 3.2}\n\\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq \\frac{r}{4}+C^2(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})r^2+\\frac{4}{3}C^2N\\theta^\\alpha r^2+\\frac{4}{3}C^2N\\theta^\\alpha\\|a\\|_{\\mathcal{X}_{s,T}}^2.\n\\end{align}",
    "post_theorem_intro_text_len": 2359,
    "post_theorem_intro_text": "The proof of the previous theorem relies on several analytical tools combined with a fixed-point argument. In addition, a new notion of $s$-compatibility adapted to the graph structure is introduced (see Section \\ref{sec2}), inspired by the single-edge KdV framework in \\cite{BSZ 2003}. This theorem provides a sharp local well-posedness result, along with regularity properties, for the associated functions. Furthermore, we establish the existence of global solutions at the $H^s$-norm, which enables applications to control problems on star graphs and resolves several open questions that will be addressed in a forthcoming paper. The second main result of this work can be stated as follows.\n\n\\begin{theorem}\\label{global_(0,3)}\nLet $T>0$ and $\\lambda\\geq 1$ be. Consider $(u^0,g_0,g,p)\\in \\mathbb{X}_{s,T}^\\lambda$, where $\\mathbb{X}_{s,T}^\\lambda$ is the set of $s$-compatible data\n$$u^0 \\in \\mathbb{H}^{s}(\\mathcal{T}), g_0 \\in H^{\\frac{s+\\lambda-1}{3}}(0,T),\\ g \\in [H^\\frac{s}{3}(0,T)]^N\\quad \\text{and}\\quad p\\in [H^\\frac{s+\\lambda+1}{3}(0,T)]^N.$$\nThus, the problem \\eqref{kdV} has a unique solution $u\\in \\mathcal{X}_{s,T}$ which satisfies\n\\begin{align*}\n\t\\|u\\|_{\\mathcal{X}_{s,T}}&\\leq C\\left(\\|(u^0,g_0,g,p)\\|_{0,\\lambda}\\right)\\left(\n\t\\begin{array}{c}\n\t\t\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s+\\lambda-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}+\\|p\\|_{[H^\\frac{s+\\lambda+1}{3}(0,T)]^N}\n\t\\end{array}\n\t\\right)\n\\end{align*}\nwhere $C>0$ is a positive constant. \n\\end{theorem}\nTogether, these two theorems provide, for the first time, a comprehensive framework for the well-posedness of general graph structures with boundary controls. Moreover, they extend the classical result for a single KdV equation \\cite{BSZ 2003} to star-shaped graphs composed of $N$ equations.\n\\vspace{0.2cm}\n\nThe paper is organized as follows. In Section \\ref{sec2}, we introduce a general notion of $s$-compatibility for star graphs. Building on this framework, Section \\ref{sec3} addresses the well-posedness of the linear system associated with \\eqref{kdV}. After that, Section \\ref{sec4} is devoted to proving the local well-posedness result for the system \\eqref{kdV}, giving the proof of Theorem \\ref{main}. The second main result, concerning global well-posedness, in the $H^s$-regime, for the system \\eqref{kdV} is established in Section \\ref{sec6}.",
    "sketch": "To prove Theorem~\\ref{main}, the introduction states that it \"relies on several analytical tools combined with a fixed-point argument.\" It also mentions that \"a new notion of $s$-compatibility adapted to the graph structure is introduced\" (Section~\\ref{sec2}), \"inspired by the single-edge KdV framework in \\cite{BSZ 2003}.\" The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\\ref{sec2}); then treat \"the well-posedness of the linear system associated with \\eqref{kdV}\" (Section~\\ref{sec3}); and then Section~\\ref{sec4} \"is devoted to proving the local well-posedness result for the system \\eqref{kdV}, giving the proof of Theorem~\\ref{main}.\"",
    "expanded_sketch": "To prove the main theorem, the introduction states that it “relies on several analytical tools combined with a fixed-point argument.” It also mentions that “a new notion of $s$-compatibility adapted to the graph structure is introduced” (Section~\\ref{sec2}), “inspired by the single-edge KdV framework in \\cite{BSZ 2003}.” The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\\ref{sec2}); then treat “the well-posedness of the linear system associated with\n\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\n” (Section~\\ref{sec3}); and then Section~\\ref{sec4} “is devoted to proving the local well-posedness result for the system above, giving the proof of the main theorem.”",
    "expanded_theorem": "\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP\n\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\nadmits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}",
    "theorem_type": [
      "Existential–Universal",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(T>0\\), \\(l_j>0\\) for \\(j=1,\\dots,N\\), and \\(s\\in[0,3]\\). Let \\(\\mathcal T\\) be the star-shaped network with edges \\((0,l_j)\\), and write \\(\\mathbb L^2(\\mathcal T)=\\prod_{j=1}^N L^2(0,l_j)\\) and \\(\\mathbb H^s(\\mathcal T)=\\prod_{j=1}^N H^s(0,l_j)\\). Assume\n\\[\nf\\in W^{\\frac{s}{3},1}(0,T;\\mathbb L^2(\\mathcal T))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb H^{\\frac{s}{3}}(\\mathcal T)),\n\\]\nand\n\\[\n(u^0,g_0,g,p)\\in \\mathbb H^s(\\mathcal T)\\times H^{\\frac{s-1}{3}}(0,T)\\times [H^{\\frac{s}{3}}(0,T)]^N\\times [H^{\\frac{s+1}{3}}(0,T)]^N\n\\]\nsatisfy the \\(s\\)-compatibility conditions. Consider, for \\(j=1,\\dots,N\\), the initial-boundary value problem\n\\[\n\\begin{cases}\n\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,& t\\in(0,T),\\ x\\in(0,l_j),\\\\\nu_j(t,0)=u_1(t,0),& t\\in(0,T),\\\\\n\\displaystyle\\sum_{j=1}^N \\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),& t\\in(0,T),\\\\\nu_j(t,l_j)=p_j(t),\\quad \\partial_x u_j(t,l_j)=g_j(t),& t\\in(0,T),\\\\\nu_j(0,x)=u_j^0(x),& x\\in(0,l_j).\n\\end{cases}\n\\]\nAlso set\n\\[\n\\mathbb B_{s,\\tau}=C([0,\\tau],\\mathbb H^s(\\mathcal T))\\cap L^2(0,\\tau;\\mathbb H^{s+1}(\\mathcal T)),\n\\]\n\\[\n\\mathbb B^*_{s,\\tau}=\\mathbb B_{s,\\tau}\\cap H^{\\frac{s}{3}}(0,\\tau;\\mathbb H^1(\\mathcal T)),\n\\]\nand\n\\[\n\\mathcal X_{s,\\tau}=\\left\\{v\\in \\mathbb B^*_{s,\\tau}:\\ \\partial_x^k v_j\\in L_x^{\\infty}(0,l_j;H^{\\frac{s+1-k}{3}}(0,\\tau)),\\ k=0,1,2,\\ j=1,\\dots,N\\right\\}.\n\\]\nWhich statement about existence and uniqueness is valid for this problem?",
      "correct_choice": {
        "label": "A",
        "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T^*}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists \\(T^*\\in(0,T)\\), depending only on \\(T,s,N\\) and the lengths \\(l_j\\), such that for every admissible \\((u^0,g_0,g,p,f)\\) satisfying the \\(s\\)-compatibility conditions, the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T^*}\\)."
        },
        {
          "label": "C",
          "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits at most one solution in \\(\\mathcal X_{s,T^*}\\)."
        },
        {
          "label": "D",
          "text": "For every \\(T>0\\), the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T}\\)."
        },
        {
          "label": "E",
          "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits a unique solution \\(u\\in \\mathbb B^*_{s,T^*}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "dependence_of_local_time_on_data_size",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_existence_retaining_uniqueness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "local_vs_global_time_interval",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "solution_space_X_vs_Bstar_trace_regularity",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct conclusion. It lists hypotheses and asks which well-posedness statement follows; the key features of the answer (local time interval, uniqueness, and the exact solution space) are not directly stated in the stem."
      },
      "TAS": {
        "score": 1,
        "justification": "This is essentially a theorem-recall item: the stem reproduces the full hypotheses and the options are slight variants of the theorem's conclusion. It is not a pure verbatim restatement because the choices alter quantifiers, uniqueness, and function spaces, but it remains very close to the source theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish local vs global existence, uniqueness vs mere existence, and the precise solution space. However, the task mainly tests recognition of the exact theorem statement rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: global-in-time instead of local, existence without uniqueness, incorrect dependence of the lifespan, and a weaker/incorrect solution space. These reflect common ways a theorem statement can be misstated."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it is still close to theorem restatement and only moderately tests genuine generative reasoning."
    }
  },
  {
    "id": "2512.15682v1",
    "paper_link": "http://arxiv.org/abs/2512.15682v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.",
      "start_pos": 993002,
      "end_pos": 993229,
      "label": "thm:1.1"
    },
    "ref_dict": {
      "thm:1.1": "\\begin{thm}\\label{thm:1.1}\nGiven $\\phi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\phi$ on $\\Omega$ and an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ such that $\\Phi-u_\\phi\\in H^1_0(\\Omega)$.\n\\end{thm}",
      "eq:D": "\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 764,
    "pre_theorem_intro_text": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.",
    "context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}",
    "full_context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\n\nThroughout we suppose that $\\Omega$ is a connected bounded open subset of $\\RR^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\phi\\in C(\\partial\\Omega)$, the \\emphdef{Dirichlet problem} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\emphdef{classical solution} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\\cite[Corollary~4.3]{AtES24}.\n\\begin{thm}[Extension result]\\label{thm:2.1}\nLet $\\phi\\in C(\\partial\\Omega)$. Then there exist an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ and $M \\ge 0$ such that\n\\begin{equation}\\label{eq:2.1}\n \\abs*{\\int_\\Omega\\Phi\\Delta w\\dx} \\le M\\norm{w}_{H^1(\\Omega)}\n\\end{equation}\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $\\Delta\\Phi\\in H^{-1}(\\Omega)$, the dual space of $H^1_0(\\Omega)$.\n\\end{thm}\nNote that for $u\\in H^1(\\Omega)$ one has\n\\[\n \\abs*{\\int_\\Omega u\\Delta w\\dx} = \\abs*{-\\int_\\Omega\\nabla u\\nabla w\\dx}\\le\\norm{u}_{H^1(\\Omega)}\\norm{w}_{H^1(\\Omega)}\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$ and therefore $\\Delta u\\in H^{-1}(\\Omega)$.\n\nLet $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying~\\eqref{eq:2.1}.\nBy the Riesz--Fréchet representation theorem there exists a unique $v\\in H^1_0(\\Omega)$ such that\n\\[\n \\int_\\Omega \\Phi\\Delta w\\dx = -\\int_\\Omega\\nabla v\\nabla w\\dx = \\int_\\Omega v\\Delta w\\dx\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $u:=\\Phi-v$ satisfies $\\Delta u=0$ weakly. Hence $u\\in C^\\infty(\\Omega)$ by Weyl's theorem and $\\Delta u=0$ in the classical sense;\nthat is, $u$ is a harmonic function.\nWe call $u$ the \\emphdef{variational solution} of~\\ref{eq:D} and set $u_\\phi := u$.\nTo do so, we have to show that the function does not depend on the extension $\\Phi$ of $\\phi$. This follows from the next lemma.\n\nNext we talk about the Dirichlet principle.\nWe start by mentioning that the variational solution $u_\\phi$ is obtained by a minimising process.\nIn fact, let $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying the estimate~\\eqref{eq:2.1}, i.e.\\ $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nThe solution $v\\in H^1_0(\\Omega)$ of $\\Delta v=\\Delta\\Phi$ is the unique function $v\\in H^1_0(\\Omega)$ such that\n\\begin{equation}\\label{eq:2.5}\n \\frac{1}{2}\\int_\\Omega\\abs{\\nabla v}^2\\dx + \\langle\\Delta\\Phi,v\\rangle = \\min\\Bigl\\{\\frac{1}{2}\\int_\\Omega\\abs{\\nabla w}^2\\dx + \\langle\\Delta\\Phi,w\\rangle \\setcolon w\\in H^1_0(\\Omega)\\Bigr\\}\n\\end{equation}\n(see e.g.~\\cite[Theorem~4.24]{AU23}).\nNow we can characterise when $\\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx<\\infty$; i.e.\\ when $u_\\phi$ has finite energy.\nSince $u_\\phi\\in C^\\infty(\\Omega)\\cap C_\\bdd(\\Omega)$, this is equivalent to saying that $u_\\phi\\in H^1(\\Omega)$.\n\n\\begin{thm}\\label{thm:2.7}\nLet $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item\\label{en:t2.7-i}\nThe variational solution $u_\\phi$ of~\\ref{eq:D} is an element of $H^1(\\Omega)$.\n\\item\\label{en:t2.7-ii}\nThere exists $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ with $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\n\\end{romanenum}\nIn that case\n\\[\n \\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx = \\min\\Bigl\\{\\int_\\Omega\\abs{\\nabla w}^2\\dx \\setcolon w\\in H^1(\\Omega),\\ \\Phi-w\\in H^1_0(\\Omega)\\Bigr\\}.\n\\]\n\\end{thm}\n\\begin{proof}\n\\begin{parenum}\n\\item[\\ref{en:t2.7-ii}$\\Rightarrow$\\ref{en:t2.7-i}.]\nSince $\\Phi\\in H^1(\\Omega)$, $u_\\phi=\\Phi-v\\in H^1(\\Omega)$, where $v\\in H^1_0(\\Omega)$ is such that $\\Delta v=\\Delta\\Phi$.\nMoreover, by~\\eqref{eq:2.5} and considering $W:=\\{w\\in H^1(\\Omega)\\setcolon \\Phi-w\\in H^1_0(\\Omega)\\}$,\n\\begin{align*}\n \\frac{1}{2}[u_\\phi,u_\\phi] -\\frac{1}{2}[\\Phi,\\Phi] &= \\frac{1}{2}[v,v]-[\\Phi,v] \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[\\Phi-w,\\Phi-w] - [\\Phi,\\Phi-w] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[w,w] - \\frac{1}{2}[\\Phi,\\Phi] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\frac{1}{2} \\min\\Bigl\\{[w,w] \\setcolon w\\in W\\Bigr\\}-\\frac{1}{2}[\\Phi,\\Phi].\n\\end{align*}\nThus $[u_\\phi,u_\\phi] = \\min\\{[w,w]\\setcolon w\\in W\\}$ and this property characterises $u_\\phi$.\n\\item[\\ref{en:t2.7-i}$\\Rightarrow$\\ref{en:t2.7-ii}.]\nLet $\\Phi\\in C(\\clos{\\Omega})$ be such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$ and $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nLet $v\\in H^1_0(\\Omega)$ be such that $\\Delta v=\\Delta\\Phi$.\nThen $u_\\phi=\\Phi-v$.\nSince $u_\\phi\\in H^1(\\Omega)$ by hypothesis, it follows that $\\Phi\\in H^1(\\Omega)$.\\qedhere\n\\end{parenum}\n\\end{proof}\n\nIf $\\Omega$ is more regular, the following holds.\n\\begin{thm}\\label{thm:2.9}\nAssume that $\\Omega$ has Lipschitz boundary and let $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item $u_\\phi\\in H^1(\\Omega)$\n\\item $\\phi\\in H^{1/2}(\\partial\\Omega)$\n\\end{romanenum}\n\\end{thm}\nHere, since $\\Omega$ has Lipschitz boundary, $H^{1/2}(\\partial\\Omega)=\\{\\operatorname{tr}\\Phi \\setcolon \\Phi\\in H^1(\\Omega)\\}$, where $\\operatorname{tr}\\colon H^1(\\Omega)\\to L^2(\\partial\\Omega,\\mathcal{H}^{d-1})$ is the trace operator and $\\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.\nIf $\\phi\\in C(\\partial\\Omega)\\cap H^{1/2}(\\partial\\Omega)$, there even exists a $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\nWe refer to~\\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\\ref{thm:2.9}.\nA more general result on very rough domains is~\\cite[Theorem~5.8]{AtES24}.\n\nTheorem~\\ref{thm:5.1} is contained in Landkof~\\cite[p.\\,243, lines 1--5]{La72}.\nWe will give a very different proof based on the following result. We assume that $d\\ge 3$.\n\\begin{thm}\\label{thm:5.2}\nLet $z_0\\in \\partial\\Omega$ be a singular point of $\\Omega$.\nThen there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$, $z_0\\in\\partial\\Omega_0$ and $z_0$ is a singular point of $\\Omega_0$.\n\\end{thm}\nWe postpone the proof of Theorem~\\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.\nMore precisely, given the potential $V\\colon D\\to(0,\\infty)$ from Lemma~\\ref{lem:lebesgue-potential-basics}, we had $\\Omega=\\{x\\in D\\setcolon \\frac{1}{2}<V(x)<2\\}$. We can choose e.g.~$\\Omega_0=\\{x\\in D\\setcolon \\frac{1}{c}<V(r,z)<c\\}$ for any $c\\in (1,2)$.\nNow we can easily prove Theorem~\\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\\cite[Theorem~6.74]{AU23}.\n\\begin{prop}\\label{prop:5.4}\nLet $z_0\\in \\partial\\Omega$. Then $z_0$ is regular if there exists a \\emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\\ a function $b\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $b(x)>0$ for all $x\\in\\clos{\\Omega}\\setminus\\{z_0\\}$, $b(z_0)=0$ and $-\\Delta b\\ge 0$ weakly on $\\Omega$.\n\\end{prop}\n\\begin{proof}[Proof of Theorem~\\ref{thm:5.1}]\nLet $z_0\\in \\partial\\Omega$ be a singular point for $\\Omega$.\nBy Theorem~\\ref{thm:5.2} there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$ and $z_0\\in\\partial\\Omega_0$ is a singular point with respect to $\\Omega_0$.\nLet $0\\le\\phi\\in C(\\partial\\Omega)$ be such that $\\phi(z_0)=0$ and $\\phi(w)>0$ at a regular point $w\\in \\partial\\Omega$.\nThen there exists $0\\le\\Phi_0\\in C^\\infty_\\cpt(\\RR^d)$ such that $\\phi_0:=\\restrict{\\Phi_0}{\\partial\\Omega}$ satisfies $\\phi_0\\le\\phi$ and $\\phi_0(w)>0$.\nLet $u_\\phi$ and $u_{\\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\\Omega$.\nThen $u_\\phi\\ge u_{\\phi_0}\\ge 0$ on $\\Omega$ by the maximum principle, and $u_{\\phi_0}\\ne 0$.\nMoreover, $u_{\\phi_0}\\in H^1(\\Omega)$ by Theorem~\\ref{thm:2.7}.",
    "post_theorem_intro_text_len": 4394,
    "post_theorem_intro_text": "We call $u_\\varphi$ the \\textbf{\\boldmath variational solution\\unboldmath} of~\\ref{eq:D}.\nIt attains the given boundary value $\\varphi$ in the Sobolev sense.\nThe existence of such a variational solution is difficult to prove and was an open problem for general $\\varphi\\in C(\\partial\\Omega)$ until recently.\n\nIn the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.\nWe then establish diverse properties of the variational solution defined as above.\nWe show that it is well-defined; i.e.\\ $u_\\varphi$ does not depend on the extension $\\Phi$ of $\\varphi$ and it coincides with the classical solution of~\\ref{eq:D} whenever a classical solution exists.\nMoreover, we show that the weak maximum principle is valid for variational solutions.\nIn Section~\\ref{sec:3} we will see that $u_\\varphi$ coincides with the Perron solution.\nThis notion was defined by Perron in 1923~\\cite{Per1923} and has been a subject of intense research.\nOne subject is the way in which $u_\\varphi(x)$ attains the value $\\varphi(z)$ as $x\\to z$.\nSo Theorem~\\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.\n\nIn the second part of the paper, from Section~\\ref{sec:4} on, we investigate the behaviour of $u_\\varphi(x)$ at a fixed singular point $z_0\\in\\partial\\Omega$, i.e.\\ a point where $\\lim_{x\\to z_0} u_\\varphi(x)=\\varphi(z_0)$ is violated for some $\\varphi\\in C(\\partial\\Omega)$.\n\nIn a single page paper Lebesgue~\\cite{Leb1913} considered a potential in $\\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.\nHe chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.\nWe call this region \\textbf{\\boldmath Lebesgue's domain\\unboldmath}.\n\nLebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \\textbf{\\boldmath Lebesgue cusp\\unboldmath}.\nTaking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.\nThis is of great interest since in $\\mathbb{R}^2$ each simply connected domain is \\textbf{\\boldmath Dirichlet regular\\unboldmath}; i.e.\\ $u_\\varphi$ is a classical solution for all $\\varphi\\in C(\\partial\\Omega)$.\nOur point is to consider the original Lebesgue domain, whose boundary has two connected components.\nIt demonstrates two striking features of the generalised solution $u_\\varphi$.\n\n\\begin{enumerate}[1.]\n\\item It is impossible to deduce from the regularity properties of $\\varphi$ whether or not $u_\\varphi$ is a classical solution.\nThe boundary of Lebesgue's domain is the disjoint union of two closed sets $\\Gamma_2$ and $\\Gamma_{\\frac{1}{2}}$ (which are level sets of the considered potential).\nIf $\\varphi$ is constant on each of these two closed sets with two different constants, $u_\\varphi$ is not a classical solution.\nTo prove this, the variational definition of $u_\\varphi$ will be most useful.\n\n\\item The irregular behaviour of $u_\\varphi(x)$ as $x\\to z_0$ is a non-local and unstable phenomenon.\nFor $\\phi_0\\equiv 0$ one has $u_{\\phi_0}\\equiv 0$, but if we perturb $\\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\\varphi$ is no longer a classical solution.\nThis can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\\ref{sec:5}.\n\\end{enumerate}\n\nAs explained above, there seems to be no reasonable condition on the boundary function $\\varphi$ to ensure that $u_\\varphi$ is a classical solution of~\\ref{eq:D} -- even though there are always many classical solutions.\nThe situation is different if we instead consider the question of when $u_\\varphi$ has finite energy $\\int_\\Omega\\abs{\\nabla u_\\varphi}^2$.\nThis is the case if and only if $\\varphi$ has an extension $\\Phi\\in C(\\overline{\\Omega})\\cap H^1(\\Omega)$, and then $u_\\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.\nAs an easy consequence, if $\\varphi$ is Lipschitz continuous, then $u_\\varphi$ has finite energy.\nThese results are proved in Section~\\ref{sec:2} along with other properties of the variational solution.",
    "sketch": "In the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a \\emph{nontrivial extension result} that we shall take as given.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.,",
    "theorem_type": [
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    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb{R}^d\\) be a connected bounded open set with \\(d\\ge 2\\) and boundary \\(\\partial\\Omega\\). Given a boundary function \\(\\varphi\\in C(\\partial\\Omega)\\), which existence-and-uniqueness statement holds? (Here \\(H^1_0(\\Omega)\\) denotes the Sobolev space of \\(H^1\\)-functions with zero trace on \\(\\partial\\Omega\\).)",
      "correct_choice": {
        "label": "A",
        "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that there exists at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) with \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that for every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) one has \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "C",
          "text": "There exists at least one harmonic function \\(u\\) on \\(\\Omega\\) and at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "D",
          "text": "There exists exactly one function \\(u_\\varphi\\in C(\\overline{\\Omega})\\) that is harmonic on \\(\\Omega\\) and satisfies \\(u_\\varphi|_{\\partial\\Omega}=\\varphi\\); equivalently, one may choose \\(\\Phi=u_\\varphi\\), so that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "E",
          "text": "For every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\), there exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that \\(\\Phi-u_\\varphi\\in H^1(\\Omega)\\)."
        }
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          "D",
          "E"
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        "wildcard_false_label": "D"
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          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "existential_extension_replaced_by_universal_extension",
          "template_used": "quantifier_dependence"
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        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_uniqueness_of_harmonic_function",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "upgraded_variational_solution_to_classical_boundary_attainment",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "replaced_H1_0_by_H1_and_made_solution_depend_on_every_extension",
          "template_used": "property_confusion"
        }
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    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the setup and definitions; it does not explicitly state the correct conclusion or provide a direct hint toward choice A over the nearby variants."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to theorem-recognition: the correct option is essentially a precise theorem statement under the given hypotheses. However, it is not a pure restatement because the alternatives vary quantifiers, uniqueness, and regularity in meaningful ways."
      },
      "GPS": {
        "score": 2,
        "justification": "To choose correctly, the solver must distinguish subtle logical differences: existence vs uniqueness, existential vs universal choice of extension, and classical boundary regularity vs weaker harmonicity statements. That creates substantial reasoning pressure."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: overclaiming boundary continuity (B), dropping uniqueness (C), mishandling quantifiers (D), and imposing unnecessary H^1/weak-solution structure (E). They are distinct and well-designed."
      },
      "total_score": 7,
      "overall_assessment": "A strong MCQ with subtle, high-quality distractors and little answer leakage. Its main weakness is that it still functions partly as theorem-statement recognition rather than fully generative problem solving."
    }
  },
  {
    "id": "2512.15682v1",
    "paper_link": "http://arxiv.org/abs/2512.15682v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.",
      "start_pos": 993002,
      "end_pos": 993229,
      "label": "thm:1.1"
    },
    "ref_dict": {
      "thm:1.1": "\\begin{thm}\\label{thm:1.1}\nGiven $\\phi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\phi$ on $\\Omega$ and an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ such that $\\Phi-u_\\phi\\in H^1_0(\\Omega)$.\n\\end{thm}",
      "eq:D": "\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 764,
    "pre_theorem_intro_text": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.",
    "context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}",
    "full_context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\n\nThroughout we suppose that $\\Omega$ is a connected bounded open subset of $\\RR^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\phi\\in C(\\partial\\Omega)$, the \\emphdef{Dirichlet problem} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\emphdef{classical solution} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\\cite[Corollary~4.3]{AtES24}.\n\\begin{thm}[Extension result]\\label{thm:2.1}\nLet $\\phi\\in C(\\partial\\Omega)$. Then there exist an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ and $M \\ge 0$ such that\n\\begin{equation}\\label{eq:2.1}\n \\abs*{\\int_\\Omega\\Phi\\Delta w\\dx} \\le M\\norm{w}_{H^1(\\Omega)}\n\\end{equation}\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $\\Delta\\Phi\\in H^{-1}(\\Omega)$, the dual space of $H^1_0(\\Omega)$.\n\\end{thm}\nNote that for $u\\in H^1(\\Omega)$ one has\n\\[\n \\abs*{\\int_\\Omega u\\Delta w\\dx} = \\abs*{-\\int_\\Omega\\nabla u\\nabla w\\dx}\\le\\norm{u}_{H^1(\\Omega)}\\norm{w}_{H^1(\\Omega)}\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$ and therefore $\\Delta u\\in H^{-1}(\\Omega)$.\n\nLet $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying~\\eqref{eq:2.1}.\nBy the Riesz--Fréchet representation theorem there exists a unique $v\\in H^1_0(\\Omega)$ such that\n\\[\n \\int_\\Omega \\Phi\\Delta w\\dx = -\\int_\\Omega\\nabla v\\nabla w\\dx = \\int_\\Omega v\\Delta w\\dx\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $u:=\\Phi-v$ satisfies $\\Delta u=0$ weakly. Hence $u\\in C^\\infty(\\Omega)$ by Weyl's theorem and $\\Delta u=0$ in the classical sense;\nthat is, $u$ is a harmonic function.\nWe call $u$ the \\emphdef{variational solution} of~\\ref{eq:D} and set $u_\\phi := u$.\nTo do so, we have to show that the function does not depend on the extension $\\Phi$ of $\\phi$. This follows from the next lemma.\n\nNext we talk about the Dirichlet principle.\nWe start by mentioning that the variational solution $u_\\phi$ is obtained by a minimising process.\nIn fact, let $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying the estimate~\\eqref{eq:2.1}, i.e.\\ $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nThe solution $v\\in H^1_0(\\Omega)$ of $\\Delta v=\\Delta\\Phi$ is the unique function $v\\in H^1_0(\\Omega)$ such that\n\\begin{equation}\\label{eq:2.5}\n \\frac{1}{2}\\int_\\Omega\\abs{\\nabla v}^2\\dx + \\langle\\Delta\\Phi,v\\rangle = \\min\\Bigl\\{\\frac{1}{2}\\int_\\Omega\\abs{\\nabla w}^2\\dx + \\langle\\Delta\\Phi,w\\rangle \\setcolon w\\in H^1_0(\\Omega)\\Bigr\\}\n\\end{equation}\n(see e.g.~\\cite[Theorem~4.24]{AU23}).\nNow we can characterise when $\\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx<\\infty$; i.e.\\ when $u_\\phi$ has finite energy.\nSince $u_\\phi\\in C^\\infty(\\Omega)\\cap C_\\bdd(\\Omega)$, this is equivalent to saying that $u_\\phi\\in H^1(\\Omega)$.\n\n\\begin{thm}\\label{thm:2.7}\nLet $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item\\label{en:t2.7-i}\nThe variational solution $u_\\phi$ of~\\ref{eq:D} is an element of $H^1(\\Omega)$.\n\\item\\label{en:t2.7-ii}\nThere exists $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ with $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\n\\end{romanenum}\nIn that case\n\\[\n \\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx = \\min\\Bigl\\{\\int_\\Omega\\abs{\\nabla w}^2\\dx \\setcolon w\\in H^1(\\Omega),\\ \\Phi-w\\in H^1_0(\\Omega)\\Bigr\\}.\n\\]\n\\end{thm}\n\\begin{proof}\n\\begin{parenum}\n\\item[\\ref{en:t2.7-ii}$\\Rightarrow$\\ref{en:t2.7-i}.]\nSince $\\Phi\\in H^1(\\Omega)$, $u_\\phi=\\Phi-v\\in H^1(\\Omega)$, where $v\\in H^1_0(\\Omega)$ is such that $\\Delta v=\\Delta\\Phi$.\nMoreover, by~\\eqref{eq:2.5} and considering $W:=\\{w\\in H^1(\\Omega)\\setcolon \\Phi-w\\in H^1_0(\\Omega)\\}$,\n\\begin{align*}\n \\frac{1}{2}[u_\\phi,u_\\phi] -\\frac{1}{2}[\\Phi,\\Phi] &= \\frac{1}{2}[v,v]-[\\Phi,v] \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[\\Phi-w,\\Phi-w] - [\\Phi,\\Phi-w] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[w,w] - \\frac{1}{2}[\\Phi,\\Phi] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\frac{1}{2} \\min\\Bigl\\{[w,w] \\setcolon w\\in W\\Bigr\\}-\\frac{1}{2}[\\Phi,\\Phi].\n\\end{align*}\nThus $[u_\\phi,u_\\phi] = \\min\\{[w,w]\\setcolon w\\in W\\}$ and this property characterises $u_\\phi$.\n\\item[\\ref{en:t2.7-i}$\\Rightarrow$\\ref{en:t2.7-ii}.]\nLet $\\Phi\\in C(\\clos{\\Omega})$ be such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$ and $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nLet $v\\in H^1_0(\\Omega)$ be such that $\\Delta v=\\Delta\\Phi$.\nThen $u_\\phi=\\Phi-v$.\nSince $u_\\phi\\in H^1(\\Omega)$ by hypothesis, it follows that $\\Phi\\in H^1(\\Omega)$.\\qedhere\n\\end{parenum}\n\\end{proof}\n\nIf $\\Omega$ is more regular, the following holds.\n\\begin{thm}\\label{thm:2.9}\nAssume that $\\Omega$ has Lipschitz boundary and let $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item $u_\\phi\\in H^1(\\Omega)$\n\\item $\\phi\\in H^{1/2}(\\partial\\Omega)$\n\\end{romanenum}\n\\end{thm}\nHere, since $\\Omega$ has Lipschitz boundary, $H^{1/2}(\\partial\\Omega)=\\{\\operatorname{tr}\\Phi \\setcolon \\Phi\\in H^1(\\Omega)\\}$, where $\\operatorname{tr}\\colon H^1(\\Omega)\\to L^2(\\partial\\Omega,\\mathcal{H}^{d-1})$ is the trace operator and $\\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.\nIf $\\phi\\in C(\\partial\\Omega)\\cap H^{1/2}(\\partial\\Omega)$, there even exists a $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\nWe refer to~\\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\\ref{thm:2.9}.\nA more general result on very rough domains is~\\cite[Theorem~5.8]{AtES24}.\n\nTheorem~\\ref{thm:5.1} is contained in Landkof~\\cite[p.\\,243, lines 1--5]{La72}.\nWe will give a very different proof based on the following result. We assume that $d\\ge 3$.\n\\begin{thm}\\label{thm:5.2}\nLet $z_0\\in \\partial\\Omega$ be a singular point of $\\Omega$.\nThen there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$, $z_0\\in\\partial\\Omega_0$ and $z_0$ is a singular point of $\\Omega_0$.\n\\end{thm}\nWe postpone the proof of Theorem~\\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.\nMore precisely, given the potential $V\\colon D\\to(0,\\infty)$ from Lemma~\\ref{lem:lebesgue-potential-basics}, we had $\\Omega=\\{x\\in D\\setcolon \\frac{1}{2}<V(x)<2\\}$. We can choose e.g.~$\\Omega_0=\\{x\\in D\\setcolon \\frac{1}{c}<V(r,z)<c\\}$ for any $c\\in (1,2)$.\nNow we can easily prove Theorem~\\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\\cite[Theorem~6.74]{AU23}.\n\\begin{prop}\\label{prop:5.4}\nLet $z_0\\in \\partial\\Omega$. Then $z_0$ is regular if there exists a \\emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\\ a function $b\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $b(x)>0$ for all $x\\in\\clos{\\Omega}\\setminus\\{z_0\\}$, $b(z_0)=0$ and $-\\Delta b\\ge 0$ weakly on $\\Omega$.\n\\end{prop}\n\\begin{proof}[Proof of Theorem~\\ref{thm:5.1}]\nLet $z_0\\in \\partial\\Omega$ be a singular point for $\\Omega$.\nBy Theorem~\\ref{thm:5.2} there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$ and $z_0\\in\\partial\\Omega_0$ is a singular point with respect to $\\Omega_0$.\nLet $0\\le\\phi\\in C(\\partial\\Omega)$ be such that $\\phi(z_0)=0$ and $\\phi(w)>0$ at a regular point $w\\in \\partial\\Omega$.\nThen there exists $0\\le\\Phi_0\\in C^\\infty_\\cpt(\\RR^d)$ such that $\\phi_0:=\\restrict{\\Phi_0}{\\partial\\Omega}$ satisfies $\\phi_0\\le\\phi$ and $\\phi_0(w)>0$.\nLet $u_\\phi$ and $u_{\\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\\Omega$.\nThen $u_\\phi\\ge u_{\\phi_0}\\ge 0$ on $\\Omega$ by the maximum principle, and $u_{\\phi_0}\\ne 0$.\nMoreover, $u_{\\phi_0}\\in H^1(\\Omega)$ by Theorem~\\ref{thm:2.7}.",
    "post_theorem_intro_text_len": 4394,
    "post_theorem_intro_text": "We call $u_\\varphi$ the \\textbf{\\boldmath variational solution\\unboldmath} of~\\ref{eq:D}.\nIt attains the given boundary value $\\varphi$ in the Sobolev sense.\nThe existence of such a variational solution is difficult to prove and was an open problem for general $\\varphi\\in C(\\partial\\Omega)$ until recently.\n\nIn the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.\nWe then establish diverse properties of the variational solution defined as above.\nWe show that it is well-defined; i.e.\\ $u_\\varphi$ does not depend on the extension $\\Phi$ of $\\varphi$ and it coincides with the classical solution of~\\ref{eq:D} whenever a classical solution exists.\nMoreover, we show that the weak maximum principle is valid for variational solutions.\nIn Section~\\ref{sec:3} we will see that $u_\\varphi$ coincides with the Perron solution.\nThis notion was defined by Perron in 1923~\\cite{Per1923} and has been a subject of intense research.\nOne subject is the way in which $u_\\varphi(x)$ attains the value $\\varphi(z)$ as $x\\to z$.\nSo Theorem~\\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.\n\nIn the second part of the paper, from Section~\\ref{sec:4} on, we investigate the behaviour of $u_\\varphi(x)$ at a fixed singular point $z_0\\in\\partial\\Omega$, i.e.\\ a point where $\\lim_{x\\to z_0} u_\\varphi(x)=\\varphi(z_0)$ is violated for some $\\varphi\\in C(\\partial\\Omega)$.\n\nIn a single page paper Lebesgue~\\cite{Leb1913} considered a potential in $\\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.\nHe chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.\nWe call this region \\textbf{\\boldmath Lebesgue's domain\\unboldmath}.\n\nLebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \\textbf{\\boldmath Lebesgue cusp\\unboldmath}.\nTaking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.\nThis is of great interest since in $\\mathbb{R}^2$ each simply connected domain is \\textbf{\\boldmath Dirichlet regular\\unboldmath}; i.e.\\ $u_\\varphi$ is a classical solution for all $\\varphi\\in C(\\partial\\Omega)$.\nOur point is to consider the original Lebesgue domain, whose boundary has two connected components.\nIt demonstrates two striking features of the generalised solution $u_\\varphi$.\n\n\\begin{enumerate}[1.]\n\\item It is impossible to deduce from the regularity properties of $\\varphi$ whether or not $u_\\varphi$ is a classical solution.\nThe boundary of Lebesgue's domain is the disjoint union of two closed sets $\\Gamma_2$ and $\\Gamma_{\\frac{1}{2}}$ (which are level sets of the considered potential).\nIf $\\varphi$ is constant on each of these two closed sets with two different constants, $u_\\varphi$ is not a classical solution.\nTo prove this, the variational definition of $u_\\varphi$ will be most useful.\n\n\\item The irregular behaviour of $u_\\varphi(x)$ as $x\\to z_0$ is a non-local and unstable phenomenon.\nFor $\\phi_0\\equiv 0$ one has $u_{\\phi_0}\\equiv 0$, but if we perturb $\\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\\varphi$ is no longer a classical solution.\nThis can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\\ref{sec:5}.\n\\end{enumerate}\n\nAs explained above, there seems to be no reasonable condition on the boundary function $\\varphi$ to ensure that $u_\\varphi$ is a classical solution of~\\ref{eq:D} -- even though there are always many classical solutions.\nThe situation is different if we instead consider the question of when $u_\\varphi$ has finite energy $\\int_\\Omega\\abs{\\nabla u_\\varphi}^2$.\nThis is the case if and only if $\\varphi$ has an extension $\\Phi\\in C(\\overline{\\Omega})\\cap H^1(\\Omega)$, and then $u_\\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.\nAs an easy consequence, if $\\varphi$ is Lipschitz continuous, then $u_\\varphi$ has finite energy.\nThese results are proved in Section~\\ref{sec:2} along with other properties of the variational solution.",
    "sketch": "In the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a \\emph{nontrivial extension result} that we shall take as given.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.,",
    "theorem_type": [
      "Uniqueness",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(\\Omega\\subset \\mathbb{R}^d\\) be a connected bounded open set with \\(d\\ge 2\\) and boundary \\(\\partial\\Omega\\). Given a boundary function \\(\\varphi\\in C(\\partial\\Omega)\\), which existence-and-uniqueness statement holds? (Here \\(H^1_0(\\Omega)\\) denotes the Sobolev space of \\(H^1\\)-functions with zero trace on \\(\\partial\\Omega\\).)",
      "correct_choice": {
        "label": "A",
        "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that there exists at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) with \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that for every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) one has \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "C",
          "text": "There exists at least one harmonic function \\(u\\) on \\(\\Omega\\) and at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "D",
          "text": "There exists exactly one function \\(u_\\varphi\\in C(\\overline{\\Omega})\\) that is harmonic on \\(\\Omega\\) and satisfies \\(u_\\varphi|_{\\partial\\Omega}=\\varphi\\); equivalently, one may choose \\(\\Phi=u_\\varphi\\), so that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."
        },
        {
          "label": "E",
          "text": "For every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\), there exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that \\(\\Phi-u_\\varphi\\in H^1(\\Omega)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "existential_extension_replaced_by_universal_extension",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_uniqueness_of_harmonic_function",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "upgraded_variational_solution_to_classical_boundary_attainment",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "replaced_H1_0_by_H1_and_made_solution_depend_on_every_extension",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the correct theorem or its key quantifier structure. It gives standard setup and notation, but the correct choice is not leaked; the test-taker must distinguish among subtly different existence/uniqueness formulations."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a standard theorem statement about the weak Dirichlet problem, so it has some theorem-restatement flavor. However, it is not purely tautological because the options introduce meaningful variations in quantifiers, regularity, and uniqueness."
      },
      "GPS": {
        "score": 1,
        "justification": "The question requires moderate reasoning: the solver must recognize why the exact weak formulation is correct and why stronger or weaker alternatives fail. Still, it mainly tests precise theorem recall/recognition rather than deeper generative inference."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible. They target common failure modes: over-strong universal quantification, loss of uniqueness, unjustified upgrade to classical boundary attainment, and confusion between H^1 and H^1_0."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no major answer leakage and high-quality distractors, but it remains somewhat close to theorem recognition rather than a strongly generative reasoning task."
    }
  },
  {
    "id": "2512.15909v1",
    "paper_link": "http://arxiv.org/abs/2512.15909v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\n    Let $Q=\\Spec(A)$ be an affine algebraic rack with unit $e\\in Q$, and let $\\mathfrak{q}\\coloneq \\Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\\cdot,\\cdot]\\colon\\mathfrak{q}\\times\\mathfrak{q}\\to\\mathfrak{q}$ by convolution with respect to the corack operation $\\nabla$ of $A$ (see Section \\ref{sec:nabla}):\n    \\[\n    [D,E]\\coloneq (D\\otimes E)\\circ\\nabla.\n    \\]\n    Then the assignment $Q\\mapsto(\\mathfrak{q},[\\cdot,\\cdot])$ defines a covariant functor $\\Rack(\\Aff)\\to\\leib$ sending morphisms of affine algebraic racks $\\varphi\\colon Q\\to R$ to their derivatives $(d\\varphi)_e\\colon \\mathfrak{q}\\to\\mathfrak{r}$ at $e$.\n\n    Furthermore, if $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\mathfrak{q},[\\cdot,\\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\\mathbb{R}$ or $\\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\\mathfrak{q},[\\cdot,\\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.",
      "start_pos": 7206,
      "end_pos": 8146,
      "label": "thm:main"
    },
    "ref_dict": {
      "rmk:recover": "\\begin{rmk}\\label{rmk:recover}\n    If $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\q,[\\cdot,\\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\\q$ at $e$. By the definition of $\\Conj(G)$, the adjoint representations $\\Ad$ of $Q$ and $\\ad$ of $\\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \\cite{milne}*{Rem.\\ 10.24}). Hence, the claim follows from Proposition \\ref{prop:adj}.\\footnote{Alternatively, one can use Example \\ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ agrees with the usual Lie bracket $(D\\otimes E-E\\otimes D)\\circ\\Delta$ on $\\q=\\operatorname{Lie}(G)$ (see \\cite{waterhouse}*{Sec.\\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\\eps\\circ S=\\eps$ from the axiom $\\mu\\circ (S\\otimes\\id)\\circ\\Delta=\\eta\\circ\\eps$ by post-composing with $\\eps$, and then deduce from these two equalities that $D\\circ S=-D$.}\n\n    Similarly, if $k=\\R$ or $\\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\\Ad$ and $\\ad$ coincide with the ones for Lie racks in \\cite{kinyon}, so $(\\q,[\\cdot,\\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.\n\\end{rmk}",
      "prop:adj": "\\begin{prop}\\label{prop:adj}\n    When considered as a map $\\ad\\colon\\q\\to\\mathfrak{gl}(\\q)$, the adjoint representation of $\\q$ is given by the left argument of the Leibniz bracket $D\\mapsto [D,\\cdot]$. That is, $\\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\\in\\q$. In particular, $\\q$ is closed under $[\\cdot,\\cdot]$.\n\\end{prop}",
      "prop:jacobi": "\\begin{prop}\\label{prop:jacobi}\n    The convolution operation $[\\cdot,\\cdot]$ satisfies the Jacobi identity.\n\\end{prop}",
      "prop:functor": "\\begin{prop}\\label{prop:functor}\n    The assignments $Q\\mapsto\\q$ and $\\phi\\mapsto \\phi'\\coloneq (d\\phi)_e$ define a covariant functor $\\Rack(\\Aff)\\to\\leib$.\n\\end{prop}",
      "thm:main": "\\begin{thm}\\label{thm:main}\n    Let $Q=\\Spec(A)$ be an affine algebraic rack with unit $e\\in Q$, and let $\\q\\coloneq \\Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\\cdot,\\cdot]\\colon\\q\\times\\q\\to\\q$ by convolution with respect to the corack operation $\\nabla$ of $A$ (see Section \\ref{sec:nabla}):\n    \\[\n    [D,E]\\coloneq (D\\otimes E)\\circ\\nabla.\n    \\]\n    Then the assignment $Q\\mapsto(\\q,[\\cdot,\\cdot])$ defines a covariant functor $\\Rack(\\Aff)\\to\\leib$ sending morphisms of affine algebraic racks $\\phi\\colon Q\\to R$ to their derivatives $(d\\phi)_e\\colon \\q\\to\\mathfrak{r}$ at $e$.\n\n    Furthermore, if $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\q,[\\cdot,\\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\\R$ or $\\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\\q,[\\cdot,\\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 2148,
    "pre_theorem_intro_text": "This paper introduces \\emph{algebraic racks}, which are pointed racks internal to the category of schemes of finite type over a field $k$. We show that affine algebraic racks provide a solution to a version of the well-known \\emph{coquecigrue} problem for Leibniz algebras over arbitrary fields $k$. \n\n\\subsubsection{}\nLoday's \\cite{loday} original version of the \\emph{coquecigrue} problem from 1993 seeks algebraic structures on smooth manifolds that differentiate to Leibniz algebras in a way that recovers the Lie group--Lie algebra correspondence. \\emph{Leibniz algebras} are vector spaces $\\mathfrak{q}$ equipped with a bilinear mapping $[\\cdot,\\cdot]\\colon \\mathfrak{q}\\times\\mathfrak{q}\\to\\mathfrak{q}$ called the \\emph{Leibniz bracket} that satisfies the derivation form of the Jacobi identity\n    \\[\n    [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]].\n    \\]\nBloh \\cite{bloh} introduced Leibniz algebras in 1965 to generalize Lie algebras, which are precisely those Leibniz algebras whose Leibniz bracket is alternating. See \\cite{leibniz} for an introduction to Leibniz algebras. \n\nAs a solution to the \\emph{coquecigrue} problem over $\\mathbb{R}$ and $\\mathbb{C}$, Kinyon \\cite{kinyon} in 2007 introduced \\emph{Lie racks}, which are pointed rack objects (see Section \\ref{sec:pro}) in the category of smooth manifolds; see the survey \\cite{ongay} for further discussion. \\emph{Racks} \\cite{fenn} and similar structures called \\emph{quandles} \\citelist{\\cite{joyce}\\cite{matveev}} were originally introduced as invariants of knots and 3-manifolds; see \\citelist{\\cite{book}\\cite{quandlebook}} for introductions to the theory. \n\n\\subsection{Main theorem}\nKinyon also posed an analogue of the \\emph{coquecigrue} problem over arbitrary fields $k$ that replaces Lie groups with affine algebraic groups. Save for unpublished work of Ba\\v{s}i\\'{c} and \\v{S}koda \\cite{skoda} in 2008 in the case that $\\operatorname{char}(k)=0$, the author is unaware of any literature in this direction. \nTo address this, we prove the following. Let $\\Rack(\\Aff)$ and $\\leib$ denote the categories of affine algebraic racks over $k$ and Leibniz algebras over $k$, respectively.",
    "context": "This paper introduces \\emph{algebraic racks}, which are pointed racks internal to the category of schemes of finite type over a field $k$. We show that affine algebraic racks provide a solution to a version of the well-known \\emph{coquecigrue} problem for Leibniz algebras over arbitrary fields $k$.\n\n\\subsubsection{}\nLoday's \\cite{loday} original version of the \\emph{coquecigrue} problem from 1993 seeks algebraic structures on smooth manifolds that differentiate to Leibniz algebras in a way that recovers the Lie group--Lie algebra correspondence. \\emph{Leibniz algebras} are vector spaces $\\mathfrak{q}$ equipped with a bilinear mapping $[\\cdot,\\cdot]\\colon \\mathfrak{q}\\times\\mathfrak{q}\\to\\mathfrak{q}$ called the \\emph{Leibniz bracket} that satisfies the derivation form of the Jacobi identity\n \\[\n [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]].\n \\]\nBloh \\cite{bloh} introduced Leibniz algebras in 1965 to generalize Lie algebras, which are precisely those Leibniz algebras whose Leibniz bracket is alternating. See \\cite{leibniz} for an introduction to Leibniz algebras.\n\nAs a solution to the \\emph{coquecigrue} problem over $\\mathbb{R}$ and $\\mathbb{C}$, Kinyon \\cite{kinyon} in 2007 introduced \\emph{Lie racks}, which are pointed rack objects (see Section \\ref{sec:pro}) in the category of smooth manifolds; see the survey \\cite{ongay} for further discussion. \\emph{Racks} \\cite{fenn} and similar structures called \\emph{quandles} \\citelist{\\cite{joyce}\\cite{matveev}} were originally introduced as invariants of knots and 3-manifolds; see \\citelist{\\cite{book}\\cite{quandlebook}} for introductions to the theory.\n\n\\subsection{Main theorem}\nKinyon also posed an analogue of the \\emph{coquecigrue} problem over arbitrary fields $k$ that replaces Lie groups with affine algebraic groups. Save for unpublished work of Ba\\v{s}i\\'{c} and \\v{S}koda \\cite{skoda} in 2008 in the case that $\\operatorname{char}(k)=0$, the author is unaware of any literature in this direction. \nTo address this, we prove the following. Let $\\Rack(\\Aff)$ and $\\leib$ denote the categories of affine algebraic racks over $k$ and Leibniz algebras over $k$, respectively.",
    "full_context": "This paper introduces \\emph{algebraic racks}, which are pointed racks internal to the category of schemes of finite type over a field $k$. We show that affine algebraic racks provide a solution to a version of the well-known \\emph{coquecigrue} problem for Leibniz algebras over arbitrary fields $k$.\n\n\\subsubsection{}\nLoday's \\cite{loday} original version of the \\emph{coquecigrue} problem from 1993 seeks algebraic structures on smooth manifolds that differentiate to Leibniz algebras in a way that recovers the Lie group--Lie algebra correspondence. \\emph{Leibniz algebras} are vector spaces $\\mathfrak{q}$ equipped with a bilinear mapping $[\\cdot,\\cdot]\\colon \\mathfrak{q}\\times\\mathfrak{q}\\to\\mathfrak{q}$ called the \\emph{Leibniz bracket} that satisfies the derivation form of the Jacobi identity\n \\[\n [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]].\n \\]\nBloh \\cite{bloh} introduced Leibniz algebras in 1965 to generalize Lie algebras, which are precisely those Leibniz algebras whose Leibniz bracket is alternating. See \\cite{leibniz} for an introduction to Leibniz algebras.\n\nAs a solution to the \\emph{coquecigrue} problem over $\\mathbb{R}$ and $\\mathbb{C}$, Kinyon \\cite{kinyon} in 2007 introduced \\emph{Lie racks}, which are pointed rack objects (see Section \\ref{sec:pro}) in the category of smooth manifolds; see the survey \\cite{ongay} for further discussion. \\emph{Racks} \\cite{fenn} and similar structures called \\emph{quandles} \\citelist{\\cite{joyce}\\cite{matveev}} were originally introduced as invariants of knots and 3-manifolds; see \\citelist{\\cite{book}\\cite{quandlebook}} for introductions to the theory.\n\n\\subsection{Main theorem}\nKinyon also posed an analogue of the \\emph{coquecigrue} problem over arbitrary fields $k$ that replaces Lie groups with affine algebraic groups. Save for unpublished work of Ba\\v{s}i\\'{c} and \\v{S}koda \\cite{skoda} in 2008 in the case that $\\operatorname{char}(k)=0$, the author is unaware of any literature in this direction. \nTo address this, we prove the following. Let $\\Rack(\\Aff)$ and $\\leib$ denote the categories of affine algebraic racks over $k$ and Leibniz algebras over $k$, respectively.\n\n\\begin{proof}\n We prove the theorem piecemeal in Propositions \\ref{prop:adj}, \\ref{prop:jacobi}, and \\ref{prop:functor} and Remark \\ref{rmk:recover}.\n\\end{proof}\n\n\\subsubsection{}\nTo make Definition \\ref{def:corack} more concrete, let $A$ be a $k$-algebra with (commutative and associative) multiplication $\\mu\\colon A\\otimes A\\to A$, and let $\\tau\\colon A\\otimes A\\to A\\otimes A$ denote the transposition $a\\otimes b\\mapsto b\\otimes a$. A corack algebra structure on $A$ consists of two $k$-algebra homomorphisms $\\nabla,\\nabla\\inv\\colon A\\to A\\otimes A$ called the \\emph{corack operations} and a $k$-algebra homomorphism $\\eps\\colon A\\to k$ called the \\emph{counit}, \\emph{coidentity}, or \\emph{augmentation} satisfying the following five identities:\n\\begin{enumerate}[(C1)]\n \\item (Co-invertability) $(\\mathord{\\id}\\otimes \\mu)\\circ(\\tau\\otimes\\mu)\\circ(\\mathrm{\\id}\\otimes\\nabla\\inv\\otimes\\id)\\circ(\\nabla\\circ\\id)=(\\mathord{\\id}\\otimes \\id)=(\\mu\\otimes\\id)\\circ(\\mu\\otimes\\tau)\\circ(\\mathord{\\id}\\otimes\\nabla\\otimes\\id)\\circ(\\mathord{\\id}\\otimes\\nabla\\inv)$.\n \\item\\label{ax:cold} (Co-left distributivity) $(\\mathord{\\id}\\otimes\\nabla)\\circ\\nabla=(\\mu\\otimes\\mathord{\\id}\\otimes\\id)\\circ(\\mathord{\\id}\\otimes\\tau\\otimes\\id)\\circ(\\nabla\\otimes\\nabla)\\circ\\nabla$.\n \\item (Co-right distributivity) $(\\nabla\\inv\\otimes\\id)\\circ\\nabla\\inv=(\\mathord{\\id}\\otimes\\mathord{\\id}\\otimes\\mu)\\circ(\\mathord{\\id}\\otimes\\tau\\otimes\\id)\\circ(\\nabla\\inv\\otimes\\nabla\\inv)\\circ\\nabla\\inv$.\n \\item\\label{ax:co-fixing} (Co-fixing) $(\\eps\\otimes \\id)\\circ\\nabla=\\id$.\n \\item\\label{ax:co-fixed} (Co-fixedness) $(\\mathord{\\id}\\otimes \\eps)\\circ\\nabla=\\eps$.\n\\end{enumerate}\nWe identify $k\\otimes A\\cong A\\cong A\\otimes k$ in identities \\ref{ax:co-fixing} and \\ref{ax:co-fixed}. Actually, this paper will only have occasion to use identities \\ref{ax:cold} and \\ref{ax:co-fixing}; the other three identities are less important for our purposes than the corresponding axioms of algebraic racks.\n\nFor example, let $Q=\\Spec(A)$ be an affine algebraic rack. Then for all $k$-algebras $R$ and $R$-points $x\\in Q(R)$ (thought of as $k$-algebra homomorphisms $x^\\sharp\\colon A\\to R$), the automorphism $L_x\\colon Q(R)\\bij Q(R)$ from \\eqref{eq:lx} induces an automorphism\n\\begin{align}\nL_x^\\sharp\\colon R\\otimes_k A &\\bij R\\otimes_k A\\nonumber\\\\\n r\\otimes f&\\mapsto(r\\otimes 1)\\cdot ((x^\\sharp\\otimes\\id_{A})\\circ \\nabla)(f).\\label{eq:lx-sharp-full}\n\\end{align}\n\n\\begin{definition}\n The \\emph{Leibniz algebra of $Q$} is the vector space\n \\[\n \\q\\coloneq \\Der_k(A,k)\n \\]\n equipped with the Leibniz bracket $[\\cdot,\\cdot]\\colon \\q\\times\\q\\to \\q$ defined by \n \\[\n [D,E]\\coloneq (D\\otimes E)\\circ\\nabla,\n \\]\n the \\emph{convolution} operation with respect to $\\nabla$.\n\\end{definition}\nWriting $\\nabla f=\\sum \\fo\\otimes\\ft$, we thus have\n\\[\n[D,E](f)=\\sum D(\\fo)E(\\ft)\n\\]\nfor all $f\\in A$. Note that $[\\cdot,\\cdot]$ is $k$-bilinear.\n\n\\subsubsection{}\nSimilarly, since $\\partial D$ is a $k[\\delta]$-point of $Q$ and $\\Ad\\colon Q\\to\\GL(\\q)$ is a morphism of schemes, we obtain a $k[\\delta]$-point of $\\GL(\\q)$ \\[\\Ad_{\\partial D}\\in\\GL(\\q)(k[\\delta])=\\GL(k[\\delta]\\otimes_k \\q).\\] In particular, under the identification $k[\\delta]\\otimes_kA \\bij A[\\delta]$, the definition of the derivative shows that \n\\begin{equation}\\label{eq:adj-partial}\n \\Ad_{\\partial D}( 1\\otimes E)= E\\circ L_{\\partial D}^\\sharp\\colon A[\\delta]\\to k[\\delta]\n\\end{equation}\nfor all $k$-derivations $E\\in\\q$. We use this to show the following. \\begin{prop}\\label{prop:adj}\n When considered as a map $\\ad\\colon\\q\\to\\mathfrak{gl}(\\q)$, the adjoint representation of $\\q$ is given by the left argument of the Leibniz bracket $D\\mapsto [D,\\cdot]$. That is, $\\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\\in\\q$. In particular, $\\q$ is closed under $[\\cdot,\\cdot]$.\n\\end{prop}\n\n\\begin{rmk}\\label{rmk:recover}\n If $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\q,[\\cdot,\\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\\q$ at $e$. By the definition of $\\Conj(G)$, the adjoint representations $\\Ad$ of $Q$ and $\\ad$ of $\\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \\cite{milne}*{Rem.\\ 10.24}). Hence, the claim follows from Proposition \\ref{prop:adj}.\\footnote{Alternatively, one can use Example \\ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ agrees with the usual Lie bracket $(D\\otimes E-E\\otimes D)\\circ\\Delta$ on $\\q=\\operatorname{Lie}(G)$ (see \\cite{waterhouse}*{Sec.\\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\\eps\\circ S=\\eps$ from the axiom $\\mu\\circ (S\\otimes\\id)\\circ\\Delta=\\eta\\circ\\eps$ by post-composing with $\\eps$, and then deduce from these two equalities that $D\\circ S=-D$.}\n\n\\begin{enumerate}\n \\item Does the Leibniz algebra of a (not necessarily affine) algebraic rack have a natural Leibniz algebra structure? That is, does the functor $\\Rack(\\Aff)\\to\\leib$ from Theorem \\ref{thm:main} extend to a functor $\\Rack(\\Sch)\\to\\leib$?\n \\item If $G$ is an algebraic group, then its Lie algebra $\\mathrm{Lie}(G)$ is isomorphic to the space of left-invariant $k$-derivations $D\\colon \\O(G)\\to\\O(G)$, that is, $k$-derivations that satisfy the identity $\\Delta\\circ D=(\\mathord{\\id}\\otimes D)\\circ\\Delta$. (See, for example, \\cite{milne}*{Sec.\\ 10e}.) Do Leibniz algebras of algebraic racks have a similar interpretation?\n \\item Mirroring the representation theory of algebraic groups and comodules of commutative Hopf algebras (see, for example, \\cite{milne}*{Sec.\\ 4e}), develop a theory of affine representations of affine algebraic racks $Q$ and $\\O(Q)$-comodules. Also, study their relationships with the induced omni-representations (see \\cite{omni}) of Leibniz algebras. See Comment \\ref{com:rep} for further discussion.\n \\item Building upon the previous question, introduce a cohomology theory for algebraic racks and their representations. Study its relationships with Leibniz algebra cohomology \\cite{lp}, traditional rack and quandle cohomology \\cite{covez2}, and (in the case of conjugation quandle schemes) the cohomology of algebraic groups and their representations (see \\cite{bendel}).\n \\item More broadly, which theorems about affine algebraic groups and their Lie algebras extend to affine algebraic racks and their Leibniz algebras? (Cf.\\ Remark \\ref{rmk:triv}.)\n \\item Determine which Leibniz algebras are \\emph{algebraic} or ``integrate to an algebraic rack.'' That is, given $\\q$, determine whether there exists an algebraic rack whose Leibniz algebra is $\\q$.\n \\item La Rosa and Mancini \\cite{larosa}*{Thm.\\ 4.6} proved that if $Q$ is a Lie rack with Leibniz algebra $\\q$, then $\\q$ is a Lie algebra if and only if $Q$ is a quandle. Does the same hold for algebraic racks?\n \\item Study non-pointed algebraic racks (that is, \\emph{rack objects} in $\\Sch$; cf.\\ \\cite{rack-roll}). In particular, generalize the results of Takahashi \\citelist{\\cite{takahashi}\\cite{takahashi2}} from quandle varieties to rack schemes.\n \\item Develop the general theory of corack algebras, perhaps in analogy to the theory of Hopf algebras. What relationships do they have with rack bialgebras (see \\citelist{\\cite{bardakov}\\cite{alexandre}})?\n \\item Racks \\cite{fenn} and quandles \\citelist{\\cite{joyce}\\cite{matveev}} were originally introduced to construct invariants of knots and 3-manifolds; see \\citelist{\\cite{book}\\cite{quandlebook}}. In the past decade, racks (for example, \\citelist{\\cite{randal}\\cite{shusterman}}) and quandles (for example, \\citelist{\\cite{fan}\\cite{bianchi}\\cite{bianchi2}\\cite{takahashi1}}) have also enjoyed applications as invariants of various algebro-geometric objects. Use algebraic racks to enhance these invariants, and compare them to other algebro-geometric invariants.\n\\end{enumerate}",
    "post_theorem_intro_text_len": 6843,
    "post_theorem_intro_text": "\\begin{proof}\n    We prove the theorem piecemeal in Propositions \\ref{prop:adj}, \\ref{prop:jacobi}, and \\ref{prop:functor} and Remark \\ref{rmk:recover}.\n\\end{proof}\n\n\\subsubsection{}\nTheorem \\ref{thm:main} is an algebraic analogue of a result of Kinyon's result for Lie racks in \\cite{kinyon}*{Thm.\\ 3.4}. \nCompared to that result, two advantages of Theorem \\ref{thm:main} are that it holds for all fields $k$ and gives an explicit formula for the Leibniz bracket $[\\cdot,\\cdot]$ that does not require computing any derivatives. That said, the Leibniz bracket agrees with the derivative of the \\emph{adjoint representation} of $Q$ at the identity; see Proposition \\ref{prop:adj}. \n\nThe last part of Theorem \\ref{thm:main} states that our construction recovers the usual functors sending affine algebraic groups to their Lie algebras (see, for example, \\cite{waterhouse}*{Sec.\\ 12.2}) and affine algebraic Lie racks over $\\mathbb{R}$ or $\\mathbb{C}$ to their Leibniz algebras (see \\cite{kinyon}). As racks and quandles have enjoyed increasing interest in both quantum algebra and algebraic geometry in the past decade, this paper lays the foundations for what we expect will be a very rich area of study; cf.\\ Section \\ref{sec:open}.\n\n\\subsection{Literature review}\nWhile there does not appear to be any published work on the algebraic \\emph{coquecigrue} problem over arbitrary fields $k$---see \\cite{skoda} for a preliminary report of unpublished work of Ba\\v{s}i\\'{c} and \\v{S}koda from 2008 in the case that $\\operatorname{char}(k)=0$---there is a rich literature on the \\emph{coquecigrue} problem for smooth manifolds over $\\mathbb{R}$ and $\\mathbb{C}$. See \\cite{ongay} for a discussion of the literature up to 2014.\n\nAlthough the first work on the \\emph{coquecigrue} problem is due to Datuashvili \\cite{Datuashvili} in 2004, most modern approaches to the problem employ Lie racks, which Kinyon \\cite{kinyon} introduced in 2007 in response to the problem. \nFor example, Covez \\cite{covez} in 2013 introduced \\emph{local augmented Lie racks}, which address the \\emph{coquecigrue} problem with the caveat that an algebraic structure is only defined in a neighborhood of the identity. By contrast, Bordemann and Wagemann \\cite{bordemann} in 2016 introduced a global rack-theoretic solution with the caveat of not being functorial. \n\nWithin the last decade, various authors have classified special types of Lie racks \\citelist{\\cite{abchir2}\\cite{benayadi}\\cite{dibartolo}}. These include Lie racks corresponding to certain kinds of nilpotent Leibniz algebras \\citelist{\\cite{larosa}\\cite{larosa2}} and symmetric Leibniz algebras \\cite{abchir}, as well as all Lie racks corresponding to Lie algebras \\cite{larosa}*{Thm.\\ 4.6}. For further work on the \\emph{coquecigrue} problem, see \\citelist{\\cite{alexandre}\\cite{dherin}}. For non-rack-theoretic approaches, see \\citelist{\\cite{smith}\\cite{rodriguez}\\cite{uslu}}.\n\n\\subsubsection{}\nAlthough there does not appear to be existing literature on rack schemes, we note that quandle varieties and ordinary set-theoretic racks have enjoyed increasing attention in algebraic geometry in the past decade. In particular, quandle varieties provide algebro-geometric analogues of Riemannian symmetric spaces \\citelist{\\cite{takahashi}\\cite{takahashi2}} and applications to geometric invariant theory \\cite{fan}. \n\nOrdinary racks and quandles provide invariants of not only knots and 3-manifolds but also various objects of study in algebraic geometry, particularly arithmetic algebraic geometry. For example, quandles provide invariants of arithmetic schemes \\cite{takahashi1} and categorical constructions of Hurwitz spaces \\citelist{\\cite{bianchi}\\cite{bianchi2}}. Racks have similar applications in the theory of Hurwitz spaces and algebraic function fields \\citelist{\\cite{randal}\\cite{shusterman}}. The increasing attention racks and quandles have received in algebraic geometry provides another motivation for developing a modern scheme-theoretic treatment of algebraic racks. \n\n\\begin{notationC}\n    We work over a field $k$ of arbitrary characteristic. \n    By a \\emph{scheme}, we mean a scheme of finite type over $k$. Unless we are specifically discussing Leibniz algebras or Lie algebras, by a \\emph{$k$-algebra} we mean a finitely generated commutative associative $k$-algebra. \n    Let $\\Sch$, $\\Aff$, $\\Alg$, $\\leib$, $\\mathsf{Grp}$, $\\mathsf{Grp}(\\Sch)$, and $\\mathsf{Set}$ denote the categories of schemes, affine schemes, $k$-algebras, Leibniz algebras over $k$, groups, algebraic groups (that is, group objects in $\\Sch$), and sets, respectively. Denote the $k$-algebra of \\emph{dual numbers} by $k[\\delta]\\coloneq k[\\delta]/(\\delta^2)$. Given a morphism of affine schemes $\\varphi\\colon\\Spec(B)\\to\\Spec(A)$, let $\\varphi^\\sharp$ denote the induced homomorphism $A\\to B$.\n\\end{notationC}\n\n\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:ptd-racks}, we recall the definition of the category of pointed racks $\\Rack$ and introduce pointed rack objects internal to cartesian monoidal categories.\n\nIn Section \\ref{sec:alg-racks}, we introduce the categories of algebraic racks $\\Rack(\\Sch)$ and corack algebras $\\Corack(\\Alg)$. We also discuss the functor of points approach for algebraic racks.\n\nIn Section \\ref{sec:corack}, we specialize our attention to the category of affine algebraic racks $\\Rack(\\Aff)$, which is equivalent to the opposite category $(\\Corack(\\Alg))\\op$. As examples, we compute the corack algebras corresponding to three classes of affine algebraic racks. We also show that corack algebras are almost never coassociative or cocommutative, unlike Hopf algebras of algebraic groups.\n\nIn Section \\ref{sec:pre-pf}, we discuss our approach to proving Theorem \\ref{thm:main}, and we recall various facts about $k$-derivations and tangent spaces.\n\nIn Section \\ref{sec:pf}, we prove Theorem \\ref{thm:main}. A major step in the proof is to show that the Leibniz bracket $[\\cdot,\\cdot]$ in Theorem \\ref{thm:main} agrees with the scheme-theoretic derivative $\\ad\\coloneq (d\\Ad)_e$ of the \\emph{adjoint representation} of $Q$ at the identity $e$.\n\nIn Section \\ref{sec:ol}, we illustrate Theorem \\ref{thm:main} with a class of affine algebraic racks $\\OL_n$ whose Leibniz algebras $\\ol_n(k)$ are not generally Lie algebras. \nThen we provide several results relating the structure of an affine algebraic rack to that of its Leibniz algebra, including sufficient conditions for the latter to be abelian. These results recover the analogous results for Lie algebras of affine algebraic groups.\n\nIn Section \\ref{sec:open}, we propose directions for future research.\n\n\\begin{ack}\n    An early draft of this paper was submitted as the final project for a course on affine algebraic groups taught by Prof.\\ Bogdan Ion at the University of Pittsburgh in fall 2025.\n\\end{ack}",
    "sketch": "We \"prove the theorem piecemeal in Propositions \\ref{prop:adj}, \\ref{prop:jacobi}, and \\ref{prop:functor} and Remark \\ref{rmk:recover}.\" A \"major step in the proof\" is \"to show that the Leibniz bracket $[\\cdot,\\cdot]$ in Theorem \\ref{thm:main} agrees with the scheme-theoretic derivative $\\ad\\coloneq (d\\Ad)_e$ of the \\emph{adjoint representation} of $Q$ at the identity $e$.\"",
    "expanded_sketch": "We prove the theorem piecemeal in the following results. \\begin{prop}\\label{prop:adj}\n    When considered as a map $\\ad\\colon\\q\\to\\mathfrak{gl}(\\q)$, the adjoint representation of $\\q$ is given by the left argument of the Leibniz bracket $D\\mapsto [D,\\cdot]$. That is, $\\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\\in\\q$. In particular, $\\q$ is closed under $[\\cdot,\\cdot]$.\n\\end{prop} \\begin{prop}\\label{prop:jacobi}\n    The convolution operation $[\\cdot,\\cdot]$ satisfies the Jacobi identity.\n\\end{prop} \\begin{prop}\\label{prop:functor}\n    The assignments $Q\\mapsto\\q$ and $\\phi\\mapsto \\phi'\\coloneq (d\\phi)_e$ define a covariant functor $\\Rack(\\Aff)\\to\\leib$.\n\\end{prop} We also use the following remark. \\begin{rmk}\\label{rmk:recover}\n    If $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\q,[\\cdot,\\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\\q$ at $e$. By the definition of $\\Conj(G)$, the adjoint representations $\\Ad$ of $Q$ and $\\ad$ of $\\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, J. S. Milne, \\emph{Algebraic Groups, Lie Groups, and their Arithmetic Subgroups}*{Rem.\\ 10.24}). Hence, the claim follows from the preceding proposition.\\footnote{Alternatively, one can use Example \\ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ agrees with the usual Lie bracket $(D\\otimes E-E\\otimes D)\\circ\\Delta$ on $\\q=\\operatorname{Lie}(G)$ (see W. C. Waterhouse, \\emph{Introduction to Affine Group Schemes}*{Sec.\\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\\eps\\circ S=\\eps$ from the axiom $\\mu\\circ (S\\otimes\\id)\\circ\\Delta=\\eta\\circ\\eps$ by post-composing with $\\eps$, and then deduce from these two equalities that $D\\circ S=-D$.}\n\n    Similarly, if $k=\\R$ or $\\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\\Ad$ and $\\ad$ coincide with the ones for Lie racks in \\cite{kinyon}, so $(\\q,[\\cdot,\\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.\n\\end{rmk} A major step in the proof is to show that the Leibniz bracket $[\\cdot,\\cdot]$ in the main theorem agrees with the scheme-theoretic derivative $\\ad\\coloneq (d\\Ad)_e$ of the \\emph{adjoint representation} of $Q$ at the identity $e$.",
    "expanded_theorem": "\\label{thm:main}\n    Let $Q=\\Spec(A)$ be an affine algebraic rack with unit $e\\in Q$, and let $\\mathfrak{q}\\coloneq \\Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\\cdot,\\cdot]\\colon\\mathfrak{q}\\times\\mathfrak{q}\\to\\mathfrak{q}$ by convolution with respect to the corack operation $\\nabla$ of $A$ (see below):\n    \\[\n    [D,E]\\coloneq (D\\otimes E)\\circ\\nabla.\n    \\]\n    Then the assignment $Q\\mapsto(\\mathfrak{q},[\\cdot,\\cdot])$ defines a covariant functor $\\Rack(\\Aff)\\to\\leib$ sending morphisms of affine algebraic racks $\\varphi\\colon Q\\to R$ to their derivatives $(d\\varphi)_e\\colon \\mathfrak{q}\\to\\mathfrak{r}$ at $e$.\n\n    Furthermore, if $G$ is an affine algebraic group and $Q=\\Conj(G)$ is its conjugation quandle scheme, then $(\\mathfrak{q},[\\cdot,\\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\\mathbb{R}$ or $\\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\\mathfrak{q},[\\cdot,\\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.,",
    "theorem_type": [
      "Algorithmic or Constructive",
      "Universal"
    ],
    "mcq": {
      "question": "Let $k$ be a field, and let $Q=\\operatorname{Spec}(A)$ be an affine algebraic rack over $k$ with unit $e\\in Q$. Write $\\nabla\\colon A\\to A\\otimes_k A$ for the corack operation of $A$, and define\n\\[\n\\mathfrak q:=\\operatorname{Der}_k(A,k),\\qquad [D,E]:=(D\\otimes E)\\circ \\nabla\\quad(D,E\\in\\mathfrak q).\n\\]\nHere $\\Rack(\\Aff)$ is the category of affine algebraic racks over $k$, and $\\leib$ is the category of Leibniz algebras over $k$. Also, for an affine algebraic group $G$, let $\\operatorname{Conj}(G)$ denote $G$ equipped with its conjugation quandle structure. Which statement holds for every such $Q$ (and, when applicable, in the special cases just described)?",
      "correct_choice": {
        "label": "A",
        "text": "The bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ makes $\\mathfrak q=\\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\\mapsto (\\mathfrak q,[\\cdot,\\cdot])$ extends to a covariant functor $\\Rack(\\Aff)\\to\\leib$ that sends each morphism of affine algebraic racks $\\varphi\\colon Q\\to R$ to its derivative at the unit, $(d\\varphi)_e\\colon \\mathfrak q\\to\\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\\operatorname{Conj}(G)$, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the usual Lie algebra of $G$; and if $k=\\mathbb R$ or $\\mathbb C$ and $Q(k)$ is a Lie rack, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ makes $\\mathfrak q=\\operatorname{Der}_k(A,k)$ into a Lie algebra, and the assignment $Q\\mapsto (\\mathfrak q,[\\cdot,\\cdot])$ extends to a contravariant functor $\\Rack(\\Aff)\\to\\leib$ that sends each morphism of affine algebraic racks $\\varphi\\colon Q\\to R$ to the pullback on derivations induced by $\\varphi^\\sharp$. Moreover, if $G$ is an affine algebraic group and $Q=\\operatorname{Conj}(G)$, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the usual Lie algebra of $G$; and if $k=\\mathbb R$ or $\\mathbb C$ and $Q(k)$ is a Lie rack, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$."
        },
        {
          "label": "C",
          "text": "The bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ makes $\\mathfrak q=\\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and if $G$ is an affine algebraic group with $Q=\\operatorname{Conj}(G)$, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the usual Lie algebra of $G$. Moreover, if $k=\\mathbb R$ or $\\mathbb C$ and $Q(k)$ is a Lie rack, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$."
        },
        {
          "label": "D",
          "text": "The bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ makes $\\mathfrak q=\\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\\mapsto (\\mathfrak q,[\\cdot,\\cdot])$ extends to a covariant functor $\\Rack(\\Aff)\\to\\leib$ that sends each morphism of affine algebraic racks $\\varphi\\colon Q\\to R$ to its derivative at the unit, $(d\\varphi)_e\\colon \\mathfrak q\\to\\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\\operatorname{Conj}(G)$, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the usual Lie algebra of $G$ if and only if $\\operatorname{char}(k)=0$; and if $k=\\mathbb R$ or $\\mathbb C$ and $Q(k)$ is a Lie rack, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$."
        },
        {
          "label": "E",
          "text": "The bracket $[D,E]=(D\\otimes E)\\circ\\nabla$ makes $\\mathfrak q=\\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\\mapsto (\\mathfrak q,[\\cdot,\\cdot])$ extends to a covariant functor $\\Rack(\\Aff)\\to\\leib$ that sends each morphism of affine algebraic racks $\\varphi\\colon Q\\to R$ to its derivative at the unit, $(d\\varphi)_e\\colon \\mathfrak q\\to\\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\\operatorname{Conj}(G)$, then $(\\mathfrak q,[\\cdot,\\cdot])$ identifies with the Lie algebra of $G$ only after replacing the convolution bracket by its antisymmetrization $(D\\otimes E-E\\otimes D)\\circ\\nabla$; and if $k=\\mathbb R$ or $\\mathbb C$ and $Q(k)$ is a Lie rack, then $(\\mathfrak q,[\\cdot,\\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "covariant derivative-at-unit functoriality and non-antisymmetric Leibniz nature",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "drops the functorial-extension assertion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "adds a spurious characteristic-zero restriction in the conjugation-case recovery",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "misstates the recovery step by requiring antisymmetrization instead of equality of the convolution bracket with the Lie bracket in the conjugation case",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the setup and definitions but does not reveal the correct conclusion. It does not explicitly state the functoriality, covariance, or the special-case identifications that distinguish the correct option."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-statement recognition item: the correct choice is a near-complete restatement of the full result, with distractors formed by small perturbations. It tests recall of the theorem more than selection among independently motivated conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in mathematically meaningful ways (Leibniz vs. Lie, covariant vs. contravariant, exact vs. weaker statement, characteristic restriction, antisymmetrization). However, the task is still mainly to recognize the exact theorem statement rather than generate or derive it."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and plausible: they target common confusions about duality/functoriality, antisymmetry, characteristic assumptions, and omission of a stronger functorial claim. The weaker-true option is especially effective."
      },
      "total_score": 5,
      "overall_assessment": "A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and uses high-quality distractors, but it is largely a direct restatement of a result rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.16062v1",
    "paper_link": "http://arxiv.org/abs/2512.16062v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:diagonal_to_ratio}\n    If Conjecture~\\ref{conj:weak.mult.RDC} holds, and $\\lim\\limits_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}$ exists and is equal to $\\mathcal{L}$, then\n    \\begin{equation*}     \n        f(n)=\\big(\\mathcal{L}^2+o(1)\\big)\\frac{n}{(\\log n)^2} .\n    \\end{equation*}",
      "start_pos": 8526,
      "end_pos": 8845,
      "label": "thm:diagonal_to_ratio"
    },
    "ref_dict": {
      "limit1": "\\begin{equation}\\label{limit1}\n    \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation}",
      "thm:lower": "\\begin{theorem}\\label{thm:lower}\n    The function $g(n)$ satisfies\n    \\begin{equation*}\n         \\liminf_{n \\to \\infty} g(n) \\ge L^2 \n    \\quad \\text{ and } \\quad \\limsup_{n \\to \\infty} g(n) \\ge M^2.\n    \\end{equation*}\n\\end{theorem}",
      "thm:diagonal_to_ratio": "\\begin{theorem}\\label{thm:diagonal_to_ratio}\n    If Conjecture~\\ref{conj:weak.mult.RDC} holds, and $\\lim\\limits_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}$ exists and is equal to $\\mathcal{L}$, then\n    \\begin{equation*}     \n        f(n)=\\big(\\mathcal{L}^2+o(1)\\big)\\frac{n}{(\\log n)^2} .\n    \\end{equation*} \n\\end{theorem}",
      "conj:weak.mult.RDC": "\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\le k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\le R(k, k).\n    \\end{equation*}\n\\end{conjecture}",
      "thm:g_upper": "\\begin{theorem}\\label{thm:g_upper}\n    The function $f(n)$ satisfies\n\\begin{equation}\\label{eq:thm.bound.f.1}\n f(n) \\le \\big( 3.71943+o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\nMoreover, if Conjecture~\\ref{conj:weak.mult.RDC} holds, then\n \\begin{equation}\\label{eq:thm.bound.f.2}\n    f(n) \\le \\big( 3.70831 + o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\n\\end{theorem}",
      "limit2": "\\begin{equation}\\label{limit2}\n    \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation}",
      "thm:upper": "\\begin{theorem}\\label{thm:upper}\n    The function $g(n)$ satisfies\n    \\begin{equation*}\n         \\limsup_{n \\to \\infty} g(n) \\le M^2 .\n    \\end{equation*}\n\\end{theorem}",
      "subsec:numerics": "\\begin{equation}\\label{eq:bound.logm}\n    \\log m \\le \\big(M+o_m(1)\\big) \\sqrt{st}.\n\\end{equation}\n\nNotice that any graph $H$ on $m$ vertices provides a red/blue edge-coloring of $K_m$ implying that $m < R(\\omega(H)+1, \\alpha(H)+1)$. Thus, by \\eqref{eq:bound.logm}, for every subgraph $G'$ of $G$ on $m$ vertices, we have\n\\begin{equation*}\n     \\omega(G') \\cdot \\alpha(G') \\ge \n\\Big( \\frac{1}{M^2} + o_m(1) \\Big) (\\log m)^2. \n\\end{equation*}\nSince $\\omega(G') \\le \\omega(G)$, we obtain that \\begin{equation*}\n    \\alpha(G') \\ge\n\\Big( \\frac{1}{M^2} + o_m(1) \\Big) \\frac{(\\log m)^2}{\\omega(G)}.\n\\end{equation*} \nHence, together with the fact that $\\log m = \\big(1+o_n(1)\\big) \\log n$, \\Cref{lem:erdos} yields\n\\begin{equation*}\n    \\chi(G) \\le \\big(M^2 +o_n(1)\\big)\\frac{n\\cdot\\omega(G)}{(\\log n)^2} + \\frac{n}{(\\log n)^3}.\n\\end{equation*}\nFinally, we conclude that\n\\begin{equation*}\n    \\frac{\\chi(G)}{\\omega(G)} \\le \\big(M^2 +o_n(1)\\big)\\frac{n}{(\\log n)^2}.\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{thm:g_upper}} \\label{subsec:numerics}\nIn this section, we prove Theorem~\\ref{thm:g_upper}. The currently best known upper bound for $R(k,k)$ is\n\\begin{equation}\\label{eq:diagonal_upper_bound}\n    R(k,k) \\le \\left( 4e^{-0.14e^{-1}} \\right)^{k+o(k)}.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3264,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nRamsey Theory, often described as the investigation of when order arises from chaos, is one of the most influential areas in Graph Theory. The \\emph{Ramsey number} $R(s,t)$ is the minimum $n$ such that every red/blue edge-coloring of the complete graph on $n$ vertices contains either a clique on $s$ vertices colored red or a clique on $t$ vertices colored blue. \n\nThe first explicit upper bound for Ramsey numbers was obtained by Erd\\H{o}s and Szekeres~\\cite{E-Sz} in 1935, who showed that $R(s,t) \\leqslant \\binom{s+t-2}{s-1}$. Note that in the \\emph{diagonal} case, this shows that $R(k,k)\\leqslant 4^{k+o(k)}$. \nOn the other hand, Erd\\H{o}s~\\cite{Ramsey.lower} established the lower bound $R(k,k)\\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\\'asz Local Lemma~\\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit\n\\begin{equation}\\label{limit1}\n    \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation} \n exists and, if so, what is its value. This question appears as Problem 1 in~\\cite{chung}, as Problem 77 in~\\url{https://www.erdosproblems.com/77}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}.\n\nIn 1967, Erd\\H{o}s~\\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\\frac{\\chi(G)}{\\omega(G)}$, where $\\chi(G)$ denotes the chromatic number of $G$ and $\\omega(G)$ the clique number of $G$. Formally, we have\n\\begin{equation*}\n    f(n)\\coloneq \\max \\Big\\{ \\ \\frac{\\chi(G)}{\\omega(G)} \\ : \n    \\ G \\text{ is a graph on } n \\text{ vertices } \\Big\\}.\n\\end{equation*}\n\nErd\\H{o}s showed that $f(n) = \\Theta(n/(\\log n)^2)$ and asked whether the following limit exists: \n\\begin{equation}\\label{limit2}\n    \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation} \nThis question also appears in~\\cite{erdos2}, as Problem 627 in~\\url{https://www.erdosproblems.com/627}, as Problem 53 in~\\cite{chung}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\leqslant k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\leqslant R(k, k).\n    \\end{equation*}\n\\end{conjecture}\n\nWe remark that Conjecture~\\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \\eqref{limit1} and \\eqref{limit2}, should they exist, under the assumption of Conjecture~\\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.",
    "context": "\\label{sec:intro}\n\nThe first explicit upper bound for Ramsey numbers was obtained by Erd\\H{o}s and Szekeres~\\cite{E-Sz} in 1935, who showed that $R(s,t) \\leqslant \\binom{s+t-2}{s-1}$. Note that in the \\emph{diagonal} case, this shows that $R(k,k)\\leqslant 4^{k+o(k)}$. \nOn the other hand, Erd\\H{o}s~\\cite{Ramsey.lower} established the lower bound $R(k,k)\\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\\'asz Local Lemma~\\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit\n\\begin{equation}\\label{limit1}\n \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation} \n exists and, if so, what is its value. This question appears as Problem 1 in~\\cite{chung}, as Problem 77 in~\\url{https://www.erdosproblems.com/77}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}.\n\nIn 1967, Erd\\H{o}s~\\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\\frac{\\chi(G)}{\\omega(G)}$, where $\\chi(G)$ denotes the chromatic number of $G$ and $\\omega(G)$ the clique number of $G$. Formally, we have\n\\begin{equation*}\n f(n)\\coloneq \\max \\Big\\{ \\ \\frac{\\chi(G)}{\\omega(G)} \\ : \n \\ G \\text{ is a graph on } n \\text{ vertices } \\Big\\}.\n\\end{equation*}\n\nErd\\H{o}s showed that $f(n) = \\Theta(n/(\\log n)^2)$ and asked whether the following limit exists: \n\\begin{equation}\\label{limit2}\n \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation} \nThis question also appears in~\\cite{erdos2}, as Problem 627 in~\\url{https://www.erdosproblems.com/627}, as Problem 53 in~\\cite{chung}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n For every $s, t, k \\in \\N$ such that $st \\leqslant k^2$ we have that \n \\begin{equation*} \n R(s,t) \\leqslant R(k, k).\n \\end{equation*}\n\\end{conjecture}\n\nWe remark that Conjecture~\\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \\eqref{limit1} and \\eqref{limit2}, should they exist, under the assumption of Conjecture~\\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\le k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\le R(k, k).\n    \\end{equation*}\n\\end{conjecture}\n\n\\begin{equation}\\label{limit1}\n    \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation}\n\n\\begin{equation}\\label{limit2}\n    \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation}",
    "full_context": "\\label{sec:intro}\n\nThe first explicit upper bound for Ramsey numbers was obtained by Erd\\H{o}s and Szekeres~\\cite{E-Sz} in 1935, who showed that $R(s,t) \\leqslant \\binom{s+t-2}{s-1}$. Note that in the \\emph{diagonal} case, this shows that $R(k,k)\\leqslant 4^{k+o(k)}$. \nOn the other hand, Erd\\H{o}s~\\cite{Ramsey.lower} established the lower bound $R(k,k)\\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\\'asz Local Lemma~\\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit\n\\begin{equation}\\label{limit1}\n \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation} \n exists and, if so, what is its value. This question appears as Problem 1 in~\\cite{chung}, as Problem 77 in~\\url{https://www.erdosproblems.com/77}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}.\n\nIn 1967, Erd\\H{o}s~\\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\\frac{\\chi(G)}{\\omega(G)}$, where $\\chi(G)$ denotes the chromatic number of $G$ and $\\omega(G)$ the clique number of $G$. Formally, we have\n\\begin{equation*}\n f(n)\\coloneq \\max \\Big\\{ \\ \\frac{\\chi(G)}{\\omega(G)} \\ : \n \\ G \\text{ is a graph on } n \\text{ vertices } \\Big\\}.\n\\end{equation*}\n\nErd\\H{o}s showed that $f(n) = \\Theta(n/(\\log n)^2)$ and asked whether the following limit exists: \n\\begin{equation}\\label{limit2}\n \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation} \nThis question also appears in~\\cite{erdos2}, as Problem 627 in~\\url{https://www.erdosproblems.com/627}, as Problem 53 in~\\cite{chung}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n For every $s, t, k \\in \\N$ such that $st \\leqslant k^2$ we have that \n \\begin{equation*} \n R(s,t) \\leqslant R(k, k).\n \\end{equation*}\n\\end{conjecture}\n\nWe remark that Conjecture~\\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \\eqref{limit1} and \\eqref{limit2}, should they exist, under the assumption of Conjecture~\\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\le k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\le R(k, k).\n    \\end{equation*}\n\\end{conjecture}\n\n\\begin{equation}\\label{limit1}\n    \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation}\n\n\\begin{equation}\\label{limit2}\n    \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation}\n\nErd\\H{o}s showed that $f(n) = \\Theta(n/(\\log n)^2)$ and asked whether the following limit exists: \n\\begin{equation}\\label{limit2}\n \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation} \nThis question also appears in~\\cite{erdos2}, as Problem 627 in~\\url{https://www.erdosproblems.com/627}, as Problem 53 in~\\cite{chung}, and on the webpage~\\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.\n\nWe remark that Conjecture~\\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \\eqref{limit1} and \\eqref{limit2}, should they exist, under the assumption of Conjecture~\\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.\n\nIn~\\cite{erdos}, Erd\\H{o}s mentioned that by their method, it would be easy to prove that \n\\begin{equation*}\n \\big(c+o(1)\\big) \\frac{n}{(\\log n)^2} \\le f(n) \n \\le \\big(C+o(1) \\big) \\frac{n}{(\\log n)^2},\n\\end{equation*}\nfor constants $c=\\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.\n\n\\begin{theorem}\\label{thm:g_upper}\n The function $f(n)$ satisfies\n\\begin{equation}\\label{eq:thm.bound.f.1}\n f(n) \\le \\big( 3.71943+o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\nMoreover, if Conjecture~\\ref{conj:weak.mult.RDC} holds, then\n \\begin{equation}\\label{eq:thm.bound.f.2}\n f(n) \\le \\big( 3.70831 + o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\n\\end{theorem}\n\n\\section{Proof of Theorem~\\ref{thm:diagonal_to_ratio}}\n\\label{sec:proof_1.2}\nIn this section, we establish upper and lower bounds sufficient to prove Theorem~\\ref{thm:diagonal_to_ratio}. We write $h = o_n(t)$ if the functions $t,h\\colon \\N \\to \\RR$ satisfy $\\lim\\limits_{n \\to \\infty} \\frac{h(n)}{t(n)} = 0$. When the variable $n$ is clear from the context, we simply write $h=o(t)$. Let us define \n\\begin{equation*}\ng(n) \\coloneqq \\frac{(\\log n)^2}{n} f(n),\\quad\nL \\coloneqq \\liminf\\limits_{k \\to \\infty} \\; \\frac{\\log (R(k,k))}{k}\n\\quad \\text{ and } \\quad \nM \\coloneq \\limsup_{t \\to \\infty } \\; \\max_{s\\le t} \\ \\frac{\\log(R(s,t))}{\\sqrt{st \\ }}\n\\end{equation*}\n\n\\begin{fact}\\label{fact} Under the assumption of Conjecture~\\ref{conj:weak.mult.RDC}, equality holds in \\eqref{ineq:limsups}.\n\\end{fact}\n\\begin{proof}\nFor $0 < s\\le t$, let $k$ be the positive integer such that $(k-1)^2 < st \\le k^2$. Let \n\\[D\\coloneq\\limsup_{k \\to \\infty} \\;\\frac{\\log (R(k,k))}{k}.\\]\nConjecture~\\ref{conj:weak.mult.RDC} implies that \n\\begin{equation*}\n R(s,t) \\le R(k,k) \\le 2^{(D+o_k(1))k}\n\\end{equation*}\n and, therefore, we have\n\\begin{equation*}\n \\frac{\\log(R(s,t))}{\\sqrt{st \\ }} \\le \\frac{\\log(R(k,k))}{k-1} \\le D +o_k(1) = D +o_t(1). \\qedhere\n\\end{equation*}\n\\end{proof}\n\nLet $G$ be a graph on $n$ vertices with no clique or independent set of size $k_n+1$. In other words, $\\alpha(G) \\le k_n$ and $\\omega(G) \\le k_n$. As $\\chi(G) \\ge n/\\alpha(G)$, we conclude that\n\\begin{equation}\\label{eq:lower.bound.inf.g}\n g(n)=\\frac{f(n)}{n/(\\log n)^2} \\ge \\frac{\\chi(G)}{\\omega(G)}\\cdot\\frac{(\\log n)^2}{n} \\ge \n\\frac{(\\log n)^2}{\\alpha(G)\\omega(G)}\\ge\n\\frac{(\\log n)^2}{k_n^2} \\ge\nL^2 + o_n(1),\n\\end{equation}\nwhere the last inequality is true by \\eqref{eq:bound.logn}. Hence, $\\liminf\\limits_{n \\to \\infty} g(n) \\ge L^2$.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:g_upper}]\nIf Conjecture~\\ref{conj:weak.mult.RDC} holds, then Theorem~\\ref{thm:upper}, Fact~\\ref{fact}, and \\eqref{eq:diagonal_upper_bound} imply~\\eqref{eq:thm.bound.f.2}. Indeed, \n\\begin{equation*}\n f(n) \\le \\big( ( \\log 4e^{-0.14e^{-1}} )^2 + o(1) \\big) \\frac{n}{(\\log n)^2} < \\big( 3.70831 + o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation*}\n\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\le k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\le R(k, k).\n    \\end{equation*}\n\\end{conjecture}\n\n\\begin{equation}\\label{limit1}\n    \\lim_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}\n\\end{equation}\n\n\\begin{equation}\\label{limit2}\n    \\lim_{n \\to \\infty} \\frac{f(n)}{n/(\\log n)^2}.\n\\end{equation}\n\n\\begin{theorem}\\label{thm:diagonal_to_ratio}\n    If Conjecture~\\ref{conj:weak.mult.RDC} holds, and $\\lim\\limits_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}$ exists and is equal to $\\mathcal{L}$, then\n    \\begin{equation*}     \n        f(n)=\\big(\\mathcal{L}^2+o(1)\\big)\\frac{n}{(\\log n)^2} .\n    \\end{equation*} \n\\end{theorem}\n\n\\begin{theorem}\\label{thm:g_upper}\n    The function $f(n)$ satisfies\n\\begin{equation}\\label{eq:thm.bound.f.1}\n f(n) \\le \\big( 3.71943+o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\nMoreover, if Conjecture~\\ref{conj:weak.mult.RDC} holds, then\n \\begin{equation}\\label{eq:thm.bound.f.2}\n    f(n) \\le \\big( 3.70831 + o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:upper}\n    The function $g(n)$ satisfies\n    \\begin{equation*}\n         \\limsup_{n \\to \\infty} g(n) \\le M^2 .\n    \\end{equation*}\n\\end{theorem}",
    "post_theorem_intro_text_len": 1699,
    "post_theorem_intro_text": "In~\\cite{erdos}, Erd\\H{o}s mentioned that by their method, it would be easy to prove that \n\\begin{equation*}\n    \\big(c+o(1)\\big) \\frac{n}{(\\log n)^2} \\leqslant f(n) \n    \\leqslant \\big(C+o(1) \\big)  \\frac{n}{(\\log n)^2},\n\\end{equation*}\nfor constants $c=\\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.\n\n\\begin{theorem}\\label{thm:g_upper}\n    The function $f(n)$ satisfies\n\\begin{equation}\\label{eq:thm.bound.f.1}\n f(n) \\leqslant \\big( 3.71943+o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\nMoreover, if Conjecture~\\ref{conj:weak.mult.RDC} holds, then\n \\begin{equation}\\label{eq:thm.bound.f.2}\n    f(n) \\leqslant \\big( 3.70831 + o(1) \\big) \\frac{n}{(\\log n)^2}.\n\\end{equation}\n\\end{theorem}\n\nThis upper bound on $f(n)$ is the first improvement in the asymptotics of $f(n)$ since 1967.   \n\n\\subsection*{Organization of the paper} In Section~\\ref{sec:proof_1.2}, we introduce some notation used throughout the paper and state bounds on $f(n)$, namely Theorems~\\ref{thm:lower} and~\\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. Theorem~\\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\\ref{sec:proof_1.2}. In Section~\\ref{subsec:numerics}, we establish the numerical bounds of Theorem~\\ref{thm:g_upper}. In Appendix~\\ref{sec:DiagonalConjecture}, we discuss versions of the Ramsey Diagonal Conjecture and how they relate to Conjecture~\\ref{conj:weak.mult.RDC}.",
    "sketch": "The post-theorem introduction says that in Section~\\ref{sec:proof_1.2} the authors \"state bounds on $f(n)$, namely Theorems~\\ref{thm:lower} and~\\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$.\" It then states that \"Theorem~\\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\\ref{sec:proof_1.2}.\"",
    "expanded_sketch": "The post-theorem introduction says that next the authors state bounds on $f(n)$, namely the following results.\n\n\\begin{theorem}\\label{thm:lower}\n    The function $g(n)$ satisfies\n    \\begin{equation*}\n         \\liminf_{n \\to \\infty} g(n) \\ge L^2 \n    \\quad \\text{ and } \\quad \\limsup_{n \\to \\infty} g(n) \\ge M^2.\n    \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:upper}\n    The function $g(n)$ satisfies\n    \\begin{equation*}\n         \\limsup_{n \\to \\infty} g(n) \\le M^2 .\n    \\end{equation*}\n\\end{theorem}\n\nThese establish a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. It then states that, in establishing the main theorem, it is an immediate consequence of these results, whose proofs appear later.",
    "expanded_theorem": "\\label{thm:diagonal_to_ratio}\nIf the following conjecture holds:\n\\begin{conjecture}\\label{conj:weak.mult.RDC}\n    For every $s, t, k \\in \\N$ such that $st \\le k^2$ we have that    \n    \\begin{equation*}    \n        R(s,t) \\le R(k, k).\n    \\end{equation*}\n\\end{conjecture}\nand $\\lim\\limits_{k \\to \\infty} \\frac{\\log(R(k,k))}{k}$ exists and is equal to $\\mathcal{L}$, then\n\\begin{equation*}     \n    f(n)=\\big(\\mathcal{L}^2+o(1)\\big)\\frac{n}{(\\log n)^2} .\n\\end{equation*}",
    "theorem_type": [
      "Implication",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $R(s,t)$ denote the Ramsey number, i.e. the least integer $N$ such that every graph on $N$ vertices contains either a clique of size $s$ or an independent set of size $t$. For each $n$, define\n\\[\nf(n)\\coloneq \\max\\left\\{\\frac{\\chi(G)}{\\omega(G)}: G \\text{ is a graph on } n \\text{ vertices}\\right\\},\n\\]\nwhere $\\chi(G)$ is the chromatic number of $G$ and $\\omega(G)$ is the clique number of $G$. Assume that for every $s,t,k\\in\\mathbb N$ with $st\\le k^2$, one has\n\\[\nR(s,t)\\le R(k,k),\n\\]\nand also that the limit\n\\[\n\\lim_{k\\to\\infty}\\frac{\\log(R(k,k))}{k}\n\\]\nexists and is equal to $\\mathcal L$ (with logarithms in base $2$). Under these assumptions, which statement about $f(n)$ holds as $n\\to\\infty$?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\nf(n)=\\big(\\mathcal L^2+o(1)\\big)\\frac{n}{(\\log n)^2},\n\\]\nas $n\\to\\infty$."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\nf(n)=\\big(\\mathcal L+o(1)\\big)\\frac{n}{(\\log n)^2},\n\\]\nas $n\\to\\infty$."
        },
        {
          "label": "C",
          "text": "One has\n\\[\nf(n)=O\\!\\left(\\frac{n}{(\\log n)^2}\\right),\n\\]\nas $n\\to\\infty$."
        },
        {
          "label": "D",
          "text": "One has\n\\[\n\\limsup_{n\\to\\infty}\\frac{f(n)}{n/(\\log n)^2}=\\mathcal L^2,\n\\]\nas $n\\to\\infty$."
        },
        {
          "label": "E",
          "text": "One has\n\\[\nf(n)=\\big(\\mathcal L^2+o(1)\\big)\\frac{n}{\\log n},\n\\]\nas $n\\to\\infty$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "square-on-the-constant from combining lower/upper bounds",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the precise asymptotic constant \\mathcal L^2 and retained only the growth order",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "full limit/equality conclusion replaced by only a limsup statement",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "normalization by (\\log n)^2 in g(n)=f(n)/(n/(\\log n)^2)",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or uniquely signal choice A. It gives definitions and hypotheses, then asks for the asymptotic consequence for f(n)."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: under the stated assumptions, the student is asked to identify the theorem's exact conclusion. It does not substantially reframe the result into a new setting or inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact asymptotic from nearby variants such as a weaker O-bound, a limsup statement, or an incorrect constant/normalization. However, the main task is still recognition of the precise theorem statement rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: they vary the constant, weaken the mode of convergence, or alter the logarithmic normalization. These reflect realistic failure modes in asymptotic reasoning."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is close to a direct restatement of a result rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.16220v2",
    "paper_link": "http://arxiv.org/abs/2512.16220v2",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\mathbb{Z}[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\varepsilon(p)\\right\\}\n\\,.\n$$\nHere $\\varepsilon(p)\\to 0$ and is effectively computable.",
      "start_pos": 20703,
      "end_pos": 21260,
      "label": "T:1"
    },
    "ref_dict": {
      "S:EH": "\\label{S:EH}\n\nLet $f\\in\\Z[x]$ and write\n$$\n  f(x)=a_nx^n+a_{n-1}x^{n-1}+\\dots+a_1x+a_0\n  \\,.\n$$\nWe say that $f$ is Eisenstein at the prime $p$\n(or $p$-Eisenstein for short) if \n$p \\mid a_i$\nfor $i=0,\\",
      "S:reallyFinal": "\\label{S:reallyFinal}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:2}]\n\nLet $Y>0$ be a parameter to be chosen later.\nLet $\\mathcal{F}^*_{p,n}$ denote the subset of \n$\\mathcal{F}_{p,n}$ consisting of polynom",
      "S:gotime": "\\begin{proof}[Proof of Lemma~\\ref{L:heilbronn}]\nSuppose $p$ is totally ramified and we can write $p=a+b$ as in\nthe statement of the result.\nBy way of contradiction, suppose $K$ is norm-Euclidean.\nBy hypothesis, $(p)=\\mathfrak{p}^n$, where $N(\\mathfrak{p})=p$.\nIt follows that $p=u\\pi^n$ in $K$,\nfor some $u\\in\\mathcal{O}_K^\\times$ and some\n$\\pi\\in\\mathcal{O}_K$ irreducible with $N(\\pi)=p$.\nSince $a$ is an $n$-th power residue modulo $n$, there exists $x\\in\\Z$ such that $x^n\\equiv a\\pmod{p}$.\nUsing the norm-Euclidean condition, we find\n$x=\\gamma\\pi+\\rho$ for some $\\gamma,\\rho\\in\\mathcal{O}_K$\nwith $|N(\\rho)|<|N(\\pi)|=p$.  We have $x\\equiv \\rho\\pmod{\\pi}$\nand hence $x\\equiv \\rho^\\sigma\\pmod{\\pi}$ for any\nembedding $\\sigma:K\\to\\C$.  This leads to\n$a\\equiv x^n\\equiv N(\\rho)\\pmod{p}$.\nWe have $0<a<p$ and $-p<N(\\rho)<p$.\nThus either $N(\\rho)=a$ or $N(\\rho)+p=a$.\nThis implies one of $\\{a,-b\\}$ is a norm, \na contradiction.\n\\end{proof}\n\n\\section{Counting $p$-Eisenstein polynomials with local specifications}\\label{S:gotime}\n\nGiven $a=(a_0,\\dots,a_{n-1})\\in\\Z^n$, we define\n$$\n\\Ht(a)=\\max\\{|a_0|,\\dots,|a_{n-1}|\\}\n$$\nso that it agrees  with our polynomial height.\nThe next result is what allows us to count polynomials \nof bounded height with local specifications.\nThis type of result is certainly well-known,\nbut the proof is short and so we include it for the \nsake of completeness.\n\\begin{lemma}\\label{L:sieve}\nLet $S$ be a subset of $(\\Z/m\\Z)^n$.  We have\n$$\n\\#\\{a\\in\\Z^n\\mid a\\hspace{-1ex}\\mod{m}\\in S\\,,\\;\\Ht(a)\\leq X\\}\n=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n2^n m(2X)^{n-1}\\right)\n\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{A}\\colonequals [-X,X]^n\\cap\\Z^n$.\nChoose the maximal $k\\in\\Z^+$ such that\n$$\n  \\mathcal{B}\\colonequals (-km,km]^n\\cap\\Z^n\\subseteq \\mathcal{A}\n  \\,.\n$$\nThe set $\\mathcal{B}$ is chosen so that\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}\n=\n|S|(2k)^n\n$$\nUsing the Binomial Theorem,\none has\n$$\n\\left(\\frac{X}{m}\\right)^n \\ge k^n \\ge \\left(\\frac{X}{m}-1\\right)^n\n=\n\\left(\\frac{X}{m}\\right)^n+O\\left(n2^n\\left(\\frac{X}{m}\\right)^{n-1}\\right)\n\\,,\n$$\nand therefore\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n 2^n m(2X)^{n-1}\\right)\n\\,.\n$$\nNext one bounds\n$$\n  \\#(\\mathcal{A}\\setminus\\mathcal{B})\n  \\leq\n  (2X+1)^n-(2km)^n\n  \\ll\n  n2^n m(2X)^{n-1}\n  \\,.\n$$\nThe result follows.\n\\end{proof}",
      "T:2": "\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\eps}(p)\n\\,,\n$$\nwhere $\\hat{\\eps}(p)\\to 0$ and is effectively computable.\n\\end{theorem}",
      "S:final": "\\begin{proof}\nFirst note that \n    $f(k) = {p\\choose k}/p^k$\n    is a decreasing function. Indeed, for $k\\le p-1$\n    $$\\frac{f(k)}{f(k+1)} = \\frac{(k+1)p}{p-k} > 0,$$\n    so $f(k) > f(k+1)$ for all $k\\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\\ge 2$,\n    $$\\frac{{p\\choose 2}}{p^2} - \\frac{{p\\choose 3}}{p^3} \\le \\frac{A_p(n)}{p^n} \\le \\frac{{p\\choose 2}}{p^2}.$$\n    \\end{proof}\n\n\\section{Proof of the main result}\\label{S:final}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:1}]\nAdopt the notation from the statement of the theorem.\nFurther, write $N_{p,n}(X)$ to denote the number of \n$f\\in\\mathcal{F}_{p,n}$ with $\\Ht(f)\\leq X$.\nWe have $N_{p,n}(X)\\sim E(p)(2X)^n$ where $E(p)$\nis given in (\\ref{E:eisenstein}).\nThis follows from~\\cite{MR1995564} or alternatively,\nfrom Proposition~\\ref{P:count} with $t=0$.\n\nFirst we will establish the lower bound\n$\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\geq 2/27$.\nAssume $p\\neq 7,11,19$.\nBy Lemma~\\ref{lem: Froby}, we can write $p=2u+3v$ with\n$(2,u)=1$ and $(3,v)=1$ for $p\\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.\nTherefore, provided $f(x)$ has no roots in $\\F_2$\nor $\\F_3$, Heilbronn's criterion applies.\n(Recall Lemma~\\ref{L:heilbronn} and the discussion in~\\S\\ref{S:EH}.)\nWrite $N^*_{p,n}(X)$ to denote the counting function for this collection\nof polynomials.\nUsing Proposition~\\ref{P:count}\nwith $q_1=2$, $q_2=3$, we have\n$$\n  N^*_{p,n}(X)\\sim E_p(n)C_2(n)C_3(n)(2X)^n\n  \\,,\n$$\nand therefore,\nvia Lemma~\\ref{lem: A_p formula} with $n\\geq 3$,\nwe find\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{n,p}(X)\\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27\n\\,.\n$$\n\nIn the case where $p\\in\\{7,11,19\\}$,\nwe can write $p=2u+5v$ with\n$(2,u)=1$ and $(5,v)=1$.\nThe same approach proves\n$$\n\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\\geq C_2(n)C_5(n)\\geq(1/4)(8/25)>2/27\n\\,.\n$$\nIndeed, by Lemma~\\ref{L:Cbounds},\nas $n\\geq 3$ is odd,\none has $C_5(n)\\geq 8/25$.\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n  D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1  \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n  &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n  \\notag\\\\\n  &\\quad\\leq\n  \\prod_{i=1}^t(1-C(q_i))+\n  \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n  \\notag\\\\\n  &\\quad\\leq\n  (1+tD)\\prod_{i=1}^t(1-C(q_i))\n  \\notag\\\\\n  &\\quad\\leq\n  \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}",
      "S:local": "\\label{S:local}\n\nThe following lemma gives a formula for our local density\n$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity\ndefined immediately before Proposition~\\ref{P:count}\nand $A_p(n)$ is defi",
      "T:1": "\\begin{theorem}\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,.\n$$\nHere $\\eps(p)\\to 0$ and is effectively computable.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5964,
    "pre_theorem_intro_text": "We are interested in proving statistical results concerning\npolynomials with integer coefficents.  Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite.  \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n  N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n  \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n  \\delta(X)=\\frac{N^*(X)}{N(X)}\n  \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n   N(X)\\sim (2X)^n\n   \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}).  Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n  \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n  \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields.  As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$.  This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps.  The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\nOne knows that \nall norm-Euclidean fields have class number one, but the converse is not\ntrue in general. Recall that $K$ having class number one is equivalent to $\\O_K$ being a unique factorization domain.\nWe expect the norm-Euclidean property to be rather rare in general, but proving this is quite difficult.\nOn the other hand, if $K$ is generated by an Eisenstein polynomial, we actually have a tool available to us, due to the existence of a totally ramified prime.  More specifically, if $f$ is Eisenstein at the prime $p$, then $p$ is totally ramified in $K$; that is, $(p)=\\mathfrak{p}^n$ for some prime ideal $\\mathfrak{p}$.  The tool we refer to is as follows\n(see~\\cite{MR1574525, MR35313, MR43134, MR752120}):\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above.  Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is:  How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant.  \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said.  For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$.  We establish the following result:",
    "context": "We are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields. As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$. This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above. Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$. We establish the following result:",
    "full_context": "We are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields. As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$. This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above. Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$. We establish the following result:\n\nSuppose $n\\geq 3$ is odd and $p\\geq 5$ is prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nWe order our polynomials by the natural height $\\Ht(f)$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,,\n$$\nwhere $\\eps(p)\\to 0$ and is effectively computable.\nIn particular, the lower density tends to $1$ as $p\\to\\infty$\nuniformly in $n$.\nWe also give a version of this result where we weaken the\ncondition on $\\gcd(p-1,n)$.\n\nWe are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\Z[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\Z[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\Z[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\Z[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\Z[x]$ if the criterion applies to the number field\n$K=\\Q[x]/(f)$. We establish the following result:\n\nWhen $p$ is large, there are many ways to write $p=a+b$ and hence\ndetermining the exact density of polynomials that satisfy Heilbronn's\ncriterion is likely very difficult. However, the previous\ntheorem says that the lower density tends to $1$ as $p\\to\\infty$,\nuniformly in the variable $n$.\n\n\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\eps}(p)\n\\,,\n$$\nwhere $\\hat{\\eps}(p)\\to 0$ and is effectively computable.\n\\end{theorem}\n\nBefore proceeding, we require a couple definitions.\nFirst we define\n\\begin{equation}\\label{E:eisenstein}\nE_p(n)\\colonequals\n\\frac{1}{p^n}-\\frac{1}{p^{n+1}}\n\\,,\n\\end{equation}\nwhich is the $p$-Eisenstein density that contributes\nto (\\ref{E:dub}).\nSecondly, we write $C_q(n)$ to denote --- the number of\n$(a_0,\\dots,a_{n-1})\\in(\\Z/q\\Z)^n$ such that the corresponding polynomial\nin $\\F_q[x]$ has no roots in $\\F_q$ ---\ndivided by $q^n$.\nWe are now ready to prove the following result, which\nwill be used in the proof of Theorem~\\ref{T:1}.\n\\begin{prop}\\label{P:count}\nLet $n\\in\\Z^+$.\nSuppose $q_1<\\dots<q_t<p$ are primes.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nThe number of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which\n$f$ has no roots in $\\F_q$ for all $q\\in\\{q_1,\\dots,q_t\\}$\nequals\n$$\n E_p(n)\\left(\\prod_{i=1}^t C_{q_i}(n)\\right)(2X)^n+O\\left(n 2^n q_1\\dots q_t p^2 (2X)^{n-1}\\right)\n \\,.\n$$\n\\end{prop}\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1 \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n \\notag\\\\\n &\\quad\\leq\n \\prod_{i=1}^t(1-C(q_i))+\n \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n \\notag\\\\\n &\\quad\\leq\n (1+tD)\\prod_{i=1}^t(1-C(q_i))\n \\notag\\\\\n &\\quad\\leq\n \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}\n\n\\begin{theorem}\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,.\n$$\nHere $\\eps(p)\\to 0$ and is effectively computable.\n\\end{theorem}",
    "post_theorem_intro_text_len": 2022,
    "post_theorem_intro_text": "When $p$ is large, there are many ways to write $p=a+b$ and hence\ndetermining the exact density of polynomials that satisfy Heilbronn's\ncriterion is likely very difficult.  However, the previous\ntheorem says that the lower density tends to $1$ as $p\\to\\infty$,\nuniformly in the variable $n$.\n\nAs a result, we immediately obtain the following consequence:\nWhen $n\\geq 3$ is odd,\na positive proportion of Eisenstein polynomials of degree $n$\nfail to generate norm-Euclidean fields.\n\nThe condition from Theorem~\\ref{T:1} that $\\mathrm{gcd}(p-1,n) = 1$\nis not a severe restriction as there are infinitely \nmany primes $p$ that satisfy this condition for each fixed $n$.\nNonetheless, we show that we can\nrelax this condition at the expense of considering larger primes.\nThis will allow us to deal with the case where $n$ is even.\n\n\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\varepsilon}(p)\n\\,,\n$$\nwhere $\\hat{\\varepsilon}(p)\\to 0$ and is effectively computable.\n\\end{theorem}\n\n\\begin{corollary}\\label{C:1}\nFix an integer $n\\geq 2$.\nA positive proportion of Eisenstein polynomials of degree $n$\nfail to generate norm-Euclidean fields.\n\\end{corollary}\n\nIn \\S\\ref{S:EH} we discuss Eisenstein polynomials and Heilbronn's criterion,\nand outline the strategy of our proof.  In addition, we establish some preliminary \nlemmas.  In the case where the proofs are very short, we err on the side of giving\nproofs of known results for the sake of completeness.\nIn \\S\\ref{S:gotime} we prove our main counting result for Eisenstein polynomials\nsatisfying certain other local conditions.  In \\S\\ref{S:local} we prove a formula\nfor the relevant local densities, which is then used to establish the necessary bounds.\nIn \\S\\ref{S:final}\nwe put everything together and prove Theorem~\\ref{T:1}. Finally, in \\S\\ref{S:reallyFinal}, we prove Theorem \\ref{T:2}.",
    "sketch": "In \\,\\S\\ref{S:EH} the authors “discuss Eisenstein polynomials and Heilbronn's criterion, and outline the strategy of our proof,” and “establish some preliminary lemmas.” Then “in \\,\\S\\ref{S:gotime} we prove our main counting result for Eisenstein polynomials satisfying certain other local conditions.” Next, “in \\,\\S\\ref{S:local} we prove a formula for the relevant local densities, which is then used to establish the necessary bounds.” Finally, “in \\,\\S\\ref{S:final} we put everything together and prove Theorem~\\ref{T:1}.”",
    "expanded_sketch": "\\label{S:EH}\n\nLet $f\\in\\Z[x]$ and write\n$$\n  f(x)=a_nx^n+a_{n-1}x^{n-1}+\\dots+a_1x+a_0\n  \\,.\n$$\nWe say that $f$ is Eisenstein at the prime $p$\n(or $p$-Eisenstein for short) if \n$p \\mid a_i$\nfor $i=0,\\,\n\n\\begin{proof}[Proof of Lemma~\\ref{L:heilbronn}]\nSuppose $p$ is totally ramified and we can write $p=a+b$ as in\nthe statement of the result.\nBy way of contradiction, suppose $K$ is norm-Euclidean.\nBy hypothesis, $(p)=\\mathfrak{p}^n$, where $N(\\mathfrak{p})=p$.\nIt follows that $p=u\\pi^n$ in $K$,\nfor some $u\\in\\mathcal{O}_K^\\times$ and some\n$\\pi\\in\\mathcal{O}_K$ irreducible with $N(\\pi)=p$.\nSince $a$ is an $n$-th power residue modulo $n$, there exists $x\\in\\Z$ such that $x^n\\equiv a\\pmod{p}$.\nUsing the norm-Euclidean condition, we find\n$x=\\gamma\\pi+\\rho$ for some $\\gamma,\\rho\\in\\mathcal{O}_K$\nwith $|N(\\rho)|<|N(\\pi)|=p$.  We have $x\\equiv \\rho\\pmod{\\pi}$\nand hence $x\\equiv \\rho^\\sigma\\pmod{\\pi}$ for any\nembedding $\\sigma:K\\to\\C$.  This leads to\n$a\\equiv x^n\\equiv N(\\rho)\\pmod{p}$.\nWe have $0<a<p$ and $-p<N(\\rho)<p$.\nThus either $N(\\rho)=a$ or $N(\\rho)+p=a$.\nThis implies one of $\\{a,-b\\}$ is a norm, \na contradiction.\n\\end{proof}\n\n\\section{Counting $p$-Eisenstein polynomials with local specifications}\\label{S:gotime}\n\nGiven $a=(a_0,\\dots,a_{n-1})\\in\\Z^n$, we define\n$$\n\\Ht(a)=\\max\\{|a_0|,\\dots,|a_{n-1}|\\}\n$$\nso that it agrees  with our polynomial height.\nThe next result is what allows us to count polynomials \nof bounded height with local specifications.\nThis type of result is certainly well-known,\nbut the proof is short and so we include it for the \nsake of completeness.\n\\begin{lemma}\\label{L:sieve}\nLet $S$ be a subset of $(\\Z/m\\Z)^n$.  We have\n$$\n\\#\\{a\\in\\Z^n\\mid a\\hspace{-1ex}\\mod{m}\\in S\\,,\\;\\Ht(a)\\leq X\\}\n=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n2^n m(2X)^{n-1}\\right)\n\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{A}\\colonequals [-X,X]^n\\cap\\Z^n$.\nChoose the maximal $k\\in\\Z^+$ such that\n$$\n  \\mathcal{B}\\colonequals (-km,km]^n\\cap\\Z^n\\subseteq \\mathcal{A}\n  \\,.\n$$\nThe set $\\mathcal{B}$ is chosen so that\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}\n=\n|S|(2k)^n\n$$\nUsing the Binomial Theorem,\none has\n$$\n\\left(\\frac{X}{m}\\right)^n \\ge k^n \\ge \\left(\\frac{X}{m}-1\\right)^n\n=\n\\left(\\frac{X}{m}\\right)^n+O\\left(n2^n\\left(\\frac{X}{m}\\right)^{n-1}\\right)\n\\,,\n$$\nand therefore\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n 2^n m(2X)^{n-1}\\right)\n\\,.\n$$\nNext one bounds\n$$\n  \\#(\\mathcal{A}\\setminus\\mathcal{B})\n  \\leq\n  (2X+1)^n-(2km)^n\n  \\ll\n  n2^n m(2X)^{n-1}\n  \\,.\n$$\nThe result follows.\n\\end{proof}\n\n\\label{S:local}\n\nThe following lemma gives a formula for our local density\n$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity\ndefined immediately before Proposition~\\ref{P:count}\nand $A_p(n)$ is defi\n\n\\begin{proof}\nFirst note that \n    $f(k) = {p\\choose k}/p^k$\n    is a decreasing function. Indeed, for $k\\le p-1$\n    $$\\frac{f(k)}{f(k+1)} = \\frac{(k+1)p}{p-k} > 0,$$\n    so $f(k) > f(k+1)$ for all $k\\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\\ge 2$,\n    $$\\frac{{p\\choose 2}}{p^2} - \\frac{{p\\choose 3}}{p^3} \\le \\frac{A_p(n)}{p^n} \\le \\frac{{p\\choose 2}}{p^2}.$$\n    \\end{proof}\n\n\\section{Proof of the main result}\\label{S:final}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:1}]\nAdopt the notation from the statement of the theorem.\nFurther, write $N_{p,n}(X)$ to denote the number of \n$f\\in\\mathcal{F}_{p,n}$ with $\\Ht(f)\\leq X$.\nWe have $N_{p,n}(X)\\sim E(p)(2X)^n$ where $E(p)$\nis given in (\\ref{E:eisenstein}).\nThis follows from~\\cite{MR1995564} or alternatively,\nfrom Proposition~\\ref{P:count} with $t=0$.\n\nFirst we will establish the lower bound\n$\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\geq 2/27$.\nAssume $p\\neq 7,11,19$.\nBy Lemma~\\ref{lem: Froby}, we can write $p=2u+3v$ with\n$(2,u)=1$ and $(3,v)=1$ for $p\\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.\nTherefore, provided $f(x)$ has no roots in $\\F_2$\nor $\\F_3$, Heilbronn's criterion applies.\n(Recall Lemma~\\ref{L:heilbronn} and the discussion above.)\nWrite $N^*_{p,n}(X)$ to denote the counting function for this collection\nof polynomials.\nUsing Proposition~\\ref{P:count}\nwith $q_1=2$, $q_2=3$, we have\n$$\n  N^*_{p,n}(X)\\sim E_p(n)C_2(n)C_3(n)(2X)^n\n  \\,,\n$$\nand therefore,\nvia Lemma~\\ref{lem: A_p formula} with $n\\geq 3$,\nwe find\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{n,p}(X)\\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27\n\\,.\n$$\n\nIn the case where $p\\in\\{7,11,19\\}$,\nwe can write $p=2u+5v$ with\n$(2,u)=1$ and $(5,v)=1$.\nThe same approach proves\n$$\n\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\\geq C_2(n)C_5(n)\\geq(1/4)(8/25)>2/27\n\\,.\n$$\nIndeed, by Lemma~\\ref{L:Cbounds},\nas $n\\geq 3$ is odd,\none has $C_5(n)\\geq 8/25$.\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n  D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1  \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n  &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n  \\notag\\\\\n  &\\quad\\leq\n  \\prod_{i=1}^t(1-C(q_i))+\n  \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n  \\notag\\\\\n  &\\quad\\leq\n  (1+tD)\\prod_{i=1}^t(1-C(q_i))\n  \\notag\\\\\n  &\\quad\\leq\n  \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}\n\nThis completes the proof of the main theorem.",
    "expanded_theorem": "\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\mathbb{Z}[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\varepsilon(p)\\right\\}\n\\,.\n$$\nHere $\\varepsilon(p)\\to 0$ and is effectively computable.",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(n\\ge 3\\) be an odd integer and \\(p\\ge 5\\) a prime such that \\(\\gcd(p-1,n)=1\\). Let \\(\\mathcal F_{p,n}\\) be the set of monic polynomials\n\\[\nf(x)=x^n+a_{n-1}x^{n-1}+\\cdots+a_0\\in \\mathbb Z[x]\n\\]\nof degree \\(n\\) that are Eisenstein at \\(p\\), i.e. \\(p\\mid a_i\\) for every \\(0\\le i\\le n-1\\) and \\(p^2\\nmid a_0\\). For such a polynomial, define its height by\n\\[\n\\operatorname{Ht}(f)=\\max\\{1,|a_{n-1}|,\\dots,|a_0|\\}.\n\\]\nFor each \\(X>0\\), let \\(\\delta_{p,n}(X)\\) be the proportion of polynomials \\(f\\in \\mathcal F_{p,n}\\) with \\(\\operatorname{Ht}(f)\\le X\\) for which Heilbronn's criterion applies to the number field \\(K=\\mathbb Q[x]/(f)\\), meaning that there exist integers \\(a,b\\) with \\(p=a+b\\), such that \\(a\\) is an \\(n\\)-th power residue modulo \\(p\\) and neither \\(a\\) nor \\(-b\\) is a norm from \\(\\mathcal O_K\\). Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(p)\\right\\},\n\\]\nwhere \\(\\varepsilon(p)\\) is effectively computable and satisfies \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\n\\lim_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(p)\\right\\},\n\\]\nwhere \\(\\varepsilon(p)\\) is effectively computable and satisfies \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
        },
        {
          "label": "C",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\frac{2}{27}.\n\\]"
        },
        {
          "label": "D",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge 1-\\varepsilon(p),\n\\]\nwhere \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
        },
        {
          "label": "E",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(n)\\right\\},\n\\]\nwhere \\(\\varepsilon(n)\\) is effectively computable and satisfies \\(\\varepsilon(n)\\to 0\\) as \\(n\\to\\infty\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "liminf_vs_limit",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped_the_1-\\varepsilon(p)_term_from_the_maximum_bound",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "uniform_large-p_bound_stated_for_all_allowed_p_without_the_2/27_fallback",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "quantifier_dependence",
          "tampered_component": "dependence_of_error_term_on_p_not_n",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not disclose the conclusion; it only gives the setup and asks which asymptotic statement is valid. There is no explicit answer leakage, though the phrasing strongly signals that one option is the exact theorem statement."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the question asks for the precise limiting statement that holds under the stated hypotheses. It does not substantially transform the result into an application or independent inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle logical strength (liminf vs limit vs limsup, retaining or dropping the max term, weaker true consequence). However, the main task is still recognizing the exact theorem statement rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically sharp: they test common errors involving quantifier strength, asymptotic notions, and weakening/strengthening of bounds. They are distinct and aligned with realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed distractors and no real answer leakage, but the item is primarily a theorem-restatement/recall question rather than a genuine generative reasoning task."
    }
  },
  {
    "id": "2512.16220v2",
    "paper_link": "http://arxiv.org/abs/2512.16220v2",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\mathbb{Z}[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\varepsilon(p)\\right\\}\n\\,.\n$$\nHere $\\varepsilon(p)\\to 0$ and is effectively computable.",
      "start_pos": 20703,
      "end_pos": 21260,
      "label": "T:1"
    },
    "ref_dict": {
      "S:EH": "\\label{S:EH}\n\nLet $f\\in\\Z[x]$ and write\n$$\n  f(x)=a_nx^n+a_{n-1}x^{n-1}+\\dots+a_1x+a_0\n  \\,.\n$$\nWe say that $f$ is Eisenstein at the prime $p$\n(or $p$-Eisenstein for short) if \n$p \\mid a_i$\nfor $i=0,\\",
      "S:reallyFinal": "\\label{S:reallyFinal}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:2}]\n\nLet $Y>0$ be a parameter to be chosen later.\nLet $\\mathcal{F}^*_{p,n}$ denote the subset of \n$\\mathcal{F}_{p,n}$ consisting of polynom",
      "S:gotime": "\\begin{proof}[Proof of Lemma~\\ref{L:heilbronn}]\nSuppose $p$ is totally ramified and we can write $p=a+b$ as in\nthe statement of the result.\nBy way of contradiction, suppose $K$ is norm-Euclidean.\nBy hypothesis, $(p)=\\mathfrak{p}^n$, where $N(\\mathfrak{p})=p$.\nIt follows that $p=u\\pi^n$ in $K$,\nfor some $u\\in\\mathcal{O}_K^\\times$ and some\n$\\pi\\in\\mathcal{O}_K$ irreducible with $N(\\pi)=p$.\nSince $a$ is an $n$-th power residue modulo $n$, there exists $x\\in\\Z$ such that $x^n\\equiv a\\pmod{p}$.\nUsing the norm-Euclidean condition, we find\n$x=\\gamma\\pi+\\rho$ for some $\\gamma,\\rho\\in\\mathcal{O}_K$\nwith $|N(\\rho)|<|N(\\pi)|=p$.  We have $x\\equiv \\rho\\pmod{\\pi}$\nand hence $x\\equiv \\rho^\\sigma\\pmod{\\pi}$ for any\nembedding $\\sigma:K\\to\\C$.  This leads to\n$a\\equiv x^n\\equiv N(\\rho)\\pmod{p}$.\nWe have $0<a<p$ and $-p<N(\\rho)<p$.\nThus either $N(\\rho)=a$ or $N(\\rho)+p=a$.\nThis implies one of $\\{a,-b\\}$ is a norm, \na contradiction.\n\\end{proof}\n\n\\section{Counting $p$-Eisenstein polynomials with local specifications}\\label{S:gotime}\n\nGiven $a=(a_0,\\dots,a_{n-1})\\in\\Z^n$, we define\n$$\n\\Ht(a)=\\max\\{|a_0|,\\dots,|a_{n-1}|\\}\n$$\nso that it agrees  with our polynomial height.\nThe next result is what allows us to count polynomials \nof bounded height with local specifications.\nThis type of result is certainly well-known,\nbut the proof is short and so we include it for the \nsake of completeness.\n\\begin{lemma}\\label{L:sieve}\nLet $S$ be a subset of $(\\Z/m\\Z)^n$.  We have\n$$\n\\#\\{a\\in\\Z^n\\mid a\\hspace{-1ex}\\mod{m}\\in S\\,,\\;\\Ht(a)\\leq X\\}\n=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n2^n m(2X)^{n-1}\\right)\n\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{A}\\colonequals [-X,X]^n\\cap\\Z^n$.\nChoose the maximal $k\\in\\Z^+$ such that\n$$\n  \\mathcal{B}\\colonequals (-km,km]^n\\cap\\Z^n\\subseteq \\mathcal{A}\n  \\,.\n$$\nThe set $\\mathcal{B}$ is chosen so that\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}\n=\n|S|(2k)^n\n$$\nUsing the Binomial Theorem,\none has\n$$\n\\left(\\frac{X}{m}\\right)^n \\ge k^n \\ge \\left(\\frac{X}{m}-1\\right)^n\n=\n\\left(\\frac{X}{m}\\right)^n+O\\left(n2^n\\left(\\frac{X}{m}\\right)^{n-1}\\right)\n\\,,\n$$\nand therefore\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n 2^n m(2X)^{n-1}\\right)\n\\,.\n$$\nNext one bounds\n$$\n  \\#(\\mathcal{A}\\setminus\\mathcal{B})\n  \\leq\n  (2X+1)^n-(2km)^n\n  \\ll\n  n2^n m(2X)^{n-1}\n  \\,.\n$$\nThe result follows.\n\\end{proof}",
      "T:2": "\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\eps}(p)\n\\,,\n$$\nwhere $\\hat{\\eps}(p)\\to 0$ and is effectively computable.\n\\end{theorem}",
      "S:final": "\\begin{proof}\nFirst note that \n    $f(k) = {p\\choose k}/p^k$\n    is a decreasing function. Indeed, for $k\\le p-1$\n    $$\\frac{f(k)}{f(k+1)} = \\frac{(k+1)p}{p-k} > 0,$$\n    so $f(k) > f(k+1)$ for all $k\\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\\ge 2$,\n    $$\\frac{{p\\choose 2}}{p^2} - \\frac{{p\\choose 3}}{p^3} \\le \\frac{A_p(n)}{p^n} \\le \\frac{{p\\choose 2}}{p^2}.$$\n    \\end{proof}\n\n\\section{Proof of the main result}\\label{S:final}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:1}]\nAdopt the notation from the statement of the theorem.\nFurther, write $N_{p,n}(X)$ to denote the number of \n$f\\in\\mathcal{F}_{p,n}$ with $\\Ht(f)\\leq X$.\nWe have $N_{p,n}(X)\\sim E(p)(2X)^n$ where $E(p)$\nis given in (\\ref{E:eisenstein}).\nThis follows from~\\cite{MR1995564} or alternatively,\nfrom Proposition~\\ref{P:count} with $t=0$.\n\nFirst we will establish the lower bound\n$\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\geq 2/27$.\nAssume $p\\neq 7,11,19$.\nBy Lemma~\\ref{lem: Froby}, we can write $p=2u+3v$ with\n$(2,u)=1$ and $(3,v)=1$ for $p\\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.\nTherefore, provided $f(x)$ has no roots in $\\F_2$\nor $\\F_3$, Heilbronn's criterion applies.\n(Recall Lemma~\\ref{L:heilbronn} and the discussion in~\\S\\ref{S:EH}.)\nWrite $N^*_{p,n}(X)$ to denote the counting function for this collection\nof polynomials.\nUsing Proposition~\\ref{P:count}\nwith $q_1=2$, $q_2=3$, we have\n$$\n  N^*_{p,n}(X)\\sim E_p(n)C_2(n)C_3(n)(2X)^n\n  \\,,\n$$\nand therefore,\nvia Lemma~\\ref{lem: A_p formula} with $n\\geq 3$,\nwe find\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{n,p}(X)\\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27\n\\,.\n$$\n\nIn the case where $p\\in\\{7,11,19\\}$,\nwe can write $p=2u+5v$ with\n$(2,u)=1$ and $(5,v)=1$.\nThe same approach proves\n$$\n\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\\geq C_2(n)C_5(n)\\geq(1/4)(8/25)>2/27\n\\,.\n$$\nIndeed, by Lemma~\\ref{L:Cbounds},\nas $n\\geq 3$ is odd,\none has $C_5(n)\\geq 8/25$.\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n  D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1  \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n  &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n  \\notag\\\\\n  &\\quad\\leq\n  \\prod_{i=1}^t(1-C(q_i))+\n  \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n  \\notag\\\\\n  &\\quad\\leq\n  (1+tD)\\prod_{i=1}^t(1-C(q_i))\n  \\notag\\\\\n  &\\quad\\leq\n  \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}",
      "S:local": "\\label{S:local}\n\nThe following lemma gives a formula for our local density\n$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity\ndefined immediately before Proposition~\\ref{P:count}\nand $A_p(n)$ is defi",
      "T:1": "\\begin{theorem}\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,.\n$$\nHere $\\eps(p)\\to 0$ and is effectively computable.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5964,
    "pre_theorem_intro_text": "We are interested in proving statistical results concerning\npolynomials with integer coefficents.  Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite.  \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n  N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n  \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n  \\delta(X)=\\frac{N^*(X)}{N(X)}\n  \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n   N(X)\\sim (2X)^n\n   \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}).  Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n  \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n  \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields.  As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$.  This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps.  The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\nOne knows that \nall norm-Euclidean fields have class number one, but the converse is not\ntrue in general. Recall that $K$ having class number one is equivalent to $\\O_K$ being a unique factorization domain.\nWe expect the norm-Euclidean property to be rather rare in general, but proving this is quite difficult.\nOn the other hand, if $K$ is generated by an Eisenstein polynomial, we actually have a tool available to us, due to the existence of a totally ramified prime.  More specifically, if $f$ is Eisenstein at the prime $p$, then $p$ is totally ramified in $K$; that is, $(p)=\\mathfrak{p}^n$ for some prime ideal $\\mathfrak{p}$.  The tool we refer to is as follows\n(see~\\cite{MR1574525, MR35313, MR43134, MR752120}):\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above.  Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is:  How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant.  \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said.  For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$.  We establish the following result:",
    "context": "We are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields. As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$. This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above. Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$. We establish the following result:",
    "full_context": "We are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\mathbb{Z}[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\mathbb{Z}[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nOn the other hand, one of the central objects of study in algebraic number theory\nare number fields. As an Eisenstein polynomial\n$f\\in\\mathcal{F}^*_n$ is always irreducible,\nit generates a number field $K$ of degree $n$.\nThat is, if we define $K=\\mathbb{Q}[x]/(f)$ then one has $[K:\\mathbb{Q}]=n$.\nWe can ask various questions about properties of\nthis number field as we vary the polynomial.\nAs is standard, we let $\\O_K$ denote the ring of integers in $K$,\nand let $N:K\\to\\mathbb{Q}$ denote the norm map.\n\nOne classical question is regarding when the Euclidean algorithm holds in $K$.\nWe call a number field $K$ norm-Euclidean if for every $\\alpha,\\beta\\in\\O_K$, $\\beta\\neq 0$,\nthere exists $\\gamma,\\rho\\in\\O_K$ such that $\\alpha=\\gamma\\beta+\\rho$\nand $|N(\\rho)|<|N(\\gamma)|$. This is the generalization of the\ndivision property of $\\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\\O_K$ being a Euclidean domain with respect to the function $\\partial(\\alpha)=|N(\\alpha)|$.\nThe quadratic fields that are norm-Euclidean have been completely determined\n(see~\\cite{MR41883, MR41885, MR53162, MR1191702}),\nbut this has not been accomplished in full for any degree $n>2$.\nThe reader may be interested to consult the survey article~\\cite{MR1362867}.\n\n\\begin{lemma}[Heilbronn's criterion]\\label{L:heilbronn}\nLet $K$ be a number field of degree $n$.\nLet $p$ be a prime that is totally ramified in $K$.\nIf we can write $p=a+b$ where $\\{a,-b\\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.\n\\end{lemma}\nFor clarity, recall that we refer to $a\\in \\mathbb{Z}$ as a norm (from $\\O_K$) when\nthere exists $\\alpha\\in \\O_K$ such that $N(\\alpha)=a$.\nNote that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence\none can simply write $\\{a,b\\}$ where we have written $\\{a,-b\\}$\nin the lemma above. Additionally, note that the criterion never applies\nwhen $p=2$ or $p=3$, so we should always be considering $p\\geq 5$ in our applications\nof this result.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\mathbb{Z}[x]$ if the criterion applies to the number field\n$K=\\mathbb{Q}[x]/(f)$. We establish the following result:\n\nSuppose $n\\geq 3$ is odd and $p\\geq 5$ is prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nWe order our polynomials by the natural height $\\Ht(f)$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,,\n$$\nwhere $\\eps(p)\\to 0$ and is effectively computable.\nIn particular, the lower density tends to $1$ as $p\\to\\infty$\nuniformly in $n$.\nWe also give a version of this result where we weaken the\ncondition on $\\gcd(p-1,n)$.\n\nWe are interested in proving statistical results concerning\npolynomials with integer coefficents. Let $\\mathcal{F}\\subseteq\\Z[x]$.\nBefore proceeding, we must have an ordering on our set,\nwhich is typically endowed by a height function.\nLet $\\mathcal{F}\\subseteq\\Z[x]$ be a collection of polynomials.\nMost importantly, a height function $H:\\mathcal{F}\\to\\R_{\\geq 0}$ should satisfy the following property:\nFor any $X>0$, the set\n$$\n\\{f\\in\\mathcal{F}\\mid H(f)\\leq X\\}\n$$\nis finite. \nGiven $$f(x)=a_n x^n+\\dots+a_1 x+a_0\\in\\Z[x]\\,,$$\nwe define the standard height function\n$$\n\\Ht(f)=\\max\\{|a_0|,|a_1|,\\dots,|a_n|\\}\n\\,.\n$$\nOne might consider other height functions, but here we will always\nuse $\\Ht(f)$ as defined above.\nGiven a subset $\\mathcal{F}\\subseteq\\mathbb{Z}[x]$, we define\nthe associated counting function\n$$\n N(X)=\\#\\{f\\in\\mathcal{F}\\mid \\Ht(f)\\leq X\\}\n \\,.\n$$\nGiven a distinguished subset $\\mathcal{F}^*\\subseteq\\mathcal{F}$,\nwe let $\\delta(X)$ denote the proportion of elements of $\\mathcal{F}$ that belong to $\\mathcal{F}^*$; that is\n$$\n \\delta(X)=\\frac{N^*(X)}{N(X)}\n \\,,\n$$\nwhere we have written $N^*(X)$ to denote the counting function associated to $\\mathcal{F}^*$.\nOne then defines the density of $\\mathcal{F}^*$ in $\\mathcal{F}$ to be\n$$\n\\delta\\colonequals \\lim_{X\\to\\infty}\\delta(X)\n\\,,\n$$\nprovided the limit exists.\nIn cases where it is difficult to prove this limit exists, \none might seek bounds on the lower and upper densities:\n$$\n\\underline{\\delta}\\colonequals\n\\liminf_{X\\to\\infty}\\delta(X)\n\\,,\\;\\; \n\\overline{\\delta}\\colonequals\n\\limsup_{X\\to\\infty}\\delta(X)\n\\,.$$\n\nWe begin with an important example.\nFix a positive integer $n$.\nDenote by $\\mathcal{F}_n$ the collection\nof all monic polynomials in $\\Z[x]$ of degree $n$.\nIt is easy to show that\n$$\n N(X)\\sim (2X)^n\n \\,.\n$$\nLet $\\mathcal{F}_n^*$ be the \nsubset of $\\mathcal{F}_n$ consisting of Eisenstein polynomials.\nDubickas considers the probability that a randomly chosen \npolynomial of degree $n$ is Eisenstein (see~\\cite{MR1995564}). Indeed, he\nproves that the associated density exists and that it equals\n\\begin{equation}\\label{E:dub}\n \\delta=1-\\prod_p\\left(1-\\frac{1}{p^n}+\\frac{1}{p^{n+1}}\\right)\n \\,.\n\\end{equation}\nThis appears to be a topic of interest, as this paper led to\nthe papers~\\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.\n\nA natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?\nOne could ask this question in the case of all number fields of fixed degree,\nordered by their discriminant. \nThere are some results in this direction when\n$n=3$ and $n=5$ (see~\\cite{MR4613609, MR4649640}).\nHowever, such questions are intractable \nat present when $n>5$ (see, for example, the introduction to~\\cite{MR2745272}).\nOn the other hand, if we consider the corresponding\nstatistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\\in\\Z[x]$ if the criterion applies to the number field\n$K=\\Q[x]/(f)$. We establish the following result:\n\nWhen $p$ is large, there are many ways to write $p=a+b$ and hence\ndetermining the exact density of polynomials that satisfy Heilbronn's\ncriterion is likely very difficult. However, the previous\ntheorem says that the lower density tends to $1$ as $p\\to\\infty$,\nuniformly in the variable $n$.\n\n\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\eps}(p)\n\\,,\n$$\nwhere $\\hat{\\eps}(p)\\to 0$ and is effectively computable.\n\\end{theorem}\n\nBefore proceeding, we require a couple definitions.\nFirst we define\n\\begin{equation}\\label{E:eisenstein}\nE_p(n)\\colonequals\n\\frac{1}{p^n}-\\frac{1}{p^{n+1}}\n\\,,\n\\end{equation}\nwhich is the $p$-Eisenstein density that contributes\nto (\\ref{E:dub}).\nSecondly, we write $C_q(n)$ to denote --- the number of\n$(a_0,\\dots,a_{n-1})\\in(\\Z/q\\Z)^n$ such that the corresponding polynomial\nin $\\F_q[x]$ has no roots in $\\F_q$ ---\ndivided by $q^n$.\nWe are now ready to prove the following result, which\nwill be used in the proof of Theorem~\\ref{T:1}.\n\\begin{prop}\\label{P:count}\nLet $n\\in\\Z^+$.\nSuppose $q_1<\\dots<q_t<p$ are primes.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nThe number of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which\n$f$ has no roots in $\\F_q$ for all $q\\in\\{q_1,\\dots,q_t\\}$\nequals\n$$\n E_p(n)\\left(\\prod_{i=1}^t C_{q_i}(n)\\right)(2X)^n+O\\left(n 2^n q_1\\dots q_t p^2 (2X)^{n-1}\\right)\n \\,.\n$$\n\\end{prop}\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1 \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n \\notag\\\\\n &\\quad\\leq\n \\prod_{i=1}^t(1-C(q_i))+\n \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n \\notag\\\\\n &\\quad\\leq\n (1+tD)\\prod_{i=1}^t(1-C(q_i))\n \\notag\\\\\n &\\quad\\leq\n \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}\n\n\\begin{theorem}\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\Z[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\eps(p)\\right\\}\n\\,.\n$$\nHere $\\eps(p)\\to 0$ and is effectively computable.\n\\end{theorem}",
    "post_theorem_intro_text_len": 2022,
    "post_theorem_intro_text": "When $p$ is large, there are many ways to write $p=a+b$ and hence\ndetermining the exact density of polynomials that satisfy Heilbronn's\ncriterion is likely very difficult.  However, the previous\ntheorem says that the lower density tends to $1$ as $p\\to\\infty$,\nuniformly in the variable $n$.\n\nAs a result, we immediately obtain the following consequence:\nWhen $n\\geq 3$ is odd,\na positive proportion of Eisenstein polynomials of degree $n$\nfail to generate norm-Euclidean fields.\n\nThe condition from Theorem~\\ref{T:1} that $\\mathrm{gcd}(p-1,n) = 1$\nis not a severe restriction as there are infinitely \nmany primes $p$ that satisfy this condition for each fixed $n$.\nNonetheless, we show that we can\nrelax this condition at the expense of considering larger primes.\nThis will allow us to deal with the case where $n$ is even.\n\n\\begin{theorem}\\label{T:2}\nSuppose $n\\geq 2$ and $p\\geq 5$ is a prime with\n$$\n\\gcd(p-1,n)<\\frac{p^{1/2}}{(\\log p)^2}\n\\,.\n$$\nDefine $\\delta_{p,n}(X)$ as in Theorem~\\ref{T:1}.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq 1-\\hat{\\varepsilon}(p)\n\\,,\n$$\nwhere $\\hat{\\varepsilon}(p)\\to 0$ and is effectively computable.\n\\end{theorem}\n\n\\begin{corollary}\\label{C:1}\nFix an integer $n\\geq 2$.\nA positive proportion of Eisenstein polynomials of degree $n$\nfail to generate norm-Euclidean fields.\n\\end{corollary}\n\nIn \\S\\ref{S:EH} we discuss Eisenstein polynomials and Heilbronn's criterion,\nand outline the strategy of our proof.  In addition, we establish some preliminary \nlemmas.  In the case where the proofs are very short, we err on the side of giving\nproofs of known results for the sake of completeness.\nIn \\S\\ref{S:gotime} we prove our main counting result for Eisenstein polynomials\nsatisfying certain other local conditions.  In \\S\\ref{S:local} we prove a formula\nfor the relevant local densities, which is then used to establish the necessary bounds.\nIn \\S\\ref{S:final}\nwe put everything together and prove Theorem~\\ref{T:1}. Finally, in \\S\\ref{S:reallyFinal}, we prove Theorem \\ref{T:2}.",
    "sketch": "In \\,\\S\\ref{S:EH} the authors “discuss Eisenstein polynomials and Heilbronn's criterion, and outline the strategy of our proof,” and “establish some preliminary lemmas.” Then “in \\,\\S\\ref{S:gotime} we prove our main counting result for Eisenstein polynomials satisfying certain other local conditions.” Next, “in \\,\\S\\ref{S:local} we prove a formula for the relevant local densities, which is then used to establish the necessary bounds.” Finally, “in \\,\\S\\ref{S:final} we put everything together and prove Theorem~\\ref{T:1}.”",
    "expanded_sketch": "\\label{S:EH}\n\nLet $f\\in\\Z[x]$ and write\n$$\n  f(x)=a_nx^n+a_{n-1}x^{n-1}+\\dots+a_1x+a_0\n  \\,.\n$$\nWe say that $f$ is Eisenstein at the prime $p$\n(or $p$-Eisenstein for short) if \n$p \\mid a_i$\nfor $i=0,\\,\n\n\\begin{proof}[Proof of Lemma~\\ref{L:heilbronn}]\nSuppose $p$ is totally ramified and we can write $p=a+b$ as in\nthe statement of the result.\nBy way of contradiction, suppose $K$ is norm-Euclidean.\nBy hypothesis, $(p)=\\mathfrak{p}^n$, where $N(\\mathfrak{p})=p$.\nIt follows that $p=u\\pi^n$ in $K$,\nfor some $u\\in\\mathcal{O}_K^\\times$ and some\n$\\pi\\in\\mathcal{O}_K$ irreducible with $N(\\pi)=p$.\nSince $a$ is an $n$-th power residue modulo $n$, there exists $x\\in\\Z$ such that $x^n\\equiv a\\pmod{p}$.\nUsing the norm-Euclidean condition, we find\n$x=\\gamma\\pi+\\rho$ for some $\\gamma,\\rho\\in\\mathcal{O}_K$\nwith $|N(\\rho)|<|N(\\pi)|=p$.  We have $x\\equiv \\rho\\pmod{\\pi}$\nand hence $x\\equiv \\rho^\\sigma\\pmod{\\pi}$ for any\nembedding $\\sigma:K\\to\\C$.  This leads to\n$a\\equiv x^n\\equiv N(\\rho)\\pmod{p}$.\nWe have $0<a<p$ and $-p<N(\\rho)<p$.\nThus either $N(\\rho)=a$ or $N(\\rho)+p=a$.\nThis implies one of $\\{a,-b\\}$ is a norm, \na contradiction.\n\\end{proof}\n\n\\section{Counting $p$-Eisenstein polynomials with local specifications}\\label{S:gotime}\n\nGiven $a=(a_0,\\dots,a_{n-1})\\in\\Z^n$, we define\n$$\n\\Ht(a)=\\max\\{|a_0|,\\dots,|a_{n-1}|\\}\n$$\nso that it agrees  with our polynomial height.\nThe next result is what allows us to count polynomials \nof bounded height with local specifications.\nThis type of result is certainly well-known,\nbut the proof is short and so we include it for the \nsake of completeness.\n\\begin{lemma}\\label{L:sieve}\nLet $S$ be a subset of $(\\Z/m\\Z)^n$.  We have\n$$\n\\#\\{a\\in\\Z^n\\mid a\\hspace{-1ex}\\mod{m}\\in S\\,,\\;\\Ht(a)\\leq X\\}\n=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n2^n m(2X)^{n-1}\\right)\n\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{A}\\colonequals [-X,X]^n\\cap\\Z^n$.\nChoose the maximal $k\\in\\Z^+$ such that\n$$\n  \\mathcal{B}\\colonequals (-km,km]^n\\cap\\Z^n\\subseteq \\mathcal{A}\n  \\,.\n$$\nThe set $\\mathcal{B}$ is chosen so that\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}\n=\n|S|(2k)^n\n$$\nUsing the Binomial Theorem,\none has\n$$\n\\left(\\frac{X}{m}\\right)^n \\ge k^n \\ge \\left(\\frac{X}{m}-1\\right)^n\n=\n\\left(\\frac{X}{m}\\right)^n+O\\left(n2^n\\left(\\frac{X}{m}\\right)^{n-1}\\right)\n\\,,\n$$\nand therefore\n$$\n\\#\\{a\\in\\mathcal{B}\\mid a\\hspace{-1ex}\\mod{m}\\in S\\}=\n\\frac{|S|}{m^n}(2X)^n+O\\left(n 2^n m(2X)^{n-1}\\right)\n\\,.\n$$\nNext one bounds\n$$\n  \\#(\\mathcal{A}\\setminus\\mathcal{B})\n  \\leq\n  (2X+1)^n-(2km)^n\n  \\ll\n  n2^n m(2X)^{n-1}\n  \\,.\n$$\nThe result follows.\n\\end{proof}\n\n\\label{S:local}\n\nThe following lemma gives a formula for our local density\n$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity\ndefined immediately before Proposition~\\ref{P:count}\nand $A_p(n)$ is defi\n\n\\begin{proof}\nFirst note that \n    $f(k) = {p\\choose k}/p^k$\n    is a decreasing function. Indeed, for $k\\le p-1$\n    $$\\frac{f(k)}{f(k+1)} = \\frac{(k+1)p}{p-k} > 0,$$\n    so $f(k) > f(k+1)$ for all $k\\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\\ge 2$,\n    $$\\frac{{p\\choose 2}}{p^2} - \\frac{{p\\choose 3}}{p^3} \\le \\frac{A_p(n)}{p^n} \\le \\frac{{p\\choose 2}}{p^2}.$$\n    \\end{proof}\n\n\\section{Proof of the main result}\\label{S:final}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:1}]\nAdopt the notation from the statement of the theorem.\nFurther, write $N_{p,n}(X)$ to denote the number of \n$f\\in\\mathcal{F}_{p,n}$ with $\\Ht(f)\\leq X$.\nWe have $N_{p,n}(X)\\sim E(p)(2X)^n$ where $E(p)$\nis given in (\\ref{E:eisenstein}).\nThis follows from~\\cite{MR1995564} or alternatively,\nfrom Proposition~\\ref{P:count} with $t=0$.\n\nFirst we will establish the lower bound\n$\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\geq 2/27$.\nAssume $p\\neq 7,11,19$.\nBy Lemma~\\ref{lem: Froby}, we can write $p=2u+3v$ with\n$(2,u)=1$ and $(3,v)=1$ for $p\\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.\nTherefore, provided $f(x)$ has no roots in $\\F_2$\nor $\\F_3$, Heilbronn's criterion applies.\n(Recall Lemma~\\ref{L:heilbronn} and the discussion above.)\nWrite $N^*_{p,n}(X)$ to denote the counting function for this collection\nof polynomials.\nUsing Proposition~\\ref{P:count}\nwith $q_1=2$, $q_2=3$, we have\n$$\n  N^*_{p,n}(X)\\sim E_p(n)C_2(n)C_3(n)(2X)^n\n  \\,,\n$$\nand therefore,\nvia Lemma~\\ref{lem: A_p formula} with $n\\geq 3$,\nwe find\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{n,p}(X)\\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27\n\\,.\n$$\n\nIn the case where $p\\in\\{7,11,19\\}$,\nwe can write $p=2u+5v$ with\n$(2,u)=1$ and $(5,v)=1$.\nThe same approach proves\n$$\n\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\\geq C_2(n)C_5(n)\\geq(1/4)(8/25)>2/27\n\\,.\n$$\nIndeed, by Lemma~\\ref{L:Cbounds},\nas $n\\geq 3$ is odd,\none has $C_5(n)\\geq 8/25$.\n\nIt remains to establish a lower bound $\\delta_{p,n}(X)\\geq 1-\\eps(p)$\nwhere $\\eps(p)\\to 0$.\nWe may assume that $p$ is large.\nChoose $Y$ maximal such that \\mbox{$Y^4<p$}.\nLet $q_1<\\dots<q_t\\leq Y$ be all the primes less than $Y$.\nIn particular, Lemma~\\ref{lem: Froby} applies\nto any pair of primes in $Q\\colonequals\\{q_1,\\dots,q_t\\}$\nand we have $t=\\pi(Y)\\leq \\pi(p^{1/4})$.\nBefore proceeding with the final estimate,\nobserve that by Lemma~\\ref{L:Cbounds}, one has\n$$\n  D\\colonequals\\max_{j=1,\\dots,t}\\left\\{\\frac{C(q_j)}{1-C(q_j)}\\right\\} \\le 1  \\,.\n$$\nWe wish to bound from above the\nnumber of $p$-Eisenstein polynomials with $\\Ht(f)\\leq X$\nsuch that Heilbronn's criterion does not apply for any\npair of primes in $Q$.\nThis means that $f$ has no root modulo $q$ for at most one $q\\in Q$.\nUsing Proposition~\\ref{P:count}\nin the situation where\n$\\{q\\in Q\\mid f \\text{ has a root in }\\F_q\\}$\nequals $Q$, $Q\\setminus\\{q_1\\}$, $Q\\setminus\\{q_2\\}$, and so on,\nwe obtain\n\\begin{align}\\label{E:liminf}\n  &1-\\liminf_{X\\to\\infty}\\delta_{n,p}(X)\n  \\notag\\\\\n  &\\quad\\leq\n  \\prod_{i=1}^t(1-C(q_i))+\n  \\sum_{j=1}^tC(q_j)\\left((1-C(q_j))^{-1}\\prod_{i=1}^t(1-C(q_i)\\right)\n  \\notag\\\\\n  &\\quad\\leq\n  (1+tD)\\prod_{i=1}^t(1-C(q_i))\n  \\notag\\\\\n  &\\quad\\leq\n  \\left(1+\\pi\\left(p^{1/4}\\right)\\right)\\left(\\frac{3}{4}\\right)^{\\pi(p^{1/4})}=:\\eps(p)\\,.\n\\end{align}\nIn the last step, we have used\nthe fact that $C_p(n)\\ge 1/4$ for $p\\geq 2$\nwhich follows from Lemma~\\ref{L:Cbounds}.\n\\end{proof}\n\nThis completes the proof of the main theorem.",
    "expanded_theorem": "\\label{T:1}\nSuppose $n\\geq 3$ is an odd integer and $p\\geq 5$ is a prime with\n$\\gcd(p-1,n)=1$.\nLet $\\mathcal{F}_{p,n}$ denote the collection\nof monic polynomials $f\\in\\mathbb{Z}[x]$ of degree $n$\nthat are Eisenstein at the prime $p$.\nDefine $\\delta_{p,n}(X)$ to be the \nproportion of polynomials $f\\in\\mathcal{F}_{p,n}$\nwith $\\Ht(f)\\leq X$\nfor which Heilbronn's criterion applies.\nOne has\n$$\n\\liminf_{X\\to\\infty}\n\\delta_{p,n}(X)\\geq \\max\\left\\{\\frac{2}{27}\\,,\\;1-\\varepsilon(p)\\right\\}\n\\,.\n$$\nHere $\\varepsilon(p)\\to 0$ and is effectively computable.",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(n\\ge 3\\) be an odd integer and \\(p\\ge 5\\) a prime such that \\(\\gcd(p-1,n)=1\\). Let \\(\\mathcal F_{p,n}\\) be the set of monic polynomials\n\\[\nf(x)=x^n+a_{n-1}x^{n-1}+\\cdots+a_0\\in \\mathbb Z[x]\n\\]\nof degree \\(n\\) that are Eisenstein at \\(p\\), i.e. \\(p\\mid a_i\\) for every \\(0\\le i\\le n-1\\) and \\(p^2\\nmid a_0\\). For such a polynomial, define its height by\n\\[\n\\operatorname{Ht}(f)=\\max\\{1,|a_{n-1}|,\\dots,|a_0|\\}.\n\\]\nFor each \\(X>0\\), let \\(\\delta_{p,n}(X)\\) be the proportion of polynomials \\(f\\in \\mathcal F_{p,n}\\) with \\(\\operatorname{Ht}(f)\\le X\\) for which Heilbronn's criterion applies to the number field \\(K=\\mathbb Q[x]/(f)\\), meaning that there exist integers \\(a,b\\) with \\(p=a+b\\), such that \\(a\\) is an \\(n\\)-th power residue modulo \\(p\\) and neither \\(a\\) nor \\(-b\\) is a norm from \\(\\mathcal O_K\\). Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(p)\\right\\},\n\\]\nwhere \\(\\varepsilon(p)\\) is effectively computable and satisfies \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has\n\\[\n\\lim_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(p)\\right\\},\n\\]\nwhere \\(\\varepsilon(p)\\) is effectively computable and satisfies \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
        },
        {
          "label": "C",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\frac{2}{27}.\n\\]"
        },
        {
          "label": "D",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge 1-\\varepsilon(p),\n\\]\nwhere \\(\\varepsilon(p)\\to 0\\) as \\(p\\to\\infty\\)."
        },
        {
          "label": "E",
          "text": "One has\n\\[\n\\liminf_{X\\to\\infty}\\delta_{p,n}(X)\\ge \\max\\left\\{\\frac{2}{27},\\,1-\\varepsilon(n)\\right\\},\n\\]\nwhere \\(\\varepsilon(n)\\) is effectively computable and satisfies \\(\\varepsilon(n)\\to 0\\) as \\(n\\to\\infty\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "liminf_vs_limit",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped_the_1-\\varepsilon(p)_term_from_the_maximum_bound",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "uniform_large-p_bound_stated_for_all_allowed_p_without_the_2/27_fallback",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "quantifier_dependence",
          "tampered_component": "dependence_of_error_term_on_p_not_n",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It sets up hypotheses and asks for the valid estimate, without directly embedding the exact conclusion."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the hypotheses are stated in full and the choices are near-verbatim variants of the theorem’s conclusion. It tests recognition of the exact statement more than mathematical inference."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in distinguishing liminf vs. limit, the max bound vs. weaker sub-bounds, and the dependence on p rather than n. However, the item mainly rewards recall of the precise theorem statement rather than generative reasoning from the setup."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are plausible theorem variants involving common errors such as strengthening liminf to limit, dropping a fallback term, or changing parameter dependence. They are distinct and mathematically meaningful."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than recall."
    }
  },
  {
    "id": "2512.16358v1",
    "paper_link": "http://arxiv.org/abs/2512.16358v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.",
      "start_pos": 6591,
      "end_pos": 7104,
      "label": "thm:main"
    },
    "ref_dict": {
      "lem1": "\\begin{lemma} \\label{lem1}\nFor $k\\geq 2$ we have that \n$$f_k = (p_{k}-1) f_{k-1}.$$\n\\end{lemma}",
      "thm:main": "\\begin{theorem} \\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2166,
    "pre_theorem_intro_text": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq  1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots  & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots &  \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}.  $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots  & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots &  \\vdots & \\ddots & \\vdots  & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}.  $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:",
    "context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:",
    "full_context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\n\\begin{abstract}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.\nWe relate this sequence to a concrete counting problem.\nChoose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \\leq i \\leq k$ and set $P_k = \\prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \\emph{at most} one of the $k$ chosen residue classes.\nInterestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.\n\n\\section{Introduction}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\nMoreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\nWe aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \\ldots, p_k$ and arbitrary choice of residue classes $r_1, \\ldots, r_k$ with $P_k = \\prod_{i=1}^k p_i$.\nIn particular, we want to understand whether an integer $1\\leq n \\leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.\n\nAssume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets\n$s_i$, $1\\leq i \\leq k$.\nFor each integer $1\\leq n \\leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\\gamma(n)$. \nWe denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.\n\\begin{align*}\nav(k) &:= \\# \\{n : \\gamma(n) \\leq 1 \\} \\\\\nfree(k) &:=\\# \\{n : \\gamma(n) = 0 \\} \\\\\nocc(k) &:=\\# \\{n : \\gamma(n) > 1 \\} \n\\end{align*}\nTo illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.\nIt is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.\n\n\\begin{theorem} \\label{thm2}\nLet $p_{j_1}, \\ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\\prod_{i=1}^k p_{j_i}$ be their product. \nFor $k\\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \\ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \\ldots, p_{j_k})$.\nIn other words, for every choice of residue classes, there are $A_k(p_{j_1}, \\ldots, p_{j_k})$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $F_k(p_{j_1}, \\ldots, p_{j_k})$ integers with $\\gamma(n)=0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1536,
    "post_theorem_intro_text": "Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\n\\begin{remark}\nLemma \\ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \\cite{oeis2} in terms of determinants.\n\\end{remark}\n\n\\begin{remark}\nAs a side product, the proof of Theorem \\ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.\n\\end{remark}\n\nThe paper is organized as follows. In Section \\ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \\ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \\ref{thm:main} is proven in Section \\ref{sec4} and generalised in Section \\ref{sec5}.",
    "sketch": "The post-theorem introduction does not give an explicit proof outline for Theorem~\\ref{thm:main}, but it indicates the structure used to prove it: Section~\\ref{sec2} \"derive[s] structural results about $a_k$ and $f_k$\"; Section~\\ref{sec3} \"analyse[s] the counting problem\" and \"obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions\"; then \"Theorem~\\ref{thm:main} is proven in Section~\\ref{sec4}\" (and \"generalised in Section~\\ref{sec5}\").",
    "expanded_sketch": "The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).",
    "expanded_theorem": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.,\n",
    "theorem_type": [
      "Universal",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Fix a positive integer $k$. Let $p_1,\\dots,p_k$ be the first $k$ prime numbers and let $P_k=\\prod_{i=1}^k p_i$. For each $i$, choose one residue class $r_i \\pmod{p_i}$, and for each integer $n$ with $1\\le n\\le P_k$ define\n$$\\gamma(n)=\\#\\{i\\in\\{1,\\dots,k\\}: n\\equiv r_i \\pmod{p_i}\\}.$$ \nCall $n$ \\emph{available} if $\\gamma(n)\\le 1$ and \\emph{free} if $\\gamma(n)=0$. Also define\n$$a_k=\\det\\begin{bmatrix}\n2 & 1 & \\ldots & 1\\\\\n1 & 3 & \\ldots & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n1 & 1 & \\ldots & p_k\n\\end{bmatrix},$$\nand\n$$f_k=(-1)^k\\det\\begin{bmatrix}\n1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1 & 1\\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_k & 1\n\\end{bmatrix}.$$ \nWhich statement holds for every choice of the residue classes $r_1,\\dots,r_k$?",
      "correct_choice": {
        "label": "A",
        "text": "The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=a_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k.$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$, provided the chosen residue classes $r_1,\\dots,r_k$ are pairwise distinct as integers; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=a_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k$$\nfor every such choice of pairwise distinct residues."
        },
        {
          "label": "C",
          "text": "The number of free integers in $[1,P_k]$ is exactly $f_k$ for every choice of the residue classes $r_1,\\dots,r_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k.$$"
        },
        {
          "label": "D",
          "text": "For every choice of the residue classes $r_1,\\dots,r_k$, the number of available integers in $[1,P_k]$ is at least $a_k$, and equality holds if and only if no integer of $[1,P_k]$ lies in more than two of the chosen residue classes; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}\\ge a_k.$$"
        },
        {
          "label": "E",
          "text": "The number of available integers in $[1,P_k]$ is exactly $f_k$, and the number of free integers in $[1,P_k]$ is exactly $a_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=f_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=a_k.$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity in the residue-class choice",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the available-count conclusion while keeping the free-count conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "exact recurrence-derived equality for available numbers",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "distinction between available and free counts",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the claimed counting result. It introduces the objects and asks which conclusion is valid, without directly revealing that the counts must equal the determinant-defined quantities."
      },
      "TAS": {
        "score": 1,
        "justification": "The intended correct option appears to state the theorem almost verbatim: exact formulas for the counts of available and free integers. The alternatives add some variation in quantifier strength and count interpretation, but the item is still close to a direct theorem restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "A student must distinguish universal from existential claims and separate counts of free, available, and singly covered integers. However, this is still mostly theorem recognition rather than substantial generative reasoning or derivation."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible mathematical confusions: swapping free and singly covered counts, altering the quantifier, or changing the relation between the two counts. But choice C is a weaker statement that would also be true if A is true, which creates ambiguity and lowers distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "Moderate-quality MCQ: it avoids overt answer leakage and has some logical discrimination, but it leans heavily on theorem recall and is weakened by an apparently true weaker distractor."
    }
  },
  {
    "id": "2512.16358v1",
    "paper_link": "http://arxiv.org/abs/2512.16358v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.",
      "start_pos": 6591,
      "end_pos": 7104,
      "label": "thm:main"
    },
    "ref_dict": {
      "lem1": "\\begin{lemma} \\label{lem1}\nFor $k\\geq 2$ we have that \n$$f_k = (p_{k}-1) f_{k-1}.$$\n\\end{lemma}",
      "thm:main": "\\begin{theorem} \\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2166,
    "pre_theorem_intro_text": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq  1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots  & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots &  \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}.  $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots  & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots &  \\vdots & \\ddots & \\vdots  & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}.  $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:",
    "context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:",
    "full_context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\n\\begin{abstract}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.\nWe relate this sequence to a concrete counting problem.\nChoose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \\leq i \\leq k$ and set $P_k = \\prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \\emph{at most} one of the $k$ chosen residue classes.\nInterestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.\n\n\\section{Introduction}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\nMoreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\nWe aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \\ldots, p_k$ and arbitrary choice of residue classes $r_1, \\ldots, r_k$ with $P_k = \\prod_{i=1}^k p_i$.\nIn particular, we want to understand whether an integer $1\\leq n \\leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.\n\nAssume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets\n$s_i$, $1\\leq i \\leq k$.\nFor each integer $1\\leq n \\leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\\gamma(n)$. \nWe denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.\n\\begin{align*}\nav(k) &:= \\# \\{n : \\gamma(n) \\leq 1 \\} \\\\\nfree(k) &:=\\# \\{n : \\gamma(n) = 0 \\} \\\\\nocc(k) &:=\\# \\{n : \\gamma(n) > 1 \\} \n\\end{align*}\nTo illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.\nIt is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.\n\n\\begin{theorem} \\label{thm2}\nLet $p_{j_1}, \\ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\\prod_{i=1}^k p_{j_i}$ be their product. \nFor $k\\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \\ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \\ldots, p_{j_k})$.\nIn other words, for every choice of residue classes, there are $A_k(p_{j_1}, \\ldots, p_{j_k})$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $F_k(p_{j_1}, \\ldots, p_{j_k})$ integers with $\\gamma(n)=0$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1536,
    "post_theorem_intro_text": "Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\n\\begin{remark}\nLemma \\ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \\cite{oeis2} in terms of determinants.\n\\end{remark}\n\n\\begin{remark}\nAs a side product, the proof of Theorem \\ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.\n\\end{remark}\n\nThe paper is organized as follows. In Section \\ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \\ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \\ref{thm:main} is proven in Section \\ref{sec4} and generalised in Section \\ref{sec5}.",
    "sketch": "The post-theorem introduction does not give an explicit proof outline for Theorem~\\ref{thm:main}, but it indicates the structure used to prove it: Section~\\ref{sec2} \"derive[s] structural results about $a_k$ and $f_k$\"; Section~\\ref{sec3} \"analyse[s] the counting problem\" and \"obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions\"; then \"Theorem~\\ref{thm:main} is proven in Section~\\ref{sec4}\" (and \"generalised in Section~\\ref{sec5}\").",
    "expanded_sketch": "The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).",
    "expanded_theorem": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.,\n",
    "theorem_type": [
      "Universal",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Fix a positive integer $k$. Let $p_1,\\dots,p_k$ be the first $k$ prime numbers and let $P_k=\\prod_{i=1}^k p_i$. For each $i$, choose one residue class $r_i \\pmod{p_i}$, and for each integer $n$ with $1\\le n\\le P_k$ define\n$$\\gamma(n)=\\#\\{i\\in\\{1,\\dots,k\\}: n\\equiv r_i \\pmod{p_i}\\}.$$ \nCall $n$ \\emph{available} if $\\gamma(n)\\le 1$ and \\emph{free} if $\\gamma(n)=0$. Also define\n$$a_k=\\det\\begin{bmatrix}\n2 & 1 & \\ldots & 1\\\\\n1 & 3 & \\ldots & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n1 & 1 & \\ldots & p_k\n\\end{bmatrix},$$\nand\n$$f_k=(-1)^k\\det\\begin{bmatrix}\n1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1 & 1\\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_k & 1\n\\end{bmatrix}.$$ \nWhich statement holds for every choice of the residue classes $r_1,\\dots,r_k$?",
      "correct_choice": {
        "label": "A",
        "text": "The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=a_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k.$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$, provided the chosen residue classes $r_1,\\dots,r_k$ are pairwise distinct as integers; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=a_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k$$\nfor every such choice of pairwise distinct residues."
        },
        {
          "label": "C",
          "text": "The number of free integers in $[1,P_k]$ is exactly $f_k$ for every choice of the residue classes $r_1,\\dots,r_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=f_k.$$"
        },
        {
          "label": "D",
          "text": "For every choice of the residue classes $r_1,\\dots,r_k$, the number of available integers in $[1,P_k]$ is at least $a_k$, and equality holds if and only if no integer of $[1,P_k]$ lies in more than two of the chosen residue classes; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}\\ge a_k.$$"
        },
        {
          "label": "E",
          "text": "The number of available integers in $[1,P_k]$ is exactly $f_k$, and the number of free integers in $[1,P_k]$ is exactly $a_k$; equivalently,\n$$\\#\\{n\\in[1,P_k]:\\gamma(n)\\le 1\\}=f_k\\quad\\text{and}\\quad \\#\\{n\\in[1,P_k]:\\gamma(n)=0\\}=a_k.$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity in the residue-class choice",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the available-count conclusion while keeping the free-count conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "exact recurrence-derived equality for available numbers",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "distinction between available and free counts",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem introduces the objects and asks for a universal conclusion, but it does not state or strongly hint at the exact counting formula in choice A. The determinant definitions alone do not leak the answer."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is essentially the full theorem packaged as a choice, so the item is close to theorem recognition rather than an independently formulated problem. However, the presence of weaker, stronger, and tampered variants makes it more than a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the student must distinguish the exact universal claim from a weaker true statement, an overstrong inequality/equality claim, and a swapped-count trap. Still, the item mainly tests recognition of the strongest correct statement rather than substantial generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and target realistic failure modes: adding an unnecessary restriction (B), selecting a weaker true consequence (C), overstrengthening the conclusion (D), and confusing the two counts (E). They are distinct and plausible."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but only moderate generative depth because the correct choice is essentially the theorem itself."
    }
  },
  {
    "id": "2512.16503v1",
    "paper_link": "http://arxiv.org/abs/2512.16503v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "[see \\Cref{cor:ascending_faithfully}]\n    \\label{introthm:fully_faithfulness_equivalence}\n    Consider affine morphism $t\\colon T \\to S$ locally of finite tor-dimension of Noetherian schemes whose target is semiseparated. Suppose $Y_1$ and $Y_2$ are proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$.",
      "start_pos": 23264,
      "end_pos": 23909,
      "label": "introthm:fully_faithfulness_equivalence"
    },
    "ref_dict": {
      "prop:intro_preservation_adjoint": "\\begin{proposition}\n    \\label{prop:intro_preservation_adjoint}\n    Suppose $t\\colon T \\to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Consider $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. \n    \\begin{enumerate}\n        \\item \\label{prop:intro_preservation_adjoint1} If $t$ is affine and $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$ (resp.\\ $\\operatorname{Perf}$), then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$ (see \\Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \\Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).\n        \\item \\label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension. \n        \\begin{enumerate}\n            \\item \\label{prop:intro_preservation_adjoint2a} If $\\Phi_K$ restricts to $\\operatorname{Perf}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its right adjoint (see  \\Cref{prop:right_adjoint_pullback}).\n            \\item \\label{prop:intro_preservation_adjoint2b} If $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its left adjoint (see \\Cref{prop:left_adjoint_pullback}).\n        \\end{enumerate}\n    \\end{enumerate}\n\\end{proposition}",
      "prop:CM_or_Gorensteinnes_morphisn_derived_invariance": "\\begin{proposition}\n    \\label{prop:CM_or_Gorensteinnes_morphisn_derived_invariance}\n    Let $f_i \\colon Y_i \\to S$ be projective flat morphisms to a separated Noetherian scheme with geometrically integral fibers. Assume that $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$. Then $f_1$ is Cohen--Macaulay (resp.\\ Gorenstein) if, and only if, $f_2$ satisfies the same condition.\n\\end{proposition}",
      "rmk:equivalences_induced": "\\begin{remark}\n    \\label{rmk:equivalences_induced}\n    Let $\\Phi\\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ be an exact functor where each $Y_i$ is Noetherian. Then the following are equivalent:\n    \\begin{enumerate}\n        \\item \\label{rmk:equivalences_induced1} $\\Phi$ is an equivalence\n        \\item \\label{rmk:equivalences_induced2} $\\Phi$ restricts to an equivalence on $D^-_{\\operatorname{coh}}$\n        \\item \\label{rmk:equivalences_induced3} $\\Phi$ restricts to an equivalence on $D^b_{\\operatorname{coh}}$\n        \\item \\label{rmk:equivalences_induced4} $\\Phi$ restricts to an equivalence on $\\operatorname{Perf}$.\n    \\end{enumerate}\n    Indeed, by \\cite[Theorem 3.2(iv)]{Neeman:2022} and \\cite[Theorem B]{Canonaco/Haesemeyer/Neeman/Stellari:2024}, $\\eqref{rmk:equivalences_induced1} \\implies \\eqref{rmk:equivalences_induced2} \\implies \\eqref{rmk:equivalences_induced3} \\implies \\eqref{rmk:equivalences_induced4}$. Additionally, if $\\eqref{rmk:equivalences_induced4}$ holds, then $\\Phi$ admits a right adjoint that preserves small coproducts \\cite[Theorem 5.1]{Neeman:1996}. So, \\Cref{lem:equivalence_or_fully_faithful_via_compacts} tells us $\\eqref{rmk:equivalences_induced4} \\implies \\eqref{rmk:equivalences_induced1}$.\n\\end{remark}",
      "prop:right_adjoint_pullback": "\\begin{proposition}\n    \\label{prop:right_adjoint_pullback}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension (e.g.\\ $t=1_S$). Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Then $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$ is right adjoint to $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ on $D_{\\operatorname{qc}}$ where $K^\\prime:= \\mathbf{R}\\operatorname{\\mathcal{H}\\! \\mathit{om}} (  K,  (f^\\prime_1)^! \\mathcal{O}_{Y_2})$. In particular, $K^\\prime\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$.\n\\end{proposition}",
      "prop:invariance_under_unit_preserve_adjointness": "\\begin{proposition}\n    \\label{prop:invariance_under_unit_preserve_adjointness}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\\prime$ for the kernel of the integral transform obtained in \\Cref{prop:right_adjoint_pullback} which is right adjoint to $\\Phi_K$ on $D_{\\operatorname{qc}}$. If $\\Phi_K$ is fully faithful on $D_{\\operatorname{qc}}$, then $\\mathbf{L} t^\\ast_1$ preserves the unit $\\eta$ of the adjoint pair $\\Phi_K$ and $\\Phi_{K^\\prime}$ on $D_{\\operatorname{qc}}$. That is, $\\mathbf{L}t_1^\\ast \\eta_E$ is the unit of $\\mathbf{L}t_1^\\ast E$ for the adjoint pair $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$ and $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{proposition}",
      "cor:special_fibers_genii_invariance": "\\begin{corollary}\n    \\label{cor:special_fibers_genii_invariance}\n    Let $S$ be a separated Noetherian $\\mathbb{Q}$-scheme. Suppose $f_i\\colon Y_i \\to S$ are proper flat morphisms whose special fibers are geometrically integral curves. Assume $f_1$ is locally projective. If $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$, then then genii of the special fibers of $Y_1$ coincide with that of $Y_2$.\n\\end{corollary}",
      "prop:left_adjoint_pullback": "\\begin{proposition}\n    \\label{prop:left_adjoint_pullback}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$ is left adjoint to $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ on $D_{\\operatorname{qc}}$ where $K^\\prime:= \\mathbf{R}\\operatorname{\\mathcal{H}\\! \\mathit{om}} (  K,  (f^\\prime_2)^! \\mathcal{O}_{Y_1}) $. In particular, $K^\\prime\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$.\n\\end{proposition}",
      "prop:smoothness_derived_invariance": "\\begin{proposition}\n    \\label{prop:smoothness_derived_invariance}\n    Let $Y_1$ and $Y_2$ be proper schemes over a field $k$ (which need not be perfect). Suppose $Y_1$ and $Y_2$ are Fourier--Mukai partners. If $Y_1$ is geometrically regular, then so is $Y_2$. In other words, smoothness is a derived invariance.\n\\end{proposition}",
      "cor:smooth_morphisn_derived_invariance": "\\begin{corollary}\n    \\label{cor:smooth_morphisn_derived_invariance}\n    Let $f_i \\colon Y_i \\to S$ be locally projective flat morphisms to a separated Noetherian scheme. Assume that $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$. Then $f_1$ is smooth if, and only if, $f_2$ is smooth.\n\\end{corollary}",
      "prop:adjoints_generalized_affine": "\\begin{proposition}\n    \\label{prop:adjoints_generalized_affine}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\\prime$ the kernel of the integral transform associated to the right adjoint of $\\Phi_K \\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ obtained in \\Cref{prop:right_adjoint_pullback}. If $\\Phi_K$ is fully faithful on $D_{\\operatorname{qc}}$, then $\\mathbf{L} t^\\ast_1$ preserves the unit $\\eta$ of the adjoint pair $\\Phi_K$ and $\\Phi_{K^\\prime}$ on $D_{\\operatorname{qc}}$. That is, $\\mathbf{L}t_1^\\ast \\eta_E$ is the unit of $\\mathbf{L}t_1^\\ast E$ for the adjoint pair $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$ and $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$.\n\\end{proposition}",
      "introthm:fully_faithfulness_equivalence": "\\begin{theorem}\n    [see \\Cref{cor:ascending_faithfully}]\n    \\label{introthm:fully_faithfulness_equivalence}\n    Consider affine morphism $t\\colon T \\to S$ locally of finite tor-dimension of Noetherian schemes whose target is semiseparated. Suppose $Y_1$ and $Y_2$ are proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$.\n\\end{theorem}",
      "cor:ascending_faithfully_affine": "\\begin{corollary}\n    \\label{cor:ascending_faithfully_affine}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}",
      "ex:curves_isomorphic_after_base_change": "\\begin{example}\n    \\label{ex:curves_isomorphic_after_base_change}\n    There is a pair of non-isomorphic elliptic curves $E_1$ and $E_2$ over $\\mathbb{Q}$ that are Fourier--Mukai partners (see \\cite[Example 2.3.11]{Ward:2014}). From \\cite[Theorem 2.2]{Orlov:2002}, it follows that any triangulated equivalence $D^b_{\\operatorname{coh}}(E_1) \\to D^b_{\\operatorname{coh}}(E_2)$ is given by an integral transform. \n    Note that an elliptic curve is geometrically integral and projective. \n    By \\Cref{prop:curves_isomorphic_after_base_change}, there is a finite field extension $L/\\mathbb{Q}$ such that $E_1 \\times_{\\mathbb{Q}} \\operatorname{Spec}(L) \\cong E_2 \\times_{\\mathbb{Q}} \\operatorname{Spec}(L)$.\n\\end{example}",
      "introthm:ascending_faithfully_affine": "\\begin{theorem}\n    [see \\Cref{cor:ascending_faithfully_affine}]\n    \\label{introthm:ascending_faithfully_affine}\n    Consider affine morphism $t\\colon T \\to S$ to a semiseparated Noetherian scheme. Suppose $Y_1\\to S$ is flat, locally projective and $Y_2\\to S$ is flat, proper. Denote by $t^\\prime \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}",
      "cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine": "\\begin{corollary}\n    \\label{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine faithfully flat morphism. Let $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. Then $\\mathbf{L}(t^\\prime)^\\ast K$ is relatively perfect over $Y_1\\times_S T$ (resp.\\ $Y_2\\times_S T$) if, and only if, $K$ is relatively perfect over $Y_1$ (resp.\\ $Y_2$).\n\\end{corollary}",
      "thm:ascending": "\\begin{theorem}\n    \\label{thm:ascending}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. If $\\Phi_K$ is fully faithful (resp.\\ an equivalence) on $D_{\\operatorname{qc}}$, then so is $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}",
      "prop:curves_isomorphic_after_base_change": "\\begin{proposition}\n    \\label{prop:curves_isomorphic_after_base_change}\n    Let $k$ be a field of characteristic zero. Suppose $C$ and $C^\\prime$ are geometrically integral projective curves over $k$. If $C$ and $C^\\prime$ are Fourier--Mukai partners over $k$, then $\\operatorname{Spec}(L)\\times_k C \\cong \\operatorname{Spec}(L)\\times_k C^\\prime$ for some finite field extension $L/k$ (i.e.\\ are $k$-forms).\n\\end{proposition}",
      "lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine": "\\begin{lemma}\n    \\label{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}\n    Consider \\Cref{setup:fm_cube_pullback} with $t$ an affine morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ restricts to an exact functor on $D^b_{\\operatorname{coh}}$. Additionally, a similar statement holds for $\\operatorname{Perf}$ if $K$ is relatively perfect over $Y_2$.\n\\end{lemma}",
      "cor:ascending_faithfully": "\\begin{corollary}\n    \\label{cor:ascending_faithfully}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}",
      "prop:fully_faithful_on_small_categories": "\\begin{proposition}\n    \\label{prop:fully_faithful_on_small_categories}\n    Consider \\Cref{setup:fm_cube_pullback} where $t=1_S$ and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\\prime$ for the kernel of the right adjoint for $\\Phi_K$ obtained in \\Cref{prop:right_adjoint_pullback}. Set $\\eta$ to be the unit of the adjunction $\\Phi_K$ and $\\Phi_{K^\\prime}$ on $D_{\\operatorname{qc}}$. Then the following are equivalent:\n    \\begin{enumerate}\n        \\item \\label{prop:fully_faithful_on_small_categories1} $\\Phi_K$ is fully faithful on $D_{\\operatorname{qc}}$\n        \\item \\label{prop:fully_faithful_on_small_categories2} $\\eta_E$ is an isomorphism for all $E\\in D^-_{\\operatorname{coh}}(Y_1)$\n        \\item \\label{prop:fully_faithful_on_small_categories3} $\\eta_E$ is an isomorphism for all $E\\in D^b_{\\operatorname{coh}}(Y_1)$\n        \\item \\label{prop:fully_faithful_on_small_categories4} $\\eta_E$ is an isomorphism for all $E\\in \\operatorname{Perf}(Y_1)$\n    \\end{enumerate}\n    Additionally, if $K$ is relatively perfect over $Y_1$, then these are equivalent to $\\Phi_K$ is fully faithful on $D^b_{\\operatorname{coh}}$.\n\\end{proposition}"
    },
    "pre_theorem_intro_text_len": 8538,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\n\\subsection{Setting the stage}\n\\label{sec:intro_stage}\n\nThe context of our work starts with Mukai \\cite{Mukai:1981}. Loc.\\ cit.\\ introduced the notion of integral transforms to study triangulated equivalences between derived categories associated with abelian varieties. Over time, integral transforms have played a central role in the interplay between derived categories and algebraic geometry (e.g.\\ \\cite{Bridgeland:2002, Kawamata:2002a}), with further applications in string theory \\cite{Andres/HernandezRuiperez:2005, Andres/Scheidegger/Sharpe:2005}.\n\nMuch of this progress has focused on smooth varieties, where geometric results rely on categorical methods. However, extending these methods to singular varieties remains challenging, partly because bounded coherent complexes are typically not perfect. Thus, an important task is to develop new results or techniques that improve our understanding of integral transforms in such singular settings.\n\nRecent work has begun to address these difficulties. Criteria determining when an integral transform is fully faithful or an equivalence have been established for varieties over algebraically closed fields, generalizing results from the smooth case \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009} and building upon foundational work of Bondal--Orlov \\cite{Bondal/Orlov:1995}. Likewise, Bondal--Orlov's reconstruction theorem for smooth projective varieties has been extended to projective curves \\cite{Spence:2023} and (anti-)Fano Gorenstein varieties \\cite{Ballard:2011}.\n\nThere is also growing interest in studying integral transforms in the relative setting. Specifically, let $Y_1$ and $Y_2$ be proper $S$-schemes where $S$ is Noetherian. An \\textit{integral transform} is an exact functor $\\Phi_K := \\mathbf{R}(p_2)_\\ast(\\mathbf{L}p_1^\\ast(-)\\otimes^{\\mathbf{L}} K) \\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ where $K \\in D_{\\operatorname{qc}}(Y_1\\times_S Y_2)$ and $p_i \\colon Y_1\\times_S Y_2 \\to Y_i$ are the projection morphisms. If $\\Phi_K$ is an equivalence, then we say $Y_1$ and $Y_2$ are \\textit{Fourier--Mukai partners (over $S$)}.\n\nSuch transforms naturally arise in the study of \\textit{fibrations}; that is, in situations involving proper and flat morphisms. See e.g.\\ \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, which dealt with families whose special fibers have mild singularities. In particular, loc.\\ cit.\\ investigated when an integral transform between locally projective fibrations $Y_i \\to S$ is an equivalence for $S$ of finite type over an algebraically closed field. It was shown that the induced integral transform on special fibers (i.e.\\ closed points) over $S$ controls the behavior of equivalences and full faithfulness.\n\nIn our work, we reveal additional depth to this perspective of exploiting special fibers. Specifically, we study the behavior of integral transforms under changes of the base scheme in more general settings that includes local algebra. Moreover, our approach naturally extends several results in the literature to arbitrary fields, and at the same time yields new insight for singularities of fibrations.\n\n\\subsection{Preservation of adjointness}\n\\label{sec:intro_preservation_adjoint}\n\nNeeman has developed a new framework called approximable triangulated categories (see e.g.\\ \\cite{Canonaco/Neeman/Stellari:2025,Neeman:2023}), which has resolved open conjectures in algebraic geometry \\cite{Neeman:2021a,Neeman:2022}. As a consequence of this machinery, any integral transform $\\Phi_K$ inducing an equivalence on $D_{\\operatorname{qc}}$ restricts to equivalences on the subcategories of bounded complexes with coherent cohomology $D^b_{\\operatorname{coh}}$ and perfect complexes $\\operatorname{Perf}$. Moreover, an equivalence on these `smaller' categories implies an equivalence on the `larger' category. See \\Cref{rmk:equivalences_induced} for details.\n\nFor those concerned with equivalences, this allows the study of integral transforms to be reduced to their behavior on these subcategories.  Whether an integral transform restricts to these smaller categories is now well-understood \\cite{Rizzardo:2017,Ballard:2009,Dutta/Lank/ManaliRahul:2025}. These results shed light to the existence of left or right adjoints to integral transforms and the conditions under which such adjoints also restrict to these subcategories.\n\nHowever, the behavior of adjointness under base change is not yet fully understood. Our focus is on base changes induced by affine morphisms of finite tor-dimension. In particular, this leads us to the following, which summarizes \\Cref{sec:preserve_adjoints}.\n\n\\begin{proposition}\n    \\label{prop:intro_preservation_adjoint}\n    Suppose $t\\colon T \\to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Consider $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. \n    \\begin{enumerate}\n        \\item \\label{prop:intro_preservation_adjoint1} If $t$ is affine and $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$ (resp.\\ $\\operatorname{Perf}$), then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$ (see \\Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \\Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).\n        \\item \\label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension. \n        \\begin{enumerate}\n            \\item \\label{prop:intro_preservation_adjoint2a} If $\\Phi_K$ restricts to $\\operatorname{Perf}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its right adjoint (see  \\Cref{prop:right_adjoint_pullback}).\n            \\item \\label{prop:intro_preservation_adjoint2b} If $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its left adjoint (see \\Cref{prop:left_adjoint_pullback}).\n        \\end{enumerate}\n    \\end{enumerate}\n\\end{proposition}\n\nIn short, \\Cref{prop:intro_preservation_adjoint} describes the behavior of integral transforms under base change when restricted to the small categories, specifically addressing whether the preservation of adjoints occurs. Key instances where it applies are morphisms $t$ arising from field extensions or finite surjective morphisms to a Dedekind scheme (i.e.\\ smooth projective curves).\n\n\\subsection{Fully faithfulness \\& equivalences}\n\\label{sec:intro_fully_faithfulness_equivalence}\n\nNow, we have a clearer understanding of how integral transforms behave under base change. Particularly, regarding the preservation of adjointness and the restriction to smaller categories. Next, we shift gears to studying fully faithfulness and equivalences under base change; specifically, the ascent and descent of these properties. Before doing so, we review the relevant literature a bit more.\n\nAs mentioned earlier, the work of \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009} studied these questions for fibrations whose base is of finite type over an algebraically closed field. A key ingredient in loc.\\ cit.\\ was to look at the behavior on special fibers. More recently, such questions have also been studied for smooth proper morphisms over a Noetherian base scheme (see \\cite[Proposition/Definition 2.3]{Kurama:2024}). However, smoothness constraints can restrict the examples which one might be interested in, e.g.\\ fibrations admitting singular fibers.\n\nA result of Orlov addresses similar questions in the case of smooth varieties over a field. Specifically, for smooth projective varieties over a field, an integral transform is fully faithful or an equivalence if, and only if, the induced integral transform satisfies the same property after any base field extension \\cite[Lemma 2.12]{Orlov:2002}. The strategy of loc.\\ cit.\\ leverages `canonical morphisms' between the kernels of integral transforms that describe the (co)units of adjunction. However, it remains unclear whether such canonical morphisms exist in the singular setting. This uncertainty is another source of motivation for our work, as we aimed to bypass the lack of these tools in singular cases using different techniques.\n\nThe brings attention to the following, which summarizes \\Cref{sec:fully_faithful_equivalence}.",
    "context": "There is also growing interest in studying integral transforms in the relative setting. Specifically, let $Y_1$ and $Y_2$ be proper $S$-schemes where $S$ is Noetherian. An \\textit{integral transform} is an exact functor $\\Phi_K := \\mathbf{R}(p_2)_\\ast(\\mathbf{L}p_1^\\ast(-)\\otimes^{\\mathbf{L}} K) \\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ where $K \\in D_{\\operatorname{qc}}(Y_1\\times_S Y_2)$ and $p_i \\colon Y_1\\times_S Y_2 \\to Y_i$ are the projection morphisms. If $\\Phi_K$ is an equivalence, then we say $Y_1$ and $Y_2$ are \\textit{Fourier--Mukai partners (over $S$)}.\n\nSuch transforms naturally arise in the study of \\textit{fibrations}; that is, in situations involving proper and flat morphisms. See e.g.\\ \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, which dealt with families whose special fibers have mild singularities. In particular, loc.\\ cit.\\ investigated when an integral transform between locally projective fibrations $Y_i \\to S$ is an equivalence for $S$ of finite type over an algebraically closed field. It was shown that the induced integral transform on special fibers (i.e.\\ closed points) over $S$ controls the behavior of equivalences and full faithfulness.\n\nNeeman has developed a new framework called approximable triangulated categories (see e.g.\\ \\cite{Canonaco/Neeman/Stellari:2025,Neeman:2023}), which has resolved open conjectures in algebraic geometry \\cite{Neeman:2021a,Neeman:2022}. As a consequence of this machinery, any integral transform $\\Phi_K$ inducing an equivalence on $D_{\\operatorname{qc}}$ restricts to equivalences on the subcategories of bounded complexes with coherent cohomology $D^b_{\\operatorname{coh}}$ and perfect complexes $\\operatorname{Perf}$. Moreover, an equivalence on these `smaller' categories implies an equivalence on the `larger' category. See \\Cref{rmk:equivalences_induced} for details.\n\n\\begin{proposition}\n \\label{prop:intro_preservation_adjoint}\n Suppose $t\\colon T \\to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Consider $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint1} If $t$ is affine and $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$ (resp.\\ $\\operatorname{Perf}$), then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$ (see \\Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \\Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).\n \\item \\label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint2a} If $\\Phi_K$ restricts to $\\operatorname{Perf}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its right adjoint (see \\Cref{prop:right_adjoint_pullback}).\n \\item \\label{prop:intro_preservation_adjoint2b} If $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its left adjoint (see \\Cref{prop:left_adjoint_pullback}).\n \\end{enumerate}\n \\end{enumerate}\n\\end{proposition}\n\nA result of Orlov addresses similar questions in the case of smooth varieties over a field. Specifically, for smooth projective varieties over a field, an integral transform is fully faithful or an equivalence if, and only if, the induced integral transform satisfies the same property after any base field extension \\cite[Lemma 2.12]{Orlov:2002}. The strategy of loc.\\ cit.\\ leverages `canonical morphisms' between the kernels of integral transforms that describe the (co)units of adjunction. However, it remains unclear whether such canonical morphisms exist in the singular setting. This uncertainty is another source of motivation for our work, as we aimed to bypass the lack of these tools in singular cases using different techniques.\n\nThe brings attention to the following, which summarizes \\Cref{sec:fully_faithful_equivalence}.",
    "full_context": "There is also growing interest in studying integral transforms in the relative setting. Specifically, let $Y_1$ and $Y_2$ be proper $S$-schemes where $S$ is Noetherian. An \\textit{integral transform} is an exact functor $\\Phi_K := \\mathbf{R}(p_2)_\\ast(\\mathbf{L}p_1^\\ast(-)\\otimes^{\\mathbf{L}} K) \\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ where $K \\in D_{\\operatorname{qc}}(Y_1\\times_S Y_2)$ and $p_i \\colon Y_1\\times_S Y_2 \\to Y_i$ are the projection morphisms. If $\\Phi_K$ is an equivalence, then we say $Y_1$ and $Y_2$ are \\textit{Fourier--Mukai partners (over $S$)}.\n\nSuch transforms naturally arise in the study of \\textit{fibrations}; that is, in situations involving proper and flat morphisms. See e.g.\\ \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, which dealt with families whose special fibers have mild singularities. In particular, loc.\\ cit.\\ investigated when an integral transform between locally projective fibrations $Y_i \\to S$ is an equivalence for $S$ of finite type over an algebraically closed field. It was shown that the induced integral transform on special fibers (i.e.\\ closed points) over $S$ controls the behavior of equivalences and full faithfulness.\n\nNeeman has developed a new framework called approximable triangulated categories (see e.g.\\ \\cite{Canonaco/Neeman/Stellari:2025,Neeman:2023}), which has resolved open conjectures in algebraic geometry \\cite{Neeman:2021a,Neeman:2022}. As a consequence of this machinery, any integral transform $\\Phi_K$ inducing an equivalence on $D_{\\operatorname{qc}}$ restricts to equivalences on the subcategories of bounded complexes with coherent cohomology $D^b_{\\operatorname{coh}}$ and perfect complexes $\\operatorname{Perf}$. Moreover, an equivalence on these `smaller' categories implies an equivalence on the `larger' category. See \\Cref{rmk:equivalences_induced} for details.\n\n\\begin{proposition}\n \\label{prop:intro_preservation_adjoint}\n Suppose $t\\colon T \\to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Consider $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint1} If $t$ is affine and $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$ (resp.\\ $\\operatorname{Perf}$), then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$ (see \\Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \\Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).\n \\item \\label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint2a} If $\\Phi_K$ restricts to $\\operatorname{Perf}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its right adjoint (see \\Cref{prop:right_adjoint_pullback}).\n \\item \\label{prop:intro_preservation_adjoint2b} If $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its left adjoint (see \\Cref{prop:left_adjoint_pullback}).\n \\end{enumerate}\n \\end{enumerate}\n\\end{proposition}\n\nA result of Orlov addresses similar questions in the case of smooth varieties over a field. Specifically, for smooth projective varieties over a field, an integral transform is fully faithful or an equivalence if, and only if, the induced integral transform satisfies the same property after any base field extension \\cite[Lemma 2.12]{Orlov:2002}. The strategy of loc.\\ cit.\\ leverages `canonical morphisms' between the kernels of integral transforms that describe the (co)units of adjunction. However, it remains unclear whether such canonical morphisms exist in the singular setting. This uncertainty is another source of motivation for our work, as we aimed to bypass the lack of these tools in singular cases using different techniques.\n\nThe brings attention to the following, which summarizes \\Cref{sec:fully_faithful_equivalence}.\n\n\\begin{proposition}\n \\label{prop:intro_preservation_adjoint}\n Suppose $t\\colon T \\to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Consider $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint1} If $t$ is affine and $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$ (resp.\\ $\\operatorname{Perf}$), then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$ (see \\Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \\Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).\n \\item \\label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension. \n \\begin{enumerate}\n \\item \\label{prop:intro_preservation_adjoint2a} If $\\Phi_K$ restricts to $\\operatorname{Perf}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its right adjoint (see \\Cref{prop:right_adjoint_pullback}).\n \\item \\label{prop:intro_preservation_adjoint2b} If $\\Phi_K$ restricts to $D^b_{\\operatorname{coh}}$, then $\\mathbf{L}\\pi^\\ast$ preserves the adjunction of $\\Phi_K$ with its left adjoint (see \\Cref{prop:left_adjoint_pullback}).\n \\end{enumerate}\n \\end{enumerate}\n\\end{proposition}\n\nThe brings attention to the following, which summarizes \\Cref{sec:fully_faithful_equivalence}.\n\nIn practice, many schemes we may consider are separated, and hence, semiseparated. As a pleasant consequence, this allows us to partially generalize Orlov's result to the singular setting.\n\n\\begin{theorem}\n [see \\Cref{cor:ascending_faithfully_affine}]\n \\label{introthm:ascending_faithfully_affine}\n Consider affine morphism $t\\colon T \\to S$ to a semiseparated Noetherian scheme. Suppose $Y_1\\to S$ is flat, locally projective and $Y_2\\to S$ is flat, proper. Denote by $t^\\prime \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}\n\n\\begin{theorem}\n \\label{thm:ascending}\n Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. If $\\Phi_K$ is fully faithful (resp.\\ an equivalence) on $D_{\\operatorname{qc}}$, then so is $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}\n\n\\begin{corollary}\n \\label{cor:ascending_faithfully}\n Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}\n\n\\begin{proposition}\n \\label{prop:adjoints_generalized_affine}\n Consider \\Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\\prime$ the kernel of the integral transform associated to the right adjoint of $\\Phi_K \\colon D_{\\operatorname{qc}}(Y_1) \\to D_{\\operatorname{qc}}(Y_2)$ obtained in \\Cref{prop:right_adjoint_pullback}. If $\\Phi_K$ is fully faithful on $D_{\\operatorname{qc}}$, then $\\mathbf{L} t^\\ast_1$ preserves the unit $\\eta$ of the adjoint pair $\\Phi_K$ and $\\Phi_{K^\\prime}$ on $D_{\\operatorname{qc}}$. That is, $\\mathbf{L}t_1^\\ast \\eta_E$ is the unit of $\\mathbf{L}t_1^\\ast E$ for the adjoint pair $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$ and $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$.\n\\end{proposition}\n\n\\begin{corollary}\n \\label{cor:ascending_faithfully_affine}\n Consider \\Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}\n\n\\begin{corollary}\n    \\label{cor:ascending_faithfully}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}\n\n\\begin{corollary}\n    \\label{cor:ascending_faithfully_affine}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}\n\n\\begin{corollary}\n    \\label{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine faithfully flat morphism. Let $K\\in D^-_{\\operatorname{coh}}(Y_1\\times_S Y_2)$. Then $\\mathbf{L}(t^\\prime)^\\ast K$ is relatively perfect over $Y_1\\times_S T$ (resp.\\ $Y_2\\times_S T$) if, and only if, $K$ is relatively perfect over $Y_1$ (resp.\\ $Y_2$).\n\\end{corollary}\n\n\\begin{lemma}\n    \\label{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}\n    Consider \\Cref{setup:fm_cube_pullback} with $t$ an affine morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ restricts to an exact functor on $D^b_{\\operatorname{coh}}$. Additionally, a similar statement holds for $\\operatorname{Perf}$ if $K$ is relatively perfect over $Y_2$.\n\\end{lemma}\n\n\\begin{proposition}\n    \\label{prop:left_adjoint_pullback}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$ is left adjoint to $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ on $D_{\\operatorname{qc}}$ where $K^\\prime:= \\mathbf{R}\\operatorname{\\mathcal{H}\\! \\mathit{om}} (  K,  (f^\\prime_2)^! \\mathcal{O}_{Y_1}) $. In particular, $K^\\prime\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$.\n\\end{proposition}\n\n\\begin{proposition}\n    \\label{prop:right_adjoint_pullback}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension (e.g.\\ $t=1_S$). Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Then $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K^\\prime}$ is right adjoint to $\\Phi_{\\mathbf{L} (t^\\prime)^\\ast K}$ on $D_{\\operatorname{qc}}$ where $K^\\prime:= \\mathbf{R}\\operatorname{\\mathcal{H}\\! \\mathit{om}} (  K,  (f^\\prime_1)^! \\mathcal{O}_{Y_2})$. In particular, $K^\\prime\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$.\n\\end{proposition}",
    "post_theorem_intro_text_len": 7417,
    "post_theorem_intro_text": "In practice, many schemes we may consider are separated, and hence, semiseparated. As a pleasant consequence, this allows us to partially generalize Orlov's result to the singular setting.\n\n\\begin{corollary}\n    [cf.\\ {\\cite[Lemma 2.12]{Orlov:2002}}]\n    \\label{cor:orlov_varieties}\n    Consider proper schemes $Y_1$ and $Y_2$ over a field $k$. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_k Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{K_L}$ for any field extension $L/k$.\n\\end{corollary}\n\nOur proofs require careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change. However, additional subtleties also arise in this context. In particular, we need to work in the larger category $ D_{\\operatorname{qc}} $ in order to have a firm grasp on ascending fully faithfulness; see e.g.\\ \\Cref{prop:fully_faithful_on_small_categories} and \\Cref{thm:ascending}. Notably, the techniques we use are independent of those appearing in \\cite[Lemma 2.12]{Orlov:2002}.\n\n\\subsection{Affine variations}\n\\label{sec:intro_affine_variations}\n\nNaturally, we sought analogs of \\Cref{introthm:fully_faithfulness_equivalence} in which $t$ need not be locally of finite tor-dimension. Ideally, such a framework would include morphisms from special fibers as in \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. Accordingly, we work in the setting of affine morphisms to the base, while imposing a mild restriction on one of the $Y_i \\to S$.\n\nThe following result is a slight generalization of \\cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. In particular, we perform base change along affine morphisms rather than closed immersions of closed points. Note that a key difference between what follows and \\Cref{sec:preserve_adjoints,sec:fully_faithful_equivalence} is that we impose one of the $Y_i \\to S$ be locally projective (as opposed to proper like our earlier results).\n\n\\begin{theorem}\n    [see \\Cref{cor:ascending_faithfully_affine}]\n    \\label{introthm:ascending_faithfully_affine}\n    Consider affine morphism $t\\colon T \\to S$ to a semiseparated Noetherian scheme. Suppose $Y_1\\to S$ is flat, locally projective and $Y_2\\to S$ is flat, proper. Denote by $t^\\prime \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}\n\nIt is worthwhile to note that \\Cref{introthm:ascending_faithfully_affine} yields an independent proof of the ascending full faithfulness or equivalences in \\cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. To do so, we show in suitable cases that the unit of an adjoint pair of integral transforms is preserved under base change when the left adjoint is fully faithful (see \\Cref{prop:adjoints_generalized_affine} and \\Cref{prop:invariance_under_unit_preserve_adjointness}). However, it is not clear to us whether this remains true when the left adjoint is not fully faithful a priori. This condition seems necessary (at least to us) to establish the `descent' part of \\cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009} in the case of special fibers.\n\n\\subsection{Applications}\n\\label{sec:intro_applications}\n\nTo conclude our work, we discuss the ramifications of \\Cref{introthm:fully_faithfulness_equivalence,introthm:ascending_faithfully_affine}. Our results extend the literature on schemes of finite type over algebraically closed fields to the more general setting of arbitrary fields. They reveal an interplay between the intuition of \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009} concerning special fibers and our work on faithfully flat descent along affine morphisms. In particular, certain singularities are shown to be derived invariants for fibrations over Noetherian schemes, broadening the scope to include arithmetically flavored situations of potential interest for future research.\n\nFor a pair of geometrically integral projective curves over a field $k$ of characteristic zero, Fourier--Mukai partnership implies they are $k$-forms (see \\Cref{prop:curves_isomorphic_after_base_change}). Here, a $k$-form means there exists a finite extension $L/k$ such that their base changes along $L/k$ become isomorphic. The result holds for singular curves and enriches algebraically closed case of \\cite{Spence:2023}. In \\Cref{ex:curves_isomorphic_after_base_change}, we exhibit a pair of non-isomorphic elliptic curves over $\\mathbb{Q}$ that are nevertheless $\\mathbb{Q}$-forms. For fibrations of relative dimension one over a Noetherian $\\mathbb{Q}$-scheme, Fourier--Mukai partnership forces the genera of special fibers to coincide (see \\Cref{cor:special_fibers_genii_invariance}).\n\nSmoothness for proper schemes over an arbitrary field is likewise invariant under Fourier--Mukai partnership (see \\Cref{prop:smoothness_derived_invariance}). This subtlety arises especially over imperfect fields. Since regularity is characterized by $D^b_{\\operatorname{coh}} = \\operatorname{Perf}$, one might expect this condition to persist after a change of field extensions, but there exist regular projective curves that become singular after such a base change (see e.g.\\ \\cite[Remark 16]{Kollar:2011}).\n\nIn the setting of projective fibrations, smoothness of the total space is also a derived invariant (see \\Cref{cor:smooth_morphisn_derived_invariance}). Finally, generalizing \\cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, we show that the Cohen--Macaulay and Gorenstein properties of a morphism remain invariant under Fourier--Mukai partnership for Noetherian base schemes (see \\Cref{prop:CM_or_Gorensteinnes_morphisn_derived_invariance}).\n\n\\subsection*{Notation}\n\\label{sec:intro_notation}\n\nLet $X$ be a Noetherian scheme. We work with the following triangulated categories:\n\\begin{enumerate}\n    \\item $D(X):=D(\\operatorname{Mod}(X))$ is the derived category of $\\mathcal{O}_X$-modules.\n    \\item $D_{\\operatorname{qc}}(X)$ is the (strictly full) subcategory of $D(X)$ consisting of complexes with quasi-coherent cohomology.\n    \\item $D_{\\operatorname{coh}}^b(X)$ (resp.\\ $D_{\\operatorname{qc}}^b(X)$) is the (strictly full) subcategory of $D(X)$ consisting of complexes having bounded and coherent (resp.\\ quasi-coherent) cohomology.\n    \\item $\\operatorname{Perf}(X)$ is the (strictly full) subcategory of $D_{\\operatorname{qc}}(X)$ consisting of the perfect complexes on $X$.\n\\end{enumerate}\nAt times if $X$ is affine, we abuse notation and write $D(R):=D_{\\operatorname{qc}}(X)$ where $R:=H^0(X,\\mathcal{O}_X)$ are the global sections; similar conventions will occur for the other categories.\n\n\\begin{ack}\n    Lank was supported under the ERC Advanced Grant 101095900-TriCatApp. Part of this work was completed while Lank visited the Basque Center for Applied Mathematics (BCAM) and thanks them for their hospitality. The authors thank Dylan Spence, Ana Cristina L\\'{o}pez-Mart\\'{i}n, and El\\'{i}as Guisado Villalgordo for discussions in an earlier version of the work.\n\\end{ack}",
    "sketch": "The post-theorem introduction does not give a step-by-step proof of \\Cref{introthm:fully_faithfulness_equivalence}, but it indicates the main technical ingredients: the proofs involve “careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change,” and “additional subtleties” mean one “need[s] to work in the larger category $D_{\\operatorname{qc}}$ in order to have a firm grasp on ascending fully faithfulness; see e.g.\\ \\Cref{prop:fully_faithful_on_small_categories} and \\Cref{thm:ascending}.” It is also noted that “the techniques we use are independent of those appearing in \\cite[Lemma 2.12]{Orlov:2002}.”",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof of \\Cref{introthm:fully_faithfulness_equivalence}, but it indicates the main technical ingredients: the proofs involve “careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change,” and “additional subtleties” mean one “need[s] to work in the larger category $D_{\\operatorname{qc}}$ in order to have a firm grasp on ascending fully faithfulness.” In particular, one uses the following proposition.\n\n\\begin{proposition}\n    \\label{prop:fully_faithful_on_small_categories}\n    Consider \\Cref{setup:fm_cube_pullback} where $t=1_S$ and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\\prime$ for the kernel of the right adjoint for $\\Phi_K$ obtained in \\Cref{prop:right_adjoint_pullback}. Set $\\eta$ to be the unit of the adjunction $\\Phi_K$ and $\\Phi_{K^\\prime}$ on $D_{\\operatorname{qc}}$. Then the following are equivalent:\n    \\begin{enumerate}\n        \\item \\label{prop:fully_faithful_on_small_categories1} $\\Phi_K$ is fully faithful on $D_{\\operatorname{qc}}$\n        \\item \\label{prop:fully_faithful_on_small_categories2} $\\eta_E$ is an isomorphism for all $E\\in D^-_{\\operatorname{coh}}(Y_1)$\n        \\item \\label{prop:fully_faithful_on_small_categories3} $\\eta_E$ is an isomorphism for all $E\\in D^b_{\\operatorname{coh}}(Y_1)$\n        \\item \\label{prop:fully_faithful_on_small_categories4} $\\eta_E$ is an isomorphism for all $E\\in \\operatorname{Perf}(Y_1)$\n    \\end{enumerate}\n    Additionally, if $K$ is relatively perfect over $Y_1$, then these are equivalent to $\\Phi_K$ is fully faithful on $D^b_{\\operatorname{coh}}$.\n\\end{proposition}\n\nOne also uses the following ascent result.\n\n\\begin{theorem}\n    \\label{thm:ascending}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ is semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over $Y_2$. If $\\Phi_K$ is fully faithful (resp.\\ an equivalence) on $D_{\\operatorname{qc}}$, then so is $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{theorem}\n\nIt is also noted that “the techniques we use are independent of those appearing in \\cite[Lemma 2.12]{Orlov:2002}.”",
    "expanded_theorem": "\\begin{corollary}\n    \\label{cor:ascending_faithfully}\n    Consider \\Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}(t^\\prime)^\\ast K}$.\n\\end{corollary}\n\n\\label{introthm:fully_faithfulness_equivalence}\nConsider affine morphism $t\\colon T \\to S$ locally of finite tor-dimension of Noetherian schemes whose target is semiseparated. Suppose $Y_1$ and $Y_2$ are proper and flat $S$-schemes. Denote by $\\pi \\colon Y_1\\times_S Y_2 \\times_S T \\to Y_1\\times_S Y_2$ the natural morphism. Let $K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\\Phi_K$ restricts to a fully faithful functor (resp.\\ an equivalence) on $D^b_{\\operatorname{coh}}$, then so does $\\Phi_{\\mathbf{L}\\pi^\\ast K}$.,",
    "theorem_type": [
      "Implication"
    ],
    "mcq": {
      "question": "Let $t\\colon T\\to S$ be an affine morphism locally of finite tor-dimension between Noetherian schemes, and assume that $S$ is semiseparated. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Write\n$$\\pi\\colon Y_1\\times_S Y_2\\times_S T\\to Y_1\\times_S Y_2$$\nfor the natural morphism, and let\n$$K\\in D^b_{\\operatorname{coh}}(Y_1\\times_S Y_2)$$\nbe relatively perfect over each $Y_i$ (with respect to the natural projections $Y_1\\times_S Y_2\\to Y_i$). For a kernel $L$ on a fiber product, let the associated integral transform be\n$$\\Phi_L:=\\mathbf{R}(p_2)_*\\big(\\mathbf{L}p_1^*(-)\\otimes^{\\mathbf L} L\\big),$$\nwhere $p_1,p_2$ are the projection morphisms. Suppose that\n$$\\Phi_K\\colon D_{\\operatorname{qc}}(Y_1)\\to D_{\\operatorname{qc}}(Y_2)$$\nrestricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)\n$$D^b_{\\operatorname{coh}}(Y_1)\\to D^b_{\\operatorname{coh}}(Y_2).$$\nWhich conclusion about the pullback kernel $\\mathbf{L}\\pi^*K$ is valid?",
      "correct_choice": {
        "label": "A",
        "text": "The base-changed integral transform with kernel $\\mathbf{L}\\pi^*K$,\n$$\\Phi_{\\mathbf{L}\\pi^*K}\\colon D_{\\operatorname{qc}}(Y_1\\times_S T)\\to D_{\\operatorname{qc}}(Y_2\\times_S T),$$\nalso restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence):\n$$D^b_{\\operatorname{coh}}(Y_1\\times_S T)\\to D^b_{\\operatorname{coh}}(Y_2\\times_S T).$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "The base-changed integral transform with kernel $\\mathbf{L}\\pi^*K$,\n$$\\Phi_{\\mathbf{L}\\pi^*K}\\colon D_{\\operatorname{qc}}(Y_1\\times_S T)\\to D_{\\operatorname{qc}}(Y_2\\times_S T),$$\nalso restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)\n$$D^b_{\\operatorname{coh}}(Y_1\\times_S T)\\to D^b_{\\operatorname{coh}}(Y_2\\times_S T),$$\nand the converse implication also holds for every affine morphism $t$ locally of finite tor-dimension."
        },
        {
          "label": "C",
          "text": "The base-changed integral transform with kernel $\\mathbf{L}\\pi^*K$,\n$$\\Phi_{\\mathbf{L}\\pi^*K}\\colon D_{\\operatorname{qc}}(Y_1\\times_S T)\\to D_{\\operatorname{qc}}(Y_2\\times_S T),$$\nrestricts on bounded coherent complexes to an exact functor\n$$D^b_{\\operatorname{coh}}(Y_1\\times_S T)\\to D^b_{\\operatorname{coh}}(Y_2\\times_S T).$$"
        },
        {
          "label": "D",
          "text": "The base-changed integral transform with kernel $\\mathbf{L}\\pi^*K$,\n$$\\Phi_{\\mathbf{L}\\pi^*K}\\colon D_{\\operatorname{qc}}(Y_1\\times_S T)\\to D_{\\operatorname{qc}}(Y_2\\times_S T),$$\nis fully faithful (respectively, an equivalence) on all of $D_{\\operatorname{qc}}(Y_1\\times_S T)\\to D_{\\operatorname{qc}}(Y_2\\times_S T)$ whenever $\\Phi_K$ restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)\n$$D^b_{\\operatorname{coh}}(Y_1)\\to D^b_{\\operatorname{coh}}(Y_2).$$"
        },
        {
          "label": "E",
          "text": "The same conclusion holds after base change for $\\Phi_{\\mathbf{L}\\pi^*K}$ provided only that $t\\colon T\\to S$ is locally of finite tor-dimension; the assumptions that $t$ be affine and that $S$ be semiseparated are unnecessary."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "faithfully-flat converse replaced by converse for all affine lftd base changes",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped fully faithful/equivalence conclusion, keeping only preservation of $D^b_{\\operatorname{coh}}$",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "small-category hypothesis upgraded directly to a $D_{\\operatorname{qc}}$ conclusion after base change",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "removed affineness and semiseparatedness assumptions needed in ascent/bookkeeping of adjunctions",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option explicitly or by wording patterns. It presents hypotheses and asks for the valid conclusion, without signaling which strengthening/weakening is intended."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: under the stated hypotheses, the correct choice is the theorem’s main base-change conclusion almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning required to reject subtle overstatements and understatements (converse for all affine base changes, extension to all of D_qc, dropping hypotheses, merely exactness). But the item mainly tests recognition of the precise theorem statement rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: unjustified converse, weakening to a merely exact functor, upgrading from bounded coherent to all quasi-coherent derived categories, and removing needed hypotheses."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically careful MCQ with strong distractors, but it is largely a direct theorem-restatement question rather than a high-generative-reasoning item."
    }
  },
  {
    "id": "2512.16505v1",
    "paper_link": "http://arxiv.org/abs/2512.16505v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{global_existence} For $d=2, 3$,\nthere exists a diffeomorphism $\\xi_0: \\mathbb{R}^d \\to \\mathbb{R}^d$  {and a small constant $\\varepsilon>0$} such that if the initial data $(\\rho_0, u_0, F_0)$, with\n$(\\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\\in H^3_x(\\mathbb{R}^d)$, satisfies\n{\\begin{eqnarray*}\\label{decay_estiwmates_1}\n    \\begin{cases}\n      \\rho_0(x) \\mathrm{det}F_0(x)=1,\\,\\,\\,\\,F_0(x) = \\nabla_y \\xi_0(y),\\,\\,\\,\\,  y =\\xi_0^{-1}(x), \\ \\forall x\\in\\mathbb{R}^d, \\\\[2mm]\n      \\|\\rho_0(x) -1\\|_{H^3_x} + \\|u_0(x) -x\\|_{H^3_x} + \\|F_0(x) -I\\|_{H^3_x} + \\|\\xi_0^{-1}(x)-x\\|_{H^4_x}\\le \\varepsilon,\n    \\end{cases}\n\\end{eqnarray*}}then the Cauchy problem \\eqref{elasticity_equation_1} admits a unique global strong solution \n$(\\rho, u, F)$ such that $\\big(\\rho(x,t) -\\frac{1}{(1+t)^d}, u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big) \\in C\\big([0, \\infty); H^3_x(\\mathbb{R}^d)\\big)$. Moreover, $(\\rho, u, F)$ satisfies the following estimates:\n\\begin{eqnarray*}\\label{decay_estimates_1}\n  \\begin{cases}\n    \\|\\nabla_x^{i}(\\rho(x,t) -\\frac{1}{(1+t)^d})\\|_{L^2_x}\\le C\\varepsilon(1+t)^{-\\frac{d}{2}-\\frac{1}{2}-i},\\\\[1mm]\n     \\|\\nabla_x^{i}\\big(u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big)\\|_{L^2_x} \\le C\\varepsilon(1+t)^{\\frac{d}{2}+\\frac{1}{2}-i}\n  \\end{cases}\n\\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\\varepsilon$ and $t$.}",
      "start_pos": 10002,
      "end_pos": 11452,
      "label": "global_existence"
    },
    "ref_dict": {
      "global_existence_Lagrangian": "\\begin{theorem}\\label{global_existence_Lagrangian}\n       For $d=2,3$,  there exists a sufficiently small  constant $\\varepsilon_1>0$ such that the Cauchy problem \\eqref{compressible_elasticity_equation_3} admits a global strong solution \n       $(V,\\xi)$ satisfying $(V(y,t)-y, \\xi(y,t) - (1+t)y)$ in $C\\big([0, \\infty); H^3_y(\\mathbb{R}^d)\\times H^4_y(\\mathbb{R}^d)\\big)$,  provided that\n          \\begin{equation*} \\label{initial_assumption_Lagrangian}\n              \\begin{split}\n              \\|V_0(y)-y\\|_{H^3_y} + \\|\\xi_0(y) -y\\|_{H^4_y}\\le \\varepsilon_1.\n              \\end{split}\n          \\end{equation*}\nMoreover, $(V,\\xi)$ satisfies the following estimates:\n\\begin{eqnarray}\\label{decay_estimates_Lagrangian}\n  \\begin{cases}\n    \\|V(y,t) -y\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{1}{2}},\\\\[1mm]\n    \\|\\xi(y,t) - (1+t)y\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{3}{2}},\\\\[1mm]\n     \\|\\nabla_y \\xi(y,t) - (1+t)I\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{1}{2}}\n  \\end{cases}\n\\end{eqnarray}\nfor all $t>0$, where $C$ is a generic positive constant depending on some known constants but independent of $\\varepsilon_1$ and $t$.\n\\end{theorem}",
      "global_existence": "\\begin{theorem}\\label{global_existence} For $d=2, 3$,\nthere exists a diffeomorphism $\\xi_0: \\mathbb{R}^d \\to \\mathbb{R}^d$  {and a small constant $\\varepsilon>0$} such that if the initial data $(\\rho_0, u_0, F_0)$, with\n$(\\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\\in H^3_x(\\mathbb{R}^d)$, satisfies\n{\\begin{eqnarray*}\\label{decay_estiwmates_1}\n    \\begin{cases}\n      \\rho_0(x) \\mathrm{det}F_0(x)=1,\\,\\,\\,\\,F_0(x) = \\nabla_y \\xi_0(y),\\,\\,\\,\\,  y =\\xi_0^{-1}(x), \\ \\forall x\\in\\mathbb{R}^d, \\\\[2mm]\n      \\|\\rho_0(x) -1\\|_{H^3_x} + \\|u_0(x) -x\\|_{H^3_x} + \\|F_0(x) -I\\|_{H^3_x} + \\|\\xi_0^{-1}(x)-x\\|_{H^4_x}\\le \\varepsilon,\n    \\end{cases}\n\\end{eqnarray*}}then the Cauchy problem \\eqref{elasticity_equation_1} admits a unique global strong solution \n$(\\rho, u, F)$ such that $\\big(\\rho(x,t) -\\frac{1}{(1+t)^d}, u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big) \\in C\\big([0, \\infty); H^3_x(\\mathbb{R}^d)\\big)$. Moreover, $(\\rho, u, F)$ satisfies the following estimates:\n\\begin{eqnarray*}\\label{decay_estimates_1}\n  \\begin{cases}\n    \\|\\nabla_x^{i}(\\rho(x,t) -\\frac{1}{(1+t)^d})\\|_{L^2_x}\\le C\\varepsilon(1+t)^{-\\frac{d}{2}-\\frac{1}{2}-i},\\\\[1mm]\n     \\|\\nabla_x^{i}\\big(u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big)\\|_{L^2_x} \\le C\\varepsilon(1+t)^{\\frac{d}{2}+\\frac{1}{2}-i}\n  \\end{cases}\n\\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\\varepsilon$ and $t$.}\n\\end{theorem}",
      "L": "\\begin{equation}\\label{L}\n\t\\begin{split}\n    \\mathcal{L}(t) := &\\|v(t)\\|_{H^3}^2 + \\frac{2}{(1+t)^2}\\|\\widetilde{\\eta}(t)\\|_{H^3}^2 + \\|\\nabla \\widetilde{\\eta}(t)\\|_{H^3}^2+ \\frac{2}{1+t}\\sum_{i = 0}^{3} \\langle\\nabla^{i} v(t),\\nabla^{i} \\widetilde{\\eta}(t)\\rangle \\\\ &+ \\frac{1}{\\gamma(1+t)^{d(\\gamma -1)+2}} \\int_{\\mathbb{R}^d}J^{\\gamma+1}(t) |\\nabla^3 q(t)|^2 \\,dy.\n\t\\end{split}\n\\end{equation}",
      "1-1": "\\begin{equation}\\label{1-1}\n    \\bar{\\rho}=\\frac{1}{(1+t)^{3}}, \\bar{u} = \\frac{x}{1+t},  \\bar{F}= (1+t)I,\n  \\end{equation}",
      "constitution_laws": "\\begin{align}\\label{constitution_laws}\n\\begin{cases}\n\\rho\\  \\mathrm{det}F=1,\\\\\nF_{j}\\cdot\\nabla F_{k}=F_{k}\\cdot\\nabla F_{j}, \\quad 1\\leq   j, k\\leq d\n\\end{cases}\n\\end{align}",
      "compressible_elasticity_equation_5": "\\begin{eqnarray} \\label{compressible_elasticity_equation_5}\n\t\\begin{cases}\n\t\t\\partial_t \\widetilde{\\eta}=v,\\\\\n\t\t\\partial_t v_{i} + \\frac{2}{1+t} v_{i} + \\frac{1}{(1+t)^{d(\\gamma -1)+2}} a_{il}\\partial_l q -\\Delta \\widetilde{\\eta}_{i} =0,\\\\\n\t\t(v,\\widetilde{\\eta})|_{t=0} = \\big(v_0, \\widetilde{\\eta}_0\\big)\n\t\\end{cases}\n  \\end{eqnarray}",
      "est_L_1": "\\begin{equation}\\label{est_L_1}\n  \\begin{split}\n  \\udt \\mathcal{L}(t) + \\frac{1}{1+t} \\mathcal{L}(t)\\le \\frac{C}{(1+t)^{2}}\\big(\\mathcal{L}(t) + \\mathcal{L}^{\\frac{7}{2}}(t)\\big).\n  \\end{split}\n\\end{equation}",
      "elasticity_equation_1": "\\begin{eqnarray} \\label{elasticity_equation_1}\n    \\begin{cases}\n        \\partial_t\\rho+{\\rm div} (\\rho u) =0,\\\\\n        \\partial_t(\\rho u)+{\\rm div} (\\rho u \\otimes u)  + \\nabla P(\\rho)={\\rm div}(\\rho F F^\\top),\\\\\n        \\partial_t F + u\\cdot\\nabla F=\\nabla u F,\\\\\n        (\\rho,u,F)|_{t=0} = (\\rho_0, u_0, F_0)\n    \\end{cases}\n\\end{eqnarray}"
    },
    "pre_theorem_intro_text_len": 6124,
    "pre_theorem_intro_text": "\\label{section_1}\nWe consider the compressible {elastodynamics} in Eulerian coordinate \\cite{Gurtin_1981,Dafermos_2010}:\n\\begin{eqnarray} \\label{elasticity_equation_1}\n    \\begin{cases}\n        \\partial_t\\rho+{\\rm div} (\\rho u) =0,\\\\\n        \\partial_t(\\rho u)+{\\rm div} (\\rho u \\otimes u)  + \\nabla P(\\rho)={\\rm div}(\\rho F F^\\top),\\\\\n        \\partial_t F + u\\cdot\\nabla F=\\nabla u F,\\\\\n        (\\rho,u,F)|_{t=0} = (\\rho_0, u_0, F_0)\n    \\end{cases}\n\\end{eqnarray}\non $\\mathbb{R}^d \\times (0, +\\infty)$, {with} $d = 2, 3${, where} $\\rho=\\rho(x,t)$, $u=u(x,t)$, $F=F(x,t)$ and $P(\\rho)=R\\rho^{\\gamma}$ denotes the fluid density, the velocity field, the deformation gradient and the pressure, respectively. The $i$th component of ${\\rm div}(\\rho F F^\\top)$ on the right-hand side of the momentum equation is $\\partial_j(\\rho F_{ik} F_{jk})$ and $(\\nabla u F)_{ij}=\\partial_k u_i F_{kj}$. The physical coefficients $R, \\gamma$ satisfy $R>0$ and $\\gamma>1$. Without loss of generality, we set $R=1$ throughout this paper. For smooth solutions $(\\rho, u, F)$ of \\eqref{elasticity_equation_1}, the physical constitution relation\n\\begin{align}\\label{constitution_laws}\n\\begin{cases}\n\\rho\\  \\mathrm{det}F=1,\\\\\nF_{j}\\cdot\\nabla F_{k}=F_{k}\\cdot\\nabla F_{j}, \\quad 1\\leq   j, k\\leq d\n\\end{cases}\n\\end{align}\nholds true for all $t\\ge 0$, see \\cite{Hu_2011, Qian_2010}. As a consequence of  \\eqref{constitution_laws}, it is verified in \\cite{Hu_2020, Hu_2020_1} that\n\\begin{align*}\\label{constitution_laws_1}\n{\\rm div}(\\rho F^\\top)=0.\n\\end{align*}\n\nWe begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \\cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \\cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \\cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \\cite{Sideris_1996,Sideris_2000} and Agemi \\cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \\cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \\cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \\cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \\cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \\cite{Wang_2017}. {All these references studied the stability issue of \\eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\\hat{\\rho}, \\hat{u}, \\hat{F})=(1,0, I).$$ Indeed, the linearized system of \\eqref{elasticity_equation_1}\nat $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form\n\\begin{equation}\\label{wave}\n\\partial_t^2 \\omega-\\Delta \\omega=0,\n\\end{equation}\n where $\\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \\cite{Sideris_2005, Lei_2016} or the space-time resonance method \\cite{Wang_2017}.}\n\n{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \\eqref{elasticity_equation_1} around some non-constant equilibrium in \nthe communities of both physics and mathematics. As an example of non-constant equilibrium, in }\\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form\n$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\\in\\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \\cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by\n\\begin{equation}\\label{1-1}\n    \\bar{\\rho}=\\frac{1}{(1+t)^{3}}, \\bar{u} = \\frac{x}{1+t},  \\bar{F}= (1+t)I,\n  \\end{equation}\n with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \\eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \\eqref{1-1} is justified in \\cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \\cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \\eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer\n by establishing}",
    "context": "We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \\cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \\cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \\cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \\cite{Sideris_1996,Sideris_2000} and Agemi \\cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \\cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \\cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \\cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \\cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \\cite{Wang_2017}. {All these references studied the stability issue of \\eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\\hat{\\rho}, \\hat{u}, \\hat{F})=(1,0, I).$$ Indeed, the linearized system of \\eqref{elasticity_equation_1}\nat $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form\n\\begin{equation}\\label{wave}\n\\partial_t^2 \\omega-\\Delta \\omega=0,\n\\end{equation}\n where $\\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \\cite{Sideris_2005, Lei_2016} or the space-time resonance method \\cite{Wang_2017}.}\n\n{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \\eqref{elasticity_equation_1} around some non-constant equilibrium in \nthe communities of both physics and mathematics. As an example of non-constant equilibrium, in }\\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form\n$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\\in\\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \\cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by\n\\begin{equation}\\label{1-1}\n \\bar{\\rho}=\\frac{1}{(1+t)^{3}}, \\bar{u} = \\frac{x}{1+t}, \\bar{F}= (1+t)I,\n \\end{equation}\n with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \\eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \\eqref{1-1} is justified in \\cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \\cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \\eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer\n by establishing}",
    "full_context": "We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \\cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \\cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \\cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \\cite{Sideris_1996,Sideris_2000} and Agemi \\cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \\cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \\cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \\cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \\cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \\cite{Wang_2017}. {All these references studied the stability issue of \\eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\\hat{\\rho}, \\hat{u}, \\hat{F})=(1,0, I).$$ Indeed, the linearized system of \\eqref{elasticity_equation_1}\nat $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form\n\\begin{equation}\\label{wave}\n\\partial_t^2 \\omega-\\Delta \\omega=0,\n\\end{equation}\n where $\\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \\cite{Sideris_2005, Lei_2016} or the space-time resonance method \\cite{Wang_2017}.}\n\n{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \\eqref{elasticity_equation_1} around some non-constant equilibrium in \nthe communities of both physics and mathematics. As an example of non-constant equilibrium, in }\\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form\n$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\\in\\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \\cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by\n\\begin{equation}\\label{1-1}\n \\bar{\\rho}=\\frac{1}{(1+t)^{3}}, \\bar{u} = \\frac{x}{1+t}, \\bar{F}= (1+t)I,\n \\end{equation}\n with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \\eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \\eqref{1-1} is justified in \\cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \\cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \\eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer\n by establishing}\n\n{With the absence of the elastic stress $\\textrm{div}(\\rho FF^\\top)$ in the momentum equation, the system \\eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \\cite{Sideris_2017}. Hadzic and Jang in \\cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\\gamma\\in(1,\\frac53]$, and the full range of $\\gamma\\in (1,\\infty)$ is studied in \\cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \\cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\\rightarrow\\infty$. In this sense, Theorem \\ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \\cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.\n\n\\section{Global well-posedness in Eulerian coordinate}\\label{section_4}\nThis section is devoted to the completion of the proof of Theorem \\ref{global_existence}. Throughout this section, C denotes a generic positive constant that depends on\nsome known constants, but independent of $\\varepsilon$, $\\varepsilon_1$ and $t$.\n\\subsection{Regularity estimates}\\label{subsection_7}\nWith Theorem \\ref{global_existence_Lagrangian}, we are going to construct a global-in-time solution to the system\n\\eqref{elasticity_equation_1} in Eulerian coordinate. For any $(u_0, \\xi_0)$ given in Theorem \\ref{global_existence}, we define $V_0$ as follows\n \\begin{equation}\\label{V_0}\n V_0(y)=u_0(\\xi_0(y)).\n\\end{equation}\n{Combining \\eqref{V_0} with the assumptions of Theorem \\ref{global_existence}, we deduce\n\\begin{equation*}\\label{1}\n \\begin{split}\n \\|V_0(y)-y\\|_{H^3_y} + \\|\\xi_0(y) -y\\|_{H^4_y}\\le C\\varepsilon + C\\varepsilon^3 \\le C\\varepsilon.\n \\end{split}\n\\end{equation*}\nThen, letting $\\varepsilon = \\frac{\\varepsilon_1}{C}$ yields\n\\begin{equation}\\label{2}\n \\begin{split}\n \\|V_0(y)-y\\|_{H^3_y} + \\|\\xi_0(y) -y\\|_{H^4_y}\\le \\varepsilon_1.\n \\end{split}\n\\end{equation}\nSubstituting $V_0(y)$ and $\\xi_0(y)$, that satisfy \\eqref{V_0} and \\eqref{2}, into Theorem \\ref{global_existence_Lagrangian}, we obtain the existence of a unique global solution $(\\xi,V)$ to the system \\eqref{compressible_elasticity_equation_3}.\n\nFor the second term, we similarly obtain\n \\begin{equation}\\label{Euler_part_17}\n \\begin{split}\n &\\int_{\\mathbb{R}^d} \\Big|{\\rm adj}[\\nabla_y \\xi] - (1+t)^{d-1}I\\Big|^2\\,dy\\\\ &\\le C \\Big(\\|\\nabla_y \\xi\\|_{L^\\infty_y}^{2(d-2)} + (1+t)^{2(d-2)}\\Big)\\|\\nabla_y \\xi -(1+t)I\\|_{L^2_y}^2\\\\ & \\le C\\varepsilon^2 (1+t)^{2d-3}.\n \\end{split}\n \\end{equation}\nSubstituting \\eqref{Euler_part_16} and \\eqref{Euler_part_17} into \\eqref{Euler_part_14} and \\eqref{Euler_part_15} gives\n \\begin{equation}\\label{Euler_part_18}\n \\begin{split}\n \\|\\rho(x,t) -\\frac{1}{(1+t)^d}\\|_{L^2_x}^2 \\le C\\varepsilon^2 (1+t)^{-d-1}.\n \\end{split}\n \\end{equation}\n Next, we consider the estimate of $\\|\\nabla \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}$ and $\\|\\nabla^2 \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}$. Combining \\eqref{decay_estimates_Lagrangian}, \\eqref{Euler_part_4_1}, \\eqref{Euler_part_nn_2}-\\eqref{Euler_part_nn_3}, {$\\varepsilon = \\frac{\\varepsilon_1}{C}$} with $\\nabla_y B_{ij}= -B_{il}\\partial_l\\nabla_y\\xi_{k}B_{kj}$ and $\\nabla_y \\mathcal{J}= \\mathcal{J} B_{ij}\\partial_j\\nabla_y \\xi_{i}$ in \\eqref{flow_map_5_1}, we have\n \\begin{equation}\\label{Euler_part_10}\n \\begin{split}\n &\\|\\nabla_x \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}^2 \\\\ &\\le C(1+t)^d \\int_{\\mathbb{R}^d} |B\\nabla_y(\\mathcal{J}^{-1})|^2\\,dy\\\\\n &\\le C(1+t)^d\\int_{\\mathbb{R}^d} |B\\mathcal{J}^{-2}\\nabla_y\\mathcal{J}|^2\\,dy\\\\\n &\\le C(1+t)^d|B|^4 |\\mathcal{J}|^{-2} \\int_{\\mathbb{R}^d} |\\nabla^2_y \\xi|^2\\,dy\\\\\n &\\le C \\varepsilon^2 (1+t)^{-d-3}\n \\end{split}\n \\end{equation}\n and\n \\begin{equation}\\label{Euler_part_11}\n \\begin{split}\n &\\|\\nabla^2_x \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}^2\\\\ \n &\\le C(1+t)^d\\int_{\\mathbb{R}^d} \\big|B\\nabla_y[B\\nabla_y(\\mathcal{J}^{-1})]\\big|^2\\,dy \\\\\n &\\le C(1+t)^d\\Big[\\int_{\\mathbb{R}^d} |B\\nabla_y B\\mathcal{J}^{-2}\\nabla_y\\mathcal{J}|^2 \\,dy + \\int_{\\mathbb{R}^d} |BB \\mathcal{J}^{-3}\\nabla_y\\mathcal{J}\\nabla_y\\mathcal{J}|^2 \\,dy\\\\ \n & +\\int_{\\mathbb{R}^d} |BB\\mathcal{J}^{-2}\\nabla_y^2\\mathcal{J}|^2 \\,dy\\Big]\\\\\n &\\le C \\varepsilon^2 (1+t)^{-d-6} + C \\varepsilon^2 (1+t)^{-d-6} +C \\varepsilon^2 (1+t)^{-d-5} \\\\ \n &\\le C \\varepsilon^2 (1+t)^{-d-5}.\n \\end{split}\n \\end{equation}\n Finally, we estimate $\\|\\nabla^3_x \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}$. Combining \\eqref{decay_estimates_Lagrangian}, \\eqref{Euler_part_4_1}, \\eqref{Euler_part_nn_4}, $\\varepsilon = \\frac{\\varepsilon_1}{C}$ with $\\nabla_y B_{ij}= -B_{il}\\partial_l\\nabla_y\\xi_{k}B_{kj}$ and $\\nabla_y \\mathcal{J}= \\mathcal{J} B_{ij}\\partial_j\\nabla_y \\xi_{i}$ in \\eqref{flow_map_5_1}, we have\n \\begin{align}\\label{Euler_part_11_1}\n &\\notag\\|\\nabla^3_x \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}^2\\\\ \\notag &\\le C(1+t)^d\\int_{\\mathbb{R}^d} \\Big|B\\nabla_y\\big\\{B\\nabla_y\\big[B\\nabla_y(\\mathcal{J}^{-1})\\big]\\big\\}\\Big|^2\\,dy\\\\ &\n \\le C(1+t)^d\\Big[\\int_{\\mathbb{R}^d} |B\\nabla_y B\\nabla_y B\\mathcal{J}^{-2}\\nabla_y\\mathcal{J}|^2 \\,dy + \\int_{\\mathbb{R}^d} |BB \\nabla_y^2 B\\mathcal{J}^{-2}\\nabla_y\\mathcal{J}|^2 \\,dy\\notag\\\\ & +\\int_{\\mathbb{R}^d} |BB\\nabla_y B\\mathcal{J}^{-3}\\nabla_y\\mathcal{J}\\nabla_y\\mathcal{J}|^2 \\,dy +\\int_{\\mathbb{R}^d} |BB\\nabla_y B\\mathcal{J}^{-2}\\nabla_y^2\\mathcal{J}|^2 \\,dy\\\\ & +\\int_{\\mathbb{R}^d} |BBB\\mathcal{J}^{-4}\\nabla_y\\mathcal{J}\\nabla_y\\mathcal{J}\\nabla_y\\mathcal{J}|^2 \\,dy +\\int_{\\mathbb{R}^d} |BBB\\mathcal{J}^{-3}\\nabla_y\\mathcal{J}\\nabla_y^2\\mathcal{J}|^2 \\,dy\\notag\\\\ & \\notag +\\int_{\\mathbb{R}^d} |BBB\\mathcal{J}^{-2}\\nabla_y^3\\mathcal{J}|^2\\,dy\\Big]\\\\ &\\notag\n \\le\\, C \\varepsilon^2 (1+t)^{-d-9} + C \\varepsilon^2 (1+t)^{-d-8} + C \\varepsilon^2 (1+t)^{-d-9}+ C \\varepsilon^2 (1+t)^{-d-8} \\notag \\\\ &+ C \\varepsilon^2 (1+t)^{-d-9} + C \\varepsilon^2 (1+t)^{-d-8} + C \\varepsilon^2 (1+t)^{-d-7}\\notag\\\\ &\\le \\,C \\varepsilon^2 (1+t)^{-d-7}.\\notag\n\\end{align}\n From \\eqref{Euler_part_ww_1}-\\eqref{Euler_part_ww_4}, \\eqref{Euler_part_ee_1}-\\eqref{Euler_part_ee_4}, and \\eqref{Euler_part_18}-\\eqref{Euler_part_11_1}, we conclude that\n \\begin{eqnarray*}\\label{decay_1}\n \\begin{cases}\n \\|\\nabla_x^i \\big(\\rho(x,t) -\\frac{1}{(1+t)^d}\\big)\\|_{L^2_x}\\le C\\varepsilon (1+t)^{-\\frac{d}{2}-\\frac{1}{2}-i},\\\\[1mm]\n \\|\\nabla_x^i\\big(u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big)\\|_{L^2_x} \\le C\\varepsilon(1+t)^{\\frac{d}{2}+\\frac{1}{2}-i}\n \\end{cases}\n \\end{eqnarray*}\n for all $t \\ge 0$ and $i=0,1,2,3$. The proof of Theorem \\ref{global_existence} is complete.",
    "post_theorem_intro_text_len": 5211,
    "post_theorem_intro_text": "We prefer to work in the framework of $H^3$-spaces in order to keep the strategy as clear as possible. With our strategy in hand, it is possible to reduce the regularity of solutions to some Besov spaces, however the presentation would be more technical and complicated, and we leave it for the interested readers.\n\n{With the absence of the elastic stress $\\textrm{div}(\\rho FF^\\top)$ in the momentum equation, the system \\eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \\cite{Sideris_2017}. Hadzic and Jang in \\cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\\gamma\\in(1,\\frac53]$, and the full range of $\\gamma\\in (1,\\infty)$ is studied in \\cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \\cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\\rightarrow\\infty$. In this sense, Theorem \\ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \\cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.\n\nThe presence of the elastic stress $\\textrm{div}(\\rho FF^\\top)$ in the momentum equation makes the wave operator non-degenerate. Motivated by the study of compressible Euler equations in \\cite{Shkoller_2019}, we turn to employing a Lagrangian transformation and consider a different class of affine solutions that nevertheless share the same scaling property, namely $\\det A(t)\\sim (1+t)^3$. Compared with the Euler equations, the presence of elastic stress introduces an additional diffusion mechanism, see~\\eqref{compressible_elasticity_equation_5}. However, this feature requires a different analytical framework, and the framework developed in \\cite{Shkoller_2019} cannot be applied in a straightforward manner. Fortunately, the expanding factor on $t$ in the affine solution \\eqref{1-1} contributes a damping term for the velocity in linearized system of \\eqref{elasticity_equation_1} in Lagrangian coordinate, see \\eqref{compressible_elasticity_equation_5}. However, due to the expanding factor in $t$ of the affine solution,  the coefficients of the linearized system of  \\eqref{elasticity_equation_1} involves functions depending on $t$. This kind of\nhyperbolic systems with variable coefficients excludes the applicability of both the vector field method and the space-time resonance method, which are designed for quasilinear wave equations with constant variables. In order to overcome this difficulty, we turn to the $t-$dependent damping effect of the velocity. Indeed, with this damping at hand, different weights for unknowns will be constructed in the form of $\\mathcal{L}(t)$, and  the differential inequality \\eqref{est_L_1} is verified. More precisely, since we are concerned with the stability of a non-constant equilibrium, it is necessary to introduce different temporal weights in a series of uniform estimates. For instance, the terms in $\\mathcal{L}(t)$ in \\eqref{L} carry distinct weights {depending on $\\gamma$ and the dimension}. Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all {unknowns} become uniform. This uniformity plays a key role in establishing the global well-posedness of the compressible elastodynamics system in both two and three dimensions for all $\\gamma>1$ within a unified framework. As a consequence, the global existence in Lagrangian coordinate would be established in Theorem \\ref{global_existence_Lagrangian} with the help of }the local well-posedness  and the global-in-time estimate of the quantity $\\mathcal{L}(t)$. \n\nThe global well-posedness in Lagrangian coordinate will be transferred back into the Eulerian coordinate as stated in Theorem \\ref{global_existence} via the flow map. We remark that in Lagrangian coordinate, the impact of differentiation on time decay is largely concealed, whereas in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$. Moreover, the $L^2$ norms of $\\rho(x,t) -\\frac{1}{(1+t)^d}$, $u(x,t)-\\frac{x}{1+t}$ and  $F(x,t) -(1+t)I$ decay at faster rates compared with those obtained in \\cite{Hu_preprint}.\n\nThe rest of the paper is organized as follows. In Section \\ref{section_3},\nwe will reformulate the Cauchy problem \\eqref{elasticity_equation_1} in the Lagrangian coordinate and establish the global well-posedness of system \\eqref{compressible_elasticity_equation_5} through energy estimates. Building upon Theorem \\ref{global_existence_Lagrangian}, Section \\ref{section_4} is devoted to the construction of the global solution of system \\eqref{elasticity_equation_1} \nand the proof of Theorem \\ref{global_existence}.",
    "sketch": "To prove Theorem~\\ref{global_existence}, the authors:\n\\begin{itemize}\n\\item \\emph{Employ a Lagrangian transformation} (\"we turn to employing a Lagrangian transformation\") and \"consider a different class of affine solutions that nevertheless share the same scaling property, namely $\\det A(t)\\sim (1+t)^3$.\"\n\\item Use that the affine expansion yields a \\emph{$t$-dependent damping} in the Lagrangian linearized system (\"the expanding factor on $t$ in the affine solution \\eqref{1-1} contributes a damping term for the velocity in linearized system ... see \\eqref{compressible_elasticity_equation_5}\").\n\\item Since the coefficients depend on $t$ (\"the coefficients ... involves functions depending on $t$\"), they do not use vector-field/space-time resonance methods; instead they \"turn to the $t$-dependent damping effect of the velocity\" and construct \"different weights for unknowns ... in the form of $\\mathcal{L}(t)$\" and verify \"the differential inequality \\eqref{est_L_1}.\"\n\\item Introduce \"different temporal weights in a series of uniform estimates\"; \"the terms in $\\mathcal{L}(t)$ in \\eqref{L} carry distinct weights\". \"Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform,\" and this \"plays a key role in establishing the global well-posedness ... in both two and three dimensions\". This yields global existence in Lagrangian variables (\"the global existence in Lagrangian coordinate would be established in Theorem~\\ref{global_existence_Lagrangian} with the help of the local well-posedness and the global-in-time estimate of the quantity $\\mathcal{L}(t)$\").\n\\item \\emph{Transfer back to Eulerian coordinates} (\"transferred back into the Eulerian coordinate as stated in Theorem~\\ref{global_existence} via the flow map\"), and obtain the stated decay, noting that \"in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$.\"\n\\end{itemize}",
    "expanded_sketch": "To prove the main theorem, the authors:\n\\begin{itemize}\n\\item \\emph{Employ a Lagrangian transformation} (\"we turn to employing a Lagrangian transformation\") and \"consider a different class of affine solutions that nevertheless share the same scaling property, namely $\\det A(t)\\sim (1+t)^3$.\"\n\\item Use that the affine expansion yields a \\emph{$t$-dependent damping} in the Lagrangian linearized system (\"the expanding factor on $t$ in the affine solution\n\\begin{equation}\\label{1-1}\n    \\bar{\\rho}=\\frac{1}{(1+t)^{3}}, \\bar{u} = \\frac{x}{1+t},  \\bar{F}= (1+t)I,\n  \\end{equation}\ncontributes a damping term for the velocity in linearized system ... see\n\\begin{eqnarray} \\label{compressible_elasticity_equation_5}\n\t\\begin{cases}\n\t\t\\partial_t \\widetilde{\\eta}=v,\\\\\n\t\t\\partial_t v_{i} + \\frac{2}{1+t} v_{i} + \\frac{1}{(1+t)^{d(\\gamma -1)+2}} a_{il}\\partial_l q -\\Delta \\widetilde{\\eta}_{i} =0,\\\\\n\t\t(v,\\widetilde{\\eta})|_{t=0} = \\big(v_0, \\widetilde{\\eta}_0\\big)\n\t\\end{cases}\n  \\end{eqnarray}\n).\n\\item Since the coefficients depend on $t$ (\"the coefficients ... involves functions depending on $t$\"), they do not use vector-field/space-time resonance methods; instead they \"turn to the $t$-dependent damping effect of the velocity\" and construct \"different weights for unknowns ... in the form of $\\mathcal{L}(t)$\" and verify the differential inequality\n\\begin{equation}\\label{est_L_1}\n  \\begin{split}\n  \\udt \\mathcal{L}(t) + \\frac{1}{1+t} \\mathcal{L}(t)\\le \\frac{C}{(1+t)^{2}}\\big(\\mathcal{L}(t) + \\mathcal{L}^{\\frac{7}{2}}(t)\\big).\n  \\end{split}\n\\end{equation}\n\\item Introduce \"different temporal weights in a series of uniform estimates\"; \"the terms in $\\mathcal{L}(t)$ in\n\\begin{equation}\\label{L}\n\t\\begin{split}\n    \\mathcal{L}(t) := &\\|v(t)\\|_{H^3}^2 + \\frac{2}{(1+t)^2}\\|\\widetilde{\\eta}(t)\\|_{H^3}^2 + \\|\\nabla \\widetilde{\\eta}(t)\\|_{H^3}^2+ \\frac{2}{1+t}\\sum_{i = 0}^{3} \\langle\\nabla^{i} v(t),\\nabla^{i} \\widetilde{\\eta}(t)\\rangle \\\\ &+ \\frac{1}{\\gamma(1+t)^{d(\\gamma -1)+2}} \\int_{\\mathbb{R}^d}J^{\\gamma+1}(t) |\\nabla^3 q(t)|^2 \\,dy.\n\t\\end{split}\n\\end{equation}\ncarry distinct weights\". \"Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform,\" and this \"plays a key role in establishing the global well-posedness ... in both two and three dimensions\". This yields global existence in Lagrangian variables; more precisely, they establish the following theorem.\n\n\\begin{theorem}\\label{global_existence_Lagrangian}\n       For $d=2,3$,  there exists a sufficiently small  constant $\\varepsilon_1>0$ such that the Cauchy problem \\eqref{compressible_elasticity_equation_3} admits a global strong solution \n       $(V,\\xi)$ satisfying $(V(y,t)-y, \\xi(y,t) - (1+t)y)$ in $C\\big([0, \\infty); H^3_y(\\mathbb{R}^d)\\times H^4_y(\\mathbb{R}^d)\\big)$,  provided that\n          \\begin{equation*} \\label{initial_assumption_Lagrangian}\n              \\begin{split}\n              \\|V_0(y)-y\\|_{H^3_y} + \\|\\xi_0(y) -y\\|_{H^4_y}\\le \\varepsilon_1.\n              \\end{split}\n          \\end{equation*}\nMoreover, $(V,\\xi)$ satisfies the following estimates:\n\\begin{eqnarray}\\label{decay_estimates_Lagrangian}\n  \\begin{cases}\n    \\|V(y,t) -y\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{1}{2}},\\\\[1mm]\n    \\|\\xi(y,t) - (1+t)y\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{3}{2}},\\\\[1mm]\n     \\|\\nabla_y \\xi(y,t) - (1+t)I\\|_{H^3_y} \\le C\\varepsilon_1(1+t)^{\\frac{1}{2}}\n  \\end{cases}\n\\end{eqnarray}\nfor all $t>0$, where $C$ is a generic positive constant depending on some known constants but independent of $\\varepsilon_1$ and $t$.\n\\end{theorem}\n\n\\item \\emph{Transfer back to Eulerian coordinates} (\"transferred back into the Eulerian coordinate as stated in the main theorem via the flow map\"), and obtain the stated decay, noting that \"in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$.\"\n\\end{itemize}",
    "expanded_theorem": "\\label{global_existence} For $d=2, 3$,\nthere exists a diffeomorphism $\\xi_0: \\mathbb{R}^d \\to \\mathbb{R}^d$  {and a small constant $\\varepsilon>0$} such that if the initial data $(\\rho_0, u_0, F_0)$, with\n$(\\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\\in H^3_x(\\mathbb{R}^d)$, satisfies\n{\\begin{eqnarray*}\\label{decay_estiwmates_1}\n    \\begin{cases}\n      \\rho_0(x) \\mathrm{det}F_0(x)=1,\\,\\,\\,\\,F_0(x) = \\nabla_y \\xi_0(y),\\,\\,\\,\\,  y =\\xi_0^{-1}(x), \\ \\forall x\\in\\mathbb{R}^d, \\\\[2mm]\n      \\|\\rho_0(x) -1\\|_{H^3_x} + \\|u_0(x) -x\\|_{H^3_x} + \\|F_0(x) -I\\|_{H^3_x} + \\|\\xi_0^{-1}(x)-x\\|_{H^4_x}\\le \\varepsilon,\n    \\end{cases}\n\\end{eqnarray*}}then, in establishing the main theorem, the Cauchy problem\n\\begin{eqnarray} \\label{elasticity_equation_1}\n    \\begin{cases}\n        \\partial_t\\rho+{\\rm div} (\\rho u) =0,\\\\\n        \\partial_t(\\rho u)+{\\rm div} (\\rho u \\otimes u)  + \\nabla P(\\rho)={\\rm div}(\\rho F F^\\top),\\\\\n        \\partial_t F + u\\cdot\\nabla F=\\nabla u F,\\\\\n        (\\rho,u,F)|_{t=0} = (\\rho_0, u_0, F_0)\n    \\end{cases}\n\\end{eqnarray}\nadmits a unique global strong solution \n$(\\rho, u, F)$ such that $\\big(\\rho(x,t) -\\frac{1}{(1+t)^d}, u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big) \\in C\\big([0, \\infty); H^3_x(\\mathbb{R}^d)\\big)$. Moreover, $(\\rho, u, F)$ satisfies the following estimates:\n\\begin{eqnarray*}\\label{decay_estimates_1}\n  \\begin{cases}\n    \\|\\nabla_x^{i}(\\rho(x,t) -\\frac{1}{(1+t)^d})\\|_{L^2_x}\\le C\\varepsilon(1+t)^{-\\frac{d}{2}-\\frac{1}{2}-i},\\\\[1mm]\n     \\|\\nabla_x^{i}\\big(u(x,t)-\\frac{x}{1+t}, F(x,t) -(1+t)I\\big)\\|_{L^2_x} \\le C\\varepsilon(1+t)^{\\frac{d}{2}+\\frac{1}{2}-i}\n  \\end{cases}\n\\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\\varepsilon$ and $t$.}",
    "theorem_type": [
      "Implication",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Let $d\\in\\{2,3\\}$, and consider the compressible elastodynamics Cauchy problem on $\\mathbb{R}^d$ for unknowns density $\\rho(x,t)$, velocity $u(x,t)$, and deformation gradient $F(x,t)$:\n\\[\n\\begin{cases}\n\\partial_t\\rho+\\operatorname{div}(\\rho u)=0,\\\\\n\\partial_t(\\rho u)+\\operatorname{div}(\\rho u\\otimes u)+\\nabla P(\\rho)=\\operatorname{div}(\\rho F F^{\\top}),\\\\\n\\partial_tF+u\\cdot\\nabla F=\\nabla u\\,F,\\\\\n(\\rho,u,F)|_{t=0}=(\\rho_0,u_0,F_0).\n\\end{cases}\n\\]\nHere $I$ denotes the $d\\times d$ identity matrix, and $H_x^k(\\mathbb{R}^d)$ is the usual Sobolev space in the $x$-variable. Assume there exist a diffeomorphism $\\xi_0:\\mathbb{R}^d\\to\\mathbb{R}^d$ and a sufficiently small constant $\\varepsilon>0$ such that\n\\[\n(\\rho_0-1,\\,u_0-x,\\,F_0-I)\\in H_x^3(\\mathbb{R}^d),\n\\]\nand, for all $x\\in\\mathbb{R}^d$,\n\\[\n\\rho_0(x)\\det F_0(x)=1,\\qquad F_0(x)=\\nabla_y\\xi_0(y),\\qquad y=\\xi_0^{-1}(x),\n\\]\nwith the smallness condition\n\\[\n\\|\\rho_0-1\\|_{H_x^3}+\\|u_0-x\\|_{H_x^3}+\\|F_0-I\\|_{H_x^3}+\\|\\xi_0^{-1}-x\\|_{H_x^4}\\le \\varepsilon.\n\\]\nUnder these hypotheses, which conclusion about the solution holds?",
      "correct_choice": {
        "label": "A",
        "text": "The Cauchy problem admits a unique global strong solution $(\\rho,u,F)$ such that\n\\[\n\\Big(\\rho(x,t)-\\frac{1}{(1+t)^d},\\;u(x,t)-\\frac{x}{1+t},\\;F(x,t)-(1+t)I\\Big)\\in C\\big([0,\\infty);H_x^3(\\mathbb{R}^d)\\big).\n\\]\nMoreover, for every $t>0$ and every $i=0,1,2,3$,\n\\[\n\\|\\nabla_x^i\\big(\\rho(x,t)-\\tfrac{1}{(1+t)^d}\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12-i},\n\\]\nand\n\\[\n\\|\\nabla_x^i\\big(u(x,t)-\\tfrac{x}{1+t},\\;F(x,t)-(1+t)I\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{\\frac d2+\\frac12-i},\n\\]\nwhere $C$ is a positive constant independent of $\\varepsilon$ and $t$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The Cauchy problem admits a unique global strong solution $(\\rho,u,F)$ such that\n\\[\n\\Big(\\rho(x,t)-\\frac{1}{(1+t)^d},\\;u(x,t)-\\frac{x}{1+t},\\;F(x,t)-(1+t)I\\Big)\\in C\\big([0,\\infty);H_x^3(\\mathbb{R}^d)\\big).\n\\]\nMoreover, for every $t>0$ and every $i=0,1,2,3$,\n\\[\n\\|\\nabla_x^i\\big(\\rho(x,t)-\\tfrac{1}{(1+t)^d}\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12-i},\n\\]\nand\n\\[\n\\|\\nabla_x^i\\big(u(x,t)-\\tfrac{x}{1+t},\\;F(x,t)-(1+t)I\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12-i},\n\\]\nwhere $C$ is a positive constant independent of $\\varepsilon$ and $t$."
        },
        {
          "label": "C",
          "text": "The Cauchy problem admits a unique global strong solution $(\\rho,u,F)$ such that\n\\[\n\\Big(\\rho(x,t)-\\frac{1}{(1+t)^d},\\;u(x,t)-\\frac{x}{1+t},\\;F(x,t)-(1+t)I\\Big)\\in C\\big([0,\\infty);H_x^3(\\mathbb{R}^d)\\big).\n\\]\nMoreover, there exists a positive constant $C$ independent of $\\varepsilon$ and $t$ such that, for every $t>0$,\n\\[\n\\|\\rho(x,t)-\\tfrac{1}{(1+t)^d}\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12},\n\\]\nand\n\\[\n\\|\\big(u(x,t)-\\tfrac{x}{1+t},\\;F(x,t)-(1+t)I\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{\\frac d2+\\frac12}.\n\\]"
        },
        {
          "label": "D",
          "text": "The Cauchy problem admits a unique global strong solution $(\\rho,u,F)$ such that\n\\[\n\\Big(\\rho(x,t)-\\frac{1}{(1+t)^d},\\;u(x,t)-\\frac{x}{1+t},\\;F(x,t)-(1+t)I\\Big)\\in C\\big([0,\\infty);H_x^3(\\mathbb{R}^d)\\big).\n\\]\nMoreover, there exists a positive constant $C$ independent of $\\varepsilon$, $t$, and the initial diffeomorphism $\\xi_0$ such that, for every $t>0$ and every $i=0,1,2,3$,\n\\[\n\\|\\nabla_x^i\\big(\\rho(x,t)-\\tfrac{1}{(1+t)^d}\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12-i},\n\\]\nand\n\\[\n\\|\\nabla_x^i\\big(u(x,t)-\\tfrac{x}{1+t},\\;F(x,t)-(1+t)I\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{\\frac d2+\\frac12-i},\n\\]\nfor all sufficiently small initial data satisfying the stated assumptions."
        },
        {
          "label": "E",
          "text": "The Cauchy problem admits a unique global strong solution $(\\rho,u,F)$ such that\n\\[\n\\Big(\\rho(x,t)-\\frac{1}{(1+t)^d},\\;u(x,t)-\\frac{x}{1+t},\\;F(x,t)-(1+t)I\\Big)\\in C\\big([0,\\infty);H_x^4(\\mathbb{R}^d)\\big).\n\\]\nMoreover, for every $t>0$ and every $i=0,1,2,3,4$,\n\\[\n\\|\\nabla_x^i\\big(\\rho(x,t)-\\tfrac{1}{(1+t)^d}\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{-\\frac d2-\\frac12-i},\n\\]\nand\n\\[\n\\|\\nabla_x^i\\big(u(x,t)-\\tfrac{x}{1+t},\\;F(x,t)-(1+t)I\\big)\\|_{L_x^2}\\le C\\varepsilon(1+t)^{\\frac d2+\\frac12-i},\n\\]\nwhere $C$ is a positive constant independent of $\\varepsilon$ and $t$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "Eulerian derivative-transfer gives opposite behavior for velocity/deformation rates",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "higher-derivative decay family restricted to the zeroth-order case only",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "uniformity_effectivity",
          "tampered_component": "dependence of the generic constant on fixed auxiliary data such as the initial diffeomorphism",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "available regularity level is H^3/H^4 on flow map, not H^4 Eulerian solution with fourth-order decay estimates",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly disclose the correct option. It presents the hypotheses and asks for the resulting conclusion, without giving away the exact decay rates or regularity statement."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem states the full hypotheses of a specific result and asks for the corresponding conclusion. The correct choice is basically the theorem's conclusion verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning/verification pressure because the distractors alter decay exponents, regularity, and uniformity of constants in subtle ways. However, the task is still primarily recognition of the exact theorem statement rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: incorrect decay behavior, a weaker true statement, overly strong regularity, and an unjustified uniformity claim. They are distinct and nontrivial."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recall MCQ with strong distractors, but it is largely a restatement of a known result rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.16535v1",
    "paper_link": "http://arxiv.org/abs/2512.16535v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.",
      "start_pos": 8495,
      "end_pos": 8882,
      "label": "thm:connectedSum"
    },
    "ref_dict": {
      "example": "\\begin{example}\\label{example}\n\tLet $M^n$ be a closed aspherical spin manifold with $n\\ge 5$, e.g., the $n$-torus. Suppose that $\\pi_1(M)$ is represented by finitely many disjoint embedded circles $S^1$ in $M$, whose tubular neighborhoods are diffeomorphic to $S^1 \\times B^{n-1}$. Remove these tubular neighborhoods and attach $B^2 \\times S^{n-2}$ to obtain a closed manifold $M_1$. Then $M_1$ is simply connected.  \n\n\tFinally, set $Z \\coloneqq M_1 \\# M_1$, which is a spin surgery of $M$. Note that $Z$ is simply connected, and the Dirac operator on $Z$ has vanishing $KO$-index as $Z$ is a double. Hence, $Z$ admits a metric with positive scalar curvature \\cite{Stolz}.\n\\end{example}",
      "conj:Novikov": "\\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\\label{conj:Novikov}\n\tLet $M$ be a closed aspherical manifold and $\\Gamma=\\pi_1(M)$. Then $\\Gamma$ satisfies the rational strong Novikov conjecture if\n\t\\[\n\t\\mu\\colon K_*(M)\\otimes\\Q \\to K_*(C^*_r(\\Gamma))\\otimes\\Q\n\t\\]\n\tis injective.\n\\end{conjecture}",
      "thm:main": "\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}",
      "thm:connectedSum": "\\begin{theorem}\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1742,
    "pre_theorem_intro_text": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.",
    "context": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.",
    "full_context": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.\n\n\\begin{abstract}\n Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every complete metric with non-negative scalar curvature is Ricci-flat. In particular, on the connected sum of $M$ with a spin manifold, any complete metric with non-negative scalar curvature is Ricci-flat.\n\\end{abstract}\n\nThe study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nWe remark that in Theorem \\ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\\pi_1(M)$ \\cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\\pi_1(M)$ is rationally injective; see Conjecture \\ref{conj:Novikov} for a precise formulation. The reader may consult \\cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.\n\n\\begin{theorem}\\label{thm:main}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n Let $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\n\n\\begin{corollary}\\label{coro:remove}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map},$$\n and the rational strong Novikov conjecture holds for $\\pi_1(M)$. Then $M\\backslash\\Omega$ does not carry any complete metric with positive scalar curvature. \n\\end{corollary}\n\n\\section{Non-existence of uniformly positive scalar curvature}\\label{sec:UPSC}\nIn this section, we prove a simple version of Theorem \\ref{thm:main} to illustrate the key ideas. This version follows from classical index-theoretic tools, while the proof of the full version requires additional new ingredients and will be presented in the next section.\n\\begin{proposition}\\label{prop:noPSC}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n Let $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with uniform positive scalar curvature, i.e., scalar curvature bounded below by some positive constant.\n\\end{proposition}\n\n@article{Zeidler:2019aa,\n abstract = {Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \\times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\\sqrtσ$, where $C_n = \\sqrt{(n-1)/{n}} \\cdot C$ with $C < 8(1+\\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \\times \\mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the \"$\\mathcal{KO}$-width\" of a closed manifold and deduce that infinite $\\mathcal{KO}$-width is an obstruction to positive scalar curvature.},\n author = {Rudolf Zeidler},\n date-added = {2021-11-10 09:44:42 -0600},\n date-modified = {2021-11-10 09:44:42 -0600},\n eprint = {1905.08520},\n journal = {Journal of Differential Geometry},\n title = {Band width estimates via the {D}irac operator},\n url = {https://arxiv.org/pdf/1905.08520.pdf},\n year = {2020},\n bdsk-url-1 = {https://arxiv.org/pdf/1905.08520.pdf}}\n\n\\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\\label{conj:Novikov}\n\tLet $M$ be a closed aspherical manifold and $\\Gamma=\\pi_1(M)$. Then $\\Gamma$ satisfies the rational strong Novikov conjecture if\n\t\\[\n\t\\mu\\colon K_*(M)\\otimes\\Q \\to K_*(C^*_r(\\Gamma))\\otimes\\Q\n\t\\]\n\tis injective.\n\\end{conjecture}\n\n\\begin{theorem}\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}",
    "post_theorem_intro_text_len": 3454,
    "post_theorem_intro_text": "We remark that in Theorem \\ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\\pi_1(M)$ \\cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\\pi_1(M)$ is rationally injective; see Conjecture \\ref{conj:Novikov} for a precise formulation. The reader may consult \\cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.\n\nWe further generalize Theorem \\ref{thm:connectedSum} to spin surgeries on aspherical manifolds. \n\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\n\nTheorem \\ref{thm:connectedSum} is a special case of Theorem \\ref{thm:main} by choosing $\\Omega$ to be a small ball in $M$. The assumption that $\\pi_1(\\Omega)$ maps trivially into $\\pi_1(M)$ is essential; in Example \\ref{example} we construct a spin surgery on a closed aspherical manifold that admits a metric of positive scalar curvature when this condition fails.\n\nOur proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \\cite[Theorem~A]{XieYu14}. Inspired by the approaches in \\cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \\cite{WangZhu}.\n\nAs an immediate application, by taking $N=\\partial\\Omega\\times[0,1)$ in Theorem \\ref{thm:main}, we obtain the following corollary.\n\n\\begin{corollary}\\label{coro:remove}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map},$$\n\tand the rational strong Novikov conjecture holds for $\\pi_1(M)$. Then $M\\backslash\\Omega$ does not carry any complete metric with positive scalar curvature. \n\\end{corollary}\n\nThe paper is organized as follows. In Section \\ref{sec:preliminary}, we review the higher indices and the strong Novikov conjecture. In Section \\ref{sec:UPSC}, we prove a simplified version of Theorem \\ref{thm:main}, establishing the non-existence of uniformly positive scalar curvature metrics on spin surgeries. The full proof of Theorem \\ref{thm:main} is given in Section \\ref{sec:nnsc}.\n\n\\medskip\n\\noindent\\textbf{Acknowledgments.}  \nThe author would like to thank Zhizhang Xie for helpful discussions.",
    "sketch": "Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \\cite[Theorem~A]{XieYu14}. Inspired by the approaches in \\cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \\cite{WangZhu}. (Theorem~\\ref{thm:connectedSum} is treated as a special case of Theorem~\\ref{thm:main} by choosing $\\Omega$ to be a small ball in $M$.)",
    "expanded_sketch": "Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu (Theorem~A in Xie--Yu, \\emph{[title not provided in ref\\_dict]} (2014)). Inspired by the approaches in Su--Zhang, \\emph{[title not provided in ref\\_dict]}, and Wang--Xie--Linf, \\emph{[title not provided in ref\\_dict]}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in Wang--Zhu, \\emph{[title not provided in ref\\_dict]}. (In establishing the main theorem, we treat it as a special case of\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\nby choosing $\\Omega$ to be a small ball in $M$.)",
    "expanded_theorem": "\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.",
    "theorem_type": [
      "Implication",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let $M^n$ be a closed aspherical manifold, let $N$ be a spin manifold of the same dimension $n$, and write $M\\#N$ for their connected sum. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$, meaning that for $\\Gamma=\\pi_1(M)$ the assembly map\n\\[\n\\mu\\colon K_*(M)\\otimes\\mathbb{Q}\\to K_*(C_r^*(\\Gamma))\\otimes\\mathbb{Q}\n\\]\nis injective. Which of the following conclusions about complete Riemannian metrics on $M\\#N$ is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\\#N$ has non-negative scalar curvature everywhere, then that metric is Ricci-flat."
      },
      "choices": [
        {
          "label": "B",
          "text": "The manifold $M\\#N$ admits no complete Riemannian metric with non-negative scalar curvature. In particular, every complete metric on $M\\#N$ must have scalar curvature negative somewhere."
        },
        {
          "label": "C",
          "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature."
        },
        {
          "label": "D",
          "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\\#N$ has non-negative scalar curvature everywhere, then that metric is flat."
        },
        {
          "label": "E",
          "text": "If the manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature, then the rational strong Novikov conjecture must hold for $\\pi_1(M)$. Moreover, any complete Riemannian metric on $M\\#N$ with non-negative scalar curvature is Ricci-flat."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "nonnegative_vs_positive_case_split",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_Ricci_flatness_conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "Ricci_flatness_not_flatness",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "direction_of_Novikov_hypothesis",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal the correct option or uniquely cue its exact wording. There is no direct answer leakage beyond setting up the relevant theorem."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall/application item: the stem gives the assumptions and asks which conclusion holds. However, it is not a pure tautology because the options include stronger, weaker, and subtly altered conclusions, so the student must distinguish the exact statement."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some precision in reasoning: one distractor is a weaker true statement, others are overstatements ('non-negative scalar curvature' obstruction, 'flat' instead of 'Ricci-flat') or logical misreadings. Still, the task is mainly identifying the exact theorem conclusion rather than generating a new argument."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible. They target common failure modes: confusing positive vs non-negative scalar curvature, mistaking Ricci-flat for flat, and selecting a weaker true statement instead of the strongest correct one."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with good distractors and little answer leakage, but it is still primarily a theorem-recall item rather than a deeply generative reasoning question."
    }
  },
  {
    "id": "2512.16655v2",
    "paper_link": "http://arxiv.org/abs/2512.16655v2",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.",
      "start_pos": 64453,
      "end_pos": 64950,
      "label": "main theorem"
    },
    "ref_dict": {
      "pro ortho": "\\begin{prop}\\label{pro ortho}\n\t\tIf  Eq. \\eqref{eq-k} is solvable, then $f$ satisfies\n\n\t\t\\begin{eqnarray}\\label{ortho-condition}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\H^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n.\n\\end{eqnarray}\t\t\n\t\\end{prop}",
      "main theorem": "\\begin{thm}\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\ov{{\\RR}^{n+1}_{+}}$, such that $\\widehat{\\S}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\S$ is unique up to a horizontal translation in $\\ov{{\\RR}^{n+1}_{+}}$. \n\\end{thm}",
      "measure": "\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\H}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}",
      "c0 est": "\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of  Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}.  \nThen there exists a positive constant $C$ depending  on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n\t0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}",
      "pro-2.6": "\\begin{prop}\\label{pro-2.6}\n The capillary  Christoffel-Minkowski problem \t\tis equivalent to finding a strictly convex solution to  Eq.  \\eqref{eq-k}.\\end{prop}",
      "def1.1": "\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\n^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}",
      "eq-k": "\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}",
      "ortho-condition1": "\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray}"
    },
    "pre_theorem_intro_text_len": 12832,
    "pre_theorem_intro_text": "In convex geometry, the prescribed $k$-th area measure problem is a well-known problem. It asks whether, for a given positive function $f$ on $\\mathbb{S}^{n}$, one can find a closed, strongly convex hypersurface $\\Sigma$ with $k$-th symmetric function of the principal curvature radii equal to $f$ as a function on its normals? When $k=n$, it corresponds to the famous Minkowski problem, which has been fully solved through the works of Minkowski \\cite{Min}, Alexandrov \\cite{Alex}, Lewy \\cite{Lewy}, Nirenberg \\cite{Nire}, Pogorelov \\cite{Pog52, Pog}, Cheng-Yau \\cite{CY76} and many others. Analytically, the Minkowski problem is equivalent to solving a Monge-Amp\\`ere equation on $\\mathbb{S}^{n}$. The study of the Minkowski problem has played a significant role in advancing research on fully nonlinear equations, especially on the Monge-Amp\\`ere equation. We can also see a closely related work by Xia \\cite{Xia} for an anisotropic Minkowski problem for closed hypersurfaces.\nWhen $k=1$, it is known as the Christoffel problem, which is equivalent to finding convex solutions to the Laplace equation on $\\mathbb{S}^{n}$.  Firey \\cite{Firey, Firey1} and Berg \\cite{Berg}  found necessary and sufficient conditions using the Green function of the Laplacian on $\\mathbb{S}^{n}$. More recently, Li-Wan-Wang \\cite{LWW} expressed the Christoffel problem as a Laplace equation on the entire space $\\mathbb{R}^{n+1}$ and provided a new and simpler necessary and sufficient condition for the Christoffel problem.  Further details on Christoffel's problem can be found in \\cite{Chri,  Hilbert, Hur, Suss}. For the intermediate cases, i.e., $2\\leq k\\leq n-1$, it is referred to as the Christoffel-Minkowski problem. Guan-Ma \\cite{GM}  solved this problem under a continuous homotopy condition, which was removed later by  Sheng-Wang-Trudinger \\cite{STW}. In addition to the elliptic method, Chou-Wang \\cite{CW00} and Bryan-Ivaki-Scheuer \\cite{BIS2023} provided a flow method to solve the above problems.\nFor a more comprehensive description of the prescribed area measure problem in convex geometry, see \\cite{Guan-note} and \\cite{Sch}. \n\n\\subsection{Motivation and set up}\nIn recent years, there has been growing interest in the study of capillary hypersurfaces, including free boundary hypersurfaces, along with their geometry and topology, see, for instance, \\cite{FL14, FS16, JWXZ, MWW-GCF, SWX, WW20, WWX22, WX2019, WeX21} and references therein. \nCapillary hypersurfaces originated from physics through the work of Thomas Young. Later,\nYoung, Liouville, and Gauss formulated it as a variational problem using the so-called wetting energy.  For capillary hypersurfaces, we refer to the elegant book by Finn \\cite{Finn}, and also a book by Maggi  \\cite{Maggi}. In this paper, with a slight abuse of terminology,\na smooth compact embedded hypersurface in $\\overline{{\\mathbb{R}}_+^{n+1}}$  whose boundary lies on $\\partial\\overline{{\\mathbb{R}}_+^{n+1}}$ is called {\\it capillary hypersurface} if\nit intersects $\\partial\\overline{{{\\mathbb{R}}}_+^{n+1}}$ at a constant contact angle $\\theta\\in (0,\\pi)$. \n   Let $\\nu$ be the unit outward normal of $\\Sigma$. The contact angle $\\theta $ is defined by $$ \\cos (\\pi -\\theta)=\\langle \\nu, e \\rangle,\\quad \\rm{along~\\partial\\Sigma},$$\n\twhere $e\\coloneqq -E_{n+1}$, $E_{n+1}$ is the $(n+1)$-th unit coordinate vector in $\\overline{\\mathbb{R}^{n+1}_+}$ and thus $e$ is the unit outward normal of $\\partial \\overline{\\mathbb{R}^{n+1}_+}$.\nThis means that a capillary hypersurface in this paper is not necessarily a hypersurface of constant mean curvature.\nIf $\\Sigma$ is strictly convex, then it is clear that the image of the Gauss map, namely,  $\\nu$, lies on the spherical cap \n\\begin{eqnarray*}\n\t\t\\mathbb{S}^n_\\theta \\coloneqq  \\left\\{ x\\in \\mathbb{S}^n \\,|\\,  \\langle x, E_{n+1} \\rangle \\geq \\cos \\theta \\right\\}.\n\\end{eqnarray*}\nInstead of using the usual Gauss map $\\nu$, it is more convenient to use the following map\n$$\\tilde \\nu\\coloneqq  T \\circ \\nu: \\Sigma \\to \\C_\\theta,$$\nwhere $\\C_\\theta$ is a spherical cap, defined by\n\\begin{eqnarray*}\n\t\t\\C_{\\theta }  \\coloneqq  \\left\\{\\xi\\in \\overline{\\mathbb{R}^{n+1}_+} \\mid |\\xi-\\cos\\theta e|= 1 \\right\\},\n\\end{eqnarray*}\n\tand $T:\\mathbb{S}^n_\\theta\\to\\C_\\theta$ is defined by $ T(y)\\coloneqq y+\\cos\\theta e$,  which is a translation in the vertical direction.\n We refer to $\\tilde \\nu$  as the {\\it capillary Gauss map} of $\\Sigma$.  It turns out that $\\tilde \\nu: \\Sigma\\to \\C_\\theta$ is a diffeomorphism. Consequently,  one can reparametrize $\\Sigma$ using its inverse on $\\C_\\theta$, see, e.g., \\cite[Section 2]{MWWX}. \n\nIn \\cite{MWW-AIM}, the authors formulated a capillary Minkowski problem, which concerns the existence of a strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_{+}}$ with a prescribed Gauss-Kronecker curvature on $\\C_{\\theta}$. We have established the following result in \\cite[Theorem 1.1]{MWW-AIM}.\n\n\\\n\n\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function  satisfying  \\eqref{ortho-condition1} below.  Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n\t\t{f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\n  \\\n\nThe capillary Minkowski problem can also be viewed as a prescribed capillary area measure problem, finding a strictly convex capillary hypersurface such that the induced capillary area measure equals a prescribed Borel measure on $\\C_{\\theta}$. See \\cite{MWW-AIM}. The area measures are an important subject in the study of convex bodies, and they are the local versions of quermassintegrals in the Brunn-Minkowski theory.  For more details about the classical Christoffel-Minkowski problem, see \\cite[Chapter~8]{Sch}.\n\nThe primary objective of this article is to extend our work presented in \\cite{MWW-AIM} and study the capillary Christoffel-Minkowski problem or the prescribed  $k$-th capillary area measure problem for the convex body with a capillary boundary in $\\overline{\\mathbb{R}^{n+1}_{+}}$.  \n Firstly, we introduce a novel concept of local parallel sets for the capillary convex bodies. We say $\\widehat{\\Sigma}$ is a \\textit{capillary convex body} (a compact convex set with non-empty interior) in $\\mathbb{R}^{n+1}$ if it is a bounded closed region in $\\overline{\\mathbb{R}^{n+1}_+}$ enclosed by a strictly convex capillary hypersurface $\\Sigma$ and  $\\partial \\overline{\\mathbb{R}^{n+1}_{+}}$. The class of capillary convex bodies in $\\overline{{\\mathbb{R}}^{n+1}_+}$ is denoted by $\\K_{\\theta}$.   Given a Borel set $\\beta\\subset \\C_{\\theta}$ and a capillary convex body $\\widehat{\\Sigma}\\subset \\K_{\\theta}$, we consider the following set :\n\\begin{eqnarray*}\n    B_{s}(\\widehat{\\Sigma}, \\beta)\\coloneqq  \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+t\\tilde{\\nu}(X),~0<t<s,~\\text{for}~X\\in \\Sigma ~ \\text{and} ~\\tilde{\\nu}(X)\\in \\beta \\right\\}.\n\\end{eqnarray*}\nWhen $\\beta=\\C_{\\theta}$, the set\n\\begin{eqnarray*}\n    A_{s}(\\widehat{\\Sigma}, \\C_{\\theta}) \\coloneqq \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+s\\tilde{\\nu}(X), ~X\\in \\Sigma~\\text{and}~s>0 \\right\\}\n\\end{eqnarray*}\nis also a strictly convex capillary hypersurface in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$ (see, for instance, \\cite[Remark 2.17]{MWWX} or \\cite[Proposition~2.3]{JWXZ}).\n\nDirect calculation yields the following local Steiner-type formula (see also \\cite[Remark~2.17]{MWWX} for a global version),\n\\begin{eqnarray*}\n    \\text{Vol}(B_{s}(\\widehat{\\Sigma}, \\beta))&=&\\sum\\limits_{k=0}^{n}\\frac{s^{n+1-k}}{n+1-k}\\int_{\\Sigma \\cap \\tilde{\\nu}^{-1}(\\beta)}(1+\\cos\\theta \\<\\nu, e\\>)\\sigma_{n-k}(\\kappa)d\\mu\\notag\\\\\n    &=&\\frac{1}{n+1}\\sum\\limits_{k=0}^{n}s^{n+1-k}\\binom{n+1}{k}S_{k}^c(\\widehat{\\Sigma}, \\beta),\n\\end{eqnarray*}\nwhere $S_k^c(\\widehat{\\Sigma},\\beta)$ is given by\n\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\mathcal{H}}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}\nHere, $\\kappa$ and $r$ are the set of the principal curvatures and principal curvature radii of $\\Sigma$ respectively, $\\sigma_{k}$ is the $k$-th elementary symmetric polynomial function, $d \\mathcal{H}^n$ and $d\\mu$ are the $n$-dimensional Hausdorff measure on $\\C_\\theta$\n and  $\\Sigma$ respectively. The superscript $c$ indicates the presence of the \\textit{capillary} effect in our notation. Moreover, $S_k^c (\\widehat{\\Sigma}, \\C_{\\theta})$ $(0\\leq k\\leq n)$ coincide with the quermassintegrals of convex capillary hypersurface introduced in  \\cite[Eq. (1.8)]{WWX22}, see also \\cite[Lemma~2.14]{MWWX}. \n\n For each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce  a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given  by \\begin{eqnarray}\\notag\n     dS_k^c \\coloneqq  \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray}  \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the  $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\n \\\n\n\\noindent\t\\textbf{Capillary Christoffel-Minkowski problem:} \\textit{Given a positive, $C^{2}$ function $f$ on $\\C_{\\theta}$, when does  there exist a capillary convex body $\\widehat{\\Sigma}$ in $\\overline{\\mathbb{R}^{n+1}_+}$ such that\n\\begin{eqnarray*}\n    dS_k^c=\\ell fd\\mathcal{H}^{n}~?\n\\end{eqnarray*}}\n\n\\\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case  $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding  convex solutions to the following Hessian type equation with Robin boundary value condition,  \\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\partial \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\nwhere   $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively  and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$  satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere  $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result,  we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n  For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.",
    "context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.",
    "full_context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.\n\nFor each capillary convex body $\\widehat{\\S}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\n^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\S$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\S}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\\n\nIn this section, we establish the a priori estimates for the strictly convex solution of Eq. \\eqref{eq-k}. \nFor any solution $h$ to Eq. \\eqref{eq-k}, it is easy to verify that \n\\begin{eqnarray*}\n h(\\xi)+\\<\\xi,E_\\alpha\\> ,\\quad \\text{ for all } 1\\leq \\a\\leq n,\n\\end{eqnarray*} is still a solution. Up to a horizontal translation, we assume $h$ satisfies the following orthogonal condition:\n\\begin{eqnarray}\\label{orth cond}\n \\int_{\\C_\\theta} h(\\xi) \\<\\xi,E_\\alpha\\> d\\H^{n}=0, \\quad \\text{ for all } 1\\leq \\a\\leq n.\n\\end{eqnarray}\nUnder the condition \\eqref{orth cond}, we are able to establish the following a priori estimates.\n\\begin{thm}\\label{priori estimate}\nLet $\\theta\\in (0, \\frac{\\pi}{2})$. Suppose that $h$ is a strictly convex solution of Eq. \\eqref{eq-k} and satisfies the condition \\eqref{orth cond}. For any integer $m\\geq 2$ and $\\gamma\\in(0,1)$, there exists a constant $C>0$ depending on $n, k,$ $\\min\\limits_{\\C_{\\theta}} f$ and $\\|f\\|_{C^{m}(\\C_\\theta)}$, such that\n \\begin{eqnarray}\\label{sch est}\n \\|h\\|_{C^{m+1,\\gamma}(\\C_\\theta)}\\leq C.\n \\end{eqnarray}\n\\end{thm}\n\n\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}. \nThen there exists a positive constant $C$ depending on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n 0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}\n\\begin{proof} \nFirst of all, due to \\eqref{orth cond}, by \\cite[Lemma 3.1]{MWW-AIM}, we have $h>0$ and the origin lies in the interior of $\\widehat{\\partial\\Sigma}$. \nLet $R_{0}>0$ be the smallest number of $R$ such that \n \\begin{eqnarray}\\label{enclosed}\n \\widehat \\S\\subset \\widehat{R\\C_\\theta},\n \\end{eqnarray}where $R\\C_\\theta\\coloneqq \\{\\xi \\in \\ov{\\RR^{n+1}_+}: |\\xi-\\cos\\theta R e|=R\\}$. It is clear that there exists $X_0\\in \\Sigma \\cap R_0\\C_{\\theta } $. Denote $\\widehat{X}_0\\coloneqq \\frac{X_0 }{R_0}\\in \\C_\\theta$.\nFor any $\\xi \\in \\C_\\theta$, we have \n \\begin{eqnarray}\\label{lower bdd of h}\n h(\\xi)&=&\\max_{Y\\in \\S}\\<\\xi-\\cos\\theta e, Y\\>\\notag \\\\\n &\\geq& \\max \\left\\{0, \\<\\xi-\\cos\\theta e,X_0\\> \\right\\}\\notag \\\\\n &=& \\max \\left\\{0, R_0\\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right\\}.\n ~~~\\end{eqnarray} \nMultiplying $f$ on both sides of \\eqref{lower bdd of h} and integrating over $ \\C_\\theta$, we see\n\\begin{eqnarray}\\label{R1}\n R_0\\leq \\left(\\int_{\\C_\\theta}h(\\xi)\\cdot f(\\xi)d \\H^{n}\\right) \\left(\\int_{\\C_\\theta}\\max \\left\\{0, \\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right \\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\n\\end{eqnarray}\nNote that \n\\begin{eqnarray}\\label{R2}\n \\int_{\\C_{\\theta}}h(\\xi)\\cdot f(\\xi) d \\H^{n}=\n\\int_{\\tilde{\\nu}(\\Sigma)}h(\\xi)\\cdot f(\\xi)d\\H^{n}=\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu.\n\\end{eqnarray}\nTogether with the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$ and the Minkowski formulas for capillary hypersurfaces \\cite[Proposition~2.6]{WWX22}, we obtain\n\\begin{eqnarray}\n C_{n, k}\\left(\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu\\right)^{\\frac{1}{k+1}}&\\leq& \\left(\\int_{\\Sigma}\\<X, \\nu\\>\\sigma_{n-k+1}(\\kappa)d\\mu\\right)^{\\frac{1}{k}}\\notag \\\\\n &=&\\left(\\frac{k}{n-k+1}\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{1}{k}}.\\label{R3}\n\\end{eqnarray}\nTogether with \\eqref{R1}, \\eqref{R2} and \\eqref{R3}, we conclude that \n\\begin{eqnarray}\n R_{0}\n &\\leq& C_{n,k}\\left(\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{k+1}{k}}\\cdot\\left(\\inf\\limits_{\\eta\\in \\C_{\\theta}}\\int_{\\C_\\theta}\\max\\{0, \\<\\xi-\\cos\\theta e, \\eta\\>\\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\\quad ~~~\\label{upper bound of R0}\n\\end{eqnarray}\nUsing \\eqref{enclosed} and \\eqref{upper bound of R0}, we see \n\\begin{eqnarray*}\n h(\\xi) =\\<\\xi-\\cos\\theta e , \\tilde{\\nu}^{-1}(\\xi)\\>\\leq |\\tilde{\\nu}^{-1}(\\xi)| <2R_0\\leq C(n, \\min_{\\C_\\theta} f,\\max_{\\C_\\theta} f).\n\\end{eqnarray*}\n We complete the proof of Lemma \\ref{c0 est}.\n\\end{proof}\n\\subsection{\\texorpdfstring{$C^1$ }{}estimate}\n In this subsection, we get the gradient estimate for the strictly convex solution $h$ of Eq. \\eqref{eq-k}. This estimate is a direct consequence of the convexity and the $C^{0}$ estimate.\n\\begin{lem}\\label{C1 est}\n Let $\\theta\\in (0, \\pi)$, and let $h$ be a strictly convex solution of Eq. \\eqref{eq-k}. Then \\begin{eqnarray}\\label{gradient ps}\n \\max\\limits_{\\C_{\\theta}} |\\n h|\\leq C,\n \\end{eqnarray}\n where the constant $C$ only depends on $\\|h\\|_{C^{0}(\\C_{\\theta})}$.\n\\end{lem}\n\\begin{proof}\n Consider the function\n \\begin{eqnarray*}\n P=|\\n h|^{2}+h^{2}.\n \\end{eqnarray*}\nSuppose that $P$ attains its maximum value at some point, say $\\xi_{0}\\in \\C_{\\theta}$. If $\\xi_{0}\\in \\C_{\\theta}\\setminus \\partial\\C_{\\theta}$, by the maximal condition, we have\n\\begin{eqnarray*}\n 0=\\n_{e_i}P=2h_{k}h_{ki}+2hh_{i},\\quad \\text{for}~1\\leq i\\leq n.\n\\end{eqnarray*}\nSince $h$ is a strictly convex function, we have $\\n h(\\xi_{0})=0$. Combining with Lemma \\ref{c0 est}, we conclude that \\eqref{gradient ps} holds.\n\n\\section{Convexity}\\label{sec-convex}\nThe primary goal of this section is to obtain a strictly convex solution of Eq. \\eqref{eq-k}. Based on the celebrated result of Guan-Ma \\cite{GM} and with suitable assumptions on the function $f$, we show that the convex solution of Eq. \\eqref{eq-k} is strictly convex. To begin with, we establish the following key lemma.\n\\begin{lem}\\label{lem-def}\n Let $\\theta\\in (0, \\frac{\\pi}{2})$ and $\\tilde{F}(W) \\coloneqq \\sigma_{k}(W)$. Suppose that $h\\in C^{2}(\\C_{\\theta})$ is a solution of Eq. \\eqref{eq-k}, and assume that $W$ is positive semi-definite on $\\C_\\theta$. Suppose there exists a positive constant $C_{0}$ such that for a fixed integer $l $ $(k\\leq l\\leq n-1)$, $\\sigma_{l}(W)\\geq C_{0}$ for all $\\xi\\in \\C_{\\theta}$. If $f\\in \\mathcal{L}_{-\\frac{1}{k}}$ and $\\n_{\\mu}f\\geq 0$ on $\\partial \\C_{\\theta}$, denote $$\\phi\\coloneqq \\sigma_{l+1}(W(\\xi)),$$ then there exist constants $C_{1}, C_{2}$ depending only on $n, k, C_{0}, \\|h\\|_{C^{3}},$ and $\\|f\\|_{C^{2}}$, such that $\\phi$ satisfies \n \\begin{eqnarray}\n & \\tilde{F}^{ij}\\phi_{ij}\\leq C_{1}|\\n \\phi|+C_{2}\\phi, & \\quad \\rm{in} ~~ \\C_{\\theta} ,\\label{Diff-ine}\\\\\n &\\n_{\\mu} \\phi\\geq 0, & \\quad {\\rm{on}}~\\partial \\C_{\\theta}.\\label{boundary ine}\n \\end{eqnarray}\n \\end{lem}\n \\begin{proof}\n The differential inequality \\eqref{Diff-ine} follows from \\cite[Lemma~4.1]{GM} and $f\\in \\mathcal{L}_{-\\frac{1}{k}}$. We just need to prove the boundary condition \\eqref{boundary ine}.",
    "post_theorem_intro_text_len": 2591,
    "post_theorem_intro_text": "\\\n\nWe conclude the introduction by outlining the strategy for solving  Eq. \\eqref{eq-k}. If $\\theta=\\frac{\\pi}{2}$, this problem can be reduced to the closed case using a reflection argument. Hence, we focus on the case $\\theta<\\frac \\pi 2$ in proving Theorem \\ref{main theorem}. We will employ the method of continuity.  We need to establish a  priori estimates for solutions to Eq. \\eqref{eq-k} and show that the convexity is preserved. To establish the $C^{0}$ estimate, we modify the approach in  \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combine it with a general form of capillary Alexandrov-Fenchel inequalities for capillary hypersurfaces presented in \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$.  Hence our $C^0$-estimate for \\eqref{eq-k} holds for all contact angle $\\theta\\in(0, \\pi)$, see Lemma \\ref{c0 est}. The $C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate. In contrast to the closed case considered in \\cite[Proposition 3.2]{GM}, our global $C^2$ estimate depends on the gradient estimates, due to the specific choice of test functions used to derive the boundary $C^2$ estimate. The choice of the test functions is motivated by the works of \\cite{LTU, MQ} and \\cite{MWW-AIM}.\nFor the preservation of the convexity, we apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem.\nIt is worth noting that the assumption $\\theta\\in (0, \\frac{\\pi}{2})$ is crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.\nA similar assumption was employed in our previous works \\cite{MWW-Weingarten} and \\cite{MWW-Lp}, concerning the problems of convex capillary surfaces with prescribed $k$-th Weingarten curvature and prescribed capillary $L_p$-surface area measures for convex capillary bodies, respectively.  Nevertheless, we expect that the above results still hold for $\\theta > \\frac{\\pi}{2}$.\n\n\\\n\n\\textbf{The rest of the article is structured as follows.} \n\tIn Section \\ref{sec2}, we list some properties of elementary symmetric functions and introduce some notations and basic facts about capillary hypersurfaces. Moreover, we prove the necessary condition \\eqref{ortho-condition1} for the capillary Christoffel-Minkowski problem. \n\tIn Section \\ref{sec3}, we establish a priori estimates for the solution to Eq. \\eqref{eq-k}. In Section \\ref{sec-convex}, under certain assumptions on $f$, we investigate the convexity of the solution to Eq. \\eqref{eq-k}.\n In the final section, by using the method of continuity, we prove Theorem \\ref{main theorem}.\n\t\\vspace{.2cm}",
    "sketch": "To prove Theorem~\\ref{main theorem}, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to Eq.~\\eqref{eq-k} and show[ing] that the convexity is preserved.”\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of Theorem~\\ref{main theorem}.",
    "expanded_sketch": "To prove the main theorem, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to\n\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\n and show[ing] that the convexity is preserved.\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of the main theorem.",
    "expanded_theorem": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.,",
    "theorem_type": [
      "Existence",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(\\theta\\in(0,\\pi/2]\\), \\(1\\le k\\le n-1\\), \\(e=E_{n+1}\\), and let \\(\\mathcal C_\\theta\\subset \\mathbb S^n\\) be the spherical cap \\(\\{\\xi\\in\\mathbb S^n:\\langle \\xi,e\\rangle>-\\cos\\theta\\}\\). For \\(s\\in\\mathbb R\\), let \\(\\mathcal L_s\\) be the set of positive \\(C^2\\)-functions \\(g\\) on \\(\\mathcal C_\\theta\\) such that\n\\[\n\\int_{\\mathcal C_\\theta} g(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nand \\(g^s\\) is convex in the sense that \\(\\nabla^2(g^s)+g^s\\sigma\\) is positive semidefinite on \\(\\mathcal C_\\theta\\). Say that \\(f\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\) if there is a continuous path \\(q(t)\\in \\mathcal L_{-1/k}\\), \\(0\\le t\\le1\\), with \\(q(0)=1\\), \\(q(1)=f\\), and \\(\\partial_\\mu q(t)\\ge0\\) on \\(\\partial\\mathcal C_\\theta\\) for all \\(t\\). For a capillary convex body \\(\\widehat\\Sigma\\), its \\(k\\)-th capillary area measure on \\(\\mathcal C_\\theta\\) is\n\\[\ndS_k^c=\\ell(\\xi)\\,\\sigma_k(\\nabla^2h+h\\sigma)\\,d\\mathcal H^n,\n\\qquad \\ell(\\xi)=\\sin^2\\theta+\\cos\\theta\\langle \\xi,e\\rangle,\n\\]\nwhere \\(h\\) is the support function of \\(\\Sigma\\). If \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, such a hypersurface is unique in \\(\\overline{\\mathbb R^{n+1}_+}\\) without allowing any horizontal translation."
        },
        {
          "label": "C",
          "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\theta\\in(0,\\pi)\\), if \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), then there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta,\n\\]\nand any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
        },
        {
          "label": "E",
          "text": "If \\(f\\in C^2(\\mathcal C_\\theta)\\) satisfies the orthogonality condition\n\\[\n\\int_{\\mathcal C_\\theta} f(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nthen there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "translation-invariant uniqueness",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "uniqueness up to horizontal translation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "angle range restricted to \\((0,\\pi/2]\\)",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "homotopy/convexity-preserving connectedness in \\(\\mathcal L_{-1/k}\\)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion. It supplies hypotheses and definitions, but the exact existence/uniqueness claim must still be selected from the options."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to theorem-recall: it asks which existence statement holds under a long list of assumptions, and the correct option is essentially the theorem’s conclusion. Still, the alternatives vary meaningfully in uniqueness, strict convexity, and hypothesis strength, so it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to reject overstrong or underqualified conclusions, especially regarding uniqueness up to horizontal translation and the need for the path-connectedness assumption. However, the task mainly tests recognition of the precise theorem statement rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one ignores translation invariance, one gives only a weaker existence claim, and others improperly weaken the hypotheses. These reflect common failure modes in reading existence/uniqueness theorems."
      },
      "total_score": 6,
      "overall_assessment": "A reasonably well-constructed theorem-identification MCQ with strong distractors and no direct answer leakage, but it leans more toward precise recall of a stated result than toward genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.16655v2",
    "paper_link": "http://arxiv.org/abs/2512.16655v2",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.",
      "start_pos": 64453,
      "end_pos": 64950,
      "label": "main theorem"
    },
    "ref_dict": {
      "pro ortho": "\\begin{prop}\\label{pro ortho}\n\t\tIf  Eq. \\eqref{eq-k} is solvable, then $f$ satisfies\n\n\t\t\\begin{eqnarray}\\label{ortho-condition}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\H^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n.\n\\end{eqnarray}\t\t\n\t\\end{prop}",
      "main theorem": "\\begin{thm}\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\ov{{\\RR}^{n+1}_{+}}$, such that $\\widehat{\\S}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\S$ is unique up to a horizontal translation in $\\ov{{\\RR}^{n+1}_{+}}$. \n\\end{thm}",
      "measure": "\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\H}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}",
      "c0 est": "\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of  Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}.  \nThen there exists a positive constant $C$ depending  on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n\t0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}",
      "pro-2.6": "\\begin{prop}\\label{pro-2.6}\n The capillary  Christoffel-Minkowski problem \t\tis equivalent to finding a strictly convex solution to  Eq.  \\eqref{eq-k}.\\end{prop}",
      "def1.1": "\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\n^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}",
      "eq-k": "\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}",
      "ortho-condition1": "\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray}"
    },
    "pre_theorem_intro_text_len": 12832,
    "pre_theorem_intro_text": "In convex geometry, the prescribed $k$-th area measure problem is a well-known problem. It asks whether, for a given positive function $f$ on $\\mathbb{S}^{n}$, one can find a closed, strongly convex hypersurface $\\Sigma$ with $k$-th symmetric function of the principal curvature radii equal to $f$ as a function on its normals? When $k=n$, it corresponds to the famous Minkowski problem, which has been fully solved through the works of Minkowski \\cite{Min}, Alexandrov \\cite{Alex}, Lewy \\cite{Lewy}, Nirenberg \\cite{Nire}, Pogorelov \\cite{Pog52, Pog}, Cheng-Yau \\cite{CY76} and many others. Analytically, the Minkowski problem is equivalent to solving a Monge-Amp\\`ere equation on $\\mathbb{S}^{n}$. The study of the Minkowski problem has played a significant role in advancing research on fully nonlinear equations, especially on the Monge-Amp\\`ere equation. We can also see a closely related work by Xia \\cite{Xia} for an anisotropic Minkowski problem for closed hypersurfaces.\nWhen $k=1$, it is known as the Christoffel problem, which is equivalent to finding convex solutions to the Laplace equation on $\\mathbb{S}^{n}$.  Firey \\cite{Firey, Firey1} and Berg \\cite{Berg}  found necessary and sufficient conditions using the Green function of the Laplacian on $\\mathbb{S}^{n}$. More recently, Li-Wan-Wang \\cite{LWW} expressed the Christoffel problem as a Laplace equation on the entire space $\\mathbb{R}^{n+1}$ and provided a new and simpler necessary and sufficient condition for the Christoffel problem.  Further details on Christoffel's problem can be found in \\cite{Chri,  Hilbert, Hur, Suss}. For the intermediate cases, i.e., $2\\leq k\\leq n-1$, it is referred to as the Christoffel-Minkowski problem. Guan-Ma \\cite{GM}  solved this problem under a continuous homotopy condition, which was removed later by  Sheng-Wang-Trudinger \\cite{STW}. In addition to the elliptic method, Chou-Wang \\cite{CW00} and Bryan-Ivaki-Scheuer \\cite{BIS2023} provided a flow method to solve the above problems.\nFor a more comprehensive description of the prescribed area measure problem in convex geometry, see \\cite{Guan-note} and \\cite{Sch}. \n\n\\subsection{Motivation and set up}\nIn recent years, there has been growing interest in the study of capillary hypersurfaces, including free boundary hypersurfaces, along with their geometry and topology, see, for instance, \\cite{FL14, FS16, JWXZ, MWW-GCF, SWX, WW20, WWX22, WX2019, WeX21} and references therein. \nCapillary hypersurfaces originated from physics through the work of Thomas Young. Later,\nYoung, Liouville, and Gauss formulated it as a variational problem using the so-called wetting energy.  For capillary hypersurfaces, we refer to the elegant book by Finn \\cite{Finn}, and also a book by Maggi  \\cite{Maggi}. In this paper, with a slight abuse of terminology,\na smooth compact embedded hypersurface in $\\overline{{\\mathbb{R}}_+^{n+1}}$  whose boundary lies on $\\partial\\overline{{\\mathbb{R}}_+^{n+1}}$ is called {\\it capillary hypersurface} if\nit intersects $\\partial\\overline{{{\\mathbb{R}}}_+^{n+1}}$ at a constant contact angle $\\theta\\in (0,\\pi)$. \n   Let $\\nu$ be the unit outward normal of $\\Sigma$. The contact angle $\\theta $ is defined by $$ \\cos (\\pi -\\theta)=\\langle \\nu, e \\rangle,\\quad \\rm{along~\\partial\\Sigma},$$\n\twhere $e\\coloneqq -E_{n+1}$, $E_{n+1}$ is the $(n+1)$-th unit coordinate vector in $\\overline{\\mathbb{R}^{n+1}_+}$ and thus $e$ is the unit outward normal of $\\partial \\overline{\\mathbb{R}^{n+1}_+}$.\nThis means that a capillary hypersurface in this paper is not necessarily a hypersurface of constant mean curvature.\nIf $\\Sigma$ is strictly convex, then it is clear that the image of the Gauss map, namely,  $\\nu$, lies on the spherical cap \n\\begin{eqnarray*}\n\t\t\\mathbb{S}^n_\\theta \\coloneqq  \\left\\{ x\\in \\mathbb{S}^n \\,|\\,  \\langle x, E_{n+1} \\rangle \\geq \\cos \\theta \\right\\}.\n\\end{eqnarray*}\nInstead of using the usual Gauss map $\\nu$, it is more convenient to use the following map\n$$\\tilde \\nu\\coloneqq  T \\circ \\nu: \\Sigma \\to \\C_\\theta,$$\nwhere $\\C_\\theta$ is a spherical cap, defined by\n\\begin{eqnarray*}\n\t\t\\C_{\\theta }  \\coloneqq  \\left\\{\\xi\\in \\overline{\\mathbb{R}^{n+1}_+} \\mid |\\xi-\\cos\\theta e|= 1 \\right\\},\n\\end{eqnarray*}\n\tand $T:\\mathbb{S}^n_\\theta\\to\\C_\\theta$ is defined by $ T(y)\\coloneqq y+\\cos\\theta e$,  which is a translation in the vertical direction.\n We refer to $\\tilde \\nu$  as the {\\it capillary Gauss map} of $\\Sigma$.  It turns out that $\\tilde \\nu: \\Sigma\\to \\C_\\theta$ is a diffeomorphism. Consequently,  one can reparametrize $\\Sigma$ using its inverse on $\\C_\\theta$, see, e.g., \\cite[Section 2]{MWWX}. \n\nIn \\cite{MWW-AIM}, the authors formulated a capillary Minkowski problem, which concerns the existence of a strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_{+}}$ with a prescribed Gauss-Kronecker curvature on $\\C_{\\theta}$. We have established the following result in \\cite[Theorem 1.1]{MWW-AIM}.\n\n\\\n\n\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function  satisfying  \\eqref{ortho-condition1} below.  Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n\t\t{f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\n  \\\n\nThe capillary Minkowski problem can also be viewed as a prescribed capillary area measure problem, finding a strictly convex capillary hypersurface such that the induced capillary area measure equals a prescribed Borel measure on $\\C_{\\theta}$. See \\cite{MWW-AIM}. The area measures are an important subject in the study of convex bodies, and they are the local versions of quermassintegrals in the Brunn-Minkowski theory.  For more details about the classical Christoffel-Minkowski problem, see \\cite[Chapter~8]{Sch}.\n\nThe primary objective of this article is to extend our work presented in \\cite{MWW-AIM} and study the capillary Christoffel-Minkowski problem or the prescribed  $k$-th capillary area measure problem for the convex body with a capillary boundary in $\\overline{\\mathbb{R}^{n+1}_{+}}$.  \n Firstly, we introduce a novel concept of local parallel sets for the capillary convex bodies. We say $\\widehat{\\Sigma}$ is a \\textit{capillary convex body} (a compact convex set with non-empty interior) in $\\mathbb{R}^{n+1}$ if it is a bounded closed region in $\\overline{\\mathbb{R}^{n+1}_+}$ enclosed by a strictly convex capillary hypersurface $\\Sigma$ and  $\\partial \\overline{\\mathbb{R}^{n+1}_{+}}$. The class of capillary convex bodies in $\\overline{{\\mathbb{R}}^{n+1}_+}$ is denoted by $\\K_{\\theta}$.   Given a Borel set $\\beta\\subset \\C_{\\theta}$ and a capillary convex body $\\widehat{\\Sigma}\\subset \\K_{\\theta}$, we consider the following set :\n\\begin{eqnarray*}\n    B_{s}(\\widehat{\\Sigma}, \\beta)\\coloneqq  \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+t\\tilde{\\nu}(X),~0<t<s,~\\text{for}~X\\in \\Sigma ~ \\text{and} ~\\tilde{\\nu}(X)\\in \\beta \\right\\}.\n\\end{eqnarray*}\nWhen $\\beta=\\C_{\\theta}$, the set\n\\begin{eqnarray*}\n    A_{s}(\\widehat{\\Sigma}, \\C_{\\theta}) \\coloneqq \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+s\\tilde{\\nu}(X), ~X\\in \\Sigma~\\text{and}~s>0 \\right\\}\n\\end{eqnarray*}\nis also a strictly convex capillary hypersurface in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$ (see, for instance, \\cite[Remark 2.17]{MWWX} or \\cite[Proposition~2.3]{JWXZ}).\n\nDirect calculation yields the following local Steiner-type formula (see also \\cite[Remark~2.17]{MWWX} for a global version),\n\\begin{eqnarray*}\n    \\text{Vol}(B_{s}(\\widehat{\\Sigma}, \\beta))&=&\\sum\\limits_{k=0}^{n}\\frac{s^{n+1-k}}{n+1-k}\\int_{\\Sigma \\cap \\tilde{\\nu}^{-1}(\\beta)}(1+\\cos\\theta \\<\\nu, e\\>)\\sigma_{n-k}(\\kappa)d\\mu\\notag\\\\\n    &=&\\frac{1}{n+1}\\sum\\limits_{k=0}^{n}s^{n+1-k}\\binom{n+1}{k}S_{k}^c(\\widehat{\\Sigma}, \\beta),\n\\end{eqnarray*}\nwhere $S_k^c(\\widehat{\\Sigma},\\beta)$ is given by\n\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\mathcal{H}}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}\nHere, $\\kappa$ and $r$ are the set of the principal curvatures and principal curvature radii of $\\Sigma$ respectively, $\\sigma_{k}$ is the $k$-th elementary symmetric polynomial function, $d \\mathcal{H}^n$ and $d\\mu$ are the $n$-dimensional Hausdorff measure on $\\C_\\theta$\n and  $\\Sigma$ respectively. The superscript $c$ indicates the presence of the \\textit{capillary} effect in our notation. Moreover, $S_k^c (\\widehat{\\Sigma}, \\C_{\\theta})$ $(0\\leq k\\leq n)$ coincide with the quermassintegrals of convex capillary hypersurface introduced in  \\cite[Eq. (1.8)]{WWX22}, see also \\cite[Lemma~2.14]{MWWX}. \n\n For each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce  a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given  by \\begin{eqnarray}\\notag\n     dS_k^c \\coloneqq  \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray}  \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the  $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\n \\\n\n\\noindent\t\\textbf{Capillary Christoffel-Minkowski problem:} \\textit{Given a positive, $C^{2}$ function $f$ on $\\C_{\\theta}$, when does  there exist a capillary convex body $\\widehat{\\Sigma}$ in $\\overline{\\mathbb{R}^{n+1}_+}$ such that\n\\begin{eqnarray*}\n    dS_k^c=\\ell fd\\mathcal{H}^{n}~?\n\\end{eqnarray*}}\n\n\\\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case  $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding  convex solutions to the following Hessian type equation with Robin boundary value condition,  \\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\partial \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\nwhere   $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively  and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$  satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere  $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result,  we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n  For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.",
    "context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.",
    "full_context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.\n\nFor each capillary convex body $\\widehat{\\S}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\n^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\S$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\S}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\\n\nIn this section, we establish the a priori estimates for the strictly convex solution of Eq. \\eqref{eq-k}. \nFor any solution $h$ to Eq. \\eqref{eq-k}, it is easy to verify that \n\\begin{eqnarray*}\n h(\\xi)+\\<\\xi,E_\\alpha\\> ,\\quad \\text{ for all } 1\\leq \\a\\leq n,\n\\end{eqnarray*} is still a solution. Up to a horizontal translation, we assume $h$ satisfies the following orthogonal condition:\n\\begin{eqnarray}\\label{orth cond}\n \\int_{\\C_\\theta} h(\\xi) \\<\\xi,E_\\alpha\\> d\\H^{n}=0, \\quad \\text{ for all } 1\\leq \\a\\leq n.\n\\end{eqnarray}\nUnder the condition \\eqref{orth cond}, we are able to establish the following a priori estimates.\n\\begin{thm}\\label{priori estimate}\nLet $\\theta\\in (0, \\frac{\\pi}{2})$. Suppose that $h$ is a strictly convex solution of Eq. \\eqref{eq-k} and satisfies the condition \\eqref{orth cond}. For any integer $m\\geq 2$ and $\\gamma\\in(0,1)$, there exists a constant $C>0$ depending on $n, k,$ $\\min\\limits_{\\C_{\\theta}} f$ and $\\|f\\|_{C^{m}(\\C_\\theta)}$, such that\n \\begin{eqnarray}\\label{sch est}\n \\|h\\|_{C^{m+1,\\gamma}(\\C_\\theta)}\\leq C.\n \\end{eqnarray}\n\\end{thm}\n\n\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}. \nThen there exists a positive constant $C$ depending on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n 0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}\n\\begin{proof} \nFirst of all, due to \\eqref{orth cond}, by \\cite[Lemma 3.1]{MWW-AIM}, we have $h>0$ and the origin lies in the interior of $\\widehat{\\partial\\Sigma}$. \nLet $R_{0}>0$ be the smallest number of $R$ such that \n \\begin{eqnarray}\\label{enclosed}\n \\widehat \\S\\subset \\widehat{R\\C_\\theta},\n \\end{eqnarray}where $R\\C_\\theta\\coloneqq \\{\\xi \\in \\ov{\\RR^{n+1}_+}: |\\xi-\\cos\\theta R e|=R\\}$. It is clear that there exists $X_0\\in \\Sigma \\cap R_0\\C_{\\theta } $. Denote $\\widehat{X}_0\\coloneqq \\frac{X_0 }{R_0}\\in \\C_\\theta$.\nFor any $\\xi \\in \\C_\\theta$, we have \n \\begin{eqnarray}\\label{lower bdd of h}\n h(\\xi)&=&\\max_{Y\\in \\S}\\<\\xi-\\cos\\theta e, Y\\>\\notag \\\\\n &\\geq& \\max \\left\\{0, \\<\\xi-\\cos\\theta e,X_0\\> \\right\\}\\notag \\\\\n &=& \\max \\left\\{0, R_0\\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right\\}.\n ~~~\\end{eqnarray} \nMultiplying $f$ on both sides of \\eqref{lower bdd of h} and integrating over $ \\C_\\theta$, we see\n\\begin{eqnarray}\\label{R1}\n R_0\\leq \\left(\\int_{\\C_\\theta}h(\\xi)\\cdot f(\\xi)d \\H^{n}\\right) \\left(\\int_{\\C_\\theta}\\max \\left\\{0, \\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right \\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\n\\end{eqnarray}\nNote that \n\\begin{eqnarray}\\label{R2}\n \\int_{\\C_{\\theta}}h(\\xi)\\cdot f(\\xi) d \\H^{n}=\n\\int_{\\tilde{\\nu}(\\Sigma)}h(\\xi)\\cdot f(\\xi)d\\H^{n}=\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu.\n\\end{eqnarray}\nTogether with the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$ and the Minkowski formulas for capillary hypersurfaces \\cite[Proposition~2.6]{WWX22}, we obtain\n\\begin{eqnarray}\n C_{n, k}\\left(\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu\\right)^{\\frac{1}{k+1}}&\\leq& \\left(\\int_{\\Sigma}\\<X, \\nu\\>\\sigma_{n-k+1}(\\kappa)d\\mu\\right)^{\\frac{1}{k}}\\notag \\\\\n &=&\\left(\\frac{k}{n-k+1}\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{1}{k}}.\\label{R3}\n\\end{eqnarray}\nTogether with \\eqref{R1}, \\eqref{R2} and \\eqref{R3}, we conclude that \n\\begin{eqnarray}\n R_{0}\n &\\leq& C_{n,k}\\left(\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{k+1}{k}}\\cdot\\left(\\inf\\limits_{\\eta\\in \\C_{\\theta}}\\int_{\\C_\\theta}\\max\\{0, \\<\\xi-\\cos\\theta e, \\eta\\>\\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\\quad ~~~\\label{upper bound of R0}\n\\end{eqnarray}\nUsing \\eqref{enclosed} and \\eqref{upper bound of R0}, we see \n\\begin{eqnarray*}\n h(\\xi) =\\<\\xi-\\cos\\theta e , \\tilde{\\nu}^{-1}(\\xi)\\>\\leq |\\tilde{\\nu}^{-1}(\\xi)| <2R_0\\leq C(n, \\min_{\\C_\\theta} f,\\max_{\\C_\\theta} f).\n\\end{eqnarray*}\n We complete the proof of Lemma \\ref{c0 est}.\n\\end{proof}\n\\subsection{\\texorpdfstring{$C^1$ }{}estimate}\n In this subsection, we get the gradient estimate for the strictly convex solution $h$ of Eq. \\eqref{eq-k}. This estimate is a direct consequence of the convexity and the $C^{0}$ estimate.\n\\begin{lem}\\label{C1 est}\n Let $\\theta\\in (0, \\pi)$, and let $h$ be a strictly convex solution of Eq. \\eqref{eq-k}. Then \\begin{eqnarray}\\label{gradient ps}\n \\max\\limits_{\\C_{\\theta}} |\\n h|\\leq C,\n \\end{eqnarray}\n where the constant $C$ only depends on $\\|h\\|_{C^{0}(\\C_{\\theta})}$.\n\\end{lem}\n\\begin{proof}\n Consider the function\n \\begin{eqnarray*}\n P=|\\n h|^{2}+h^{2}.\n \\end{eqnarray*}\nSuppose that $P$ attains its maximum value at some point, say $\\xi_{0}\\in \\C_{\\theta}$. If $\\xi_{0}\\in \\C_{\\theta}\\setminus \\partial\\C_{\\theta}$, by the maximal condition, we have\n\\begin{eqnarray*}\n 0=\\n_{e_i}P=2h_{k}h_{ki}+2hh_{i},\\quad \\text{for}~1\\leq i\\leq n.\n\\end{eqnarray*}\nSince $h$ is a strictly convex function, we have $\\n h(\\xi_{0})=0$. Combining with Lemma \\ref{c0 est}, we conclude that \\eqref{gradient ps} holds.\n\n\\section{Convexity}\\label{sec-convex}\nThe primary goal of this section is to obtain a strictly convex solution of Eq. \\eqref{eq-k}. Based on the celebrated result of Guan-Ma \\cite{GM} and with suitable assumptions on the function $f$, we show that the convex solution of Eq. \\eqref{eq-k} is strictly convex. To begin with, we establish the following key lemma.\n\\begin{lem}\\label{lem-def}\n Let $\\theta\\in (0, \\frac{\\pi}{2})$ and $\\tilde{F}(W) \\coloneqq \\sigma_{k}(W)$. Suppose that $h\\in C^{2}(\\C_{\\theta})$ is a solution of Eq. \\eqref{eq-k}, and assume that $W$ is positive semi-definite on $\\C_\\theta$. Suppose there exists a positive constant $C_{0}$ such that for a fixed integer $l $ $(k\\leq l\\leq n-1)$, $\\sigma_{l}(W)\\geq C_{0}$ for all $\\xi\\in \\C_{\\theta}$. If $f\\in \\mathcal{L}_{-\\frac{1}{k}}$ and $\\n_{\\mu}f\\geq 0$ on $\\partial \\C_{\\theta}$, denote $$\\phi\\coloneqq \\sigma_{l+1}(W(\\xi)),$$ then there exist constants $C_{1}, C_{2}$ depending only on $n, k, C_{0}, \\|h\\|_{C^{3}},$ and $\\|f\\|_{C^{2}}$, such that $\\phi$ satisfies \n \\begin{eqnarray}\n & \\tilde{F}^{ij}\\phi_{ij}\\leq C_{1}|\\n \\phi|+C_{2}\\phi, & \\quad \\rm{in} ~~ \\C_{\\theta} ,\\label{Diff-ine}\\\\\n &\\n_{\\mu} \\phi\\geq 0, & \\quad {\\rm{on}}~\\partial \\C_{\\theta}.\\label{boundary ine}\n \\end{eqnarray}\n \\end{lem}\n \\begin{proof}\n The differential inequality \\eqref{Diff-ine} follows from \\cite[Lemma~4.1]{GM} and $f\\in \\mathcal{L}_{-\\frac{1}{k}}$. We just need to prove the boundary condition \\eqref{boundary ine}.",
    "post_theorem_intro_text_len": 2591,
    "post_theorem_intro_text": "\\\n\nWe conclude the introduction by outlining the strategy for solving  Eq. \\eqref{eq-k}. If $\\theta=\\frac{\\pi}{2}$, this problem can be reduced to the closed case using a reflection argument. Hence, we focus on the case $\\theta<\\frac \\pi 2$ in proving Theorem \\ref{main theorem}. We will employ the method of continuity.  We need to establish a  priori estimates for solutions to Eq. \\eqref{eq-k} and show that the convexity is preserved. To establish the $C^{0}$ estimate, we modify the approach in  \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combine it with a general form of capillary Alexandrov-Fenchel inequalities for capillary hypersurfaces presented in \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$.  Hence our $C^0$-estimate for \\eqref{eq-k} holds for all contact angle $\\theta\\in(0, \\pi)$, see Lemma \\ref{c0 est}. The $C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate. In contrast to the closed case considered in \\cite[Proposition 3.2]{GM}, our global $C^2$ estimate depends on the gradient estimates, due to the specific choice of test functions used to derive the boundary $C^2$ estimate. The choice of the test functions is motivated by the works of \\cite{LTU, MQ} and \\cite{MWW-AIM}.\nFor the preservation of the convexity, we apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem.\nIt is worth noting that the assumption $\\theta\\in (0, \\frac{\\pi}{2})$ is crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.\nA similar assumption was employed in our previous works \\cite{MWW-Weingarten} and \\cite{MWW-Lp}, concerning the problems of convex capillary surfaces with prescribed $k$-th Weingarten curvature and prescribed capillary $L_p$-surface area measures for convex capillary bodies, respectively.  Nevertheless, we expect that the above results still hold for $\\theta > \\frac{\\pi}{2}$.\n\n\\\n\n\\textbf{The rest of the article is structured as follows.} \n\tIn Section \\ref{sec2}, we list some properties of elementary symmetric functions and introduce some notations and basic facts about capillary hypersurfaces. Moreover, we prove the necessary condition \\eqref{ortho-condition1} for the capillary Christoffel-Minkowski problem. \n\tIn Section \\ref{sec3}, we establish a priori estimates for the solution to Eq. \\eqref{eq-k}. In Section \\ref{sec-convex}, under certain assumptions on $f$, we investigate the convexity of the solution to Eq. \\eqref{eq-k}.\n In the final section, by using the method of continuity, we prove Theorem \\ref{main theorem}.\n\t\\vspace{.2cm}",
    "sketch": "To prove Theorem~\\ref{main theorem}, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to Eq.~\\eqref{eq-k} and show[ing] that the convexity is preserved.”\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of Theorem~\\ref{main theorem}.",
    "expanded_sketch": "To prove the main theorem, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to\n\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\n and show[ing] that the convexity is preserved.\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of the main theorem.",
    "expanded_theorem": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.,",
    "theorem_type": [
      "Existence",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(\\theta\\in(0,\\pi/2]\\), \\(1\\le k\\le n-1\\), \\(e=E_{n+1}\\), and let \\(\\mathcal C_\\theta\\subset \\mathbb S^n\\) be the spherical cap \\(\\{\\xi\\in\\mathbb S^n:\\langle \\xi,e\\rangle>-\\cos\\theta\\}\\). For \\(s\\in\\mathbb R\\), let \\(\\mathcal L_s\\) be the set of positive \\(C^2\\)-functions \\(g\\) on \\(\\mathcal C_\\theta\\) such that\n\\[\n\\int_{\\mathcal C_\\theta} g(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nand \\(g^s\\) is convex in the sense that \\(\\nabla^2(g^s)+g^s\\sigma\\) is positive semidefinite on \\(\\mathcal C_\\theta\\). Say that \\(f\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\) if there is a continuous path \\(q(t)\\in \\mathcal L_{-1/k}\\), \\(0\\le t\\le1\\), with \\(q(0)=1\\), \\(q(1)=f\\), and \\(\\partial_\\mu q(t)\\ge0\\) on \\(\\partial\\mathcal C_\\theta\\) for all \\(t\\). For a capillary convex body \\(\\widehat\\Sigma\\), its \\(k\\)-th capillary area measure on \\(\\mathcal C_\\theta\\) is\n\\[\ndS_k^c=\\ell(\\xi)\\,\\sigma_k(\\nabla^2h+h\\sigma)\\,d\\mathcal H^n,\n\\qquad \\ell(\\xi)=\\sin^2\\theta+\\cos\\theta\\langle \\xi,e\\rangle,\n\\]\nwhere \\(h\\) is the support function of \\(\\Sigma\\). If \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), which statement holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, such a hypersurface is unique in \\(\\overline{\\mathbb R^{n+1}_+}\\) without allowing any horizontal translation."
        },
        {
          "label": "C",
          "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\theta\\in(0,\\pi)\\), if \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), then there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta,\n\\]\nand any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
        },
        {
          "label": "E",
          "text": "If \\(f\\in C^2(\\mathcal C_\\theta)\\) satisfies the orthogonality condition\n\\[\n\\int_{\\mathcal C_\\theta} f(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nthen there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "translation-invariant uniqueness",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "uniqueness up to horizontal translation",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "angle range restricted to \\((0,\\pi/2]\\)",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "homotopy/convexity-preserving connectedness in \\(\\mathcal L_{-1/k}\\)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or uniquely reveal choice A. It gives hypotheses and asks for the resulting statement, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem presents the full hypothesis set and asks for the exact conclusion. The correct option is basically the theorem statement itself."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants involving uniqueness, translation invariance, parameter range, and hypothesis weakening. However, the task is still mostly recognition of the exact theorem rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one overstates uniqueness, one gives a weaker true statement, one alters the admissible angle range, and one drops an essential connectedness hypothesis."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors, but it is largely a direct restatement of a result and only moderately tests reasoning rather than generative mathematical thinking."
    }
  },
  {
    "id": "2512.17211v1",
    "paper_link": "http://arxiv.org/abs/2512.17211v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "nonsec",
      "content": "{\\bf Approximation of quasihyperbolic geodesics.}\nIn spite of all these applications, finding the explicit value of the quasihyperbolic distance is not possible\nexcept in a few special cases.  Therefore, during the past five decades many comparison metrics  have been introduced  that are easily evaluated and, at the same time,  give either upper or lower bounds for the quasihyperbolic metric.  For a survey of these comparison metrics, see \\cite{hkv} and the recent  article \\cite{mo}.",
      "start_pos": 217172,
      "end_pos": 217688,
      "label": null
    },
    "ref_dict": {
      "qhdef": "\\begin{equation}\\label{qhdef}\nk_G(x,y)= \\inf_{\\gamma} \\int_{\\gamma} \\, \\frac{|dz|}{d(z, \\partial G)}\\,.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3803,
    "pre_theorem_intro_text": "The problem of finding a shortest path between two given locations is a classical problem in geodesy (see e.g. \\cite[Section 13.6]{VermeerRasila00}), and it has many connections to real world questions such as route planning, logistics, robotics, aviation, engineering, and communication networks. In this paper, we investigate this problem for hyperbolic type weighted metrics (see \\cite{hkv}). One of the methods to study these problems is to discretize the underlying mathematical model by using graphs. The vertices of the graph are possible intermediate locations between initial and terminal locations, whereas the edges depict all the connections between vertices. The research literature of this part of mathematical graph theory is vast due to the possible savings as a result of process speedup of algorithms.\n\nWe study this shortest path problem in a planar domain with a given metric with some\nspecial features: in particular, this metric measures not only the distance between two points\nbut also their location with respect to the boundary of the domain. In the cases we study, the\nexistence of shortest paths between two given points, i.e. length-minimizing curves or geodesics, is known by theory. A characteristic feature of the geodesics of hyperbolic type metrics is that they have a tendency to avoid the boundary of the domain as much as possible. \n\n\\subsection{Weighted metrics}\nFor a domain $G \\subsetneq \\mathbb{R}^n, n\\ge 2,$ define a continuous function $w: G \\to (0,\\infty),$ called {\\it weight function} and the {\\it weighted\nlength} of a rectifiable curve $\\gamma$ in $ G$ by\n\\begin{equation}\n\\label{w-len}\n\\ell_G^w(\\gamma) = \\int_{\\gamma} \\,w(z)\\, |dz|\\,.\n\\end{equation}\nIf $w(x)=1$ for all $x \\in G\\,,$ then $\\ell_G^w(\\gamma)$ reduces to the Euclidean length of\n$\\gamma$, and if \n\\begin{equation} \\label{qh-len}\nw(z)= \\frac{1}{d(z, \\partial G)}\\,,\n\\end{equation}\nwhere $d(z, \\partial G)$ is the distance of $z\\in G$ from the boundary of the domain, then $\\ell_G^w(\\gamma)$ is the {\\it quasihyperbolic\nlength} of $\\gamma\\,.$\nThe quasihyperbolic distance \n is defined for $x,y \\in G$ as\n\\begin{equation}\\label{qhdef}\nk_G(x,y)= \\inf_{\\gamma} \\int_{\\gamma} \\, \\frac{|dz|}{d(z, \\partial G)}\\,.\n\\end{equation}\nwhere the infimum is taken over all rectifiable curves $\\gamma$ in $G$ with\n$x,y \\in \\gamma.$  This metric was defined by F.W. Gehring and his coauthors \nB.P. Palka \\cite{gp} and B. Osgood \\cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \\eqref{qhdef} defines a metric and that there exists a length\nminimizing curve, geodesic, joining the points $x$ and $y\\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \\cite{RasilaTalponen14}). \n\nSince its introduction, the quasihyperbolic metric has become a standard tool in the study \nof geometric function theory,  quasiconformal mappings,  and  analysis on metric spaces \\cite{gh, gmp,  hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \\cite{b} at most by a constant factor \\cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \\cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.",
    "context": "The problem of finding a shortest path between two given locations is a classical problem in geodesy (see e.g. \\cite[Section 13.6]{VermeerRasila00}), and it has many connections to real world questions such as route planning, logistics, robotics, aviation, engineering, and communication networks. In this paper, we investigate this problem for hyperbolic type weighted metrics (see \\cite{hkv}). One of the methods to study these problems is to discretize the underlying mathematical model by using graphs. The vertices of the graph are possible intermediate locations between initial and terminal locations, whereas the edges depict all the connections between vertices. The research literature of this part of mathematical graph theory is vast due to the possible savings as a result of process speedup of algorithms.\n\nWe study this shortest path problem in a planar domain with a given metric with some\nspecial features: in particular, this metric measures not only the distance between two points\nbut also their location with respect to the boundary of the domain. In the cases we study, the\nexistence of shortest paths between two given points, i.e. length-minimizing curves or geodesics, is known by theory. A characteristic feature of the geodesics of hyperbolic type metrics is that they have a tendency to avoid the boundary of the domain as much as possible.\n\n\\subsection{Weighted metrics}\nFor a domain $G \\subsetneq \\mathbb{R}^n, n\\ge 2,$ define a continuous function $w: G \\to (0,\\infty),$ called {\\it weight function} and the {\\it weighted\nlength} of a rectifiable curve $\\gamma$ in $ G$ by\n\\begin{equation}\n\\label{w-len}\n\\ell_G^w(\\gamma) = \\int_{\\gamma} \\,w(z)\\, |dz|\\,.\n\\end{equation}\nIf $w(x)=1$ for all $x \\in G\\,,$ then $\\ell_G^w(\\gamma)$ reduces to the Euclidean length of\n$\\gamma$, and if \n\\begin{equation} \\label{qh-len}\nw(z)= \\frac{1}{d(z, \\partial G)}\\,,\n\\end{equation}\nwhere $d(z, \\partial G)$ is the distance of $z\\in G$ from the boundary of the domain, then $\\ell_G^w(\\gamma)$ is the {\\it quasihyperbolic\nlength} of $\\gamma\\,.$\nThe quasihyperbolic distance \n is defined for $x,y \\in G$ as\n\\begin{equation}\\label{qhdef}\nk_G(x,y)= \\inf_{\\gamma} \\int_{\\gamma} \\, \\frac{|dz|}{d(z, \\partial G)}\\,.\n\\end{equation}\nwhere the infimum is taken over all rectifiable curves $\\gamma$ in $G$ with\n$x,y \\in \\gamma.$ This metric was defined by F.W. Gehring and his coauthors \nB.P. Palka \\cite{gp} and B. Osgood \\cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \\eqref{qhdef} defines a metric and that there exists a length\nminimizing curve, geodesic, joining the points $x$ and $y\\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \\cite{RasilaTalponen14}).\n\nSince its introduction, the quasihyperbolic metric has become a standard tool in the study \nof geometric function theory, quasiconformal mappings, and analysis on metric spaces \\cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \\cite{b} at most by a constant factor \\cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \\cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.",
    "full_context": "The problem of finding a shortest path between two given locations is a classical problem in geodesy (see e.g. \\cite[Section 13.6]{VermeerRasila00}), and it has many connections to real world questions such as route planning, logistics, robotics, aviation, engineering, and communication networks. In this paper, we investigate this problem for hyperbolic type weighted metrics (see \\cite{hkv}). One of the methods to study these problems is to discretize the underlying mathematical model by using graphs. The vertices of the graph are possible intermediate locations between initial and terminal locations, whereas the edges depict all the connections between vertices. The research literature of this part of mathematical graph theory is vast due to the possible savings as a result of process speedup of algorithms.\n\nWe study this shortest path problem in a planar domain with a given metric with some\nspecial features: in particular, this metric measures not only the distance between two points\nbut also their location with respect to the boundary of the domain. In the cases we study, the\nexistence of shortest paths between two given points, i.e. length-minimizing curves or geodesics, is known by theory. A characteristic feature of the geodesics of hyperbolic type metrics is that they have a tendency to avoid the boundary of the domain as much as possible.\n\n\\subsection{Weighted metrics}\nFor a domain $G \\subsetneq \\mathbb{R}^n, n\\ge 2,$ define a continuous function $w: G \\to (0,\\infty),$ called {\\it weight function} and the {\\it weighted\nlength} of a rectifiable curve $\\gamma$ in $ G$ by\n\\begin{equation}\n\\label{w-len}\n\\ell_G^w(\\gamma) = \\int_{\\gamma} \\,w(z)\\, |dz|\\,.\n\\end{equation}\nIf $w(x)=1$ for all $x \\in G\\,,$ then $\\ell_G^w(\\gamma)$ reduces to the Euclidean length of\n$\\gamma$, and if \n\\begin{equation} \\label{qh-len}\nw(z)= \\frac{1}{d(z, \\partial G)}\\,,\n\\end{equation}\nwhere $d(z, \\partial G)$ is the distance of $z\\in G$ from the boundary of the domain, then $\\ell_G^w(\\gamma)$ is the {\\it quasihyperbolic\nlength} of $\\gamma\\,.$\nThe quasihyperbolic distance \n is defined for $x,y \\in G$ as\n\\begin{equation}\\label{qhdef}\nk_G(x,y)= \\inf_{\\gamma} \\int_{\\gamma} \\, \\frac{|dz|}{d(z, \\partial G)}\\,.\n\\end{equation}\nwhere the infimum is taken over all rectifiable curves $\\gamma$ in $G$ with\n$x,y \\in \\gamma.$ This metric was defined by F.W. Gehring and his coauthors \nB.P. Palka \\cite{gp} and B. Osgood \\cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \\eqref{qhdef} defines a metric and that there exists a length\nminimizing curve, geodesic, joining the points $x$ and $y\\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \\cite{RasilaTalponen14}).\n\nSince its introduction, the quasihyperbolic metric has become a standard tool in the study \nof geometric function theory, quasiconformal mappings, and analysis on metric spaces \\cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \\cite{b} at most by a constant factor \\cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \\cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.\n\n\\subsection{Weighted metrics}\nFor a domain $G \\subsetneq \\mathbb{R}^n, n\\ge 2,$ define a continuous function $w: G \\to (0,\\infty),$ called {\\it weight function} and the {\\it weighted\nlength} of a rectifiable curve $\\gamma$ in $ G$ by\n\\begin{equation}\n\\label{w-len}\n\\ell_G^w(\\gamma) = \\int_{\\gamma} \\,w(z)\\, |dz|\\,.\n\\end{equation}\nIf $w(x)=1$ for all $x \\in G\\,,$ then $\\ell_G^w(\\gamma)$ reduces to the Euclidean length of\n$\\gamma$, and if \n\\begin{equation} \\label{qh-len}\nw(z)= \\frac{1}{d(z, \\partial G)}\\,,\n\\end{equation}\nwhere $d(z, \\partial G)$ is the distance of $z\\in G$ from the boundary of the domain, then $\\ell_G^w(\\gamma)$ is the {\\it quasihyperbolic\nlength} of $\\gamma\\,.$\nThe quasihyperbolic distance \n is defined for $x,y \\in G$ as\n\\begin{equation}\\label{qhdef}\nk_G(x,y)= \\inf_{\\gamma} \\int_{\\gamma} \\, \\frac{|dz|}{d(z, \\partial G)}\\,.\n\\end{equation}\nwhere the infimum is taken over all rectifiable curves $\\gamma$ in $G$ with\n$x,y \\in \\gamma.$ This metric was defined by F.W. Gehring and his coauthors \nB.P. Palka \\cite{gp} and B. Osgood \\cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \\eqref{qhdef} defines a metric and that there exists a length\nminimizing curve, geodesic, joining the points $x$ and $y\\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \\cite{RasilaTalponen14}).\n\nSince its introduction, the quasihyperbolic metric has become a standard tool in the study \nof geometric function theory, quasiconformal mappings, and analysis on metric spaces \\cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \\cite{b} at most by a constant factor \\cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \\cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.\n\nIn this study, quasihyperbolic geometry is investigated from the point of view of constructive approximation.\nWe discretize the problem by forming a grid of points in the domain $G\\,.$\nThis grid must be fine enough considering the mutual locations of the points\nand the geometric features of the domain $G\\,.$ We define a graph using\nthese grid points and define the distance between two neighbouring points\nin a manner that is compatible with the quasihyperbolic geometry. The main\nstep of the process is to find the shortest path in this graph. For this purpose \nwe use Dijkstra's algorithm \\cite{Dijkstra59}. This method applies not only for the\nquasihyperbolic metric but also for many other weighted metrics for which shortest\npaths joining the points are known to exist. The method enables us to approximate the\nquasihyperbolic geodesics by means of polygonal curves and present figures of these approximations in several polygonal\ndomains. We also compare our approximate values\nof the quasihyperbolic distances to approximations\nobtained by other methods.\n\nAfter this introduction the contents of this paper is organized as follows. In Section 2 we introduce some basic notation for the metrics we use, describe how to proceed from grid points of a domain to\na graph, and some facts about special functions. In Section 3 we discuss computing distances in graphs and introduce Dijkstra's algorithm and discuss implementation of the algorithm for approximation of geodesics based on Dijkstra's algorithm. In Section 4 we use this algorithm for approximation of hyperbolic and quasihyperbolic geodesics in the unit disk, with an aim of validating the method and understanding its accuracy. We also consider conformal mappings of the upper half plane onto two polygonal domains, expressed in terms of Schwarz-Christoffel integrals. In Section 5 we investigate cases where non-uniqueness and bifurcation of geodesics is expected from theoretical considerations. In Sections 6--8 we consider other interesting examples where hyperbolic and quasihyperbolic geodesics can be approximated by using our methods. Finally, in Section 9, we give our final conclusions.\n\\section{Preliminaries}\nIn this section, we give the formulas and results needed later in our examples. We first focus on hyperbolic geometry and then present the necessary tools in order to handle conformal mappings of the upper half plane onto a quadrilateral domain.\n\n\\begin{nonsec}{\\bf Distance ratio metric.}\n A useful comparison\nmetric for the quasihyperbolic metric is the distance ratio metric \\cite{gp} \n\\begin{equation}\\label{jdef}\nj_G(x,y)= \\log \\left( 1+ \\frac{|x-y|}{\\min\\{ d(x, \\partial G), d(y, \\partial G)\\}}\\right) \\,,\n\\end{equation}\nwhere $x,y \\in G$.\nThe metrics $j_G$ and $k_G$ are not conformally invariant, and they are only \nsimilarity invariant.\nIf $G \\in \\{ \\mathbb{B}^2, \\mathbb{H}^2\\}, $ then for\n$x,y \\in G$\n \\begin{equation} \\label{jrho}\n j_G(x,y) \\le \\rho_G(x,y) \\le 2 j_G(x,y) \\,,\n \\end{equation}\n \\begin{equation} \\label{krho}\n k_G(x,y) \\le \\rho_G(x,y) \\le 2 k_G(x,y) \\,.\n \\end{equation} \nIn fact, $k_G \\equiv \\rho_G $ for $G= \\mathbb{H}^2.$\nThe geodesics of the quasihyperbolic metric are smooth\n\\cite{mar}.\n\nIn view of the above two-sided inequalities \\eqref{jrho} and \\eqref{krho}\nand the lower bound \\eqref{kjineq} it is natural to ask for which classes of\ndomains there is a majorant for $k_G$ in terms of $j_G.$\n Indeed, inequalities between\nmetrics can be used to define some important classes of domains. Thus, for instance, the class of simply connected plane domains $G$ for which there exists a constant $A \\ge 1$ such that for all $x,y \\in G$\n\\begin{equation} \\label{unifdef}\nk_G(x,y) \\le A \\, j_G(x,y)\n\\end{equation}\ndefines the class of quasidisks studied in \\cite{gh}. Domains of this class, in turn, are\nspecial cases of the so called $\\varphi$-uniform domains \\cite{hkv}.\n\n\\begin{nonsec}{\\bf Dijkstra's algorithm compared to other path finding methods.}\nDespite its age and simplicity, Dijkstra's algorithm remains one of the fastest and most used algorithms of computing distances in weighted graphs. If negative edge weights are allowed (but there are no negative cycles), the distances can be computed using the Bellman-Ford-Moore algorithm \\cite{Bellman58,Ford56,Moore59}. If the digraph is unweighted, it is faster to use a simple algorithm adapted from breadth-first-search. During the preparation of this article, Duan et al. \\cite{Duan25} showed that there exists an algorithm running in $\\mathcal{O} (m \\log^{2/3} n)$ time for computing the distances from one vertex to all others in a weighted digraph with non-negative edge weights. This is the long-awaited improvement to Dijkstra's algorithm, which has turned out difficult to improve due to the sorting step. However, Duan et al. make several assumptions to simplify the presentation of their algorithm. Most notably, the graphs are assumed to be sparse and every vertex has constant in- and out-degree. The algorithm they present is generalizable to graphs without these constraints. Dijkstra's algorithm is readily available and well-optimized for different programming languages, which is why we chose to use Dijkstra's algorithm and not implement Duan et al.'s algorithm for the purposes of our experiments. \n\\end{nonsec}",
    "post_theorem_intro_text_len": 2168,
    "post_theorem_intro_text": "In this study, quasihyperbolic geometry is investigated from the point of view of constructive approximation.\nWe discretize the problem by forming a grid of points in the domain $G\\,.$\nThis grid must be fine enough considering the mutual locations of the points\nand the geometric features of the domain $G\\,.$  We define a graph using\nthese grid points and define the distance between two neighbouring points\nin a manner that is compatible with the quasihyperbolic geometry.  The main\nstep of the process is to find the shortest path in this graph.  For this purpose \nwe use Dijkstra's algorithm \\cite{Dijkstra59}. This method applies not only for the\nquasihyperbolic metric but also for many other weighted metrics for which shortest\npaths joining the points are known to exist. The method enables us to approximate the\nquasihyperbolic geodesics by means of polygonal curves and present figures of these approximations in several polygonal\ndomains. We also compare our approximate values\nof the quasihyperbolic distances to approximations\nobtained by other methods.\n\nAfter this introduction the contents of this paper is organized as follows. In Section 2 we introduce some basic notation for the metrics we use, describe how to  proceed from grid points of a domain to\na graph, and some facts about special functions. In Section 3 we discuss computing distances in graphs and introduce Dijkstra's algorithm and discuss implementation of the algorithm for approximation of geodesics  based on Dijkstra's algorithm. In Section 4 we use this algorithm for approximation of hyperbolic and quasihyperbolic geodesics in the unit disk, with an aim of validating the method and understanding its accuracy. We also consider conformal mappings of the upper half plane onto two polygonal domains, expressed in terms of Schwarz-Christoffel integrals. In Section 5 we investigate cases where non-uniqueness and bifurcation of geodesics is expected from theoretical considerations. In Sections 6--8 we consider other interesting examples where hyperbolic and quasihyperbolic geodesics can be approximated by using our methods. Finally, in Section 9, we give our final conclusions.",
    "sketch": "We discretize the problem by forming a grid of points in the domain $G$ (``fine enough'' with respect to the locations of points and the geometric features of $G$). Using these grid points, we define a graph and define the distance between two neighbouring points ``in a manner that is compatible with the quasihyperbolic geometry.'' The main step is then to find the shortest path in this graph; for this we use Dijkstra's algorithm. This yields approximations of quasihyperbolic geodesics by polygonal curves (shortest graph paths), and corresponding approximate quasihyperbolic distances, which are then compared to approximations obtained by other methods.",
    "expanded_sketch": "We discretize the problem by forming a grid of points in the domain $G$ (``fine enough'' with respect to the locations of points and the geometric features of $G$). Using these grid points, we define a graph and define the distance between two neighbouring points ``in a manner that is compatible with the quasihyperbolic geometry.'' The main step is then to find the shortest path in this graph; for this we use Dijkstra's algorithm. This yields approximations of quasihyperbolic geodesics by polygonal curves (shortest graph paths), and corresponding approximate quasihyperbolic distances, which are then compared to approximations obtained by other methods.",
    "expanded_theorem": "{\\bf Approximation of quasihyperbolic geodesics.}\nIn spite of all these applications, finding the explicit value of the quasihyperbolic distance is not possible\nexcept in a few special cases. Therefore, during the past five decades many comparison metrics have been introduced that are easily evaluated and, at the same time, give either upper or lower bounds for the quasihyperbolic metric. For a survey of these comparison metrics, see \\cite{hkv} and the recent article \\cite{mo}.",
    "theorem_type": [
      "Inequality or Bound",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(G\\subsetneq \\mathbb{R}^n\\) be a domain, and define the quasihyperbolic distance for \\(x,y\\in G\\) by\n\\[\nk_G(x,y)=\\inf_{\\gamma}\\int_{\\gamma}\\frac{|dz|}{d(z,\\partial G)},\n\\]\nwhere the infimum is over all rectifiable curves \\(\\gamma\\subset G\\) joining \\(x\\) and \\(y\\), and \\(d(z,\\partial G)\\) denotes the Euclidean distance from \\(z\\) to the boundary of \\(G\\). Which statement about evaluating or bounding \\(k_G\\) is asserted?",
      "correct_choice": {
        "label": "A",
        "text": "Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, many comparison metrics have been introduced that are easy to evaluate and provide either upper bounds or lower bounds for the quasihyperbolic metric."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every domain \\(G\\subsetneq \\mathbb{R}^n\\), the explicit value of the quasihyperbolic distance can be found by discretizing \\(G\\) finely enough; thus comparison metrics are introduced mainly to recover the same values in a simpler closed form."
        },
        {
          "label": "C",
          "text": "Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, comparison metrics have been introduced that are easy to evaluate and are used to estimate the quasihyperbolic metric."
        },
        {
          "label": "D",
          "text": "Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, many comparison metrics have been introduced that are easy to evaluate and provide both upper bounds and lower bounds for the quasihyperbolic metric simultaneously."
        },
        {
          "label": "E",
          "text": "Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, one introduces comparison metrics that, for every pair \\(x,y\\in G\\), converge to \\(k_G(x,y)\\) under arbitrarily fine graph discretizations of \\(G\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "discretization_as_exact_evaluation",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_upper_or_lower_specificity",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "either_vs_both_bounds",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "comparison_metrics_vs_discretization_convergence_claim",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the quasihyperbolic metric but does not explicitly or implicitly reveal the correct claim about computability or comparison metrics. The answer must be identified from the options themselves."
      },
      "TAS": {
        "score": 0,
        "justification": "The question is essentially a recall/recognition task asking which textual statement is asserted, and the correct option is a near-verbatim restatement of that assertion rather than an application or consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish among closely related claims, especially A vs. C vs. D, but the task is mostly precise paraphrase matching rather than genuine mathematical generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and target common confusions: exact computability via discretization, weakening 'upper or lower bounds' to vague estimation, and incorrectly strengthening to simultaneous two-sided bounds. They are distinct and mathematically aligned."
      },
      "total_score": 5,
      "overall_assessment": "A solid recognition-style MCQ with strong distractors and no answer leakage, but it is largely tautological and tests textual recall more than generative mathematical reasoning."
    }
  },
  {
    "id": "2512.17606v1",
    "paper_link": "http://arxiv.org/abs/2512.17606v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{T_main}\n    Let $A\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a set with positive reach with $\\dim A\\leq 2$ and let $a\\in A$ be such that $\\dim(\\operatorname{Tan}(A,a))=1$. Then there exist a $\\operatorname{B}$-set $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^d$, a multirotation $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\subset\\mathbb{R}^d\\to\\mathbb{R}^d$ such that $0\\in U$, $a=\\Phi(0)$, $\\rho(B)\\subset U$ and $\\Phi(\\rho(B))$ is a neighbourhood of $a$ in $A$.\n\n    Moreover, if $\\Phi$, $B$ and $\\rho$ are as above, then $\\Phi(\\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\\subset\\mathbb{R}^d$ with positive reach satisfying $\\dim A\\leq 2$ and $\\dim(\\operatorname{Tan}(A,a))=1$.",
      "start_pos": 14419,
      "end_pos": 15088,
      "label": "T_main"
    },
    "ref_dict": {
      "T_main": "\\begin{theorem}  \\label{T_main}\n    Let $A\\subset\\R^d$ ($d\\geq 3$) be a set with positive reach with $\\dim A\\leq 2$ and let $a\\in A$ be such that $\\dim(\\Tan(A,a))=1$. Then there exist a $\\B$-set $B\\subset\\R^2\\subset\\R^d$, a multirotation $\\rho:\\R^d\\to\\R^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\subset\\R^d\\to\\R^d$ such that $0\\in U$, $a=\\Phi(0)$, $\\rho(B)\\subset U$ and $\\Phi(\\rho(B))$ is a neighbourhood of $a$ in $A$.\n\n    Moreover, if $\\Phi$, $B$ and $\\rho$ are as above, then $\\Phi(\\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\\subset\\R^d$ with positive reach satisfying $\\dim A\\leq 2$ and $\\dim(\\Tan(A,a))=1$. \n\\end{theorem}",
      "T_k": "\\begin{lemma}   \\label{T_k}\n    Let $A\\subset\\R^d$ have positive reach and $1\\leq k\\leq d$. Then\n    \\begin{enumerate}\n        \\item[(i)] $T_k(A)\\cap\\overline{\\bigcup_{i=0}^{k-1}T_i(A)}=\\emptyset$,\n        \\item[(ii)] the mapping $\\psi_k^A$ is continuous on $T_k(A)$,\n        \\item[(iii)] $T_1(A)$ is closed,\n        \\item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.\n    \\end{enumerate}\n\\end{lemma}",
      "local": "\\begin{lemma}\\label{local}\nFor $A \\subset \\R^d$ and $a \\in A$, the following are equivalent.\n\\begin{enumerate}\n\\item[(i)]  $  \\reach(A,a) > 0.$\n\\item[(ii)]  There exists $\\delta>0$ such that  $A \\cap \\overline{B}(a,\\delta)$ has positive reach.\n\\item [(iii)] \nThere exists a  set  $C$ and $\\omega>0$ such that  $\\reach C>0$ and $A \\cap B(a, \\omega)= C \\cap B(a, \\omega)$.\n\\end{enumerate}\n\\end{lemma}",
      "prorot": "\\begin{lemma}\\label{prorot} \nLet $A\\subset\\R^d$ ($d\\geq  3$) be a set with positive reach skewered on a segment $[p,q]\\subset \\R$ and let $r>0$ be such that  $0< r < \\reach A$ and \n\\begin{enumerate}\n    \\item[(a)] $\\diam A < r$, or\n    \\item[(b)] $A\\subset \\R^2 \\subset \\R^d$ is a $B$-set.\n\\end{enumerate}\t\nLet $\\rho$ be a multirotation associated with $T_1(A)$.\nThen \n\\begin{enumerate}\n\\item[(i)]\nthe mapping  $\\rho|_A$ is bi-Lipschitz,\n\\item[(ii)]\n $\\rho(A)$ is compact and $\\reach \\rho(A) > 0$,\n\\item[(iii)] $T_1(\\rho(A))=T_1(A)$ and $\\rho(A)$ is skewered on $[p,q]$.\n\\end{enumerate}  \n\\end{lemma}",
      "plochy": "\\begin{definition}\\label{plochy}  \n\t We say that $A \\subset \\R^d$ is a {\\it  $k$-dimensional $C^{1,1}$ \n\t surface} ($0<k<d$), if there exist a $k$-dimensional subspace $W$ of $\\R^d$ and a \n\t   $C^{1,1}$  mapping $\\vf: W \\to W^{\\perp}=:V$ such that  \n\t\t  $A = \\{w+\\vf(w): \\ w \\in W\\}$.\n\t\tThen we will also say that $A$ is a   $C^{1,1}$  surface {\\it associated with $V$}.\n\t\t\\end{definition}",
      "T1+": "\\begin{theorem}\\label{T1+}\n\tLet $d\\geq 2$,  $A \\subset \\R^d$ have positive reach and $a\\in T_1^+(A)$. Then there exists \n\ta  $1$-dimensional $C^{1,1}$ surface\n\t $\\Gamma$ and\n\t$\\delta >0$\n\t such that \n\t\\begin{equation}\\label{intj}\n\t T_1(A)\\cap B(a, \\delta) \\subset \\Gamma \\cap B(a, \\delta) \\subset  A.\n\t\\end{equation}\n\\end{theorem}",
      "count": "\\begin{corollary}  \\label{count}\n    Let $A\\subset\\R^d$ be a set with positive reach and $1\\leq k< d$. Then \n    \\begin{enumerate}\n        \\item[(i)] $T_k(A)$ can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces,\n        \\item[(ii)] if $\\dim A= k$ then $A$ can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces.\n    \\end{enumerate}\n\\end{corollary}",
      "bilip": "\\begin{corollary}  \\label{bilip}\n    Let $A\\subset\\R^d$ ($d\\geq 3$) be a two-dimensional set with positive reach and $a\\in A$. Then there exists a compact set $0\\in B\\subset\\R^2$ with positive reach and a bi-Lipschitz mapping $\\Psi:B\\to A^*$, where $A^*$ is a neighbourhood of $a$ in $A$.\n\\end{corollary}",
      "lipna2": "\\begin{lemma}\\label{lipna2}\n\tLet  $A\\subset \\R^d$, $0<r< \\reach A$,\n\t$\\diam A < r$ and \n\t $A \\subset T_1^-(A)  \\cup   T_2(A)$. \n\t Then  $\\psi_2:=\\psi_2^A$  is $(2^{12}\\pi/r)$-Lipschitz on $T_2(A)$.\n\\end{lemma}",
      "reach_compact": "\\begin{equation}  \\label{reach_compact}\n    \\reach(K,a)>0\\text{ for all }a\\in K\\iff\\reach K>0.\n\\end{equation}",
      "tdmnapl2": "\\begin{theorem}\\label{tdmnapl2}\n\tLet $d\\geq 2$, $1\\leq k \\leq d-1$,  $A \\subset \\R^d$ have positive reach and $a\\in T_k(A)$. \n\tThen there exists $\\delta >0$\n\t such that  $T_k(A) \\cap B(a, \\delta)$ is a subset of a   $k$-dimensional $C^{1,1}$ surface\n\t $\\Gamma$.\n\\end{theorem}",
      "B_set_PR": "\\begin{lemma} \\label{B_set_PR}\n    Any $\\B$-set $B\\subset\\R^2$ has positive reach.\n\\end{lemma}",
      "B-sets": "\\begin{definition}[B-sets]  \\label{B-sets}\n\\begin{enumerate}\n    \\item[(i)] We call $M\\subset \\R^2$  a \\emph{$\\B_-$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\vf$ on $[0,r]$ such that $\\vf$ is semiconcave on $(0,r)$, $\\psi$ is semiconvex on $(0,r)$, $\\vf(0)= \\psi(0)=0$, $\\vf'_+(0)= \\psi'_+(0)=0$ and $M= \\{(x,y):\\ x\\in [0,r],  \\psi(x) \\leq y \\leq \\vf(x)\\}$. \n    \\item[(ii)] We call $M\\subset \\R^2$  a \\emph{$\\B_+$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\vf$ on $[-r,r]$ such that $\\vf$ is semiconcave on $(-r,r)$, $\\psi$ is semiconvex on $(-r,r)$, $\\vf(0)= \\psi(0)=0$, $\\vf'(0)= \\psi'(0)=0$ and $M= \\{(x,y):\\ x\\in [-r,r],  \\psi(x) \\leq y \\leq \\vf(x)\\}$.  \n    \\item[(iii)] We say that $M\\subset\\R^2$ is a \\emph{$\\B$-set} if $M$ is a $\\B_-$-set or a $\\B_+$-set.\n\\end{enumerate}\n\\end{definition}"
    },
    "pre_theorem_intro_text_len": 9087,
    "pre_theorem_intro_text": "A set $A\\subset\\mathbb{R}^d$ is said to have \\emph{positive reach} if for some $\\varepsilon>0$, any point $x\\in\\mathbb{R}^d$ with $\\mathrm{dist}(x,A)<\\varepsilon$ have its unique nearest point $a\\in A$ (point with $\\mathrm{dist}(x,A)=|x-a|$). Sets with positive reach, which form a common extension of closed convex sets and compact $C^2$-domains with boundary,  were introduced  and applied by Federer \\cite{Fed59}. Later their natural generalization have been investigated also in Riemannian manifolds (see \\cite{Ban82}) and in general Hilbert spaces (see e.g. \\cite{CT10,ThII}), under the name \\emph{prox-regular sets}. Currently, sets with positive reach are extensively used in data analysis for manifold reconstruction (see e.g.\\ \\cite{BCM22}).\n\nIt appears that sets with positive reach may have much  more complicated geometric or topological  structure than closed convex sets or compact $C^2$-domains. Even the problem of complete\n characterization of the local structure of sets with positive reach in $\\mathbb{R}^d$ seems to be too difficult. \nHowever, satisfactory answers are known in some special cases.\n\nThe following result   is a consequence of statements claimed in \\cite[Remark 4.20]{Fed59}\n and easily follows from \\cite[Proposition 7.4]{RZ17}.\n\n\\begin{theoremF*} [Federer]\nLet  $A \\subset \\mathbb{R}^d$ be a $k$-dimensional set ($1\\leq k<d$) with positive reach, $a \\in A$,\nand let the tangent cone of $A$ at $a$ be a $k$-dimensional space. Then $A$ agrees on a neighbourhood of $a$ with  a $k$-dimensional $C^{1,1}$ surface.\n\\end{theoremF*}\n\nBy well-known results, if $P$ is a $k$-dimensional $C^{1,1}$ surface (cf.\\ Definition~\\ref{plochy}), $a\\in P$ and $r>0$ is sufficiently\n small, then $P\\cap\\overline{B}(a,r)$ has positive reach. It follows that  Theorem~F gives a complete satisfactory\n characterization of the local structure of $k$-dimensional sets $A$ with positive reach at a fixed point $a\\in\\mathbb{R}^d$ such that the tangent cone of $A$ at $a$ is a subspace of dimension $k$: it is\n the same as the local structure of $k$-dimensional $C^{1,1}$ surfaces containing $a$. (More formally, the two systems\n of the corresponding germs coincide.)\n\nRecently, A. Lytchak proved an interesting result (\\cite[Theorem 1.2]{Ly24}) which is essentially an extension of Theorem~F. Lytchak's result deals with sets in $d$-dimensional Riemannian manifold; its (essentially equivalent) Euclidean version is the following.\n\n\\begin{theoremL*} [Lytchak]\nLet  $A \\subset \\mathbb{R}^d$ be a $k$-dimensional set ($1\\leq k\\leq d$) with positive reach, $a \\in A$, and let the tangent cone of $A$ at $a$ have dimension $k$. Then there exist a convex body $K\\subset\\mathbb{R}^k\\subset \\mathbb{R}^d$ (of dimension $k$), an open set $U\\subset\\mathbb{R}^d$ containing $K$ and a $C^{1,1}$-diffeomorphism  $\\Phi: U \\to \\mathbb{R}^d$  such that $\\Phi(K)$ is a neighbourhood of $a$ in $A$.\n\\end{theoremL*}\n\nSince any set  $\\Phi(K)$ as above and containing $a$ has all properties of the set $A$ from Theorem~L, we see that Theorem~L gives a characterization of the local structure of sets $A$ with positive reach at points $a$ where the dimension of the tangent cone is the same as the dimension of $A$ (cf.\\ our remark after Theorem~F). \n\nNote that Theorem~L not only generalizes Theorem~F (which corresponds to the case $\\Phi^{-1}(a)  \\in {\\rm int}\\, K$), but also gives (for $k=1$) a local characterization of one-dimensional sets $A$ with positive reach in $\\mathbb{R}^d$ which is equivalent to that which follows from  \\cite[Corollary~8.9]{RZ17}: on a neighbourhood of $a\\in A$, $A$ is either a singleton, or a simple $C^{1,1}$ arc. \n\nIf $A\\subset\\mathbb{R}^2$ has positive reach then its local description using semiconcave and semiconvex functions (without any further restrictions) follows easily from \\cite[Theorem~6.4]{RZ17}. This result, however, cannot be easily extended to higher dimensions.\n\nThe main result of the present article is a complete characterization of the local structure of two-dimensional sets $A$ with positive reach in $\\mathbb{R}^d$ for $d\\geq 3$. Note that Theorem~L gives such a characterization at points $a\\in A$ where $\\operatorname{Tan}(A,a)$ has dimension $2$; the only nontrivial remaining case is when $\\dim\\operatorname{Tan}(A,a)=1$.  This case, however, is much  different from the case $\\dim\\operatorname{Tan}(A,a)=2$ - in particular, we cannot expect that $A$ will lie on a two-dimensional $C^{1,1}$ surface in some neighbourhood of $a$. This follows already from \\cite[Example~7.13]{RZ17}, which we will describe now in a slightly generalized version.\n\nConsider $r>0$ and Lipschitz functions\n$\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$\nis semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$, and set\n$B:= \\{(x,y):\\, x\\in [0,r],\\,  \\psi(x) \\leq y \\leq \\varphi(x)\\}$.\nThen $B$ has positive reach (see Lemma~\\ref{B_set_PR}).\nWe further perturbe $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^3$ to a set $M\\subset \\mathbb{R}^3$ by a ``multirotation'' by the following way.\nWe set $C:= \\{x\\in [0,r]: \\varphi(x)=\\psi(x)\\}$, denote by $\\Lambda$ the set of all components of $(0,r)\\setminus C$ and for each $\\lambda \\in \\Lambda$ denote $B_\\lambda = \\{(x,y) \\in B:\\ x\\in \\overline \\lambda\\}$. Now we rotate each  $B_\\lambda$  in $\\mathbb{R}^3$ around the $x$-axis by an angle $\\theta(\\lambda)$ in the positive sense to a set $M_\\lambda$ and set  \n\\begin{equation}  \\label{M}\n    M:= (B\\setminus \\bigcup _{\\lambda \\in \\Lambda} B_\\lambda)\n \\cup   \\bigcup _{\\lambda \\in \\Lambda} M_\\lambda.\n\\end{equation} \nThen  $M$ has positive reach as well (it follows from Lemma~\\ref{prorot}) and its tangent cone at $0$ is a half-line. It is easy to see that if $C$ is totally disconnected and $0$ is its accumulation point, we can choose the rotations $M_\\lambda$ of $B_\\lambda$ so that no neighbourhood of the origin in $M$ lies on a two-dimensional $C^{1,1}$ surface.\n\nOur main result shows (rather surprisingly) that the above example is in a sense universal (up to a $C^{1,1}$-diffeomorphism) for the case when the tangent cone of $A$ at $a$ is a half-line. In our proof of the main result (Theorem~\\ref{T_main} below), we need to include the ``both-sided'' version of the set $B$ above (where the tangent cone is the full line, see case (ii) in Definition~\\ref{B-sets}), and to define properly the ``multirotations'' in $\\mathbb{R}^d$. So, before stating the result, we need two definitions.\n\n\\begin{definition}[B-sets]  \\label{B-sets}\n\\begin{enumerate}\n    \\item[(i)] We call $M\\subset \\mathbb{R}^2$  a \\emph{$\\B_-$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$ is semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$ and $M= \\{(x,y):\\ x\\in [0,r],  \\psi(x) \\leq y \\leq \\varphi(x)\\}$. \n    \\item[(ii)] We call $M\\subset \\mathbb{R}^2$  a \\emph{$\\B_+$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[-r,r]$ such that $\\varphi$ is semiconcave on $(-r,r)$, $\\psi$ is semiconvex on $(-r,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'(0)= \\psi'(0)=0$ and $M= \\{(x,y):\\ x\\in [-r,r],  \\psi(x) \\leq y \\leq \\varphi(x)\\}$.  \n    \\item[(iii)] We say that $M\\subset\\mathbb{R}^2$ is a \\emph{$\\operatorname{B}$-set} if $M$ is a $\\B_-$-set or a $\\B_+$-set.\n\\end{enumerate}\n\\end{definition}\n\nAny $\\operatorname{B}$-set has positive reach (see Lemma~\\ref{B_set_PR}).\n\nLet $SO(d)$ be the group of rotations in $\\mathbb{R}^d$ (i.e., linear mappings preserving the scalar product and orientation). We call an element $R\\in SO(d)$ a \\emph{2-rotation}\nif there exists a two-dimensional subspace $T$ of $\\mathbb{R}^d$ orthogonal to $e_1$ and such that $Rv=v$ for any $v\\in T^\\perp$. (Notice that 2-rotations are determined by rotations in some 2-plane orthogonal to $e_1$.) Note that $R^{-1}$ is a 2-rotation whenever $R$ is.\n\nLet $\\pi$ denote the orthogonal projection onto the $x_1$-axis $\\operatorname{span}\\{e_1\\}$. \n\n\\begin{definition}[multirotations]  \\label{multrot}\nLet $\\emptyset\\neq C\\subset\\operatorname{span}\\{e_1\\}\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a compact set and $\\Lambda$ the family of all components (open intervals) of $\\operatorname{span}\\{e_1\\}\\setminus C$.\nWe will call a mapping $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ \\emph{multirotation} (in $\\mathbb{R}^d$) \\emph{associated with $C$} if \n$\\rho(x)=x$ whenever $\\pi(x)\\in C$ and for any $\\lambda\\in\\Lambda$ there exists a 2-rotation $R_\\lambda$ such that $\\rho|_{\\pi^{-1}(\\lambda)}=R_\\lambda|_{\\pi^{-1}(\\lambda)}$.\nWe will say that the multirotation $\\rho$ is determined by the family of $2$-rotations $(R_\\lambda)_{\\lambda\\in\\Lambda}$.\n\\end{definition}\n\nGiven a set $A\\subset\\mathbb{R}^d$ with positive reach and $k\\in\\{0,1,\\dots,d\\}$, we denote by $T_k(A)$ the set of all points of $A$ at which the tangent cone of $A$ has dimension $k$.\n\nNote that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\\mathbb{R}=\\operatorname{span}\\{e_1\\}$ by Lemma~\\ref{T_k}~(iii).",
    "context": "\\begin{theoremL*} [Lytchak]\nLet $A \\subset \\mathbb{R}^d$ be a $k$-dimensional set ($1\\leq k\\leq d$) with positive reach, $a \\in A$, and let the tangent cone of $A$ at $a$ have dimension $k$. Then there exist a convex body $K\\subset\\mathbb{R}^k\\subset \\mathbb{R}^d$ (of dimension $k$), an open set $U\\subset\\mathbb{R}^d$ containing $K$ and a $C^{1,1}$-diffeomorphism $\\Phi: U \\to \\mathbb{R}^d$ such that $\\Phi(K)$ is a neighbourhood of $a$ in $A$.\n\\end{theoremL*}\n\nConsider $r>0$ and Lipschitz functions\n$\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$\nis semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$, and set\n$B:= \\{(x,y):\\, x\\in [0,r],\\, \\psi(x) \\leq y \\leq \\varphi(x)\\}$.\nThen $B$ has positive reach (see Lemma~\\ref{B_set_PR}).\nWe further perturbe $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^3$ to a set $M\\subset \\mathbb{R}^3$ by a ``multirotation'' by the following way.\nWe set $C:= \\{x\\in [0,r]: \\varphi(x)=\\psi(x)\\}$, denote by $\\Lambda$ the set of all components of $(0,r)\\setminus C$ and for each $\\lambda \\in \\Lambda$ denote $B_\\lambda = \\{(x,y) \\in B:\\ x\\in \\overline \\lambda\\}$. Now we rotate each $B_\\lambda$ in $\\mathbb{R}^3$ around the $x$-axis by an angle $\\theta(\\lambda)$ in the positive sense to a set $M_\\lambda$ and set \n\\begin{equation} \\label{M}\n M:= (B\\setminus \\bigcup _{\\lambda \\in \\Lambda} B_\\lambda)\n \\cup \\bigcup _{\\lambda \\in \\Lambda} M_\\lambda.\n\\end{equation} \nThen $M$ has positive reach as well (it follows from Lemma~\\ref{prorot}) and its tangent cone at $0$ is a half-line. It is easy to see that if $C$ is totally disconnected and $0$ is its accumulation point, we can choose the rotations $M_\\lambda$ of $B_\\lambda$ so that no neighbourhood of the origin in $M$ lies on a two-dimensional $C^{1,1}$ surface.\n\n\\begin{definition}[B-sets] \\label{B-sets}\n\\begin{enumerate}\n \\item[(i)] We call $M\\subset \\mathbb{R}^2$ a \\emph{$\\B_-$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$ is semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$ and $M= \\{(x,y):\\ x\\in [0,r], \\psi(x) \\leq y \\leq \\varphi(x)\\}$. \n \\item[(ii)] We call $M\\subset \\mathbb{R}^2$ a \\emph{$\\B_+$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[-r,r]$ such that $\\varphi$ is semiconcave on $(-r,r)$, $\\psi$ is semiconvex on $(-r,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'(0)= \\psi'(0)=0$ and $M= \\{(x,y):\\ x\\in [-r,r], \\psi(x) \\leq y \\leq \\varphi(x)\\}$. \n \\item[(iii)] We say that $M\\subset\\mathbb{R}^2$ is a \\emph{$\\operatorname{B}$-set} if $M$ is a $\\B_-$-set or a $\\B_+$-set.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}[multirotations] \\label{multrot}\nLet $\\emptyset\\neq C\\subset\\operatorname{span}\\{e_1\\}\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a compact set and $\\Lambda$ the family of all components (open intervals) of $\\operatorname{span}\\{e_1\\}\\setminus C$.\nWe will call a mapping $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ \\emph{multirotation} (in $\\mathbb{R}^d$) \\emph{associated with $C$} if \n$\\rho(x)=x$ whenever $\\pi(x)\\in C$ and for any $\\lambda\\in\\Lambda$ there exists a 2-rotation $R_\\lambda$ such that $\\rho|_{\\pi^{-1}(\\lambda)}=R_\\lambda|_{\\pi^{-1}(\\lambda)}$.\nWe will say that the multirotation $\\rho$ is determined by the family of $2$-rotations $(R_\\lambda)_{\\lambda\\in\\Lambda}$.\n\\end{definition}\n\nGiven a set $A\\subset\\mathbb{R}^d$ with positive reach and $k\\in\\{0,1,\\dots,d\\}$, we denote by $T_k(A)$ the set of all points of $A$ at which the tangent cone of $A$ has dimension $k$.\n\nNote that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\\mathbb{R}=\\operatorname{span}\\{e_1\\}$ by Lemma~\\ref{T_k}~(iii).\n\n\\begin{lemma} \\label{B_set_PR}\n    Any $\\B$-set $B\\subset\\R^2$ has positive reach.\n\\end{lemma}\n\n\\begin{lemma}   \\label{T_k}\n    Let $A\\subset\\R^d$ have positive reach and $1\\leq k\\leq d$. Then\n    \\begin{enumerate}\n        \\item[(i)] $T_k(A)\\cap\\overline{\\bigcup_{i=0}^{k-1}T_i(A)}=\\emptyset$,\n        \\item[(ii)] the mapping $\\psi_k^A$ is continuous on $T_k(A)$,\n        \\item[(iii)] $T_1(A)$ is closed,\n        \\item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.\n    \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{prorot} \nLet $A\\subset\\R^d$ ($d\\geq  3$) be a set with positive reach skewered on a segment $[p,q]\\subset \\R$ and let $r>0$ be such that  $0< r < \\reach A$ and \n\\begin{enumerate}\n    \\item[(a)] $\\diam A < r$, or\n    \\item[(b)] $A\\subset \\R^2 \\subset \\R^d$ is a $B$-set.\n\\end{enumerate}\t\nLet $\\rho$ be a multirotation associated with $T_1(A)$.\nThen \n\\begin{enumerate}\n\\item[(i)]\nthe mapping  $\\rho|_A$ is bi-Lipschitz,\n\\item[(ii)]\n $\\rho(A)$ is compact and $\\reach \\rho(A) > 0$,\n\\item[(iii)] $T_1(\\rho(A))=T_1(A)$ and $\\rho(A)$ is skewered on $[p,q]$.\n\\end{enumerate}  \n\\end{lemma}",
    "full_context": "\\begin{theoremL*} [Lytchak]\nLet $A \\subset \\mathbb{R}^d$ be a $k$-dimensional set ($1\\leq k\\leq d$) with positive reach, $a \\in A$, and let the tangent cone of $A$ at $a$ have dimension $k$. Then there exist a convex body $K\\subset\\mathbb{R}^k\\subset \\mathbb{R}^d$ (of dimension $k$), an open set $U\\subset\\mathbb{R}^d$ containing $K$ and a $C^{1,1}$-diffeomorphism $\\Phi: U \\to \\mathbb{R}^d$ such that $\\Phi(K)$ is a neighbourhood of $a$ in $A$.\n\\end{theoremL*}\n\nConsider $r>0$ and Lipschitz functions\n$\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$\nis semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$, and set\n$B:= \\{(x,y):\\, x\\in [0,r],\\, \\psi(x) \\leq y \\leq \\varphi(x)\\}$.\nThen $B$ has positive reach (see Lemma~\\ref{B_set_PR}).\nWe further perturbe $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^3$ to a set $M\\subset \\mathbb{R}^3$ by a ``multirotation'' by the following way.\nWe set $C:= \\{x\\in [0,r]: \\varphi(x)=\\psi(x)\\}$, denote by $\\Lambda$ the set of all components of $(0,r)\\setminus C$ and for each $\\lambda \\in \\Lambda$ denote $B_\\lambda = \\{(x,y) \\in B:\\ x\\in \\overline \\lambda\\}$. Now we rotate each $B_\\lambda$ in $\\mathbb{R}^3$ around the $x$-axis by an angle $\\theta(\\lambda)$ in the positive sense to a set $M_\\lambda$ and set \n\\begin{equation} \\label{M}\n M:= (B\\setminus \\bigcup _{\\lambda \\in \\Lambda} B_\\lambda)\n \\cup \\bigcup _{\\lambda \\in \\Lambda} M_\\lambda.\n\\end{equation} \nThen $M$ has positive reach as well (it follows from Lemma~\\ref{prorot}) and its tangent cone at $0$ is a half-line. It is easy to see that if $C$ is totally disconnected and $0$ is its accumulation point, we can choose the rotations $M_\\lambda$ of $B_\\lambda$ so that no neighbourhood of the origin in $M$ lies on a two-dimensional $C^{1,1}$ surface.\n\n\\begin{definition}[B-sets] \\label{B-sets}\n\\begin{enumerate}\n \\item[(i)] We call $M\\subset \\mathbb{R}^2$ a \\emph{$\\B_-$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[0,r]$ such that $\\varphi$ is semiconcave on $(0,r)$, $\\psi$ is semiconvex on $(0,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'_+(0)= \\psi'_+(0)=0$ and $M= \\{(x,y):\\ x\\in [0,r], \\psi(x) \\leq y \\leq \\varphi(x)\\}$. \n \\item[(ii)] We call $M\\subset \\mathbb{R}^2$ a \\emph{$\\B_+$-set} if there exist $r>0$, Lipschitz functions $\\psi \\leq 0 \\leq \\varphi$ on $[-r,r]$ such that $\\varphi$ is semiconcave on $(-r,r)$, $\\psi$ is semiconvex on $(-r,r)$, $\\varphi(0)= \\psi(0)=0$, $\\varphi'(0)= \\psi'(0)=0$ and $M= \\{(x,y):\\ x\\in [-r,r], \\psi(x) \\leq y \\leq \\varphi(x)\\}$. \n \\item[(iii)] We say that $M\\subset\\mathbb{R}^2$ is a \\emph{$\\operatorname{B}$-set} if $M$ is a $\\B_-$-set or a $\\B_+$-set.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}[multirotations] \\label{multrot}\nLet $\\emptyset\\neq C\\subset\\operatorname{span}\\{e_1\\}\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a compact set and $\\Lambda$ the family of all components (open intervals) of $\\operatorname{span}\\{e_1\\}\\setminus C$.\nWe will call a mapping $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ \\emph{multirotation} (in $\\mathbb{R}^d$) \\emph{associated with $C$} if \n$\\rho(x)=x$ whenever $\\pi(x)\\in C$ and for any $\\lambda\\in\\Lambda$ there exists a 2-rotation $R_\\lambda$ such that $\\rho|_{\\pi^{-1}(\\lambda)}=R_\\lambda|_{\\pi^{-1}(\\lambda)}$.\nWe will say that the multirotation $\\rho$ is determined by the family of $2$-rotations $(R_\\lambda)_{\\lambda\\in\\Lambda}$.\n\\end{definition}\n\nGiven a set $A\\subset\\mathbb{R}^d$ with positive reach and $k\\in\\{0,1,\\dots,d\\}$, we denote by $T_k(A)$ the set of all points of $A$ at which the tangent cone of $A$ has dimension $k$.\n\nNote that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\\mathbb{R}=\\operatorname{span}\\{e_1\\}$ by Lemma~\\ref{T_k}~(iii).\n\n\\begin{lemma} \\label{B_set_PR}\n    Any $\\B$-set $B\\subset\\R^2$ has positive reach.\n\\end{lemma}\n\n\\begin{lemma}   \\label{T_k}\n    Let $A\\subset\\R^d$ have positive reach and $1\\leq k\\leq d$. Then\n    \\begin{enumerate}\n        \\item[(i)] $T_k(A)\\cap\\overline{\\bigcup_{i=0}^{k-1}T_i(A)}=\\emptyset$,\n        \\item[(ii)] the mapping $\\psi_k^A$ is continuous on $T_k(A)$,\n        \\item[(iii)] $T_1(A)$ is closed,\n        \\item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.\n    \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{prorot} \nLet $A\\subset\\R^d$ ($d\\geq  3$) be a set with positive reach skewered on a segment $[p,q]\\subset \\R$ and let $r>0$ be such that  $0< r < \\reach A$ and \n\\begin{enumerate}\n    \\item[(a)] $\\diam A < r$, or\n    \\item[(b)] $A\\subset \\R^2 \\subset \\R^d$ is a $B$-set.\n\\end{enumerate}\t\nLet $\\rho$ be a multirotation associated with $T_1(A)$.\nThen \n\\begin{enumerate}\n\\item[(i)]\nthe mapping  $\\rho|_A$ is bi-Lipschitz,\n\\item[(ii)]\n $\\rho(A)$ is compact and $\\reach \\rho(A) > 0$,\n\\item[(iii)] $T_1(\\rho(A))=T_1(A)$ and $\\rho(A)$ is skewered on $[p,q]$.\n\\end{enumerate}  \n\\end{lemma}\n\nNote that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\\R=\\spa\\{e_1\\}$ by Lemma~\\ref{T_k}~(iii).\n\n\\begin{remark}\n It is easy to see that Theorem~\\ref{T_main} holds if we replace ``$\\dim A\\leq 2$'' with ``$\\dim A=2$'' at both occurrences. The only difference is in the proof of the ``moreover'' part: While in the first case we set in our proof $A:=\\Phi(\\rho(B))$, in the latter case we could take $A:=\\Phi(\\rho(B))\\cup D$, where $\\reach D>0$, $\\dim D=2$ and $\\dist(\\Phi(\\rho(B)),D)>0$. \n\\end{remark}\n\n\\begin{corollary}\n Let $A\\subset\\R^d$ ($d\\leq 3$) be a two-dimensional compact set. Then $A$ has positive reach if and only if for each non-isolated $a\\in A$ there exist a compact set $K\\subset\\R^d$, an open set $K\\subset U\\subset\\R^d$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\to\\R^d$ such that $\\Phi(K)$ is a neighbourhood of $a$ in $A$ and either \n \\begin{enumerate}\n \\item[(a)] $K\\subset\\R^2\\subset\\R^d$ is convex two-dimensional, or\n \\item[(b)] $K=\\rho(B)$, where $B\\subset\\R^2\\subset\\R^d$ is a $\\B$-set and $\\rho:\\R^d\\to\\R^d$ is a multirotation associated with $T_1(B)$.\n \\end{enumerate} \n\\end{corollary}\n\n\\begin{proposition} \\label{T}\n Let $A\\subset\\R^d$ ($d\\geq 3$) have positive reach, $\\dim A\\leq 2$ and $a\\in T_1(A)$. Then there exists a $\\B$-set $B$ and a multirotation $\\rho$ associated with $T_1(B)$ such that $(A,a)\\approx(\\rho(B),0)$.\n\\end{proposition}\n\n\\begin{proposition} \\label{T3}\n Let $S\\subset\\R^d$ ($d\\geq 3$) with positive reach be a set skewered on $[c,d]$, $c<0<d$, with planar leaves, and $0\\in T_1(S)$. Then there exists a $\\B_+$-set $B$ and a multirotation $\\rho$ associated with $T_1(B)$ such that $(S,0)\\sim(\\rho(B),0)$.\n\\end{proposition}\n\nLet us consider now the case $a\\in T_1^-(A)$. Choose $0<r<\\reach A$ and let $u$ be the unit vector such that $\\Tan(A,a)=\\{tu:\\, t\\geq 0\\}$. Denoting $S:=\\{a+tu:\\, -\\frac r4\\leq t< 0\\}$ and $A^*:=A\\cup S$, we have clearly $a\\in T_1^+(A^*)$ and \\cite[Lemma 3.8]{RZ17} gives $\\reach A^* \\geq r/4$. We apply the already proved assertion and find a $\\B_+$-set $B^*$ and a multirotation $\\rho^*$ associated with $T_1(B^*)$ such that $(A^*,a)\\approx(\\rho^*(B^*),0)$. \nBy Remark~\\ref{oekv1}~(i) there exist a $C^{1,1}$-diffeomorphism $\\Phi$ defined on a neighbourhood $U$ of $a$ and $\\ep>0$ such that \n\\begin{equation} \\label{AB}\n\\Phi(a)=0 \\ \\text{ and }\\ \\Phi(A^*\\cap U)\\cap B(0,\\ep)=\\rho^*(B^*)\\cap B(0,\\ep).\n\\end{equation}\nFurther, there exists $0<\\delta<r/4$ such that\n\\begin{equation} \\label{S_delta}\n B(a,\\delta)\\subset U\\text{ and }\\Phi(B(a,\\delta))\\subset B(0,\\ep).\n\\end{equation}\nDenote $S_\\delta:=S\\cap B(a,\\delta)=\\{a+tu:\\, -\\delta< t< 0\\}$.\nSince $S\\subset T_1(A^*)$, we obtain\n$$\\Phi(S_\\delta)\\subset \\Phi(T_1(A^*)\\cap U)\\cap B(0,\\ep) \\subset T_1(\\rho^*(B^*))$$\nusing Lemma~\\ref{tkdif} and \\eqref{AB}.\nWe have further $T_1(\\rho^*(B^*))=T_1(B^*)\\subset W$ by Lemma~\\ref{prorot}~(iii) and Remark~\\ref{Rem_listky}~(b), hence we get $\\Phi(S_\\delta)\\subset W$. Since $\\Phi(S_\\delta)$ is homeomorphic to $(0,1)$ and $0\\in\\overline{\\Phi(S_\\delta)}\\setminus \\Phi(S_\\delta)$, we have (a) $\\Phi(S_\\delta)=(-\\ep',0)$ or (b) $\\Phi(S_\\delta)=(0,\\ep')$ for some $0<\\ep'\\leq \\ep$.\nWe claim that we can choose $B^*,\\rho^*,\\Phi,U,\\ep$ and $\\delta$ as above satisfying \\eqref{AB} and \\eqref{S_delta} such that the case (a) occurs. Indeed, if $B^*,\\rho^*,\\Phi,U,\\ep$ and $\\delta$ satisfy \\eqref{AB}, \\eqref{S_delta} and (b), consider the reflection $G:(x,z)\\mapsto(-x,z)$ ($(x,z)\\in\\R\\times\\R^{d-1}$) and set $\\tilde{B}^*:=G(B^*)$, $\\tilde{\\rho}^*:=G\\circ\\rho^*\\circ G$, $\\tilde{\\Phi}:=G\\circ\\Phi$, $\\tilde{U}:=U$, $\\tilde{\\ep}:=\\ep$ and $\\tilde{\\delta}:=\\delta$. Then both \\eqref{AB} and \\eqref{S_delta} remain true and it is easy to see that $\\tilde{B}^*$ is again a $\\B_+$-set (using Remark~\\ref{rem_semicon}) and that $\\tilde{\\rho}^*$ is a multirotation associated with\n $T_1(\\tilde{B}^*)$. Moreover, $\\tilde{\\Phi}(S_{\\tilde{\\delta}})=-\\Phi(S_\\delta)=(-\\ep',0)$, i.e., case (a) occurs.\n\nFor the second statement of Theorem~\\ref{T_main}, assume that $B\\subset\\R^2\\subset\\R^d$ is a $\\B$-set, $\\rho$ a multirotation associated with $T_1(B)$, $U$ an open neighbourhood of $\\rho(B)$ and $\\Phi:U\\to\\R^d$ a $C^{1,1}$-diffeomorphism. Then $B$ has positive reach by Lemma~\\ref{B_set_PR} and it is clearly compact, hence $\\rho(B)$ is compact and has positive reach by Lemma~\\ref{prorot}~(ii). Setting $A:=\\Phi(\\rho(B))$, we have $\\reach A>0$ by Lemma~\\ref{reach_C11} and $a\\in T_1(A)$ by Lemma~\\ref{prorot}~(iii) and Lemma~\\ref{tkdif}~(iii). This implies the second assertion of Theorem~\\ref{T_main} and completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary~\\ref{bilip}] \\hskip 2mm\n Let $A\\subset\\R^d$ be a two-dimensional set with positive reach and $a\\in A$. \n If $a\\in T_0(A)$ then $a$ is an isolated point in $A$ and the assertion is obvious.\n If $a\\in T_1(A)$ then we apply Theorem~\\ref{T_main} and obtain a $\\B$-set $B\\subset\\R^2$, a multirotation $\\rho$ associated with $T_1(B)$, a neighbourhood $U$ of $\\rho(B)$ in $\\R^d$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\to\\R^d$ such that $\\Phi(\\rho(B))$ is a neighbourhood of $a$ in $A$. \n The set $\\rho(B)$ is compact by Lemma~\\ref{prorot}~(ii), hence $\\Phi(\\rho(B))$ is compact as well. Since both $\\Phi$ and $\\Phi^{-1}$ are locally Lipschitz (see Remark~\\ref{stability}~(i)), their restrictions to compact sets $\\rho(B)$ and $\\Phi(\\rho(B))$ (respectively) are Lipschitz by \\cite[Proposition~A48]{Lee}, i.e., $\\Phi$ is bi-Lipschitz on $\\rho(B)$.\n Since $\\rho$ is bi-Lipschitz on $B$ by Lemma~\\ref{prorot}~(i), we can choose $\\Psi:=\\Phi\\circ\\rho|_B$ and the assertion follows.\n\n\\begin{lemma} \\label{B_set_PR}\n    Any $\\B$-set $B\\subset\\R^2$ has positive reach.\n\\end{lemma}\n\n\\begin{lemma}   \\label{T_k}\n    Let $A\\subset\\R^d$ have positive reach and $1\\leq k\\leq d$. Then\n    \\begin{enumerate}\n        \\item[(i)] $T_k(A)\\cap\\overline{\\bigcup_{i=0}^{k-1}T_i(A)}=\\emptyset$,\n        \\item[(ii)] the mapping $\\psi_k^A$ is continuous on $T_k(A)$,\n        \\item[(iii)] $T_1(A)$ is closed,\n        \\item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.\n    \\end{enumerate}\n\\end{lemma}\n\n\\begin{theorem}  \\label{T_main}\n    Let $A\\subset\\R^d$ ($d\\geq 3$) be a set with positive reach with $\\dim A\\leq 2$ and let $a\\in A$ be such that $\\dim(\\Tan(A,a))=1$. Then there exist a $\\B$-set $B\\subset\\R^2\\subset\\R^d$, a multirotation $\\rho:\\R^d\\to\\R^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\subset\\R^d\\to\\R^d$ such that $0\\in U$, $a=\\Phi(0)$, $\\rho(B)\\subset U$ and $\\Phi(\\rho(B))$ is a neighbourhood of $a$ in $A$.\n\n    Moreover, if $\\Phi$, $B$ and $\\rho$ are as above, then $\\Phi(\\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\\subset\\R^d$ with positive reach satisfying $\\dim A\\leq 2$ and $\\dim(\\Tan(A,a))=1$. \n\\end{theorem}\n\n\\begin{corollary}  \\label{bilip}\n    Let $A\\subset\\R^d$ ($d\\geq 3$) be a two-dimensional set with positive reach and $a\\in A$. Then there exists a compact set $0\\in B\\subset\\R^2$ with positive reach and a bi-Lipschitz mapping $\\Psi:B\\to A^*$, where $A^*$ is a neighbourhood of $a$ in $A$.\n\\end{corollary}\n\n\\begin{lemma}\\label{prorot} \nLet $A\\subset\\R^d$ ($d\\geq  3$) be a set with positive reach skewered on a segment $[p,q]\\subset \\R$ and let $r>0$ be such that  $0< r < \\reach A$ and \n\\begin{enumerate}\n    \\item[(a)] $\\diam A < r$, or\n    \\item[(b)] $A\\subset \\R^2 \\subset \\R^d$ is a $B$-set.\n\\end{enumerate}\t\nLet $\\rho$ be a multirotation associated with $T_1(A)$.\nThen \n\\begin{enumerate}\n\\item[(i)]\nthe mapping  $\\rho|_A$ is bi-Lipschitz,\n\\item[(ii)]\n $\\rho(A)$ is compact and $\\reach \\rho(A) > 0$,\n\\item[(iii)] $T_1(\\rho(A))=T_1(A)$ and $\\rho(A)$ is skewered on $[p,q]$.\n\\end{enumerate}  \n\\end{lemma}",
    "post_theorem_intro_text_len": 5240,
    "post_theorem_intro_text": "\\begin{remark}\n   It is easy to see that Theorem~\\ref{T_main} holds if we replace ``$\\dim A\\leq 2$'' with ``$\\dim A=2$'' at both occurrences. The only difference is in the proof of the ``moreover'' part: While in the first case we set in our proof  $A:=\\Phi(\\rho(B))$, in the latter case we could take $A:=\\Phi(\\rho(B))\\cup D$, where ${\\rm reach}\\, D>0$, $\\dim D=2$ and $\\mathrm{dist}(\\Phi(\\rho(B)),D)>0$. \n\\end{remark}\n\nNote that (ignoring the trivial case when $a$ is an isolated point of $A$), Theorem~\\ref{T_main} together with Theorem~L give a complete characterization of the local structure of two-dimensional sets $A\\subset\\mathbb{R}^d$ with positive reach: For a given point $a\\in\\mathbb{R}^d$, the set of all germs at $a$ of sets $A$ as above with $a\\in A$ is described ``constructively''. Also, both results together imply (using also \\eqref{reach_compact} and Lemma~\\ref{local}) the following characterization of compact two-dimensional sets with positive reach.\n\n\\begin{corollary}\n    Let $A\\subset\\mathbb{R}^d$ ($d\\leq 3$) be a two-dimensional compact set. Then $A$ has positive reach if and only if for each non-isolated $a\\in A$ there exist a compact set $K\\subset\\mathbb{R}^d$, an open set $K\\subset U\\subset\\mathbb{R}^d$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\to\\mathbb{R}^d$ such that $\\Phi(K)$ is a neighbourhood of $a$ in $A$ and either \n    \\begin{enumerate}\n        \\item[(a)] $K\\subset\\mathbb{R}^2\\subset\\mathbb{R}^d$ is convex two-dimensional, or\n        \\item[(b)] $K=\\rho(B)$, where $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^d$ is a $\\operatorname{B}$-set and $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ is a multirotation associated with $T_1(B)$.\n    \\end{enumerate} \n\\end{corollary}\n\nIn the situation of Theorem~\\ref{T_main} (Theorem~L), we can easily observe that the mapping \n$\\Psi:=\\Phi\\circ\\rho|_B$ ($\\Psi:=\\Phi|_K$, respectively) is bi-Lipschitz. Using this, we obtain a positive answer to \\cite[Question~1.7]{Ly24}:\n\n\\begin{corollary}  \\label{bilip}\n    Let $A\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a two-dimensional set with positive reach and $a\\in A$. Then there exists a compact set $0\\in B\\subset\\mathbb{R}^2$ with positive reach and a bi-Lipschitz mapping $\\Psi:B\\to A^*$, where $A^*$ is a neighbourhood of $a$ in $A$.\n\\end{corollary}\n\nThe proof of Theorem~\\ref{T_main} has two principal ingredients which come from Lytchak's proof of Theorem~L.\n\nFirst, given a set $A\\subset\\mathbb{R}^d$ with positive reach and $1\\leq k\\leq d$, we consider the mapping $\\psi_k=\\psi_k^A:x\\mapsto\\operatorname{span}\\{\\operatorname{Tan}(A,x)\\}$ from $T_k(A)$ into the Grassmannian $G(d,k)$ and prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$. Second, using the Lipschitzness of $\\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney's $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\\subset\\mathbb{R}^d\\to\\mathbb{R}^d$. Here our approach differs from that of Lytchak who also uses the Whitney's $C^{1,1}$  extension, but for a mapping from $\\mathbb{R}^k$ to $\\mathbb{R}^{d-k}$. For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see \\cite{GJ90}), which is the third principal ingredient of our proof.\n\nWhen proving the Lipschitzness of $\\psi_k$, in contrast to Lytchak who applies the theory of CAT($\\kappa$) spaces, \nwe use a more elementary method based on ``tangential regularity'' of sets with positive reach and some well-known facts on ``fullness'' (very close to the more frequently used ``thickness'') of simplices taken from Whitney \\cite{Whi57}. \nSome additional arguments are needed to prove Lemma~\\ref{lipna2} about Lipschitzness of $\\psi_2$ for special two-dimensional sets  which is important in the proof of Theorem~\\ref{T_main}.\n\nUsing our method we also provide an alternative (more elementary) proof of Theorem~L. \n\nMoreover, our method also yields the local Lipschitzness of $\\psi_k$ on $T_k(A)$ for \\emph{any} set $A\\subset\\mathbb{R}^d$ with positive reach, which implies that $T_k(A)$ lies locally on a $k$-dimensional $C^{1,1}$-surface (Theorem~\\ref{tdmnapl2}). (Note that for $A$ $k$-dimensional, this follows already from Theorem~L.)\nIn the case $k=1$ we obtain (using a different method) a stronger version (Theorem~\\ref{T1+}) which will be used substantially in the proof of our main result (Theorem~\\ref{T_main}).\nTheorem~\\ref{tdmnapl2} easily implies that for $1\\leq k<d$, each $k$-dimensional subset of $\\mathbb{R}^d$ with positive reach can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces (see Corollary~\\ref{count}), which is new up to our knowledge. \n\nThe structure of the article is the following.\nIn Section~2 (Preliminaries), we fix some basic notation and recall some (mostly well-known) facts about $C^{1,1}$ mappings and sets with positive reach.\nSection~3 is devoted to the study of Lipschitzness of the mapping\n$\\psi_k$. \nIn Section~4 we prove results on the local structure of $T_k(A)$ (Theorems~\\ref{tdmnapl2} and \\ref{T1+}) and, as a consequence, Corollary~\\ref{count}.\nIn Section~5 we prove our main results, Theorem~\\ref{T_main} and Corollary~\\ref{bilip}.  \nFinally, in Section~6 we present an alternative proof of Theorem~L.",
    "sketch": "The post-theorem introduction explicitly outlines the proof of Theorem~\\ref{T_main} via “two principal ingredients” (adapted from Lytchak’s proof of Theorem~L) plus a “third principal ingredient.”\n\n1. **Lipschitz control of the tangent-span map.** For a set $A\\subset\\mathbb{R}^d$ with positive reach and $1\\le k\\le d$, consider\n\\[\n\\psi_k=\\psi_k^A:x\\mapsto \\operatorname{span}\\{\\operatorname{Tan}(A,x)\\}\\quad \\text{from }T_k(A)\\text{ into }G(d,k).\n\\]\nThey “prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$.” For this, “in contrast to Lytchak who applies the theory of CAT($\\kappa$) spaces,” they use “a more elementary method based on ‘tangential regularity’ of sets with positive reach” and facts on “fullness” of simplices from Whitney \\cite{Whi57}. They note “additional arguments are needed to prove Lemma~\\ref{lipna2} about Lipschitzness of $\\psi_2$ for special two-dimensional sets … important in the proof of Theorem~\\ref{T_main}.”\n\n2. **Build a $C^{1,1}$-diffeomorphism from Lipschitzness of $\\psi_2$.** “Using the Lipschitzness of $\\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney’s $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\\subset\\mathbb{R}^d\\to\\mathbb{R}^d$.” They emphasize their approach differs from Lytchak’s, who applies Whitney extension “for a mapping from $\\mathbb{R}^k$ to $\\mathbb{R}^{d-k}$.”\n\n3. **Product-integration input.** “For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see \\cite{GJ90}), which is the third principal ingredient of our proof.”\n\nAdditionally, the remark gives a proof detail for the “moreover” part: for $\\dim A\\le 2$ they “set in our proof $A:=\\Phi(\\rho(B))$,” while if one assumes $\\dim A=2$ they can take $A:=\\Phi(\\rho(B))\\cup D$ with ${\\rm reach}\\, D>0$, $\\dim D=2$, and $\\mathrm{dist}(\\Phi(\\rho(B)),D)>0$.",
    "expanded_sketch": "The post-theorem introduction explicitly outlines the proof of the main theorem via “two principal ingredients” (adapted from Lytchak’s proof of Theorem~L) plus a “third principal ingredient.”\n\n1. **Lipschitz control of the tangent-span map.** For a set $A\\subset\\mathbb{R}^d$ with positive reach and $1\\le k\\le d$, consider\n\\[\n\\psi_k=\\psi_k^A:x\\mapsto \\operatorname{span}\\{\\operatorname{Tan}(A,x)\\}\\quad \\text{from }T_k(A)\\text{ into }G(d,k).\n\\]\nThey “prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$.” For this, “in contrast to Lytchak who applies the theory of CAT($\\kappa$) spaces,” they use “a more elementary method based on ‘tangential regularity’ of sets with positive reach” and facts on “fullness” of simplices from Whitney (H. Whitney, *Geometric Integration Theory*, 1957). They note “additional arguments are needed” to obtain the following auxiliary lemma about Lipschitzness of $\\psi_2$ for special two-dimensional sets, which is important for establishing the main theorem.\n\n\\begin{lemma}\\label{lipna2}\n\\tLet  $A\\subset \\R^d$, $0<r< \\reach A$,\n\\t$\\diam A < r$ and \n\\t $A \\subset T_1^-(A)  \\cup   T_2(A)$. \n\\t Then  $\\psi_2:=\\psi_2^A$  is $(2^{12}\\pi/r)$-Lipschitz on $T_2(A)$.\n\\end{lemma}\n\n2. **Build a $C^{1,1}$-diffeomorphism from Lipschitzness of $\\psi_2$.** “Using the Lipschitzness of $\\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney’s $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\\subset\\mathbb{R}^d\\to\\mathbb{R}^d$.” They emphasize their approach differs from Lytchak’s, who applies Whitney extension “for a mapping from $\\mathbb{R}^k$ to $\\mathbb{R}^{d-k}$.”\n\n3. **Product-integration input.** “For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see G. Gripenberg and S.-O. Londen, *[title not provided]*, 1990), which is the third principal ingredient of our proof.”\n\nAdditionally, the remark gives a proof detail for the “moreover” part: for $\\dim A\\le 2$ they “set in our proof $A:=\\Phi(\\rho(B))$,” while if one assumes $\\dim A=2$ they can take $A:=\\Phi(\\rho(B))\\cup D$ with ${\\rm reach}\\, D>0$, $\\dim D=2$, and $\\mathrm{dist}(\\Phi(\\rho(B)),D)>0$.",
    "expanded_theorem": "\\label{T_main}\n    Let $A\\subset\\mathbb{R}^d$ ($d\\geq 3$) be a set with positive reach with $\\dim A\\leq 2$ and let $a\\in A$ be such that $\\dim(\\operatorname{Tan}(A,a))=1$. Then there exist a $\\operatorname{B}$-set $B\\subset\\mathbb{R}^2\\subset\\mathbb{R}^d$, a multirotation $\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\\Phi:U\\subset\\mathbb{R}^d\\to\\mathbb{R}^d$ such that $0\\in U$, $a=\\Phi(0)$, $\\rho(B)\\subset U$ and $\\Phi(\\rho(B))$ is a neighbourhood of $a$ in $A$.\n\n    Moreover, if $\\Phi$, $B$ and $\\rho$ are as above, then $\\Phi(\\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\\subset\\mathbb{R}^d$ with positive reach satisfying $\\dim A\\leq 2$ and $\\dim(\\operatorname{Tan}(A,a))=1$.",
    "theorem_type": [
      "Existence",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(A\\subset \\mathbb{R}^d\\) with \\(d\\ge 3\\) have positive reach, assume \\(\\dim A\\le 2\\), and let \\(a\\in A\\) satisfy \\(\\dim(\\operatorname{Tan}(A,a))=1\\). Here a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\) means either\n\\[\nB=\\{(x,y):x\\in[0,r],\\ \\psi(x)\\le y\\le \\varphi(x)\\}\n\\]\nfor some \\(r>0\\) and Lipschitz functions \\(\\psi\\le 0\\le \\varphi\\) on \\([0,r]\\), with \\(\\varphi\\) semiconcave on \\((0,r)\\), \\(\\psi\\) semiconvex on \\((0,r)\\), \\(\\varphi(0)=\\psi(0)=0\\), and \\(\\varphi'_+(0)=\\psi'_+(0)=0\\); or\n\\[\nB=\\{(x,y):x\\in[-r,r],\\ \\psi(x)\\le y\\le \\varphi(x)\\}\n\\]\nfor some \\(r>0\\) and Lipschitz functions \\(\\psi\\le 0\\le \\varphi\\) on \\([-r,r]\\), with \\(\\varphi\\) semiconcave on \\((-r,r)\\), \\(\\psi\\) semiconvex on \\((-r,r)\\), \\(\\varphi(0)=\\psi(0)=0\\), and \\(\\varphi'(0)=\\psi'(0)=0\\). Also, \\(T_1(B)\\) denotes the set of points of \\(B\\) where the tangent cone has dimension \\(1\\). A multirotation associated with \\(T_1(B)\\) is a map \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) such that, writing \\(\\pi\\) for the orthogonal projection onto \\(\\operatorname{span}\\{e_1\\}\\), one has \\(\\rho(x)=x\\) whenever \\(\\pi(x)\\in T_1(B)\\), and on each connected component \\(\\lambda\\) of \\(\\operatorname{span}\\{e_1\\}\\setminus T_1(B)\\), the restriction \\(\\rho|_{\\pi^{-1}(\\lambda)}\\) coincides with some fixed 2-rotation. Under these hypotheses, which conclusion holds about the local structure of \\(A\\) near \\(a\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exist a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\subset \\mathbb{R}^d\\), a multirotation \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) associated with \\(T_1(B)\\), and a \\(C^{1,1}\\)-diffeomorphism \\(\\Phi:U\\subset \\mathbb{R}^d\\to \\mathbb{R}^d\\) from an open set \\(U\\) such that \\(0\\in U\\), \\(\\Phi(0)=a\\), \\(\\rho(B)\\subset U\\), and \\(\\Phi(\\rho(B))\\) is a neighbourhood of \\(a\\) in \\(A\\). Moreover, for any such \\(\\Phi\\), \\(B\\), and \\(\\rho\\), there exists a set \\(A'\\subset \\mathbb{R}^d\\) of positive reach with \\(\\dim A'\\le 2\\) and \\(\\dim(\\operatorname{Tan}(A',a))=1\\) such that \\(\\Phi(\\rho(B))\\) agrees with \\(A'\\) on a neighbourhood of \\(a\\); equivalently, there is a neighbourhood \\(V\\) of \\(a\\) with \\(V\\cap \\Phi(\\rho(B))=V\\cap A'\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\subset \\mathbb{R}^d\\), a multirotation \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) associated with \\(T_1(B)\\), and a \\(C^{1,1}\\)-diffeomorphism \\(\\Phi:U\\subset \\mathbb{R}^d\\to \\mathbb{R}^d\\) from an open set \\(U\\) such that \\(0\\in U\\), \\(\\Phi(0)=a\\), \\(\\rho(B)\\subset U\\), and \\(\\Phi(\\rho(B))\\) is a neighbourhood of \\(a\\) in \\(A\\). Moreover, for any such \\(\\Phi\\), \\(B\\), and \\(\\rho\\), one may always take \\(A'=\\Phi(\\rho(B))\\), so that \\(\\Phi(\\rho(B))\\) itself has positive reach, \\(\\dim \\Phi(\\rho(B))\\le 2\\), and \\(\\dim(\\operatorname{Tan}(\\Phi(\\rho(B)),a))=1\\)."
        },
        {
          "label": "C",
          "text": "There exist a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\subset \\mathbb{R}^d\\), a multirotation \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) associated with \\(T_1(B)\\), and a \\(C^{1,1}\\)-diffeomorphism \\(\\Phi:U\\subset \\mathbb{R}^d\\to \\mathbb{R}^d\\) from an open set \\(U\\) such that \\(0\\in U\\), \\(\\Phi(0)=a\\), \\(\\rho(B)\\subset U\\), and \\(\\Phi(\\rho(B))\\subset A\\)."
        },
        {
          "label": "D",
          "text": "There exist a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\subset \\mathbb{R}^d\\), a multirotation \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) associated with \\(T_1(B)\\), and a \\(C^1\\)-diffeomorphism \\(\\Phi:U\\subset \\mathbb{R}^d\\to \\mathbb{R}^d\\) from an open set \\(U\\) such that \\(0\\in U\\), \\(\\Phi(0)=a\\), \\(\\rho(B)\\subset U\\), and \\(\\Phi(\\rho(B))\\) is a neighbourhood of \\(a\\) in \\(A\\). Moreover, for any such \\(\\Phi\\), \\(B\\), and \\(\\rho\\), there exists a set \\(A'\\subset \\mathbb{R}^d\\) of positive reach with \\(\\dim A'\\le 2\\) and \\(\\dim(\\operatorname{Tan}(A',a))=1\\) such that \\(\\Phi(\\rho(B))\\) agrees with \\(A'\\) on a neighbourhood of \\(a\\)."
        },
        {
          "label": "E",
          "text": "There exist a \\(\\operatorname{B}\\)-set \\(B\\subset \\mathbb{R}^2\\subset \\mathbb{R}^d\\), a multirotation \\(\\rho:\\mathbb{R}^d\\to\\mathbb{R}^d\\) associated with \\(T_2(B)\\), and a \\(C^{1,1}\\)-diffeomorphism \\(\\Phi:U\\subset \\mathbb{R}^d\\to \\mathbb{R}^d\\) from an open set \\(U\\) such that \\(0\\in U\\), \\(\\Phi(0)=a\\), \\(\\rho(B)\\subset U\\), and \\(\\Phi(\\rho(B))\\) is a neighbourhood of \\(a\\) in \\(A\\). Moreover, for any such \\(\\Phi\\), \\(B\\), and \\(\\rho\\), there exists a set \\(A'\\subset \\mathbb{R}^d\\) of positive reach with \\(\\dim A'\\le 2\\) and \\(\\dim(\\operatorname{Tan}(A',a))=1\\) such that \\(\\Phi(\\rho(B))\\) agrees with \\(A'\\) on a neighbourhood of \\(a\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "moreover-part local-vs-global positive reach of image",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped neighbourhood equality/converse local-model conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "required C^{1,1} regularity from Whitney C^{1,1} construction weakened to C^1",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "multirotation must be associated with T_1(B), not T_2(B)",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives assumptions and definitions but does not explicitly reveal the correct option. The answer must be identified by comparing subtle differences in the conclusion."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem sets up the exact hypotheses and asks which existence statement holds, so the correct choice is basically the theorem’s conclusion restated."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in meaningful technical ways (local vs global diffeomorphism, C^{1,1} vs C^1, dependence on T_1(B), weaker vs exact conclusion). However, the task is still mainly precise recognition/recall rather than genuinely generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically plausible: they modify regularity, domain/globality, dependence of the multirotation, or weaken the statement. These reflect realistic failure modes when recalling a technical theorem."
      },
      "total_score": 5,
      "overall_assessment": "Technically well-constructed with strong distractors and little answer leakage, but it is largely a direct theorem-restatement question rather than a genuine reasoning task."
    }
  },
  {
    "id": "2512.17702v1",
    "paper_link": "http://arxiv.org/abs/2512.17702v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm: vis}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, and $L\\ge1$. Suppose $K\\subset \\mathbb{R}^n$ is compact. Then, the value function $v$ of \\eqref{soc problem} is an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation \n$F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition (see Definition \\ref{def:visc} below) where\n\\begin{equation} \\label{def: F}\nF(p, M):=\\inf\\bigg\\{\\!-\\frac{1}{2}\\mathrm{tr}(Ma):\\, a\\succeq 0,\\, a p = 0,\\,  \\lambda_{(n-k)}(a)\\geq 1,\\, \\lambda_{(1)}(a)\\leq L\\bigg\\}.\n\\end{equation}\nSuppose, in addition, that there are $T_\\iota\\!:\\mathbb{R}^n\\to\\mathbb{R}^n$, $\\iota\\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\\subset \\accentset{\\circ}{T_\\iota(K)}$, for which $\\lim_{\\iota\\downarrow1} T_\\iota=I$. (Here, $I$ is the identity map on $\\mathbb{R}^n$.) Then, the upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.",
      "start_pos": 24181,
      "end_pos": 25199,
      "label": "thm: vis"
    },
    "ref_dict": {
      "soc problem": "\\begin{equation}\\label{soc problem}\nv(x) := \\sup_{\\mathrm{P} \\in \\mathcal{P}_x} \\mathrm{P}\\text{-ess} \\inf \\tau_K\\,;\n\\end{equation}",
      "subsec super": "\\begin{equation}\n0 < \\delta \\mathbb{E}\\left[Z(\\theta)\\,\\mathbf{1}_{\\{X(\\theta) \\in \\partial B_{\\varepsilon}(x)\\cap K\\}} \\right] \\leq \\mathbb{E}\\left[ Z(\\theta) \\int_{0}^{\\theta} \\nabla\\varphi(X(s))^{\\top} \\,\\mathrm{d}\\widetilde{X}(s) \\right] \\leq 0, \n\\end{equation}\na contradiction. The proof of the subsolution property is complete. \\ep\n\n\\subsection{Supersolution property of the value function} \\label{subsec super}\n\nSince $v\\ge 0$, it suffices to check the supersolution inequality \\eqref{eq: supers ineq} for $x\\in\\accentset{\\circ}{K}$. Fix any $x\\in\\accentset{\\circ}{K}$, and let $\\varphi\\in C^2(\\mathbb{R}^n)$ satisfy $\\varphi \\le v_*$ on $K$ and $\\varphi(x)=v_*(x)$. Since we can study $\\varphi-\\varepsilon|\\cdot-x|^2$ and then pass to the limit $\\varepsilon\\downarrow0$, we may assume  $\\varphi<v_*$ on $K\\backslash\\{x\\}$ and that $\\nabla^2 \\varphi(x)$ is non-singular. We distinguish two cases: $\\nabla\\varphi(x)\\neq0$ and $\\nabla\\varphi(x)=0$.\n\n\\medskip\n\n\\noindent\\textbf{Case 1: $\\nabla\\varphi(x)\\neq 0$.} In this case, $F^*(\\nabla\\varphi(x),\\nabla^2\\varphi(x))=F(\\nabla\\varphi(x),\\nabla^2\\varphi(x))$. We argue by contradiction and suppose that $F(\\nabla \\varphi(x),\\nabla^2\\varphi(x))<1$. Then, there exists an $a\\in{\\mathcal A}$ with \n\\begin{equation}\\label{case 1 a cond}\n1+\\frac{1}{2}\\text{tr}\\big(a\\nabla^2\\varphi(x)\\big)>0\\quad\\text{and}\\quad a\\nabla\\varphi(x) = 0.\n\\end{equation}",
      "thm: vis": "\\begin{thm} \\label{thm: vis}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, and $L\\ge1$. Suppose $K\\subset \\mathbb{R}^n$ is compact. Then, the value function $v$ of \\eqref{soc problem} is an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation \n$F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition (see Definition \\ref{def:visc} below) where\n\\begin{equation} \\label{def: F}\nF(p, M):=\\inf\\bigg\\{\\!-\\frac{1}{2}\\mathrm{tr}(Ma):\\, a\\succeq 0,\\, a p = 0,\\,  \\lambda_{(n-k)}(a)\\geq 1,\\, \\lambda_{(1)}(a)\\leq L\\bigg\\}.\n\\end{equation}\nSuppose, in addition, that there are $T_\\iota\\!:\\rr^n\\to\\rr^n$, $\\iota\\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\\subset \\accentset{\\circ}{T_\\iota(K)}$, for which $\\lim_{\\iota\\downarrow1} T_\\iota=I$. (Here, $I$ is the identity map on $\\rr^n$.) Then, the upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.\n\\end{thm}",
      "LSS cond": "\\begin{equation}\\label{LSS cond}\n\\lambda_{(n-k)}\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\ge 1,\\quad \\text{a.e. }t\\ge0\n\\end{equation}",
      "FK cond": "\\begin{equation}\\label{FK cond}\n\\sum_{i=1}^d  \\frac{1}{\\mu_i(t)}\\,\\frac{\\mathrm{d}{\\langle\\mu_i\\rangle}(t)}{\\mathrm{d}t}  \\ge C>0,\\quad t\\geq 0\n\\end{equation}",
      "LSS cond'": "\\begin{equation}\\label{LSS cond'}\n\\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. } t\\ge0,\n\\end{equation}",
      "subsec: sub": "\\begin{equation}\\label{n' SDE}\n\\mathrm{d}X_{[n']}(t) = a(X_{[n']}(t))^{1/2}\\,\\mathrm{d}{W(t)},\n\\end{equation}\nwhere $a(y) := I - yy^{\\top}/|y|^2$ when $y\\neq 0$, $a(0):=I$, and $W$ is an $n'$-dimensional standard Brownian motion. The remaining coordinates of $X$ are chosen to be constant. Then, $a^{1/2}$ is continuous on ${\\mathbb R}^{n'} \\setminus \\{0\\}$, and $a(y)^{1/2}y = 0$. Thus, for any $\\varepsilon\\in(0,|x_{[n']}|)$, by Itô's formula,\n\\[\n|X_{[n']}(t)|^2 = |x_{[n']}|^2 + (n-k)t,\\quad t\\le\\inf\\big\\{t'\\ge0:\\,|X_{[n']}(t')|\\le\\varepsilon\\big\\}.\n\\]\nConsequently, \\eqref{n' SDE} has a global weak solution satisfying $|X_{[n']}(t)|\\ge|x_{[n']}|$, $t\\ge0$. In addition, $\\tau_K = (r^2 - |x|^2) / (n-k)$ almost surely under $\\mathrm{P}^{\\ast}$.\n\n\\medskip\n\nFor $x=0$, consider a sequence $(x^m)_{m=1}^\\infty$ in $K\\backslash\\{0\\}$ going to $x$. By the compactness of $\\mathcal{P}_0$ (Lemma \\ref{lemma: compact}), the associated sequence $(\\mathrm{P}^{\\ast}_{x^m})_{m=1}^\\infty$ has a subsequence going to a $\\mathrm{P}^{\\ast}\\in\\mathcal{P}_0$. Then, it holds $|X(t)|^2 = (n-k) t$, $t \\geq 0$, thus $\\tau_K = r^2 / (n-k)$, almost surely under $\\mathrm{P}^{\\ast}$.\n\\end{proof}\n\n\\subsection{Subsolution property of the value function} \\label{subsec: sub}\n\nWe are now ready to verify that the value function $v$ of \\eqref{soc problem} satisfies Definition \\ref{def:visc}(a). Since $v^*=v$ (Proposition \\ref{prop: dpp}) and $F_*=F$ (Lemma \\ref{compute F}), we may replace $v^*$ by $v$ and $F_*$ by $F$ in Definition \\ref{def:visc}(a). Moreover, $v(x)>0$ for all $x\\in \\accentset{\\circ}{K}$ by Example \\ref{ex: ball}. Thus, we only need to study $x\\in K$ with $v(x)>0$. Let $\\varphi\\in C^2(\\mathbb{R}^n)$ be a test function such that $\\varphi\\ge v$ on $K$ and $\\varphi(x)=v(x)$. Since $\\varphi$ can be replaced by $\\varphi+|\\cdot-x|^4$, we may assume that $\\varphi>v$ on $K\\backslash\\{x\\}$.\n\n\\medskip\n\nWe prove the inequality \\eqref{eq: subs ineq} by contradiction. Suppose that $F(\\nabla\\varphi(x),\\nabla^2\\varphi(x))>1$. Then, we let \n\\begin{equation}\n\\mathcal{A}:=\\big\\{a\\succeq0:\\,\\lambda_{(n-k)}(a)\\ge1,\\,\\lambda_{(1)}(a)\\le L\\big\\}\n\\end{equation}",
      "def:visc": "\\begin{defn} \\label{def:visc}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, $L\\ge1$, and $K\\subset \\mathbb{R}^n$ be compact.\n\\begin{enumerate}[(a)]\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity subsolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: subs ineq}\nF_{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\leq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u^{\\ast}(x)>0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: subs ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity supersolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: supers ineq}\nF^{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\geq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u_{\\ast}(x)<0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: supers ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity solution} of $F(\\nabla u, \\nabla^2 u)=1$ on $K$ with \\textit{zero boundary condition} if it is a viscosity subsolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition and a viscosity supersolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition. \n\\end{enumerate}\n\\end{defn}",
      "lem: convex hull": "\\begin{lemma} \\label{lem: convex hull}\nLet $n\\ge2$ and $k\\in\\{1,2,\\ldots,n-1\\}$. Then, the convex hull of the set $\\{a \\in \\mathbb{S}_+^n \\!: \\lambda_{(n-k)}(a)\\geq 1\\}$ is\n\\begin{equation}\n\\big\\{a \\in \\mathbb{S}^n_+:\\,\\Pi_m(a)\\geq m-k \\text{ for }m=k+1,\\, k+2,\\,\\ldots,\\, n \\big\\}=:A.\n\\end{equation}\n\\end{lemma}",
      "LR cond": "\\begin{equation}\\label{LR cond}\n\\sum_{i=1}^d \\langle\\mu_i\\rangle(t)\\ge t,\\quad t\\geq 0\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 10080,
    "pre_theorem_intro_text": "\\label{sec1}\n\nIn \\cite[Section 3.3]{fernholz_stochastic_2002}, Fernholz has introduced the concept of portfolio domination, now more commonly referred to as \\textit{relative arbitrage}, to rigorously capture the notion of ``beating the market''. More specifically, consider a market with $d\\ge2$ assets whose vector $\\mu:=(\\mu_1,\\mu_2,\\ldots,\\mu_d)$ of market weights (i.e., market capitalizations as fractions of the total market capitalization) constitutes a continuous semimartingale with respect to some stochastic basis. A predictable $\\mu$-integrable process $\\theta$ is called a trading strategy. We refer to $V^{\\theta}:=\\theta^{\\top} \\mu := \\sum_{i=1}^d \\theta_i \\,\\mu_i$ as the value process of $\\theta$ relative to the market and only consider self-financing trading strategies, meaning:  \n\\begin{equation}\nV^{\\theta}(t) = V^\\theta(0)+\\int_0^t \\theta(s)^{\\top}\\,\\mathrm{d}\\mu(s), \\quad t\\geq 0.\n\\end{equation}\nTaking the market (i.e., the trading strategy $(1,1,\\ldots,1)$) as the benchmark, we can state the definition of relative arbitrage from the seminal monograph by Fernholz and Karatzas (\\cite[Definition 6.1]{karatzas_stochastic_2009}) as follows.\n\n\\begin{defn}\nA trading strategy $\\theta$ is called a \\textit{relative arbitrage} over a time horizon $[0, T]$ if its value process $V^{\\theta}$ relative to the market satisfies:\n\\begin{enumerate}[(a)]\n    \\item $V^\\theta\\ge0$ almost surely, \n \t\\item $V^{\\theta}(T) \\ge V^{\\theta}(0)$ almost surely,\n    \\item $V^{\\theta}(T) > V^{\\theta}(0) $ with positive probability.\n\\end{enumerate}\n\\end{defn}\n\n\\smallskip\n\nThe terms `trading strategy', `value process', `self-financing', and `arbitrage' used above\ntake on their customary meaning when one starts from a market without a bank account and employs the value of the portfolio continuously investing according to the market weights (`market portfolio') as the num\\'{e}raire. The latter is emphasized by the term `relative (to the market)'. \n\n\\medskip\n\nA central problem of stochastic portfolio theory, going back to the foundational work \\cite{fernholz_stochastic_2002} by Fernholz, is to identify classes of markets that admit relative arbitrages over suitable time horizons $[0,T]$. In particular, it is proven in \\cite[Example 3.3.3]{fernholz_stochastic_2002} that every market in which the smallest eigenvalue of the underlying assets' instantaneous covariation matrix is bounded away from $0$ and the largest market weight is bounded away from $1$ admits a relative arbitrage over all long enough time horizons $[0,T]$. Surprisingly, the existence of relative arbitrages for the same kind of markets has been shown to hold over \\textit{any} non-trivial time horizon in \\cite{fernholz_diversity_2005}. This phenomenon, now known as \\textit{short-term relative arbitrage}, has been demonstrated in \\cite{fernholz_relative_2005} also for the so-called volatility-stabilized markets. We refer further to \\cite{KR17}, \\cite{vovk18}, \\cite{Cu19}, \\cite{RX19}, \\cite{KK20}, \\cite{itkin25} for various extensions of these findings. \n\n\\medskip\n\nThe practically important distinction between relative arbitrage over long enough time horizons and short-term relative arbitrage naturally prompted the following question (see \\cite[Section 4]{BF08}). Does every sufficiently volatile market, in the sense of \n\\begin{equation}\\label{FK cond}\n\\sum_{i=1}^d  \\frac{1}{\\mu_i(t)}\\,\\frac{\\mathrm{d}{\\langle\\mu_i\\rangle}(t)}{\\mathrm{d}t}  \\ge C>0,\\quad t\\geq 0\n\\end{equation}\n-- a variant of the key assumption behind the construction in \\cite[Example 3.3.3]{fernholz_stochastic_2002} highlighted in \\cite[Section 3]{fernholz_relative_2005}\\footnote{Note that the left-hand side in \\eqref{FK cond} is precisely the `excess growth rate' in \\cite[display (3.1)]{fernholz_relative_2005}.}, admit a short-term relative arbitrage, and not only relative arbitrages over long enough time horizons? In another surprising twist, this question has been answered negatively in  \\cite[Section 6]{fernholz_volatility_2018}. Hence, the focus has shifted to finding the \\textit{smallest} $T^*>0$ such that a relative arbitrage over the time horizon $[0,T]$ is possible for any $T>T^*$ in every sufficiently volatile market, usually referred to as the \\textit{relative arbitrage problem}.      \n\n\\medskip\n\nThe relative arbitrage problem appears intractable at first glance, but two remarkable insights by Larsson and Ruf (see \\cite{larsson_relative_2021}) have allowed them to characterize $T^*$ when the sufficient volatility condition \\eqref{FK cond} is replaced by \n\\begin{equation}\\label{LR cond}\n\\sum_{i=1}^d \\langle\\mu_i\\rangle(t)\\ge t,\\quad t\\geq 0\n\\end{equation}\n(a prominent variant of \\eqref{FK cond}, see, e.g., \\cite[Example 5.5]{KR17} and apply a time change). Firstly, \\cite{larsson_relative_2021} uses the Fundamental Theorem of Asset Pricing to express $T^*$ through the value function of a stochastic optimal control problem. Secondly, \\cite{larsson_relative_2021} identifies the Hamilton-Jacobi-Bellman equation associated with the latter as the arrival time formulation of the minimum curvature flow, a geometric flow akin to the celebrated mean curvature flow (see \\cite{Mullins56}, \\cite{Brakke78}, \\cite{Huisken84}), on the probability simplex. The arrival time formulation of the minimum curvature flow on the probability simplex turns out to have a unique viscosity solution that eventually characterizes $T^*$ (see \\cite[Theorem~5.1]{larsson_relative_2021}). This resolves the relative arbitrage problem under the sufficient volatility condition \\eqref{LR cond}.\n\n\\medskip\n\nIn the setting of \\cite{larsson_relative_2021}, the process of market weights $\\mu$ \nfree of relative arbitrage over the time horizon $[0,T^*]$ is expected to have an instantaneous covariation matrix $\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$ generically of rank $1$  (cf.~\\cite[discussion following Remark 1.4]{larsson_minimum_2022}). At the same time, while instantaneous covariation matrices of large asset universes are commonly estimated with factor models in the empirical finance literature (see \\cite[Section 8]{johansson_simple_2023} for a concise overview) and the number of factors used -- equal to the number of dominant eigenvalues reliably estimated -- is indeed much smaller than $d$, this number tends to be much larger than $1$ (e.g., $20$ for $d=238$ in \\cite[Figure 8.1]{johansson_simple_2023}). For this reason, we amend the sufficient volatility condition \\eqref{LR cond} to \n\\begin{equation}\\label{LSS cond}\n\\lambda_{(n-k)}\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\ge 1,\\quad \\text{a.e. }t\\ge0\n\\end{equation}\nwhere $n:=d-1$, $k$ is any fixed element of $\\{1,2,\\ldots,n-1\\}$, and $\\lambda_{(n-k)}$ refers to the $(n-k)$-largest eigenvalue. A situation such as in \\cite[Figure 8.1]{johansson_simple_2023} then has $n=237$ and $k=217$ for example. To obtain a positive constant other than $1$ on the right-hand side of~\\eqref{LSS cond}, it suffices to apply a simple time change throughout. \n\n\\medskip\n\nThe ``null hypothesis'' \\eqref{LSS cond} is not closed with respect to the convergence in distribution of continuous processes because the lower bound of \\eqref{LSS cond} is not preserved under convex combinations of instantaneous covariation matrices. Therefore, we convexify \\eqref{LSS cond} (cf. Lemma \\ref{lem: convex hull} below): \n\\begin{equation}\\label{LSS cond'}\n\\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. } t\\ge0,\n\\end{equation}\nwhere $\\Pi_m(a):= \\inf\\{\\text{tr}(aP)\\!: P^2 = P,\\,\\text{tr}(P)=m\\}$ and $\\text{tr}$ stands for the trace of a square matrix. Hence, we study the relative arbitrage problem under the sufficient volatility condition \\eqref{LSS cond'} together with the technical condition \n$\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}\\le L$, $\\text{a.e. }t\\ge0$ for some fixed $L\\ge1$. The latter may be chosen large enough to accommodate a confidence interval around an empirical estimate of $\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$. The compactness of the resulting set of instantaneous covariation matrices allows us to establish the semicontinuity of the value function and the dynamic programming principle in the ensuing stochastic optimal control problem. \n\n\\medskip\n\nTo characterize $T^*$, we begin as in \\cite[Section 5]{larsson_relative_2021}. More specifically, we apply a linear transformation $U$ mapping the probability simplex isometrically onto a polytope $K\\subset\\mathbb{R}^n$. Then, an application of the Fundamental Theorem of Asset Pricing as in \\cite[proof of Theorem 3.1]{larsson_relative_2021} yields the representation $T^*=v(U\\mu(0))$, where  \n\\begin{equation}\\label{soc problem}\nv(x) := \\sup_{\\mathrm{P} \\in \\mathcal{P}_x} \\mathrm{P}\\text{-ess} \\inf \\tau_K\\,;\n\\end{equation}\n$\\mathcal{P}_x$ is the set of probability measures on $\\Omega:=C([0,\\infty),\\mathbb{R}^n)$, equipped with the Borel $\\sigma$-algebra for the topology of locally uniform convergence, under which the coordinate process $X$ is a martingale starting from $x$ and\n\\begin{equation}\\label{eq: constraint}\n\\begin{split}\n& \\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. }t\\ge0, \\\\\n& \\lambda_{(1)}\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\le L,\\quad \\text{a.e. } t\\ge0\n\\end{split}\n\\end{equation}\nhold almost surely; and \n\\begin{equation}\n\\tau_K := \\inf\\big\\{t \\geq 0:\\, X(t) \\notin K\\big\\}.\n\\end{equation}\nIn words: $v(x)$ is the largest, across all martingale laws $\\mathrm{P} \\in \\mathcal{P}_x$, deterministic almost sure lower bound on the exit time from $K$. The following theorem characterizes $v$, for \\textit{any compact} $K\\subset\\mathbb{R}^n$, and is our main result.",
    "context": "The relative arbitrage problem appears intractable at first glance, but two remarkable insights by Larsson and Ruf (see \\cite{larsson_relative_2021}) have allowed them to characterize $T^*$ when the sufficient volatility condition \\eqref{FK cond} is replaced by \n\\begin{equation}\\label{LR cond}\n\\sum_{i=1}^d \\langle\\mu_i\\rangle(t)\\ge t,\\quad t\\geq 0\n\\end{equation}\n(a prominent variant of \\eqref{FK cond}, see, e.g., \\cite[Example 5.5]{KR17} and apply a time change). Firstly, \\cite{larsson_relative_2021} uses the Fundamental Theorem of Asset Pricing to express $T^*$ through the value function of a stochastic optimal control problem. Secondly, \\cite{larsson_relative_2021} identifies the Hamilton-Jacobi-Bellman equation associated with the latter as the arrival time formulation of the minimum curvature flow, a geometric flow akin to the celebrated mean curvature flow (see \\cite{Mullins56}, \\cite{Brakke78}, \\cite{Huisken84}), on the probability simplex. The arrival time formulation of the minimum curvature flow on the probability simplex turns out to have a unique viscosity solution that eventually characterizes $T^*$ (see \\cite[Theorem~5.1]{larsson_relative_2021}). This resolves the relative arbitrage problem under the sufficient volatility condition \\eqref{LR cond}.\n\nIn the setting of \\cite{larsson_relative_2021}, the process of market weights $\\mu$ \nfree of relative arbitrage over the time horizon $[0,T^*]$ is expected to have an instantaneous covariation matrix $\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$ generically of rank $1$ (cf.~\\cite[discussion following Remark 1.4]{larsson_minimum_2022}). At the same time, while instantaneous covariation matrices of large asset universes are commonly estimated with factor models in the empirical finance literature (see \\cite[Section 8]{johansson_simple_2023} for a concise overview) and the number of factors used -- equal to the number of dominant eigenvalues reliably estimated -- is indeed much smaller than $d$, this number tends to be much larger than $1$ (e.g., $20$ for $d=238$ in \\cite[Figure 8.1]{johansson_simple_2023}). For this reason, we amend the sufficient volatility condition \\eqref{LR cond} to \n\\begin{equation}\\label{LSS cond}\n\\lambda_{(n-k)}\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\ge 1,\\quad \\text{a.e. }t\\ge0\n\\end{equation}\nwhere $n:=d-1$, $k$ is any fixed element of $\\{1,2,\\ldots,n-1\\}$, and $\\lambda_{(n-k)}$ refers to the $(n-k)$-largest eigenvalue. A situation such as in \\cite[Figure 8.1]{johansson_simple_2023} then has $n=237$ and $k=217$ for example. To obtain a positive constant other than $1$ on the right-hand side of~\\eqref{LSS cond}, it suffices to apply a simple time change throughout.\n\nThe ``null hypothesis'' \\eqref{LSS cond} is not closed with respect to the convergence in distribution of continuous processes because the lower bound of \\eqref{LSS cond} is not preserved under convex combinations of instantaneous covariation matrices. Therefore, we convexify \\eqref{LSS cond} (cf. Lemma \\ref{lem: convex hull} below): \n\\begin{equation}\\label{LSS cond'}\n\\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. } t\\ge0,\n\\end{equation}\nwhere $\\Pi_m(a):= \\inf\\{\\text{tr}(aP)\\!: P^2 = P,\\,\\text{tr}(P)=m\\}$ and $\\text{tr}$ stands for the trace of a square matrix. Hence, we study the relative arbitrage problem under the sufficient volatility condition \\eqref{LSS cond'} together with the technical condition \n$\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}\\le L$, $\\text{a.e. }t\\ge0$ for some fixed $L\\ge1$. The latter may be chosen large enough to accommodate a confidence interval around an empirical estimate of $\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$. The compactness of the resulting set of instantaneous covariation matrices allows us to establish the semicontinuity of the value function and the dynamic programming principle in the ensuing stochastic optimal control problem.\n\n\\medskip\n\nTo characterize $T^*$, we begin as in \\cite[Section 5]{larsson_relative_2021}. More specifically, we apply a linear transformation $U$ mapping the probability simplex isometrically onto a polytope $K\\subset\\mathbb{R}^n$. Then, an application of the Fundamental Theorem of Asset Pricing as in \\cite[proof of Theorem 3.1]{larsson_relative_2021} yields the representation $T^*=v(U\\mu(0))$, where \n\\begin{equation}\\label{soc problem}\nv(x) := \\sup_{\\mathrm{P} \\in \\mathcal{P}_x} \\mathrm{P}\\text{-ess} \\inf \\tau_K\\,;\n\\end{equation}\n$\\mathcal{P}_x$ is the set of probability measures on $\\Omega:=C([0,\\infty),\\mathbb{R}^n)$, equipped with the Borel $\\sigma$-algebra for the topology of locally uniform convergence, under which the coordinate process $X$ is a martingale starting from $x$ and\n\\begin{equation}\\label{eq: constraint}\n\\begin{split}\n& \\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. }t\\ge0, \\\\\n& \\lambda_{(1)}\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\le L,\\quad \\text{a.e. } t\\ge0\n\\end{split}\n\\end{equation}\nhold almost surely; and \n\\begin{equation}\n\\tau_K := \\inf\\big\\{t \\geq 0:\\, X(t) \\notin K\\big\\}.\n\\end{equation}\nIn words: $v(x)$ is the largest, across all martingale laws $\\mathrm{P} \\in \\mathcal{P}_x$, deterministic almost sure lower bound on the exit time from $K$. The following theorem characterizes $v$, for \\textit{any compact} $K\\subset\\mathbb{R}^n$, and is our main result.",
    "full_context": "The relative arbitrage problem appears intractable at first glance, but two remarkable insights by Larsson and Ruf (see \\cite{larsson_relative_2021}) have allowed them to characterize $T^*$ when the sufficient volatility condition \\eqref{FK cond} is replaced by \n\\begin{equation}\\label{LR cond}\n\\sum_{i=1}^d \\langle\\mu_i\\rangle(t)\\ge t,\\quad t\\geq 0\n\\end{equation}\n(a prominent variant of \\eqref{FK cond}, see, e.g., \\cite[Example 5.5]{KR17} and apply a time change). Firstly, \\cite{larsson_relative_2021} uses the Fundamental Theorem of Asset Pricing to express $T^*$ through the value function of a stochastic optimal control problem. Secondly, \\cite{larsson_relative_2021} identifies the Hamilton-Jacobi-Bellman equation associated with the latter as the arrival time formulation of the minimum curvature flow, a geometric flow akin to the celebrated mean curvature flow (see \\cite{Mullins56}, \\cite{Brakke78}, \\cite{Huisken84}), on the probability simplex. The arrival time formulation of the minimum curvature flow on the probability simplex turns out to have a unique viscosity solution that eventually characterizes $T^*$ (see \\cite[Theorem~5.1]{larsson_relative_2021}). This resolves the relative arbitrage problem under the sufficient volatility condition \\eqref{LR cond}.\n\nIn the setting of \\cite{larsson_relative_2021}, the process of market weights $\\mu$ \nfree of relative arbitrage over the time horizon $[0,T^*]$ is expected to have an instantaneous covariation matrix $\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$ generically of rank $1$ (cf.~\\cite[discussion following Remark 1.4]{larsson_minimum_2022}). At the same time, while instantaneous covariation matrices of large asset universes are commonly estimated with factor models in the empirical finance literature (see \\cite[Section 8]{johansson_simple_2023} for a concise overview) and the number of factors used -- equal to the number of dominant eigenvalues reliably estimated -- is indeed much smaller than $d$, this number tends to be much larger than $1$ (e.g., $20$ for $d=238$ in \\cite[Figure 8.1]{johansson_simple_2023}). For this reason, we amend the sufficient volatility condition \\eqref{LR cond} to \n\\begin{equation}\\label{LSS cond}\n\\lambda_{(n-k)}\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\ge 1,\\quad \\text{a.e. }t\\ge0\n\\end{equation}\nwhere $n:=d-1$, $k$ is any fixed element of $\\{1,2,\\ldots,n-1\\}$, and $\\lambda_{(n-k)}$ refers to the $(n-k)$-largest eigenvalue. A situation such as in \\cite[Figure 8.1]{johansson_simple_2023} then has $n=237$ and $k=217$ for example. To obtain a positive constant other than $1$ on the right-hand side of~\\eqref{LSS cond}, it suffices to apply a simple time change throughout.\n\nThe ``null hypothesis'' \\eqref{LSS cond} is not closed with respect to the convergence in distribution of continuous processes because the lower bound of \\eqref{LSS cond} is not preserved under convex combinations of instantaneous covariation matrices. Therefore, we convexify \\eqref{LSS cond} (cf. Lemma \\ref{lem: convex hull} below): \n\\begin{equation}\\label{LSS cond'}\n\\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le d}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. } t\\ge0,\n\\end{equation}\nwhere $\\Pi_m(a):= \\inf\\{\\text{tr}(aP)\\!: P^2 = P,\\,\\text{tr}(P)=m\\}$ and $\\text{tr}$ stands for the trace of a square matrix. Hence, we study the relative arbitrage problem under the sufficient volatility condition \\eqref{LSS cond'} together with the technical condition \n$\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle(t)}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}\\le L$, $\\text{a.e. }t\\ge0$ for some fixed $L\\ge1$. The latter may be chosen large enough to accommodate a confidence interval around an empirical estimate of $\\lambda_{(1)}\\big(\\frac{\\mathrm{d}\\langle \\mu_i,\\mu_j\\rangle}{\\mathrm{d}t}\\big)_{1\\le i,j\\le d}$. The compactness of the resulting set of instantaneous covariation matrices allows us to establish the semicontinuity of the value function and the dynamic programming principle in the ensuing stochastic optimal control problem.\n\n\\medskip\n\nTo characterize $T^*$, we begin as in \\cite[Section 5]{larsson_relative_2021}. More specifically, we apply a linear transformation $U$ mapping the probability simplex isometrically onto a polytope $K\\subset\\mathbb{R}^n$. Then, an application of the Fundamental Theorem of Asset Pricing as in \\cite[proof of Theorem 3.1]{larsson_relative_2021} yields the representation $T^*=v(U\\mu(0))$, where \n\\begin{equation}\\label{soc problem}\nv(x) := \\sup_{\\mathrm{P} \\in \\mathcal{P}_x} \\mathrm{P}\\text{-ess} \\inf \\tau_K\\,;\n\\end{equation}\n$\\mathcal{P}_x$ is the set of probability measures on $\\Omega:=C([0,\\infty),\\mathbb{R}^n)$, equipped with the Borel $\\sigma$-algebra for the topology of locally uniform convergence, under which the coordinate process $X$ is a martingale starting from $x$ and\n\\begin{equation}\\label{eq: constraint}\n\\begin{split}\n& \\Pi_m\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\geq m-k \\quad\\text{for}\\quad m = k+1,\\, k+2,\\,\\ldots,\\, n,\\quad \\text{a.e. }t\\ge0, \\\\\n& \\lambda_{(1)}\\bigg(\\frac{\\mathrm{d}\\langle X_i,X_j\\rangle(t)}{\\mathrm{d}t}\\bigg)_{1\\le i,j\\le n}\\le L,\\quad \\text{a.e. } t\\ge0\n\\end{split}\n\\end{equation}\nhold almost surely; and \n\\begin{equation}\n\\tau_K := \\inf\\big\\{t \\geq 0:\\, X(t) \\notin K\\big\\}.\n\\end{equation}\nIn words: $v(x)$ is the largest, across all martingale laws $\\mathrm{P} \\in \\mathcal{P}_x$, deterministic almost sure lower bound on the exit time from $K$. The following theorem characterizes $v$, for \\textit{any compact} $K\\subset\\mathbb{R}^n$, and is our main result.\n\n\\begin{rmk}\n\\begin{enumerate}[(a)]\n\\item Theorem \\ref{thm: vis} characterizes, in particular, the solution $T^*$ of the described relative arbitrage problem via the representation $T^*=v(U\\mu(0))$. \\vskip 5.5pt\n\\item The nonlinearity $F$ is `geometric', as defined in \\cite{BSS}, i.e., for any $p \\in \\rr^n$, symmetric $n\\times n$ matrix $M$, $c_1>0$, and $c_2\\in\\rr$,\n\\begin{equation}\nF(c_1 p, c_1 M + c_2pp^\\top) = c_1 F(p,M).\n\\end{equation}\nParabolic equations with such nonlinearities appear in weak formulations of geometric flows, and the corresponding viscosity theory was first developed in \\cite{cgg,evans_motion_1991,son} for the classical mean curvature flow, and then extended in \\cite{BSS,ambrosio_level_1996}. \\vskip 5.5pt\n\\item When $L=1$, the partial differential equation $F(\\nabla v,\\nabla^2 v)=1$ with zero boundary condition becomes the arrival time formulation of a co-dimension mean curvature flow from \\cite{ambrosio_level_1996}. For a related but different stochastic representation of these geometric flows we refer to \\cite{SoTo}. \\vskip 5.5pt\n\\item In view of the right-hand side of \\eqref{soc problem}, it is natural to conjecture that the value function $v$ does not depend on $L$, at least when $K$ is convex. We were not able to show this and leave it as a tantalizing open problem. \n\\end{enumerate}\n\\end{rmk}\n\n\\begin{prop}\\label{prop: dpp}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, and $L\\ge1$. Suppose $K\\subset \\mathbb{R}^n$ is compact. Then, the value function $v$ from \\eqref{soc problem} is upper semicontinuous on $\\mathbb{R}^n$. Moreover, it satisfies the following dynamic programming principle: For any $x\\in\\mathbb{R}^n$ and any stopping time $\\theta$ with respect to the filtration generated by the coordinate process $X$,\n\\begin{equation}\\label{eq: DPP}\nv(x) = \\sup_{\\mathrm{P}\\in\\mathcal{P}_x} \\mathrm{P}\\text{-}\\mathrm{ess} \\inf \\, \\big(\\theta\\wedge\\tau_K + v(X(\\theta))\\,\\mathbf{1}_{\\{\\theta\\leq \\tau_K\\}}\\big).\n\\end{equation}\nIn addition, the supremum in \\eqref{eq: DPP} is attained by any optimizer $\\mathrm{P}\\in \\mathcal{P}_x$ in \\eqref{soc problem}.\n\\end{prop}\n\nThis section is devoted to verifying that the value function $v$ of \\eqref{soc problem} is a viscosity solution to $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition, where \n\\begin{equation} \\label{codim k pde}\nF(p, M):=\\inf\\bigg\\{\\!-\\frac{1}{2}\\mathrm{tr}(Ma):\\, a\\succeq 0,\\, a p = 0,\\, \\lambda_{(n-k)}(a)\\geq 1,\\, \\lambda_{(1)}(a)\\leq L\\bigg\\}.\n\\end{equation}\n\n\\begin{defn} \\label{def:visc}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, $L\\ge1$, and $K\\subset \\mathbb{R}^n$ be compact.\n\\begin{enumerate}[(a)]\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity subsolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: subs ineq}\nF_{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\leq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u^{\\ast}(x)>0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: subs ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity supersolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: supers ineq}\nF^{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\geq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u_{\\ast}(x)<0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: supers ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity solution} of $F(\\nabla u, \\nabla^2 u)=1$ on $K$ with \\textit{zero boundary condition} if it is a viscosity subsolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition and a viscosity supersolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition. \n\\end{enumerate}\n\\end{defn}\n\n\\begin{prop} \\label{prop: uniq}\nIn the setting of Theorem \\ref{thm: vis}, suppose that there are $T_\\iota\\!:\\rr^n\\to\\rr^n$, $\\iota\\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\\subset \\accentset{\\circ}{T_\\iota(K)}$, for which $\\lim_{\\iota\\downarrow1} T_\\iota=I$. Then, the upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2v)=1$ on $K$ with zero boundary condition is unique. \n\\end{prop}\n\n\\begin{thm}[Comparison Principle] \\label{thm: comparison}\nIn the setting of Theorem \\ref{thm: vis}, suppose that there are $T_\\iota\\!:\\rr^n\\to\\rr^n$, $\\iota\\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\\subset \\accentset{\\circ}{T_\\iota(K)}$, for which $\\lim_{\\iota\\downarrow1} T_\\iota=I$. Then, for any upper semicontinuous viscosity subsolution $u$ of $F(\\nabla u,\\nabla^2u)=1$ on $K$ satisfying the zero boundary condition and any lower semicontinuous viscosity supersolution $w$ of $F(\\nabla w,\\nabla^2w)=1$ on $K$ satisfying the zero boundary condition, it holds $u\\le w^*$.\n\\end{thm}\n\n\\noindent\\textbf{Proof.} Consider $w^\\iota\\!:T_\\iota(K)\\to\\rr$, $x\\mapsto w(T_\\iota^{-1}x)$. We aim to show that $c_\\iota^2 w^\\iota$ is a lower semicontinuous viscosity supersolution of $F(\\nabla w,\\nabla^2w)=1$ on $T_\\iota(K)$ satisfying the zero boundary condition, where $c_\\iota>0$ is the dilation factor in $T_\\iota$. To this end, we claim that for any orthogonal $n\\times n$ matrix $O$ and for all $(p,M)\\in\\rr^n\\times\\mathbb{S}^n$: \n\\begin{equation} \\label{F transf}\nF(p, M) = c_\\iota^2\\, F(O^\\top p,\\, c_\\iota^{-2} O^{\\top} M O).\n\\end{equation}\nIndeed, let $\\overline{a}\\succeq 0$ with $\\overline{a}O^\\top p=0$,\n$\\lambda_{(n-k)}(\\overline{a})\\geq 1$ and $\\lambda_{(1)}(\\overline{a})\\leq L$. Then, $a:=O\\overline{a}O^\\top\\succeq0$ satisfies $ap=0$, $\\lambda_{(n-k)}(a)\\geq 1$ and $\\lambda_{(1)}(a)\\leq L$. Thus, the definition of $F$ in \\eqref{def: F} yields\n\\begin{equation} \\label{F transf ineq}\nF(p, M) \\le - \\frac{1}{2} \\mathrm{tr}(Ma) = - \\frac{c_\\iota^2}{2}\\,\\mathrm{tr}(c_\\iota^{-2}O^\\top M O\\overline{a}).\n\\end{equation}\nTaking the infimum over all $\\overline{a}$ as described, we arrive at \\eqref{F transf} with ``$\\le$''. Conversely, we can use the obtained inequality with $c_\\iota$, $O$, and $(p,M)$ replaced by $c_\\iota^{-1}$, $O^\\top$, and $(O^{\\top} p,\\, c_\\iota^{-2} O^{\\top} M O)$, respectively, to find that\n\\begin{equation}\nF(O^{\\top} p,\\, c_\\iota^{-2} O^{\\top} M O) \n\\le c_\\iota^{-2}\\, F(O O^{\\top} p,\\, \nc_\\iota^2 Oc_\\iota^{-2} O^{\\top} M O O^\\top)=c_\\iota^{-2}\\, F(p, M).\n\\end{equation}",
    "post_theorem_intro_text_len": 2264,
    "post_theorem_intro_text": "\\begin{rmk}\n\\begin{enumerate}[(a)]\n\\item Theorem \\ref{thm: vis} characterizes, in particular, the solution $T^*$ of the described relative arbitrage problem via the representation $T^*=v(U\\mu(0))$. \\vskip 5.5pt\n\\item The nonlinearity $F$ is `geometric', as defined in \\cite{BSS}, i.e., for any $p \\in \\mathbb{R}^n$, symmetric $n\\times n$ matrix $M$, $c_1>0$, and $c_2\\in\\mathbb{R}$,\n\\begin{equation}\nF(c_1 p, c_1 M + c_2pp^\\top) = c_1 F(p,M).\n\\end{equation}\nParabolic equations with such nonlinearities appear in weak formulations of geometric flows, and the corresponding viscosity theory was first developed in \\cite{cgg,evans_motion_1991,son} for the classical mean curvature flow, and then extended in \\cite{BSS,ambrosio_level_1996}. \\vskip 5.5pt\n\\item When $L=1$, the partial differential equation $F(\\nabla v,\\nabla^2 v)=1$ with zero boundary condition becomes the arrival time formulation of a co-dimension mean curvature flow from \\cite{ambrosio_level_1996}. For a related but different stochastic representation of these geometric flows we refer to \\cite{SoTo}. \\vskip 5.5pt\n\\item In view of the right-hand side of \\eqref{soc problem}, it is natural to conjecture that the value function $v$ does not depend on $L$, at least when $K$ is convex. We were not able to show this and leave it as a tantalizing open problem. \n\\end{enumerate}\n\\end{rmk}\n\nThe remainder of the paper is structured as follows. In Section \\ref{sec2}, we show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies. In Subsections \\ref{subsec: sub} and \\ref{subsec super} of Section \\ref{sec3}, we establish the viscosity subsolution and supersolution properties of $v$, respectively. Section \\ref{sec4} is then devoted to the uniqueness of the upper semicontinuous viscosity solution under the additional assumption in Theorem~\\ref{thm: vis}, finishing the proof of the latter. Finally, Section \\ref{sec5} examines the continuity of the value function $v$, particularly in the case that $K$ is a polytope as in the setting of stochastic portfolio theory.   \n\n\\medskip\n\n\\noindent\\textbf{Acknowledgement.} The authors would like to thank Martin Larsson and Johannes Ruf for many enlightening discussions on the subject of the paper.",
    "sketch": "In view of the paper structure described after Theorem~\\ref{thm: vis}, the proof proceeds as follows: (i) in Section~\\ref{sec2} the authors \"show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies\"; (ii) in Section~\\ref{sec3} they \"establish the viscosity subsolution and supersolution properties of $v$\" in Subsections~\\ref{subsec: sub} and~\\ref{subsec super}, respectively; (iii) Section~\\ref{sec4} proves \"the uniqueness of the upper semicontinuous viscosity solution under the additional assumption in Theorem~\\ref{thm: vis}, finishing the proof of the latter.\"",
    "expanded_sketch": "In view of the paper structure described after Theorem~\\ref{thm: vis}, the proof proceeds as follows: (i) next the authors show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies; (ii) they then establish the viscosity subsolution and supersolution properties of $v$ in the following two subsections.\n\n\\begin{equation}\\label{n' SDE}\n\\mathrm{d}X_{[n']}(t) = a(X_{[n']}(t))^{1/2}\\,\\mathrm{d}{W(t)},\n\\end{equation}\nwhere $a(y) := I - yy^{\\top}/|y|^2$ when $y\\neq 0$, $a(0):=I$, and $W$ is an $n'$-dimensional standard Brownian motion. The remaining coordinates of $X$ are chosen to be constant. Then, $a^{1/2}$ is continuous on ${\\mathbb R}^{n'} \\setminus \\{0\\}$, and $a(y)^{1/2}y = 0$. Thus, for any $\\varepsilon\\in(0,|x_{[n']}|)$, by It\u0006f4's formula,\n\\[\n|X_{[n']}(t)|^2 = |x_{[n']}|^2 + (n-k)t,\\quad t\\le\\inf\\big\\{t'\\ge0:\\,|X_{[n']}(t')|\\le\\varepsilon\\big\\}.\n\\]\nConsequently, \\eqref{n' SDE} has a global weak solution satisfying $|X_{[n']}(t)|\\ge|x_{[n']}|$, $t\\ge0$. In addition, $\\tau_K = (r^2 - |x|^2) / (n-k)$ almost surely under $\\mathrm{P}^{\\ast}$.\n\n\\medskip\n\nFor $x=0$, consider a sequence $(x^m)_{m=1}^\\infty$ in $K\\backslash\\{0\\}$ going to $x$. By the compactness of $\\mathcal{P}_0$ (Lemma \\ref{lemma: compact}), the associated sequence $(\\mathrm{P}^{\\ast}_{x^m})_{m=1}^\\infty$ has a subsequence going to a $\\mathrm{P}^{\\ast}\\in\\mathcal{P}_0$. Then, it holds $|X(t)|^2 = (n-k) t$, $t \\geq 0$, thus $\\tau_K = r^2 / (n-k)$, almost surely under $\\mathrm{P}^{\\ast}$.\n\\end{proof}\n\n\\subsection{Subsolution property of the value function} \\label{subsec: sub}\n\nWe are now ready to verify that the value function $v$ of \\eqref{soc problem} satisfies Definition \\ref{def:visc}(a). Since $v^*=v$ (Proposition \\ref{prop: dpp}) and $F_*=F$ (Lemma \\ref{compute F}), we may replace $v^*$ by $v$ and $F_*$ by $F$ in Definition \\ref{def:visc}(a). Moreover, $v(x)>0$ for all $x\\in \\accentset{\\circ}{K}$ by Example \\ref{ex: ball}. Thus, we only need to study $x\\in K$ with $v(x)>0$. Let $\\varphi\\in C^2(\\mathbb{R}^n)$ be a test function such that $\\varphi\\ge v$ on $K$ and $\\varphi(x)=v(x)$. Since $\\varphi$ can be replaced by $\\varphi+|\\cdot-x|^4$, we may assume that $\\varphi>v$ on $K\\backslash\\{x\\}$.\n\n\\medskip\n\nWe prove the inequality \\eqref{eq: subs ineq} by contradiction. Suppose that $F(\\nabla\\varphi(x),\\nabla^2\\varphi(x))>1$. Then, we let \n\\begin{equation}\n\\mathcal{A}:=\\big\\{a\\succeq0:\\,\\lambda_{(n-k)}(a)\\ge1,\\,\\lambda_{(1)}(a)\\le L\\big\\}\n\\end{equation}\n\n\\begin{equation}\n0 < \\delta \\mathbb{E}\\left[Z(\\theta)\\,\\mathbf{1}_{\\{X(\\theta) \\in \\partial B_{\\varepsilon}(x)\\cap K\\}} \\right] \\leq \\mathbb{E}\\left[ Z(\\theta) \\int_{0}^{\\theta} \\nabla\\varphi(X(s))^{\\top} \\,\\mathrm{d}\\widetilde{X}(s) \\right] \\leq 0, \n\\end{equation}\na contradiction. The proof of the subsolution property is complete. \\ep\n\n\\subsection{Supersolution property of the value function} \\label{subsec super}\n\nSince $v\\ge 0$, it suffices to check the supersolution inequality \\eqref{eq: supers ineq} for $x\\in\\accentset{\\circ}{K}$. Fix any $x\\in\\accentset{\\circ}{K}$, and let $\\varphi\\in C^2(\\mathbb{R}^n)$ satisfy $\\varphi \\le v_*$ on $K$ and $\\varphi(x)=v_*(x)$. Since we can study $\\varphi-\\varepsilon|\\cdot-x|^2$ and then pass to the limit $\\varepsilon\\downarrow0$, we may assume  $\\varphi<v_*$ on $K\\backslash\\{x\\}$ and that $\\nabla^2 \\varphi(x)$ is non-singular. We distinguish two cases: $\\nabla\\varphi(x)\\neq0$ and $\\nabla\\varphi(x)=0$.\n\n\\medskip\n\n\\noindent\\textbf{Case 1: $\\nabla\\varphi(x)\\neq 0$.} In this case, $F^*(\\nabla\\varphi(x),\\nabla^2\\varphi(x))=F(\\nabla\\varphi(x),\\nabla^2\\varphi(x))$. We argue by contradiction and suppose that $F(\\nabla \\varphi(x),\\nabla^2\\varphi(x))<1$. Then, there exists an $a\\in{\\mathcal A}$ with \n\\begin{equation}\\label{case 1 a cond}\n1+\\frac{1}{2}\\text{tr}\\big(a\\nabla^2\\varphi(x)\\big)>0\\quad\\text{and}\\quad a\\nabla\\varphi(x) = 0.\n\\end{equation}\n\n(iii) Finally, under the additional assumption stated in the main theorem, they prove uniqueness of the upper semicontinuous viscosity solution, thereby completing the proof of the main theorem.",
    "expanded_theorem": "\\label{thm: vis}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, and $L\\ge1$. Suppose $K\\subset \\mathbb{R}^n$ is compact. Then, the value function $v$ of\n\\begin{equation}\\label{soc problem}\nv(x) := \\sup_{\\mathrm{P} \\in \\mathcal{P}_x} \\mathrm{P}\\text{-ess} \\inf \\tau_K\\,;\n\\end{equation}\nis an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation \n$F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition, in the following sense.\n\n\\begin{defn} \\label{def:visc}\nLet $n\\geq 2$, $k\\in\\{1,2,\\ldots,n-1\\}$, $L\\ge1$, and $K\\subset \\mathbb{R}^n$ be compact.\n\\begin{enumerate}[(a)]\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity subsolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: subs ineq}\nF_{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\leq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u^{\\ast}(x)>0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u^{\\ast}-\\varphi)(x)=\\max_K (u^{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: subs ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity supersolution} of $F(\\nabla u, \\nabla^2 u)=1$ in $\\accentset{\\circ}{K}$ if for any $x\\in\\accentset{\\circ}{K}$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, it holds\n\\begin{equation}\\label{eq: supers ineq}\nF^{\\ast}(\\nabla \\varphi(x), \\nabla^2 \\varphi(x))\\geq 1.\n\\end{equation}\nThe function $u$ satisfies the \\textit{zero boundary condition} if for any $x\\in\\partial K$ with $u_{\\ast}(x)<0$ and $\\varphi\\in C^2(\\mathbb{R}^n)$ such that $(u_{\\ast}-\\varphi)(x)=\\min_K (u_{\\ast}-\\varphi)$, one has the inequality \\eqref{eq: supers ineq}. \\\\\n\\item A bounded function $u\\!: K \\to \\mathbb{R}$ is a \\textit{viscosity solution} of $F(\\nabla u, \\nabla^2 u)=1$ on $K$ with \\textit{zero boundary condition} if it is a viscosity subsolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition and a viscosity supersolution in $\\accentset{\\circ}{K}$ satisfying the zero boundary condition. \n\\end{enumerate}\n\\end{defn}\n\nHere\n\\begin{equation} \\label{def: F}\nF(p, M):=\\inf\\bigg\\{\\!-\\frac{1}{2}\\mathrm{tr}(Ma):\\, a\\succeq 0,\\, a p = 0,\\,  \\lambda_{(n-k)}(a)\\geq 1,\\, \\lambda_{(1)}(a)\\leq L\\bigg\\}.\n\\end{equation}\nSuppose, in addition, that there are $T_\\iota\\!:\\mathbb{R}^n\\to\\mathbb{R}^n$, $\\iota\\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\\subset \\accentset{\\circ}{T_\\iota(K)}$, for which $\\lim_{\\iota\\downarrow1} T_\\iota=I$. (Here, $I$ is the identity map on $\\mathbb{R}^n$.) Then, the upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.,",
    "theorem_type": [
      "Implication",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let $n\\ge 2$, $k\\in\\{1,\\dots,n-1\\}$, $L\\ge 1$, and let $K\\subset\\mathbb{R}^n$ be compact. For $p\\in\\mathbb{R}^n$ and $M\\in\\mathbb{S}^n$, define\n\\[\nF(p,M):=\\inf\\left\\{-\\frac12\\operatorname{tr}(Ma):\\ a\\succeq 0,\\ ap=0,\\ \\lambda_{(n-k)}(a)\\ge 1,\\ \\lambda_{(1)}(a)\\le L\\right\\},\n\\]\nwhere $\\lambda_{(j)}(a)$ denotes the $j$-th largest eigenvalue of $a$. Let\n\\[\nv(x):=\\sup_{\\mathrm P\\in\\mathcal P_x}\\ \\mathrm P\\text{-ess}\\inf\\tau_K.\n\\]\nAssume additionally that for each $\\iota\\in(1,2]$ there is a map $T_\\iota:\\mathbb{R}^n\\to\\mathbb{R}^n$, obtained by composing a rotation, a dilation, and a translation, such that $K\\subset \\operatorname{int}(T_\\iota(K))$ and $T_\\iota\\to I$ as $\\iota\\downarrow 1$, where $I$ is the identity map. Which statement holds about existence and uniqueness for the problem $F(\\nabla u,\\nabla^2u)=1$ on $K$ with zero boundary condition in the viscosity sense?",
      "correct_choice": {
        "label": "A",
        "text": "The value function $v$ is an upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition, and it is the unique upper semicontinuous viscosity solution of that boundary value problem on $K$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The value function $v$ is a continuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $\\operatorname{int}(K)$ with the classical boundary condition $v=0$ on $\\partial K$, and under the stated assumption on $T_\\iota$ it is the unique continuous viscosity solution of this boundary value problem."
        },
        {
          "label": "C",
          "text": "The value function $v$ is an upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition."
        },
        {
          "label": "D",
          "text": "The value function $v$ is an upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition, and under the stated assumption on $T_\\iota$ it is the unique viscosity solution of that boundary value problem among all bounded viscosity solutions on $K$."
        },
        {
          "label": "E",
          "text": "The value function $v$ is an upper semicontinuous viscosity solution of $F(\\nabla v,\\nabla^2 v)=1$ on $K$ with zero boundary condition, and the same conclusion remains valid without the assumption that $K\\subset \\operatorname{int}(T_\\iota(K))$ and $T_\\iota\\to I$ as $\\iota\\downarrow1$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
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          "D",
          "E"
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      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "upper_semicontinuous_zero_boundary_vs_continuous_classical_boundary",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_uniqueness_conclusion_under_extra_T_iota_assumption",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniqueness_only_for_upper_semicontinuous_solutions",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "necessity_of_T_iota_domain_expansion_hypothesis_for_comparison_uniqueness",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives the full PDE setup and hypotheses, but the correct conclusion is not stated outright; the test-taker still has to distinguish among several nearby existence/uniqueness claims."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to theorem-statement recall: the hypotheses and conclusion align very directly with a specific result. However, it is not a pure verbatim restatement, since the options differ in regularity, scope of uniqueness, and necessity of the geometric assumption."
      },
      "GPS": {
        "score": 1,
        "justification": "The item requires some reasoning about subtle distinctions (upper semicontinuous vs continuous solutions, uniqueness class, and role of the dilation/rotation hypothesis), but it mainly tests recognition of the exact theorem rather than deeper derivation or synthesis."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and target common overstatements, such as demanding continuity or dropping a needed hypothesis. But choice C is a weaker true statement implied by A, so the options are not fully mutually exclusive, which weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-targeted theorem-recognition MCQ with plausible distractors, but it is somewhat theorem-restatement-heavy and weakened by the inclusion of a weaker true option alongside the intended correct answer."
    }
  },
  {
    "id": "2512.17798v1",
    "paper_link": "http://arxiv.org/abs/2512.17798v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theo1}\nThe map\n$$(v,w) \\in \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}\\mapsto (v\\cdot w)_H \\in \\mathcal{D'},$$\nis weakly continuous. \nMore precisely, let $(v_n,w_n)$ be  a bounded sequence in $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ converging in the sense of distributions to $(v,w)$:\n$$ v_n \\overset{\\mathcal{D}'}{\\rightharpoonup} v, \\qquad w_n \\overset{\\mathcal{D}'}{\\rightharpoonup}  w,$$\nso in particular $(v,w)\\in  \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}$. \nThen we have the product convergence:\n$$ (v_n\\cdot w_n)_H \\overset{\\mathcal{D}'}{\\rightharpoonup}  (v\\cdot  w)_H,$$\nwhere the products are defined by Proposition~\\ref{prop1}.\nThe result is valid also with $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ replaced by:\\par\n\\,\\hspace{3mm}{\\it{(i) $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$,}} \\par\n\\,\\hspace{1mm}{\\it{(ii)$^*$ $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp+\\epsilon, p}$, for $d<p< \\infty$, $\\epsilon>0$,}}\\par\n\\,\\hspace{1mm}\\it{(iii)}  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(iv)} $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$ and $\\delta\\in(0,1),\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(v)$^*$}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $1< p< \\infty $, $\\epsilon>0$, $\\alpha\\in \\mathbb R$, \\par\n\\,\\hspace{2mm}\\it{(vi)}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.",
      "start_pos": 13615,
      "end_pos": 15224,
      "label": "theo1"
    },
    "ref_dict": {
      "remdecv": "\\begin{remark}\\label{remdecv} Note that we have also the following regularity properties of $Y_\\chi v$ and $Z_\\chi v$:\n \\begin{equation}\\label{YZregbis} \n\\left\\{\\begin{array}{c}Y_\\chi w\\in \\Psi^{0} w,\\quad \\divb Y_\\chi w\\in \\Psi^{-\\infty} w,\\\\  Z_\\chi w\\in \\Psi^{-2}\\divb w,\\end{array}\\right.\n  \\end{equation}\nthat are useful if we have regularity information on $\\divb w$ or if we need regularity on $\\divb Y_\\chi w$. Typically when the regularity is different for $v$ and $\\divb v$, as in \\cite{Br}, both decompositions of $v$ and $w$ are needed to define the product $vw$. \nIn our case the Hodge decomposition \\eqref{Hodge} is not on free divergence but on smooth divergence  fields, and the product formula \\eqref{prod} can be extended to\n\\begin{equation}\\label{prodbis}\nv\\cdot w =Y_\\chi v\\cdot Y_\\chi w +\\divb (Z_\\chi v Y_\\chi w)-(\\divb Y_\\chi w)\\,Z_\\chi v-(\\divb Y_\\chi v)\\,Z_\\chi w+\\divb(Y_\\chi v\\,Z_\\chi w)+\\nabla Z_\\chi v\\cdot \\nabla Z_\\chi w.\n\\end{equation}\nIn this note we stick to the classical framework of same regularity for $v$ and $\\divb v$ as well as for $w$ and $\\curl w$, and we will need to decompose only $w$ and to use \\eqref{YZreg}. \n\\end{remark}",
      "app": "\\label{app}\nIn this appendix we recall the definition of pseudodifferential operators and describe their action on $L^p$ spaces and on $\\mathcal M_0$. We refer to~\\cite[Chapter XVIII]{Ho} for a genera",
      "Hodge": "\\begin{equation}\\label{Hodge}\n Z_\\chi w := -\\chi(D)(-\\Delta)^{-1}\\divb w \\in \\mathcal D ', \\quad Y_\\chi w:=w-\\nabla Z_\\chi w\\in\\mathcal  D '\n \\end{equation}",
      "theo1": "\\begin{theorem}\\label{theo1}\nThe map\n$$(v,w) \\in \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}\\mapsto (v\\cdot w)_H \\in \\mathcal{D'},$$\nis weakly continuous. \nMore precisely, let $(v_n,w_n)$ be  a bounded sequence in $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ converging in the sense of distributions to $(v,w)$:\n$$ v_n \\overset{\\mathcal{D}'}{\\rightharpoonup} v, \\qquad w_n \\overset{\\mathcal{D}'}{\\rightharpoonup}  w,$$\nso in particular $(v,w)\\in  \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}$. \nThen we have the product convergence:\n$$ (v_n\\cdot w_n)_H \\overset{\\mathcal{D}'}{\\rightharpoonup}  (v\\cdot  w)_H,$$\nwhere the products are defined by Proposition~\\ref{prop1}.\nThe result is valid also with $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ replaced by:\\par\n\\,\\hspace{3mm}{\\it{(i) $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$,}} \\par\n\\,\\hspace{1mm}{\\it{(ii)$^*$ $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp+\\epsilon, p}$, for $d<p< \\infty$, $\\epsilon>0$,}}\\par\n\\,\\hspace{1mm}\\it{(iii)}  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(iv)} $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$ and $\\delta\\in(0,1),\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(v)$^*$}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $1< p< \\infty $, $\\epsilon>0$, $\\alpha\\in \\mathbb R$, \\par\n\\,\\hspace{2mm}\\it{(vi)}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\n\\end{theorem}",
      "prop1": "\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of  the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item  $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$. \n\nMoreover, in all space $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\\infty$ for which $C^\\infty_0$ is not dense, the classical product map\n$$ (v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0 \\mapsto v \\cdot w  \\in \\mathcal C ^\\infty_0 $$\nadmits $(vw)_H$ as a unique continuous extension from  $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ to $\\mathcal{D}' $. \n\\end{prop}"
    },
    "pre_theorem_intro_text_len": 9769,
    "pre_theorem_intro_text": "Div-curl lemmas have been introduced by Murat and Tartar while establishing the compensated compactness theory in the end of the seventies (\\cite{Mu1},\\cite{Mu2},\\cite{Ta1},\\cite{Ta2}, see also \\S 7,17 of \\cite{Tartarbook}, and \\cite{Mu3}). \n\nThe classical result states, for $1<p<\\infty$ and $p'$ its conjugate, that given two sequences $(v_n)$ and $(w_n)$ uniformly bounded and weakly converging in $L^{p}$ and $L^{p'}$respectively, such that $(\\divb\\, v_n)$ is uniformly bounded in $L^p$ and $(\\curl w_n)$ is uniformly bounded in $L^{p'}$, their product $(v_n\\cdot w_n)$ converges in the sense of distributions\\footnote{Another slightly different classical setting is the case, based mainly on the case $p=2$, where the sequences $(v_n)$ and $(w_n)$ are bounded and weakly converging in $L^{p}$ and $L^{p'}$respectively, and such that $(\\divb\\, v_n)$ and $(\\curl w_n)$ are only compact in $W^{-1,p}$ and $W^{-1,p'}$ respectively.}. The classical div-curl lemma has several different proofs\\footnote{In the case $p=2$ an approach based on default measures can be used. However these objects do not have an obvious extension to the $L^1/L^\\infty$ regularity.}, see for instance \\cite{Pr} for a review. Its classical proof uses Hodge's decomposition on divergence-free fields $w_n=y_n+\\nabla z_n$, retrieving for instance uniform boundedness of $z_n$ in $W^{1,p'}$, which allows for compactness in $L^{p'}$. We note that here $1<p<\\infty$, thus Hodge's decomposition on divergence-free fields is classical. We also note that in the conjugate-exponent spaces $L^p-L^{p'}$ there is no issue with defining the products $v_n\\cdot w_n$. \n\nA recent extension has been obtained by Briane, Casado-D\\'iaz and Murat in \\cite{Br} by considering for the regularity of $v_n$ and $w_n$, instead of $L^p-L^{p'}$, the spaces $L^p-L^q$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$ and $1<p,q<\\infty$, and by considering for $\\divb v_n$ and $\\curl w_n$ some other appropriate regularity, see Theorem 2.3 in \\cite{Br}. They also considered the spaces $\\mathcal M-L^d$ and $L^d-\\mathcal M$, where $\\mathcal{M}$ is the set of Radon measures, see Theorems 3.1 and 4.1 respectively in \\cite{Br}. In the first case $L^p-L^q$ the authors give a definition of the product $v_nw_n$ in the sense of distributions, based on Hodge's decompositions on divergence-free fields for both $v_n$ and $w_n$. In the remaining two cases, involving $\\mathcal M$, firstly the product is defined by proving moreover an extension to measures of the representation of divergence free functions in $L^1$ of Br\\'ezis and Van Schaftingen from \\cite{BrVS}. \nThen, once the product is well defined, for instance in the first case $L^p-L^q$, the uniform boundedness of $z_n$ in $W^{1,q}$ does not allow for compactness in $L^{p'}$, since $W^{1,q}$ is embedded in $L^{q^*}=L^{p'}$ but without compactness. The authors use the defect of compactness of the embedding, with the help of the celebrated concentration compactness principle of Lions \\cite{Li}, to get a $\\mathcal D'$-limit for $v_nz_n$, involving $v\\cdot w$ and a combination of Dirac measures. \\\\\n\nThe main aim\\footnote{The motivation to look at div-curl lemmas came to us, after having obtained in \\cite{BBWFL1} results of $L^1$-regularity in linear situations, by having in mind nonlinear situations.} of this note is to give a short proof for the limit conjugate case $\\mathcal M-L^\\infty$. We shall use a non unique microlocal Hodge's decomposition that, contrary to the classical one, does not involve a divergence free field part. It will be our main ingredient, that will allow us to define the products $v_n\\cdot w_n$ and $v\\cdot w$, and modifying slightly the standard approach to obtain very easily a limit in the sense of distributions for $v_n\\cdot w_n$. The method applies to much more spaces than $\\mathcal M-L^\\infty$, as stated in Proposition \\ref{prop1} and Theorem \\ref{theo1}.\n\nLet $d\\geq 2$, $1\\leq  p \\leq +\\infty$, and $\\mathcal F$ a function space among $L^p$, Sobolev spaces and $\\mathcal M_0$, the set of finite Radon measures. We denote  \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\curl}&:= \\{ u \\in \\mathcal F( \\mathbb{R}^d, \\mathbb{R}^d), \\curl u \\in \\mathcal F( \\mathbb{R}^d, \\mathbb{R}^{d^2})\\},\\\\\n\\mathcal{F}_{\\divb}&:= \\{ u \\in \\mathcal F( \\mathbb{R}^d, \\mathbb{R}^d),\\divb u \\in \\mathcal F( \\mathbb{R}^d, \\mathbb{R})\\},\\\\\n\\end{aligned}\n\\end{equation} \nand we endow these spaces with their natural norms. For sake of simplicity we do not specify anymore the natural domain and range of the functions. The spaces $\\mathcal{F}_{\\divb}$ and $\\mathcal{F}_{\\curl}$ that we shall consider in the sequel are continuously stable by multiplication by a $\\mathcal C^\\infty_0$ localization.\n\n\\begin{remark} Since the conclusions in Proposition \\ref{prop1} and Theorem~\\ref{theo1} below are in $\\mathcal{D}'$, thus local, we can weaken the global hypothesis and assume only local conditions as for instance\n$ \\mathcal{M}_{\\divb}, (L^\\infty_{\\curl})_{\\text{loc}}$ instead of $\\mathcal{M}_{0,\\divb},L^\\infty_{\\curl}$. In particular we could place ourselves in the case of an open set $\\Omega$ of $\\mathbb R^d$ instead of $\\mathbb R^d$ and consider local spaces in $\\Omega$. \n\\end{remark}\n\nWe start with a result ensuring the existence of products in the sense of distributions between various spaces. \n\n\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of  the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item  $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$. \n\nMoreover, in all space $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\\infty$ for which $C^\\infty_0$ is not dense, the classical product map\n$$ (v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0 \\mapsto v \\cdot w  \\in \\mathcal C ^\\infty_0 $$\nadmits $(vw)_H$ as a unique continuous extension from  $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ to $\\mathcal{D}' $. \n\\end{prop}\n\n\\begin{remark}\n\nIn the remaining spaces $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (i)-(iii)-(vi), involving $L^\\infty$, the product $(v\\cdot w)_H$ is still the natural one. More precisely, if $(v_n,w_n)$\nis any sequence bounded in $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$  converging in the sense of distributions to $(v,w)$, such that the product $v_n\\cdot w_n$ is well-defined in the classical sense, for instance a mollified approximation of $(v,w)$, then we will obtain by Theorem \\ref{theo1} that\n$$(v\\cdot w)_H\\overset{\\mathcal D'}{=}\\lim_{n \\rightarrow \\infty}v_n\\cdot  w_n,$$\nand consequently, the product $(v\\cdot w)_H$ is in this cases the unique extension from $\\mathcal F_{\\divb}\\cap\\mathcal C^0\\times\\mathcal {\\tilde F}_{\\curl}\\cap\\mathcal C^0$ to $\\mathcal D'$ which enjoys this weak continuity property. \n\\end{remark}\n\nThe proof of Proposition \\ref{prop1} is based on the Hodge decomposition defined in \\eqref{Hodge} and on the action of pseudo-differential operators on $L^p$ and $\\mathcal M_0$. The definition, notations and properties of pseudo-differential operators that will be of use in this note are given in Appendix \\ref{app}.\n\nWe note that the space from {\\em{(v)}} with $\\alpha=0$, i.e. $W_{\\divb}^{p}\\times W_{\\curl}^{-1, p'}$ for $1< p<\\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\\divb}\\times L^q_{\\curl}$ with  $\\frac 1p+\\frac 1q=1+\\frac 1d$, since\\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\\alpha}\\subset L^p$ for $k\\in(0,1), k<\\frac d\\alpha, \\alpha\\in [1,\\infty), p\\in[\\alpha,\\frac{d\\alpha}{d-k\\alpha}]$,  \n$W^{k,\\alpha}\\subset \\mathcal C^{0,\\frac{k\\alpha-d}{\\alpha}}$ for $k\\in (0,1), k>\\frac d\\alpha, \\alpha\\in [1,\\infty]$, and $W^{\\delta,1}\\subset L^p$ for $p\\in [1,\\frac d{d-\\delta}]$.}\n in this case $L^q=L^{(p^*)'}\\subset W^{-1,p'}$. \nWe also note that from {\\em{(ii)}}, since $L^d\\subset W^{-1+\\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\\mathcal M_{0,\\divb}\\times L^d_{\\curl}$, and similarly we obtain the space $L^d_{\\divb}\\times \\mathcal M_{0,\\curl}$ as a subcase of {\\em{(iv)}}. These last three cases corresponds to the ones in \\cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\\divb v_n$ and for $w_n$ and $\\curl w_n$. To extend to different appropriate regularities for $\\divb v_n$ and $\\curl w_n$ as in \\cite{Br} the proof could be extended in the spirit of Remark \\ref{remdecv}. \n\nNow we can state the div-curl lemma.",
    "context": "A recent extension has been obtained by Briane, Casado-D\\'iaz and Murat in \\cite{Br} by considering for the regularity of $v_n$ and $w_n$, instead of $L^p-L^{p'}$, the spaces $L^p-L^q$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$ and $1<p,q<\\infty$, and by considering for $\\divb v_n$ and $\\curl w_n$ some other appropriate regularity, see Theorem 2.3 in \\cite{Br}. They also considered the spaces $\\mathcal M-L^d$ and $L^d-\\mathcal M$, where $\\mathcal{M}$ is the set of Radon measures, see Theorems 3.1 and 4.1 respectively in \\cite{Br}. In the first case $L^p-L^q$ the authors give a definition of the product $v_nw_n$ in the sense of distributions, based on Hodge's decompositions on divergence-free fields for both $v_n$ and $w_n$. In the remaining two cases, involving $\\mathcal M$, firstly the product is defined by proving moreover an extension to measures of the representation of divergence free functions in $L^1$ of Br\\'ezis and Van Schaftingen from \\cite{BrVS}. \nThen, once the product is well defined, for instance in the first case $L^p-L^q$, the uniform boundedness of $z_n$ in $W^{1,q}$ does not allow for compactness in $L^{p'}$, since $W^{1,q}$ is embedded in $L^{q^*}=L^{p'}$ but without compactness. The authors use the defect of compactness of the embedding, with the help of the celebrated concentration compactness principle of Lions \\cite{Li}, to get a $\\mathcal D'$-limit for $v_nz_n$, involving $v\\cdot w$ and a combination of Dirac measures. \\\\\n\n\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$.\n\nIn the remaining spaces $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (i)-(iii)-(vi), involving $L^\\infty$, the product $(v\\cdot w)_H$ is still the natural one. More precisely, if $(v_n,w_n)$\nis any sequence bounded in $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ converging in the sense of distributions to $(v,w)$, such that the product $v_n\\cdot w_n$ is well-defined in the classical sense, for instance a mollified approximation of $(v,w)$, then we will obtain by Theorem \\ref{theo1} that\n$$(v\\cdot w)_H\\overset{\\mathcal D'}{=}\\lim_{n \\rightarrow \\infty}v_n\\cdot w_n,$$\nand consequently, the product $(v\\cdot w)_H$ is in this cases the unique extension from $\\mathcal F_{\\divb}\\cap\\mathcal C^0\\times\\mathcal {\\tilde F}_{\\curl}\\cap\\mathcal C^0$ to $\\mathcal D'$ which enjoys this weak continuity property. \n\\end{remark}\n\nWe note that the space from {\\em{(v)}} with $\\alpha=0$, i.e. $W_{\\divb}^{p}\\times W_{\\curl}^{-1, p'}$ for $1< p<\\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\\divb}\\times L^q_{\\curl}$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$, since\\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\\alpha}\\subset L^p$ for $k\\in(0,1), k<\\frac d\\alpha, \\alpha\\in [1,\\infty), p\\in[\\alpha,\\frac{d\\alpha}{d-k\\alpha}]$, \n$W^{k,\\alpha}\\subset \\mathcal C^{0,\\frac{k\\alpha-d}{\\alpha}}$ for $k\\in (0,1), k>\\frac d\\alpha, \\alpha\\in [1,\\infty]$, and $W^{\\delta,1}\\subset L^p$ for $p\\in [1,\\frac d{d-\\delta}]$.}\n in this case $L^q=L^{(p^*)'}\\subset W^{-1,p'}$. \nWe also note that from {\\em{(ii)}}, since $L^d\\subset W^{-1+\\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\\mathcal M_{0,\\divb}\\times L^d_{\\curl}$, and similarly we obtain the space $L^d_{\\divb}\\times \\mathcal M_{0,\\curl}$ as a subcase of {\\em{(iv)}}. These last three cases corresponds to the ones in \\cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\\divb v_n$ and for $w_n$ and $\\curl w_n$. To extend to different appropriate regularities for $\\divb v_n$ and $\\curl w_n$ as in \\cite{Br} the proof could be extended in the spirit of Remark \\ref{remdecv}.\n\nNow we can state the div-curl lemma.",
    "full_context": "A recent extension has been obtained by Briane, Casado-D\\'iaz and Murat in \\cite{Br} by considering for the regularity of $v_n$ and $w_n$, instead of $L^p-L^{p'}$, the spaces $L^p-L^q$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$ and $1<p,q<\\infty$, and by considering for $\\divb v_n$ and $\\curl w_n$ some other appropriate regularity, see Theorem 2.3 in \\cite{Br}. They also considered the spaces $\\mathcal M-L^d$ and $L^d-\\mathcal M$, where $\\mathcal{M}$ is the set of Radon measures, see Theorems 3.1 and 4.1 respectively in \\cite{Br}. In the first case $L^p-L^q$ the authors give a definition of the product $v_nw_n$ in the sense of distributions, based on Hodge's decompositions on divergence-free fields for both $v_n$ and $w_n$. In the remaining two cases, involving $\\mathcal M$, firstly the product is defined by proving moreover an extension to measures of the representation of divergence free functions in $L^1$ of Br\\'ezis and Van Schaftingen from \\cite{BrVS}. \nThen, once the product is well defined, for instance in the first case $L^p-L^q$, the uniform boundedness of $z_n$ in $W^{1,q}$ does not allow for compactness in $L^{p'}$, since $W^{1,q}$ is embedded in $L^{q^*}=L^{p'}$ but without compactness. The authors use the defect of compactness of the embedding, with the help of the celebrated concentration compactness principle of Lions \\cite{Li}, to get a $\\mathcal D'$-limit for $v_nz_n$, involving $v\\cdot w$ and a combination of Dirac measures. \\\\\n\n\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$.\n\nIn the remaining spaces $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (i)-(iii)-(vi), involving $L^\\infty$, the product $(v\\cdot w)_H$ is still the natural one. More precisely, if $(v_n,w_n)$\nis any sequence bounded in $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ converging in the sense of distributions to $(v,w)$, such that the product $v_n\\cdot w_n$ is well-defined in the classical sense, for instance a mollified approximation of $(v,w)$, then we will obtain by Theorem \\ref{theo1} that\n$$(v\\cdot w)_H\\overset{\\mathcal D'}{=}\\lim_{n \\rightarrow \\infty}v_n\\cdot w_n,$$\nand consequently, the product $(v\\cdot w)_H$ is in this cases the unique extension from $\\mathcal F_{\\divb}\\cap\\mathcal C^0\\times\\mathcal {\\tilde F}_{\\curl}\\cap\\mathcal C^0$ to $\\mathcal D'$ which enjoys this weak continuity property. \n\\end{remark}\n\nWe note that the space from {\\em{(v)}} with $\\alpha=0$, i.e. $W_{\\divb}^{p}\\times W_{\\curl}^{-1, p'}$ for $1< p<\\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\\divb}\\times L^q_{\\curl}$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$, since\\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\\alpha}\\subset L^p$ for $k\\in(0,1), k<\\frac d\\alpha, \\alpha\\in [1,\\infty), p\\in[\\alpha,\\frac{d\\alpha}{d-k\\alpha}]$, \n$W^{k,\\alpha}\\subset \\mathcal C^{0,\\frac{k\\alpha-d}{\\alpha}}$ for $k\\in (0,1), k>\\frac d\\alpha, \\alpha\\in [1,\\infty]$, and $W^{\\delta,1}\\subset L^p$ for $p\\in [1,\\frac d{d-\\delta}]$.}\n in this case $L^q=L^{(p^*)'}\\subset W^{-1,p'}$. \nWe also note that from {\\em{(ii)}}, since $L^d\\subset W^{-1+\\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\\mathcal M_{0,\\divb}\\times L^d_{\\curl}$, and similarly we obtain the space $L^d_{\\divb}\\times \\mathcal M_{0,\\curl}$ as a subcase of {\\em{(iv)}}. These last three cases corresponds to the ones in \\cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\\divb v_n$ and for $w_n$ and $\\curl w_n$. To extend to different appropriate regularities for $\\divb v_n$ and $\\curl w_n$ as in \\cite{Br} the proof could be extended in the spirit of Remark \\ref{remdecv}.\n\nNow we can state the div-curl lemma.\n\nWe note that the space from {\\em{(v)}} with $\\alpha=0$, i.e. $W_{\\divb}^{p}\\times W_{\\curl}^{-1, p'}$ for $1< p<\\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\\divb}\\times L^q_{\\curl}$ with $\\frac 1p+\\frac 1q=1+\\frac 1d$, since\\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\\alpha}\\subset L^p$ for $k\\in(0,1), k<\\frac d\\alpha, \\alpha\\in [1,\\infty), p\\in[\\alpha,\\frac{d\\alpha}{d-k\\alpha}]$, \n$W^{k,\\alpha}\\subset \\mathcal C^{0,\\frac{k\\alpha-d}{\\alpha}}$ for $k\\in (0,1), k>\\frac d\\alpha, \\alpha\\in [1,\\infty]$, and $W^{\\delta,1}\\subset L^p$ for $p\\in [1,\\frac d{d-\\delta}]$.}\n in this case $L^q=L^{(p^*)'}\\subset W^{-1,p'}$. \nWe also note that from {\\em{(ii)}}, since $L^d\\subset W^{-1+\\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\\mathcal M_{0,\\divb}\\times L^d_{\\curl}$, and similarly we obtain the space $L^d_{\\divb}\\times \\mathcal M_{0,\\curl}$ as a subcase of {\\em{(iv)}}. These last three cases corresponds to the ones in \\cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\\divb v_n$ and for $w_n$ and $\\curl w_n$. To extend to different appropriate regularities for $\\divb v_n$ and $\\curl w_n$ as in \\cite{Br} the proof could be extended in the spirit of Remark \\ref{remdecv}.\n\nThe proof of Theorem \\ref{theo1} is based on Proposition \\ref{prop1} and on the fact that the smooth localization $u\\in\\ W^{\\epsilon,p}\\to \\chi u \\in L^p$ is compact for $\\epsilon>0$ and $1\\leq p\\leq \\infty$. Thus a bit of regularity is lost with respect to Proposition \\ref{prop1}. This explains why Theorem \\ref{theo1} is valid in all the functional settings listed in Proposition \\ref{prop1} with slight modifications for the spaces in {\\em{(ii)}} and {\\em{(v)}}, i.e. for instance $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p<\\infty $, $\\alpha\\in \\mathbb R$, is replaced by $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$.\n\nNow we are ready to define the product in the statement of Proposition \\ref{prop1}. We use the Hodge decomposition \\eqref{Hodgedec} of $w$ to define, whenever each product term in the following is well defined in the sense of distributions, the quantity:\n\\begin{equation}\\label{prod}\n(v\\cdot w)_\\chi =v\\cdot Y_\\chi w -(\\divb\\,v)Z_\\chi w+\\divb\\,(v\\,Z_\\chi w).\n\\end{equation}\nWe notice that for $ (v,w) \\in \\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$, this quantity identifies with the usual product:\n$$v\\cdot w=(v\\cdot w)_\\chi.$$\n\nTo prove the last extension result in Proposition \\ref{prop1}, say for $\\mathcal{M}_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ with $d<p<\\infty$ (the other cases follow similarly), we use the same arguments before to get that the map \n$$ (v,w) \\in \\mathcal C^\\infty_0\\times \\mathcal C^\\infty_0\\mapsto v w \\in \\mathcal{D}'$$\nis continuous for the $\\mathcal{M}_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ topology.\nIndeed, as before, we get from \\eqref{YZreg} that along with \n$$\nY_\\chi w, Z_\\chi w\\in \\Psi^{-1}W^{-1+\\frac dp+\\epsilon,p}\\subset W^{\\frac dp+\\epsilon,p}\\subset \\mathcal C^0,\n$$\nwe have also the continuity for the $W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ topology of the maps \n$$\nw\\in C^\\infty_0 \\mapsto Y_\\chi w \\in \\mathcal C^0,\\quad w\\in C^\\infty_0 \\mapsto Z_\\chi w \\in \\mathcal C^0.\n$$\nAs a consequence we deduce that the maps \n\\begin{equation}\\label{maps} \n (v,w) \\in \\mathcal C^\\infty_0\\times \\mathcal C^\\infty_0 \\mapsto \\begin{cases}v\\cdot Y_\\chi w \\in \\mathcal{D}',\\\\ (\\divb v) Z_\\chi w \\in \\mathcal{D}',\\\\\nv\\,Z_\\chi w \\in \\mathcal{D}',\n\\end{cases}\n\\end{equation}\nare continuous for the $\\mathcal{M}_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ topology, thus so is the map \n$$ (w\\cdot v) \\in \\mathcal C^\\infty_0 \\times \\mathcal C^\\infty_0 \\mapsto w\\cdot v =v\\cdot Y_\\chi w-(\\divb\\, v)Z_\\chi w+\\divb\\,(v\\,Z_\\chi w) \\in \\mathcal{D}'.$$ \nConsequently the product map has a unique extension from $\\mathcal{M}_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ to $\\mathcal D'$. This extension defines the product on $ \\mathcal{M}_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}$ as a distribution, and it does not depend on the choice of the cut-off function $\\chi$ because it coincides, whatever this choice, with the usual product on the dense subspace $\\mathcal C^\\infty_0\\times \\mathcal C^\\infty _0$.\\\\\n\n\\appendix\n\\section{}\\label{app}\nIn this appendix we recall the definition of pseudodifferential operators and describe their action on $L^p$ spaces and on $\\mathcal M_0$. We refer to~\\cite[Chapter XVIII]{Ho} for a general presentation or~\\cite[\\S VI.6]{Stein} for a presentation closer to our needs of the pseudodifferential calculus (see also \\cite{AlGe}).\nLet us first recall the definitions of the class of symbol of order $\\delta\\in\\mathbb R$: \n$$\n S^\\delta( \\mathbb{R}^d) =\\{a\\in C^\\infty( \\mathbb{R}^{d}); \\\\\n \\forall \\alpha, \\beta \\in \\mathbb{N}^d, \\sup_{x,\\xi \\in \\mathbb{R}^d} |\\partial_x^\\alpha \\partial_\\xi^\\beta a(x, \\xi)| (1+ |\\xi|)^{-\\delta+|\\gamma|}=: \\|a\\|_{\\delta, \\alpha, \\beta}<+\\infty\\}.\n$$\nTo any symbol $a \\in S^\\delta_{\\text{cl}}(\\mathbb{R}^d)$ we can associate an operator on the temperate distributions set $\\mathcal{S}'( \\mathbb{R}^d)$ by the formula\n$$ a(x,D_x) u (x)= \\text{Op} (a) u(x)= \\frac{ 1 }{(2\\pi)^d} \\int e^{i(x-y) \\cdot \\xi} a(x, \\xi) u(y) dy d\\xi.$$\nThe set of such operators, that are called pseudodifferential operators of order $\\delta$, is denoted by $\\Psi^\\delta$. The set $\\Psi^{-\\infty}$ is defined as $\\cap_{\\delta<0}\\Psi^\\delta$. In the following proposition we gather the results needed in this note regarding to the action of pseudodifferential operators.\n\n\\begin{proof}\nThe results (i)-(ii) are classical: for the first one see for instance \\cite[Section VI.5.2]{Stein}, and the second follows by the dual estimate of the Lemma in~\\cite[Section VI.5.3.1]{Stein}. The result (iii) is less classical and we give here a complete (simple) proof. \nWe use a dyadic partition of unity \n$$ 1 =\\sum_{j\\geq 0} \\phi_j (\\xi) , \\, \\phi_0 \\in C^\\infty _ 0 (\\mathbb{R}^d), \\,\\forall j \\geq 1, \\phi_j (\\xi) = \\phi( 2^{-j} \\xi), \\,\\phi \\in C^\\infty_0 ( \\{ \\frac 1 2 < \\| \\xi\\| <2 \\} ),$$\n to decompose\n $$ A = \\sum_{j\\geq 0} A\\, \\phi_j (D).$$\n Each operator of the sum has the kernel\n$$\nK_j(x,y):= \\frac 1 {(2\\pi)^d} \\int e^{i(x-y) \\cdot \\xi } a(x,\\xi) \\phi_j(\\xi) d\\xi.$$\nIn view of the decay of $a$ and of the localization of $\\phi_j$ we have\n$$|K_j(x,y)|\\leq C 2^{j(d+\\delta)}\\|a\\|_{\\delta,0,0}.$$\nAlso, integrating by parts $N$ times using the identity\n$$ L(e^{i(x-y) \\cdot \\xi} )= - e^{i(x-y) \\cdot \\xi}, \\qquad L= \\frac {i (x-y) \\cdot \\nabla_\\xi} {\\| x-y\\|^2} ,$$\nwe get for $N>1$ if $\\|x-y\\|\\neq 0$, that\n$$\n|K_j(x,y)|=\\Big| \\frac {1} {(2\\pi)^d} \\int e^{i(x-y) \\cdot \\xi } L^N (a(x,\\xi) \\phi_j(\\xi) ) d\\xi\\Big| \\leq C 2^{j(d+\\delta)}\\frac{\\|a\\|_{\\delta,0,N}}{(2^j\\|x-y\\|)^N}.\n$$\nTherefore by taking $N=d+1$ we obtain\n$$\\int|K_j(x,y)|dx=\\int_{2^j\\|x-y\\|<1}|K_j(x,y)|dx+\\int_{2^j\\|x-y\\|>1}|K_j(x,y)|dx\n\\leq C 2^{j\\delta}(\\|a\\|_{\\delta,0,0}+\\|a\\|_{\\delta,0,N}),\n$$\nand similarly \n$$\\int|K_j(x,y)|dy\\leq C(a)2^{j\\delta}.$$\nFor $\\mu\\in\\mathcal M_0$ we have, since $\\delta<0$,\n$$\\|A\\mu\\|_{L^1}\\leq \\sum_{j\\geq 0}\\Big\\|\\int K_j(x,y)d\\mu(y)\\Big\\|\\leq C(a) \\|\\mu\\|\\sum_{j\\geq 0}2^{j\\delta}\\leq C(a)\\|\\mu\\|.$$\nIn conclusion $\\Psi^\\delta$ acts continuously from $\\mathcal M_0(\\mathbb R^n)$ to $L^1(\\mathbb R^n)$ for any $\\delta<0$. Combining with (ii) we obtain that $\\Psi^\\delta$ acts continuously from $\\mathcal M_0(\\mathbb R^n)$ to $W^{-\\delta-\\epsilon,1}(\\mathbb R^n)$ for any $\\delta<0$ and $\\epsilon>0$.\n\\end{proof}",
    "post_theorem_intro_text_len": 1798,
    "post_theorem_intro_text": "The proof of Theorem \\ref{theo1} is based on Proposition \\ref{prop1} and on  the fact that the smooth localization $u\\in\\ W^{\\epsilon,p}\\to \\chi u \\in L^p$ is compact for $\\epsilon>0$ and $1\\leq p\\leq \\infty$. Thus a bit of regularity is lost with respect to Proposition \\ref{prop1}. This explains why Theorem \\ref{theo1} is valid in all the functional settings listed in Proposition \\ref{prop1} with slight modifications for the spaces in {\\em{(ii)}} and {\\em{(v)}}, i.e. for instance $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p<\\infty $, $\\alpha\\in \\mathbb R$, is replaced by $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$. \n\nTheorem \\ref{theo1} is not considering the spaces $W_{\\divb}^{-\\alpha,p}\\times W_{\\curl}^{-1+\\alpha, p'}$, for $1< p<\\infty$, and in particular the subset $L^p_{\\divb}\\times L^q_{\\curl}$ with $\\frac 1p+\\frac 1q=1+\\frac 1d, 1<p<\\infty$. This is normal, since in this case concentration effects in terms of Dirac measures can appear in the limit of the product, as in Example 2.10 in \\cite{Br}, and more generally in Theorem 2.1 in \\cite{Br} that describes the limit of $v_nw_n$ in a more involved than simply $vw$. To get such a result here one could continue the analysis in the spirit of \\cite{Br} by exploiting the defect of compactness of the embedding $W^{1,q}\\subset L^{p'}$.  \\\\\n\n{\\bf{Aknowledgements: }} The authors thank Anne-Laure Dalibard, Gilles Francfort, Antoine Gloria and Fran\\c{c}ois Murat for enriching discussions on the div-curl lemma. \nBoth authors acknowledge the funding from the European Research Council (ERC) under the\nEuropean Union's Horizon 2020\nresearch and innovation programme (Grant agreement 101097172 -- GEOEDP). The first author was also partially supported by the French ANR project BOURGEONS.",
    "sketch": "The proof of Theorem~\\ref{theo1} is based on Proposition~\\ref{prop1} and on the fact that the smooth localization $u\\in W^{\\epsilon,p}\\mapsto \\chi u\\in L^p$ is compact for $\\epsilon>0$ and $1\\le p\\le \\infty$. As a consequence, “a bit of regularity is lost with respect to Proposition~\\ref{prop1}$,” which explains why Theorem~\\ref{theo1} holds in the functional settings of Proposition~\\ref{prop1} but with “slight modifications” in cases \\emph{(ii)} and \\emph{(v)}, e.g. replacing $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha,p}$ by $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon,p}$.",
    "expanded_sketch": "The proof of Theorem~\\ref{theo1} is based on the following proposition.\n\n\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of  the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item  $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$. \n\nMoreover, in all space $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\\infty$ for which $C^\\infty_0$ is not dense, the classical product map\n$$ (v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0 \\mapsto v \\cdot w  \\in \\mathcal C ^\\infty_0 $$\nadmits $(vw)_H$ as a unique continuous extension from  $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ to $\\mathcal{D}' $. \n\\end{prop}\n\nIt also uses the fact that the smooth localization $u\\in W^{\\epsilon,p}\\mapsto \\chi u\\in L^p$ is compact for $\\epsilon>0$ and $1\\le p\\le \\infty$. As a consequence, “a bit of regularity is lost with respect to the proposition above,” which explains why, in establishing the main theorem, the result holds in the functional settings of the proposition above but with “slight modifications” in cases \\emph{(ii)} and \\emph{(v)}, e.g. replacing $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha,p}$ by $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon,p}$.,",
    "expanded_theorem": "\\label{theo1}\nThe map\n$$(v,w) \\in \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}\\mapsto (v\\cdot w)_H \\in \\mathcal{D'},$$\nis weakly continuous. \nMore precisely, let $(v_n,w_n)$ be  a bounded sequence in $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ converging in the sense of distributions to $(v,w)$:\n$$ v_n \\overset{\\mathcal{D}'}{\\rightharpoonup} v, \\qquad w_n \\overset{\\mathcal{D}'}{\\rightharpoonup}  w,$$\nso in particular $(v,w)\\in  \\mathcal{M}_{0,\\divb}\\times L^\\infty_{\\curl}$. \nThen we have the product convergence:\n$$ (v_n\\cdot w_n)_H \\overset{\\mathcal{D}'}{\\rightharpoonup}  (v\\cdot  w)_H,$$\nwhere the products are defined as follows.\n\n\\begin{prop}\\label{prop1} Let $ (v,w) \\in \\mathcal M_{0,\\divb} \\times L_{\\curl}^\\infty$ or belonging to one of  the following spaces:\n\\begin{enumerate}\n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$, \n\\item $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp, p}$, for $d<p< \\infty$, \n\\item  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \n\\item  $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$\\footnote{The condition $1<p\\leq \\frac d{d-1+\\delta}$ is equivalent to $\\frac{d}{1-\\delta}\\leq p'< \\infty$, thus varying $\\delta$ in $(0,1)$ we have the range $d<p'< \\infty$}. and $\\delta\\in(0,1),\\epsilon>0$, \n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha, p}$, for $1< p< \\infty $, $\\alpha\\in \\mathbb R$,\n\\item  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.\n\\end{enumerate}\nThen $w$ admits a class of Hodge-type decompositions $w=y+\\nabla z$ such that the following quantity\n\\begin{equation}\\label{Hodgeprod}\n(v\\cdot w)_H:=v\\cdot y -(\\divb\\,v)z+\\divb\\,(vz).\n\\end{equation}\nis well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0$ or if $(v,w) \\in L_{\\divb}^{p'}\\times L_{\\curl}^p$ for $1\\leq p\\leq \\infty$. \n\nMoreover, in all space $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\\infty$ for which $C^\\infty_0$ is not dense, the classical product map\n$$ (v,w) \\in\\mathcal C^\\infty_0\\times \\mathcal C ^\\infty_0 \\mapsto v \\cdot w  \\in \\mathcal C ^\\infty_0 $$\nadmits $(vw)_H$ as a unique continuous extension from  $\\mathcal{F}_{\\divb}\\times\\mathcal{\\tilde F}_{\\curl}$ to $\\mathcal{D}' $. \n\\end{prop}\n\nThe result is valid also with $\\mathcal{M}_{0,\\divb} \\times L^\\infty_{\\curl}$ replaced by:\\par\n\\,\\hspace{3mm}{\\it{(i) $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+ \\epsilon, \\infty}$, for $\\epsilon>0$,}} \\par\n\\,\\hspace{1mm}{\\it{(ii)$^*$ $\\mathcal M_{0,\\divb} \\times W_{\\curl}^{-1+\\frac dp+\\epsilon, p}$, for $d<p< \\infty$, $\\epsilon>0$,}}\\par\n\\,\\hspace{1mm}\\it{(iii)}  $W^{-1+\\epsilon,\\infty}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(iv)} $W^{-\\delta+\\epsilon,p'}_{\\divb} \\times \\mathcal M_{0,\\curl}$, for $1< p\\leq \\frac d{d-1+\\delta}$ and $\\delta\\in(0,1),\\epsilon>0$, \\par\n\\,\\hspace{2mm}\\it{(v)$^*$}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $1< p< \\infty $, $\\epsilon>0$, $\\alpha\\in \\mathbb R$, \\par\n\\,\\hspace{2mm}\\it{(vi)}  $W_{\\divb}^{-\\alpha,p'}\\times W_{\\curl}^{-1+\\alpha+\\epsilon, p}$, for $p\\in\\{1,\\infty\\}$, $\\epsilon>0$, $\\alpha\\in \\mathbb R$.,",
    "theorem_type": [
      "Implication",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal M_0\\) denote the space of finite Radon measures on \\(\\mathbb R^d\\). For a distribution space \\(X\\), write \\(X_{\\divb}:=\\{v: v\\in X,\\ \\divb v\\in X\\}\\) and \\(X_{\\curl}:=\\{w: w\\in X,\\ \\curl w\\in X\\}\\). For any admissible pair \\((v,w)\\) in one of the classes below, choose a Hodge-type decomposition \\(w=y+\\nabla z\\) and define\n\\[\n(v\\cdot w)_H:=v\\cdot y-(\\divb v)z+\\divb(vz),\n\\]\nwhich is a well-defined distribution independent of the chosen decomposition. Let \\(p'\\) be the H\\\"older-conjugate exponent of \\(p\\) (with \\(1'=\\infty\\) and \\(\\infty'=1\\)). Suppose \\((v_n,w_n)\\) is a bounded sequence converging in \\(\\mathcal D'(\\mathbb R^d)\\) to \\((v,w)\\),\n\\[\nv_n\\rightharpoonup v,\\qquad w_n\\rightharpoonup w,\n\\]\nand assume that all \\((v_n,w_n)\\) belong to one fixed class among\n\\[\n\\mathcal M_{0,\\divb}\\times L^\\infty_{\\curl},\n\\]\nor\n\\[\n\\text{(i) }\\mathcal M_{0,\\divb}\\times W^{-1+\\epsilon,\\infty}_{\\curl}\\quad (\\epsilon>0),\n\\]\n\\[\n\\text{(ii) }\\mathcal M_{0,\\divb}\\times W^{-1+\\frac dp+\\epsilon,p}_{\\curl}\\quad (d<p<\\infty,\\ \\epsilon>0),\n\\]\n\\[\n\\text{(iii) }W^{-1+\\epsilon,\\infty}_{\\divb}\\times \\mathcal M_{0,\\curl}\\quad (\\epsilon>0),\n\\]\n\\[\n\\text{(iv) }W^{-\\delta+\\epsilon,p'}_{\\divb}\\times \\mathcal M_{0,\\curl}\\quad \\Big(1<p\\le \\frac d{d-1+\\delta},\\ \\delta\\in(0,1),\\ \\epsilon>0\\Big),\n\\]\n\\[\n\\text{(v) }W^{-\\alpha,p'}_{\\divb}\\times W^{-1+\\alpha+\\epsilon,p}_{\\curl}\\quad (1<p<\\infty,\\ \\epsilon>0,\\ \\alpha\\in\\mathbb R),\n\\]\nor\n\\[\n\\text{(vi) }W^{-\\alpha,p'}_{\\divb}\\times W^{-1+\\alpha+\\epsilon,p}_{\\curl}\\quad (p\\in\\{1,\\infty\\},\\ \\epsilon>0,\\ \\alpha\\in\\mathbb R).\n\\]\nWhich statement holds for every such bounded distribution-convergent sequence?",
      "correct_choice": {
        "label": "A",
        "text": "The Hodge products converge weakly in distributions:\n\\[\n(v_n\\cdot w_n)_H\\rightharpoonup (v\\cdot w)_H\\qquad\\text{in }\\mathcal D'(\\mathbb R^d).\n\\]\nEquivalently, on each listed class, the map \\((v,w)\\mapsto (v\\cdot w)_H\\) is weakly continuous from that class into \\(\\mathcal D'\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The Hodge products converge weakly in distributions:\n\\[\n(v_n\\cdot w_n)_H\\rightharpoonup (v\\cdot w)_H\\qquad\\text{in }\\mathcal D'(\\mathbb R^d),\n\\]\nbut this remains valid already for the larger endpoint classes obtained by deleting the \\(\\epsilon>0\\) losses in \\textup{(ii)} and \\textup{(v)}, i.e. for\n\\[\n\\mathcal M_{0,\\divb}\\times W^{-1+\\frac dp,p}_{\\curl}\\quad(d<p<\\infty)\n\\]\nand\n\\[\nW^{-\\alpha,p'}_{\\divb}\\times W^{-1+\\alpha,p}_{\\curl}\\quad(1<p<\\infty,\\ \\alpha\\in\\mathbb R).\n\\]\nEquivalently, the map \\((v,w)\\mapsto (v\\cdot w)_H\\) is weakly continuous on those endpoint classes as well."
        },
        {
          "label": "C",
          "text": "For every such bounded distribution-convergent sequence, after passing to a subsequence one has\n\\[\n(v_{n_k}\\cdot w_{n_k})_H\\rightharpoonup (v\\cdot w)_H\\qquad\\text{in }\\mathcal D'(\\mathbb R^d).\n\\]\nIn particular, every bounded sequence in any one fixed listed class has at least one distributionally convergent subsequence of Hodge products with limit equal to the Hodge product of the distributional limits."
        },
        {
          "label": "D",
          "text": "For every such bounded distribution-convergent sequence, the conclusion improves from weak distributional convergence to strong convergence in the local Sobolev topology: for every compactly supported \\(\\chi\\in C_c^\\infty(\\mathbb R^d)\\),\n\\[\n\\chi (v_n\\cdot w_n)_H\\to \\chi (v\\cdot w)_H\n\\]\nstrongly in some fixed negative Sobolev space depending only on the chosen class. Equivalently, the map \\((v,w)\\mapsto (v\\cdot w)_H\\) is compact from each listed class into \\(\\mathcal D'\\) after localization."
        },
        {
          "label": "E",
          "text": "For every such bounded distribution-convergent sequence and for every choice of Hodge-type decompositions \\(w_n=y_n+\\nabla z_n\\) and \\(w=y+\\nabla z\\), one has termwise weak convergence in \\(\\mathcal D'(\\mathbb R^d)\\):\n\\[\nv_n\\cdot y_n\\rightharpoonup v\\cdot y,\n\\qquad\n(\\divb v_n)z_n\\rightharpoonup (\\divb v)z,\n\\qquad\n\\divb(v_n z_n)\\rightharpoonup \\divb(vz),\n\\]\nand hence\n\\[\n(v_n\\cdot w_n)_H\\rightharpoonup (v\\cdot w)_H.\n\\]\nThus weak continuity holds separately for each term appearing in the definition of \\((v\\cdot w)_H\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "epsilon_loss_in_cases_ii_v",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "full_sequence_convergence_dropped_to_subsequence_convergence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "compact_localization_misread_as_compactness_of_product_map",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "cancellation_between_hodge_terms_vs_termwise_continuity",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the marked answer. It states hypotheses and asks for the valid conclusion, without giving away the distinctive clauses that separate A from the other options."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall/restatement item: the stem lists assumptions in detail and the choices restate candidate conclusions of the same theorem with small perturbations. It is not purely tautological, since the options differ in subtle claims, but it is close to direct theorem matching."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle regularity losses, uniqueness-extension clauses, and decomposition-independence claims. However, the task mainly rewards precise theorem recall/comparison rather than generating a conclusion from first principles. Also, choice C appears to be a weaker true statement, which reduces the force of the reasoning task."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and target realistic failure modes: overextending continuity, removing epsilon losses, or strengthening decomposition claims. But the distractor set is weakened because C is apparently also true (just weaker than A), so the item is not cleanly single-correct under the wording 'Which conclusion ... is valid?'"
      },
      "total_score": 5,
      "overall_assessment": "Technically sophisticated and mostly free of answer leakage, but it is heavily theorem-restatement based and suffers from ambiguity because at least one distractor appears genuinely true as a weaker conclusion. This limits its quality as a single-best-answer MCQ testing generative reasoning."
    }
  },
  {
    "id": "2512.17821v1",
    "paper_link": "http://arxiv.org/abs/2512.17821v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}",
      "start_pos": 10443,
      "end_pos": 10986,
      "label": "thm:main thm"
    },
    "ref_dict": {
      "eq:ternary cubic eqn": "\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}",
      "lem:factorization of terms": "\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}",
      "eq:generalized ErdSelf eqn": "\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}",
      "defn: coeff vector": "\\begin{definition}\\label{defn: coeff vector}\nAssume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Let $n+jd=a_jx_j^3$ be the factorization as described in Lemma \\ref{lem:factorization of terms}. We call $(a_0,a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$ the \\emph{coefficient vector} corresponding to $y,n,d$.\n\\end{definition}",
      "cor:rat pts": "\\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}",
      "thm:main thm": "\\begin{theorem}\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}\n\\end{theorem}",
      "prop:only solns to pair of cubic eqns": "\\begin{proposition}\\label{prop:only solns to pair of cubic eqns}\nThe only solutions $(x,y,z,w) \\in \\mathbb{Z}^4$ to the pair of equations\n\\begin{align*}\nx^3+y^3&=9z^3 \\\\\n5x^3-y^3&=3w^3\n\\end{align*}\nwith $xyzw \\neq 0$ and $x,y$ coprime are $(x,y,z,w) = \\pm (1,2,1,-1)$.\n\\end{proposition}",
      "eq:main eqn of study": "\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}",
      "lem:EC over K": "\\begin{lemma}\\label{lem:EC over K}\nLet $\\alpha = \\sqrt[3]{5}$, and set $K = \\mathbb{Q}(\\alpha)$. Write $\\mathcal{O}_K = \\mathbb{Z}[\\alpha]$ for the ring of integers of $K$. The only solutions $(x,y,u) \\in \\mathbb{Z} \\times \\mathbb{Z}\\times \\mathcal{O}_K$ to\n\\begin{align}\\label{eq:cubic over K}\n(\\alpha x - y)(x^2-xy+y^2) = 3(2-\\alpha) u^3\n\\end{align}\nwith $xy \\neq 0$ and $x,y$ coprime are $(x,y,u) = \\pm (1,2,-1)$.\n\\end{lemma}",
      "eq:ErdSelf eqn": "\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}",
      "eq:dlss curve": "\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}",
      "lem:selmer": "\\begin{lemma}\\label{lem:selmer}\nLet\n\\begin{align*}\n\\mathcal{L} = \\{&1, 3, 4, 5, 10, 14, 18, 21, 25, 36, 45, 60, 100, 147, 150, \\\\\n                &175, 196, 225, 245, 252, 300, 315, 350, 882, 980, \\\\\n                &1050, 1470, 1575, 1764, 2940, 7350, 14700\\}.\n\\end{align*}\nLet $a,b,c$ be nonzero integers, and let $D$ denote the cube-free part of $abc$. If $D \\in \\mathcal{L}$, then the equation $ax^3+by^3+cz^3=0$ has no solutions in nonzero integers $x,y,z$.\n\\end{lemma}",
      "eq:solns for main thm": "\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}",
      "eq:Bennett eqn": "\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}"
    },
    "pre_theorem_intro_text_len": 2628,
    "pre_theorem_intro_text": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$. \n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large). \n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$.  However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.",
    "context": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$.\n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large).\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.",
    "full_context": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$.\n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large).\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.\n\n\\begin{abstract}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. We determine all solutions to the equation\n\\begin{align*}\nn(n+d)(n+2d)\\cdots(n+(i-1)d)(n+(i+1)d) \\cdots (n+(k-1)d) = y^3\n\\end{align*}\nin integers $n,d,y$ with $ny \\neq 0$, $d\\geq 1$, and $\\textup{gcd}(n,d) = 1$. Our method relies on the theory of elliptic curves, including elliptic curve Chabauty over a number field. As an application, we answer a question of Das, Laishram, Saradha, and Sharma concerning rational points on a certain superelliptic curve.\n\\end{abstract}\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.\n\nIt would be interesting to extend Theorem \\ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.\n\n\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nLet $0 \\leq j < \\ell \\leq k-1$ with $j,\\ell \\neq i$, and let $g = \\text{gcd}(n+jd,n+\\ell d)$. Then $g$ divides $(n+\\ell d)-(n+jd) = (\\ell-j)d$, and since $\\text{gcd}(n,d)=1$ we find $g$ is coprime to $d$. Hence $g \\mid (\\ell - j)$. It follows that the greatest common divisor of $n+jd,n+\\ell d$ is at most $k-1$. Therefore, if $p\\geq k$ is a prime and $p \\mid y$, then $p$ divides a unique term $n+jd$.\n\nOur general strategy is to determine the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. Since the integers $a_j$ are cube-free and their largest prime factors are bounded, there are only a finite number of potential coefficient vectors to consider for given values of $k$ and $i$. Given a potential coefficient vector $\\textbf{v} = (a_0,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$, we may try to show $\\textbf{v}$ cannot correspond to solutions of \\eqref{eq:main eqn of study} by reducing to various cubic equations. More particularly, given distinct integers $r,s,t$ with $0 \\leq r,s,t \\leq k-1$ and $r,s,t \\neq i$, we have\n\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}\nIf the cubic equation $(s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3 = 0$ has no solutions in nonzero integers $x_r,x_s,x_t$, then $\\textbf{v}$ cannot be a coefficient vector corresponding to any solutions to \\eqref{eq:main eqn of study}.\n\n\\begin{lemma}\\label{lem:HTT}\nSuppose we have a solution to the equation\n\\begin{align*}\n\\prod_{j=0}^{k-1}(n+jd) = y^3\n\\end{align*}\nin integers $k,n,d,y$ with $4 \\leq k \\leq 38, d \\geq 1, ny \\neq 0$, and $\\textup{gcd}(n,d)=1$. Then $(k,n,d) = (4,-9,5)$ or $(4,-6,5)$.\n\\end{lemma}\n\nHence, we may assume $1 \\leq i \\leq k-2$. Suppose we have a solution to \\eqref{eq:main eqn of study}. If we set $m=n+(k-1)d$, then\n\\begin{align*}\ny^3 &= (-1)^{k-1}\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(-m+(k-1-j)d) = (-1)^{k-1} \\prod_{\\substack{\\ell = 0 \\\\ \\ell \\neq k-1-i}}^{k-1} (-m+\\ell d).\n\\end{align*}\nIf we absorb the sign $(-1)^{k-1}$ into $y^3$, then this is an equation of the form \\eqref{eq:main eqn of study} with $i$ replaced by $k-1-i$. In fact, this yields a bijective involution between different solutions to \\eqref{eq:main eqn of study} given by\n\\begin{align}\\label{eq:flipping bijection}\n(k,i,n,d) \\leftrightarrow (k,k-1-i,-n-(k-1)d,d).\n\\end{align}\nIt therefore suffices to consider $1 \\leq i \\leq \\frac{k-1}{2}$ in what follows.\n\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\n\n\\begin{theorem}\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}\n\\end{theorem}",
    "post_theorem_intro_text_len": 3396,
    "post_theorem_intro_text": "It would be interesting to extend Theorem \\ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.\n\nTheorem \\ref{thm:main thm} allows us to answer a question raised by Das, Laishram, Saradha, and Sharma \\cite[p. 1710]{DLSS2023} concerning rational points on a certain superelliptic curve.\n\n\\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}\n\nThe points on \\eqref{eq:dlss curve} with $y=0$ are the ``trivial'' rational points\n\\begin{align*}\n(-1,0),(-2,0),(-3,0),(-5,0),(-6,0),(-7,0).\n\\end{align*}\nThe authors of \\cite{DLSS2023} noted the existence of the ``nontrivial'' rational point $\\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, but did not provably determine all of the rational points on \\eqref{eq:dlss curve}.\n\n\\subsection{Outline of the paper} \n\nIn Section \\ref{sec:prelim}, we show how Corollary \\ref{cor:rat pts} follows from Theorem \\ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \\ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \\eqref{eq:main eqn of study}. This gives rise to the notion of a \\emph{coefficient vector}  (Definition \\ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. It will transpire that solutions to \\eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \\eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \\ref{lem:selmer}). We end the section with a general description of our strategy for determining potential coefficient vectors, which relies extensively on computer calculation. (All the computer code used in this paper, written in Magma \\cite{Magma} and Sage \\cite{Sage}, is available at our ``Cubes from products with one term missing'' GitHub repository \\cite{Pratt_cubes_2025}.)\n\nIn Section \\ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \\ref{sec:prelim} to solve \\eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$. There are two coefficient vectors in this case that correspond to solutions to \\eqref{eq:main eqn of study} (see \\eqref{eq:solns for main thm}). All the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \\ref{sec:prelim} are inapplicable.\n\nIn the last section of the paper, Section \\ref{sec:finishing proof of main thm}, we finish the analysis in the case $k=7,i=3$ and give the proof of Theorem \\ref{thm:main thm}, using a result on the simultaneous solutions to a pair of ternary cubic equations (Proposition \\ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K = \\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$. All the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method (see Lemma \\ref{lem:EC over K}).",
    "sketch": "In Section \\ref{sec:prelim}, we show how Corollary \\ref{cor:rat pts} follows from Theorem \\ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \\ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \\eqref{eq:main eqn of study}. This gives rise to the notion of a \\emph{coefficient vector} (Definition \\ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. Solutions to \\eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \\eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \\ref{lem:selmer}).\n\nIn Section \\ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \\ref{sec:prelim} to solve \\eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \\ref{sec:prelim} are inapplicable.”\n\nIn Section \\ref{sec:finishing proof of main thm}, we finish the analysis for $k=7,i=3$ and thereby prove Theorem \\ref{thm:main thm}, “using a result on the simultaneous solutions to a pair of ternary cubic equations” (Proposition \\ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K=\\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method” (Lemma \\ref{lem:EC over K}).",
    "expanded_sketch": "Next we show how \\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}\nfollows from the main theorem. We then prove an elementary but important lemma detailing the structure of the terms $n+jd$ in any solution to\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nnamely:\n\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}\nThis gives rise to the notion of a \\emph{coefficient vector}:\n\\begin{definition}\\label{defn: coeff vector}\nAssume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Let $n+jd=a_jx_j^3$ be the factorization as described in Lemma \\ref{lem:factorization of terms}. We call $(a_0,a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$ the \\emph{coefficient vector} corresponding to $y,n,d$.\n\\end{definition}\nA large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to the equation above. Solutions to that equation give rise to many different ternary cubic equations, for instance\n\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}\nand these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero. Specifically, we use the following lemma:\n\\begin{lemma}\\label{lem:selmer}\nLet\n\\begin{align*}\n\\mathcal{L} = \\{&1, 3, 4, 5, 10, 14, 18, 21, 25, 36, 45, 60, 100, 147, 150, \\\\\n                &175, 196, 225, 245, 252, 300, 315, 350, 882, 980, \\\\\n                &1050, 1470, 1575, 1764, 2940, 7350, 14700\\}.\n\\end{align*}\nLet $a,b,c$ be nonzero integers, and let $D$ denote the cube-free part of $abc$. If $D \\in \\mathcal{L}$, then the equation $ax^3+by^3+cz^3=0$ has no solutions in nonzero integers $x,y,z$.\n\\end{lemma}\n\nNext, we use elementary arguments and the results established earlier to solve\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results established earlier are inapplicable.”\n\nFinally, we finish the analysis for $k=7,i=3$ and thereby prove the main theorem, using the following result on the simultaneous solutions to a pair of ternary cubic equations:\n\\begin{proposition}\\label{prop:only solns to pair of cubic eqns}\nThe only solutions $(x,y,z,w) \\in \\mathbb{Z}^4$ to the pair of equations\n\\begin{align*}\nx^3+y^3&=9z^3 \\\\\n5x^3-y^3&=3w^3\n\\end{align*}\nwith $xyzw \\neq 0$ and $x,y$ coprime are $(x,y,z,w) = \\pm (1,2,1,-1)$.\n\\end{proposition}\nComputations in the pure cubic field $K=\\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method,” using the following lemma:\n\\begin{lemma}\\label{lem:EC over K}\nLet $\\alpha = \\sqrt[3]{5}$, and set $K = \\mathbb{Q}(\\alpha)$. Write $\\mathcal{O}_K = \\mathbb{Z}[\\alpha]$ for the ring of integers of $K$. The only solutions $(x,y,u) \\in \\mathbb{Z} \\times \\mathbb{Z}\\times \\mathcal{O}_K$ to\n\\begin{align}\\label{eq:cubic over K}\n(\\alpha x - y)(x^2-xy+y^2) = 3(2-\\alpha) u^3\n\\end{align}\nwith $xy \\neq 0$ and $x,y$ coprime are $(x,y,u) = \\pm (1,2,-1)$.\n\\end{lemma}",
    "expanded_theorem": "\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}.",
    "theorem_type": [
      "Classification or Bijection",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let integers $k,i,n,d,y$ satisfy $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$. Consider the equation obtained by deleting the $i$th term from a length-$k$ arithmetic progression product,\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3.\n\\]\nWhich statement correctly classifies all integer solutions?",
      "correct_choice": {
        "label": "A",
        "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand there are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\ntogether with their images under the involution\n\\[\n(k,i,n,d)\\mapsto (k,k-1-i,-n-(k-1)d,d),\n\\]\nand there are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."
        },
        {
          "label": "C",
          "text": "If\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3\n\\]\nhas an integer solution with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$, then necessarily $k\\in\\{5,7\\}$; in particular, there are no solutions for $k\\in\\{6,8,9,10,11\\}$."
        },
        {
          "label": "D",
          "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand, moreover, every integer solution with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$ must satisfy $1\\le i\\le \\frac{k-1}{2}$."
        },
        {
          "label": "E",
          "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand for each such quadruple there is exactly one corresponding integer $y$ satisfying\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3.\n\\]\nThere are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "flipping-bijection-used-only-for-reduction-not-for-adjoining-new-solutions",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped-complete-enumeration-of-the-eight-quadruples",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "range-restriction-on-i-applies-only-after-symmetry-reduction",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "theorem-classifies-quadruples-not-uniqueness-of-y",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly or implicitly reveal the correct tuple list. It asks for an exact classification, and no wording in the question itself singles out choice A over the alternatives."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-classification recall question: it asks for the exact exceptional tuples and 'no others.' However, it is not a pure verbatim restatement, since the options vary in completeness, logical strength, and misuse of symmetry."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish an exact classification from a weaker true statement, an incomplete list, and overgenerated cases. Still, the task mainly rewards recognition/recall of the classification rather than substantial fresh mathematical derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong and mathematically meaningful: one omits exceptional cases, one is a weaker true statement, one overextends a special case, and one misuses an involution. These reflect plausible failure modes and are clearly distinct."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no stem-based answer leakage and high-quality distractors, but it is somewhat theorem-restatement-like and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.17872v1",
    "paper_link": "http://arxiv.org/abs/2512.17872v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]",
      "start_pos": 8037,
      "end_pos": 8836,
      "label": "thmPoincare"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 3515,
    "pre_theorem_intro_text": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n    W^{1,p}(M,\\mathbb{R})$,\n\\[\n    \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n    \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n    \\mathbb{E}[f]\n    = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n    \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n    = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n    C(M,g,p,\\omega)$ such that\n\\[\n    \\|f - \\bE_\\omega[f]\\|_{L^p}\n    \\leq C \\|df\\|_{L^p}\n    \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.",
    "context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.",
    "full_context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.\n\nTo the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nWe now pass to polar coordinates for $y-x$. Note that $\\omega$ can be extended\n to zero outside of $U$. We also extend $df$ by zero outside of $U$ without\n changing the notation. This allows us not to care about restricting the domain\n of integration. Let also $d = \\diam(U)$. We have\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{r=0}^d \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) \\int_0^r |df_{x + s \\theta}|^t ds d\\theta dr,\n \\]\n where $d\\theta$ is the surface measure on the unit sphere $\\bS^{n-1} \\subset\n \\bR^n$. Hence,\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds.\n \\]\n Integrating against $x \\in U$, we get\n \\begin{align*}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq \\int_U \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds dx \\\\\n & \\leq \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} r^{n+t-2} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx \\\\\n & \\leq d^{n+t-2} \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx.\n \\end{align*}\n We change the order of integration and perform the change of variable $z = x + s \\theta$ to get:\n \\begin{align}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{x \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t dx d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{z \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr \\right) |df_{z}|^t dz d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr ds d\\theta\\right) |df_{z}|^t dz\\nonumber \\\\\n & \\leq d^{n+t-1} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{\\rho=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta\\right) |df_{z}|^t dz,\\label{eqPoincare3}\n \\end{align}\n where we also performed the change of variable $\\rho = r-s$ to obtain the last line. Now remark that the inner integral can be rewritten as follows:\n \\begin{align*}\n \\omegatil(z)\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta \\\\\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) \\rho^{-n-1} \\rho^{n-1} d\\rho d\\theta \\\\\n & = \\int_{\\bR^n} \\omega(z - x) \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}} dx,\n \\end{align*}\n where we have set $x = \\rho \\theta$. Note that the function $x \\mapsto \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}}$ belongs to $L^r(\\bR^n, \\bR)$ for any $r < \\frac{n}{n-1}$\\footnote{This function is actually a truncated Riesz potential for which we could use the Hardy-Littlewood-Sobolev inequality~\\cite[Section 7.8]{GilbargTrudinger}. This is not needed here.} so we use Young's inequality together with the fact that $q > \\frac{n}{2}$ to get\n \\[\n \\|\\omegatil\\|_{L^n} \\leq c \\|\\omega\\|_{L^q}\n \\]\n for some constant $c = c(n, q, d)$. We finally conclude\n from~\\eqref{eqPoincare3} that\n \\[\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx \\leq c d^{n+p-2} d^{n+t-1} \\int_{z \\in U} \\omegatil(z) |df_{z}|^t dz \\leq \\|\\omegatil\\|_{L^n} \\left(\\int_U |df|^p dx\\right)^{\\frac{n-1}{n}}.\n \\]\n\\end{proof}\n\nThe next step is to extend the previous result to a broader range for $t$ by\nmeans of the Sobolev inequality:\n\\begin{lemma}\\label{lmPoincare2}\n Let $U$ be an open non-empty bounded convex subset of $\\bR^n$ with smooth boundary. Let $p, q \\in (1, \\infty)$ be such that $p \\geq \\frac{n}{n-1}$ and $q > \\frac{n}{2}$. Let $r \\in [1, \\infty]$ be such that\n \\[\n \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n} \\quad\\text{if } p < n.\n \\]\n There exists a constant $c' = c'(U, p, q, r)$ such that, for any $\\omega \\in\n L^q(U, \\bR)$, $\\omega \\geq 0$ a.e., $\\bE[\\omega] = 1$ and any function $f \\in\n W^{1, p}(U, \\bR)$,\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^r} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n\\end{lemma}\n\n\\begin{proof}\n Let $t = \\frac{n-1}{n} p$ be as in Lemma~\\ref{lmPoincare1}. The conclusion of this lemma can be written as follows:\n \\begin{equation}\\label{eqPoincare4}\n \\|f - \\bE_\\omega[f]\\|_{L^t} \\leq (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\end{equation}\n As $U$ has finite volume, the conclusion of the lemma is immediate if $r \\leq t$ as there is a continuous inclusion $L^t(U, \\bR) \\hookrightarrow L^r(U, \\bR)$. If, instead, $r > t$, we let $p'$ be defined as follows:\n \\[\n p' =\n \\left\\lbrace\n \\begin{aligned}\n \\frac{np}{n-p} & \\quad\\text{if } p < n, \\\\\n \\infty & \\quad\\text{if } p > n, \\\\\n \\text{arbitrary} > p & \\quad\\text{if } p = n.\n \\end{aligned}\n \\right.\n \\]\n Let $\\ftil = f - \\bE_\\omega[f]$. The Sobolev embedding theorem implies that\n there exists a constant $s > 0$ such that\n \\begin{equation}\\label{eqPoincare5}\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\|\\ftil\\|_{L^p}\\right).\n \\end{equation}\n As $t < p < p'$, we have\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\|\\ftil\\|_{L^t}^\\theta \\|\\ftil\\|_{L^{p'}}^{1-\\theta}\\quad\\text{with } \\theta \\in (0, 1) \\text{ such that } \\frac{1}{p} = \\frac{\\theta}{t} + \\frac{1-\\theta}{p'}.\n \\]\n By the $\\epsilon$-Young inequality~\\cite[Equation $(7.6)$]{GilbargTrudinger},\n we deduce that, for any $\\epsilon > 0$,\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} \\|\\ftil\\|_{L^t} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n From~\\eqref{eqPoincare4}, we deduce that\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n Hence,~\\eqref{eqPoincare5} implies\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}\\right),\n \\]\n which we rewite as follows:\n \\[\n \\left(\\frac{1}{s} - (1-\\theta) \\epsilon\\right) \\|\\ftil\\|_{L^{p'}} \\leq \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n Choosing $\\epsilon$ such that $(1-\\theta) \\epsilon = \\frac{1}{2s}$, we conclude\n that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq 2s \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n As $1 = \\bE[\\omega] = \\|\\omega\\|_{L^1} \\leq \\vol(U)^{1 - \\frac{1}{q}}\n \\|\\omega\\|_{L^q}$, we can replace the constant $1$ in the previous inequality\n by\n \\[\n 1 = 1^{\\frac{n}{(n-1)p}} \\leq \\vol(U)^{\\left(1 - \\frac{1}{q}\\right)\\frac{n}{(n-1)p}} \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}.\n \\]\n Thus, we conclude that there is a constant $c'$ depending only on $U$ and $p$\n but not on $f$ such that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|d\\ftil\\|_{L^p},\n \\]\n i.e.\\ such that\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n Using once again that, if $r \\leq p'$, we have a continuous embedding\n $L^{p'}(U, \\bR) \\hookrightarrow L^r(U, \\bR)$, we obtain the conclusion of the\n lemma.\n\\end{proof}\n\nLet $f \\in W^{1,p}(M)$ and $\\omega \\in L^q(M)$ be non-negative with\n $\\mathbb{E}[\\omega] = 1$. We define $\\tilde{f} := f \\circ \\Psi$, and wish to\n construct a weight $\\tilde{\\omega}$ on $B$ such that\n $\\mathbb{E}[\\tilde{\\omega}] = 1$. For this, we use the coarea formula (see\n e.g.~\\cite[Chapter 5.2]{KrantzParks}):\n \\begin{equation}\\label{eqCoarea}\n \\int_B h(x) \\, dx = \\int_M \\left(\\sum_{x \\in \\Psi^{-1}(y)} \\frac{h(x)}{|\\det d\\Psi_x|} \\right) d\\mu^g(y),\n \\end{equation}\n which holds for any integrable function $h : B \\to \\bR$.",
    "post_theorem_intro_text_len": 650,
    "post_theorem_intro_text": "To the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nThe outline of the paper is as follows. In Section~\\ref{secRn}, we address the\ncase of bounded convex open subsets of $\\mathbb{R}^n$. In Section~\\ref{secDiffeo}, we\nconstruct a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in\n$\\mathbb{R}^n$ onto the manifold $M$. Finally, in Section~\\ref{secProof}, we use the\ncoarea formula to lift the function and $\\omega$ from $M$ to $B$ to prove our\nmain result.\n\n\\subsection*{Acknowledgments}\n\nI am grateful to Laurent Véron for useful discussion about the paper.",
    "sketch": "To prove the main result (Theorem~\\ref{thmPoincare}), the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, in Section~\\ref{secProof}, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main result.",
    "expanded_sketch": "To prove the main theorem, the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main theorem.",
    "expanded_theorem": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]",
    "theorem_type": [
      "Inequality or Bound",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $p,q\\in(1,\\infty)$ satisfy\n\\[\np\\ge \\frac{n}{n-1}\n\\qquad\\text{and}\\qquad\nq>\\frac{n}{2}.\n\\]\nLet $r\\in(1,\\infty]$, and assume that if $p<n$, then\n\\[\n\\frac{1}{r}\\ge \\frac{1}{p}-\\frac{1}{n}.\n\\]\nFor a nonnegative function $\\omega\\in L^q(M)$ with\n\\[\n\\int_M \\omega\\,d\\mu^g = 1,\n\\]\ndefine the weighted average of $f$ by\n\\[\n\\mathbb E_\\omega[f] := \\int_M f\\,\\omega\\,d\\mu^g.\n\\]\nHere $f\\in W^{1,p}(M,\\mathbb R)$, $df$ denotes its differential, and all $L^r$ and $L^q$ norms are taken with respect to the Riemannian volume measure $d\\mu^g$. Which quantitative estimate holds for every such $f$ and $\\omega$?",
      "correct_choice": {
        "label": "A",
        "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nprovided that, if $p<n$, then\n\\[\n\\frac{1}{r}> \\frac{1}{p}-\\frac{1}{n}.\n\\]"
        },
        {
          "label": "C",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\bigl(\\|df\\|_{L^p}+\\|f\\|_{L^p}\\bigr).\n\\]"
        },
        {
          "label": "D",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nwhere the constant $C$ may depend on $\\omega$ but is independent of $f$."
        },
        {
          "label": "E",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)q}}\\,\\|df\\|_{L^p}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical Sobolev endpoint for r",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "pure gradient control by adding an extra $\\|f\\|_{L^p}$ term",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "uniformity of the constant with respect to the density after lifting via coarea",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "power of $\\|\\omega\\|_{L^q}$ coming from the Euclidean estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and definitions but does not reveal the specific inequality, exponent of \\|\\omega\\|_{L^q}, or the precise uniformity claim. The correct answer is not stated or strongly signaled."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-identification question: it asks for the correct uniform conclusion under the stated assumptions. The options do introduce meaningful competing formulations, but the task remains close to recalling the exact statement rather than applying it in a new setting."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the sharp weight exponent, the role of uniform dependence on \\omega, and the admissible target space L^r versus L^\\infty. However, the question mostly tests recognition/recall of the precise result rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they vary the \\|\\omega\\|_{L^q} exponent, weaken uniformity by allowing dependence on \\omega, or overstate the target norm. These reflect common failure modes and are clearly distinct."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recall MCQ with no answer leakage and strong, plausible distractors, but it only moderately avoids tautology and does not strongly test generative reasoning."
    }
  },
  {
    "id": "2512.17872v1",
    "paper_link": "http://arxiv.org/abs/2512.17872v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]",
      "start_pos": 8037,
      "end_pos": 8836,
      "label": "thmPoincare"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 3515,
    "pre_theorem_intro_text": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n    W^{1,p}(M,\\mathbb{R})$,\n\\[\n    \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n    \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n    \\mathbb{E}[f]\n    = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n    \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n    = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n    C(M,g,p,\\omega)$ such that\n\\[\n    \\|f - \\bE_\\omega[f]\\|_{L^p}\n    \\leq C \\|df\\|_{L^p}\n    \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.",
    "context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.",
    "full_context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.\n\nTo the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nWe now pass to polar coordinates for $y-x$. Note that $\\omega$ can be extended\n to zero outside of $U$. We also extend $df$ by zero outside of $U$ without\n changing the notation. This allows us not to care about restricting the domain\n of integration. Let also $d = \\diam(U)$. We have\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{r=0}^d \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) \\int_0^r |df_{x + s \\theta}|^t ds d\\theta dr,\n \\]\n where $d\\theta$ is the surface measure on the unit sphere $\\bS^{n-1} \\subset\n \\bR^n$. Hence,\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds.\n \\]\n Integrating against $x \\in U$, we get\n \\begin{align*}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq \\int_U \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds dx \\\\\n & \\leq \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} r^{n+t-2} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx \\\\\n & \\leq d^{n+t-2} \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx.\n \\end{align*}\n We change the order of integration and perform the change of variable $z = x + s \\theta$ to get:\n \\begin{align}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{x \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t dx d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{z \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr \\right) |df_{z}|^t dz d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr ds d\\theta\\right) |df_{z}|^t dz\\nonumber \\\\\n & \\leq d^{n+t-1} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{\\rho=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta\\right) |df_{z}|^t dz,\\label{eqPoincare3}\n \\end{align}\n where we also performed the change of variable $\\rho = r-s$ to obtain the last line. Now remark that the inner integral can be rewritten as follows:\n \\begin{align*}\n \\omegatil(z)\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta \\\\\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) \\rho^{-n-1} \\rho^{n-1} d\\rho d\\theta \\\\\n & = \\int_{\\bR^n} \\omega(z - x) \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}} dx,\n \\end{align*}\n where we have set $x = \\rho \\theta$. Note that the function $x \\mapsto \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}}$ belongs to $L^r(\\bR^n, \\bR)$ for any $r < \\frac{n}{n-1}$\\footnote{This function is actually a truncated Riesz potential for which we could use the Hardy-Littlewood-Sobolev inequality~\\cite[Section 7.8]{GilbargTrudinger}. This is not needed here.} so we use Young's inequality together with the fact that $q > \\frac{n}{2}$ to get\n \\[\n \\|\\omegatil\\|_{L^n} \\leq c \\|\\omega\\|_{L^q}\n \\]\n for some constant $c = c(n, q, d)$. We finally conclude\n from~\\eqref{eqPoincare3} that\n \\[\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx \\leq c d^{n+p-2} d^{n+t-1} \\int_{z \\in U} \\omegatil(z) |df_{z}|^t dz \\leq \\|\\omegatil\\|_{L^n} \\left(\\int_U |df|^p dx\\right)^{\\frac{n-1}{n}}.\n \\]\n\\end{proof}\n\nThe next step is to extend the previous result to a broader range for $t$ by\nmeans of the Sobolev inequality:\n\\begin{lemma}\\label{lmPoincare2}\n Let $U$ be an open non-empty bounded convex subset of $\\bR^n$ with smooth boundary. Let $p, q \\in (1, \\infty)$ be such that $p \\geq \\frac{n}{n-1}$ and $q > \\frac{n}{2}$. Let $r \\in [1, \\infty]$ be such that\n \\[\n \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n} \\quad\\text{if } p < n.\n \\]\n There exists a constant $c' = c'(U, p, q, r)$ such that, for any $\\omega \\in\n L^q(U, \\bR)$, $\\omega \\geq 0$ a.e., $\\bE[\\omega] = 1$ and any function $f \\in\n W^{1, p}(U, \\bR)$,\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^r} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n\\end{lemma}\n\n\\begin{proof}\n Let $t = \\frac{n-1}{n} p$ be as in Lemma~\\ref{lmPoincare1}. The conclusion of this lemma can be written as follows:\n \\begin{equation}\\label{eqPoincare4}\n \\|f - \\bE_\\omega[f]\\|_{L^t} \\leq (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\end{equation}\n As $U$ has finite volume, the conclusion of the lemma is immediate if $r \\leq t$ as there is a continuous inclusion $L^t(U, \\bR) \\hookrightarrow L^r(U, \\bR)$. If, instead, $r > t$, we let $p'$ be defined as follows:\n \\[\n p' =\n \\left\\lbrace\n \\begin{aligned}\n \\frac{np}{n-p} & \\quad\\text{if } p < n, \\\\\n \\infty & \\quad\\text{if } p > n, \\\\\n \\text{arbitrary} > p & \\quad\\text{if } p = n.\n \\end{aligned}\n \\right.\n \\]\n Let $\\ftil = f - \\bE_\\omega[f]$. The Sobolev embedding theorem implies that\n there exists a constant $s > 0$ such that\n \\begin{equation}\\label{eqPoincare5}\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\|\\ftil\\|_{L^p}\\right).\n \\end{equation}\n As $t < p < p'$, we have\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\|\\ftil\\|_{L^t}^\\theta \\|\\ftil\\|_{L^{p'}}^{1-\\theta}\\quad\\text{with } \\theta \\in (0, 1) \\text{ such that } \\frac{1}{p} = \\frac{\\theta}{t} + \\frac{1-\\theta}{p'}.\n \\]\n By the $\\epsilon$-Young inequality~\\cite[Equation $(7.6)$]{GilbargTrudinger},\n we deduce that, for any $\\epsilon > 0$,\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} \\|\\ftil\\|_{L^t} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n From~\\eqref{eqPoincare4}, we deduce that\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n Hence,~\\eqref{eqPoincare5} implies\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}\\right),\n \\]\n which we rewite as follows:\n \\[\n \\left(\\frac{1}{s} - (1-\\theta) \\epsilon\\right) \\|\\ftil\\|_{L^{p'}} \\leq \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n Choosing $\\epsilon$ such that $(1-\\theta) \\epsilon = \\frac{1}{2s}$, we conclude\n that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq 2s \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n As $1 = \\bE[\\omega] = \\|\\omega\\|_{L^1} \\leq \\vol(U)^{1 - \\frac{1}{q}}\n \\|\\omega\\|_{L^q}$, we can replace the constant $1$ in the previous inequality\n by\n \\[\n 1 = 1^{\\frac{n}{(n-1)p}} \\leq \\vol(U)^{\\left(1 - \\frac{1}{q}\\right)\\frac{n}{(n-1)p}} \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}.\n \\]\n Thus, we conclude that there is a constant $c'$ depending only on $U$ and $p$\n but not on $f$ such that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|d\\ftil\\|_{L^p},\n \\]\n i.e.\\ such that\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n Using once again that, if $r \\leq p'$, we have a continuous embedding\n $L^{p'}(U, \\bR) \\hookrightarrow L^r(U, \\bR)$, we obtain the conclusion of the\n lemma.\n\\end{proof}\n\nLet $f \\in W^{1,p}(M)$ and $\\omega \\in L^q(M)$ be non-negative with\n $\\mathbb{E}[\\omega] = 1$. We define $\\tilde{f} := f \\circ \\Psi$, and wish to\n construct a weight $\\tilde{\\omega}$ on $B$ such that\n $\\mathbb{E}[\\tilde{\\omega}] = 1$. For this, we use the coarea formula (see\n e.g.~\\cite[Chapter 5.2]{KrantzParks}):\n \\begin{equation}\\label{eqCoarea}\n \\int_B h(x) \\, dx = \\int_M \\left(\\sum_{x \\in \\Psi^{-1}(y)} \\frac{h(x)}{|\\det d\\Psi_x|} \\right) d\\mu^g(y),\n \\end{equation}\n which holds for any integrable function $h : B \\to \\bR$.",
    "post_theorem_intro_text_len": 650,
    "post_theorem_intro_text": "To the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nThe outline of the paper is as follows. In Section~\\ref{secRn}, we address the\ncase of bounded convex open subsets of $\\mathbb{R}^n$. In Section~\\ref{secDiffeo}, we\nconstruct a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in\n$\\mathbb{R}^n$ onto the manifold $M$. Finally, in Section~\\ref{secProof}, we use the\ncoarea formula to lift the function and $\\omega$ from $M$ to $B$ to prove our\nmain result.\n\n\\subsection*{Acknowledgments}\n\nI am grateful to Laurent Véron for useful discussion about the paper.",
    "sketch": "To prove the main result (Theorem~\\ref{thmPoincare}), the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, in Section~\\ref{secProof}, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main result.",
    "expanded_sketch": "To prove the main theorem, the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main theorem.",
    "expanded_theorem": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]",
    "theorem_type": [
      "Inequality or Bound",
      "Existential–Universal"
    ],
    "mcq": {
      "question": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $p,q\\in(1,\\infty)$ satisfy\n\\[\np\\ge \\frac{n}{n-1}\n\\qquad\\text{and}\\qquad\nq>\\frac{n}{2}.\n\\]\nLet $r\\in(1,\\infty]$, and assume that if $p<n$, then\n\\[\n\\frac{1}{r}\\ge \\frac{1}{p}-\\frac{1}{n}.\n\\]\nFor a nonnegative function $\\omega\\in L^q(M)$ with\n\\[\n\\int_M \\omega\\,d\\mu^g = 1,\n\\]\ndefine the weighted average of $f$ by\n\\[\n\\mathbb E_\\omega[f] := \\int_M f\\,\\omega\\,d\\mu^g.\n\\]\nHere $f\\in W^{1,p}(M,\\mathbb R)$, $df$ denotes its differential, and all $L^r$ and $L^q$ norms are taken with respect to the Riemannian volume measure $d\\mu^g$. Which quantitative estimate holds for every such $f$ and $\\omega$?",
      "correct_choice": {
        "label": "A",
        "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nprovided that, if $p<n$, then\n\\[\n\\frac{1}{r}> \\frac{1}{p}-\\frac{1}{n}.\n\\]"
        },
        {
          "label": "C",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\bigl(\\|df\\|_{L^p}+\\|f\\|_{L^p}\\bigr).\n\\]"
        },
        {
          "label": "D",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nwhere the constant $C$ may depend on $\\omega$ but is independent of $f$."
        },
        {
          "label": "E",
          "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)q}}\\,\\|df\\|_{L^p}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "critical Sobolev endpoint for r",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "pure gradient control by adding an extra $\\|f\\|_{L^p}$ term",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "uniformity of the constant with respect to the density after lifting via coarea",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "power of $\\|\\omega\\|_{L^q}$ coming from the Euclidean estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or reproduce the target estimate itself. It specifies hypotheses and asks for the valid conclusion, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem lists the precise assumptions and the correct choice is the theorem's conclusion almost verbatim. The task is mainly to identify the exact stated estimate rather than derive a new consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish endpoint admissibility, the exponent of \\(\\|\\omega\\|_{L^q}\\), uniformity of the constant, and the weaker true variant with an added \\(\\|f\\|_{L^p}\\) term. However, the item primarily tests recognition of the exact theorem statement rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: excluding the critical Sobolev endpoint, weakening the estimate by adding an extra norm term, allowing non-uniform dependence on \\(\\omega\\), and altering the exponent incorrectly. They are distinct and well aligned with common confusions."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the result and only moderately tests genuine reasoning."
    }
  },
  {
    "id": "2512.18590v1",
    "paper_link": "http://arxiv.org/abs/2512.18590v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}",
      "start_pos": 5163,
      "end_pos": 5713,
      "label": "Mth"
    },
    "ref_dict": {
      "Mth": "\\begin{thm}\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\m(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}\n\t\\end{thm}",
      "subsec:bundle": "\\label{subsec:bundle}\nFor the sake of simplicity,\tin the sequel we will use the same notation for (the equivalence class of) a vector bundle and (the homotopy class of) its classifying map from it bas",
      "subsec:gen": "\\label{subsec:gen}\n\nA natural idea to study the mapping class group is to relate it to the group of homotopy equivalences. To be precise, let $M$ be a simply closed connected manifold with $\\dim M>5$.",
      "subsec:det": "\\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.    \n\n\t\tBy Lemma \\ref{t7} and the exactness of $S_\\partial(N_r\\times I)\\to I(N_r)\\to I^h(N_r)$, the kernel of the forgetful homomorphism $I(N_r) \\to I^h(N_r)$ is generated by $g_1$. Thus we only need to show that the image of $g_2$ is a generator of $I^h(N_r)$. This will be done by showing that the image of $g_2$ generates  the image of $\\mu_N^*$ (see Lemma \\ref{Ih}).\n\n\t\tLet $\\mu_3:S^3\\times D^3 \\to (S^3\\times D^3)\\vee S^6$ be the pinch map. It induces a map\n\t\t$$\\mu_3^* \\colon \\pi_6(S^3)\\to \\pi_0(G_\\partial(S^3\\times D^3)), \\ \\alpha \\mapsto (id\\vee\\alpha)\\circ\\mu_3.$$\nwhere we represent $\\alpha \\in \\pi_6(S^3)$ by a map $S^6 \\to S^3 \\times \\{0\\}$. \nView $S^3\\times D^3$ as a submanifold of $N_r$ through the embedding $i\\circ i_\\eta \\colon S^3 \\times D^3 \\to S^2 \\times D^4 \\to N_r$, extending the map $\\mu^*(\\alpha)$ by the identity on the complement we have a homotopy equivalence $\\mu^*(\\alpha)$ of $N_r$. For $\\alpha\\in\\pi_6(S^3)$, consider the composite $i \\circ \\eta \\circ \\alpha \\colon S^6 \\to S^3 \\to S^2 \\to N_r$. From the construction of the maps one readily verifies that $\\mu_3^*(\\alpha)=\\mu_N^*(i \\circ\\eta \\circ \\alpha)\\in I^h(N_r)$, i.e., there is a commutative diagram\n$$\\xymatrix{\\pi_6(S^3) \\ar[r]^{(i \\circ \\eta)_*} \\ar[d]_{\\mu_3^*} & \\pi_6(N_r) \\ar[d]^{\\mu_N^*} \\\\\n \\pi_0(G_\\partial(S^3\\times D^3)) \\ar[r]^{\\ \\ \\ \\ \\ \\Phi} & I^h(N_r)}$$\n where the map $\\Phi$ at the bottom is the extension by the identity on the complement of $i\\circ i_{\\eta}(S^3 \\times D^3)$.\n\nIt can be verified by elementary homotopy theory (compare \\cite[p.98]{BeMa}) that the homotopy Dehn twist map $De^h:\\Omega^3 G(S^3)\\to G_\\partial(S^3\\times D^3)$ defined by $De^h(\\alpha)(x,y)=(\\alpha(y)x,y)$ \ninduces an isomorphism \n$\\pi_0(\\Omega^3G(S^3)) \\to \\pi_0(G_\\partial(S^3\\times D^3))$.  Under the natural isomorphisms\n$$\\pi_0(\\Omega^3G(S^3)) \\cong \\pi_3(G(S^3), \\mathrm{id}) \\cong \\pi_3((\\Omega^3S^3)_{\\mathrm{id}}) \\cong \\pi_6(S^3),$$\nwhere $(\\Omega^3S^3)_{\\mathrm{id}}$ is the connected component containing $\\mathrm{id}_{S^3}$, this map can be identified with $\\mu_3^* \\colon \\pi_6(S^3) \\to \\pi_0(G_\\partial(S^3\\times D^3))$. The following commutative diagram\n$$\\xymatrix{\\pi_3(SO(3))\\ar[d]\\ar[rr]^J&&\\pi_3(\\Omega^3S^3)\\ar[d]_{\\mu_3^*}\\\\\n\t\t\t\\pi_3(SO(4))\\ar[r]^{De \\ \\ \\ \\ }&\\m_{\\partial}(S^3\\times D^3)\\ar[r]&\\pi_0(G_\\partial(S^3\\times D^3))\n\t\t\t}$$ \nshows that the image of $g_2$ in $I^h(N_r)$ equals $\\Phi(\\mu_3^*(J(t_1)))$.\nThe $J$-homomorphism $J \\colon \\pi_3(SO(3))\\to\\pi_6(S^3)$ is surjective. Therefore by Corollary \\ref{htg} and Lemma \\ref{Ih},  the image of $g_2$ generates $I^h(N_r)$.   This finishes the proof of this proposition.\t\n\n\t\\end{proof}\n\n\t\\subsection{determination of $I(N_r)$}\\label{subsec:det}\nWe have shown in last subsection that the Torelli group $I(N_r)$ is generated by the diffeomorphisms $g_1$ and $g_2$. They obviously commute with each other since they are supported in disjoint regions. In this subsection we determine the orders of them using the generalized Kreck-Stolz invariant developed in \\cite{KreSu}.\n\nLet  $B=\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin$, $p \\colon B\\to BO$ be the composition of the projection $B \\to BSpin$ and the canonical projection $BSpin \\to BO$.    \n\nSince $r \\in 8\\z +5$, we may identify $N_r$ with $\\mathbb{P}(\\xi_{1,l})$ for some $l \\in \\z$. By Proposition \\ref{Hcp}, $w_2(N_r)$ vanishes, hence $N_r$ is spin. The normal Gauss map  $g \\colon N_r\\to BO$ of $N_r$ admits a lifting $g_{spin}$ to $BSpin$, which is  unique (up to fiber homotopy) since $N_r$ is simply connected. Let $\\{s, t\\}$ be the basis of $H^2(N_r)$ given in Proposition \\ref{Hcp}, then the map $\\hat{g}=(s,t,g_{spin}) \\colon N_r \\to \\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin=B$ is the normal $B$-structure corresponding to the polarization $((s,t),0)$ (see \\cite{Kre18} for definitions). \n\nLet $x=c_1(O(-1)\\times0\\times0)$, $y=c_1(0\\times O(-1)\\times0)$ be a basis of $H^2(B)$. \tOur normal $B$-structure satisfies the condition of Theorem 5 of \\cite{Kre18}, since $H^2(B)=\\mathbb{Z}\\{x,y\\}, H^4(B)=\\mathbb{Z}\\{x^2,xy,y^2,\\hat{p}_1\\}$, where $\\hat{p}_1$ is the first spin Potrjagin class of the universal spin stable vector bundle. The homomorphism $g^* \\colon H^4(B) \\to H^4(N_r)$ is given by  \n\t\t$$\\hat{g}^* \\colon H^4(B)\\to H^4(N_r), (x^2,xy,y^2,\\hat{p}_1)\\to (s^2,st,st-ls^2,(2l-2)s^2),$$\nand in $H^4(N_r)$ we have the relations (see Proposition \\ref{Hcp})\n$$\\hat{g}^*(\\hat{p}_1)=-\\hat{p}_1(N_r)=(2l-2)s^2, \\ \\ t^2-ts+ls^2=0.$$\nTherefore \n\t$$\\mathrm{Ker}(\\hat{g}^* \\colon H^4(B)\\to H^4(N_r))=\\mathbb{Z}\\{y^2-xy+lx^2,\\hat{p}_1+2(1-l)x^2\\}$$\nLet $\\alpha=y^2-xy+lx^2$,  $\\beta=\\hat{p}_1+2(1-l)x^2\\in H^4(B)$ and $s_1=\\alpha^2$, $s_2=\\alpha^2+\\alpha\\beta$, $ s_3=\\beta^2\\in H^8(B)$.\n\n\tFix an embedded $6$-disk $D \\subset N_r$, let $I(N_r,D)$ be the group of $D$-fixing isotopy classes of $D$-fixing diffeomorphisms which induce the identity on cohomology. By the disk theorem of Palais and the isotopy extension theorem, the forgetful homomorphism $I(N_r,D)\\to I(N_r)$ is surjective. By a theorem from (\\cite{Wa63} p.265), $\\mathrm{Ker}(I(N_r,D)\\to I(N_r))$ is trivial or isomorphic to $\\mathbb{Z}_2$. \n\nGiven a diffeomorphism $f \\colon N_r \\to N_r$ with $f|_D = \\mathrm{id}$, let $T_f$ be the mapping torus of $f$ with the preferred embedding $S^1\\times D \\subset T_f$. Assume that the action of $f$ on the cohomology groups is trivial,  then $f$ preserves the normal $B$-structure, and henceforth there is a normal $B$-structure $\\hat{G}$ on $T_f$, $\\hat G \\colon T_f \\to B$,  \nextending $\\hat{g}$ and the restriction of $\\hat G$ on $S^1 \\times D$ is the ordinary normal $B$ structure on $S^1\\times D$. It's ready to see by obstruction theory that the normal $B$-structure $\\hat G$ is a unique up to homotopy. It is shown in \\cite[Theorem 6]{Kre18} that the $7$-dimensional $B$-bordism group $\\Omega_7(B,p)$ is trivial, therefore $(T_f,\\hat{G})$ admits a coboundary $(W,\\bar{G})$ whose signature $\\mathrm{sign}(W)$  is $0$. Notice that the restriction of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta$ on $\\partial W= T_f$ vanish, let $\\bar{\\alpha}$, $\\bar{\\beta}\\in H^4(W,\\partial W)$ be a preimage of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta\\in H^4(W)$, respectively. We have characteristic numbers $\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle \\in \\z^3$. For $8$-dimensional $B$-bordism classes $[X,\\varphi] \\in \\Omega_8(B,p)$, we have characteristic numbers $\\langle (s_1, s_2,s_3), \\varphi_*[X] \\rangle \\in \\z^3$ in a similar way. Let $L=(s_1,s_2,s_3)\\Omega_8^0(B)\\subset \\mathbb{Z}^3$ be the lattice generated by the characteristic numbers on the bordism group $\\Omega^0_8(B,p)=\\mathrm{Ker}(\\mathrm{sign} \\colon \\Omega_8(B,p)\\to \\mathbb{Z})$. The $KS$-invariant is defined by\n\t$$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$$\n\t$$KS(f)=(s_1,s_2,s_3)(f):=\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle.$$\n\n\t\\begin{prop}\\label{KS}\n\t\t$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$ is a well defined injective homomorphism. \n\t\\end{prop}\n\t\\begin{proof}\n\t\tThis proof is along the same line of the proof of Proposition 6.2 of \\cite{KreSu}, with more details to be checked. \n\n\t\tTo prove $KS$ is well defined, suppose $(W_1,\\bar{G}_1),(W_2,\\bar{G}_2)$ are two $B$ coboundary of $(T_f,\\hat{G})$ chose embeddings $i_j: W_j\\to \\mathbb{R}^N\\times (-1)^j[0,+\\infty)$ good enough such that $i_j(W_j)\\cap\\mathbb{R}^N=T_f$. Define $W:=i_1(W_1)\\cup i_2(W_2)\\cong W_1\\cup_\\partial W_2$ then $(W,G_w:=\\bar{G}_1\\cup \\bar{G}_2)$ is a well defined element in $\\Omega_8(B,p)$, by \\cite{Wa69} $sign(W)=0$. Consider the following diagram\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_f)\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_f)\\\\\n\t\t\tH^4(W,T_f)\\ar[r]\\ar[u]^\\cong &H^4(W)\\ar[r]\\ar[u]&H^4(T_f)\\ar[u]^{injective}\n\t\t}$$\n\t\tThis implies the preimage of $G_w^*\\alpha\\in H^4(W)$ in $H^4(W,T_f)\\cong\\oplus_j H^4(W_j,T_f)$ is a preimage of $(\\bar{G}_1\\alpha, \\bar{G}_2\\alpha)$ ($\\beta$ resp.), and the difference of $KS(f)$ defined by $W_2$ and $W_1$ equals $(s_1,s_1+s_2,s_3)(W,G_w)\\in L$. Thus we have proved $KS$ is well defined. \n\n\t\tTo prove $KS$ is a homomorphism, suppose $f_3,f_4\\in I(N_r,D)$ and $(W_j,\\bar{G}_j)$ is a $B$ coboundary of $(T_{f_j},\\hat{G})$. Identify $(n,t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_3}$ and $(n,1-t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_4}$, define $W':=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$, then $\\partial W'=T_{f_3\\circ f_4}$. By the $M-V$ sequence of $(W';W_3,W_4)$ there are $(s',t')\\in [W',\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty]$ extending the cohomology factor of $\\bar{G}_j$. Since the pasting $W'=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$ preserve the spin structure of $\\bar{G}_j$ there is spin structure on $W'$ extending the spin factor of $\\bar{G}_j$. Thus we can define a $B$ normal structure $G'_w$ on $W'$ extending $\\bar{G}_j$. Consider the following diagram\n\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_{f_j})\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_{f_j})\\\\\n\t\t\tH^4(W',T_{f_3}\\cup T_{f_4})\\ar[r]\\ar[u]^\\cong \\ar[d]&H^4(W')\\ar[r]\\ar[u]\\ar[d]&H^4(T_{f_3}\\cup T_{f_4})\\ar[u]^{injective}\\ar[d]\\\\\n\t\t\tH^4(W',T_{f_3\\circ f_4})\\ar[r]&H^4(W')\\ar[r]&H^4(T_{f_3\\circ f_4})\\\\\n\t\t}$$\n\t\tArguments analog to the proof that $KS$ is well defined, one can show $KS$ is a homomorphism through this diagram. Note that we need $H^4(T_{f_3}\\cup T_{f_4})\\to \\oplus_j H^4(T_{f_j}))$ being injective since we need a preimage of $G'_w\\alpha\\in H^4(W')$ in $H^4(W',T_{f_3}\\cup T_{f_4})$. \n\n\t\tBy Theorem 5 of \\cite{Kre18}, if $KS(f)=0$ then there is a $B$ coboundary $(W'',\\hat{G})$ of $(T_f,\\hat{G})$ such that $(W'';N_r\\times [0,\\frac{1}{2}],N_r\\times [\\frac{1}{2},1])$ is\n\t\tan $h$-cobordism. Since the spin structure of $W''$ is compatible to $S^1\\times D$ we may extend $S^1\\times D$ to be $D^2\\times D\\subset W$ and define a relative $h$-cobordism. Hence by the peudo-isotopy theorem of Cerf, $f=id\\in I(N_r, D)$. \n\n\t\\end{proof}",
      "prop:real3": "\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}",
      "ntb": "\\begin{prop}\\label{ntb}\n\t\t$h^*\\xi_r$ is nontrivial if and only if $r\\in 8\\mathbb{Z}+5$. \n\t\\end{prop}"
    },
    "pre_theorem_intro_text_len": 2243,
    "pre_theorem_intro_text": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$.  In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds. \n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp  \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$. \n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as  projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale  \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently,  such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$. \n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.",
    "context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}",
    "full_context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\m(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\nThe projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\p^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\p^2$.\n\n\\begin{cor}\\label{gmhs}\n Let $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum_i x^k_iy_i=0\\}$, then\n \\begin{enumerate}\n \\item there is a short exact sequence \n $$ 1 \\to I(M_k) \\to \\m(M_k) \\to A \\to 1,$$\n where $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n \\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$. \n \\end{enumerate}\n \\end{cor}\n\n\\begin{eg} \\label{eg:1}\n Let $\\zeta_k$ be the complex vector bundle over $\\p^2$ with total space \n $$E_r=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{C}^3 \\ | \\ \\sum z^k_iz'_i=0\\}$$\n and bundle projection $\\pi \\colon E_r \\to \\p^2$, $\\pi(z,z')=z$.\n One can identify $\\zeta_k$ as conjugation of the orthogonal complement of canonical embedding $O(-k)\\to\\mathbb{C}^3$. So $\\zeta_k\\cong \\xi_{k,k^2}$ and $p_1(\\phi( \\zeta_k))=-3k^2$. Notice that $\\mathbb{P}(\\zeta_1)$ is the Milnor hypersurface $\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z_iz'_i=0\\}$. We call $M_k=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z^k_iz'_i=0\\}=\\mathbb{P}(\\zeta_k)$ a generalized Milnor hypersurface. One can easily show that the canonical generators of $M_k$ induced from the bundle $\\zeta_k$ are $(\\pi^* x,\\pi'^* x)$. Here $\\pi(z,z')=z, \\pi'(z,z')=z'$. \n\\end{eg}\n\n\\begin{lemma}\\label{bh}\nLet $\\xi$ be the nontrivial $3$-dimensional real vector bundle over $S^5$, $\\partial \\colon \\pi_{i+1}(S^5)\\to \\pi_i(S^2)$ be the boundary homomorphism in the long exact sequence of homotopy groups associated to the sphere bundle of $\\xi$, $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5)$ be the map induced by the adjoint of the identity map $S^5= \\Sigma S^4 \\to S^5$. Then the composite $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5) = \\pi_{i+1}(S^5) \\to \\pi_i(S^2)$ is identified with the homomorphism $\\eta^2 \\colon \\pi_i(S^4) \\to \\pi_i(S^2)$ induced by the Hopf map $\\eta \\colon S^3 \\to S^2$.\n\n\\begin{cor}\\label{htg}\nLet $N_r$ be the $S^2$-bundle over $ \\mathbb{P}^2$ defined as above. Assume $r\\in 8\\mathbb{Z}+5$. Then\n\\begin{enumerate}\n\\item the inclusion of a fiber induces an isomorphism $\\pi_3(S^2)\\to\\pi_3(N_r)$; \n\\item$\\pi_4(N_r)=0$; \n\\item the bundle projection induces an injective homomorphism $\\pi_5(N_r)\\to \\pi_5(\\mathbb{P}^2)$ with image $2 \\cdot \\pi_5(\\mathbb{P}^2)$;\n\\item $\\pi_6(S^2)\\to\\pi_6(N_r)$ is surjective and $\\pi_6(N_r)\\cong \\mathbb{Z}_6$.\n\\end{enumerate}\n \\end{cor}\n\n\\begin{lemma}\\label{t7}\n Let $\\Theta_7\\cong \\mathbb{Z}_{28}$ be the Milnor-Kervaire group of oriented differential structures on the $7$-sphere. Then connected sum operation induces a surjective map $cs:\\Theta_7\\to S_\\partial(N_r\\times I)$. \n \\end{lemma} \n \\begin{proof}\nConsider the smooth surgery exact sequence ( Theorem 11.22 of \\cite{LuTi})\n$$L_8(\\z) \\to S_\\partial(N_r\\times I) \\to [\\Sigma N_r, G/O] \\to L_7(\\z).$$\nA direct calculation shows that the group $H^i(\\Sigma N_r;\\pi_i(G/O))$ vanishes for all $i$. Therefore by obstruction theory, the homotopy set $[\\Sigma N_r, G/O]$ consists of a single element. Therefore\nthe map $L_8(\\z) \\to S_\\partial(N_r\\times I)$ is surjective. By the construction, the Wall realization map $L_8(\\mathbb Z) \\to S_{\\partial}(N_r \\times I)$ factors through $L_8(\\z) \\to \\Theta_7 \\to S_\\partial(N_r\\times I)$.\n \\end{proof}\n\nIn \\cite[Theorem 7.9 and 7.12]{KreSu}, the group $\\m_\\partial(S^2\\times D^4)$ is computed and a system of generator is given. Let $g_1$ be a diffeomorphism of $D^6$ relative to the boundary, representing a generator of $ \\m_\\partial(D^6)\\cong \\Theta_7$, $t_1\\in\\pi_3(SO(4))$ be a generator of $\\mathrm{Im}\\, \\pi_3(SO(3))\\cong\\mathbb{Z}$, $t_2$ be a generator of $\\pi_4(SO(3))\\cong \\mathbb{Z}_2$ and $i_\\eta:S^3\\times D^3\\to S^2\\times D^4$ be an embedding representing the Hopf map $\\eta\\in\\pi_3(S^2)$. Fix an embedded $6$-disk in $S^2 \\times D^4$ and extend $g_1$ to the complement by the identity; let $g_2$ and $g_3$ be the Dehn twists in $i_{\\eta}(S^3 \\times D^3)$ with parameters $t_1$ and $t_2$, respectively. Then $$\\m_\\partial(S^2\\times D^4)=\\mathbb{Z}_{28}\\{g_1\\}\\oplus\\mathbb{Z}_{12}\\{g_2\\}\\oplus\\mathbb{Z}_2\\{g_3\\}$$\nFor any given embedding of $S^2 \\times D^4$ in a $6$-manifold, we extend these diffeomorphisms by the identity on the complement and still denote them by the same notation. \n \\begin{prop}\\label{sur}\n Let $i \\colon S^2\\times D^4\\to N_r$ be a local trivialization of the $S^2$-bundle. Then $I(N_r)$ is generate by the disk-supported diffeomorphism $g_1$ and the Dehn twist $g_2$. \n \\end{prop}\n \\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.",
    "post_theorem_intro_text_len": 3664,
    "post_theorem_intro_text": "The projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\mathbb P^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\mathbb P^2$. \n\nRank $2$-complex vector bundles over $\\mathbb P^2$ are classified by their Chern classes $c_1$ and $c_2$. For such a vector bundle $\\eta$ with Chern classes $c_1(\\eta)$ and $c_2(\\eta)$, the corresponding $3$-dimensional real vector bundle $\\xi$ satisfies \n$$p_1(\\xi) = c_1(\\eta)^2 - 4 c_2(\\eta).$$ \n(see Proposition \\ref{prop:real3}). Consequently, Theorem \\ref{Mth} also determines the mapping class group of the projective bundle $\\mathbb P(\\eta)$ of rank $2$ complex vector bundles $\\eta$ over $\\mathbb P^2$, provided that $\\langle c_1(\\eta)^2 - 4c_2(\\eta), [\\mathbb P^2] \\rangle = 8n+5$.\n\nAn interesting class of examples is given by the hypersurfaces $M_k \\subset \\mathbb P^2 \\times \\mathbb P^2$ defined by \n$$M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}.$$\nNote that  $M_1=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x_iy_i=0\\}$ is the Milnor hypersurface, a Fano 3-fold. The hypersurfaces $M_k$ can be identified with $N_{-3k^2}$ (see \\S \\ref{subsec:bundle}). We  therefore obtain the following corollary. \n\n\t\\begin{cor}\\label{gmhs}\n\t\tLet $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}$, then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(M_k) \\to \\mathrm{MCG}(M_k) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n\t\t\\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$.    \n\t\t\\end{enumerate}\n\t\\end{cor}\n\nTo conclude this introduction, we briefly describe the strategy used in our computations. It combines methods from homotopy theory with classical and modified surgery theory. The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}. \n\nThe determination of the Torelli group $I(M)$ is more subtle. By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\t$$ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),$$\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group (see \\S \\ref{subsec:gen}) For the manifolds $N_r$, the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$. On the other hand, when $r \\in 8 \\mathbb Z +5$,\n homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.  Consequently, in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$ (see \\S \\ref{subsec:gen}). To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory, which is an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$ (see \\S \\ref{subsec:det}). \n\n \\begin{ack}\nThe author thanks Professor Yang Su for entrusting him with this project and for patiently guiding him through the research and writing process. The author is also grateful to Professor Stephen Theriault, who pointed out a key direction for proving Proposition \\ref{ntb}. This research was partially supported by NSFC 12471069.\n \\end{ack}",
    "sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in Theorem~\\ref{Mth}).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”",
    "expanded_sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in the main theorem).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out next.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”",
    "expanded_theorem": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(\\xi\\) be a 3-dimensional real vector bundle over the complex projective plane \\(\\mathbb{P}^2\\) with first Pontrjagin class \\(p_1(\\xi)=r\\omega\\), where \\(\\omega\\in H^4(\\mathbb{P}^2)\\) is the positive generator, and let \\(N_r=S(\\xi)\\) be the total space of the associated sphere bundle. Let \\(\\mathrm{MCG}(N_r)\\) be the mapping class group of \\(N_r\\), i.e. the group of isotopy classes of orientation-preserving self-diffeomorphisms of \\(N_r\\), and let \\(I(N_r)\\subset \\mathrm{MCG}(N_r)\\) be the Torelli group consisting of classes acting trivially on the cohomology of \\(N_r\\). Which statement holds for these manifolds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+1\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
        },
        {
          "label": "C",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\)."
        },
        {
          "label": "D",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) for every \\(r\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
        },
        {
          "label": "E",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}\\oplus \\mathbb{Z}_{28}.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "congruence class required for the homotopy-theoretic description of I(N_r)",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "the explicit computation of the Torelli group when r=8n+5",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "exceptional cohomological action at r=-3 giving a larger quotient",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "generation by g_1 and g_2 versus incorrectly adjoining the full smooth-structure contribution as a direct summand",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct statement. It names the objects and asks for the correct structural description, without giving away the exceptional case or the formula for the Torelli group."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct option is a near-verbatim statement of the classification result for \\(\\mathrm{MCG}(N_r)\\) and \\(I(N_r)\\). It does not meaningfully require selecting among independently derived consequences."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish close variants, such as the exceptional value \\(r=-3\\), the congruence condition \\(r=8n+5\\), and the orders \\(14\\) versus \\(28\\). However, this is mainly precision recall rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically close to the target theorem and reflect realistic failure modes: swapping the exceptional parameter, weakening by omission, altering a numerical invariant, or overgeneralizing a special-case statement. They are distinct and plausible."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-constructed theorem-discrimination MCQ with strong distractors and no answer leakage, but it is largely a recall/restatement question rather than one that tests genuine generative reasoning."
    }
  },
  {
    "id": "2512.18590v1",
    "paper_link": "http://arxiv.org/abs/2512.18590v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}",
      "start_pos": 5163,
      "end_pos": 5713,
      "label": "Mth"
    },
    "ref_dict": {
      "Mth": "\\begin{thm}\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\m(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}\n\t\\end{thm}",
      "subsec:bundle": "\\label{subsec:bundle}\nFor the sake of simplicity,\tin the sequel we will use the same notation for (the equivalence class of) a vector bundle and (the homotopy class of) its classifying map from it bas",
      "subsec:gen": "\\label{subsec:gen}\n\nA natural idea to study the mapping class group is to relate it to the group of homotopy equivalences. To be precise, let $M$ be a simply closed connected manifold with $\\dim M>5$.",
      "subsec:det": "\\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.    \n\n\t\tBy Lemma \\ref{t7} and the exactness of $S_\\partial(N_r\\times I)\\to I(N_r)\\to I^h(N_r)$, the kernel of the forgetful homomorphism $I(N_r) \\to I^h(N_r)$ is generated by $g_1$. Thus we only need to show that the image of $g_2$ is a generator of $I^h(N_r)$. This will be done by showing that the image of $g_2$ generates  the image of $\\mu_N^*$ (see Lemma \\ref{Ih}).\n\n\t\tLet $\\mu_3:S^3\\times D^3 \\to (S^3\\times D^3)\\vee S^6$ be the pinch map. It induces a map\n\t\t$$\\mu_3^* \\colon \\pi_6(S^3)\\to \\pi_0(G_\\partial(S^3\\times D^3)), \\ \\alpha \\mapsto (id\\vee\\alpha)\\circ\\mu_3.$$\nwhere we represent $\\alpha \\in \\pi_6(S^3)$ by a map $S^6 \\to S^3 \\times \\{0\\}$. \nView $S^3\\times D^3$ as a submanifold of $N_r$ through the embedding $i\\circ i_\\eta \\colon S^3 \\times D^3 \\to S^2 \\times D^4 \\to N_r$, extending the map $\\mu^*(\\alpha)$ by the identity on the complement we have a homotopy equivalence $\\mu^*(\\alpha)$ of $N_r$. For $\\alpha\\in\\pi_6(S^3)$, consider the composite $i \\circ \\eta \\circ \\alpha \\colon S^6 \\to S^3 \\to S^2 \\to N_r$. From the construction of the maps one readily verifies that $\\mu_3^*(\\alpha)=\\mu_N^*(i \\circ\\eta \\circ \\alpha)\\in I^h(N_r)$, i.e., there is a commutative diagram\n$$\\xymatrix{\\pi_6(S^3) \\ar[r]^{(i \\circ \\eta)_*} \\ar[d]_{\\mu_3^*} & \\pi_6(N_r) \\ar[d]^{\\mu_N^*} \\\\\n \\pi_0(G_\\partial(S^3\\times D^3)) \\ar[r]^{\\ \\ \\ \\ \\ \\Phi} & I^h(N_r)}$$\n where the map $\\Phi$ at the bottom is the extension by the identity on the complement of $i\\circ i_{\\eta}(S^3 \\times D^3)$.\n\nIt can be verified by elementary homotopy theory (compare \\cite[p.98]{BeMa}) that the homotopy Dehn twist map $De^h:\\Omega^3 G(S^3)\\to G_\\partial(S^3\\times D^3)$ defined by $De^h(\\alpha)(x,y)=(\\alpha(y)x,y)$ \ninduces an isomorphism \n$\\pi_0(\\Omega^3G(S^3)) \\to \\pi_0(G_\\partial(S^3\\times D^3))$.  Under the natural isomorphisms\n$$\\pi_0(\\Omega^3G(S^3)) \\cong \\pi_3(G(S^3), \\mathrm{id}) \\cong \\pi_3((\\Omega^3S^3)_{\\mathrm{id}}) \\cong \\pi_6(S^3),$$\nwhere $(\\Omega^3S^3)_{\\mathrm{id}}$ is the connected component containing $\\mathrm{id}_{S^3}$, this map can be identified with $\\mu_3^* \\colon \\pi_6(S^3) \\to \\pi_0(G_\\partial(S^3\\times D^3))$. The following commutative diagram\n$$\\xymatrix{\\pi_3(SO(3))\\ar[d]\\ar[rr]^J&&\\pi_3(\\Omega^3S^3)\\ar[d]_{\\mu_3^*}\\\\\n\t\t\t\\pi_3(SO(4))\\ar[r]^{De \\ \\ \\ \\ }&\\m_{\\partial}(S^3\\times D^3)\\ar[r]&\\pi_0(G_\\partial(S^3\\times D^3))\n\t\t\t}$$ \nshows that the image of $g_2$ in $I^h(N_r)$ equals $\\Phi(\\mu_3^*(J(t_1)))$.\nThe $J$-homomorphism $J \\colon \\pi_3(SO(3))\\to\\pi_6(S^3)$ is surjective. Therefore by Corollary \\ref{htg} and Lemma \\ref{Ih},  the image of $g_2$ generates $I^h(N_r)$.   This finishes the proof of this proposition.\t\n\n\t\\end{proof}\n\n\t\\subsection{determination of $I(N_r)$}\\label{subsec:det}\nWe have shown in last subsection that the Torelli group $I(N_r)$ is generated by the diffeomorphisms $g_1$ and $g_2$. They obviously commute with each other since they are supported in disjoint regions. In this subsection we determine the orders of them using the generalized Kreck-Stolz invariant developed in \\cite{KreSu}.\n\nLet  $B=\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin$, $p \\colon B\\to BO$ be the composition of the projection $B \\to BSpin$ and the canonical projection $BSpin \\to BO$.    \n\nSince $r \\in 8\\z +5$, we may identify $N_r$ with $\\mathbb{P}(\\xi_{1,l})$ for some $l \\in \\z$. By Proposition \\ref{Hcp}, $w_2(N_r)$ vanishes, hence $N_r$ is spin. The normal Gauss map  $g \\colon N_r\\to BO$ of $N_r$ admits a lifting $g_{spin}$ to $BSpin$, which is  unique (up to fiber homotopy) since $N_r$ is simply connected. Let $\\{s, t\\}$ be the basis of $H^2(N_r)$ given in Proposition \\ref{Hcp}, then the map $\\hat{g}=(s,t,g_{spin}) \\colon N_r \\to \\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin=B$ is the normal $B$-structure corresponding to the polarization $((s,t),0)$ (see \\cite{Kre18} for definitions). \n\nLet $x=c_1(O(-1)\\times0\\times0)$, $y=c_1(0\\times O(-1)\\times0)$ be a basis of $H^2(B)$. \tOur normal $B$-structure satisfies the condition of Theorem 5 of \\cite{Kre18}, since $H^2(B)=\\mathbb{Z}\\{x,y\\}, H^4(B)=\\mathbb{Z}\\{x^2,xy,y^2,\\hat{p}_1\\}$, where $\\hat{p}_1$ is the first spin Potrjagin class of the universal spin stable vector bundle. The homomorphism $g^* \\colon H^4(B) \\to H^4(N_r)$ is given by  \n\t\t$$\\hat{g}^* \\colon H^4(B)\\to H^4(N_r), (x^2,xy,y^2,\\hat{p}_1)\\to (s^2,st,st-ls^2,(2l-2)s^2),$$\nand in $H^4(N_r)$ we have the relations (see Proposition \\ref{Hcp})\n$$\\hat{g}^*(\\hat{p}_1)=-\\hat{p}_1(N_r)=(2l-2)s^2, \\ \\ t^2-ts+ls^2=0.$$\nTherefore \n\t$$\\mathrm{Ker}(\\hat{g}^* \\colon H^4(B)\\to H^4(N_r))=\\mathbb{Z}\\{y^2-xy+lx^2,\\hat{p}_1+2(1-l)x^2\\}$$\nLet $\\alpha=y^2-xy+lx^2$,  $\\beta=\\hat{p}_1+2(1-l)x^2\\in H^4(B)$ and $s_1=\\alpha^2$, $s_2=\\alpha^2+\\alpha\\beta$, $ s_3=\\beta^2\\in H^8(B)$.\n\n\tFix an embedded $6$-disk $D \\subset N_r$, let $I(N_r,D)$ be the group of $D$-fixing isotopy classes of $D$-fixing diffeomorphisms which induce the identity on cohomology. By the disk theorem of Palais and the isotopy extension theorem, the forgetful homomorphism $I(N_r,D)\\to I(N_r)$ is surjective. By a theorem from (\\cite{Wa63} p.265), $\\mathrm{Ker}(I(N_r,D)\\to I(N_r))$ is trivial or isomorphic to $\\mathbb{Z}_2$. \n\nGiven a diffeomorphism $f \\colon N_r \\to N_r$ with $f|_D = \\mathrm{id}$, let $T_f$ be the mapping torus of $f$ with the preferred embedding $S^1\\times D \\subset T_f$. Assume that the action of $f$ on the cohomology groups is trivial,  then $f$ preserves the normal $B$-structure, and henceforth there is a normal $B$-structure $\\hat{G}$ on $T_f$, $\\hat G \\colon T_f \\to B$,  \nextending $\\hat{g}$ and the restriction of $\\hat G$ on $S^1 \\times D$ is the ordinary normal $B$ structure on $S^1\\times D$. It's ready to see by obstruction theory that the normal $B$-structure $\\hat G$ is a unique up to homotopy. It is shown in \\cite[Theorem 6]{Kre18} that the $7$-dimensional $B$-bordism group $\\Omega_7(B,p)$ is trivial, therefore $(T_f,\\hat{G})$ admits a coboundary $(W,\\bar{G})$ whose signature $\\mathrm{sign}(W)$  is $0$. Notice that the restriction of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta$ on $\\partial W= T_f$ vanish, let $\\bar{\\alpha}$, $\\bar{\\beta}\\in H^4(W,\\partial W)$ be a preimage of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta\\in H^4(W)$, respectively. We have characteristic numbers $\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle \\in \\z^3$. For $8$-dimensional $B$-bordism classes $[X,\\varphi] \\in \\Omega_8(B,p)$, we have characteristic numbers $\\langle (s_1, s_2,s_3), \\varphi_*[X] \\rangle \\in \\z^3$ in a similar way. Let $L=(s_1,s_2,s_3)\\Omega_8^0(B)\\subset \\mathbb{Z}^3$ be the lattice generated by the characteristic numbers on the bordism group $\\Omega^0_8(B,p)=\\mathrm{Ker}(\\mathrm{sign} \\colon \\Omega_8(B,p)\\to \\mathbb{Z})$. The $KS$-invariant is defined by\n\t$$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$$\n\t$$KS(f)=(s_1,s_2,s_3)(f):=\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle.$$\n\n\t\\begin{prop}\\label{KS}\n\t\t$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$ is a well defined injective homomorphism. \n\t\\end{prop}\n\t\\begin{proof}\n\t\tThis proof is along the same line of the proof of Proposition 6.2 of \\cite{KreSu}, with more details to be checked. \n\n\t\tTo prove $KS$ is well defined, suppose $(W_1,\\bar{G}_1),(W_2,\\bar{G}_2)$ are two $B$ coboundary of $(T_f,\\hat{G})$ chose embeddings $i_j: W_j\\to \\mathbb{R}^N\\times (-1)^j[0,+\\infty)$ good enough such that $i_j(W_j)\\cap\\mathbb{R}^N=T_f$. Define $W:=i_1(W_1)\\cup i_2(W_2)\\cong W_1\\cup_\\partial W_2$ then $(W,G_w:=\\bar{G}_1\\cup \\bar{G}_2)$ is a well defined element in $\\Omega_8(B,p)$, by \\cite{Wa69} $sign(W)=0$. Consider the following diagram\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_f)\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_f)\\\\\n\t\t\tH^4(W,T_f)\\ar[r]\\ar[u]^\\cong &H^4(W)\\ar[r]\\ar[u]&H^4(T_f)\\ar[u]^{injective}\n\t\t}$$\n\t\tThis implies the preimage of $G_w^*\\alpha\\in H^4(W)$ in $H^4(W,T_f)\\cong\\oplus_j H^4(W_j,T_f)$ is a preimage of $(\\bar{G}_1\\alpha, \\bar{G}_2\\alpha)$ ($\\beta$ resp.), and the difference of $KS(f)$ defined by $W_2$ and $W_1$ equals $(s_1,s_1+s_2,s_3)(W,G_w)\\in L$. Thus we have proved $KS$ is well defined. \n\n\t\tTo prove $KS$ is a homomorphism, suppose $f_3,f_4\\in I(N_r,D)$ and $(W_j,\\bar{G}_j)$ is a $B$ coboundary of $(T_{f_j},\\hat{G})$. Identify $(n,t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_3}$ and $(n,1-t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_4}$, define $W':=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$, then $\\partial W'=T_{f_3\\circ f_4}$. By the $M-V$ sequence of $(W';W_3,W_4)$ there are $(s',t')\\in [W',\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty]$ extending the cohomology factor of $\\bar{G}_j$. Since the pasting $W'=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$ preserve the spin structure of $\\bar{G}_j$ there is spin structure on $W'$ extending the spin factor of $\\bar{G}_j$. Thus we can define a $B$ normal structure $G'_w$ on $W'$ extending $\\bar{G}_j$. Consider the following diagram\n\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_{f_j})\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_{f_j})\\\\\n\t\t\tH^4(W',T_{f_3}\\cup T_{f_4})\\ar[r]\\ar[u]^\\cong \\ar[d]&H^4(W')\\ar[r]\\ar[u]\\ar[d]&H^4(T_{f_3}\\cup T_{f_4})\\ar[u]^{injective}\\ar[d]\\\\\n\t\t\tH^4(W',T_{f_3\\circ f_4})\\ar[r]&H^4(W')\\ar[r]&H^4(T_{f_3\\circ f_4})\\\\\n\t\t}$$\n\t\tArguments analog to the proof that $KS$ is well defined, one can show $KS$ is a homomorphism through this diagram. Note that we need $H^4(T_{f_3}\\cup T_{f_4})\\to \\oplus_j H^4(T_{f_j}))$ being injective since we need a preimage of $G'_w\\alpha\\in H^4(W')$ in $H^4(W',T_{f_3}\\cup T_{f_4})$. \n\n\t\tBy Theorem 5 of \\cite{Kre18}, if $KS(f)=0$ then there is a $B$ coboundary $(W'',\\hat{G})$ of $(T_f,\\hat{G})$ such that $(W'';N_r\\times [0,\\frac{1}{2}],N_r\\times [\\frac{1}{2},1])$ is\n\t\tan $h$-cobordism. Since the spin structure of $W''$ is compatible to $S^1\\times D$ we may extend $S^1\\times D$ to be $D^2\\times D\\subset W$ and define a relative $h$-cobordism. Hence by the peudo-isotopy theorem of Cerf, $f=id\\in I(N_r, D)$. \n\n\t\\end{proof}",
      "prop:real3": "\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}",
      "ntb": "\\begin{prop}\\label{ntb}\n\t\t$h^*\\xi_r$ is nontrivial if and only if $r\\in 8\\mathbb{Z}+5$. \n\t\\end{prop}"
    },
    "pre_theorem_intro_text_len": 2243,
    "pre_theorem_intro_text": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$.  In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds. \n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp  \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$. \n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as  projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale  \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently,  such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$. \n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.",
    "context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}",
    "full_context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\m(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\nThe projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\p^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\p^2$.\n\n\\begin{cor}\\label{gmhs}\n Let $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum_i x^k_iy_i=0\\}$, then\n \\begin{enumerate}\n \\item there is a short exact sequence \n $$ 1 \\to I(M_k) \\to \\m(M_k) \\to A \\to 1,$$\n where $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n \\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$. \n \\end{enumerate}\n \\end{cor}\n\n\\begin{eg} \\label{eg:1}\n Let $\\zeta_k$ be the complex vector bundle over $\\p^2$ with total space \n $$E_r=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{C}^3 \\ | \\ \\sum z^k_iz'_i=0\\}$$\n and bundle projection $\\pi \\colon E_r \\to \\p^2$, $\\pi(z,z')=z$.\n One can identify $\\zeta_k$ as conjugation of the orthogonal complement of canonical embedding $O(-k)\\to\\mathbb{C}^3$. So $\\zeta_k\\cong \\xi_{k,k^2}$ and $p_1(\\phi( \\zeta_k))=-3k^2$. Notice that $\\mathbb{P}(\\zeta_1)$ is the Milnor hypersurface $\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z_iz'_i=0\\}$. We call $M_k=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z^k_iz'_i=0\\}=\\mathbb{P}(\\zeta_k)$ a generalized Milnor hypersurface. One can easily show that the canonical generators of $M_k$ induced from the bundle $\\zeta_k$ are $(\\pi^* x,\\pi'^* x)$. Here $\\pi(z,z')=z, \\pi'(z,z')=z'$. \n\\end{eg}\n\n\\begin{lemma}\\label{bh}\nLet $\\xi$ be the nontrivial $3$-dimensional real vector bundle over $S^5$, $\\partial \\colon \\pi_{i+1}(S^5)\\to \\pi_i(S^2)$ be the boundary homomorphism in the long exact sequence of homotopy groups associated to the sphere bundle of $\\xi$, $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5)$ be the map induced by the adjoint of the identity map $S^5= \\Sigma S^4 \\to S^5$. Then the composite $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5) = \\pi_{i+1}(S^5) \\to \\pi_i(S^2)$ is identified with the homomorphism $\\eta^2 \\colon \\pi_i(S^4) \\to \\pi_i(S^2)$ induced by the Hopf map $\\eta \\colon S^3 \\to S^2$.\n\n\\begin{cor}\\label{htg}\nLet $N_r$ be the $S^2$-bundle over $ \\mathbb{P}^2$ defined as above. Assume $r\\in 8\\mathbb{Z}+5$. Then\n\\begin{enumerate}\n\\item the inclusion of a fiber induces an isomorphism $\\pi_3(S^2)\\to\\pi_3(N_r)$; \n\\item$\\pi_4(N_r)=0$; \n\\item the bundle projection induces an injective homomorphism $\\pi_5(N_r)\\to \\pi_5(\\mathbb{P}^2)$ with image $2 \\cdot \\pi_5(\\mathbb{P}^2)$;\n\\item $\\pi_6(S^2)\\to\\pi_6(N_r)$ is surjective and $\\pi_6(N_r)\\cong \\mathbb{Z}_6$.\n\\end{enumerate}\n \\end{cor}\n\n\\begin{lemma}\\label{t7}\n Let $\\Theta_7\\cong \\mathbb{Z}_{28}$ be the Milnor-Kervaire group of oriented differential structures on the $7$-sphere. Then connected sum operation induces a surjective map $cs:\\Theta_7\\to S_\\partial(N_r\\times I)$. \n \\end{lemma} \n \\begin{proof}\nConsider the smooth surgery exact sequence ( Theorem 11.22 of \\cite{LuTi})\n$$L_8(\\z) \\to S_\\partial(N_r\\times I) \\to [\\Sigma N_r, G/O] \\to L_7(\\z).$$\nA direct calculation shows that the group $H^i(\\Sigma N_r;\\pi_i(G/O))$ vanishes for all $i$. Therefore by obstruction theory, the homotopy set $[\\Sigma N_r, G/O]$ consists of a single element. Therefore\nthe map $L_8(\\z) \\to S_\\partial(N_r\\times I)$ is surjective. By the construction, the Wall realization map $L_8(\\mathbb Z) \\to S_{\\partial}(N_r \\times I)$ factors through $L_8(\\z) \\to \\Theta_7 \\to S_\\partial(N_r\\times I)$.\n \\end{proof}\n\nIn \\cite[Theorem 7.9 and 7.12]{KreSu}, the group $\\m_\\partial(S^2\\times D^4)$ is computed and a system of generator is given. Let $g_1$ be a diffeomorphism of $D^6$ relative to the boundary, representing a generator of $ \\m_\\partial(D^6)\\cong \\Theta_7$, $t_1\\in\\pi_3(SO(4))$ be a generator of $\\mathrm{Im}\\, \\pi_3(SO(3))\\cong\\mathbb{Z}$, $t_2$ be a generator of $\\pi_4(SO(3))\\cong \\mathbb{Z}_2$ and $i_\\eta:S^3\\times D^3\\to S^2\\times D^4$ be an embedding representing the Hopf map $\\eta\\in\\pi_3(S^2)$. Fix an embedded $6$-disk in $S^2 \\times D^4$ and extend $g_1$ to the complement by the identity; let $g_2$ and $g_3$ be the Dehn twists in $i_{\\eta}(S^3 \\times D^3)$ with parameters $t_1$ and $t_2$, respectively. Then $$\\m_\\partial(S^2\\times D^4)=\\mathbb{Z}_{28}\\{g_1\\}\\oplus\\mathbb{Z}_{12}\\{g_2\\}\\oplus\\mathbb{Z}_2\\{g_3\\}$$\nFor any given embedding of $S^2 \\times D^4$ in a $6$-manifold, we extend these diffeomorphisms by the identity on the complement and still denote them by the same notation. \n \\begin{prop}\\label{sur}\n Let $i \\colon S^2\\times D^4\\to N_r$ be a local trivialization of the $S^2$-bundle. Then $I(N_r)$ is generate by the disk-supported diffeomorphism $g_1$ and the Dehn twist $g_2$. \n \\end{prop}\n \\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.",
    "post_theorem_intro_text_len": 3664,
    "post_theorem_intro_text": "The projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\mathbb P^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\mathbb P^2$. \n\nRank $2$-complex vector bundles over $\\mathbb P^2$ are classified by their Chern classes $c_1$ and $c_2$. For such a vector bundle $\\eta$ with Chern classes $c_1(\\eta)$ and $c_2(\\eta)$, the corresponding $3$-dimensional real vector bundle $\\xi$ satisfies \n$$p_1(\\xi) = c_1(\\eta)^2 - 4 c_2(\\eta).$$ \n(see Proposition \\ref{prop:real3}). Consequently, Theorem \\ref{Mth} also determines the mapping class group of the projective bundle $\\mathbb P(\\eta)$ of rank $2$ complex vector bundles $\\eta$ over $\\mathbb P^2$, provided that $\\langle c_1(\\eta)^2 - 4c_2(\\eta), [\\mathbb P^2] \\rangle = 8n+5$.\n\nAn interesting class of examples is given by the hypersurfaces $M_k \\subset \\mathbb P^2 \\times \\mathbb P^2$ defined by \n$$M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}.$$\nNote that  $M_1=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x_iy_i=0\\}$ is the Milnor hypersurface, a Fano 3-fold. The hypersurfaces $M_k$ can be identified with $N_{-3k^2}$ (see \\S \\ref{subsec:bundle}). We  therefore obtain the following corollary. \n\n\t\\begin{cor}\\label{gmhs}\n\t\tLet $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}$, then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(M_k) \\to \\mathrm{MCG}(M_k) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n\t\t\\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$.    \n\t\t\\end{enumerate}\n\t\\end{cor}\n\nTo conclude this introduction, we briefly describe the strategy used in our computations. It combines methods from homotopy theory with classical and modified surgery theory. The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}. \n\nThe determination of the Torelli group $I(M)$ is more subtle. By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\t$$ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),$$\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group (see \\S \\ref{subsec:gen}) For the manifolds $N_r$, the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$. On the other hand, when $r \\in 8 \\mathbb Z +5$,\n homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.  Consequently, in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$ (see \\S \\ref{subsec:gen}). To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory, which is an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$ (see \\S \\ref{subsec:det}). \n\n \\begin{ack}\nThe author thanks Professor Yang Su for entrusting him with this project and for patiently guiding him through the research and writing process. The author is also grateful to Professor Stephen Theriault, who pointed out a key direction for proving Proposition \\ref{ntb}. This research was partially supported by NSFC 12471069.\n \\end{ack}",
    "sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in Theorem~\\ref{Mth}).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”",
    "expanded_sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in the main theorem).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out next.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”",
    "expanded_theorem": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}",
    "theorem_type": [
      "Classification or Bijection",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(\\xi\\) be a 3-dimensional real vector bundle over the complex projective plane \\(\\mathbb{P}^2\\) with first Pontrjagin class \\(p_1(\\xi)=r\\omega\\), where \\(\\omega\\in H^4(\\mathbb{P}^2)\\) is the positive generator, and let \\(N_r=S(\\xi)\\) be the total space of the associated sphere bundle. Let \\(\\mathrm{MCG}(N_r)\\) be the mapping class group of \\(N_r\\), i.e. the group of isotopy classes of orientation-preserving self-diffeomorphisms of \\(N_r\\), and let \\(I(N_r)\\subset \\mathrm{MCG}(N_r)\\) be the Torelli group consisting of classes acting trivially on the cohomology of \\(N_r\\). Which statement holds for these manifolds?",
      "correct_choice": {
        "label": "A",
        "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+1\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
        },
        {
          "label": "C",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\)."
        },
        {
          "label": "D",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) for every \\(r\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}.\\]"
        },
        {
          "label": "E",
          "text": "There exists a short exact sequence \\[1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1,\\] where \\(A\\cong \\mathbb{Z}_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\[I(N_r)\\cong \\mathbb{Z}_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\gcd(14,n+1)}\\oplus \\mathbb{Z}_{28}.\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "congruence class required for the homotopy-theoretic description of I(N_r)",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "the explicit computation of the Torelli group when r=8n+5",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "exceptional cohomological action at r=-3 giving a larger quotient",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "generation by g_1 and g_2 versus incorrectly adjoining the full smooth-structure contribution as a direct summand",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the geometric objects and asks which global statement is true, but it does not reveal the exceptional case, the quotient group, or the explicit formula for the Torelli group. There is no direct answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall question: the correct option states a specific classification/result about the mapping class group and Torelli group. Although the alternatives force comparison among nearby claims, the task is still close to recognizing the theorem rather than deriving a new conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "The question requires some discrimination among subtly different statements, especially the exceptional case r = -3 and the congruence condition r = 8n+5. However, the main burden is precise recall of a known result, not substantial generative reasoning from the stem."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one weakens the true statement, one alters the congruence class, one removes the exceptional S3 case, and one adds an unjustified extra summand. These are plausible and reflect realistic mathematical confusion modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid high-level theorem-recognition MCQ with good distractors and no answer leakage, but it tests precise recall more than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.18638v1",
    "paper_link": "http://arxiv.org/abs/2512.18638v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm1.1}\nLet $X$ be a smooth projective $3$-fold of general type. If $\\Vol(X)>12^6$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.",
      "start_pos": 7669,
      "end_pos": 7836,
      "label": "thm1.1"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 2665,
    "pre_theorem_intro_text": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type. \n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large. \n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.",
    "context": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type.\n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.",
    "full_context": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type.\n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nWe briefly explain the difficulties and techniques in the proof. As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\\Delta\\sim_{\\bQ}\\delta K_X$ for some $\\delta<2$\nand cut down the dimension of non-klt centers by the Anghern--Siu type method.\nThe issue in dealing with $\\Phi_{|2K_X|}$ instead of $\\Phi_{|5K_X|}$ or $\\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center\nis small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$. \nHere we remark that in \\cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration.\n\nAs a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \\cite[Theorem~1.4, Theorem~1.5]{XuJ14}:\n\\begin{theorem}\\label{thm1.3}\nLet $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\\to C$ to a curve with a general fiber denoted by $F$.\n\\begin{enumerate}\n \\item If $F$ is a $(1,2)$-surface and $\\Vol(X)> 60$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 4$ and is birational for $m\\geq 5$.\n \\item If $F$ is a $(2,3)$-surface and $\\Vol(X)>456$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 3$ and is birational for $m\\geq 4$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{lemma}\\label{lem separete Z1Z2}\nLet $X$ be a normal projective variety and let $D$ be an effective Cartier divisor on $X$.\nLet $C_1$ and $C_2$ be closed subvarieties on $X$.\nSuppose that the natural restriction map \n\\[\n H^0(X,D) \\to H^0(C_1, D|_{C_1} ) \\oplus H^0(C_2, D|_{C_2})\n\\]\nis surjective.\nThen \\[\n\\dim\\overline{\\Phi_{{|D|}}(X)}\\geq \\min\\{\\dim\\overline{\\Phi_{{|D|_{C_1}|}}(C_1)}, \\dim\\overline{\\Phi_{{|D|_{C_2}|}}(C_2)}\\}+1.\\]\nIn particular,\nif $h^0(C_i, D|_{C_i})\\geq 2$ for $i=1,2$, then $\\dim\\overline{\\Phi_{{|D|}}(X)}\\geq 2$. \n\\end{lemma}\n\nAs an application of Theorem~\\ref{thm:ext for surface}, we prove Theorem~\\ref{thm1.3} on fibrations of $(1,2)$-surfaces and $(2,3)$-surfaces. \nFirst we recall properties of pluricanonical systems of $(1,2)$-surfaces and $(2,3)$-surfaces.\n\\begin{theorem}\\label{surface} Let $S$ be a smooth projective surface of general type. \n\\begin{itemize}\n\\item If $S$ is a $(1,2)$-surface, then $\\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\\leq m\\leq 4$ and is a birational map for $m\\geq 5$.\n\\item If $S$ is a $(2,3)$-surface, then $\\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\\leq m\\leq 3$ and is a birational map for $m\\geq 4$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\n This follows from \\cite[Theorem~VII.5.1, Proposition~VII.7.1, Proposition~VII.7.2]{BHPV}, \\cite[Lemma~1.1]{Horikawa1}, and \\cite[Lemma~2.1]{Horikawa2}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:whenvolgeq12}\nLet $X$ be a minimal $3$-fold of general type.\nAssume that there exists a resolution $\\pi:W\\to X$ and a fibration $f: W\\to C$ to a curve with a general fiber denoted by $F$. \nAssume that $\\pi^*K_X-bF$\n is $\\mathbb{Q}$-effective for some rational number $b>2$. \nIf \\[\\frac{(b-2)^2}{(b+2)^2} \\Vol(F)>8,\\] then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.\n\\end{proposition}\n\nNow suppose that there exists an irreducible curve $C$ passing through a very general closed point $P$ such that $(L\\cdot C)< 4<\\frac{1}{2}L^2$. Since $P$ is very general, $C$ is a nef curve with $p_g(C)\\geq 2$. Moreover, after taking a higher model of $F$ and replacing $L$ by its pullback, we may assume that $C$ is smooth; this step does not change $L^2$ or $(L\\cdot C)$ by the projection formula. As $L^2>2(L\\cdot C)$, there exists a sufficiently small positive number $\\epsilon$ such that $L-(1+\\epsilon)C$ is big by \\cite[Theorem~2.2.15]{Positivity1}. In particular, we may write \n$$L\\sim_{\\mathbb{Q}}(1+\\epsilon)C+T$$ where $T$ is an effective $\\mathbb{Q}$-divisor.\nThen by Sakai's lemma \\cite[Example~9.4.12]{Positivity2},\n\\[\nH^1\\left(F, K_F+\\left\\lceil{L-C-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)=0,\n\\]\nwhich implies that the natural map\n\\[\nH^0\\left(F, K_F+\\left\\lceil{L-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)\\to H^0\\left(C, K_C+\\left. \\left\\lceil{L-\\frac{1}{1+\\epsilon}T-C}\\right\\rceil \\right|_{C}\\right)\n\\]\nis surjective.\nSince $\\deg(\\roundup{L-\\frac{1}{1+\\epsilon}T-C}|_C)>0$, it is clear that\n\\[h^0\\left(C, K_C+\\left. \\left\\lceil{L-\\frac{1}{1+\\epsilon}T-C}\\right\\rceil \\right|_C\\right)\\geq 2\\]\nby the Riemann--Roch formula. Therefore, \n\\[\nh^0(F, K_F+\\roundup{L})\\geq h^0\\left(F, K_F+\\left\\lceil{L-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)\\geq 2. \\qedhere\\]\n\\end{proof}\nNow we are ready to study the bicanonical maps. \n\\begin{corollary}\\label{cor X/C}\nLet $X$ be a minimal $3$-fold of general type.\nAssume that there exists a resolution $\\pi:W\\to X$ and a fibration $f:W\\to C$ to a curve with a general fiber denoted by $F$. \nAssume that $ \\Vol(X)>2.9\\times10^4$ and $\\pi^*K_X-bF$ is $\\mathbb{Q}$-effective\nfor some rational number $b> 24$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.\n\\end{corollary}",
    "post_theorem_intro_text_len": 2223,
    "post_theorem_intro_text": "We briefly explain the difficulties and techniques in the proof. As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$\nand cut down the dimension of non-klt centers by the Anghern--Siu type method.\nThe issue in dealing with $\\Phi_{|2K_X|}$ instead of $\\Phi_{|5K_X|}$ or $\\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center\nis small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$. \nHere we remark that in \\cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration. \n\nAs a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \\cite[Theorem~1.4, Theorem~1.5]{XuJ14}:\n\\begin{theorem}\\label{thm1.3}\nLet $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\\to C$ to a curve with a general fiber denoted by $F$.\n\\begin{enumerate}\n \\item If $F$ is a $(1,2)$-surface and $\\Vol(X)> 60$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 4$ and is birational for $m\\geq 5$.\n \\item If $F$ is a $(2,3)$-surface and $\\Vol(X)>456$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 3$ and is birational for $m\\geq 4$.\n\\end{enumerate}\n\\end{theorem}",
    "sketch": "As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove Theorem~\\ref{thm1.1}) is to \"creat non-klt centers\" by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$ and then \"cut down the dimension of non-klt centers by the Anghern--Siu type method.\" The difficulty for $\\Phi_{|2K_X|}$ is that one is often \"not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\\delta$ will exceed $2$.\" In this situation, \"instead of cutting down the dimension of the non-klt center,\" the approach is to \"prove certain extension theorems which allow us to lift global sections from the centers to $X$.\" The paper claims an \"effective version\" of such extension theorems \"by using the Nadel vanishing theorem,\" and \"for curve centers\" it does not require the centers \"to form a fibration.\"",
    "expanded_sketch": "As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove the main theorem) is to \"creat non-klt centers\" by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$ and then \"cut down the dimension of non-klt centers by the Anghern--Siu type method.\" The difficulty for $\\Phi_{|2K_X|}$ is that one is often \"not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\\delta$ will exceed $2$.\" In this situation, \"instead of cutting down the dimension of the non-klt center,\" the approach is to \"prove certain extension theorems which allow us to lift global sections from the centers to $X$.\" The paper claims an \"effective version\" of such extension theorems \"by using the Nadel vanishing theorem,\" and \"for curve centers\" it does not require the centers \"to form a fibration.\"",
    "expanded_theorem": "\\label{thm1.1}\nLet $X$ be a smooth projective $3$-fold of general type. If $\\Vol(X)>12^6$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $X$ be a smooth projective $3$-fold of general type. Write $\\Vol(X)=\\Vol(K_X)$ for its canonical volume, and let $\\Phi_{|2K_X|}$ be the rational map induced by the bicanonical linear system $|2K_X|$. If $\\Vol(X)>12^6$, which quantitative statement about the image of the bicanonical map is valid?",
      "correct_choice": {
        "label": "A",
        "text": "One has $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ provided only that $\\Vol(X)\\geq 12^6$."
        },
        {
          "label": "C",
          "text": "One has $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 1$."
        },
        {
          "label": "D",
          "text": "The map $\\Phi_{|2K_X|}$ is generically finite onto its image."
        },
        {
          "label": "E",
          "text": "One has $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$, and moreover this follows by cutting down non-klt centers all the way to points using a boundary $\\Delta\\sim_{\\mathbb Q}\\delta K_X$ with some uniform $\\delta<2$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "strict volume threshold",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "drop the conclusion from image dimension at least 2 to mere non-triviality",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "confusing non-pencil with generically finite bicanonical map",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "ability to cut down all non-klt centers uniformly with a single effective $\\delta<2$",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the correct conclusion beyond giving the hypothesis. The exact answer is not leaked; the options must still be compared."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a direct recall of a theorem with the same numerical threshold and conclusion. However, it is not purely tautological because the choices include nearby variants: a weaker true statement, a threshold perturbation, and stronger but invalid conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the strongest valid conclusion from weaker or overly strong alternatives, especially against choices C and D. Still, the question mainly tests theorem recall rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: B tests strict versus non-strict threshold, C is a weaker true conclusion, D confuses non-pencil behavior with generic finiteness, and E overstates the proof mechanism. These reflect realistic failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no major answer leakage and strong distractors, but it leans heavily on recalling a specific theorem rather than eliciting deep generative reasoning."
    }
  },
  {
    "id": "2512.18759v1",
    "paper_link": "http://arxiv.org/abs/2512.18759v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.",
      "start_pos": 10053,
      "end_pos": 10441,
      "label": "mainthm"
    },
    "ref_dict": {
      "mainthm": "\\begin{theorem}\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.   \n\\end{theorem}",
      "prop:solvable_example": "\\begin{prop}\\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\\Q$ with $\\Gal(M | \\Q)\\cong\\Gal(M' | \\Q)$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\\Q$ have infinite torsion over $M'$. \n\\end{prop}",
      "rmk: counterexample1": "\\begin{remark}\\label{rmk: counterexample1}\n One may ask whether Theorem \\ref{thm: reduction} holds with $K'=K$, i.e.~whether it is sufficient to check finiteness over abelian subextensions of $M|K$. The answer is negative, even when $i=j=1$ and $X$ is an abelian variety,\nas the following example shows. \n\nRecall from \nthe introduction that this case amounts to understanding the torsion subgroups of abelian varieties.\nConsider the elliptic curve $E$ with Weierstrass equation $y^2=x^3+x$, with complex multiplication by $\\mathbb{Z}[i]$ over $\\Q(i)$. By the theory of complex multiplication the field $M$ obtained by adjoining all torsion points of $E$ to $\\Q(i)$ is the maximal abelian extension of $\\Q(i)$ (it is easy to show that $M| \\Q(i)$ is abelian, and this is all we need). This implies that $M| \\Q$ is a solvable extension, because both $M| \\Q(i)$ and $\\Q(i)| \\Q$ are abelian. By construction, the torsion subgroup $E(M)_{\\mathrm{tors}}\\subset E(M)$ is infinite. On the other hand, since $\\Q^{\\operatorname{ab}}=\\Q^{\\operatorname{cyc}}$ by the Kronecker--Weber theorem,  Ribet's result \\cite{Ribet1981CyclotomicTorsion} shows that $E(\\Q^{\\operatorname{ab}})$ is finite, hence there is no abelian subextension $F$ of $M| \\Q$ such that $E(F)_{\\mathrm{tors}}$ is infinite.\n\\end{remark}",
      "thm: reduction": "\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}",
      "lemma: p-adic nilpotent implies algebraic nilpotent": "\\begin{lem}\\label{lemma: p-adic nilpotent implies algebraic nilpotent}\nLet $G \\subset\\GL_{n}(\\Q_p)$ be a connected reductive algebraic subgroup and $H \\subset G(\\Q_p)$ be an abstract subgroup, dense in the Zariski topology. Suppose that $H$ is solvable of finite class as an abstract group. Then $G$ is a torus.\n\\end{lem}"
    },
    "pre_theorem_intro_text_len": 1745,
    "pre_theorem_intro_text": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.",
    "context": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.",
    "full_context": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.\n\nLet $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\Gal(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\Q/\\Z(1))$ identifies with the torsion subgroup of $A(\\Kbar)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.\n\nAs already pointed out in \\cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite. On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.\n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}\n\n\\begin{prop}\\label{prop:31-general}\nIf $i$ is odd, then\n$\\left( H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Q}_p(j))^{\\operatorname{ss}} \\right)^{G_{L^{\\rm cyc}}}=0$ for all primes $p$.\n\\end{prop}\n\\begin{proof}\nThe proof of \\cite[Proposition~3.1]{RoesslerSzamuely} goes through verbatim with Corollary~\\ref{cor:cyclo} replacing Ribet's Galois-theoretic lemma cited above. We briefly review the argument. Pick a simple nonzero\n$G_L$-submodule $W\\subset \\left( H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Q}_p(j))^{\\operatorname{ss}} \\right)^{G_{L^{\\rm cyc}}}$. Note that $G_L$ acts on $W$ via $G_L/G_{L^{\\rm cyc}}$ which is abelian, and that $W$ is semisimple for this action. Using Grothendieck's local monodromy theorem\none then shows that after taking a suitable finite\nextension of $L$ the action of $G_L$ on $W$ is unramified outside $p$; thus by\nCorollary~\\ref{cor:cyclo} it factors through $G_p:=\\mathrm{Gal}(L(\\mu_{p^\\infty})|L)$.\nFor a place $w\\mid p$ of $L$ the extension $L_w|\\mathbf{Q}_p$ is finite by our assumption on $L$ and the local\nextension $L_w(\\mu_{p^\\infty})|L_w$ is totally ramified, hence the local Galois group of $L_w$ surjects onto\n$G_p$. As in the original proof, after taking a further finite field extension the existence of the Hodge–Tate decomposition and the Weil conjectures (Deligne's theorem) force a Frobenius at a place of good reduction $w\\nmid p$ to have eigenvalues that are both integral powers of $Nw$ and of absolute value $(Nw)^{i/2-j}$, which is impossible for odd $i$.\n\\end{proof}\n\n\\begin{prop}\\label{prop:34-general} \nIf $i$ is odd, then\n$\\left(H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}}\\right)^{G_{L^{\\rm cyc}}}=0$\nfor all but finitely many primes $p$.\n\\end{prop}\n\\begin{proof} Here we adapt the proof of \\cite[Proposition~3.4]{RoesslerSzamuely}; note some similarities with the proof of Theorem \\ref{thm: reduction}.\n\n\\begin{proof}[Proof of Theorem \\ref{mainthm}]\nSuppose by contradiction that $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\rm Kum}}}$ is infinite. By \\Cref{thm: reduction}, there exist a finite extension $K' | K$ and an abelian subextension $F$ of $K' \\Kkum | K'$ such that at least one of the following holds:\n\\begin{enumerate}\n \\item $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F} \\neq (0)$ for some prime $p$;\n \\item $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F} \\neq (0)$ for infinitely many primes $p$.\n\\end{enumerate}\nReplacing $K$ with $K'$ (which is again a number field), we have an abelian extension $F|K$ contained in $K^{\\operatorname{Kum}}$ for which (1) or (2) above holds. Applying \\Cref{prop: abelianisation Kummer}, we find that $F$ is the compositum of $K^{\\operatorname{cyc}}$ and an abelian extension $L|K$ with bounded local degrees. Further replacing $K$ with $L$, we obtain an extension $F|L$, where $F$ is contained in $L^{\\operatorname{cyc}}$ and (1) or (2) above holds. The two cases contradict \\Cref{prop:31-general} and \\Cref{prop:34-general}, respectively.\n\\end{proof}\n\n\\begin{theorem}\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.   \n\\end{theorem}\n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}",
    "post_theorem_intro_text_len": 6209,
    "post_theorem_intro_text": "As already pointed out in \\cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite.  On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.\n\nThe Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2. The proof of Theorem \\ref{mainthm} will proceed by a general reduction to abelian extensions from solvable extensions of finite class: \n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\overline K$ such that $\\mathrm{Gal}(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\overline K$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ is finite.\n\\end{theorem}\n\nHere $H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}}$ (resp.~$H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}}$) denotes the semisimplication of the module $H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))$ (resp.~$H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))$) with respect to the action of $\\mathrm{Gal}(\\overline K|K)$. It is part of the Tate conjecture (at least with $\\Q_p$-coefficients) that these actions are semisimple and thus the superscripts `ss' can be omitted. If we assume semisimplicity, then in view of well-known arguments about abelian groups (see (\\cite[Lemma 2.1]{RoesslerSzamuely}) conditions (1) and (2) above can be replaced by the simpler condition that the group $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.\n\n\\begin{remarks} ${}$\n\\begin{enumerate}\n\\item In fact, we prove a slightly stronger statement: there exists a finite extension $K'|K$, depending only on $i$, $j$ and $X$, such that if $M$ is as in the theorem and all abelian extensions $F|K'$ contained in $K'M$ satisfy conditions (1) and (2), then $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ is finite. On the other hand, there may exist $K''|K$ finite (often we may take $K''=K$; see Remark \\ref{rmk: counterexample1}) such that (1) and (2) hold for all abelian $F|K''$ contained in $K''M$ but nevertheless finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ fails. \n\\item The semisimplicity conjecture with finite coefficients is perhaps less widely known. However, it is part of general motivic expectations and Corollary 1.3.4 (4) of \\cite{Cadoretetal} provides strong evidence for it: Cadoret, Hui and Tamagawa prove that over a finitely generated field of positive characteristic semisimplicity of cohomology with $\\Q_p$-coefficients for $p$ large enough (prime to the characteristic) implies semisimplicity with $\\mathbf{Z}/p\\mathbf{Z}$-coefficients for $p$ large enough. The characteristic 0 analogue of this result would suffice for our purposes. Note also that by an observation of Moonen \\cite{Moonen} semisimplicity in characteristic 0 with $\\Q_p$-coefficients follows from the usual Tate conjecture on the image of the cycle map.\n\\end{enumerate}\n\\end{remarks}\n\nFor an abelian variety $A$ over $K$ semisimplicity of the Galois action on the rational Tate module $V_p(A)$ and on $p$-torsion points (for $p$ large enough) is known by classical work of Faltings and Zarhin (see e.g.~Chapter IV by Schappacher in \\cite{FaltingsWustholz}). So in this case finiteness of torsion points of $A$ over the maximal abelian extension $(K')^{\\rm ab}$ of every finite extension $K' | K$ implies finiteness over solvable extensions of finite class. In particular, plugging in a well-known result of Zarhin (\\cite{ZarhinCM}, Corollary of Theorem 1) we obtain: \n\n\\begin{cor} Let $A$ be an abelian variety over $K$ having no simple isogeny factor of CM type over the algebraic closure $\\overline K$. Then $A$ has finitely many torsion points over Galois extensions $M|K$ such that $\\mathrm{Gal}(M| K)$ is solvable of finite class. \\end{cor} This statement was also obtained using arguments similar to ours in very recent work of Huryn \\cite{Huryn}. Huryn's preprint appeared while we were writing up the present note and we took the opportunity to borrow one of his arguments to simplify the proof of one of our lemmas (Lemma \\ref{lemma: p-adic nilpotent implies algebraic nilpotent} below).\n\nObserve that the conditions on the extensions of $K$ appearing in Theorem \\ref{thm: reduction} are purely Galois-theoretic. This prompts the question whether for fixed $X$, $i$ and $j$ there exists a Galois-theoretic criterion on extensions $M|K$ that ensures the finiteness of  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$. Unfortunately, this is not the case: \n\n\\begin{prop}\\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\\mathbf{Q}$ with $\\mathrm{Gal}(M | \\mathbf{Q})\\cong\\mathrm{Gal}(M' | \\mathbf{Q})$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\\mathbf{Q}$ have infinite torsion over $M'$. \n\\end{prop}\n\n{After some preliminaries, Theorem \\ref{thm: reduction} will be proven in Section \\ref{secred}. Theorem \\ref{mainthm} will be deduced from \\Cref{thm: reduction} in Section \\ref{secmainthm} by means of an adjustment of arguments from \\cite{RoesslerSzamuely}. The last section contains the proof of Proposition \\ref{prop:solvable_example}.\n  }",
    "sketch": "The proof of Theorem \\ref{mainthm} is described as follows. First, one uses that “the Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via Theorem \\ref{thm: reduction}. After “some preliminaries,” Theorem \\ref{thm: reduction} is proven in Section \\ref{secred}. Then “Theorem \\ref{mainthm} will be deduced from \\Cref{thm: reduction} in Section \\ref{secmainthm} by means of an adjustment of arguments from \\cite{RoesslerSzamuely}.” The reduction theorem says that for a Galois extension $M|K$ with $\\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\\bigl(H^i_{\\textup{ét}}(\\overline{X},\\Q_p(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all primes $p$, and (2) $\\bigl(H^i_{\\textup{ét}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\\textup{ét}}(\\overline{X},\\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.)",
    "expanded_sketch": "The proof of the main theorem is described as follows. First, one uses that “the Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via the following theorem. \n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}\n\nAfter “some preliminaries,” the reduction theorem is proven later. Then the main theorem is deduced from the reduction theorem later by means of an adjustment of arguments from \\cite{RoesslerSzamuely}. The reduction theorem says that for a Galois extension $M|K$ with $\\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\\bigl(H^i_{\\textup{\\u00e9t}}(\\overline{X},\\Q_p(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all primes $p$, and (2) $\\bigl(H^i_{\\textup{\\u00e9t}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\\textup{\\u00e9t}}(\\overline{X},\\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.)",
    "expanded_theorem": "\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.",
    "theorem_type": [
      "Universal",
      "Existence"
    ],
    "mcq": {
      "question": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, let $X$ be a smooth projective geometrically connected variety over $K$, and write $\\overline X:=X\\times_K \\overline K$. For any intermediate field $M$ with $K\\subset M\\subset \\overline K$, set $G_M:=\\mathrm{Gal}(\\overline K/M)$. Let $K^{\\rm Kum}$ denote the maximal Kummer extension of $K$, obtained by adjoining to $K$ all roots of all elements of $K$. Which statement holds for every odd integer $i$ and every integer $j$?",
      "correct_choice": {
        "label": "A",
        "text": "The subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\\rm Kum})$ is finite."
      },
      "choices": [
        {
          "label": "B",
          "text": "The subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite for every integer $i$ and every integer $j$. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\\rm Kum})$ is finite."
        },
        {
          "label": "C",
          "text": "For every odd integer $i$ and every integer $j$, the subgroup $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite."
        },
        {
          "label": "D",
          "text": "For every odd integer $i$ and every integer $j$, there exists a finite extension $K'\\!/K$ such that the subgroup of $G_{K'K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K'K^{\\rm Kum}}}$ is finite. In particular, after a finite extension of the ground field, the torsion subgroup of an abelian variety over $K$ becomes finite over $K^{\\rm Kum}$."
        },
        {
          "label": "E",
          "text": "For every odd integer $i$ and every integer $j$, the subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite provided that, for every abelian subextension $F/K$ contained in $K^{\\rm Kum}$, one has $\\left(H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}_p(j))\\right)^{G_F}=0$ for all primes $p$ and $\\left(H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Z}/p\\mathbf{Z}(j))\\right)^{G_F}=0$ for all but finitely many primes $p$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "odd_i_restriction",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "abelian_variety_corollary",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "field_of_invariants_fixed_as_KKum_not_after_base_change",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "semisimplification_and_all_finite_Kprime_requirement",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and asks for the valid conclusion, but it does not explicitly reveal the finiteness statement or uniquely hint at choice A. There is no direct answer leakage beyond the topic itself."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-statement recognition question: the correct option essentially states the target result verbatim. However, the alternatives introduce meaningful changes in parity, strength, and quantifiers, so it is not a pure tautological restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact theorem from stronger, weaker, or mis-quantified variants, especially among A/C/D/E. Still, the task is mainly precise theorem recall/recognition rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common error modes: dropping the oddness restriction, weakening finiteness to mere torsion, introducing an unnecessary finite base change, and overstrengthening finiteness to vanishing. They are distinct and well-calibrated."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward recall of the precise statement than toward deep generative reasoning."
    }
  },
  {
    "id": "2512.18798v1",
    "paper_link": "http://arxiv.org/abs/2512.18798v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ in $H^{2n}(X', \\mathbb{Z})$,\n$X' \\ensuremath{\\rightarrow} X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\mathbb{Z}) = H^{2n}(X', \\mathbb{Z})$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ inside  $H^{2n}(X', \\mathbb{Z})$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\mathbb{Z}) \\cong H^{2n}(\\Ga, \\mathbb{Z})$.",
      "start_pos": 11789,
      "end_pos": 12797,
      "label": "isogenous-char"
    },
    "ref_dict": {
      "PVIP": "\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}",
      "PIP": "\\begin{theorem}\\label{PIP}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ and \n$$\\psi :  \\pi_1^{alg} (X): = \\widehat{\\pi_1(X) }  \\ra  \\widehat{\\Pi}_{g}$$\nbe a surjective homomorphism, where $g \\geq 2$.\n\nThen there exists a holomorphic map $f: X \\ra C$ with connected fibres, where $C$ is a complex curve of genus $ g_2 \\geq g$,\nsuch that  $\\psi$ factors through the surjection $\\widehat{\\pi_1(f)}$. And $\\psi = \\widehat{\\pi_1(f)}$ if the homomorphism\n$\\psi$  is maximal\nwith respect to such  factorizations  $ \\widehat{\\pi_1(X)}   \\ra  \\widehat{\\Pi}_{g_2} \\ra  \\widehat{\\Pi}_{g}$.\\end{theorem}",
      "isogenous-char": "\\begin{theorem}\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\ZZ)$ has index $d$ inside the image of $H^{2n}(\\Ga', \\ZZ) $ in $H^{2n}(X', \\ZZ)$,\n$X' \\ra X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\ZZ) = H^{2n}(X', \\ZZ)$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\ZZ) $ inside  $H^{2n}(X', \\ZZ)$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\ZZ) \\cong H^{2n}(\\Ga, \\ZZ)$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 462,
    "pre_theorem_intro_text": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a  higher product}, or a VIP,  see \\cite{isogenous},   if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are  smooth curves of genus  $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting  freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:",
    "context": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:",
    "full_context": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:\n\n\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:\n\n(i) $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga' \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$,\n\n\\begin{remark}\nConditions (ii), (ii') are verified if $X$ is aspherical, that is, $X$ is a $ K(\\Ga,1)$ space.\n\\end{remark}\n\n(I) condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called \n the algebraic fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\nAs already observed, it suffices to prove Theorem \\ref{PVIP} in the case where we assume that \n \\begin{equation}\\label{product}\n \\widehat{\\pi_1(X) } \\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n }.\n \\end{equation}\n\n\\begin{lemma}\\label{sg}\n (a) the fundamental group $\\Pi_g$ of a compact complex curve\n $C$ of genus $g \\geq 2$ cannot be a nontrivial semidirect\n product $ K \\rtimes H$.\n\nSuppose on the contrary that we have such a splitting $f:\\widehat{\\Pi}_{g}\\longrightarrow \\widehat{\\Pi}_{g'}$. Since $\\widehat{\\Pi}_{g}$ is $2g$ -generated, so is $\\widehat{\\Pi}^{(p)}_{g}$ and $f(\\widehat{\\Pi}_{g})$. It follows that $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ is a 2g-generated pro-$p$ group and $\\pi_{g'}f$ factors through $\\pi_g$. Thus the lower horizontal map splits. Since $\\widehat{\\Pi}^{(p)}_{g'}$ is $2g'$-generated, $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ has infinite index in $\\widehat{\\Pi}^{(p)}_{g'}$. But every subgroup of infinite index of\na pro-$p$ surface group is free pro-$p$ (this is true for Demushkin groups, see \\cite[Exercise 5 on page 44]{S 65}, and a pro-$p$ surface group is a particular case of it), so $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ must be a free pro-$p$ group, contradicting the fact that\n $\\pi_{g'}(f(\\widehat{\\Pi}_{g})) \\cong {\\widehat{\\Pi}^{(p)}}_{g}$ has cohomological dimension 2. This finishes the proof.\n\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}",
    "post_theorem_intro_text_len": 2023,
    "post_theorem_intro_text": "\\begin{remark}\nConditions (ii), (ii')  are verified  if $X$ is aspherical, that is, $X$ is a $ K(\\Ga,1)$ space.\n\\end{remark} \n\nThe heart of the proof boils down to treating the case where $\\Ga' = \\Ga$, \nshowing that then $X$ is biholomorphic to a product of curves.\n\nThe aim of this short note is to give a short proof of the following result. It was  proven in  \\cite{authors} as Theorem A using  assumption (I), and \nthe conditions: $\\pi_1(X)$ is residually finite,  and \n$X$ is aspherical.\n\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}\n\nIt suffices then to prove Theorem \\ref{PVIP} in the case where we assume that \n $$ \\widehat{\\pi_1 (X) } \\cong {\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n }.$$\n\n As in \\cite{isogenous} the main point is to show that there exist holomorphic maps $f_i :  X\\  \\ensuremath{\\rightarrow} C_i$,\n where $C_i$ is a complex curve of genus $g_i$, which induce the \n i-th homomorphism $ (f_i)_*\\colon  \\widehat{\\pi_1 (X) }   \\ensuremath{\\rightarrow}  \\widehat{\\pi_1 (C_i) } \\cong \\widehat{\\Pi}_{g_i}$.\n\n This will be done in the next section, in Theorem \\ref{PIP},\n and is based on earlier results about irrational pencils, contained\n in \\cite{alb-gen}, \\cite{gl}.\n\n We shall then conclude the proof of Theorem \\ref{PVIP} in the final section.\n\n The arguments used concern properties of the fundamental\n groups $\\Pi_g$,  of  their profinite completions,\n and especially non splitting properties of surjections between them.\n\n One such result is due to Pavel Zalesskii, and is proven in the\n appendix.",
    "sketch": "Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\\Ga,1)$). The ``heart of the proof'' is said to be the case $\\Ga'=\\Ga$, where one shows that $X$ is biholomorphic to a product of curves.\n\nFor Theorem~\\ref{PVIP}, it is stated that it suffices to treat the case where\n\\[ \\widehat{\\pi_1(X)}\\cong \\widehat{\\Pi}_{g_1}\\times\\cdots\\times \\widehat{\\Pi}_{g_n}. \\]\nFollowing \\cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \\widehat{\\pi_1(X)}\\to \\widehat{\\pi_1(C_i)}\\cong \\widehat{\\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section (Theorem~\\ref{PIP}) and is said to be based on earlier results on irrational pencils (\\cite{alb-gen}, \\cite{gl}).\n\nAfter constructing these maps, the text says they ``shall then conclude the proof of Theorem~\\ref{PVIP} in the final section'', using properties of surface groups $\\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.",
    "expanded_sketch": "Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\\Ga,1)$). The ``heart of the proof'' is said to be the case $\\Ga'=\\Ga$, where one shows that $X$ is biholomorphic to a product of curves.\n\nWe first recall the following theorem.\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}\n\nFor this theorem, it is stated that it suffices to treat the case where\n\\[ \\widehat{\\pi_1(X)}\\cong \\widehat{\\Pi}_{g_1}\\times\\cdots\\times \\widehat{\\Pi}_{g_n}. \\]\nFollowing \\cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \\widehat{\\pi_1(X)}\\to \\widehat{\\pi_1(C_i)}\\cong \\widehat{\\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section, where the following theorem is proved.\n\n\\begin{theorem}\\label{PIP}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ and \n$$\\psi :  \\pi_1^{alg} (X): = \\widehat{\\pi_1(X) }  \\ra  \\widehat{\\Pi}_{g}$$\nbe a surjective homomorphism, where $g \\geq 2$.\n\nThen there exists a holomorphic map $f: X \\ra C$ with connected fibres, where $C$ is a complex curve of genus $ g_2 \\geq g$,\nsuch that  $\\psi$ factors through the surjection $\\widehat{\\pi_1(f)}$. And $\\psi = \\widehat{\\pi_1(f)}$ if the homomorphism\n$\\psi$  is maximal\nwith respect to such  factorizations  $ \\widehat{\\pi_1(X)}   \\ra  \\widehat{\\Pi}_{g_2} \\ra  \\widehat{\\Pi}_{g}$.\\end{theorem}\n\nThis is said to be based on earlier results on irrational pencils (\\cite{alb-gen}, \\cite{gl}).\n\nAfter constructing these maps, the text says they ``shall then conclude the proof'' in the final section, using properties of surface groups $\\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.",
    "expanded_theorem": "\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ in $H^{2n}(X', \\mathbb{Z})$,\n$X' \\ensuremath{\\rightarrow} X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\mathbb{Z}) = H^{2n}(X', \\mathbb{Z})$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ inside  $H^{2n}(X', \\mathbb{Z})$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\mathbb{Z}) \\cong H^{2n}(\\Ga, \\mathbb{Z})$.,\n",
    "theorem_type": [
      "Implication",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$, and write $\\Pi_g$ for the fundamental group of a compact complex curve of genus $g$. A variety isogenous to a higher product (VIP) means a smooth projective variety of the form\n$$W\\cong (C_1\\times \\cdots \\times C_n)/G,$$\nwhere each $C_i$ is a smooth curve of genus $g(C_i)=g_i\\ge 2$ and $G$ is a finite group acting freely on $C_1\\times \\cdots \\times C_n$.\n\nAssume that $\\pi_1(X)\\cong \\Gamma$ admits a subgroup $\\Gamma'$ of index $d$ such that\n$$\\Gamma'\\cong \\Pi_{g_1}\\times \\cdots \\times \\Pi_{g_n},$$\nwith all $g_j\\ge 2$. Let $X'\\to X$ be the unramified covering associated to $\\Gamma'$.\n\nUnder these hypotheses, suppose moreover that one of the following holds:\n\n1. $H^{2n}(X,\\mathbb Z)$ has index $d$ inside the image of $H^{2n}(\\Gamma',\\mathbb Z)$ in $H^{2n}(X',\\mathbb Z)$ (as stated, $H^{2n}(\\Gamma',\\mathbb Z)=H^{2n}(X',\\mathbb Z)$);\n\n2. the image of $H^{2n}(\\Gamma',\\mathbb Z)$ inside $H^{2n}(X',\\mathbb Z)$ is nonzero, and\n$$K_X^n=\\frac1d\\,2^n n!\\prod_j (g_j-1);$$\n\n3. $K_X$ is ample and\n$$H^{2n}(X,\\mathbb Z)\\cong H^{2n}(\\Gamma,\\mathbb Z).$$\n\nWhich conclusion about $X$ is valid?",
      "correct_choice": {
        "label": "A",
        "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\\cong W$ (so $X$ itself is isomorphic to that quotient of a product of curves). If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
      },
      "choices": [
        {
          "label": "B",
          "text": "If condition 1 holds, then $X$ is a VIP; moreover, if $K_X$ is ample, then $X\\cong W$ for some VIP $W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is the blow-up of a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "C",
          "text": "If condition 1 holds and $K_X$ is ample, then $X$ is a VIP. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "D",
          "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is nef, then actually $X\\cong W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "E",
          "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\\cong W$. If instead the image of $H^{2n}(\\Gamma',\\mathbb Z)$ inside $H^{2n}(X',\\mathbb Z)$ is nonzero, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "distinction between blow-up conclusion in (1) and VIP conclusion in (2)/(3)",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the unconditional blow-up conclusion under condition 1, retaining only the ample-case consequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "positivity hypothesis strengthened in theorem as ampleness, not merely nefness",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "numerical equality $K_X^n=\\frac1d\\,2^n n!\\prod_j (g_j-1)$ required in case (2)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the conclusion explicitly; it only lists hypotheses and asks for the correct geometric outcome. There is no direct answer leakage, though the structure strongly signals that a specific theorem is being recalled."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem gives a detailed package of hypotheses and asks for the corresponding conclusion case-by-case. It is very close to a direct restatement rather than an independently motivated problem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle alternatives such as 'blow-up of a VIP' versus 'VIP', and when ampleness is required. However, the main task is matching theorem hypotheses to its precise conclusion, not generating a conclusion from broader principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic confusions: weakening the conclusion, swapping conclusions between cases, adding unnecessary ampleness assumptions, or replacing exact cohomological conditions by weaker finite-index statements."
      },
      "total_score": 5,
      "overall_assessment": "A technically strong but theorem-centric MCQ. It avoids obvious answer leakage and has good distractors, but it is largely a precise restatement/matching exercise rather than a genuinely generative reasoning problem."
    }
  },
  {
    "id": "2512.18798v1",
    "paper_link": "http://arxiv.org/abs/2512.18798v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ in $H^{2n}(X', \\mathbb{Z})$,\n$X' \\ensuremath{\\rightarrow} X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\mathbb{Z}) = H^{2n}(X', \\mathbb{Z})$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ inside  $H^{2n}(X', \\mathbb{Z})$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\mathbb{Z}) \\cong H^{2n}(\\Ga, \\mathbb{Z})$.",
      "start_pos": 11789,
      "end_pos": 12797,
      "label": "isogenous-char"
    },
    "ref_dict": {
      "PVIP": "\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}",
      "PIP": "\\begin{theorem}\\label{PIP}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ and \n$$\\psi :  \\pi_1^{alg} (X): = \\widehat{\\pi_1(X) }  \\ra  \\widehat{\\Pi}_{g}$$\nbe a surjective homomorphism, where $g \\geq 2$.\n\nThen there exists a holomorphic map $f: X \\ra C$ with connected fibres, where $C$ is a complex curve of genus $ g_2 \\geq g$,\nsuch that  $\\psi$ factors through the surjection $\\widehat{\\pi_1(f)}$. And $\\psi = \\widehat{\\pi_1(f)}$ if the homomorphism\n$\\psi$  is maximal\nwith respect to such  factorizations  $ \\widehat{\\pi_1(X)}   \\ra  \\widehat{\\Pi}_{g_2} \\ra  \\widehat{\\Pi}_{g}$.\\end{theorem}",
      "isogenous-char": "\\begin{theorem}\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\ZZ)$ has index $d$ inside the image of $H^{2n}(\\Ga', \\ZZ) $ in $H^{2n}(X', \\ZZ)$,\n$X' \\ra X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\ZZ) = H^{2n}(X', \\ZZ)$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\ZZ) $ inside  $H^{2n}(X', \\ZZ)$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\ZZ) \\cong H^{2n}(\\Ga, \\ZZ)$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 462,
    "pre_theorem_intro_text": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a  higher product}, or a VIP,  see \\cite{isogenous},   if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are  smooth curves of genus  $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting  freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:",
    "context": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:",
    "full_context": "\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\n\\end{defin}\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:\n\n\\begin{defin}Let $X$ be a smooth projective variety.\n$X$ is said to be a {\\bf Variety isogenous to a higher product}, or a VIP, see \\cite{isogenous}, if\n$$W \\cong (C_1 \\times C_2 \\times \\dots C_n)/G,$$\nwhere the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \\ge 2$ and $G$ is a finite group acting freely on\nthe product variety $(C_1 \\times C_2 \\times \\dots C_n)$.\n\nIn \\cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:\n\n(i) $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga' \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$,\n\n\\begin{remark}\nConditions (ii), (ii') are verified if $X$ is aspherical, that is, $X$ is a $ K(\\Ga,1)$ space.\n\\end{remark}\n\n(I) condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called \n the algebraic fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\nAs already observed, it suffices to prove Theorem \\ref{PVIP} in the case where we assume that \n \\begin{equation}\\label{product}\n \\widehat{\\pi_1(X) } \\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n }.\n \\end{equation}\n\n\\begin{lemma}\\label{sg}\n (a) the fundamental group $\\Pi_g$ of a compact complex curve\n $C$ of genus $g \\geq 2$ cannot be a nontrivial semidirect\n product $ K \\rtimes H$.\n\nSuppose on the contrary that we have such a splitting $f:\\widehat{\\Pi}_{g}\\longrightarrow \\widehat{\\Pi}_{g'}$. Since $\\widehat{\\Pi}_{g}$ is $2g$ -generated, so is $\\widehat{\\Pi}^{(p)}_{g}$ and $f(\\widehat{\\Pi}_{g})$. It follows that $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ is a 2g-generated pro-$p$ group and $\\pi_{g'}f$ factors through $\\pi_g$. Thus the lower horizontal map splits. Since $\\widehat{\\Pi}^{(p)}_{g'}$ is $2g'$-generated, $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ has infinite index in $\\widehat{\\Pi}^{(p)}_{g'}$. But every subgroup of infinite index of\na pro-$p$ surface group is free pro-$p$ (this is true for Demushkin groups, see \\cite[Exercise 5 on page 44]{S 65}, and a pro-$p$ surface group is a particular case of it), so $\\pi_{g'}(f(\\widehat{\\Pi}_{g}))$ must be a free pro-$p$ group, contradicting the fact that\n $\\pi_{g'}(f(\\widehat{\\Pi}_{g})) \\cong {\\widehat{\\Pi}^{(p)}}_{g}$ has cohomological dimension 2. This finishes the proof.\n\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}",
    "post_theorem_intro_text_len": 2023,
    "post_theorem_intro_text": "\\begin{remark}\nConditions (ii), (ii')  are verified  if $X$ is aspherical, that is, $X$ is a $ K(\\Ga,1)$ space.\n\\end{remark} \n\nThe heart of the proof boils down to treating the case where $\\Ga' = \\Ga$, \nshowing that then $X$ is biholomorphic to a product of curves.\n\nThe aim of this short note is to give a short proof of the following result. It was  proven in  \\cite{authors} as Theorem A using  assumption (I), and \nthe conditions: $\\pi_1(X)$ is residually finite,  and \n$X$ is aspherical.\n\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}\n\nIt suffices then to prove Theorem \\ref{PVIP} in the case where we assume that \n $$ \\widehat{\\pi_1 (X) } \\cong {\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n }.$$\n\n As in \\cite{isogenous} the main point is to show that there exist holomorphic maps $f_i :  X\\  \\ensuremath{\\rightarrow} C_i$,\n where $C_i$ is a complex curve of genus $g_i$, which induce the \n i-th homomorphism $ (f_i)_*\\colon  \\widehat{\\pi_1 (X) }   \\ensuremath{\\rightarrow}  \\widehat{\\pi_1 (C_i) } \\cong \\widehat{\\Pi}_{g_i}$.\n\n This will be done in the next section, in Theorem \\ref{PIP},\n and is based on earlier results about irrational pencils, contained\n in \\cite{alb-gen}, \\cite{gl}.\n\n We shall then conclude the proof of Theorem \\ref{PVIP} in the final section.\n\n The arguments used concern properties of the fundamental\n groups $\\Pi_g$,  of  their profinite completions,\n and especially non splitting properties of surjections between them.\n\n One such result is due to Pavel Zalesskii, and is proven in the\n appendix.",
    "sketch": "Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\\Ga,1)$). The ``heart of the proof'' is said to be the case $\\Ga'=\\Ga$, where one shows that $X$ is biholomorphic to a product of curves.\n\nFor Theorem~\\ref{PVIP}, it is stated that it suffices to treat the case where\n\\[ \\widehat{\\pi_1(X)}\\cong \\widehat{\\Pi}_{g_1}\\times\\cdots\\times \\widehat{\\Pi}_{g_n}. \\]\nFollowing \\cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \\widehat{\\pi_1(X)}\\to \\widehat{\\pi_1(C_i)}\\cong \\widehat{\\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section (Theorem~\\ref{PIP}) and is said to be based on earlier results on irrational pencils (\\cite{alb-gen}, \\cite{gl}).\n\nAfter constructing these maps, the text says they ``shall then conclude the proof of Theorem~\\ref{PVIP} in the final section'', using properties of surface groups $\\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.",
    "expanded_sketch": "Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\\Ga,1)$). The ``heart of the proof'' is said to be the case $\\Ga'=\\Ga$, where one shows that $X$ is biholomorphic to a product of curves.\n\nWe first recall the following theorem.\n\\begin{theorem}\\label{PVIP}\nIn Theorem \\ref{isogenous-char} one obtains the same conclusions  replacing\n\n(I)  condition (i) by condition (i-alg) $$\\widehat{\\Ga' } \n\\cong \\widehat{\\Pi}_{g_1} \\times \\dots \\widehat{\\Pi}_{g_n } ,$$\nwhere  $\\widehat{\\Ga}$ denotes the profinite completion of $\\Ga$ ($\\widehat{\\Ga'} = \\widehat{\\pi_1(X')}$ may be called  \n the algebraic  fundamental group of $X'$ and be denoted $ \\pi_1^{alg} (X') $),\n\n (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\\Ga$ is residually finite.\n\\end{theorem}\n\nFor this theorem, it is stated that it suffices to treat the case where\n\\[ \\widehat{\\pi_1(X)}\\cong \\widehat{\\Pi}_{g_1}\\times\\cdots\\times \\widehat{\\Pi}_{g_n}. \\]\nFollowing \\cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \\widehat{\\pi_1(X)}\\to \\widehat{\\pi_1(C_i)}\\cong \\widehat{\\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section, where the following theorem is proved.\n\n\\begin{theorem}\\label{PIP}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ and \n$$\\psi :  \\pi_1^{alg} (X): = \\widehat{\\pi_1(X) }  \\ra  \\widehat{\\Pi}_{g}$$\nbe a surjective homomorphism, where $g \\geq 2$.\n\nThen there exists a holomorphic map $f: X \\ra C$ with connected fibres, where $C$ is a complex curve of genus $ g_2 \\geq g$,\nsuch that  $\\psi$ factors through the surjection $\\widehat{\\pi_1(f)}$. And $\\psi = \\widehat{\\pi_1(f)}$ if the homomorphism\n$\\psi$  is maximal\nwith respect to such  factorizations  $ \\widehat{\\pi_1(X)}   \\ra  \\widehat{\\Pi}_{g_2} \\ra  \\widehat{\\Pi}_{g}$.\\end{theorem}\n\nThis is said to be based on earlier results on irrational pencils (\\cite{alb-gen}, \\cite{gl}).\n\nAfter constructing these maps, the text says they ``shall then conclude the proof'' in the final section, using properties of surface groups $\\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.",
    "expanded_theorem": "\\label{isogenous-char}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that \n\n(i)  $\\pi_1(X) \\cong \\Ga$ \nadmits an index $d$ subgroup $\\Ga'$ such that\n$$\\Ga'  \n\\cong \\Pi_{g_1} \\times \\dots \\Pi_{g_n } ,$$\nwhere $\\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \\geq 2$, \n\n(ii) $H^{2n}(X, \\mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ in $H^{2n}(X', \\mathbb{Z})$,\n$X' \\ensuremath{\\rightarrow} X$ being the unramified covering associated to $\\Ga'$\n(i.e.,  $H^{2n}(\\Ga', \\mathbb{Z}) = H^{2n}(X', \\mathbb{Z})$).\n\nThen $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \\cong W $)\nif $K_X$ is ample.\n\nAlternatively, $X$ is a VIP if \n\n(i), (ii bis) and (iii) hold true, where\n\n(ii bis) the image of $H^{2n}(\\Ga', \\mathbb{Z}) $ inside  $H^{2n}(X', \\mathbb{Z})$ is nonzero, \n\n(iii) $K_X^n =  \\frac{1}{d} 2^n n! \\prod_j (g_j-1)$.\n\nAlternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,\nwhere\n\n(ii') $H^{2n}(X, \\mathbb{Z}) \\cong H^{2n}(\\Ga, \\mathbb{Z})$.,\n",
    "theorem_type": [
      "Implication",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$, and write $\\Pi_g$ for the fundamental group of a compact complex curve of genus $g$. A variety isogenous to a higher product (VIP) means a smooth projective variety of the form\n$$W\\cong (C_1\\times \\cdots \\times C_n)/G,$$\nwhere each $C_i$ is a smooth curve of genus $g(C_i)=g_i\\ge 2$ and $G$ is a finite group acting freely on $C_1\\times \\cdots \\times C_n$.\n\nAssume that $\\pi_1(X)\\cong \\Gamma$ admits a subgroup $\\Gamma'$ of index $d$ such that\n$$\\Gamma'\\cong \\Pi_{g_1}\\times \\cdots \\times \\Pi_{g_n},$$\nwith all $g_j\\ge 2$. Let $X'\\to X$ be the unramified covering associated to $\\Gamma'$.\n\nUnder these hypotheses, suppose moreover that one of the following holds:\n\n1. $H^{2n}(X,\\mathbb Z)$ has index $d$ inside the image of $H^{2n}(\\Gamma',\\mathbb Z)$ in $H^{2n}(X',\\mathbb Z)$ (as stated, $H^{2n}(\\Gamma',\\mathbb Z)=H^{2n}(X',\\mathbb Z)$);\n\n2. the image of $H^{2n}(\\Gamma',\\mathbb Z)$ inside $H^{2n}(X',\\mathbb Z)$ is nonzero, and\n$$K_X^n=\\frac1d\\,2^n n!\\prod_j (g_j-1);$$\n\n3. $K_X$ is ample and\n$$H^{2n}(X,\\mathbb Z)\\cong H^{2n}(\\Gamma,\\mathbb Z).$$\n\nWhich conclusion about $X$ is valid?",
      "correct_choice": {
        "label": "A",
        "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\\cong W$ (so $X$ itself is isomorphic to that quotient of a product of curves). If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
      },
      "choices": [
        {
          "label": "B",
          "text": "If condition 1 holds, then $X$ is a VIP; moreover, if $K_X$ is ample, then $X\\cong W$ for some VIP $W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is the blow-up of a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "C",
          "text": "If condition 1 holds and $K_X$ is ample, then $X$ is a VIP. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "D",
          "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is nef, then actually $X\\cong W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        },
        {
          "label": "E",
          "text": "If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\\cong W$. If instead the image of $H^{2n}(\\Gamma',\\mathbb Z)$ inside $H^{2n}(X',\\mathbb Z)$ is nonzero, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "distinction between blow-up conclusion in (1) and VIP conclusion in (2)/(3)",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the unconditional blow-up conclusion under condition 1, retaining only the ample-case consequence",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "positivity hypothesis strengthened in theorem as ampleness, not merely nefness",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "numerical equality $K_X^n=\\frac1d\\,2^n n!\\prod_j (g_j-1)$ required in case (2)",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It states hypotheses and asks for the valid conclusion; the answer must be identified from the choices rather than being directly embedded in the wording."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem reproduces the assumptions in detail and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some moderate pressure to distinguish subtle variants (blow-up vs VIP, ample vs nef, presence of the numerical equality), but the task is mainly recognition/recall rather than genuine mathematical generation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are close to the correct theorem, differ in mathematically meaningful ways, and reflect plausible failure modes such as weakening hypotheses, swapping conclusions, or omitting a required numerical condition."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors, but it is largely a direct restatement/recall question rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.19335v1",
    "paper_link": "http://arxiv.org/abs/2512.19335v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{spinc-thm-intro}\nLet $M$ be an \\(n\\)-dimensional almost flat spin\\(^c\\) manifold. Then $M$ bounds an orientable manifold. Moreover, if \\(n\\) is odd, then $M$ with any spin$^c$ structure bounds a spin\\(^c\\) manifold.",
      "start_pos": 8586,
      "end_pos": 8837,
      "label": "spinc-thm-intro"
    },
    "ref_dict": {
      "simple-ex": "\\begin{example}\\label{simple-ex}\nLet $A$ be a finite simple group. The Schur multiplier $H_2(A;\\mathbb{Z})$ has odd order precisely in the following cases:\n\\begin{itemize}\n\\item[(1).] Cyclic groups: $\\mathbb{Z}/p$ with $p$ a prime number;\n\\item[(2).] Groups of Lie type (with $q=p^k$ for a prime number $p$ and positive $k$):\n  \\begin{itemize}\n  \\item Chevalley groups: \n     \\begin{itemize} \n     \\item[(Ch1).] linear groups $A_n(q)$ with $(n+1,q-1)$ odd and $(n, q)\\neq (1, 4)$, $(2, 2)$, $(2, 4)$, $(3, 2)$; \n     \\item[(Ch2).] orthogonal groups $B_n(q)$ with $n>1$, $q$ even and $(n, q)\\neq (2, 2)$, $(3, 2)$; \n     \\item[(Ch3).] symplectic groups $C_n(q)$ with $n>2$, $q$ even and $(n, q)\\neq (3, 2)$; \n     \\item[(Ch4).] orthogonal groups $D_n(q)$ with $n>3$, $(4,q^n-1)=1$ and $(n, q)\\neq (4, 2)$; \n     \\item[(Ch5).] $E_6(q)$; \n     \\item[(Ch6).] $E_7(q)$ with $q$ even; \n     \\item[(Ch7).] $E_8(q)$; \n     \\item[(Ch8).] $F_4(q)$ with $q\\neq 2$;\n     \\item[(Ch9).] $G_2(q)$ with $q\\neq 4$;\n     \\end{itemize}\n \\item Steinberg groups: \n   \\begin{itemize} \n     \\item[(S1).] unitary groups ${}^{2}A_n(q^2)$ with $n>1$, $(n+1,q+1)$ odd and $(n, q)\\neq (3, 2)$, $(5,2)$; \n     \\item[(S2).] orthogonal groups ${}^{2}D_n(q^2)$ with $n>3$, $(4,q^n+1)=1$; \n     \\item[(S3).] ${}^{2}E_6(q^2)$ with $q\\neq 2$;\n     \\item[(S4).] ${}^{3}D_4(q^3)$;\n     \\end{itemize}\n \\item Suzuki groups: ${}^{2}B_2(2^{2n+1})$ with $n\\geq 2$;\n  \\item Ree groups and Tits group: \n    \\begin{itemize} \n     \\item[(R1)] Ree groups ${}^{2}F_4(2^{2n+1})$ with $n\\geq 1$; \n     \\item[(R2)] Tits group ${}^{2}F_4(2)'$;\n     \\item[(R3)] Ree groups ${}^{2}G_2(3^{2n+1})$ with $n\\geq 1$;\n     \\item[(R4)] the derived group ${}^{2}G_2(3)'$;\n     \\end{itemize}\n     \\end{itemize}\n\\item[(3).] Sporadic groups:\n  \\begin{itemize}\n  \\item Mathieu groups:   \n     \\begin{itemize} \n     \\item[(M1)] $M_{11}$;\n     \\item[(M2)] $M_{23}$;\n     \\item[(M3)] $M_{24}$;\n     \\end{itemize}\n  \\item Janko groups:   \n     \\begin{itemize} \n     \\item[(J1)] $J_{1}$;\n     \\item[(J2)] $J_{3}$;\n     \\item[(J3)] $J_{4}$;\n     \\end{itemize}\n  \\item Conway groups:   \n     \\begin{itemize} \n     \\item[(C1)] $Co_{2}$;\n     \\item[(C2)] $Co_{3}$;\n     \\end{itemize}\n  \\item Fischer groups:\n    \\begin{itemize} \n     \\item[(F1)] $Fi_{23}$;\n     \\item[(F2)] $Fi_{24}'$;\n     \\end{itemize}\n  \\item McLaughlin group: $McL$;\n  \\item Held group: $He$;\n  \\item O'Nan group: $O'N$;\n  \\item Harada-Norton group: $HN$;\n  \\item Lyons group: $Ly$;\n  \\item Thompson group: $Th$;\n  \\item Fischer-Griess Monster group, $M$.\n   \\end{itemize}\n\\end{itemize}\n\\end{example}",
      "4-thm-intro": "\\begin{theorem}\\label{4-thm-intro}\nLet \\(M\\) be an almost flat orientable \\(4\\)-manifold. Then \\(M\\) bounds a spin\\(^c\\) manifold. Furthermore, if \\(M\\) is spin, then \\(M\\) with any spin structure bounds a spin manifold.\n\\end{theorem}",
      "G-thm-intro": "\\begin{theorem}\\label{G-thm-intro}\nLet \\(M\\) be an almost flat oriented manifold with holonomy group \\(G\\).\nIf the Schur multiplier of \\(G\\) is of odd order, then \\(M\\) bounds a compact manifold.  \n\nIn particular, if the \\(2\\)-Sylow subgroup of \\(G\\) is cyclic, generalized quaternionic, or coincides with the \\(2\\)-Sylow subgroup of any finite simple group listed in Example~\\ref{simple-ex}, then \\(M\\) is the boundary of a compact manifold.\n\\end{theorem}",
      "denom-lemma": "\\begin{lemma}\\label{denom-lemma}\nIn the expression of the universal integral spin$^c$ Wu class $\\mu_{2n}^{Spin^c}$, the denominator of the coefficient $a_n$ of $c^{n}$ satisfies \n\\[\n{\\rm denom}~(a_n)~|~2^{n-s_2(n)},\n\\] \nwhere $s_2(n)$ denotes the sum of the binary digits of $n$.\n\\end{lemma}",
      "spinc-thm-intro": "\\begin{theorem}\\label{spinc-thm-intro}\nLet $M$ be an \\(n\\)-dimensional almost flat spin\\(^c\\) manifold. Then $M$ bounds an orientable manifold. Moreover, if \\(n\\) is odd, then $M$ with any spin$^c$ structure bounds a spin\\(^c\\) manifold.\n\\end{theorem}",
      "nasen": "\\begin{remark}\\label{nasen}\n\tThe proof in fact yields a more general result than Theorem \\ref{spinc-thm-intro}. Indeed, we show that if a spin$^c$ manifold $M$ has all rational Pontryagin classes vanishing, then it bounds an orientable manifold. Moreover, if $n$ is odd, then $M$, equipped with any spin$^c$ structure, bounds a spin$^c$ manifold.\n\t\\end{remark}",
      "Xio-ex": "\\begin{example}\\label{Xio-ex}\nTheorem \\ref{spinc-thm-intro} also recovers \\cite[Theorem 1.2]{Xio16}, where $L$ is assumed to be a simply connected $2$-step nilpotent Lie group with ${\\rm dim}[L,L] = 1$ and $G$ acts trivially on the center of $L$. \nIndeed, as shown in the proof of \\cite[Theorem 1.2]{Xio16} there, the classifying map $\\varphi: M\\larrow BSO(n)$ factors as \n\\[\n\\varphi: M\\stackrel{\\eta}{\\larrow} BG\\stackrel{}{\\larrow} BU(m)\\stackrel{}{\\larrow} BSO(n)\n\\]\nfor a suitable unitary group $U(m)$. Consequently, $M$ is spin$^c$ and Theorem \\ref{spinc-thm-intro} applies to show that $M$ bounds. \n\\end{example}"
    },
    "pre_theorem_intro_text_len": 2342,
    "pre_theorem_intro_text": "A closed manifold \\( M \\) is said to be \\emph{almost flat} if, for every \\( \\epsilon > 0 \\), there exists a Riemannian metric \\( g_\\epsilon \\) on \\( M \\) such that \n\\[\n\\operatorname{diam}(M, g_\\epsilon) \\le 1, \\qquad |K_{g_\\epsilon}| < \\epsilon ,\n\\]\nwhere \\( K_{g_\\epsilon} \\) denotes the sectional curvature of \\( g_\\epsilon \\). \nThus, \\( M \\) admits metrics of arbitrarily small curvature and uniformly bounded diameter. For each dimension \\( n \\), there exists a constant \\( \\epsilon_n > 0 \\) with the property that if an \\( n \\)-dimensional manifold admits an \\( \\epsilon_n \\)-flat metric of diameter at most one, then it is almost flat \\cite{CG86}.\n\nA classical theorem of Gromov~\\cite{Gro78} asserts that every almost flat manifold is finitely covered by a nilmanifold. Conversely, Farrell and Hsiang~\\cite{FH83} proved that any manifold finitely covered by a nilmanifold is homeomorphic to an almost flat manifold. Ruh~\\cite{Ruh82} subsequently strengthened Gromov’s result, showing that every almost flat manifold is in fact diffeomorphic to an infranilmanifold.\n\nA long-standing conjecture, proposed independently by Farrell--Zdravkovska~\\cite{FZ83} and Yau~\\cite{Yau93}, predicts that every almost flat manifold bounds a compact manifold. \nIn the special case of flat manifolds, this was established by Hamrick and Royster~\\cite{HR82} in a striking theorem. \nMore recently, Davis and Fang~\\cite{DF16} confirmed the conjecture under the additional assumption that the \\(2\\)-Sylow subgroup of the holonomy group is cyclic or generalized quaternionic; their result was later rederived by Xiong~\\cite{Xio16} through a different argument. \nIn particular, the theorem of Davis and Fang encompasses earlier advances due to Farrell--Zdravkovska~\\cite{FZ83} and Upadhyay~\\cite{Upa01}.\n\nThe general case of the conjecture remains open. Moreover, Davis and Fang~\\cite{DF16} pointed out that determining whether every almost flat \\emph{spin} manifold, possibly after changing the spin structure, bounds a spin manifold is a subtle and difficult problem. It is therefore natural, and of considerable interest, to ask the corresponding question for almost flat \\(\\mathrm{spin}^c\\) manifolds.\n\nThe main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\\(^c\\) as follows.",
    "context": "A closed manifold \\( M \\) is said to be \\emph{almost flat} if, for every \\( \\epsilon > 0 \\), there exists a Riemannian metric \\( g_\\epsilon \\) on \\( M \\) such that \n\\[\n\\operatorname{diam}(M, g_\\epsilon) \\le 1, \\qquad |K_{g_\\epsilon}| < \\epsilon ,\n\\]\nwhere \\( K_{g_\\epsilon} \\) denotes the sectional curvature of \\( g_\\epsilon \\). \nThus, \\( M \\) admits metrics of arbitrarily small curvature and uniformly bounded diameter. For each dimension \\( n \\), there exists a constant \\( \\epsilon_n > 0 \\) with the property that if an \\( n \\)-dimensional manifold admits an \\( \\epsilon_n \\)-flat metric of diameter at most one, then it is almost flat \\cite{CG86}.\n\nA classical theorem of Gromov~\\cite{Gro78} asserts that every almost flat manifold is finitely covered by a nilmanifold. Conversely, Farrell and Hsiang~\\cite{FH83} proved that any manifold finitely covered by a nilmanifold is homeomorphic to an almost flat manifold. Ruh~\\cite{Ruh82} subsequently strengthened Gromov’s result, showing that every almost flat manifold is in fact diffeomorphic to an infranilmanifold.\n\nA long-standing conjecture, proposed independently by Farrell--Zdravkovska~\\cite{FZ83} and Yau~\\cite{Yau93}, predicts that every almost flat manifold bounds a compact manifold. \nIn the special case of flat manifolds, this was established by Hamrick and Royster~\\cite{HR82} in a striking theorem. \nMore recently, Davis and Fang~\\cite{DF16} confirmed the conjecture under the additional assumption that the \\(2\\)-Sylow subgroup of the holonomy group is cyclic or generalized quaternionic; their result was later rederived by Xiong~\\cite{Xio16} through a different argument. \nIn particular, the theorem of Davis and Fang encompasses earlier advances due to Farrell--Zdravkovska~\\cite{FZ83} and Upadhyay~\\cite{Upa01}.\n\nThe general case of the conjecture remains open. Moreover, Davis and Fang~\\cite{DF16} pointed out that determining whether every almost flat \\emph{spin} manifold, possibly after changing the spin structure, bounds a spin manifold is a subtle and difficult problem. It is therefore natural, and of considerable interest, to ask the corresponding question for almost flat \\(\\mathrm{spin}^c\\) manifolds.\n\nThe main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\\(^c\\) as follows.",
    "full_context": "A closed manifold \\( M \\) is said to be \\emph{almost flat} if, for every \\( \\epsilon > 0 \\), there exists a Riemannian metric \\( g_\\epsilon \\) on \\( M \\) such that \n\\[\n\\operatorname{diam}(M, g_\\epsilon) \\le 1, \\qquad |K_{g_\\epsilon}| < \\epsilon ,\n\\]\nwhere \\( K_{g_\\epsilon} \\) denotes the sectional curvature of \\( g_\\epsilon \\). \nThus, \\( M \\) admits metrics of arbitrarily small curvature and uniformly bounded diameter. For each dimension \\( n \\), there exists a constant \\( \\epsilon_n > 0 \\) with the property that if an \\( n \\)-dimensional manifold admits an \\( \\epsilon_n \\)-flat metric of diameter at most one, then it is almost flat \\cite{CG86}.\n\nA classical theorem of Gromov~\\cite{Gro78} asserts that every almost flat manifold is finitely covered by a nilmanifold. Conversely, Farrell and Hsiang~\\cite{FH83} proved that any manifold finitely covered by a nilmanifold is homeomorphic to an almost flat manifold. Ruh~\\cite{Ruh82} subsequently strengthened Gromov’s result, showing that every almost flat manifold is in fact diffeomorphic to an infranilmanifold.\n\nA long-standing conjecture, proposed independently by Farrell--Zdravkovska~\\cite{FZ83} and Yau~\\cite{Yau93}, predicts that every almost flat manifold bounds a compact manifold. \nIn the special case of flat manifolds, this was established by Hamrick and Royster~\\cite{HR82} in a striking theorem. \nMore recently, Davis and Fang~\\cite{DF16} confirmed the conjecture under the additional assumption that the \\(2\\)-Sylow subgroup of the holonomy group is cyclic or generalized quaternionic; their result was later rederived by Xiong~\\cite{Xio16} through a different argument. \nIn particular, the theorem of Davis and Fang encompasses earlier advances due to Farrell--Zdravkovska~\\cite{FZ83} and Upadhyay~\\cite{Upa01}.\n\nThe general case of the conjecture remains open. Moreover, Davis and Fang~\\cite{DF16} pointed out that determining whether every almost flat \\emph{spin} manifold, possibly after changing the spin structure, bounds a spin manifold is a subtle and difficult problem. It is therefore natural, and of considerable interest, to ask the corresponding question for almost flat \\(\\mathrm{spin}^c\\) manifolds.\n\nThe main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\\(^c\\) as follows.\n\nThe main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\\(^c\\) as follows.\n\nThe existence of a spin\\(^c\\) structure on an almost flat manifold is intimately connected with the nature of its holonomy group (which is finite). \nWe now consider a particular instance of this relationship and obtain the following theorem as a consequence of Theorem~\\ref{spinc-thm-intro}. \nRecall that the classical \\emph{Schur multiplier} of a group \\(G\\) is its second group homology \\(H_2(G;\\mathbb{Z})\\), introduced by Issai Schur in 1904. It is clear that the Schur multiplier of a finite group is finite.\n\n\\begin{theorem}\\label{G-thm-intro}\nLet \\(M\\) be an almost flat oriented manifold with holonomy group \\(G\\).\nIf the Schur multiplier of \\(G\\) is of odd order, then \\(M\\) bounds a compact manifold.\n\nIn dimension 4, we can give a complete verification of the Farrell--Zdravkovska--Yau conjecture, both in the orientable and the spin cases.\n\\begin{theorem}\\label{4-thm-intro}\nLet \\(M\\) be an almost flat orientable \\(4\\)-manifold. Then \\(M\\) bounds a spin\\(^c\\) manifold. Furthermore, if \\(M\\) is spin, then \\(M\\) with any spin structure bounds a spin manifold.\n\\end{theorem}\n\n\\begin{lemma}\\label{sw-lemma}\nLet $(M, c)$ be an $n$-dimensional almost flat spin$^c$ manifold. Then its Stiefel-Whitney numbers all vanish.\n\\end{lemma}\n\\begin{proof}\nSince the rational Pontryagin classes of the almost flat manifold $M$ all vanish by Lemma \\ref{chern-lemma}, each integral spin$^c$ Wu classes $\\mu _{2i}^{Spin^c}(M)=a_i c^i$, where $a_i$ is the coefficient of $c^i$ in the expression of the universal integral spin$^c$ Wu class $\\mu _{2i}^{Spin^c}$. Hence, any integral spin$^c$ Wu number vanishes if $n$ is odd, or is of the form $a_I c^{m}$ with $I=\\{j_1,\\ldots, j_r\\}$ a partition of $m$ if $n=2m$ is even. In the latter case, recall the classical Legendre formula \n$$m!=2^{m-s_2(m)}\\cdot (\\text{an odd number})$$\n and the fact $s_2(m)\\leq s_2(j_1)+\\cdots+s_2(j_r)$. Then by Lemmas \\ref{denom-lemma} and \\ref{index-lemma}, \n\\[\n\\langle a_I c^{m}, [M]\\rangle \\in 2^m m! a_I \\mathbb{Z}\\subseteq 2^{s_2(j_1)+\\cdots+s_2(j_r)}m! \\mathbb{Z} \\subseteq \n2^{m}\\mathbb{Z}.\n\\]\nIn particular, the mod-$2$ reduction of any integral spin$^c$ Wu number of $M$ vanishes, or equivalently, the mod-$2$ Wu numbers of $M$ all vanish. Since Stiefel-Whitney numbers are equivalent to mod-$2$ Wu numbers, the Stiefel-Whitney numbers of $M$ all vanish as well. \n\\end{proof}\n\nThe following proposition provides a simple criterion when almost flat spin$^c$ manifold bounds. \n\\begin{proposition}\\label{cn=0-prop}\nLet $(M, c)$ be an $n$-dimensional almost flat spin$^c$ manifold. Then $(M, c)$ bounds a spin$^c$ manifold if and only if $\\langle c^{\\lfloor \\frac{n}{2} \\rfloor }, [M]\\rangle =0$.\n\\end{proposition}\n\\begin{proof}\nBy Lemma \\ref{spinc-bord-lemma}, $(M, c)$ bounds a spin$^c$ manifold if and only if the Stiefel-Whitney numbers and $c$-Pontryagin numbers all vanish. By Lemma \\ref{sw-lemma}, its Stiefel-Whitney numbers all vanish. By Lemma \\ref{chern-lemma}, only the number $\\langle c^{\\lfloor \\frac{n}{2} \\rfloor }, [M]\\rangle$ can be nontrivial among all the $c$-Pontryagin numbers. Hence, the proposition follows. \n\\end{proof}\n\n\\begin{remark}\\label{nasen}\n The proof in fact yields a more general result than Theorem \\ref{spinc-thm-intro}. Indeed, we show that if a spin$^c$ manifold $M$ has all rational Pontryagin classes vanishing, then it bounds an orientable manifold. Moreover, if $n$ is odd, then $M$, equipped with any spin$^c$ structure, bounds a spin$^c$ manifold.\n \\end{remark}\n\nThe following proposition recovers the main theorem of \\cite{DF16} for the {\\em oriented case}. \n\\begin{proposition}\\label{DFprop}\nLet $M$ be an $n$-dimensional almost flat oriented manifold. If the $2$-Sylow subgroup ${\\rm Syl}_2(G)$ of the holonomy group $G$ of $M$ is cyclic or generalized quaternionic, then $M$ bounds an orientable manifold.\n\\end{proposition}\n\\begin{proof}\nThe assumption that ${\\rm Syl}_2(G)$ is cyclic or generalized quaternionic implies $H^3({\\rm Syl}_2(G);\\mathbb{Z})=0$ by Lemma \\ref{DFlemma}, and then implies that $M$ is spin$^c$ by Lemma \\ref{H3lemma}. Hence, $M$ bounds by Theorem \\ref{spinc-thm-intro}.\n\\end{proof}\n\n\\begin{lemma}\\label{denom-lemma}\nIn the expression of the universal integral spin$^c$ Wu class $\\mu_{2n}^{Spin^c}$, the denominator of the coefficient $a_n$ of $c^{n}$ satisfies \n\\[\n{\\rm denom}~(a_n)~|~2^{n-s_2(n)},\n\\] \nwhere $s_2(n)$ denotes the sum of the binary digits of $n$.\n\\end{lemma}\n\n\\begin{theorem}\\label{spinc-thm-intro}\nLet $M$ be an \\(n\\)-dimensional almost flat spin\\(^c\\) manifold. Then $M$ bounds an orientable manifold. Moreover, if \\(n\\) is odd, then $M$ with any spin$^c$ structure bounds a spin\\(^c\\) manifold.\n\\end{theorem}",
    "post_theorem_intro_text_len": 5577,
    "post_theorem_intro_text": "The existence of a spin\\(^c\\) structure on an almost flat manifold is intimately connected with the nature of its holonomy group (which is finite). \nWe now consider a particular instance of this relationship and obtain the following theorem as a consequence of Theorem~\\ref{spinc-thm-intro}. \nRecall that the classical \\emph{Schur multiplier} of a group \\(G\\) is its second group homology \\(H_2(G;\\mathbb{Z})\\), introduced by Issai Schur in 1904. It is clear that the Schur multiplier of a finite group is finite.\n\n\\begin{theorem}\\label{G-thm-intro}\nLet \\(M\\) be an almost flat oriented manifold with holonomy group \\(G\\).\nIf the Schur multiplier of \\(G\\) is of odd order, then \\(M\\) bounds a compact manifold.  \n\nIn particular, if the \\(2\\)-Sylow subgroup of \\(G\\) is cyclic, generalized quaternionic, or coincides with the \\(2\\)-Sylow subgroup of any finite simple group listed in Example~\\ref{simple-ex}, then \\(M\\) is the boundary of a compact manifold.\n\\end{theorem}\n\nTheorem~\\ref{G-thm-intro} combined with Theorem \\ref{spinc-thm-intro} extends the result of Davis and Fang~\\cite{DF16} for almost flat {\\em oriented} manifolds. Additionally, as noted in Example~\\ref{Xio-ex}, the result of Xiong~\\cite{Xio16} appears as a special case of Theorem~\\ref{spinc-thm-intro}.\n\n\\medskip\n\nIn dimension 4, we can give a complete verification of the Farrell--Zdravkovska--Yau conjecture, both in the orientable and the spin cases.\n\\begin{theorem}\\label{4-thm-intro}\nLet \\(M\\) be an almost flat orientable \\(4\\)-manifold. Then \\(M\\) bounds a spin\\(^c\\) manifold. Furthermore, if \\(M\\) is spin, then \\(M\\) with any spin structure bounds a spin manifold.\n\\end{theorem}\n\nOntaneda~\\cite{Ont20} established an important converse to the classical Cheeger--Fukaya--Gromov theorem~\\cite{CFG92}: \nif an almost flat manifold \\(M\\) bounds, then there exists a compact manifold \\(W\\) with boundary \\(M\\) such that \\(W \\setminus M\\) carries a complete, finite-volume Riemannian metric with negatively pinched sectional curvature.  \n\nFarrell and Zdravkovska~\\cite{FZ83} went further to conjecture that an almost flat (respectively, flat) manifold bounds a compact manifold whose interior admits a complete, finite-volume metric with negative (respectively, constant negative) sectional curvature. \nIn the flat case this conjecture was disproved by Long and Reid~\\cite{LR20}, while in the almost flat case it was verified by Davis and Fang~\\cite{DF16} when the holonomy group is cyclic or quaternionic.  \n\nIn the present work, combining Theorem~\\ref{spinc-thm-intro} with Ontaneda’s theorem~\\cite{Ont20}, we confirm the conjecture for almost flat spin\\(^c\\) manifolds.\n\n\\begin{theorem}\nAn almost flat spin\\(^c\\) manifold bounds a compact manifold whose interior admits a complete, finite-volume Riemannian metric with negatively pinched sectional curvature. \n\\end{theorem}\n\n$\\, $\n\nOur approach to the proofs of the foregoing theorems departs in a substantial way from earlier work. Whereas infranilmanifolds have traditionally been studied largely through group-theoretic methods, we adopt a more topological point of view. The central tool in our proof is the integral Wu class for \nspin\\(^c\\)\n manifolds, developed in Section~\\ref{sec: spincWu}. This construction is inspired by the integral spin Wu class introduced by Hopkins and Singer~\\cite{HS05}. The guiding idea is that, while the Stiefel–Whitney classes themselves are only indirectly related to geometry, the integral lift of Wu classes admits a more transparent geometric interpretation. As a result, questions concerning the vanishing of Stiefel–Whitney numbers may be reduced to parity properties of these integral Wu classes.\n\nThe essential point in the proof of the main theorem for \nspin\\(^c\\) almost flat manifolds is that the Atiyah–Singer index theorem yields strong divisibility properties for the top power of the \nspin\\(^c\\) characteristic class $c$. This divisibility, in turn, guarantees the evenness of the integral Wu numbers, thereby forcing the vanishing of the relevant Stiefel–Whitney numbers.\n\n$\\, $\n\nThe paper is organized as follows.  \nIn Section~\\ref{sec: spincWu}, we recall the integral spin Wu class of Hopkins and Singer and introduce the corresponding integral Wu class for spin\\(^c\\) vector bundles, establishing along the way an estimate (Lemma~\\ref{denom-lemma}) that will be used later.  \nSection~\\ref{sec: spincbound} contains the proof of our main theorem, Theorem~\\ref{spinc-thm-intro}, which confirms the Farrell--Zdravkovska--Yau conjecture for almost flat spin\\(^c\\) manifolds.  \nIn Section~\\ref{sec: holo}, we derive Theorem~\\ref{G-thm-intro} as an application of Theorem~\\ref{spinc-thm-intro}.  \nFinally, Section~\\ref{sec: 4} is devoted to the 4-dimensional case, where Theorem~\\ref{4-thm-intro} is proved.\n\n\\bigskip \n\n\\noindent{\\bf Acknowledgements.} \nFei Han was partially supported by the grant AcRF A-8000451-00-00 from National University of Singapore. Ruizhi Huang was supported in part by the National Natural Science Foundation of China (Grant nos. 12331003 and 12288201), the National Key R\\&D Program of China (No. 2021YFA1002300) and the Youth Innovation Promotion Association of Chinese Academy Sciences. Weiping Zhang was partially supported by NSFC Grant No. 11931007, National Key R\\&D Program of China (2024YFA1013202) and Nankai Zhide Foundation.\n\nThe authors would like to thank Haibao Duan for helpful discussions on integral Wu classes. Ruizhi Huang also thanks Xiaolei Wu for valuable conversations on group homology, and Nansen Petrosyan for pointing out Remark \\ref{nasen}.",
    "sketch": "The post-theorem introduction indicates that the proof of Theorem~\\ref{spinc-thm-intro} uses a topological approach whose \"central tool\" is the integral Wu class for spin$^c$ manifolds (introduced in Section~\\ref{sec: spincWu}), motivated by Hopkins--Singer’s integral spin Wu class. The \"guiding idea\" is that the integral lift of Wu classes has a clearer geometric interpretation, so that \"questions concerning the vanishing of Stiefel--Whitney numbers may be reduced to parity properties of these integral Wu classes.\" The \"essential point\" is that the Atiyah--Singer index theorem gives \"strong divisibility properties\" for the top power of the spin$^c$ characteristic class $c$; this divisibility \"guarantees the evenness of the integral Wu numbers,\" which in turn \"forc[es] the vanishing of the relevant Stiefel--Whitney numbers.\" The paper roadmap adds that Section~\\ref{sec: spincbound} contains the proof of Theorem~\\ref{spinc-thm-intro}.",
    "expanded_sketch": "The post-theorem introduction indicates that, in establishing the main theorem, the proof uses a topological approach whose \"central tool\" is the integral Wu class for spin$^c$ manifolds (introduced next), motivated by Hopkins--Singer’s integral spin Wu class. The \"guiding idea\" is that the integral lift of Wu classes has a clearer geometric interpretation, so that \"questions concerning the vanishing of Stiefel--Whitney numbers may be reduced to parity properties of these integral Wu classes.\" The \"essential point\" is that the Atiyah--Singer index theorem gives \"strong divisibility properties\" for the top power of the spin$^c$ characteristic class $c$; this divisibility \"guarantees the evenness of the integral Wu numbers,\" which in turn \"forc[es] the vanishing of the relevant Stiefel--Whitney numbers.\" The paper roadmap adds that later we give the proof of the main theorem.",
    "expanded_theorem": "\\label{spinc-thm-intro}\nLet $M$ be an \\(n\\)-dimensional almost flat spin\\(^c\\) manifold. Then $M$ bounds an orientable manifold. Moreover, if \\(n\\) is odd, then $M$ with any spin$^c$ structure bounds a spin\\(^c\\) manifold.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(M\\) be an \\(n\\)-dimensional almost flat spin\\(^c\\) manifold, meaning that \\(M\\) is a closed manifold admitting a spin\\(^c\\) structure and, for every \\(\\epsilon>0\\), there exists a Riemannian metric \\(g_\\epsilon\\) on \\(M\\) such that\n\\[\n\\operatorname{diam}(M,g_\\epsilon)\\le 1,\\qquad |K_{g_\\epsilon}|<\\epsilon,\n\\]\nwhere \\(K_{g_\\epsilon}\\) denotes the sectional curvature. Which of the following conclusions holds for \\(M\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\(M\\) is the boundary of a compact orientable manifold. Moreover, if \\(n\\) is odd, then for any chosen spin\\(^c\\) structure on \\(M\\), there exists a compact spin\\(^c\\) manifold whose induced boundary spin\\(^c\\) structure is that given one on \\(M\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "\\(M\\) is the boundary of a compact orientable manifold. Moreover, for any \\(n\\) and for any chosen spin\\(^c\\) structure on \\(M\\), there exists a compact spin\\(^c\\) manifold whose induced boundary spin\\(^c\\) structure is that given one on \\(M\\)."
        },
        {
          "label": "C",
          "text": "If \\(n\\) is odd, then for any chosen spin\\(^c\\) structure on \\(M\\), there exists a compact spin\\(^c\\) manifold whose induced boundary spin\\(^c\\) structure is that given one on \\(M\\)."
        },
        {
          "label": "D",
          "text": "\\(M\\) is the boundary of a compact orientable manifold if and only if, when \\(n\\) is odd, for any chosen spin\\(^c\\) structure on \\(M\\), there exists a compact spin\\(^c\\) manifold whose induced boundary spin\\(^c\\) structure is that given one on \\(M\\)."
        },
        {
          "label": "E",
          "text": "\\(M\\) is the boundary of a compact spin\\(^c\\) manifold. Moreover, if \\(n\\) is odd, then there exists a choice of spin\\(^c\\) structure on \\(M\\) for which some compact spin\\(^c\\) manifold induces that boundary spin\\(^c\\) structure."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "odd-dimensional restriction for spin^c bounding with any chosen structure",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "characteristic",
          "tampered_component": "dropped the unconditional orientable-bounding conclusion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "logical linkage between unconditional orientable bounding and the odd-dimensional spin^c enhancement",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "replaced orientable bounding by unconditional spin^c bounding and weakened quantifier from any structure to some structure",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the definition of an almost flat spin^c manifold and does not explicitly or implicitly reveal the odd-dimensional restriction or the precise bounding conclusion."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a direct theorem-recall question: it essentially asks for the correct conclusion under the theorem's hypotheses. The nearby variants add some comparison, but it is still largely a reformulation of the result."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact scope of the conclusion, especially the odd-dimensional spin^c-bounding clause versus stronger or weaker alternatives. However, it mainly tests precise recall of the statement rather than deeper generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are plausible overgeneralizations or weakened variants, which is good. However, choice C is a weaker statement that is still true, so it functions poorly as a distractor and creates ambiguity in a single-answer format."
      },
      "total_score": 5,
      "overall_assessment": "Good at avoiding answer leakage, but the question is mostly theorem-statement recognition rather than substantive reasoning. Its main flaw is the inclusion of a weaker true option, which weakens distractor quality and makes the item somewhat ambiguous."
    }
  },
  {
    "id": "2512.19461v1",
    "paper_link": "http://arxiv.org/abs/2512.19461v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:twistor}\nThe twistor bundle $q:\\mathbb{C} P^5\\to \\mathbb{H} P^2$ has \n\\[\n2= \\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_2)>\\mathrm{Mwgt}(q;\\F_2)=1>\\mathrm{wgt}(q;\\F_2)=0.\n\\]",
      "start_pos": 11658,
      "end_pos": 11809,
      "label": "thm:twistor"
    },
    "ref_dict": {
      "thm:twistor": "\\begin{thm}\\label{thm:twistor}\nThe twistor bundle $q:\\C P^5\\to \\Ha P^2$ has \n\\[\n2= \\secat(q)=\\Swgt(q;\\F_2)>\\Mwgt(q;\\F_2)=1>\\wgt(q;\\F_2)=0.\n\\]\n\\end{thm}",
      "thm:2cell": "\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}",
      "def:Swgt": "\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}"
    },
    "pre_theorem_intro_text_len": 5813,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section. \n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*}  \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\nThe ideas in the previous two paragraphs originated in the work of Fadell and Husseini \\cite{FH} on category weight, and were developed in the context of $q$ being the based path fibration $P_0 X\\to X$ by Rudyak \\cite{Rudyak99} and Strom \\cite{Strom}. We have given here the straightforward generalisation to arbitrary fibrations $q: E\\to B$.\n\nIn this paper we will improve on the module weight using secondary cohomology operations. Recall that whenever stable cohomology operations $\\theta$ and $\\varphi$ satisfy a relation $\\varphi\\circ\\theta\\equiv 0$, this gives rise to a \\emph{secondary cohomology operation}\n\\[\n\\Phi=\\Phi(\\varphi,\\theta) : \\operatorname{ker} \\,\\theta \\to \\mathrm{coker} \\,\\varphi\n\\]\nwith \\emph{indeterminacy} $\\operatorname{im} \\varphi$. Such secondary operations are natural, in the following sense: if $f: X\\to Y$ is a map of spaces, then the diagram\n \\[\n \\xymatrix{\n \\operatorname{ker}\\, \\theta_Y \\ar[r]^-{\\Phi} \\ar[d]_{f^*} & \\mathrm{coker}\\, \\varphi_Y \\ar[d]^{f^*} \\\\\n \\operatorname{ker}\\, \\theta_X \\ar[r]^-{\\Phi} & \\mathrm{coker}\\, \\varphi_X\n }\n \\]\n commutes. (This makes sense because $f^* (\\operatorname{ker}\\,\\theta_Y)\\subseteq \\operatorname{ker}\\,\\theta_X$ and $f^*(\\operatorname{im} \\varphi_Y)\\subseteq \\operatorname{im} \\varphi_X$.) If we consider only stable cohomology operations with $\\F_p$-coefficients, we can say that $f^*:H^*(Y;\\F_p)\\to H^*(X;\\F_p)$ is a map of $\\A_p$-modules which commutes with all secondary operations.  This leads us to our main definition.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.",
    "context": "The {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.\n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*} \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}",
    "full_context": "The {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.\n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*} \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\n\\def\\Ab{\\mathrm{Ab}}\n\\def\\Rel{\\mathrm{Rel}}\n\\def\\cat{\\mathsf{cat}}\n\\def\\secat{\\mathsf{secat}}\n\\def\\wgt{\\mathrm{wgt}}\n\\def\\rank{\\mathrm{rank}}\n\\def\\ind{\\mathrm{ind}}\n\\def\\dim{\\mathrm{dim}}\n\\def\\ker{\\operatorname{ker}}\n\\def\\nil{\\operatorname{nil}}\n\\def\\coker{\\mathrm{coker}}\n\\def\\im{\\mathrm{im}}\n\\def\\aut{\\mathrm{aut}_1\\,}\n\\def\\ev{\\mathrm{ev}}\n\\def\\cupQ{\\mathrm{cup}_0}\n\\def\\rk{\\mathrm{rank}}\n\\def\\eQ{\\mathrm{e}_0}\n\\def\\res{\\mathrm{res}}\n\\def\\tto{\\twoheadrightarrow}\n\n\\newcommand{\\co}{\\colon\\thinspace}\n\\newcommand{\\nsub}{\\mathrel{\\unlhd}}\n\\newcommand{\\PB}{\\mathcal{P}}\n\\newcommand{\\TC}{{\\sf TC}}\n\\newcommand{\\MTC}{{\\sf MTC}}\n\\newcommand{\\TCS}{\\mathbf{TC}^S}\n\\newcommand{\\swgt}{\\mathrm{swgt}}\n\\def\\Mwgt{\\mathrm{Mwgt}}\n\\def\\Swgt{\\mathrm{Swgt}}\n\\newcommand{\\cld}{{\\sf cd}}\n\\newcommand{\\gld}{{\\sf gd}}\n\\newcommand{\\catG}{{\\sf cat}_G}\n\\newcommand{\\secatG}{{\\sf secat}_G}\n\\newcommand{\\cldG}{{\\sf cd}_\\mathcal{G}}\n\\newcommand{\\gldG}{{\\sf gd}_\\mathcal{G}}\n\\newcommand{\\OG}{\\mathcal{O}G}\n\\newcommand{\\OF}{\\mathcal{O}_\\mathcal{F}}\n\\newcommand{\\G}{\\Gamma}\n\\newcommand{\\GG}{\\pi\\rtimes G}\n\\newcommand{\\gen}{\\mathfrak{genus}}\n\\newcommand{\\stab}{\\stackrel{\\simeq}{\\longrightarrow}}\n\\newcommand{\\Aut}{\\operatorname{Aut}}\n\\newcommand{\\OFG}{\\mathcal{O}_\\F\\Gamma}\n\\newcommand{\\OFGG}{\\mathcal{O}_\\FG (\\pi\\rtimes G)}\n\\newcommand{\\pspan}{\\operatorname{pspan}}\n\\newcommand{\\spn}{\\operatorname{span}}\n\\newcommand{\\paths}[1]{P#1}\n\\newcommand{\\eval}[1]{\\pi_{#1}}\n\\newcommand{\\evalG}[1]{p_{#1}}\n\\newcommand{\\evaln}[1]{\\pi_{n,#1}}\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\secat(q)=\\Swgt(q;\\F_p)>\\Mwgt(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\C P^5 \\ar[r]^q & \\Ha P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\C^6\\cong\\Ha^3$ to the quaternionic line in $\\Ha^3$ containing it. Standard techniques give $1\\le \\secat(q)\\le 2$, but are inconclusive in determining the exact value of $\\secat(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\nOur second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\\to X$ is the based path fibration on a space $X$ we denote $\\wgt(q;\\F_p)$, $\\Mwgt(q;\\F_p)$ and $\\Swgt(q;\\F_p)$ by $\\wgt(X;\\F_p)$, $\\Mwgt(X;\\F_p)$ and $\\Swgt(X;\\F_p)$, respectively.\n\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}\n\n\\begin{lem}\\label{lem:Mwgttwistor}\nFor $q$ the twistor bundle, $\\Mwgt(q;\\F_2)=1$.\n\\end{lem}\n\n\\begin{lem}\\label{lem:Swgttwistor}\n For $q$ the twistor bundle, $\\Swgt(q;\\F_2)=2$.\n \\end{lem}",
    "post_theorem_intro_text_len": 2032,
    "post_theorem_intro_text": "Theorem \\ref{thm:twistor} answers a question on MathOverflow asked by Prateep Chakraborty \\cite{Chakra}. Note that the rational sectional category $\\secat_0(q)$ in this example is $1$, as follows from results of Stanley \\cite{Stanley}.\n\nOur second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\\to X$ is the based path fibration on a space $X$ we denote $\\mathrm{wgt}(q;\\F_p)$, $\\mathrm{Mwgt}(q;\\F_p)$ and $\\mathrm{Swgt}(q;\\F_p)$ by $\\mathrm{wgt}(X;\\F_p)$, $\\mathrm{Mwgt}(X;\\F_p)$ and $\\mathrm{Swgt}(X;\\F_p)$, respectively.\n\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\mathsf{cat}(X)=\\mathrm{Swgt}(X;\\F_2)>\\mathrm{Mwgt}(X;\\F_2)=\\mathrm{wgt}(X;\\F_2)=1.\n\\]\n\\end{thm}\n\nIn the proofs of both Theorems \\ref{thm:twistor} and \\ref{thm:2cell}, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \\cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences\" in his book \\cite{Harper}. This method sometimes allows to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$. We review the method in Section 2 below. Thereafter the proofs diverge: Theorem \\ref{thm:twistor} uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel-Whitney classes, while Theorem \\ref{thm:2cell} uses results of Kono--Kozima \\cite{KonoKozima} on Steenrod squares in the loop space of $\\operatorname{Spin}(n)$.",
    "sketch": "In the proofs of both Theorems \\ref{thm:twistor} and \\ref{thm:2cell}, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \\cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences\" \\cite{Harper}, allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed in Section 2). For Theorem \\ref{thm:twistor} specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.",
    "expanded_sketch": "In establishing the main theorem, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. We also use the following theorem.\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}\nIn each case, this operation detects the Hopf invariant of the attaching map of the top cell Berrick--Hess, Cohen--Lada--May (Title not available, year not available), and Gugenheim--May--Vigu\\'{e}-Poirrier (Title not available, year not available), which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences'' Harper (Title not available, year not available), allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed later). For the main theorem specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.",
    "expanded_theorem": "\\label{thm:twistor}\nThe twistor bundle $q:\\mathbb{C} P^5\\to \\mathbb{H} P^2$ has \n\\[\n2= \\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_2)>\\mathrm{Mwgt}(q;\\F_2)=1>\\mathrm{wgt}(q;\\F_2)=0.\n\\]",
    "theorem_type": [
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let q:\\mathbb{C}P^5\\to\\mathbb{H}P^2 be the twistor bundle, where q sends a complex line in \\mathbb{C}^6\\cong\\mathbb{H}^3 to the quaternionic line in \\mathbb{H}^3 containing it. Write \\mathsf{secat}(q) for the normalized sectional category of q, i.e. the least k such that \\mathbb{H}P^2 can be covered by open sets U_0,\\dots,U_k on each of which q admits a continuous local section. For the (k+1)-fold fibred join q(k):E(k)\\to \\mathbb{H}P^2, define \\mathrm{wgt}(q;\\mathbb{F}_2)=\\min\\{k\\mid q(k)^*:H^*(\\mathbb{H}P^2;\\mathbb{F}_2)\\to H^*(E(k);\\mathbb{F}_2)\\text{ is injective}\\}, \\mathrm{Mwgt}(q;\\mathbb{F}_2) as the least k for which q(k)^* admits a retraction in the category of \\mathcal{A}_2-modules, and \\mathrm{Swgt}(q;\\mathbb{F}_2) as the least k for which q(k)^* admits such a retraction commuting with all secondary operations. Which quantitative statement holds for this bundle?",
      "correct_choice": {
        "label": "A",
        "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)=1>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."
      },
      "choices": [
        {
          "label": "B",
          "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)=\\mathrm{Mwgt}(q;\\mathbb{F}_2)>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."
        },
        {
          "label": "C",
          "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)\\ge \\mathrm{wgt}(q;\\mathbb{F}_2)."
        },
        {
          "label": "D",
          "text": "1=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)=\\mathrm{wgt}(q;\\mathbb{F}_2)=0."
        },
        {
          "label": "E",
          "text": "2=\\mathsf{secat}(q)=\\mathrm{Mwgt}(q;\\mathbb{F}_2)>\\mathrm{Swgt}(q;\\mathbb{F}_2)=1>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "nontrivial secondary operation obstructs only secondary-compatible retraction, not ordinary module retraction",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "exact numerical values of \\mathrm{Mwgt} and \\mathrm{wgt}",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "nonzero secondary obstruction at the first fibred join rules out a section of q(1)",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "ordering between \\mathrm{Swgt} and \\mathrm{Mwgt} forced by secondary-operation obstruction",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the invariants but does not reveal the numerical values or the correct ordering. There is no explicit or near-explicit statement of choice A in the prompt."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not tautological in the strict sense, since the student must choose among several competing quantitative statements. However, it is very close to asking for recall of a specific known theorem/result about this bundle rather than deriving a conclusion from supplied data."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare the invariants and reject impossible orderings, but the problem mainly tests knowledge/recall of the exact computed values rather than substantial generative mathematical reasoning from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic confusions: collapsing Swgt and Mwgt, giving only a weaker true inequality, misstating the sectional category, or reversing the secondary-operation obstruction. They are distinct and well aligned with common failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid high-level recall question with strong distractors and no answer leakage, but it leans more toward theorem recollection than genuine generative reasoning."
    }
  },
  {
    "id": "2512.19500v1",
    "paper_link": "http://arxiv.org/abs/2512.19500v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.",
      "start_pos": 26726,
      "end_pos": 27068,
      "label": "t:main"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 1849,
    "pre_theorem_intro_text": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n  Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n  $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n  an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n  point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.",
    "context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.",
    "full_context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.\n\\section{Quartic hyperelliptic curves}\nIf $D$ is a plane algebraic curve over a field $F$ given by\n$P(X, Y)\\in F[X, Y]$, then $D(F)=\\defset{(x, y)\\in F^2}{P(x,y)=0}$\ndenotes the set of affine $F$-points on $D$.\n\nFinally assume that $f(X)=(aX^2-2X+d)(aX^2+2X+d)$ is\n inseparable. The two factors are relatively prime (because the\n difference is $4X$). Furthermore, the discriminant of both factors\n is $4(1-ad)$, hence $ad=1$ and we are done.\n\\end{proof}\n\\begin{remark}\n One can check that $q=13$, $17$, and $19$ do not give\n exceptions. However, $q=3$, $5$, $7$, $9$, and $11$ are indeed\n exceptions. This makes it unlikely that there is a ``cleaner'' proof\n of the lemma for odd $q$.\n\\end{remark}\n\\begin{corollary}\\label{c:split}\n Let $q\\ge4$ be a prime power and $a, b, c, d\\in\\FF_q$ with\n $ad-bc=1$. Set\n $T=\\defset{r+1/r}{r\\in\\Fqs}\\cup\\set{0}$. Suppose that\n $ar+d/r\\in T$ and $-cr+b/r\\in T$ for all $r\\in\\Fqs$. Then one\n of the following holds:\n \\begin{itemize}\n \\item[(a)] $a=d=0$ or $b=c=0$;\n \\item[(b)] $q=9$ and $ar+d/r=2$ for some $r\\in\\Fqs$.\n \\end{itemize}\n\\end{corollary}\n\\begin{proof}\n Using the previous lemma, one only need to check the cases where\n $q\\le19$ is an odd prime power. This is most conveniently done with\n the SageMath code at \\cite{dera_psl2_verify}.\n\\end{proof}\n\\begin{lemma}\n Let $q$ be a prime power with $q\\ge23$ if $q$ is odd. For\n $F=\\FF_{q^2}$ set $N=\\defset{r\\in F}{r^{q+1}=1}$. Suppose that for\n $0\\ne a\\in F$ the following holds: For all $r\\in N$ with\n $ar+a^q/r\\ne0$ there is $s\\in N$ such that $ar+a^q/r = s+1/s$. Then\n $a\\in N$.\n\\end{lemma}\n\\begin{proof} We first handle the easier case that $q$ is even.\nAs the square map is an automorphism of $F$, there is $r_0\\in F$ with\n$a^{q-1}=r_0^2$. Note that $a^{q-1}\\in N$, hence $r_0\\in N$.\n\nWe see that the curve $Y^2=\\gamma P(X)Q(X)$ has at least $2(q-6)$\n affine points over $\\FF_q$. On the other hand, by Lemma\n \\ref{l:hasse}, this number is at most $q+1+2\\sqrt{q}$. From this one\n quickly obtains $q\\le19$.\n\\end{proof}\n\\begin{remark}\n One can check that $q=11$, $17$, and $19$ do not give exceptions.\n However, the cases $q=3$, $5$, $7$, $9$, $13$ are indeed\n exceptions. This is most easily seen for $q=3$, because then\n $\\defset{s+1/s}{s\\in N}=\\FF_q$, so the condition on $a$ is vacuous.\n\\end{remark}\n\\begin{corollary}\\label{c:nonsplit}\n Let $q$ be a prime power. Set $N=\\defset{r\\in\\FF_{q^2}}{r^{q+1}=1}$\n and $T=\\defset{r+1/r}{r\\in N}$. For $a, b\\in\\FF_{q^2}$ with\n $a^{q+1} + b^{q+1} = 1$ suppose that $ar+a^q/r\\in T$ and\n $br+b^q/r\\in T$ for all $r\\in\\Fqs$. Then $a=0$ or $b=0$.\n\\end{corollary}\n\\begin{proof}\n If $q\\ge23$, then the previous lemma shows that $a^{q+1}=1$, hence\n $b=0$, or $b^{q+1}=1$, hence $a=0$. The cases $q\\le19$ are\n checked in \\cite{dera_psl2_verify}.\n\\end{proof}\n\\section{The groups of quaternions $2A_4$, $2S_4$, and $2A_5$}\n\\begin{lemma}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Let $H$ be the preimage of $\\bH$ in\n $\\SL_2(F)$ and $x, y\\in F^{2\\times 2}$. If the characteristic of $F$\n is $0$ or relatively prime to $\\abs{\\bH}$, then\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m\\cdot (\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y))\\cdot\\det(x)^m,\n \\]\n where $m\\in\\set{0, 1}$ if $\\bH\\cong A_4$, $m\\in\\set{0, 1, 2}$ if\n $\\bH\\cong S_4$, $m\\in\\set{0, 1, 2, 3, 4}$ if $\\bH\\cong A_5$, and\n $c_0, c_1, c_2, c_3, c_4$ = $1, 2, 5, 14, 42$ are the first five\n Catalan numbers (with index shifted by $1$).\n\\end{lemma}\n\\begin{proof}\n We assume that the characteristic of $F$ is $0$ or relatively prime\n to $\\abs{\\bH}$. Let $\\bar F$ be an algebraic closure of $F$. It is\n well known that $\\PSL_2(\\bar F)$ contains exactly one conjugacy\n class of subgroups isomorphic to $\\bH$, see e.g.~\\cite[Proposition\n 4.1]{beauville___pgl2}. (In fact more is true. By \\cite[Proposition\n 4.2]{beauville___pgl2} $\\PGL_2(F)$ contains at most one conjugacy\n class of subgroups isomorphic to $\\bH$.)\n\nNote that for $h\\in H$ (recall that $H\\le\\SL_2(\\bar F)$), we\n have $f(h'x, h'y)=f(x, y)$, so $f$ is an invariant of\n $H$. Working out the low degree invariants of $H$ could\n provide a more conceptual explanation. (At a first glance, it\n might not be obvious that the right hand side is an invariant\n under simultaneous left multiplication of $x$ and $y$ by\n $h$. But this follows from the identity\n $\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=\\trace(\\adj(y)\\cdot\n x)$, where $\\adj(y)$ denotes the adjugate of $y$.)\n \\end{itemize}\n\\end{remark}\n\\begin{lemma}\\label{l:polyhedral}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Assume that the characteristic of $F$ is\n $0$ or relatively prime to $\\abs{\\bH}$. Let $H$ be the preimage of\n $\\bH$ in $\\SL_2(F)$. Set\n $T=\\defset{\\trace(h)}{h\\in H}\\setminus\\set{-2, 2}$, and for\n $x = \\begin{pmatrix}a & b\\\\c & d\\end{pmatrix}\\in\\SL_2(F)$ set\n $y = \\begin{pmatrix}0 & 0\\\\c & d\\end{pmatrix}$. Then\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in T}(\\trace(hx)-t)\\right) =\\abs{\\bH}.\n \\]\n\\end{lemma}\n\\begin{proof}\n With this choice of $x$ and $y$ we have\n \\[\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=ad-bc=\\det(x)=1,\\]\n so the previous lemma yields\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m.\n \\]\n Let $Z$ be a variable over $F$. Then \\cite{dera_psl2_verify} verifies\n \\[\n \\prod_{t\\in T}(Z-t) =\n \\begin{cases}\n Z^3 - Z,&\\text{ if }\\bH\\cong A_4\\\\\n Z^5 - 3Z^3 + 2Z,&\\text{ if }\\bH\\cong S_4\\\\\n Z^7 - 4Z^5 + 4Z^3 - Z,&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\nThis yields\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in\n T}(\\trace(hx)-t)\\right) =\n \\begin{cases}\n \\abs{\\bH}(c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_4\\\\\n \\abs{\\bH}(c_2-3c_1+2c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong S_4\\\\\n \\abs{\\bH}(c_3-4c_2+4c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\n\\end{proof}\n\\section{Proof of Theorem \\ref{t:main}}\nThe maximal subgroups of almost simple groups with socle $\\PSL_2(q)$\nare known; see \\cite[Table~8.1, Table~8.2]{bray_etal___maximal} or\n\\cite{giudici___psl2}. The results in \\cite{bray_etal___maximal} and\n\\cite{giudici___psl2} are more detailed than those in the following\nlemma, which lists only the restrictions on $q$ that we shall use.\n\\begin{lemma}\\label{l:giudici}\n Let $q\\ge4$ be a prime power and $S=\\PSL_2(q)\\le\n G\\le\\PGammaL_2(q)$. Let $M$ be a maximal subgroup of $G$ with\n $S\\not\\le M$. Set $\\bH=M\\cap S$. Then one of the following holds.\n \\begin{itemize}\n \\item[(a)] $[S:\\bH] = q+1$,\n \\item[(b)] $[S:\\bH] = q(q+1)/2$,\n \\item[(c)] $[S:\\bH] = q(q-1)/2$,\n \\item[(d)] $\\bH\\cong A_4$, and $q$ is and odd prime,\n \\item[(e)] $\\bH\\cong S_4$, and $q$ is a prime with $q\\equiv\\pm1\\mod{8}$,\n \\item[(f)] $\\bH\\cong A_5$, and $q\\equiv\\pm1\\mod{10}$ and $q=9$ if $q$ is\n power of $3$,\n \\item[(g)] $\\bH\\cong\\PSL_2(r)$ or $\\bH\\cong\\PGL_2(r)$, where $q=r^e$\n for $e\\ge2$.\n \\end{itemize}\n Furthermore, in each case $\\bH$ is unique up to conjugacy in\n $\\aut(S)$ (with two classes in case (g) if $r$ is odd).\n\\end{lemma}\nLet $S$ and $G$ be as in Theorem \\ref{t:main}, $M$ be a stabilizer of\na point in $\\Omega$, and $\\bH=M\\cap S$. Then $M$ is a maximal subgroup\nof $G$ by primitivity. We handle the cases (a) to (g) separately.\n\nThe remaining cases (b) to (g) require some work. Suppose that there\nis a counterexample. Then there exist distinct\n$\\alpha, \\beta\\in\\Omega$ such that no derangement in $S$ moves\n$\\alpha$ to $\\beta$. Let $\\bH$ be the stabilizer in $S$ of $\\alpha$\nand $\\bx\\in S$ with $\\alpha^{\\bx}=\\beta$. Then $\\bH\\bx$ is the set of\nelements which moves $\\alpha$ to $\\beta$. As this set does not contain\na derangement, every element in the coset $\\bH\\bx$ is conjugate to an\nelement from $\\bH$.",
    "post_theorem_intro_text_len": 702,
    "post_theorem_intro_text": "If $G$ acts primitively on $\\Omega$, then the possibilities for a point\nstabilizer $S_\\omega < S$ are known. While some cases are relatively\neasy to handle, the more difficult ones arise when $S_\\omega$ is\ndihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$,\n$S_4$, or $A_5$. In the former case we require Hasse’s bound on the\nnumber of points on elliptic curves over finite fields, and to rule out\nthe latter cases we derive trace identities for the binary polyhedral\ngroups $2A_4$, $2S_4$, and $2A_5$.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.",
    "sketch": "To prove Theorem~\\ref{t:main}, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”",
    "expanded_sketch": "To prove the main theorem, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”",
    "expanded_theorem": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.",
    "theorem_type": [
      "Universal–Existential",
      "Implication"
    ],
    "mcq": {
      "question": "Let $q\\ge 4$ be a prime power, let $S=\\PSL_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning $S\\le G\\le \\operatorname{Aut}(S)$. Assume that $G$ acts primitively on a set $\\Omega$ (that is, $G$ preserves no nontrivial partition of $\\Omega$). For distinct points $\\alpha,\\beta\\in\\Omega$, which conclusion holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$; equivalently, $s$ is a derangement in the action on $\\Omega$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no point of $\\Omega$; equivalently, one may always choose a derangement from the whole almost simple group $G$ sending $\\alpha$ to $\\beta$."
        },
        {
          "label": "C",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$."
        },
        {
          "label": "D",
          "text": "For every point $\\alpha\\in\\Omega$, there exists an element $s\\in S$ that fixes no point of $\\Omega$ and moves $\\alpha$; in particular, $S$ contains derangements in this primitive action."
        },
        {
          "label": "E",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, every element $s\\in S$ with $\\alpha^s=\\beta$ fixes no point of $\\Omega$; equivalently, the entire transporter from $\\alpha$ to $\\beta$ inside $S$ consists of derangements."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "socle-membership of the witnessing element",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the derangement requirement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "pairwise transporter conclusion weakened to mere existence of some derangements",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "existential statement on a coset replaced by universal statement on the whole transporter coset",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem sets up the group-action context but does not explicitly state the conclusion about derangements in S. The correct answer is not leaked directly, though the focus on elements of S narrows the target domain."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is very close to a theorem-style conclusion stated almost verbatim from the setup. However, the presence of nearby variants involving quantifiers, weakening, and ambient-group confusion makes it slightly more than a pure restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact claim from stronger, weaker, or mis-scoped alternatives, especially between A, B, and C. Still, the item mainly tests recall/recognition of the precise theorem statement rather than substantial generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are meaningful and target common errors: confusing S with G, dropping the derangement condition, or altering quantifiers. But one option (E) is fairly implausible, so overall distractor quality is mixed."
      },
      "total_score": 5,
      "overall_assessment": "A decent theorem-recognition MCQ with limited answer leakage and some plausible distractors, but it leans heavily on recall of the exact statement rather than deeper mathematical generation or derivation."
    }
  },
  {
    "id": "2512.19500v1",
    "paper_link": "http://arxiv.org/abs/2512.19500v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.",
      "start_pos": 26726,
      "end_pos": 27068,
      "label": "t:main"
    },
    "ref_dict": {},
    "pre_theorem_intro_text_len": 1849,
    "pre_theorem_intro_text": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n  Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n  $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n  an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n  point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.",
    "context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.",
    "full_context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.\n\\section{Quartic hyperelliptic curves}\nIf $D$ is a plane algebraic curve over a field $F$ given by\n$P(X, Y)\\in F[X, Y]$, then $D(F)=\\defset{(x, y)\\in F^2}{P(x,y)=0}$\ndenotes the set of affine $F$-points on $D$.\n\nFinally assume that $f(X)=(aX^2-2X+d)(aX^2+2X+d)$ is\n inseparable. The two factors are relatively prime (because the\n difference is $4X$). Furthermore, the discriminant of both factors\n is $4(1-ad)$, hence $ad=1$ and we are done.\n\\end{proof}\n\\begin{remark}\n One can check that $q=13$, $17$, and $19$ do not give\n exceptions. However, $q=3$, $5$, $7$, $9$, and $11$ are indeed\n exceptions. This makes it unlikely that there is a ``cleaner'' proof\n of the lemma for odd $q$.\n\\end{remark}\n\\begin{corollary}\\label{c:split}\n Let $q\\ge4$ be a prime power and $a, b, c, d\\in\\FF_q$ with\n $ad-bc=1$. Set\n $T=\\defset{r+1/r}{r\\in\\Fqs}\\cup\\set{0}$. Suppose that\n $ar+d/r\\in T$ and $-cr+b/r\\in T$ for all $r\\in\\Fqs$. Then one\n of the following holds:\n \\begin{itemize}\n \\item[(a)] $a=d=0$ or $b=c=0$;\n \\item[(b)] $q=9$ and $ar+d/r=2$ for some $r\\in\\Fqs$.\n \\end{itemize}\n\\end{corollary}\n\\begin{proof}\n Using the previous lemma, one only need to check the cases where\n $q\\le19$ is an odd prime power. This is most conveniently done with\n the SageMath code at \\cite{dera_psl2_verify}.\n\\end{proof}\n\\begin{lemma}\n Let $q$ be a prime power with $q\\ge23$ if $q$ is odd. For\n $F=\\FF_{q^2}$ set $N=\\defset{r\\in F}{r^{q+1}=1}$. Suppose that for\n $0\\ne a\\in F$ the following holds: For all $r\\in N$ with\n $ar+a^q/r\\ne0$ there is $s\\in N$ such that $ar+a^q/r = s+1/s$. Then\n $a\\in N$.\n\\end{lemma}\n\\begin{proof} We first handle the easier case that $q$ is even.\nAs the square map is an automorphism of $F$, there is $r_0\\in F$ with\n$a^{q-1}=r_0^2$. Note that $a^{q-1}\\in N$, hence $r_0\\in N$.\n\nWe see that the curve $Y^2=\\gamma P(X)Q(X)$ has at least $2(q-6)$\n affine points over $\\FF_q$. On the other hand, by Lemma\n \\ref{l:hasse}, this number is at most $q+1+2\\sqrt{q}$. From this one\n quickly obtains $q\\le19$.\n\\end{proof}\n\\begin{remark}\n One can check that $q=11$, $17$, and $19$ do not give exceptions.\n However, the cases $q=3$, $5$, $7$, $9$, $13$ are indeed\n exceptions. This is most easily seen for $q=3$, because then\n $\\defset{s+1/s}{s\\in N}=\\FF_q$, so the condition on $a$ is vacuous.\n\\end{remark}\n\\begin{corollary}\\label{c:nonsplit}\n Let $q$ be a prime power. Set $N=\\defset{r\\in\\FF_{q^2}}{r^{q+1}=1}$\n and $T=\\defset{r+1/r}{r\\in N}$. For $a, b\\in\\FF_{q^2}$ with\n $a^{q+1} + b^{q+1} = 1$ suppose that $ar+a^q/r\\in T$ and\n $br+b^q/r\\in T$ for all $r\\in\\Fqs$. Then $a=0$ or $b=0$.\n\\end{corollary}\n\\begin{proof}\n If $q\\ge23$, then the previous lemma shows that $a^{q+1}=1$, hence\n $b=0$, or $b^{q+1}=1$, hence $a=0$. The cases $q\\le19$ are\n checked in \\cite{dera_psl2_verify}.\n\\end{proof}\n\\section{The groups of quaternions $2A_4$, $2S_4$, and $2A_5$}\n\\begin{lemma}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Let $H$ be the preimage of $\\bH$ in\n $\\SL_2(F)$ and $x, y\\in F^{2\\times 2}$. If the characteristic of $F$\n is $0$ or relatively prime to $\\abs{\\bH}$, then\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m\\cdot (\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y))\\cdot\\det(x)^m,\n \\]\n where $m\\in\\set{0, 1}$ if $\\bH\\cong A_4$, $m\\in\\set{0, 1, 2}$ if\n $\\bH\\cong S_4$, $m\\in\\set{0, 1, 2, 3, 4}$ if $\\bH\\cong A_5$, and\n $c_0, c_1, c_2, c_3, c_4$ = $1, 2, 5, 14, 42$ are the first five\n Catalan numbers (with index shifted by $1$).\n\\end{lemma}\n\\begin{proof}\n We assume that the characteristic of $F$ is $0$ or relatively prime\n to $\\abs{\\bH}$. Let $\\bar F$ be an algebraic closure of $F$. It is\n well known that $\\PSL_2(\\bar F)$ contains exactly one conjugacy\n class of subgroups isomorphic to $\\bH$, see e.g.~\\cite[Proposition\n 4.1]{beauville___pgl2}. (In fact more is true. By \\cite[Proposition\n 4.2]{beauville___pgl2} $\\PGL_2(F)$ contains at most one conjugacy\n class of subgroups isomorphic to $\\bH$.)\n\nNote that for $h\\in H$ (recall that $H\\le\\SL_2(\\bar F)$), we\n have $f(h'x, h'y)=f(x, y)$, so $f$ is an invariant of\n $H$. Working out the low degree invariants of $H$ could\n provide a more conceptual explanation. (At a first glance, it\n might not be obvious that the right hand side is an invariant\n under simultaneous left multiplication of $x$ and $y$ by\n $h$. But this follows from the identity\n $\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=\\trace(\\adj(y)\\cdot\n x)$, where $\\adj(y)$ denotes the adjugate of $y$.)\n \\end{itemize}\n\\end{remark}\n\\begin{lemma}\\label{l:polyhedral}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Assume that the characteristic of $F$ is\n $0$ or relatively prime to $\\abs{\\bH}$. Let $H$ be the preimage of\n $\\bH$ in $\\SL_2(F)$. Set\n $T=\\defset{\\trace(h)}{h\\in H}\\setminus\\set{-2, 2}$, and for\n $x = \\begin{pmatrix}a & b\\\\c & d\\end{pmatrix}\\in\\SL_2(F)$ set\n $y = \\begin{pmatrix}0 & 0\\\\c & d\\end{pmatrix}$. Then\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in T}(\\trace(hx)-t)\\right) =\\abs{\\bH}.\n \\]\n\\end{lemma}\n\\begin{proof}\n With this choice of $x$ and $y$ we have\n \\[\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=ad-bc=\\det(x)=1,\\]\n so the previous lemma yields\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m.\n \\]\n Let $Z$ be a variable over $F$. Then \\cite{dera_psl2_verify} verifies\n \\[\n \\prod_{t\\in T}(Z-t) =\n \\begin{cases}\n Z^3 - Z,&\\text{ if }\\bH\\cong A_4\\\\\n Z^5 - 3Z^3 + 2Z,&\\text{ if }\\bH\\cong S_4\\\\\n Z^7 - 4Z^5 + 4Z^3 - Z,&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\nThis yields\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in\n T}(\\trace(hx)-t)\\right) =\n \\begin{cases}\n \\abs{\\bH}(c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_4\\\\\n \\abs{\\bH}(c_2-3c_1+2c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong S_4\\\\\n \\abs{\\bH}(c_3-4c_2+4c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\n\\end{proof}\n\\section{Proof of Theorem \\ref{t:main}}\nThe maximal subgroups of almost simple groups with socle $\\PSL_2(q)$\nare known; see \\cite[Table~8.1, Table~8.2]{bray_etal___maximal} or\n\\cite{giudici___psl2}. The results in \\cite{bray_etal___maximal} and\n\\cite{giudici___psl2} are more detailed than those in the following\nlemma, which lists only the restrictions on $q$ that we shall use.\n\\begin{lemma}\\label{l:giudici}\n Let $q\\ge4$ be a prime power and $S=\\PSL_2(q)\\le\n G\\le\\PGammaL_2(q)$. Let $M$ be a maximal subgroup of $G$ with\n $S\\not\\le M$. Set $\\bH=M\\cap S$. Then one of the following holds.\n \\begin{itemize}\n \\item[(a)] $[S:\\bH] = q+1$,\n \\item[(b)] $[S:\\bH] = q(q+1)/2$,\n \\item[(c)] $[S:\\bH] = q(q-1)/2$,\n \\item[(d)] $\\bH\\cong A_4$, and $q$ is and odd prime,\n \\item[(e)] $\\bH\\cong S_4$, and $q$ is a prime with $q\\equiv\\pm1\\mod{8}$,\n \\item[(f)] $\\bH\\cong A_5$, and $q\\equiv\\pm1\\mod{10}$ and $q=9$ if $q$ is\n power of $3$,\n \\item[(g)] $\\bH\\cong\\PSL_2(r)$ or $\\bH\\cong\\PGL_2(r)$, where $q=r^e$\n for $e\\ge2$.\n \\end{itemize}\n Furthermore, in each case $\\bH$ is unique up to conjugacy in\n $\\aut(S)$ (with two classes in case (g) if $r$ is odd).\n\\end{lemma}\nLet $S$ and $G$ be as in Theorem \\ref{t:main}, $M$ be a stabilizer of\na point in $\\Omega$, and $\\bH=M\\cap S$. Then $M$ is a maximal subgroup\nof $G$ by primitivity. We handle the cases (a) to (g) separately.\n\nThe remaining cases (b) to (g) require some work. Suppose that there\nis a counterexample. Then there exist distinct\n$\\alpha, \\beta\\in\\Omega$ such that no derangement in $S$ moves\n$\\alpha$ to $\\beta$. Let $\\bH$ be the stabilizer in $S$ of $\\alpha$\nand $\\bx\\in S$ with $\\alpha^{\\bx}=\\beta$. Then $\\bH\\bx$ is the set of\nelements which moves $\\alpha$ to $\\beta$. As this set does not contain\na derangement, every element in the coset $\\bH\\bx$ is conjugate to an\nelement from $\\bH$.",
    "post_theorem_intro_text_len": 702,
    "post_theorem_intro_text": "If $G$ acts primitively on $\\Omega$, then the possibilities for a point\nstabilizer $S_\\omega < S$ are known. While some cases are relatively\neasy to handle, the more difficult ones arise when $S_\\omega$ is\ndihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$,\n$S_4$, or $A_5$. In the former case we require Hasse’s bound on the\nnumber of points on elliptic curves over finite fields, and to rule out\nthe latter cases we derive trace identities for the binary polyhedral\ngroups $2A_4$, $2S_4$, and $2A_5$.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.",
    "sketch": "To prove Theorem~\\ref{t:main}, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”",
    "expanded_sketch": "To prove the main theorem, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”",
    "expanded_theorem": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.",
    "theorem_type": [
      "Universal–Existential",
      "Implication"
    ],
    "mcq": {
      "question": "Let $q\\ge 4$ be a prime power, let $S=\\PSL_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning $S\\le G\\le \\operatorname{Aut}(S)$. Assume that $G$ acts primitively on a set $\\Omega$ (that is, $G$ preserves no nontrivial partition of $\\Omega$). For distinct points $\\alpha,\\beta\\in\\Omega$, which conclusion holds?",
      "correct_choice": {
        "label": "A",
        "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$; equivalently, $s$ is a derangement in the action on $\\Omega$."
      },
      "choices": [
        {
          "label": "B",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no point of $\\Omega$; equivalently, one may always choose a derangement from the whole almost simple group $G$ sending $\\alpha$ to $\\beta$."
        },
        {
          "label": "C",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$."
        },
        {
          "label": "D",
          "text": "For every point $\\alpha\\in\\Omega$, there exists an element $s\\in S$ that fixes no point of $\\Omega$ and moves $\\alpha$; in particular, $S$ contains derangements in this primitive action."
        },
        {
          "label": "E",
          "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, every element $s\\in S$ with $\\alpha^s=\\beta$ fixes no point of $\\Omega$; equivalently, the entire transporter from $\\alpha$ to $\\beta$ inside $S$ consists of derangements."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "socle-membership of the witnessing element",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the derangement requirement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "pairwise transporter conclusion weakened to mere existence of some derangements",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "existential statement on a coset replaced by universal statement on the whole transporter coset",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only the group-action setup and asks which conclusion follows; it does not explicitly state or strongly hint at the derangement-transporter conclusion in choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The correct option is essentially a theorem-style conclusion stated in full, so the item is close to a direct restatement. However, the presence of nearby quantifier/strength variants means the task is not purely tautological."
      },
      "GPS": {
        "score": 1,
        "justification": "The student must distinguish subtle logical variations: witness in S versus G, existence versus universality, and pairwise transport versus mere existence of derangements. This creates moderate reasoning demand, though it is still mainly recognition of the precise theorem statement."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common errors: overstrengthening to G, weakening by dropping the derangement condition, replacing pairwise transport with mere existence, and switching existential to universal quantification."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it remains fairly close to theorem recognition rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.19505v1",
    "paper_link": "http://arxiv.org/abs/2512.19505v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.",
      "start_pos": 18681,
      "end_pos": 18878,
      "label": "thm:Main"
    },
    "ref_dict": {
      "thm:ProjectiveVersion": "\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}",
      "ssec:FuchsianAut": "\\begin{rem}\\label{rem:DMexs}\nSome nonarithmetic Deligne--Mostow lattices \\cite{DeligneMostow} also admit homomorphisms onto cocompact Fuchsian groups from \\emph{forgetful maps} associated with contracting weights. See for example \\cite[Thm.\\ 3.1(iv)]{Deraux}.\n\\end{rem}\n\n\\subsection{Automorphisms of Fuchsian groups}\\label{ssec:FuchsianAut}\n\nThe purpose of this subsection is to prove \\Cref{thm:Fixpsubs}.\n\n\\begin{pf}[Proof of \\Cref{thm:Fixpsubs}]\nLet $p$ be a prime, $T$ be a surface group of genus $g \\ge2$, and $\\phi \\in \\Aut(T)$ have the property that $\\phi(N) = N$ for every finite index normal subgroup with index a power of $p$. Consider the pro-$p$ completion $T_{\\wh{p}}$ of $T$. Then $\\varphi$ extends to a continuous automorphism of $T_{\\wh{p}}$ that is \\emph{normal}, i.e., so that $\\varphi(\\wh{N})$ for every closed normal subgroup of $T_{\\wh{p}}$. Indeed, every such normal subgroup is an intersection of finite index normal subgroups of $T_{\\wh{p}}$, and finite index normal subgroups of $T_{\\wh{p}}$ are in natural one-to-one correspondence with the normal $p$-power index subgroups of $T$.\n\nThen $T_{\\wh{p}}$ is topologically finitely presented, since $T$ is finitely presented, hence $T_{\\wh{p}}$ is \\emph{psuedo $p$-free} in the sense of Jarden--Ritter (see \\cite[\\S 1]{JardenRitter} for the definition) by \\cite[Thm. 4]{JardenRitter}. Thus the extension of $\\varphi$ to $T_{\\wh{p}}$ is inner by \\cite[Thm.\\ 1]{JardenRitter}. Moreover, Paris proved that $T$ is $p$-conjugacy separable \\cite[Thm.\\ 1.7]{Paris}, which implies that $\\varphi$ must preserve conjugacy classes of $T$ (i.e., it is \\emph{pointwise inner}). Therefore $\\varphi$ is a normal automorphism of $T$, meaning that it preserves all normal subgroups of $T$. Finally, it is a theorem of Bogopolski, Kudryavtseva, and Zieschang that normal automorphisms of higher-genus surface groups are inner \\cite[Thm.\\ 1.3]{BKZ}, which completes the proof of the theorem.\n\\end{pf}\n\n\\section{The main result}\\label{sec:Main}\n\nSuppose that $\\Lam < \\PU(2,1)$ is a cocompact nonarithmetic lattice with nontrivial universal homomorphism $\\rho$. Following \\Cref{ssec:PU21}, let $\\Gam < \\PU(2,1)$ be the unique maximal lattice containing $\\Lam$. By \\Cref{lem:TFFuchsian} and by passing to a finite index subgroup if necessary, we can assume that the universal homomorphism for $\\Lam$ has a torsion-free factor and that $\\Lam$ is a characteristic subgroup of $\\Gam$. If $M_{\\mathrm{tf}} \\coloneqq \\ker(\\rho_{\\mathrm{tf}})$ denotes the kernel of the torsion-free universal homomorphism for $\\Lam$, then $M_{\\mathrm{tf}}$ is an infinite index characteristic subgroup of $\\Gam$ by \\Cref{lem:UniversalChar} and because a characteristic subgroup of a characteristic subgroup is characteristic in the larger group. Set $T \\coloneqq \\Lam / M_{\\mathrm{tf}}$, so $T = \\rho_{\\mathrm{tf}}(\\Lam)$.\n\nAs described in \\Cref{ssec:PU21}, away from finitely many prime ideals $\\frakp$ of the ring $\\Tr\\Ad \\Gam$, there is a well-defined congruence quotient $\\wh{Q} \\coloneqq \\Gam / \\Gam(\\frakp)$ with trivial center and commutator subgroup $Q$ that is a nonabelian finite simple group. Moreover, $\\Lam$ and $M_{\\mathrm{tf}}$ are Zariski dense in $\\PU(2,1)$, hence after possibly passing to a finite index subgroup of $\\Lam$, we can assume that they both map onto $Q$ for infinitely many $\\frakp$. Fix one such $\\frakp$ for the remainder of this section. Then define $\\Lam(\\frakp) \\coloneqq \\Gam(\\frakp) \\cap \\Lam$ and $M_{\\mathrm{tf}}(\\frakp) \\coloneqq \\Gam(\\frakp) \\cap M_{\\mathrm{tf}}$, noting that both $\\Lam(\\frakp)$ and $M_{\\mathrm{tf}}(\\frakp)$ are normal subgroups of $\\Gam$, since each is the intersection of normal subgroups of $\\Gam$.\n\n\\begin{lem}\\label{lem:QuoSetup}\nWith the notation established in this section,\n\\begin{align*}\n\\rho_{\\mathrm{tf}}(\\Lam(\\frakp)) &= T \\\\\n\\Lam / M_{\\mathrm{tf}}(\\frakp) &\\cong Q \\times T \\\\\n\\Gam / M_{\\mathrm{tf}}(\\frakp) &\\cong \\wh{Q} \\times \\wh{T}\n\\end{align*}\nwhere $\\wh{T} \\coloneqq \\Gam / M_{\\mathrm{tf}}$ contains $T$ as a finite index subgroup.\n\\end{lem}\n\\begin{pf}\nThe first equality holds since $M_{\\mathrm{tf}}$ contains coset representatives for $\\Lam(\\frakp)$ in $\\Lam$. The second holds since $M_{\\mathrm{tf}} / M_{\\mathrm{tf}}(\\frakp)$ and $\\Lam(\\frakp) / M_{\\mathrm{tf}}(\\frakp)$ are disjoint normal subgroups of $\\Lam / M_{\\mathrm{tf}}(\\frakp)$, and the third is similar.\n\\end{pf}\n\n\\begin{rem}\nNote that $T$ is then a product of factors of the torsion-free universal homomorphism of $\\Lam(\\frakp)$, but there may be additional factors.\n\\end{rem}",
      "thm:Examples": "\\begin{thm}\\label{thm:Examples}\nThere are cocompact nonarithmetic lattices $\\Gam$ in $\\PU(2,1)$ with nontrivial universal homomorphism.\n\\end{thm}",
      "ssec:KahlerGroups": "\\begin{equation}\\label{eq:ReductionModp}\n\\pi_\\frakp : \\wt{\\bfG}(\\calO_k) \\lra \\wt{\\bfG}(\\bbF_\\frakp)\n\\end{equation}\nto any $\\wt{\\Del} \\le \\wt{\\bfG}(\\calO_k)$ has kernel the \\emph{congruence subgroup} $\\wt{\\Del}(\\frakp) \\trianglelefteq \\wt{\\Del}$. Strong approximation for Zariski dense subgroups of simply connected algebraic groups says that if $\\wt{\\Del}$ is Zariski dense in $\\wt{\\bfG}(k)$, then $\\pi_\\frakp(\\wt{\\Del}) = \\wt{\\bfG}(\\bbF_\\frakp)$ for all but finitely many $\\frakp$ \\cite[Thm.\\ 1.1]{Weisfeiler}; also see Cor.\\ 3 in Window 9 of \\cite{LubotzkySegal} along with discussion on p.\\ 392 implying the more general statement for number fields other than $\\bbQ$. If $\\wt{\\Del}$ is the preimage of $\\Del$ in $\\SU(2,1)$, then congruence subgroups of $\\Del$ are those whose preimage in $\\wt{\\Del}$ are congruence subgroups.\n\nSince $\\PU(2,1)$ has type ${}^2 \\mathrm{A}_2$, there is a quadratic extension $\\ell$ of $k$ canonically associated with $\\bfG$. If $\\Del \\le \\Gam$ is Zariski dense in $\\PU(2,1)$, then away from a finite set of exceptional primes, the group $\\Del / \\Del(\\frakp)$ has commutator subgroup $\\PSL_3(\\bbF_\\frakp)$ if $\\frakp$ splits in $\\ell$ or $\\PU(3; \\bbF_\\frakp)$ if $\\frakp$ is inert in $\\ell$. In particular, the congruence quotient of $\\Del$ has commutator subgroup a nonabelian finite simple group for all but finitely many primes $\\frakp$. For example in the split case, observe that $\\Del / \\Del(\\frakp)$ is then naturally a subgroup of $\\PGL_3(\\bbF_\\frakp)$ containing $\\PSL_3(\\bbF_\\frakp)$, so it has trivial center.\n\n\\subsection{K\\\"ahler groups}\\label{ssec:KahlerGroups}\n\nA \\emph{K\\\"ahler group} is the fundamental group of a compact K\\\"ahler manifold. See \\cite{KahlerBook, Py} for basics on K\\\"ahler groups. Note that finite index subgroups of K\\\"ahler groups are naturally K\\\"ahler groups.\n\nLet $\\Gam < \\PU(2,1)$ be a cocompact lattice, possibly with torsion. Since $\\PU(2,1)$ is a linear Lie group and $\\Gam$ is finitely generated, $\\Gam$ is residually finite and thus there is a finite index subgroup $\\Lam < \\Gam$ that is torsion-free. Since smooth projective varieties are compact K\\\"ahler manifolds, by pullback of the Fubini--Study metric on $\\bbP^N$ under a projective embedding, it follows that any such $\\Lam$ is a K\\\"ahler group. It then follows from \\cite[Lem.\\ 1.1.15]{KahlerBook} that all cocompact lattices in $\\PU(2,1)$ are K\\\"ahler groups.\n\n\\medskip\n\nFundamental to this paper is the \\emph{universal homomorphism} attached to any K\\\"ahler group. See \\cite[\\S 2.C]{HLPSV} for discussion and references for the following result defining the universal homomorphism.\n\n\\begin{thm}\\label{thm:Universal}\nSuppose $\\Lam$ is a K\\\"ahler group. Then there is a (possibly empty) finite collection of cocompact Fuchsian groups $\\{T_\\al\\}_{\\al \\in \\frakU}$ and a homomorphism\n\\begin{equation}\\label{eq:UniversalHom}\n\\rho : \\Lam \\lra T \\coloneqq \\prod_{\\al \\in \\frakU} T_\\al\n\\end{equation}",
      "thm:Main": "\\begin{thm}\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{thm}",
      "thm:Fixpsubs": "\\begin{thm}\\label{thm:Fixpsubs}\nLet $T$ be a surface group of genus $g \\ge 2$, $p$ be a prime, and suppose that $\\varphi \\in \\Aut(T)$ has the property that $\\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\\varphi$ is inner.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 190,
    "pre_theorem_intro_text": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.",
    "context": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.",
    "full_context": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.\n\n\\begin{abstract}\nIn response to a question raised by Belolipetsky and the first author in \\cite{BelolipetskyLubotzky}, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{abstract}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.\n\nGreenberg \\cite{Greenberg} proved \\Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \\cite{BelolipetskyLubotzky} proved the analogue of \\Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \\ge 3$, and this paper answers a question raised in \\cite[\\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.\n\n\\begin{rem}\\label{rem:Highern}\nWhether or not there are cocompact nonarithmetic lattices in $\\PU(n,1)$ for all $n \\ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \\ge 3$, if there is a cocompact nonarithmetic lattice in $\\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \\emph{verbatim}.\n\\end{rem}\n\n\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}\n\nIn what follows, $\\pi : T \\to T_r$ will denote projection onto the $r^{th}$ surface group factor of $T$ for an arbitrary $r$ that will be fixed for the remainder of this section. Let $\\sig$ denote projection of $\\Gam$ onto $\\Gam / M_{\\mathrm{tf}}(\\frakp)$ and also the restriction of $\\sig$ to any finite index subgroup of $\\Gam$. This leads to the diagram of groups and surjective homomorphisms given in \\Cref{fig:BigDiagram}.\n\\begin{figure}[h]\n\\centering\n\\begin{tikzcd}\n& \\Gam \\arrow[ddr, -] \\arrow[dl, -] \\arrow[rr, ->, \"\\sig\"] \\arrow[rrrr, bend left = 35, ->, \"\\textrm{mod}\\, M_{\\mathrm{tf}}\"] & & \\wh{Q} \\times \\wh{T} \\arrow[d, -] \\arrow[rr, ->, \"\\textrm{mod}\\, \\wh{Q}\" below] & & \\wh{T} \\arrow[d, -] \\\\\n\\Gam(\\frakp) \\arrow[dd, -] & & & Q \\times T \\arrow[rr, ->, \"\\textrm{mod}\\, Q\"] && T \\arrow[rr, ->, \"\\pi\"] & & T_r \\\\\n& & \\Lam \\arrow[dll, -] \\arrow[ur, ->, \"\\sig\"] \\arrow[urrr, bend right=20, ->, \"\\rho_{\\mathrm{tf}}\" {xshift=0.75em, yshift=-1.25em}] \\\\\n\\Lam(\\frakp) \\arrow[ddd, -] \\arrow[uurrrrr, ->, bend right=35, \"\\sig\\, =\\, \\rho_{\\mathrm{tf}}|_{\\Lam(\\frakp)}\" {xshift=6em, yshift=-0.5em}] & & \\\\\n& & \\\\\n& & M_{\\mathrm{tf}} \\arrow[dll, -] \\arrow[uuu, -, crossing over] & \\\\\nM_{\\mathrm{tf}}(\\frakp) & &\n\\end{tikzcd}\n\\caption{The relationships between various groups}\\label{fig:BigDiagram}\n\\end{figure}\nDefine\n\\begin{equation}\\label{eq:TauDef}\n\\tau \\coloneqq \\pi \\circ \\rho_{\\mathrm{tf}} : \\Lam \\lra T_r\n\\end{equation}\nand $M \\coloneqq \\ker(\\tau|_\\Del)$. For a finite index subgroup $\\Del \\le \\Gam$, let $N_\\Gam(\\Del)$ denote its normalizer in $\\Gam$. If $\\Del$ contains $M$, then any $\\gam \\in N_\\Gam(\\Del)$ normalizing $M$ induces an automorphism of $\\Del / M$ through $\\pi \\circ \\rho_{\\mathrm{tf}}$. The following proposition/construction is essential for proving \\Cref{thm:Main}.\n\n\\begin{pf}[Proof of \\Cref{thm:Main}]\nThe only input to begin this section of the paper was a cocompact nonarithmetic lattice $\\Lam < \\PU(2,1)$ with nontrivial universal homomorphism. Such examples exist by \\Cref{thm:Examples}. If $G$ is a nontrivial group (resp.\\ the trivial group), then \\Cref{prop:FinalNontrivialG} (resp.\\ \\Cref{prop:TrivialCase}) produces a torsion-free lattice $B \\le \\Lam$ whose normalizer $N(B) < \\PU(2,1)$, which equals $N_\\Gam(B)$ for $\\Gam$ the unique maximal lattice in $\\PU(2,1)$ containing $B$ by the Margulis dichotomy, has $N(B) / B \\cong G$. Mostow rigidity now implies that $G$ is the automorphism group (equivalently, holomorphic isometry group) of the smooth compact complex hyperbolic $2$-manifold $B \\bs \\bbB^n$. Varying the primes $\\{p_j\\}$ in our construction gives infinitely many lattices with covolume tending to infinity for any fixed finite group $G$, hence another application of Mostow rigidity implies that there are infinitely many isomorphism classes of lattices. This proves the theorem.\n\\end{pf}\n\nComplex hyperbolic manifolds are of general type \\cite[\\S V.20]{BPV}, hence \\Cref{thm:Main} immediately implies \\Cref{thm:ProjectiveVersion}. Surfaces of general type all have finite automorphism group, which is directly analogous to the classical Hurwitz bound $|\\Aut(C)| \\le 84(g-1)$ for a smooth projective curve of genus $g$. In fact, $|\\Aut(X)|$ can be bounded from above in terms of the self-intersection of the first Chern class of the canonical bundle; see for example \\cite{Xiao}. Thus it is natural that the volume (or Euler characteristic, by Gauss--Bonnet) goes to infinity with $|G|$ for our examples.\n\n\\begin{thm}\\label{thm:Examples}\nThere are cocompact nonarithmetic lattices $\\Gam$ in $\\PU(2,1)$ with nontrivial universal homomorphism.\n\\end{thm}\n\n\\begin{thm}\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{thm}",
    "post_theorem_intro_text_len": 6774,
    "post_theorem_intro_text": "Greenberg \\cite{Greenberg} proved \\Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \\cite{BelolipetskyLubotzky} proved the analogue of \\Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \\ge 3$, and this paper answers a question raised in \\cite[\\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.\n\nRecall that a compact complex hyperbolic $n$-manifold is a quotient $\\Gamma \\backslash \\mathbb{B}^n$ of the unit ball in $\\mathbb{C}^n$ by a torsion-free cocompact lattice $\\Gamma$ in the Lie group $\\PU(n,1)$. Note that $\\mathbb{B}^1$ is simply the Poincar\\'e disk with the hyperbolic metric. For $n \\ge 2$, Mostow rigidity then implies that the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group, and the proof of \\Cref{thm:Main} is through this interpretation.\n\nOne strong similarity between the proofs is in the input: a cocompact nonarithmetic lattice in the isometry group of the symmetric space. Moreover, \\cite{BelolipetskyLubotzky} uses a nonabelian free quotient to begin the construction, whereas we use a quotient that is a cocompact Fuchsian group. The subgroup counting methods in both papers are nearly identical. Note that it is quite easy, given a free or surface group quotient of a finite index subgroup $\\Lambda$ of a lattice $\\Gamma$, to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$, where $N_\\Gamma(B)$ is the normalizer of $B$ in $\\Gamma$, but ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle.\n\nThe nonarithmetic lattices used in \\cite{BelolipetskyLubotzky} are the famous examples of Gromov and Piatetski-Shapiro \\cite{GPS}. These lattices naturally have the structure of a free product with amalgamation over the fundamental group of a codimension one totally geodesic subspace, and the nonabelian free quotient of a finite index subgroup is derived from this (see \\cite{LubotzkyFreeQuo}). In the complex hyperbolic setting, there are no real codimension one totally geodesic subspaces \\cite[\\S 3.1]{Goldman}, hence no such splittings.\n\nOur replacement, from the theory of K\\\"ahler groups, is the \\emph{universal homomorphism} to a direct product of cocompact Fuchsian groups; see \\cite{HLPSV} for an introduction. Cocompact nonarithmetic lattices in $\\PU(2,1)$ with nontrivial universal homomorphism are found among the famous examples of Livn\\'e \\cite{Livne} and Deligne--Mostow \\cite{DeligneMostow}; see \\Cref{thm:Examples}. With this different input, a key lattice $\\Delta$ constructed early in the outline of \\cite{BelolipetskyLubotzky} now fails to be normal in the maximal lattice $\\Gamma$ in $\\PU(2,1)$. In a certain sense, much of the work in \\Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.\n\n\\begin{rem}\\label{rem:Highern}\nWhether or not there are cocompact nonarithmetic lattices in $\\PU(n,1)$ for all $n \\ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \\ge 3$, if there is a cocompact nonarithmetic lattice in $\\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \\emph{verbatim}.\n\\end{rem}\n\nIn this paper, a genus $g$ surface group will always mean the fundamental group of a closed Riemann surface of genus $g$, despite the slight clash of terminology with fundamental groups of complex surfaces also appearing. In \\Cref{ssec:FuchsianAut} we will prove the following result on automorphisms of surface groups, which is an analogue of a theorem of the first author in the free group case \\cite[Thm.\\ 1]{LubotzkyFree} that may be of independent interest. \n\n\\begin{thm}\\label{thm:Fixpsubs}\nLet $T$ be a surface group of genus $g \\ge 2$, $p$ be a prime, and suppose that $\\varphi \\in \\Aut(T)$ has the property that $\\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\\varphi$ is inner.\n\\end{thm}\n\n\\noindent\\textbf{Interpretation as a complex projective surface}\n\n\\medskip\n\nCompact complex hyperbolic $2$-manifolds are also minimal smooth complex projective surfaces of general type \\cite[\\S V.20]{BPV}. The appendix to this paper recasts \\Cref{thm:Main} in that context to prove the following result.\n\n\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}\n\nA surface is irreducible if it is not finitely (\\'etale) covered by a product of complex curves. Greenberg's theorem easily produces products of complex curves with any finite automorphism group, so only the irreducible case is of interest. As described in the appendix, one may be able to construct irreducible examples using other methods. For example, there may exist a nonsingular fibration $X \\to C$ over a curve $C$ (e.g., a ruled surface) where $\\Aut(C) \\cong G$ by Greenberg and there is a natural isomorphism $\\Aut(X) \\cong \\Aut(C)$. However, we were not able to find a reference for \\Cref{thm:ProjectiveVersion} in the literature.\n\n\\begin{rem}\\label{rem:HardPart}\nOur construction is also related to a holomorphic map to a smooth projective curve (see \\Cref{ssec:KahlerGroups}). However, the bulk of the proof is precisely in showing that our surface has no extra automorphisms, which is highly unclear from the nature of the construction.\n\\end{rem}\n\n\\medskip\n\n\\noindent\\textbf{Organization of the paper}\n\n\\medskip\n\n\\noindent\nThis paper is organized as follows. \\Cref{sec:Prelim} contains preliminaries about lattices in $\\PU(2,1)$, universal homomorphisms for K\\\"ahler groups, and the proof of \\Cref{thm:Fixpsubs}. \\Cref{sec:Main} proves \\Cref{thm:Main}. The appendix discussions automorphism groups of smooth complex projective surfaces.\n\n\\medskip\n\n\\noindent \\textbf{Acknowledgments}\n\n\\medskip\n\n\\noindent\nMuch of the work on this paper was done while the authors were at the conference `Discrete and profinite groups' at the Isaac Newton Institute for Mathematical Sciences, and thank the institute for its hospitality. The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 (N.\\ 882751). The second author was supported by Grants DMS-2203555 and DMS-2506896 from the National Science Foundation and the Simons Foundation [SFI-MPS-TSM-00014184, MS].",
    "sketch": "The proof of \\Cref{thm:Main} is “through” the fact that for $n\\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \\cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\\Lambda<\\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \\cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\\Gamma$.\n\nA key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”\n\nBecause in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\\Delta$…now fails to be normal in the maximal lattice $\\Gamma$,” and “much of the work in \\Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.”",
    "expanded_sketch": "To prove the main theorem, the argument is “through” the fact that for $n\\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \\cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\\Lambda<\\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \\cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\\Gamma$.\n\nA key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”\n\nBecause in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\\Delta$…now fails to be normal in the maximal lattice $\\Gamma$,” and “much of the work later is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.”",
    "expanded_theorem": "\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.",
    "theorem_type": [
      "Universal–Existential",
      "Existence"
    ],
    "mcq": {
      "question": "Let an automorphism of a complex manifold mean a biholomorphism. For an arbitrary finite group $G$, which statement holds about closed complex hyperbolic $2$-manifolds (that is, compact complex manifolds of complex dimension $2$ carrying a complex hyperbolic structure)?",
      "correct_choice": {
        "label": "A",
        "text": "There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group is isomorphic to $G$."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group contains a subgroup isomorphic to $G$."
        },
        {
          "label": "C",
          "text": "There exists at least one closed complex hyperbolic $2$-manifold whose automorphism group is isomorphic to $G$."
        },
        {
          "label": "D",
          "text": "There exists a closed complex hyperbolic $2$-manifold $X$ such that every finite group $G$ occurs as a subgroup of $\\operatorname{Aut}(X)$."
        },
        {
          "label": "E",
          "text": "For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose full holomorphic isometry group is isomorphic to $G$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "exactly_G_vs_contains_G",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "infinitely_many_isomorphism_classes",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "for_each_G_vs_single_universal_manifold",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "automorphism_group_vs_holomorphic_isometry_group_statement_level",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct choice and gives only definitions plus a broad existence question. There are no direct textual cues favoring the strongest statement."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to theorem recognition: it asks which existence statement holds among nearby formulations. It is not a pure verbatim restatement because the options vary in quantifier strength and exactness, but it still mainly tests recall of the precise theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare 'isomorphic to G' versus 'contains G,' and 'exists one' versus 'infinitely many,' but the question mostly rewards knowing the exact statement rather than generating a conclusion from underlying mathematical arguments."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and target common failure modes: weakening the theorem (C), confusing exact automorphism group with containing a subgroup (B, D), and introducing an overreaching uniform-family claim (E). They are distinct and mathematically meaningful."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward precise statement recall than genuine generative mathematical reasoning."
    }
  },
  {
    "id": "2512.19586v1",
    "paper_link": "http://arxiv.org/abs/2512.19586v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main_window}\nLet $u\\ge 1$, $q\\ge 2$ be integers, and let $\\mathcal F$ be a finite family of\nnon-empty binary words.  Let $L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$.\nThen the set\n\\[\n   S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}\n\\]\nis either finite or ultimately periodic.  That is, there exist integers $n_0\\ge 0$ and $p\\ge 1$\nsuch that\n\\[\n   n\\in S_u^{(M)}\\ \\Longleftrightarrow\\ n+p\\in S_u^{(M)}\\qquad(n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair $(n_0,p)$ and decide whether\n$S_u^{(M)}$ is finite or infinite.  In the finite case, the same procedure\noutputs all elements of $S_u^{(M)}$.",
      "start_pos": 8813,
      "end_pos": 9482,
      "label": "thm:main_window"
    },
    "ref_dict": {
      "thm:main_window": "\\begin{theorem}\\label{thm:main_window}\nLet $u\\ge 1$, $q\\ge 2$ be integers, and let $\\mathcal F$ be a finite family of\nnon-empty binary words.  Let $L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$.\nThen the set\n\\[\n   S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}\n\\]\nis either finite or ultimately periodic.  That is, there exist integers $n_0\\ge 0$ and $p\\ge 1$\nsuch that\n\\[\n   n\\in S_u^{(M)}\\ \\Longleftrightarrow\\ n+p\\in S_u^{(M)}\\qquad(n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair $(n_0,p)$ and decide whether\n$S_u^{(M)}$ is finite or infinite.  In the finite case, the same procedure\noutputs all elements of $S_u^{(M)}$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5148,
    "pre_theorem_intro_text": "The interplay between multiplicative structures (such as geometric progressions)\nand digital structures (such as numeration systems with local constraints) is a\nfertile source of problems at the interface of number theory, combinatorics,\nand symbolic dynamics. A classical example is Erd\\H{o}s' ternary conjecture\n\\cite{Erdos1979}, asserting that for all $n\\ge 9$ the base-$3$ expansion of\n$2^n$ contains the digit $2$.  Interpreting the middle-third Cantor set as the\nset of reals whose ternary expansions avoid the digit $1$, this conjecture can\nbe viewed as a finiteness statement for the intersection of a geometric\nprogression with a Cantor-type digit-restricted set.\n\nFor standard integer bases $b\\ge 2$, Cui, Ma, and Jiang \\cite{Cui2022}\ndeveloped an algorithmic framework to study intersections of geometric\nprogressions with Cantor-like subsets of the integers defined by forbidden\nblocks in base-$b$ expansions.  Their approach encodes the carry propagation of\nmultiplication into a finite combinatorial object, leading to effective\ndecidability results.\n\nIn the present paper we pursue an analogous viewpoint in a non-standard\nnumeration system, namely the \\textit{Zeckendorf representation} associated with the\nFibonacci numbers.  Let $(F_n)_{n\\ge 0}$ be defined by $F_0=0$, $F_1=1$,\n$F_{n+2}=F_{n+1}+F_n$.  By Zeckendorf's theorem, every integer $N\\ge 1$ admits a\nunique decomposition\n\\[\n   N=\\sum_{i=2}^k \\varepsilon_i F_i,\n\\]\nwith $\\varepsilon_i\\in\\{0,1\\}$, $\\varepsilon_k=1$, and\n$\\varepsilon_i\\varepsilon_{i+1}=0$ for $2\\le i<k$.  We write\n$Z(N)=\\varepsilon_k\\varepsilon_{k-1}\\cdots \\varepsilon_2$ for this Zeckendorf\nword.  Symbolically, the set of Zeckendorf words is the \\emph{Golden Mean shift},\ni.e., the language over $\\{0,1\\}$ with the block $11$ forbidden (together with\nthe usual ``no leading zeros'' convention).\n\n\\smallskip\n\n\\noindent\\emph{Remark on digit order.}  \nThe representation $Z(N)=\\varepsilon_k \\varepsilon_{k-1}\\cdots \\varepsilon_2$ lists digits from the most significant to the least significant.  \nFor local arithmetic (carry propagation) it is more natural to work with the \\emph{reversed} order  \n\\[\n\\operatorname{rev}(Z(N))=\\varepsilon_2\\varepsilon_3\\cdots \\varepsilon_k,\n\\]  \ni.e., reading from the least significant digit upward.  \nAll subsequent notation will refer to this LSD-first stream.\n\n\\smallskip\n\nFix a finite family $\\mathcal F$ of non-empty binary words, thought of as\nadditional forbidden patterns.  This defines a Zeckendorf-restricted subset\n\\[\n   \\mathcal K_{\\mathcal F}\n   =\\{N\\ge 1:\\ \\mathrm{rev}(Z(N))\\ \\text{contains no factor from }\\mathcal F\\},\n\\]\nwhere $\\mathrm{rev}(Z(N))$ lists the Zeckendorf digits from the least significant digit\n(the coefficient of $F_2$) upward.  Given integers $u\\ge 1$ and $q\\ge 2$, we\nseek to understand the set of exponents\n\\[\n   S_u=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}\\}.\n\\]\nIf $\\mathcal F=\\{11\\}$, then $\\mathcal K_{\\mathcal F}$ coincides with the\nfull Zeckendorf language and $S_u=\\N_{\\ge 0}$, so the interesting case is when\n$\\mathcal F$ imposes genuinely new constraints.\n\nA direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion\nof $uq^n$, whose length grows linearly with $n$.  To obtain an effective result\ncapturing the local arithmetic, we introduce a window-restricted variant.  Let\n$L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window.  Let \\(\\#\\notin\\{0,1\\}\\) be a neutral padding symbol and define the right\\-infinite word  \n\\[\n\\widetilde{Z}(N):=\\operatorname{rev}(Z(N))\\,\\#^{\\omega}\\in\\{0,1,\\#\\}^{\\mathbb{N}}.\n\\]  \nThe symbol \\(\\#\\) never equals \\(0\\) or \\(1\\); thus a forbidden binary pattern cannot be created by involving a \\(\\#\\).  \nHence the prefix of length \\(M\\) of \\(\\widetilde{Z}(N)\\) consists only of genuine Zeckendorf digits whenever \\(N\\) is large enough, and otherwise contains \\(\\#\\)'s that do not affect the avoidance condition.\n\nWe say that a finite word $v\\in\\{0,1,\\#\\}^*$ \\emph{avoids} $\\mathcal F$ if no\n$f\\in\\mathcal F$ occurs as a factor of $v$ (equivalently, occurrences cannot use\nthe symbol $\\#$).  We then define\n\\[\n   \\mathcal K_{\\mathcal F}^{(M)}\n   =\\{N\\ge 1:\\ \\pref_M(\\widetilde{Z}(N))\\ \\text{avoids }\\mathcal F\\},\n\\]\nwhere for a word \\(w = w_0 w_1 w_2 \\dots \\in \\{0,1,\\#\\}^{\\mathbb N}\\) and an integer \\(M \\ge 1\\), \\(\\pref_M(w)\\) denotes its prefix of length \\(M\\), i.e., \\(w_0 w_1 \\dots w_{M-1}\\).  \nThus constraints are enforced only within the first \\(M\\) least significant Zeckendorf digits. Clearly $\\mathcal K_{\\mathcal F}\\subseteq \\mathcal\nK_{\\mathcal F}^{(M)}$ for every $M$, and with the above padding convention one\nhas\n\\[\n   \\mathcal K_{\\mathcal F}=\\bigcap_{M\\ge L}\\mathcal K_{\\mathcal F}^{(M)}.\n\\]\nAccordingly, we set\n\\[\n   S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}.\n\\]\n\nOur main result shows that, for each fixed window length $M$, the set\n$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be\ncomputed effectively.",
    "context": "The interplay between multiplicative structures (such as geometric progressions)\nand digital structures (such as numeration systems with local constraints) is a\nfertile source of problems at the interface of number theory, combinatorics,\nand symbolic dynamics. A classical example is Erd\\H{o}s' ternary conjecture\n\\cite{Erdos1979}, asserting that for all $n\\ge 9$ the base-$3$ expansion of\n$2^n$ contains the digit $2$. Interpreting the middle-third Cantor set as the\nset of reals whose ternary expansions avoid the digit $1$, this conjecture can\nbe viewed as a finiteness statement for the intersection of a geometric\nprogression with a Cantor-type digit-restricted set.\n\nIn the present paper we pursue an analogous viewpoint in a non-standard\nnumeration system, namely the \\textit{Zeckendorf representation} associated with the\nFibonacci numbers. Let $(F_n)_{n\\ge 0}$ be defined by $F_0=0$, $F_1=1$,\n$F_{n+2}=F_{n+1}+F_n$. By Zeckendorf's theorem, every integer $N\\ge 1$ admits a\nunique decomposition\n\\[\n N=\\sum_{i=2}^k \\varepsilon_i F_i,\n\\]\nwith $\\varepsilon_i\\in\\{0,1\\}$, $\\varepsilon_k=1$, and\n$\\varepsilon_i\\varepsilon_{i+1}=0$ for $2\\le i<k$. We write\n$Z(N)=\\varepsilon_k\\varepsilon_{k-1}\\cdots \\varepsilon_2$ for this Zeckendorf\nword. Symbolically, the set of Zeckendorf words is the \\emph{Golden Mean shift},\ni.e., the language over $\\{0,1\\}$ with the block $11$ forbidden (together with\nthe usual ``no leading zeros'' convention).\n\nFix a finite family $\\mathcal F$ of non-empty binary words, thought of as\nadditional forbidden patterns. This defines a Zeckendorf-restricted subset\n\\[\n \\mathcal K_{\\mathcal F}\n =\\{N\\ge 1:\\ \\mathrm{rev}(Z(N))\\ \\text{contains no factor from }\\mathcal F\\},\n\\]\nwhere $\\mathrm{rev}(Z(N))$ lists the Zeckendorf digits from the least significant digit\n(the coefficient of $F_2$) upward. Given integers $u\\ge 1$ and $q\\ge 2$, we\nseek to understand the set of exponents\n\\[\n S_u=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}\\}.\n\\]\nIf $\\mathcal F=\\{11\\}$, then $\\mathcal K_{\\mathcal F}$ coincides with the\nfull Zeckendorf language and $S_u=\\N_{\\ge 0}$, so the interesting case is when\n$\\mathcal F$ imposes genuinely new constraints.\n\nA direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion\nof $uq^n$, whose length grows linearly with $n$. To obtain an effective result\ncapturing the local arithmetic, we introduce a window-restricted variant. Let\n$L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window. Let \\(\\#\\notin\\{0,1\\}\\) be a neutral padding symbol and define the right\\-infinite word \n\\[\n\\widetilde{Z}(N):=\\operatorname{rev}(Z(N))\\,\\#^{\\omega}\\in\\{0,1,\\#\\}^{\\mathbb{N}}.\n\\] \nThe symbol \\(\\#\\) never equals \\(0\\) or \\(1\\); thus a forbidden binary pattern cannot be created by involving a \\(\\#\\). \nHence the prefix of length \\(M\\) of \\(\\widetilde{Z}(N)\\) consists only of genuine Zeckendorf digits whenever \\(N\\) is large enough, and otherwise contains \\(\\#\\)'s that do not affect the avoidance condition.\n\nWe say that a finite word $v\\in\\{0,1,\\#\\}^*$ \\emph{avoids} $\\mathcal F$ if no\n$f\\in\\mathcal F$ occurs as a factor of $v$ (equivalently, occurrences cannot use\nthe symbol $\\#$). We then define\n\\[\n \\mathcal K_{\\mathcal F}^{(M)}\n =\\{N\\ge 1:\\ \\pref_M(\\widetilde{Z}(N))\\ \\text{avoids }\\mathcal F\\},\n\\]\nwhere for a word \\(w = w_0 w_1 w_2 \\dots \\in \\{0,1,\\#\\}^{\\mathbb N}\\) and an integer \\(M \\ge 1\\), \\(\\pref_M(w)\\) denotes its prefix of length \\(M\\), i.e., \\(w_0 w_1 \\dots w_{M-1}\\). \nThus constraints are enforced only within the first \\(M\\) least significant Zeckendorf digits. Clearly $\\mathcal K_{\\mathcal F}\\subseteq \\mathcal\nK_{\\mathcal F}^{(M)}$ for every $M$, and with the above padding convention one\nhas\n\\[\n \\mathcal K_{\\mathcal F}=\\bigcap_{M\\ge L}\\mathcal K_{\\mathcal F}^{(M)}.\n\\]\nAccordingly, we set\n\\[\n S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}.\n\\]\n\nOur main result shows that, for each fixed window length $M$, the set\n$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be\ncomputed effectively.",
    "full_context": "The interplay between multiplicative structures (such as geometric progressions)\nand digital structures (such as numeration systems with local constraints) is a\nfertile source of problems at the interface of number theory, combinatorics,\nand symbolic dynamics. A classical example is Erd\\H{o}s' ternary conjecture\n\\cite{Erdos1979}, asserting that for all $n\\ge 9$ the base-$3$ expansion of\n$2^n$ contains the digit $2$. Interpreting the middle-third Cantor set as the\nset of reals whose ternary expansions avoid the digit $1$, this conjecture can\nbe viewed as a finiteness statement for the intersection of a geometric\nprogression with a Cantor-type digit-restricted set.\n\nIn the present paper we pursue an analogous viewpoint in a non-standard\nnumeration system, namely the \\textit{Zeckendorf representation} associated with the\nFibonacci numbers. Let $(F_n)_{n\\ge 0}$ be defined by $F_0=0$, $F_1=1$,\n$F_{n+2}=F_{n+1}+F_n$. By Zeckendorf's theorem, every integer $N\\ge 1$ admits a\nunique decomposition\n\\[\n N=\\sum_{i=2}^k \\varepsilon_i F_i,\n\\]\nwith $\\varepsilon_i\\in\\{0,1\\}$, $\\varepsilon_k=1$, and\n$\\varepsilon_i\\varepsilon_{i+1}=0$ for $2\\le i<k$. We write\n$Z(N)=\\varepsilon_k\\varepsilon_{k-1}\\cdots \\varepsilon_2$ for this Zeckendorf\nword. Symbolically, the set of Zeckendorf words is the \\emph{Golden Mean shift},\ni.e., the language over $\\{0,1\\}$ with the block $11$ forbidden (together with\nthe usual ``no leading zeros'' convention).\n\nFix a finite family $\\mathcal F$ of non-empty binary words, thought of as\nadditional forbidden patterns. This defines a Zeckendorf-restricted subset\n\\[\n \\mathcal K_{\\mathcal F}\n =\\{N\\ge 1:\\ \\mathrm{rev}(Z(N))\\ \\text{contains no factor from }\\mathcal F\\},\n\\]\nwhere $\\mathrm{rev}(Z(N))$ lists the Zeckendorf digits from the least significant digit\n(the coefficient of $F_2$) upward. Given integers $u\\ge 1$ and $q\\ge 2$, we\nseek to understand the set of exponents\n\\[\n S_u=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}\\}.\n\\]\nIf $\\mathcal F=\\{11\\}$, then $\\mathcal K_{\\mathcal F}$ coincides with the\nfull Zeckendorf language and $S_u=\\N_{\\ge 0}$, so the interesting case is when\n$\\mathcal F$ imposes genuinely new constraints.\n\nA direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion\nof $uq^n$, whose length grows linearly with $n$. To obtain an effective result\ncapturing the local arithmetic, we introduce a window-restricted variant. Let\n$L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window. Let \\(\\#\\notin\\{0,1\\}\\) be a neutral padding symbol and define the right\\-infinite word \n\\[\n\\widetilde{Z}(N):=\\operatorname{rev}(Z(N))\\,\\#^{\\omega}\\in\\{0,1,\\#\\}^{\\mathbb{N}}.\n\\] \nThe symbol \\(\\#\\) never equals \\(0\\) or \\(1\\); thus a forbidden binary pattern cannot be created by involving a \\(\\#\\). \nHence the prefix of length \\(M\\) of \\(\\widetilde{Z}(N)\\) consists only of genuine Zeckendorf digits whenever \\(N\\) is large enough, and otherwise contains \\(\\#\\)'s that do not affect the avoidance condition.\n\nWe say that a finite word $v\\in\\{0,1,\\#\\}^*$ \\emph{avoids} $\\mathcal F$ if no\n$f\\in\\mathcal F$ occurs as a factor of $v$ (equivalently, occurrences cannot use\nthe symbol $\\#$). We then define\n\\[\n \\mathcal K_{\\mathcal F}^{(M)}\n =\\{N\\ge 1:\\ \\pref_M(\\widetilde{Z}(N))\\ \\text{avoids }\\mathcal F\\},\n\\]\nwhere for a word \\(w = w_0 w_1 w_2 \\dots \\in \\{0,1,\\#\\}^{\\mathbb N}\\) and an integer \\(M \\ge 1\\), \\(\\pref_M(w)\\) denotes its prefix of length \\(M\\), i.e., \\(w_0 w_1 \\dots w_{M-1}\\). \nThus constraints are enforced only within the first \\(M\\) least significant Zeckendorf digits. Clearly $\\mathcal K_{\\mathcal F}\\subseteq \\mathcal\nK_{\\mathcal F}^{(M)}$ for every $M$, and with the above padding convention one\nhas\n\\[\n \\mathcal K_{\\mathcal F}=\\bigcap_{M\\ge L}\\mathcal K_{\\mathcal F}^{(M)}.\n\\]\nAccordingly, we set\n\\[\n S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}.\n\\]\n\nOur main result shows that, for each fixed window length $M$, the set\n$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be\ncomputed effectively.\n\nFix a finite family $\\mathcal F$ of non-empty binary words, thought of as\nadditional forbidden patterns. This defines a Zeckendorf-restricted subset\n\\[\n \\mathcal K_{\\mathcal F}\n =\\{N\\ge 1:\\ \\rev(Z(N))\\ \\text{contains no factor from }\\mathcal F\\},\n\\]\nwhere $\\rev(Z(N))$ lists the Zeckendorf digits from the least significant digit\n(the coefficient of $F_2$) upward. Given integers $u\\ge 1$ and $q\\ge 2$, we\nseek to understand the set of exponents\n\\[\n S_u=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}\\}.\n\\]\nIf $\\mathcal F=\\{11\\}$, then $\\mathcal K_{\\mathcal F}$ coincides with the\nfull Zeckendorf language and $S_u=\\N_{\\ge 0}$, so the interesting case is when\n$\\mathcal F$ imposes genuinely new constraints.\n\nA direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion\nof $uq^n$, whose length grows linearly with $n$. To obtain an effective result\ncapturing the local arithmetic, we introduce a window-restricted variant. Let\n$L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window. Let \\(\\#\\notin\\{0,1\\}\\) be a neutral padding symbol and define the right\\-infinite word \n\\[\n\\widetilde{Z}(N):=\\operatorname{rev}(Z(N))\\,\\#^{\\omega}\\in\\{0,1,\\#\\}^{\\mathbb{N}}.\n\\] \nThe symbol \\(\\#\\) never equals \\(0\\) or \\(1\\); thus a forbidden binary pattern cannot be created by involving a \\(\\#\\). \nHence the prefix of length \\(M\\) of \\(\\widetilde{Z}(N)\\) consists only of genuine Zeckendorf digits whenever \\(N\\) is large enough, and otherwise contains \\(\\#\\)'s that do not affect the avoidance condition.\n\nOur main result shows that, for each fixed window length $M$, the set\n$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be\ncomputed effectively.\n\nThe proof is entirely finite-state. The crucial input is the fact that\nmultiplication by a fixed integer $q$ in the Fibonacci numeration system is\nrealizable by a finite sequential transducer \\cite{Frougny1992}. Reading digits\nfrom the least significant side, this transducer induces, for each $M$, a\ndeterministic update rule on the set of length-$M$ windows\n$\\{0,1,\\#\\}^M$. After composing with the finite automaton that checks avoidance\nof $\\mathcal F$ inside the window, we obtain a deterministic dynamical system on\na finite state space. The orbit associated with the progression $(uq^n)_{n\\ge\n0}$ is therefore eventually periodic, which yields ultimate periodicity of\n$S_u^{(M)}$ as an immediate corollary. The same finite-state construction gives\nan explicit algorithm to compute the eventual period.\n\n\\textbf{Ultimate periodicity in finite-state dynamics.} A fundamental observation, used repeatedly in our proof, is that any deterministic process on a finite state space must eventually become periodic. More precisely, if \\(X\\) is a finite set and \\(f: X \\to X\\) is a function, then for any starting point \\(x_0 \\in X\\) the sequence \\(x_{n+1} = f(x_n)\\) is \\emph{ultimately periodic}: there exist integers \\(n_0 \\ge 0\\) and \\(p \\ge 1\\) such that \\(x_{n+p} = x_n\\) for all \\(n \\ge n_0\\). This elementary combinatorial fact drives the periodicity conclusion in Theorem~\\ref{thm:main_window}.\n\n\\begin{lemma}\\label{prop:frougny}\nLet $q\\ge 2$ be an integer. There exists a finite-state synchronous\n(letter-to-letter) transducer $\\mathcal T_q$ over the alphabet $\\{0,1,\\#\\}$,\nreading the input from the least significant digit upward, with the following\nproperty: for every $N\\ge 1$,\n\\[\n \\mathcal T_q\\bigl(\\widetilde Z(N)\\bigr)=\\widetilde Z(qN),\n \\qquad \\widetilde Z(N):=\\rev(Z(N))\\,\\#^{\\omega}.\n\\]\nMoreover, such a transducer can be effectively constructed from $q$.\n\\end{lemma}\n\nFix $u\\ge 1$, $q\\ge 2$, a finite family $\\mathcal F$ of non-empty binary words,\nand an integer $M\\ge 1$. Set $x_n=uq^n$ for $n\\ge 0$ and define\n\\[\n w_n:=\\pref_M(\\widetilde{Z}(x_n))\\in\\{0,1,\\#\\}^M.\n\\]\nBy Lemma~\\ref{lem:locality}, we have the recursion\n\\[\n w_{n+1}=\\Theta_{q,M}(w_n)\\qquad(n\\ge 0).\n\\]\nThus $(w_n)_{n\\ge 0}$ is the forward orbit of $w_0$ under a deterministic map on\nthe finite set $\\{0,1,\\#\\}^M$. Consequently, the sequence $(w_n)$ is ultimately\nperiodic: there exist $n_0\\ge 0$ and $p\\ge 1$ such that\n\\[\n w_{n+p}=w_n\\qquad(n\\ge n_0).\n\\]\n\n\\medskip\n\\noindent\\emph{Effectiveness.}\nLemma~\\ref{prop:frougny} provides an effective construction of the\ntransducer $\\mathcal T_q$, hence of the induced map $\\Theta_{q,M}$ in\nLemma~\\ref{lem:locality}. Starting from $w_0=\\pref_M(\\widetilde{Z}(u))$, we\niterate $w_{n+1}=\\Theta_{q,M}(w_n)$ and record the first repeated state. The\nfirst repetition yields explicit $(n_0,p)$, and the same computation decides\nwhether $S_u^{(M)}$ is finite or infinite. In the finite case, the algorithm\noutputs all elements of $S_u^{(M)}$ by checking $P(w_n)$ along the finite orbit\nsegment before the cycle.\n\\qed\n\n\\begin{theorem}\\label{thm:main_window}\nLet $u\\ge 1$, $q\\ge 2$ be integers, and let $\\mathcal F$ be a finite family of\nnon-empty binary words.  Let $L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$.\nThen the set\n\\[\n   S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}\n\\]\nis either finite or ultimately periodic.  That is, there exist integers $n_0\\ge 0$ and $p\\ge 1$\nsuch that\n\\[\n   n\\in S_u^{(M)}\\ \\Longleftrightarrow\\ n+p\\in S_u^{(M)}\\qquad(n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair $(n_0,p)$ and decide whether\n$S_u^{(M)}$ is finite or infinite.  In the finite case, the same procedure\noutputs all elements of $S_u^{(M)}$.\n\\end{theorem}",
    "post_theorem_intro_text_len": 1327,
    "post_theorem_intro_text": "The proof is entirely finite-state.  The crucial input is the fact that\nmultiplication by a fixed integer $q$ in the Fibonacci numeration system is\nrealizable by a finite sequential transducer \\cite{Frougny1992}.  Reading digits\nfrom the least significant side, this transducer induces, for each $M$, a\ndeterministic update rule on the set of length-$M$ windows\n$\\{0,1,\\#\\}^M$.  After composing with the finite automaton that checks avoidance\nof $\\mathcal F$ inside the window, we obtain a deterministic dynamical system on\na finite state space.  The orbit associated with the progression $(uq^n)_{n\\ge\n0}$ is therefore eventually periodic, which yields ultimate periodicity of\n$S_u^{(M)}$ as an immediate corollary.  The same finite-state construction gives\nan explicit algorithm to compute the eventual period.\n\nTheorem~\\ref{thm:main_window} gives a rigorous and completely effective solution\nto the window-restricted problem.  While passing from $S_u^{(M)}$ to the global\nset $S_u$ entails additional difficulties (one must control where forbidden\npatterns may first appear as $n$ varies), the window-restricted variant already\ncaptures the local dynamics of Zeckendorf multiplication and provides a\npractical method to detect all exponents for which the low-order Zeckendorf\ndigits of $uq^n$ satisfy prescribed constraints.",
    "sketch": "The proof of Theorem~\\ref{thm:main_window} is \\u201centirely finite-state\\u201d. The key input is that \\u201cmultiplication by a fixed integer $q$ in the Fibonacci numeration system is realizable by a finite sequential transducer\\u201d, and when \\u201creading digits from the least significant side\\u201d this yields (for each $M$) \\u201ca deterministic update rule on the set of length-$M$ windows $\\{0,1,\\#\\}^M$.\\u201d Composing this with \\u201cthe finite automaton that checks avoidance of $\\mathcal F$ inside the window\\u201d produces \\u201ca deterministic dynamical system on a finite state space.\\u201d The orbit corresponding to $(uq^n)_{n\\ge 0}$ is thus \\u201ceventually periodic\\u201d, and \\u201cwhich yields ultimate periodicity of $S_u^{(M)}$ as an immediate corollary.\\u201d Finally, \\u201cthe same finite-state construction gives an explicit algorithm to compute the eventual period\\u201d (hence to compute $(n_0,p)$ and decide finiteness, and list elements in the finite case).",
    "expanded_sketch": "To prove the main theorem, the argument is “entirely finite-state”. The key input is that “multiplication by a fixed integer $q$ in the Fibonacci numeration system is realizable by a finite sequential transducer”, and when “reading digits from the least significant side” this yields (for each $M$) “a deterministic update rule on the set of length-$M$ windows $\\{0,1,\\#\\}^M$.” Composing this with “the finite automaton that checks avoidance of $\\mathcal F$ inside the window” produces “a deterministic dynamical system on a finite state space.” The orbit corresponding to $(uq^n)_{n\\ge 0}$ is thus “eventually periodic”, and “which yields ultimate periodicity of $S_u^{(M)}$ as an immediate corollary.” Finally, “the same finite-state construction gives an explicit algorithm to compute the eventual period” (hence to compute $(n_0,p)$ and decide finiteness, and list elements in the finite case).",
    "expanded_theorem": "\\label{thm:main_window}\nLet $u\\ge 1$, $q\\ge 2$ be integers, and let $\\mathcal F$ be a finite family of\nnon-empty binary words.  Let $L=\\max\\{|f|:f\\in\\mathcal F\\}$ and fix $M\\ge L$.\nThen the set\n\\[\n   S_u^{(M)}=\\{n\\ge 0:\\ uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}\n\\]\nis either finite or ultimately periodic.  That is, there exist integers $n_0\\ge 0$ and $p\\ge 1$\nsuch that\n\\[\n   n\\in S_u^{(M)}\\ \\Longleftrightarrow\\ n+p\\in S_u^{(M)}\\qquad(n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair $(n_0,p)$ and decide whether\n$S_u^{(M)}$ is finite or infinite.  In the finite case, the same procedure\noutputs all elements of $S_u^{(M)}$.",
    "theorem_type": [
      "Existential–Universal",
      "Algorithmic or Constructive"
    ],
    "mcq": {
      "question": "Let \\((F_n)_{n\\ge 0}\\) be the Fibonacci sequence defined by \\(F_0=0\\), \\(F_1=1\\), and \\(F_{n+2}=F_{n+1}+F_n\\). For each integer \\(N\\ge 1\\), let \\(Z(N)=\\varepsilon_k\\varepsilon_{k-1}\\cdots \\varepsilon_2\\) be its Zeckendorf representation, meaning \\(N=\\sum_{i=2}^k \\varepsilon_i F_i\\) with \\(\\varepsilon_i\\in\\{0,1\\}\\), \\(\\varepsilon_k=1\\), and \\(\\varepsilon_i\\varepsilon_{i+1}=0\\) for all \\(2\\le i<k\\). Let \\(\\operatorname{rev}(Z(N))\\) denote the Zeckendorf digits written from least significant to most significant, and define \\(\\widetilde Z(N)=\\operatorname{rev}(Z(N))\\,\\#^{\\omega}\\), where \\(\\#\\notin\\{0,1\\}\\) is a padding symbol. For a word \\(w\\in\\{0,1,\\#\\}^{\\mathbb N}\\), let \\(\\operatorname{pref}_M(w)\\) be its prefix of length \\(M\\). If \\(\\mathcal F\\) is a finite family of non-empty binary words, say that a word in \\(\\{0,1,\\#\\}^*\\) avoids \\(\\mathcal F\\) if none of the words in \\(\\mathcal F\\) occurs as a factor. Define\n\\[\n\\mathcal K_{\\mathcal F}^{(M)}=\\{N\\ge 1:\\operatorname{pref}_M(\\widetilde Z(N))\\text{ avoids }\\mathcal F\\}\n\\]\nand, for integers \\(u\\ge 1\\), \\(q\\ge 2\\), define\n\\[\nS_u^{(M)}=\\{n\\ge 0: uq^n\\in \\mathcal K_{\\mathcal F}^{(M)}\\}.\n\\]\nLet \\(L=\\max\\{|f|:f\\in\\mathcal F\\}\\) and fix \\(M\\ge L\\). Which statement holds for \\(S_u^{(M)}\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The set \\(S_u^{(M)}\\) is either finite or ultimately periodic: there exist integers \\(n_0\\ge 0\\) and \\(p\\ge 1\\) such that\n\\[\n n\\in S_u^{(M)}\\iff n+p\\in S_u^{(M)}\\qquad (n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair \\((n_0,p)\\) and decide whether \\(S_u^{(M)}\\) is finite or infinite; in the finite case, the same procedure outputs all elements of \\(S_u^{(M)}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The set \\(S_u^{(M)}\\) is ultimately periodic: there exist integers \\(n_0\\ge 0\\) and \\(p\\ge 1\\) such that\n\\[\n n\\in S_u^{(M)}\\iff n+p\\in S_u^{(M)}\\qquad (n\\ge n_0).\n\\]\nMoreover, there is a procedure depending only on \\(q,\\mathcal F\\), and \\(M\\) which computes such a pair \\((n_0,p)\\) simultaneously for every \\(u\\ge 1\\)."
        },
        {
          "label": "C",
          "text": "The set \\(S_u^{(M)}\\) is ultimately periodic: there exist integers \\(n_0\\ge 0\\) and \\(p\\ge 1\\) such that\n\\[\n n\\in S_u^{(M)}\\iff n+p\\in S_u^{(M)}\\qquad (n\\ge n_0).\n\\]"
        },
        {
          "label": "D",
          "text": "The set \\(S_u^{(M)}\\) is either finite or ultimately periodic provided \\(M>L\\): there exist integers \\(n_0\\ge 0\\) and \\(p\\ge 1\\) such that\n\\[\n n\\in S_u^{(M)}\\iff n+p\\in S_u^{(M)}\\qquad (n\\ge n_0).\n\\]\nMoreover, one can effectively compute such a pair \\((n_0,p)\\) and decide whether \\(S_u^{(M)}\\) is finite or infinite. In the finite case, the same procedure outputs all elements of \\(S_u^{(M)}\\)."
        },
        {
          "label": "E",
          "text": "The set \\(S_u^{(M)}\\) is either finite or eventually a finite union of arithmetic progressions: there exist integers \\(n_0\\ge 0\\), \\(r\\ge 1\\), and pairs \\((a_i,p_i)\\) with \\(p_i\\ge 1\\) for \\(1\\le i\\le r\\) such that\n\\[\n n\\in S_u^{(M)}\\iff n\\in \\bigcup_{i=1}^r \\{a_i+k p_i:k\\ge 0\\}\\qquad (n\\ge n_0).\n\\]\nMoreover, one can effectively compute such data and decide whether \\(S_u^{(M)}\\) is finite or infinite; in the finite case, the same procedure outputs all elements of \\(S_u^{(M)}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dependence of computed period/preperiod on the initial state \\(u\\)",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the effectiveness and finite-case output conclusions",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "boundary condition on the window length, replacing \\(M\\ge L\\) by \\(M>L\\)",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replacing a single eventual period from one deterministic orbit by a general semilinear eventual description",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option, nor does it contain wording that uniquely points to choice A. It sets up definitions and hypotheses without directly stating the final theorem."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-recall/restatement question: after giving the full setup, it asks which conclusion holds. The alternatives introduce nearby variants, so it is not completely tautological, but it still mostly tests recognition of the theorem statement."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in comparing subtle variants (finite-or-periodic vs. always periodic, effective vs. noneffective, dependence on u, restricted vs. unrestricted set). However, it mainly rewards recall/discrimination rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and target real failure modes, especially B, D, and E. However, choice C is a weaker true statement implied by A, so as written it is not a clean distractor and creates ambiguity unless the prompt explicitly asks for the strongest valid statement."
      },
      "total_score": 5,
      "overall_assessment": "Technically sophisticated item with mostly strong distractors, but it is weakened by theorem-restatement structure and, more importantly, ambiguity from including a weaker true option alongside the intended correct answer."
    }
  },
  {
    "id": "2512.19600v1",
    "paper_link": "http://arxiv.org/abs/2512.19600v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:planar}\nFor every real number $q\\notin\\{0,1,2\\}$,\nthe evaluation spectrum $|\\mathsf{S}_n^{\\pl}(q)|$ grows exponentially in $n$. \n\nMoreover, we can show the following uniform lower bounds: \n\\[\n|\\mathsf{S}^{\\pl}_n(q)| \\;\\ge\\;\n\\begin{cases}\n2^{(n-2)/2}, & q<0,\\\\[4pt]\n2^{n/8 - o(n)}, & \\hspace{-4mm} 0 < q < 2,\\\\[4pt]\n\\sqrt{F_{n-2}}= \\Theta(\\varphi^{(n-2)/2}), & q>2,\\\n\\end{cases}\n\\]\nwhere $(F_n)_n$ is the Fibonacci sequence and $\\varphi = (1+\\sqrt{5})/2$ is the golden ratio.",
      "start_pos": 12012,
      "end_pos": 12526,
      "label": "thm:planar"
    },
    "ref_dict": {
      "thm:planar": "\\begin{theorem}\\label{thm:planar}\nFor every real number $q\\notin\\{0,1,2\\}$,\nthe evaluation spectrum $|\\mathsf{S}_n^{\\pl}(q)|$ grows exponentially in $n$. \n\nMoreover, we can show the following uniform lower bounds: \n\\[\n|\\mathsf{S}^{\\pl}_n(q)| \\;\\ge\\;\n\\begin{cases}\n2^{(n-2)/2}, & q<0,\\\\[4pt]\n2^{n/8 - o(n)}, & \\hspace{-4mm} 0 < q < 2,\\\\[4pt]\n\\sqrt{F_{n-2}}= \\Theta(\\varphi^{(n-2)/2}), & q>2,\\\n\\end{cases}\n\\]\nwhere $(F_n)_n$ is the Fibonacci sequence and $\\varphi = (1+\\sqrt{5})/2$ is the golden ratio.\n\\end{theorem}",
      "thm:main": "\\begin{theorem}\\label{thm:main}\nFor every non-integer real number $q > 0$, and $n\\ge \\lceil q\\rceil+2$, \n$$|\\mathsf{S}_n(q)| \\;\\ge\\; 2^{(n-\\lceil q\\rceil-2)/2}.$$\n\\end{theorem}",
      "rmk:2-connec": "\\begin{remark}\n    \\label{rmk:2-connec}\n    Note that the operations defined in Definition \\ref{def:ops-neg} preserve graph planarity and connectivity, and increase the number of vertices of the graph by one. In fact, if we apply a sequence of these operations starting with $(K_3, e)$ or $(K_4,e)$, $e$ being an arbitrary edge of the respective graph, then it is not hard to inductively show that the resulting graph is $2$--connected. Indeed, this follows from Whitney's theorem \\cite{WhitneyEar}, since the graph will always have an ear decomposition: after applying $S$, replace $e$ by $a-w-b$ within the ears containing $e$; after applying $B$, we simply gain a new ear.    \n\n    In particular, replacing the starting graph in Lemma~\\ref{lem:pingpong-neg} by $(K_3, e)$, Lemma~\\ref{lem:pingpong-neg}, and the subsequent Lemma ~\\ref{lem:ping_pong_02} and Lemma~\\ref{lem:pingpong-pos}, all yield $2$--connected, planar witnesses, so after applying Lemma \\ref{lem:sqrt-neg} and Lemma \\ref{lem:sqrt-pos} respectively, we get the same bound asymptotically on the evaluation spectrum even if we require $G\\in \\mG_n^{\\pl}$ to be connected.  \n\\end{remark}"
    },
    "pre_theorem_intro_text_len": 4761,
    "pre_theorem_intro_text": "\\label{sec:intro}\n\nA \\emph{proper $q$--coloring} of a finite undirected graph $G=(V,E)$ is an assignment\n$\\varphi:V\\to[q]$ such that adjacent vertices receive different colors.\nWhen $q\\in\\mathbb{Z}_{\\ge 1}$, the number of proper $q$--colorings is denoted $P_G(q)$, and it is a classical theorem of Birkhoff and Whitney that there is a unique polynomial $P_G(x)\\in\\mathbb{Z}[x]$ which encodes the number of proper $q$-colorings of $G$ when evaluated at $x=q$. This is the so-called {\\it{chromatic polynomial}} of graph $G$, a polynomial of degree $|V|$ and with leading coefficient equal to $1$. This object was first introduced by Birkhoff~\\cite{Birkhoff1912} in 1912 as part of an algebraic approach for the Four Color Problem. Birkhoff’s original motivation was to study colorings of planar maps by encoding them in a polynomial and analyzing its behavior at small integer values, particularly at $x=4$. Although this approach did not yield a proof of the Four Color Theorem, it initiated a rich line of research and established the chromatic polynomial as a fundamental graph invariant. Whitney~\\cite{Whitney1932} subsequently extended Birkhoff’s ideas to general graphs and introduced many of the structural properties of chromatic polynomials that are now standard. Among these is the deletion--contraction identity: for any edge $e$ of $G$,\n\\begin{equation}\n    P_G(x)=P_{G-e}(x)-P_{G/e}(x), \\label{eq:dc-intro}\n\\end{equation}\nwhere $G-e$ denotes deletion and $G/e$ denotes contraction.\nThis identity places chromatic polynomials within the broader framework of the Tutte polynomial~\\cite{Tutte1954}.\n\nThe chromatic polynomial encodes substantial information about $G$.\nFor instance, the chromatic number satisfies\n\\[\n\\chi(G)=\\min\\{q\\in\\mathbb{Z}_{\\ge 1}: P_G(q)>0\\}.\n\\]\nEvaluations at negative integers also have a rich combinatorial meaning:\nStanley~\\cite{Stanley1973} showed that\n\\[\n|P_G(-1)|=\\#\\{\\text{acyclic orientations of }G\\},\n\\]\nand more generally his chromatic reciprocity theorem interprets $(-1)^{|V|}P_G(-m)$, for $m\\in\\mathbb{Z}_{\\ge 1}$,\nas a weighted count of acyclic orientations together with order-preserving maps to $[m]$. \n\n\\medskip\n\n\\noindent {\\bf{Chromatic polynomial spectrum.}} In \\cite{AKTutte}, Agol and Krushkal proved that the number of chromatic polynomials of planar triangulations on $n$ vertices grows exponentially with $n$. In a few words, their argument works with duals of planar cubic graphs and considers the number of distinct values taken by the \\emph{flow polynomial} at the special value\n$Q=(3-\\sqrt5)/2$ (for a planar graph $G$, its flow polynomial is precisely $P_{G^{*}}(x)$, where $G^{*}$ is the dual of $G$). At this particular parameter $Q$, the chromatic algebra (a certain algebra over $\\mathbb{C}[Q]$ formed from linear combinations of isotopy classes of planar graphs in a rectangle with a fixed number of univalent points on top and bottom) satisfies an additional local relation (coming from Temperley--Lieb / Jones--Wenzl theory),\nwhich collapses a natural boundary module from dimension three to dimension two.\nTwo explicit local operations on planar graphs $G$ then act on $P_{G^{*}}(Q)$ by $2\\times2$ matrices whose squares generate a free semigroup,\nyielding exponentially many distinct evaluations and hence exponentially many distinct chromatic polynomials.\nSee~\\cite{AKFlow,AKTutte} for more details and further references. \n\nIn this paper we study the asymptotics of the \\emph{evaluation spectra} of the chromatic polynomial. First, let $\\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\\ge 1$, we can now define\n\\[\n\\mathsf{S}_n(q)\\;\\coloneqq \\;\\{\\,P_G(q)\\,:\\,G \\in \\mG_n\\,\\}.\n\\]\nIn words, $\\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let \n\\begin{equation*}\n    \\mathsf{S}_n^{\\pl}(q) \\; \\coloneqq \\; \\{\\, P_G(q) \\, : \\, G \\in \\mG_n^{\\pl} \\, \\}\n\\end{equation*}\nwhere $\\mG_n^{\\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\\abs{\\mathsf{S}_n(q)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(q)}$. \n\n\\medskip\n\\noindent\\textbf{Our results.}\nWe establish lower bounds for $\\mathsf{S}_n(q)$ and $\\mathsf{S}_n^{\\pl}(q)$ for arbitrary fixed $q$. For integer $q\\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\\mathsf{S}_n^{\\pl}(3)$ should also grows exponentially over $\\mG_n^{\\pl}$; see~\\cite{TreumannMO} for more context, in particular a motivation coming from the {\\sf NP}-completeness of $3$--colorability of planar graphs. \nOur main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed\n$q\\notin\\{0,1,2\\}$.",
    "context": "\\label{sec:intro}\n\nA \\emph{proper $q$--coloring} of a finite undirected graph $G=(V,E)$ is an assignment\n$\\varphi:V\\to[q]$ such that adjacent vertices receive different colors.\nWhen $q\\in\\mathbb{Z}_{\\ge 1}$, the number of proper $q$--colorings is denoted $P_G(q)$, and it is a classical theorem of Birkhoff and Whitney that there is a unique polynomial $P_G(x)\\in\\mathbb{Z}[x]$ which encodes the number of proper $q$-colorings of $G$ when evaluated at $x=q$. This is the so-called {\\it{chromatic polynomial}} of graph $G$, a polynomial of degree $|V|$ and with leading coefficient equal to $1$. This object was first introduced by Birkhoff~\\cite{Birkhoff1912} in 1912 as part of an algebraic approach for the Four Color Problem. Birkhoff’s original motivation was to study colorings of planar maps by encoding them in a polynomial and analyzing its behavior at small integer values, particularly at $x=4$. Although this approach did not yield a proof of the Four Color Theorem, it initiated a rich line of research and established the chromatic polynomial as a fundamental graph invariant. Whitney~\\cite{Whitney1932} subsequently extended Birkhoff’s ideas to general graphs and introduced many of the structural properties of chromatic polynomials that are now standard. Among these is the deletion--contraction identity: for any edge $e$ of $G$,\n\\begin{equation}\n P_G(x)=P_{G-e}(x)-P_{G/e}(x), \\label{eq:dc-intro}\n\\end{equation}\nwhere $G-e$ denotes deletion and $G/e$ denotes contraction.\nThis identity places chromatic polynomials within the broader framework of the Tutte polynomial~\\cite{Tutte1954}.\n\nThe chromatic polynomial encodes substantial information about $G$.\nFor instance, the chromatic number satisfies\n\\[\n\\chi(G)=\\min\\{q\\in\\mathbb{Z}_{\\ge 1}: P_G(q)>0\\}.\n\\]\nEvaluations at negative integers also have a rich combinatorial meaning:\nStanley~\\cite{Stanley1973} showed that\n\\[\n|P_G(-1)|=\\#\\{\\text{acyclic orientations of }G\\},\n\\]\nand more generally his chromatic reciprocity theorem interprets $(-1)^{|V|}P_G(-m)$, for $m\\in\\mathbb{Z}_{\\ge 1}$,\nas a weighted count of acyclic orientations together with order-preserving maps to $[m]$.\n\n\\noindent {\\bf{Chromatic polynomial spectrum.}} In \\cite{AKTutte}, Agol and Krushkal proved that the number of chromatic polynomials of planar triangulations on $n$ vertices grows exponentially with $n$. In a few words, their argument works with duals of planar cubic graphs and considers the number of distinct values taken by the \\emph{flow polynomial} at the special value\n$Q=(3-\\sqrt5)/2$ (for a planar graph $G$, its flow polynomial is precisely $P_{G^{*}}(x)$, where $G^{*}$ is the dual of $G$). At this particular parameter $Q$, the chromatic algebra (a certain algebra over $\\mathbb{C}[Q]$ formed from linear combinations of isotopy classes of planar graphs in a rectangle with a fixed number of univalent points on top and bottom) satisfies an additional local relation (coming from Temperley--Lieb / Jones--Wenzl theory),\nwhich collapses a natural boundary module from dimension three to dimension two.\nTwo explicit local operations on planar graphs $G$ then act on $P_{G^{*}}(Q)$ by $2\\times2$ matrices whose squares generate a free semigroup,\nyielding exponentially many distinct evaluations and hence exponentially many distinct chromatic polynomials.\nSee~\\cite{AKFlow,AKTutte} for more details and further references.\n\nIn this paper we study the asymptotics of the \\emph{evaluation spectra} of the chromatic polynomial. First, let $\\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\\ge 1$, we can now define\n\\[\n\\mathsf{S}_n(q)\\;\\coloneqq \\;\\{\\,P_G(q)\\,:\\,G \\in \\mG_n\\,\\}.\n\\]\nIn words, $\\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let \n\\begin{equation*}\n \\mathsf{S}_n^{\\pl}(q) \\; \\coloneqq \\; \\{\\, P_G(q) \\, : \\, G \\in \\mG_n^{\\pl} \\, \\}\n\\end{equation*}\nwhere $\\mG_n^{\\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\\abs{\\mathsf{S}_n(q)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(q)}$.\n\n\\medskip\n\\noindent\\textbf{Our results.}\nWe establish lower bounds for $\\mathsf{S}_n(q)$ and $\\mathsf{S}_n^{\\pl}(q)$ for arbitrary fixed $q$. For integer $q\\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\\mathsf{S}_n^{\\pl}(3)$ should also grows exponentially over $\\mG_n^{\\pl}$; see~\\cite{TreumannMO} for more context, in particular a motivation coming from the {\\sf NP}-completeness of $3$--colorability of planar graphs. \nOur main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed\n$q\\notin\\{0,1,2\\}$.",
    "full_context": "\\label{sec:intro}\n\nA \\emph{proper $q$--coloring} of a finite undirected graph $G=(V,E)$ is an assignment\n$\\varphi:V\\to[q]$ such that adjacent vertices receive different colors.\nWhen $q\\in\\mathbb{Z}_{\\ge 1}$, the number of proper $q$--colorings is denoted $P_G(q)$, and it is a classical theorem of Birkhoff and Whitney that there is a unique polynomial $P_G(x)\\in\\mathbb{Z}[x]$ which encodes the number of proper $q$-colorings of $G$ when evaluated at $x=q$. This is the so-called {\\it{chromatic polynomial}} of graph $G$, a polynomial of degree $|V|$ and with leading coefficient equal to $1$. This object was first introduced by Birkhoff~\\cite{Birkhoff1912} in 1912 as part of an algebraic approach for the Four Color Problem. Birkhoff’s original motivation was to study colorings of planar maps by encoding them in a polynomial and analyzing its behavior at small integer values, particularly at $x=4$. Although this approach did not yield a proof of the Four Color Theorem, it initiated a rich line of research and established the chromatic polynomial as a fundamental graph invariant. Whitney~\\cite{Whitney1932} subsequently extended Birkhoff’s ideas to general graphs and introduced many of the structural properties of chromatic polynomials that are now standard. Among these is the deletion--contraction identity: for any edge $e$ of $G$,\n\\begin{equation}\n P_G(x)=P_{G-e}(x)-P_{G/e}(x), \\label{eq:dc-intro}\n\\end{equation}\nwhere $G-e$ denotes deletion and $G/e$ denotes contraction.\nThis identity places chromatic polynomials within the broader framework of the Tutte polynomial~\\cite{Tutte1954}.\n\nThe chromatic polynomial encodes substantial information about $G$.\nFor instance, the chromatic number satisfies\n\\[\n\\chi(G)=\\min\\{q\\in\\mathbb{Z}_{\\ge 1}: P_G(q)>0\\}.\n\\]\nEvaluations at negative integers also have a rich combinatorial meaning:\nStanley~\\cite{Stanley1973} showed that\n\\[\n|P_G(-1)|=\\#\\{\\text{acyclic orientations of }G\\},\n\\]\nand more generally his chromatic reciprocity theorem interprets $(-1)^{|V|}P_G(-m)$, for $m\\in\\mathbb{Z}_{\\ge 1}$,\nas a weighted count of acyclic orientations together with order-preserving maps to $[m]$.\n\n\\noindent {\\bf{Chromatic polynomial spectrum.}} In \\cite{AKTutte}, Agol and Krushkal proved that the number of chromatic polynomials of planar triangulations on $n$ vertices grows exponentially with $n$. In a few words, their argument works with duals of planar cubic graphs and considers the number of distinct values taken by the \\emph{flow polynomial} at the special value\n$Q=(3-\\sqrt5)/2$ (for a planar graph $G$, its flow polynomial is precisely $P_{G^{*}}(x)$, where $G^{*}$ is the dual of $G$). At this particular parameter $Q$, the chromatic algebra (a certain algebra over $\\mathbb{C}[Q]$ formed from linear combinations of isotopy classes of planar graphs in a rectangle with a fixed number of univalent points on top and bottom) satisfies an additional local relation (coming from Temperley--Lieb / Jones--Wenzl theory),\nwhich collapses a natural boundary module from dimension three to dimension two.\nTwo explicit local operations on planar graphs $G$ then act on $P_{G^{*}}(Q)$ by $2\\times2$ matrices whose squares generate a free semigroup,\nyielding exponentially many distinct evaluations and hence exponentially many distinct chromatic polynomials.\nSee~\\cite{AKFlow,AKTutte} for more details and further references.\n\nIn this paper we study the asymptotics of the \\emph{evaluation spectra} of the chromatic polynomial. First, let $\\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\\ge 1$, we can now define\n\\[\n\\mathsf{S}_n(q)\\;\\coloneqq \\;\\{\\,P_G(q)\\,:\\,G \\in \\mG_n\\,\\}.\n\\]\nIn words, $\\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let \n\\begin{equation*}\n \\mathsf{S}_n^{\\pl}(q) \\; \\coloneqq \\; \\{\\, P_G(q) \\, : \\, G \\in \\mG_n^{\\pl} \\, \\}\n\\end{equation*}\nwhere $\\mG_n^{\\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\\abs{\\mathsf{S}_n(q)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(q)}$.\n\n\\medskip\n\\noindent\\textbf{Our results.}\nWe establish lower bounds for $\\mathsf{S}_n(q)$ and $\\mathsf{S}_n^{\\pl}(q)$ for arbitrary fixed $q$. For integer $q\\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\\mathsf{S}_n^{\\pl}(3)$ should also grows exponentially over $\\mG_n^{\\pl}$; see~\\cite{TreumannMO} for more context, in particular a motivation coming from the {\\sf NP}-completeness of $3$--colorability of planar graphs. \nOur main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed\n$q\\notin\\{0,1,2\\}$.\n\nIn this paper we study the asymptotics of the \\emph{evaluation spectra} of the chromatic polynomial. First, let $\\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\\ge 1$, we can now define\n\\[\n\\mathsf{S}_n(q)\\;\\coloneqq \\;\\{\\,P_G(q)\\,:\\,G \\in \\mG_n\\,\\}.\n\\]\nIn words, $\\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let \n\\begin{equation*}\n \\mathsf{S}_n^{\\pl}(q) \\; \\coloneqq \\; \\{\\, P_G(q) \\, : \\, G \\in \\mG_n^{\\pl} \\, \\}\n\\end{equation*}\nwhere $\\mG_n^{\\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\\abs{\\mathsf{S}_n(q)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(q)}$.\n\n\\medskip\n\\noindent\\textbf{Our results.}\nWe establish lower bounds for $\\mathsf{S}_n(q)$ and $\\mathsf{S}_n^{\\pl}(q)$ for arbitrary fixed $q$. For integer $q\\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\\mathsf{S}_n^{\\pl}(3)$ should also grows exponentially over $\\mG_n^{\\pl}$; see~\\cite{TreumannMO} for more context, in particular a motivation coming from the {\\sf NP}-completeness of $3$--colorability of planar graphs. \nOur main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed\n$q\\notin\\{0,1,2\\}$.\n\nIn particular, Theorem \\ref{thm:planar} and the trivial inequality $|\\mathcal{P}_n| \\geq \\abs{\\mathsf{S}_n(-1)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(-1)}$ together imply that $|\\mathcal{P}_n| \\geq 2^{(n-2)/2}$. This fact also improves quantitatively upon the lower bound from \\cite{AKFlow}. Furthermore, the lower bounds from Theorem \\ref{thm:planar} hold true even if we additionally require the family of (planar) graphs in $\\mG_n^{\\pl}$ to be connected. See Remark \\ref{rmk:2-connec} for more details.\n\n\\subsection{Exponential growth for every non-integer \\texorpdfstring{$q>0$}{q > 0}}\nWe can then prove the following lower bound for the chromatic polynomial evaluation spectra at non-integer positive numbers. \n\\begin{theorem}\\label{thm:noninteger-positive}\nLet $q>0$ be a non-integer real and set $m\\coloneqq \\lceil q\\rceil$.\nThen $q-m\\in(-1,0)$ and $(q)_m\\neq 0$, and for all $n\\ge m+2$,\n\\[\n|\\mathsf{S}_n (q)| \\ \\ge\\ 2^{(n-m-2)/2}.\n\\]\n\nIn this section, we prove the last bound of Theorem \\ref{thm:main} for $\\mathsf{S}_n(q)$ by showing the following\n\\begin{theorem}\\label{thm:q_greater_than_2}\n Let $q > 2$ be a real number and $n \\geq 3$, then\n \\[\\abs{\\mathsf{S}^{\\pl}_n(q)} \\geq \\sqrt{F_{n-2}} =\\Theta(\\varphi^{n/2}),\\]\n where $(F_n)_n$ is the Fibonacci sequence and $\\varphi = (1+\\sqrt{5})/2$ is the golden ratio. In particular, we get the bound for integers $q \\geq 3$.\n\\end{theorem}\n\n\\begin{lemma}[Square-root lemma]\\label{lem:sqrt-pos}\nIf there are at least $N$ distinct $n$--attainable vectors, then\n\\[\n|\\mathsf{S}^{\\pl}_n(q)| \\;\\ge\\; \\sqrt{N}.\n\\]\n\\end{lemma}\n\\begin{proof}\n Since $A_n$ is invertible, there are at least $N$ distinct $n$-feasible vectors. By Lemma~\\ref{lem:sqrt-neg}, $|\\mathsf{S}^{\\pl}_n(q)| \\ge \\sqrt{N}.$\n\\end{proof}\nThe same basic operations $B$ and $S$ defined in Section~\\ref{sec:negative} will be the building blocks of our construction of exponentially many $n$--attainable vectors. \n\\begin{lemma}[Matrix action]\\label{lem:matrices-pos}\nLet $e$ be an edge of a planar graph $G$, and write $w(G,e)=\\binom{t}{u}$.\nThen\n\\[\n(w\\circ S)(G,e)\n=\n\\begin{pmatrix}\n-1 & 1\\\\\n0 & q-1\n\\end{pmatrix}\n\\binom{t}{u},\n\\qquad\n(w\\circ B)(G,e)\n=\n\\begin{pmatrix}\nq-1 & 0\\\\\n1 & q-2\n\\end{pmatrix}\n\\binom{t}{u}.\n\\]\n\\end{lemma}\n\\begin{proof} Let $n = |V(G)|$ and observe that Lemma~\\ref{lem:matrices-neg} and \\eqref{eq:feasible-to-attainable} imply\n \\[(w\\circ S)(G,e)\n=A_{n+1}\\begin{pmatrix}\n1 & 1\\\\\n0 & -q+1\n\\end{pmatrix}A_n^{-1}\n\\binom{t}{u} = \\begin{pmatrix}\n-1 & 1\\\\\n0 & q-1\n\\end{pmatrix}\n\\binom{t}{u}\\]\nand \n\\[(w\\circ B)(G,e)\n=A_{n+1}\\begin{pmatrix}\n-q+1 & 0\\\\\n1 & -q+2\n\\end{pmatrix}A_n^{-1}\n\\binom{t}{u} = \\begin{pmatrix}\nq-1 & 0\\\\\n1 & q-2\n\\end{pmatrix}\n\\binom{t}{u}. \\]\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:planar_0_2}]\nLet $(G_0,e_0,K,L)$ be given by Lemma~\\ref{lem:ping_pong_02} and let $k=k(q)$ and $\\ell=\\ell(q)$ be the number of basic operations in blocks $K$ and $L$, respectively. Let $t = \\smash{ \\lfloor \\frac{n-|V(G_0|)}{(k+\\ell)}\\rfloor}$ and notice that $n_0 = |V(G_0)|+t(k+\\ell) \\le n$. By Lemma~\\ref{lem:ping_pong_02}, there are at least\n\\begin{equation}\\label{eq:attainable_02}\n\\binom{2t}{t} \\ge \\frac{2^{2t}}{2t+1} = 2^{\\Omega_{q}(n)} \n\\end{equation}\ndistinct $n_0$--attainable vectors, since words in $\\{K,L\\}^\\ast$ consisting of $t$ $K$'s and $t$ $L$'s produce distinct $n_0$--attainable vectors. Thus,\n\\[\n|\\mathsf{S}^{\\mathrm{pl}}_n(q)| \\;\\ge\\; |\\mathsf{S}^{\\mathrm{pl}}_{n_0}(q)| \\;=\\;2^{\\Omega_q(n)},\n\\]\nwhere the left inequality comes from the fact that adding a leaf to a connected planar graph preserves planarity and alters the chromatic polynomial by a factor of $x-1$, and the right identity is given by~\\eqref{eq:attainable_02} and Lemma~\\ref{lem:sqrt-pos}.\n\\end{proof}\n\\begin{remark}[The dependence on $q$ or lack thereof]\n \\label{rem:q_no_dependence}\n Theorem~\\ref{thm:planar_0_2} as proven only gives that for every $q \\in (0, 2) \\setminus \\set{1}$ the existence of $c(q) > 0$ and a positive integer $n_0(q)$ such that $\\abs{\\mathsf{S}_n^{\\pl} (q)} \\geq \\exp\\of{(c(q) +o(1)) n}$. However, a more detailed analysis easily can show that we may choose $c(q)$ to be an absolute constant, namely $1/8$. \n \\begin{itemize}\n \\item For $q \\in (0,1.458)$, the construction in Case~1 of Lemma~\\ref{lem:ping_pong_02} works with $m = 1$. This construction provides the bound $\\abs{\\mathsf{S}_n^{\\pl} (q)} \\geq 2^{n/4-o(n)}$;\n \\item For $q \\in (1,3/2]$, one can use the construction in Case~3 of Lemma~\\ref{lem:ping_pong_02}. This construction provides the bound $\\abs{\\mathsf{S}_n^{\\pl} (q) }\\geq 2^{n / 6 - o(n)}$;\n\n\\noindent {\\bf{Superexponential spectra.}} For every fixed real $q\\notin\\{0,1,2\\}$ we proved that the evaluation spectrum\n\\[\n\\mathsf{S}_n(q)\\;=\\;\\{P_G(q):|V(G)|=n\\}\n\\]\ngrows exponentially in $n$, with explicit lower bounds of different qualities in various regimes. For positive integers $q \\not\\in \\left\\{0,1,2\\right\\}$, we also noted that exponential growth for the evaluation spectrum of $P_{G}(q)$ is best possible (up to the correct base of the exponent), due to the fact that $|\\mathsf{S}_n(q)| \\leq q^{n}+1$. Nevertheless, at other values of $q$, the size of $\\mathsf{S}_n(q)$ could be significantly larger, in principle. For example, when $q=-1$, $|P_{G}(-1)|$ counts the number of acyclic orientations of $G$, so the natural upper bound for $|\\mathsf{S}_n(-1)|$ is $|P_{K_n}(-1)|=n!$, rather than $q^n+1$. In this direction, we would like to make the appealing following conjecture for the spectrum of acyclic orientations.",
    "post_theorem_intro_text_len": 7208,
    "post_theorem_intro_text": "In particular, Theorem \\ref{thm:planar} and the trivial inequality $|\\mathcal{P}_n| \\geq \\abs{\\mathsf{S}_n(-1)} \\geq \\abs{\\smash{\\mathsf{S}_n^{\\pl}}(-1)}$ together imply that $|\\mathcal{P}_n| \\geq 2^{(n-2)/2}$. This fact also improves quantitatively upon the lower bound from \\cite{AKFlow}. Furthermore, the lower bounds from Theorem \\ref{thm:planar} hold true even if we additionally require the family of (planar) graphs in $\\mG_n^{\\pl}$ to be connected. See Remark \\ref{rmk:2-connec} for more details. \n\nBefore we say more about the proof of Theorem \\ref{thm:planar}, let us first remark that for $q\\in\\{0,1,2\\}$ the spectrum is not exponential even if we allow $G$ to vary over {\\it{all}} graphs on $n$ vertices.\nIndeed,\n\\begin{itemize}\n\\item $P_G(0)=0$ for every graph with at least one vertex, so $|\\mathsf{S}^{\\pl}_n(0)|=|\\mathsf{S}_n(0)|=1$;\n\\item $P_G(1)=1$ if $G$ is edgeless and $P_G(1)=0$ otherwise, so $|\\mathsf{S}^{\\pl}_n(1)|=1$ and $|\\mathsf{S}_n(1)|= 2$;\n\\item $P_G(2)$ counts proper $2$--colorings, so it is $0$ for non-bipartite $G$, and equals $2^{c(G)}$ for bipartite $G$ with $c(G)$ connected components.\nHence $|\\mathsf{S}^{\\pl}_n(2)|\\leq |\\mathsf{S}_n(2)|= n+1$.\n\\end{itemize}\n\nThe proof of Theorem \\ref{thm:planar} is more delicate when $q \\in (0,2)$ (and precise quantitative bounds are also more difficult to extract), so, as a prequel to that side of the story, we will first use an elementary argument to show that $|\\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q < 0$. We record this as a second theorem, in greater generality. \n\n\\begin{theorem}\\label{thm:main}\nFor every non-integer real number $q > 0$, and $n\\ge \\lceil q\\rceil+2$, \n$$|\\mathsf{S}_n(q)| \\;\\ge\\; 2^{(n-\\lceil q\\rceil-2)/2}.$$\n\\end{theorem}\n\nIn particular, if $q \\in (0,2)$, then Theorem \\ref{thm:main} immediately implies that $|\\mathsf{S}_n(q)| \\ge 2^{(n-4)/2}$ for all $n\\ge 4$. \n\nLast but not least, let us also emphasize that for positive integers $q \\not\\in \\left\\{0,1,2\\right\\}$, exponential growth for $\\abs{\\mathsf{S}_n(q)}$ is best possible, up to the correct base of the exponent. Indeed, if $q\\in\\mathbb{Z}_{\\ge 1}$ then every $n$--vertex graph has at most $q^n$ proper $q$--colorings, so $|\\mathsf{S}_n(q)|\\le q^n+1$. \n\nWhen $q$ is \\emph{not} a nonnegative integer, there is however no comparable a priori bound of the form $|P_G(q)|\\le C^n$.\nWhen $q=-1$, Stanley's theorem~\\cite{Stanley1973} gives $|P_{K_n}(-1)|=n!$, and, more generally, it is not difficult to show that for any fixed $q\\in\\mathbb{R}\\setminus\\mathbb{Z}_{\\ge 0}$ the complete graph satisfies\n$P_{K_n}(q)=(q)_n=q(q-1)\\cdots(q-n+1)$, whose magnitude grows on the order of $n!$.\nThese factorial-scale values suggest that, at least for some non-integer or negative points, the spectrum itself should be \\emph{superexponential}. We will discuss this some more in Section \\ref{sec:conclusion}.\n\n\\vspace{+3mm}\n\\noindent\\textbf{Related work and proof ideas.}\nA closely related spectrum problem was recently resolved for the spanning-tree enumerator.\nLet $\\tau(G)$ be the number of spanning trees of $G$, and set\n\\[\n\\mathcal{T}(n)\\coloneqq \\{\\tau(G): G \\in \\mG_n \\},\n\\]\nas well as\n\\begin{equation*}\n    \\mathcal{T}^{\\pl}(n) \\coloneqq \\{\\, P_G(q) \\, : \\, G \\in \\mG_n^{\\pl} \\}.\n\\end{equation*}\n\nIn \\cite{ChanKontorovichPak24trees}, Chan, Kontorovich and Pak proved that $|\\mathcal{T}^{\\pl}(n)|$ grows exponentially in $n$, via a connection to continued fractions, thin orbits, and a deep result of Bourgain-Kontorovich \\cite{BourgainKontorovich14} on the so-called Zaremba conjecture \\cite{Zaremba72}. See \\cite{ChanKontorovichPak24trees} and the references therein for more information. Soon after, Alon--Buci\\'c--Gishboliner~\\cite{ABG} gave a much simpler alternative proof, which also came with improved quantitative bounds and results for other classes of graphs. Both arguments made crucial use of the deletion--contraction formula for the spanning-tree enumerator $\\tau$ and a handful of basic operations that increase the number of vertices of the graph, and which allow one to track the evolution of the parameters $\\tau(G-e)$ and $\\tau(G/e)$ (for various edges $e$ of $G$), as the graph $G$ undergoes these changes. In \\cite{ABG}, however, Alon, Buci\\'c and Gishboliner noted that the parameters $\\tau(G-e)$ and $\\tau(G/e)$ of $G$ can sometimes instantly uniquely decode the entire process, an observation which drastically simplified the analysis. \n\nFor the chromatic polynomial spectrum, Agol and Krushkal \\cite{AKTutte} also previously employed a similar strategy to show that $|\\mathcal{P}_n|$ grows exponentially with $n$. An important difference in their argument is that the basic operations employed are less simple than the ones in ~\\cite{ABG} or \\cite{ChanKontorovichPak24trees}, and so one cannot readily decode the process if their operations are used freely to construct their $n$--vertex graphs. In order to circumvent this barrier, they consider only graphs obtained through a sequence of block operations, each of which combines some of their set of basic operations. To decode the sequence of block operations from the final parameters obtained, Agol and Krushkal then appeal to a simple application of the ping--pong lemma, a basic tool from geometric group theory. See for example \\cite{Loh} and the references therein or \\cite{BrennerCharnow78}.\n\nOur argument, in some sense, unifies the two strategies from \\cite{AKTutte} and \\cite{ABG}. To prove Theorem \\ref{thm:planar}, we make use of block operations that consist of sequences of two of the basic operations used in~\\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.\n\n\\vspace{+2mm}\n\n\\noindent\\textbf{Organization of the paper.} The proof of Theorem \\ref{thm:planar} is most pleasant when $q<0$, so we set the stage by discussing this case first in Section \\ref{sec:negative}.\n\nAfter that, in Section \\ref{sec:noninteger}, we take a first dip into the more problematic case when $q \\in (0,2)$, which we will first address for the $|\\mathsf{S}_n(q)|$. Using a different kind of combinatorial argument, we will derive a lower bound by reducing the problem to the case when $q \\in (-1,0)$, for which the result from Section \\ref{thm:planar} applies. This argument is the content of Theorem \\ref{thm:main} and works in greater generality for every non-integer real number $q > 0$, and $n\\ge \\lceil q\\rceil+2$. It does not apply directly to lower bound $|\\mathsf{S}^{\\pl}_n(q)|$ but it provides a better quantitative bound for $|\\mathsf{S}_n(q)|$ than Theorem \\ref{thm:planar} does in this range. \n\nIn Section \\ref{sec:integers}, we continue the proof of Theorem \\ref{thm:planar} with a more complicated ping--pong argument showing that $|\\mathsf{S}^{\\pl}_n(q)|$ is exponential in $n$ for each $q >2$. \n\nIn Section \\ref{sec:planar_0_2}, we then complete the proof of Theorem \\ref{thm:planar} by showing that $|\\mathsf{S}^{\\mathrm{pl}}_n(q)|$ is exponential in $n$ for every $q\\in(0,2)\\setminus\\{1\\}$ as well, by using a more intricate combination of block constructions. \n\nIn Section \\ref{sec:conclusion}, we end with some concluding remarks and a couple of natural open problems.",
    "sketch": "The introduction indicates that the proof of Theorem~\\ref{thm:planar} uses a “block operations + decoding” strategy: “To prove Theorem~\\ref{thm:planar}, we make use of block operations that consist of sequences of two of the basic operations used in~\\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.”\n\nIt also sketches how the argument is organized by ranges of $q$:\n- “The proof of Theorem~\\ref{thm:planar} is most pleasant when $q<0$,” treated first (Section~\\ref{sec:negative}).\n- For $q\\in(0,2)$ the proof is “more delicate”; as a “prequel” they “first use an elementary argument to show that $|\\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q<0$,” recorded as Theorem~\\ref{thm:main}. Later they “complete the proof of Theorem~\\ref{thm:planar}” for planar graphs in this range “by using a more intricate combination of block constructions” (Section~\\ref{sec:planar_0_2}).\n- For $q>2$, they “continue the proof of Theorem~\\ref{thm:planar} with a more complicated ping--pong argument” to show exponential growth (Section~\\ref{sec:integers}).",
    "expanded_sketch": "The introduction indicates that the proof of the main theorem uses a “block operations + decoding” strategy: “To prove the main theorem, we make use of block operations that consist of sequences of two of the basic operations used in~\\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.”\n\nIt also sketches how the argument is organized by ranges of $q$:\n- “The proof of the main theorem is most pleasant when $q<0$,” treated first (Section~\\ref{sec:negative}).\n- For $q\\in(0,2)$ the proof is “more delicate”; as a “prequel” they “first use an elementary argument to show that $|\\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q<0$,” recorded as the following theorem.\n\n\\begin{theorem}\\label{thm:main}\nFor every non-integer real number $q > 0$, and $n\\ge \\lceil q\\rceil+2$, \n$$|\\mathsf{S}_n(q)| \\;\\ge\\; 2^{(n-\\lceil q\\rceil-2)/2}.$$\n\\end{theorem}\n\nLater they “complete the proof of the main theorem” for planar graphs in this range “by using a more intricate combination of block constructions” (Section~\\ref{sec:planar_0_2}).\n- For $q>2$, they “continue the proof of the main theorem with a more complicated ping--pong argument” to show exponential growth (Section~\\ref{sec:integers}).",
    "expanded_theorem": "\\label{thm:planar}\nFor every real number $q\\notin\\{0,1,2\\}$,\nthe evaluation spectrum $|\\mathsf{S}_n^{\\pl}(q)|$ grows exponentially in $n$. \n\nMoreover, we can show the following uniform lower bounds: \n\\[\n|\\mathsf{S}^{\\pl}_n(q)| \\;\\ge\\;\n\\begin{cases}\n2^{(n-2)/2}, & q<0,\\\\[4pt]\n2^{n/8 - o(n)}, & \\hspace{-4mm} 0 < q < 2,\\\\[4pt]\n\\sqrt{F_{n-2}}= \\Theta(\\varphi^{(n-2)/2}), & q>2,\\\n\\end{cases}\n\\]\nwhere $(F_n)_n$ is the Fibonacci sequence and $\\varphi = (1+\\sqrt{5})/2$ is the golden ratio.,",
    "theorem_type": [
      "Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "For a simple graph $G$, let $P_G(x)$ denote its chromatic polynomial. For each integer $n\\ge 1$, let $\\mathcal G_n^{\\mathrm{pl}}$ be the set of all simple planar graphs on $n$ vertices, and for a real number $q$ define the planar evaluation spectrum\n\\[\n\\mathsf S_n^{\\mathrm{pl}}(q)=\\{P_G(q):G\\in \\mathcal G_n^{\\mathrm{pl}}\\}.\n\\]\nLet $(F_n)_n$ be the Fibonacci sequence and let $\\varphi=(1+\\sqrt5)/2$. Which statement holds for every real number $q\\notin\\{0,1,2\\}$?",
      "correct_choice": {
        "label": "A",
        "text": "For every fixed real $q\\notin\\{0,1,2\\}$, the cardinality $|\\mathsf S_n^{\\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,\n\\[\n|\\mathsf S_n^{\\mathrm{pl}}(q)|\\ge\n\\begin{cases}\n2^{(n-2)/2}, & q<0,\\\\[4pt]\n2^{n/8-o(n)}, & 0<q<2\\text{ and }q\\ne 1,\\\\[4pt]\n\\sqrt{F_{n-2}}=\\Theta\\!\\big(\\varphi^{(n-2)/2}\\big), & q>2.\n\\end{cases}\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every fixed real $q\\notin\\{0,1,2\\}$, the cardinality $|\\mathsf S_n^{\\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,\n\\[\n|\\mathsf S_n^{\\mathrm{pl}}(q)|\\ge\n\\begin{cases}\n2^{(n-2)/2}, & q<0,\\\\[4pt]\n2^{(n-2)/2}, & 0<q<2\\text{ and }q\\ne 1,\\\\[4pt]\n\\sqrt{F_{n-2}}=\\Theta\\!\\big(\\varphi^{(n-2)/2}\\big), & q>2.\n\\end{cases}\n\\]"
        },
        {
          "label": "C",
          "text": "For every fixed real $q\\notin\\{0,1,2\\}$, the cardinality $|\\mathsf S_n^{\\mathrm{pl}}(q)|$ grows exponentially with $n$."
        },
        {
          "label": "D",
          "text": "For every real $q\\notin\\{0,1,2\\}$, there exists an absolute constant $c>0$ such that\n\\[\n|\\mathsf S_n^{\\mathrm{pl}}(q)|\\ge 2^{cn}\n\\]\nfor all sufficiently large $n$."
        },
        {
          "label": "E",
          "text": "For every fixed real $q\\notin\\{0,1,2\\}$, the cardinality $|\\mathsf S_n^{\\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,\n\\[\n|\\mathsf S_n^{\\mathrm{pl}}(q)|\\ge\n\\begin{cases}\n2^{(n-2)/2}, & q\\le 0,\\\\[4pt]\n2^{n/8-o(n)}, & 0<q<2,\\\\[4pt]\n\\sqrt{F_{n-2}}=\\Theta\\!\\big(\\varphi^{(n-2)/2}\\big), & q\\ge 2.\n\\end{cases}\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "delicate_0_2_rate_replaced_by_negative_case_rate",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_piecewise_uniform_lower_bounds",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "q_dependence_of_exponential_constant_removed",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "excluded_boundary_points_1_and_2_reinserted_via_closed_ranges",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct choice. Mentioning the Fibonacci sequence and  does hint that the q>2 regime involves a Fibonacci-type bound, but that feature appears in multiple options, so it does not uniquely leak the answer."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely asking the test-taker to recognize the exact theorem statement about the planar evaluation spectrum. Although the options introduce subtle variants, the question is still close to a direct restatement rather than an application or consequence-based problem."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare the piecewise bounds and detect endpoint issues or overstrong claims, but the task is mainly theorem recognition. It does not strongly force generative mathematical reasoning beyond matching the correct quantitative statement."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are plausible and mathematically aligned with common errors: including q=2, overstating the Fibonacci growth, or transplanting the negative-q bound into 0<q<2. However, choice C is a weaker statement that is also true, so the distractor set does not support a clean single-correct MCQ."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically sophisticated but only moderately well-designed MCQ: it avoids direct answer leakage and has plausible false variants, but it is too close to theorem recall and is fundamentally weakened by the presence of a weaker true alternative, making the single-answer format problematic."
    }
  },
  {
    "id": "2512.19831v2",
    "paper_link": "http://arxiv.org/abs/2512.19831v2",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\mathbb{C})$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\mathbb{C})$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}",
      "start_pos": 30242,
      "end_pos": 31127,
      "label": "conj:PTB"
    },
    "ref_dict": {
      "thm:key1": "\\begin{theorem}\\label{thm:key1}\nLet $\\pi\\in\\cusp(G)$ be symplectic. Assume that $A/F=E/F$ is quadratic. Then:\n\\begin{enumerate}\n    \\item if $J_{\\pi,E}(s)$ has a pole at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=1$,\n    \\item if $J_{\\pi,E}(s)$ is holomorphic at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=-1$. \n    \\end{enumerate} \n\\end{theorem}",
      "prop:sign dist": "\\begin{proposition}\\label{prop:sign dist}\nAssume that either $G=\\GL_n(F)$ is split and $\\pi\\in \\gadist(G)$, or that $\\pi\\in \\cadist(G)$, then $\\beta_{\\pi,A,\\psi}(0)=1$. \n\\end{proposition}",
      "eq:iff": "\\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation}",
      "lm:product of c is one": "\\begin{lemma}\\label{lm:product of c is one}\nLet $\\mathfrak{E}/\\mathfrak{F}$ be a quadratic extension of number fields, and $\\Psi:\\mathfrak{F}\\backslash \\A_\\mathfrak{F}\\to \\C^\\times$ be a non trivial character. Let $\\mathfrak{D}$ be an $\\mathfrak{F}$-central division algebra. Assume moreover that \n$\\mathfrak{E}$ is contained inside $\\CM_m(\\mathfrak{D})$. \n If $\\Pi$ is a self-dual cuspidal automorphic representation of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that $\\JL(\\Pi)$ is cuspidal, then \\[\\prod_v c(\\Pi_v,\\Psi_v)=1,\\] where the product is over all places of $F$. More specifically \\[\\prod_{v\\in S} c(\\Pi_v,\\Psi_v)=1,\\] where $S$ is the set of places such that of $\\mathfrak{F}$ outside of which $\\mathfrak{D}_v$ is split, and $\\mathfrak{E}_v/\\mathfrak{F}_v$, $\\pi_v$ and $\\Psi_v$ are all unramified.\n\\end{lemma}",
      "cor:main": "\\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}",
      "lm:globlz": "\\begin{lemma}\\label{lm:globlz}\nLet $\\pi\\in \\sympc(G)$. There exist:\n\\begin{enumerate}\n\\item a number fields $\\mathfrak{F}$ and a place $v_0$ such that $\\mathfrak{F}_{v_0}\\simeq F$,\n\\item an $\\mathfrak{F}$-central division algebra $\\mathfrak{D}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{D}_{v_0}/\\mathfrak{F}_{v_0}\\simeq D/F$, \n\\item $\\mathfrak{D}_{w_0}$ is a division algebra of Hasse invariant opposite to that of $D$, for some finite place $w_0\\neq v_0$ of $\\mathfrak{F}$ lying over an odd prime number,\n\\item $\\mathfrak{D}_v$ is a split matrix algebra for any place $v$ of $\\mathfrak{F}$ different from $v_0$ and $w_0$, \n\\end{enumerate}\n\\item a cuspidal automorphic representation $\\Pi$ of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that:\n\\begin{enumerate}\n\\item $\\JL(\\Pi)$ is cuspidal and symplectic,\n\\item $\\Pi_{v_0}\\simeq \\pi$,\n\\item $\\Pi_{w_0}$ is cuspidal and $E'$-distinguished for some quadratic extension $E'/\\mathfrak{F}_{w_0}$, \n\\end{enumerate} \n\\item a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{E}_{v_0}/\\mathfrak{F}_{v_0}\\simeq E/F$ and $\\mathfrak{E}_{w_0}/\\mathfrak{F}_{w_0} \\simeq E'/\\mathfrak{F}_{w_0}$,\n\\item $\\mathfrak{E}_v/\\mathfrak{F}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$ splits at the finite number of places $v\\notin \\{v_0,w_0\\}$ such that $\\pi_v$ is not isomorphic to a principal series induced from a Borel subgroup of $\\GL_m(\\mathfrak{D}_v)\\simeq \\GL_n(\\mathfrak{F}_v)$,\n\\item $\\mathfrak{E}/\\mathfrak{F}$ embeds into $\\CM_m(\\mathfrak{D})/\\mathfrak{F}$. \n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}",
      "thm:key": "\\begin{theorem}\\label{thm:key}\nLet $\\pi\\in \\sympc(G)$. Then \n\\[c(\\pi,\\psi)=(-1)^{m}\\eta_{E/F}(-1)^{n/2}.\\]\n\\end{theorem}",
      "conj:PTB": "\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}",
      "cor:key": "\\begin{corollary}\\label{cor:key}\nLet $\\pi$ be as in Proposition \\ref{prop:sign dist}, but assume moreover that it is unitary, and let $c(\\pi,\\psi)$ be as in Theorem \\ref{thm:almost explicit constant}. Then $c(\\pi,\\psi)=(-1)^m\\eta_{A/F}(-1)^{n/2}$.\n\\end{corollary}"
    },
    "pre_theorem_intro_text_len": 2392,
    "pre_theorem_intro_text": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi,  \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi,  \\mathbb{C})\\neq 0$.  \n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper. \n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.",
    "context": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi, \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi, \\mathbb{C})\\neq 0$.\n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper.\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\n\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}",
    "full_context": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi, \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi, \\mathbb{C})\\neq 0$.\n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper.\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\n\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\JL(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\JL(\\pi)}:\\WD_F\\to \\GL_n(\\C).\\] Let $\\psi:F\\to \\C^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\C)$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\nLet $F$ be a local field of characteristic zero, possibly Archimedean. We fix a nontrivial character $\\psi:F\\to \\C^\\times$. Let $A$ be an $F$-separable algebra of dimension two, i.e. either $A/F$ is a quadratic extension, of $A\\simeq F\\times F$. We set $\\psi_A:=\\psi\\circ \\tr_{A/F}$, and denote by $\\eta_{A/F}$ the character of $F^\\times$ with kernel the norms of $A^\\times$, in particular $\\eta_{A/F}$ has order two when $A/F$ is quadratic, whereas it is trivial when $A\\simeq F\\times F$. We note that $A$ can be written $F[u]$, with $\\tr_{A/F}(u)=0$, i.e. $u^2\\in F$. Such an element $u$ is unique up to scaling by an element in $F^\\times$. Let $D$ be a central division algebra over $F$, such that $[D:F]=d^2$ with $d\\geq 1$. We set $\\CM:=\\CM_m(D)$ for $m\\geq 1$, and $n:=md$. We put $\\psi_{\\CM}:=\\psi\\circ \\tr_{\\CM/F}$. We assume that $A$ embeds in $\\CM$ as an $F$-subalgebra, in such a way that $\\tr_{\\CM/F}(u)=0$, and we realize the embedding explicitly, as explained in \\cite[Section 2.5]{MOY}. In particular $n$ is even, and if $A/F$ is quadratic, this parity condition is also sufficient for the assumption $\\tr_{\\CM/F}(u)=0$ to hold. If $A\\simeq F\\times F$, the assumption holds if and only if $m$ is even and $u$ is conjugate to $\\lambda\\diag(I_{m/2},-I_{m/2})$ with $\\lambda\\in F^\\times$. We set $G_m=G=\\CM^\\times$, and $H_{A,m}=H_A=Z_{\\CM}(A)^\\times$. The group $H_A$ is the subgroup of $G$ fixed by the inner involution $\\theta:=\\ad(u)$ of $G$. We denote by $\\nu:G\\to \\C^\\times$ the restriction of the composition of the normalized absolute value $|\\ |$ on $F$ with the reduced norm on $\\CM$. We denote by $P_{m,m}$ the block upper triangular parabolic subgroup of $G_{2m}$ corresponding to the partitition $(m,m)$, and by $M_{m,m}$ its block diagonal Levi subgroup. We denote by $x_0$ the representative of the unique open double coset $Px_0H$ fixed in \\cite[Section 4.1]{MOY}. In particular the involution $\\theta_o(g):=x_o\\theta(x_o^{-1}gx_o)x_o^{-1}$ of $G_{2m}$ is such that $\\theta_o(P_{m,m})$ is opposite to $P_{m,m}$ with respect to $M_{m,m}$. We denote by $\\FL(G)$ the objects of the category of smooth admissible representations of $G$ of finite length, where admissible means admissible (yes) but also Fr\\'echet and of moderate growth, when $F$ is Archimedean (see \\cite[Chapter 11]{RRG2}). For $\\pi\\in \\FL(G)$, we write $\\pi^\\vee$ for its smooth contragredient, and we define $\\ell\\in \\Hom(\\pi\\otimes \\pi^\\vee,\\C)$ by \n\\[\\ell(v\\otimes v^\\vee)=\\langle v, v^\\vee \\rangle\\] for $v \\in \\pi,\\ v^\\vee \\in \\pi^\\vee$. We use the Bernstein-Zelevinsky product notation $\\times$ for normalized parabolic induction.\n\nAn element $\\pi\\in \\Irr(G)$ is called generic if it satisfies the conditions of \\cite[Definition 2.3]{Sign}. We denote by $\\gen(G)$ the set of isomorphism classes of generic representations of $G$. In particular, for $\\pi\\in \\gen(G)$, its Jacquet-Langlands transfer $\\JL(\\pi)\\in \\gen(\\GL_n(F))$ is well-defined. Let $\\WD_F$ be the Weil-Deligne group of $F$. By definition, the L-parameter \n\\[\\phi_\\pi:\\WD_F\\to \\GL_n(\\C)\\] of $\\pi$ is that of its Jacquet-Langlands transfer $\\JL(\\pi)$ (\\cite{HT},\\cite{Hen},\\cite{DKV}), and we say that $\\pi$ is symplectic if $\\phi_\\pi(\\WD_F)\\subseteq \\Sp_n(\\C)$. We denote by $\\symp(G)$, resp. $\\sdg(G)$, the subset of $\\gen(G)$, the elements of which are symplectic, resp. self-dual; $\\pi\\in \\sdg(G)$ if $\\pi^\\vee= \\pi$. In particular $\\symp(G)\\subseteq \\sdg(G)$. When $F$ is $p$-adic and only in this case, we set $\\sympc(G)$ to be the cuspidal part of $\\symp(G)$. In particular in many statements below, if we mention cuspidal representations, it will be implicit that for this part of the statement, the field $F$ has to be $p$-adic.\n\n\\begin{lemma}\\label{lm:globlz}\nLet $\\pi\\in \\sympc(G)$. There exist:\n\\begin{enumerate}\n\\item a number fields $\\mathfrak{F}$ and a place $v_0$ such that $\\mathfrak{F}_{v_0}\\simeq F$,\n\\item an $\\mathfrak{F}$-central division algebra $\\mathfrak{D}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{D}_{v_0}/\\mathfrak{F}_{v_0}\\simeq D/F$, \n\\item $\\mathfrak{D}_{w_0}$ is a division algebra of Hasse invariant opposite to that of $D$, for some finite place $w_0\\neq v_0$ of $\\mathfrak{F}$ lying over an odd prime number,\n\\item $\\mathfrak{D}_v$ is a split matrix algebra for any place $v$ of $\\mathfrak{F}$ different from $v_0$ and $w_0$, \n\\end{enumerate}\n\\item a cuspidal automorphic representation $\\Pi$ of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that:\n\\begin{enumerate}\n\\item $\\JL(\\Pi)$ is cuspidal and symplectic,\n\\item $\\Pi_{v_0}\\simeq \\pi$,\n\\item $\\Pi_{w_0}$ is cuspidal and $E'$-distinguished for some quadratic extension $E'/\\mathfrak{F}_{w_0}$, \n\\end{enumerate} \n\\item a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{E}_{v_0}/\\mathfrak{F}_{v_0}\\simeq E/F$ and $\\mathfrak{E}_{w_0}/\\mathfrak{F}_{w_0} \\simeq E'/\\mathfrak{F}_{w_0}$,\n\\item $\\mathfrak{E}_v/\\mathfrak{F}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$ splits at the finite number of places $v\\notin \\{v_0,w_0\\}$ such that $\\pi_v$ is not isomorphic to a principal series induced from a Borel subgroup of $\\GL_m(\\mathfrak{D}_v)\\simeq \\GL_n(\\mathfrak{F}_v)$,\n\\item $\\mathfrak{E}/\\mathfrak{F}$ embeds into $\\CM_m(\\mathfrak{D})/\\mathfrak{F}$. \n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(1) The existence of $\\mathfrak{F}$ is well-known.\n\n\\begin{fact}\\label{fct}\nBy construction, the cuspidal automorphic representation $\\Pi$ in Lemma \\ref{lm:globlz} is such that $\\Pi_v$ is $\\mathfrak{E}_v$-distinguished \nfor all $v\\neq v_0$. \n\\end{fact}\n\\begin{proof}\nIt suffices to consider a place $v\\neq v_0,w_0$. Note that $\\mathfrak{D}_v$ is split by our choice of $\\mathfrak{D}$, and that $\\Pi_v=\\JL(\\Pi_v)=\\JL(\\Pi)_v$ is symplectic by the discussion at the beginning of Section \\ref{sec:conv}. Hence if $\\mathfrak{E}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$, the result follows from the equality $\\gadist(\\GL_n(\\mathfrak{F}_v))=\\symp(\\GL_n(\\mathfrak{F}_v))$, with $A=\\mathfrak{F}_v\\times \\mathfrak{F}_v$, recalled at the beginning of Section \\ref{sec:not}. If not, i.e. if $\\mathfrak{E}_v/\\mathfrak{F}_v$ is quadratic, then by assumption $\\Pi_v$ is a principal series induced from a Borel subgroup. The result then follows from \\cite[Theorem 10.4]{MOY}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nWe only need to prove that (ii) implies (i). We assume that $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$. Then Theorem \\ref{thm:key} asserts that $\\beta_{\\pi,E,\\psi}(0)=1$, further Theorem \\ref{thm:key1} implies that $J_{\\pi,E}(s)$ has a pole at $s=0$, and we conclude thanks to Proposition \\ref{prop:key} that $\\pi$ is $E$-distinguished. \n\\end{proof}",
    "post_theorem_intro_text_len": 7102,
    "post_theorem_intro_text": "The above statement was predicted by Prasad and Takloo-Bighash in \\cite[Conjecture 1]{PTB}, in a slightly more general form involving a character twist. The heart of the conjecture is its cuspidal case although the reduction from the discrete to the cuspidal case, done in \\cite{SX24}, uses local intertwining periods in a non trivial way. As written above, the general case follows from the cuspidal case by \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}. \n\nWe now review the techniques involved in the previous proof of the conjecture in the cuspidal case. The proof of the direct implication is based on establishing special cases of the direct implication of the Guo--Jacquet conjecture, via the global trace formula approach proposed by Guo in \\cite{Guo96}. Once this global conjecture established, the direct implication of the Prasad and Takloo-Bighash conjecture follows from an argument of Prasad in \\cite{Pr07}, as we shall recall in Section \\ref{sec:direct}. We observe in passing that the direct implication of the Guo-Jacquet conjecture is now a consequence of the recent \\cite[Theorem 1.4]{MOY}, established by a totally new method, which can be thought of as a global analogue of the approach developed in this paper. In \\cite{FMW}, Feigon, Martin and Whitehouse prove the direct implication of the split cuspidal case of Theorem \\ref{conj:PTB}, by establishing a special case of the Guo-Jacquet conjecture. More on the Guo-Jacquet conjecture is further established by Xue in \\cite{Xue21}, which provides two proofs of the direct implication in the cuspidal case. Xue also proves the converse implication in the cuspidal case, under the assumption $d\\leq 2$, by the global trace formula approach as well. Then, by type theoretic methods, S\\'echerre \\cite{Sec24} provides a full proof of the converse implication, under the assumption that $p\\neq 2$, which we remove here with our new proof. One common point of our proof of the converse implication and that of S\\'echerre, probably the only one, is that they rely on the direct implication.  \n\nTheorem \\ref{conj:PTB} should also follow from a local trace formula approach proving a special case of the conjecture in \\cite{Wan}, as explained and started by Suzuki in \\cite{Strace}. The advantage of this approach is that the dichotomy conjecture is put into a very general framework. On the other hand it seems to be expected that the remaining open problems there are technically highly challenging. \n\nOur method here looks much simpler, and also very natural. However, it uses two local statements from \\cite{MOY}, the proof of which is quite technical. \n\nThe statement of Theorem \\ref{conj:PTB} also makes sense over Archimedean local fields, and its more general version involving the character twist was recently settled by Tamori and Suzuki \\cite{ST23}.  \n\nAs a next step, following a strategy explained to us by Miyu Suzuki, we intend to deduce the conjecture for non Archimedean local fields of odd positive characteristic, via the theory of close local fields.\n\nLet's discuss and summarize the proof of the converse implication of Theorem \\ref{conj:PTB} in the cuspidal case. It was observed in \\cite{MatJFA} that the functional equation of local intertwining periods contains much information about distinction of quotients of induced representations, and this was used in \\cite{SX24} to reduce the discrete case of Theorem \\ref{conj:PTB} to its cuspidal case. Our main and new discovery, is that this functional equation actually also encodes most of the information about the distinction of cuspidal representations. Namely, let $\\pi$ be a cuspidal representation of $G$. It is well-known to experts that $\\pi$ is $E$-distinguished if and only if the open intertwining period \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]  has a pole at $ s=0$  (see Section \\ref{sec:signFE} for this fact, and Section \\ref{sec:OIP} for unexplained notations). \nNow assume that $\\pi$ has a symplectic Langlands parameter, so that in particular $\\pi=\\pi^\\vee$. Then by multiplicity at most one, we have a functional equation \n\\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s) J_{\\pi,E}(s),\\] where \n\\[M_{\\pi}(s):\\pi[s/2]\\times \\pi[-s/2]\\to \\pi[-s/2]\\times \\pi[s/2]\\] is the standard intertwining operator. One first main observation, proved in Theorem \\ref{thm:key1}, is that if we normalize $\\alpha_{\\pi,E}(s)$ by the Shahidi exterior and symmetric square gamma factors as \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\alpha_{\\pi,E}(s),\\] then $J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$, in other words:\n\\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation} It remains to compute $\\beta_{\\pi,E,\\psi}(0)$, which we do by a local to global argument in Section \\ref{sec:conv}. In order to achieve this, in Lemma \\ref{lm:globlz}, we globalize $E/F$ as the completion at an inert place $v_0$ of a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ of number fields, and with the help of new results of Takanashi and Wakatsuki \\cite{TW}, we globalize $\\pi$ as the local component $\\pi=\\Pi_{v_0}$ of a cuspidal automorphic representation $\\Pi$ with symplectic Jacquet-Langlands transfer, in such a way that $\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$ of $\\mathfrak{F}$. Then, in order to put the globalization argument to good use, we prove in Proposition \\ref{prop:sign dist} that the direct implication in \\eqref{eq:iff} holds as soon as $\\pi$ is generic and $G$ is $F$-split, even if we  allow $E=F\\times F$ (see Sections \\ref{sec:OIP} and \\ref{sec:signFE} for the precise meaning of this latter assertion). From this we deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$ of the number field globalizing $F$ in Corollary \\ref{cor:key}. The combination of Proposition \\ref{prop:sign dist} and Corollary \\ref{cor:key}, i.e. the formula for $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$, is our second main observation, and we appeal to the main local result of \\cite{MOY} in its proof, as well as to the direct implication of Theorem \\ref{conj:PTB}. Finally, the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$ being equal to one, as we prove in Lemma \\ref{lm:product of c is one} thanks to another result of \\cite{MOY}, we deduce in Theorem \\ref{thm:key} the expected formula for $\\beta_{\\pi,E,\\psi}(0)$. The final formula that we obtain is that if $\\pi$ is cuspidal with symplectic L-parameter:\n\\[\\beta_{\\pi,E,\\psi}(0)=(-1)^m\\e_E(\\phi_\\pi).\\] The converse implication of the cuspidal case of Theorem \\ref{conj:PTB} follows; it is Corollary \\ref{cor:main}.\n\n\\begin{Ac}\nWe thank Miyu Suzuki for sharing her insights and notes on the positive characteristic case, which served as an impetus for completing the proof in characteristic zero. The idea of the proof occurred to the author when he was visiting Huajie Li at the YMSC of Tsinghua university. We thank him and the YMSC for the hospitality. \n\\end{Ac}",
    "sketch": "For Theorem~\\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.\n\n- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \\cite{Sjnt}, \\cite{SX24}, \\cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.\n\n- **Key characterization via poles of open intertwining periods.** For cuspidal $\\pi$, it is “well-known” that $\\pi$ is $E$-distinguished **iff** the open intertwining period\n  \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]\n  “has a pole at $s=0$.”\n\n- **Functional equation and normalization.** Assuming $\\pi$ has symplectic parameter (so “in particular $\\pi=\\pi^\\vee$”), multiplicity-one gives a functional equation\n  \\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s)\\,J_{\\pi,E}(s),\\]\n  with $M_{\\pi}(s)$ the standard intertwining operator. The first main observation (Theorem~\\ref{thm:key1}) is that after normalizing\n  \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\,\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\,\\alpha_{\\pi,E}(s),\\]\n  one has\n  \\[\\pi\\ \\text{distinguished}\\iff \\beta_{\\pi,E,\\psi}(0)=1\\tag{\\ref{eq:iff}}\\]\n  i.e. “$J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$.”\n\n- **Compute $\\beta_{\\pi,E,\\psi}(0)$ by local-to-global.** “It remains to compute $\\beta_{\\pi,E,\\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\\mathfrak{E}/\\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \\cite{TW}) globalize $\\pi$ as $\\pi=\\Pi_{v_0}$ for a cuspidal automorphic $\\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$.”\n\n- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove (Proposition~\\ref{prop:sign dist}) that the direct implication in \\eqref{eq:iff} holds when “$\\pi$ is generic and $G$ is $F$-split,” even allowing $E=F\\times F$. From this they “deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$” (Corollary~\\ref{cor:key}). This combination is called the “second main observation” and it uses “the main local result of \\cite{MOY} … as well as … the direct implication of Theorem~\\ref{conj:PTB}.”\n\n- **Global product formula forces the local value at $v_0$.** They show “the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$” equals $1$ (Lemma~\\ref{lm:product of c is one}, using another result of \\cite{MOY}); hence one solves for $\\beta_{\\pi,E,\\psi}(0)$ and obtains in Theorem~\\ref{thm:key} the formula (for cuspidal $\\pi$ with symplectic parameter)\n  \\[\\beta_{\\pi,E,\\psi}(0)=(-1)^m\\,\\e_E(\\phi_\\pi).\\]\n  Combined with \\eqref{eq:iff}, this yields the “converse implication of the cuspidal case of Theorem~\\ref{conj:PTB}” (Corollary~\\ref{cor:main}).",
    "expanded_sketch": "For Theorem~\\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.\n\n- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \\cite{Sjnt}, \\cite{SX24}, \\cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.\n\n- **Key characterization via poles of open intertwining periods.** For cuspidal $\\pi$, it is “well-known” that $\\pi$ is $E$-distinguished **iff** the open intertwining period\n  \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]\n  “has a pole at $s=0$.”\n\n- **Functional equation and normalization.** Assuming $\\pi$ has symplectic parameter (so “in particular $\\pi=\\pi^\\vee$”), multiplicity-one gives a functional equation\n  \\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s)\\,J_{\\pi,E}(s),\\]\n  with $M_{\\pi}(s)$ the standard intertwining operator. We first use the following theorem.\n\n  \\begin{theorem}\\label{thm:key1}\nLet $\\pi\\in\\cusp(G)$ be symplectic. Assume that $A/F=E/F$ is quadratic. Then:\n\\begin{enumerate}\n    \\item if $J_{\\pi,E}(s)$ has a pole at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=1$,\n    \\item if $J_{\\pi,E}(s)$ is holomorphic at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=-1$. \n    \\end{enumerate} \n\\end{theorem}\n\n  More precisely, after normalizing\n  \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\,\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\,\\alpha_{\\pi,E}(s),\\]\n  one has the key equivalence\n  \\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation}\n  i.e. “$J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$.”\n\n- **Compute $\\beta_{\\pi,E,\\psi}(0)$ by local-to-global.** “It remains to compute $\\beta_{\\pi,E,\\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\\mathfrak{E}/\\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \\cite{TW}) globalize $\\pi$ as $\\pi=\\Pi_{v_0}$ for a cuspidal automorphic $\\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$.”\n\n- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove the following proposition.\n\n  \\begin{proposition}\\label{prop:sign dist}\nAssume that either $G=\\GL_n(F)$ is split and $\\pi\\in \\gadist(G)$, or that $\\pi\\in \\cadist(G)$, then $\\beta_{\\pi,A,\\psi}(0)=1$. \n\\end{proposition}\n\n  From this they “deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$” via the following corollary.\n\n  \\begin{corollary}\\label{cor:key}\nLet $\\pi$ be as in Proposition \\ref{prop:sign dist}, but assume moreover that it is unitary, and let $c(\\pi,\\psi)$ be as in Theorem \\ref{thm:almost explicit constant}. Then $c(\\pi,\\psi)=(-1)^m\\eta_{A/F}(-1)^{n/2}$.\n\\end{corollary}\n\n  This combination is called the “second main observation” and it uses “the main local result of \\cite{MOY} … as well as … the direct implication in establishing the main theorem.”\n\n- **Global product formula forces the local value at $v_0$.** They show “the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$” equals $1$ by the following lemma.\n\n  \\begin{lemma}\\label{lm:product of c is one}\nLet $\\mathfrak{E}/\\mathfrak{F}$ be a quadratic extension of number fields, and $\\Psi:\\mathfrak{F}\\backslash \\A_\\mathfrak{F}\\to \\C^\\times$ be a non trivial character. Let $\\mathfrak{D}$ be an $\\mathfrak{F}$-central division algebra. Assume moreover that \n$\\mathfrak{E}$ is contained inside $\\CM_m(\\mathfrak{D})$. \n If $\\Pi$ is a self-dual cuspidal automorphic representation of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that $\\JL(\\Pi)$ is cuspidal, then \\[\\prod_v c(\\Pi_v,\\Psi_v)=1,\\] where the product is over all places of $F$. More specifically \\[\\prod_{v\\in S} c(\\Pi_v,\\Psi_v)=1,\\] where $S$ is the set of places such that of $\\mathfrak{F}$ outside of which $\\mathfrak{D}_v$ is split, and $\\mathfrak{E}_v/\\mathfrak{F}_v$, $\\pi_v$ and $\\Psi_v$ are all unramified.\n\\end{lemma}\n\n  Hence one solves for $\\beta_{\\pi,E,\\psi}(0)$ and obtains the value asserted in the following theorem:\n  \\begin{theorem}\\label{thm:key}\nLet $\\pi\\in \\sympc(G)$. Then \n\\[c(\\pi,\\psi)=(-1)^{m}\\eta_{E/F}(-1)^{n/2}.\\]\n\\end{theorem}\n  Together with the equation above \\eqref{eq:iff}, this yields the “converse implication of the cuspidal case in establishing the main theorem,” formulated as:\n\n  \\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}\n",
    "expanded_theorem": "\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\mathbb{C})$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\mathbb{C})$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $F$ be a finite extension of $\\mathbb{Q}_p$, let $D$ be a central division $F$-algebra of dimension $d^2$, let $m\\ge 1$, set $G=\\mathrm{GL}_m(D)$, and write $n=md$. Assume $n$ is even, so that a quadratic extension $E/F$ embeds in $\\mathrm{M}_m(D)$. Let $H$ be the centralizer of $E^\\times$ in $G$, and say that a smooth representation $\\pi$ of $G$ is $E$-distinguished if $\\operatorname{Hom}_H(\\pi,\\mathbb{C})\\ne 0$. Let $\\pi$ be a generic representation of $G$, and let $\\phi=\\phi_\\pi:\\mathrm{WD}_F\\to \\mathrm{GL}_n(\\mathbb{C})$ be its $L$-parameter, defined via the Jacquet--Langlands transfer. For any symplectic $L$-parameter $\\theta:\\mathrm{WD}_F\\to \\mathrm{Sp}_r(\\mathbb{C})$, define\n\\[\n\\varepsilon_E(\\theta):=\\eta_{E/F}(-1)^{r/2}\\,\\epsilon\\!\\left(\\tfrac12,\\theta_{|\\mathrm{WD}_E},\\psi_E\\right)\n=\\eta_{E/F}(-1)^{r/2}\\,\\epsilon\\!\\left(\\tfrac12,\\theta,\\psi\\right)\\epsilon\\!\\left(\\tfrac12,\\eta_{E/F}\\otimes\\theta,\\psi\\right),\n\\]\nwhere $\\eta_{E/F}$ is the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$, $\\psi:F\\to\\mathbb{C}^\\times$ is nontrivial, and $\\psi_E=\\psi\\circ\\operatorname{tr}_{E/F}$. Suppose further, when applicable, that $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and decomposes as\n\\[\n\\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n\\]\nwith $I_\\phi,J_\\phi$ finite, each $\\phi_k:\\mathrm{WD}_F\\to\\mathrm{GL}_{n_k}(\\mathbb{C})$ a discrete $L$-parameter, each $n_i$ even and each $\\phi_i$ taking values in $\\mathrm{Sp}_{n_i}(\\mathbb{C})$ for $i\\in I_\\phi$, and $\\phi_i\\not\\simeq \\phi_{i'}$ whenever $i\\ne i'$. Which statement about $E$-distinction holds?",
      "correct_choice": {
        "label": "A",
        "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "C",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished whenever\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "D",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if there exists a choice of nontrivial character $\\psi:F\\to\\mathbb{C}^\\times$ such that\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "E",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_k)=(-1)^{n_k/d}\\qquad\\text{for every }k\\in I_\\phi\\sqcup J_\\phi.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "sign exponent depends on $n_i/d$, not $n_i$",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the converse direction of the criterion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "$\\varepsilon_E(\\theta)$ is independent of $\\psi$ for symplectic parameters; no existential choice of $\\psi$ is needed",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "criterion only constrains the symplectic self-dual summands indexed by $I_\\phi$, not the paired summands in $J_\\phi$",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the setup and relevant notation, but it does not state or strongly hint at the specific criterion \\(\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\). The correct answer is not leaked by the wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: it asks for the statement equivalent to \\(E\\)-distinction and the correct option is the theorem's exact criterion. It does not substantially reframe the result into a new problem."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish between subtle variants involving exponents, quantifiers, and uniform sign twists, so careful reading is needed. However, the item primarily rewards memory of the exact theorem rather than independent mathematical generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic errors: confusing \\(n_i\\) with \\(n_i/d\\), weakening a universal condition to existence, inserting an unjustified global sign, or replacing the blockwise criterion with an ambient parity condition."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall question with strong distractors and no answer leakage, but it is largely tautological and only moderately tests reasoning."
    }
  },
  {
    "id": "2512.19831v2",
    "paper_link": "http://arxiv.org/abs/2512.19831v2",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\mathbb{C})$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\mathbb{C})$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}",
      "start_pos": 30242,
      "end_pos": 31127,
      "label": "conj:PTB"
    },
    "ref_dict": {
      "thm:key1": "\\begin{theorem}\\label{thm:key1}\nLet $\\pi\\in\\cusp(G)$ be symplectic. Assume that $A/F=E/F$ is quadratic. Then:\n\\begin{enumerate}\n    \\item if $J_{\\pi,E}(s)$ has a pole at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=1$,\n    \\item if $J_{\\pi,E}(s)$ is holomorphic at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=-1$. \n    \\end{enumerate} \n\\end{theorem}",
      "prop:sign dist": "\\begin{proposition}\\label{prop:sign dist}\nAssume that either $G=\\GL_n(F)$ is split and $\\pi\\in \\gadist(G)$, or that $\\pi\\in \\cadist(G)$, then $\\beta_{\\pi,A,\\psi}(0)=1$. \n\\end{proposition}",
      "eq:iff": "\\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation}",
      "lm:product of c is one": "\\begin{lemma}\\label{lm:product of c is one}\nLet $\\mathfrak{E}/\\mathfrak{F}$ be a quadratic extension of number fields, and $\\Psi:\\mathfrak{F}\\backslash \\A_\\mathfrak{F}\\to \\C^\\times$ be a non trivial character. Let $\\mathfrak{D}$ be an $\\mathfrak{F}$-central division algebra. Assume moreover that \n$\\mathfrak{E}$ is contained inside $\\CM_m(\\mathfrak{D})$. \n If $\\Pi$ is a self-dual cuspidal automorphic representation of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that $\\JL(\\Pi)$ is cuspidal, then \\[\\prod_v c(\\Pi_v,\\Psi_v)=1,\\] where the product is over all places of $F$. More specifically \\[\\prod_{v\\in S} c(\\Pi_v,\\Psi_v)=1,\\] where $S$ is the set of places such that of $\\mathfrak{F}$ outside of which $\\mathfrak{D}_v$ is split, and $\\mathfrak{E}_v/\\mathfrak{F}_v$, $\\pi_v$ and $\\Psi_v$ are all unramified.\n\\end{lemma}",
      "cor:main": "\\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}",
      "lm:globlz": "\\begin{lemma}\\label{lm:globlz}\nLet $\\pi\\in \\sympc(G)$. There exist:\n\\begin{enumerate}\n\\item a number fields $\\mathfrak{F}$ and a place $v_0$ such that $\\mathfrak{F}_{v_0}\\simeq F$,\n\\item an $\\mathfrak{F}$-central division algebra $\\mathfrak{D}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{D}_{v_0}/\\mathfrak{F}_{v_0}\\simeq D/F$, \n\\item $\\mathfrak{D}_{w_0}$ is a division algebra of Hasse invariant opposite to that of $D$, for some finite place $w_0\\neq v_0$ of $\\mathfrak{F}$ lying over an odd prime number,\n\\item $\\mathfrak{D}_v$ is a split matrix algebra for any place $v$ of $\\mathfrak{F}$ different from $v_0$ and $w_0$, \n\\end{enumerate}\n\\item a cuspidal automorphic representation $\\Pi$ of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that:\n\\begin{enumerate}\n\\item $\\JL(\\Pi)$ is cuspidal and symplectic,\n\\item $\\Pi_{v_0}\\simeq \\pi$,\n\\item $\\Pi_{w_0}$ is cuspidal and $E'$-distinguished for some quadratic extension $E'/\\mathfrak{F}_{w_0}$, \n\\end{enumerate} \n\\item a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{E}_{v_0}/\\mathfrak{F}_{v_0}\\simeq E/F$ and $\\mathfrak{E}_{w_0}/\\mathfrak{F}_{w_0} \\simeq E'/\\mathfrak{F}_{w_0}$,\n\\item $\\mathfrak{E}_v/\\mathfrak{F}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$ splits at the finite number of places $v\\notin \\{v_0,w_0\\}$ such that $\\pi_v$ is not isomorphic to a principal series induced from a Borel subgroup of $\\GL_m(\\mathfrak{D}_v)\\simeq \\GL_n(\\mathfrak{F}_v)$,\n\\item $\\mathfrak{E}/\\mathfrak{F}$ embeds into $\\CM_m(\\mathfrak{D})/\\mathfrak{F}$. \n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}",
      "thm:key": "\\begin{theorem}\\label{thm:key}\nLet $\\pi\\in \\sympc(G)$. Then \n\\[c(\\pi,\\psi)=(-1)^{m}\\eta_{E/F}(-1)^{n/2}.\\]\n\\end{theorem}",
      "conj:PTB": "\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}",
      "cor:key": "\\begin{corollary}\\label{cor:key}\nLet $\\pi$ be as in Proposition \\ref{prop:sign dist}, but assume moreover that it is unitary, and let $c(\\pi,\\psi)$ be as in Theorem \\ref{thm:almost explicit constant}. Then $c(\\pi,\\psi)=(-1)^m\\eta_{A/F}(-1)^{n/2}$.\n\\end{corollary}"
    },
    "pre_theorem_intro_text_len": 2392,
    "pre_theorem_intro_text": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi,  \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi,  \\mathbb{C})\\neq 0$.  \n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper. \n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.",
    "context": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi, \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi, \\mathbb{C})\\neq 0$.\n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper.\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\n\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}",
    "full_context": "Let $F$ be a a finite extension of $\\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\\eta_{E/F}$ be the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$. \nAssume that $n:=md$ is even so that $E$ embeds in $\\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.\nLet $H$ denote the centralizer of $E^\\times$ in $G$.\nWe say that a smooth representation $\\pi$ of $G$ is \\emph{$E$-distinguished} if $\\Hom_H(\\pi, \\mathbb{C})\\neq 0$. We also say that $\\pi$ is \\emph{$F\\times F$-distinguished} if $m$ is even and $\\Hom_{\\GL_{m/2}(D)\\times \\GL_{m/2}(D)}(\\pi, \\mathbb{C})\\neq 0$.\n\nWhen $p>2$, Theorem \\ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \\cite{Sai} and \\cite{Tu} for local toric periods on $\\GL_2(F)$, follows from the works of Suzuki \\cite{Sjnt}, Xue \\cite{Xue21}, S\\'echerre \\cite{Sec24}, Suzuki and Xue \\cite{SX24}, with an input from \\cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper.\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\mathrm{JL}(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\mathrm{JL}(\\pi)}:\\WD_F\\to \\GL_n(\\mathbb{C}).\\] Let $\\psi:F\\to \\mathbb{C}^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\mathbb{C})$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\n\\begin{theorem}\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\C)$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\C)$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\C)$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\C)$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}\n\\end{theorem}\n\nWe refer to \\cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\\JL(\\pi)$, which is a generic representation of $\\GL_n(F)$, and we define the L-parameter $\\phi_\\pi$ of $\\pi$ by setting \\[\\phi_\\pi:=\\phi_{\\JL(\\pi)}:\\WD_F\\to \\GL_n(\\C).\\] Let $\\psi:F\\to \\C^\\times$ be a non trivial character, and $\\psi_E:=\\psi\\circ \\tr_{E/F}$. If $\\phi:\\WD_F\\to \\Sp_n(\\C)$ is a symplectic L-parameter, we set \\[\\e_E(\\phi):=\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi_{|\\WD_E},\\psi_E)=\n\\eta_{E/F}(-1)^{n/2}\\epsilon(1/2,\\phi,\\psi)\\epsilon(1/2,\\eta_{E/F}\\otimes\\phi,\\psi),\\] where the $\\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \\cite{Tate}. This quantity does not depend on the choice of the character $\\psi$ since $\\phi$ is symplectic.\n\nLet $F$ be a local field of characteristic zero, possibly Archimedean. We fix a nontrivial character $\\psi:F\\to \\C^\\times$. Let $A$ be an $F$-separable algebra of dimension two, i.e. either $A/F$ is a quadratic extension, of $A\\simeq F\\times F$. We set $\\psi_A:=\\psi\\circ \\tr_{A/F}$, and denote by $\\eta_{A/F}$ the character of $F^\\times$ with kernel the norms of $A^\\times$, in particular $\\eta_{A/F}$ has order two when $A/F$ is quadratic, whereas it is trivial when $A\\simeq F\\times F$. We note that $A$ can be written $F[u]$, with $\\tr_{A/F}(u)=0$, i.e. $u^2\\in F$. Such an element $u$ is unique up to scaling by an element in $F^\\times$. Let $D$ be a central division algebra over $F$, such that $[D:F]=d^2$ with $d\\geq 1$. We set $\\CM:=\\CM_m(D)$ for $m\\geq 1$, and $n:=md$. We put $\\psi_{\\CM}:=\\psi\\circ \\tr_{\\CM/F}$. We assume that $A$ embeds in $\\CM$ as an $F$-subalgebra, in such a way that $\\tr_{\\CM/F}(u)=0$, and we realize the embedding explicitly, as explained in \\cite[Section 2.5]{MOY}. In particular $n$ is even, and if $A/F$ is quadratic, this parity condition is also sufficient for the assumption $\\tr_{\\CM/F}(u)=0$ to hold. If $A\\simeq F\\times F$, the assumption holds if and only if $m$ is even and $u$ is conjugate to $\\lambda\\diag(I_{m/2},-I_{m/2})$ with $\\lambda\\in F^\\times$. We set $G_m=G=\\CM^\\times$, and $H_{A,m}=H_A=Z_{\\CM}(A)^\\times$. The group $H_A$ is the subgroup of $G$ fixed by the inner involution $\\theta:=\\ad(u)$ of $G$. We denote by $\\nu:G\\to \\C^\\times$ the restriction of the composition of the normalized absolute value $|\\ |$ on $F$ with the reduced norm on $\\CM$. We denote by $P_{m,m}$ the block upper triangular parabolic subgroup of $G_{2m}$ corresponding to the partitition $(m,m)$, and by $M_{m,m}$ its block diagonal Levi subgroup. We denote by $x_0$ the representative of the unique open double coset $Px_0H$ fixed in \\cite[Section 4.1]{MOY}. In particular the involution $\\theta_o(g):=x_o\\theta(x_o^{-1}gx_o)x_o^{-1}$ of $G_{2m}$ is such that $\\theta_o(P_{m,m})$ is opposite to $P_{m,m}$ with respect to $M_{m,m}$. We denote by $\\FL(G)$ the objects of the category of smooth admissible representations of $G$ of finite length, where admissible means admissible (yes) but also Fr\\'echet and of moderate growth, when $F$ is Archimedean (see \\cite[Chapter 11]{RRG2}). For $\\pi\\in \\FL(G)$, we write $\\pi^\\vee$ for its smooth contragredient, and we define $\\ell\\in \\Hom(\\pi\\otimes \\pi^\\vee,\\C)$ by \n\\[\\ell(v\\otimes v^\\vee)=\\langle v, v^\\vee \\rangle\\] for $v \\in \\pi,\\ v^\\vee \\in \\pi^\\vee$. We use the Bernstein-Zelevinsky product notation $\\times$ for normalized parabolic induction.\n\nAn element $\\pi\\in \\Irr(G)$ is called generic if it satisfies the conditions of \\cite[Definition 2.3]{Sign}. We denote by $\\gen(G)$ the set of isomorphism classes of generic representations of $G$. In particular, for $\\pi\\in \\gen(G)$, its Jacquet-Langlands transfer $\\JL(\\pi)\\in \\gen(\\GL_n(F))$ is well-defined. Let $\\WD_F$ be the Weil-Deligne group of $F$. By definition, the L-parameter \n\\[\\phi_\\pi:\\WD_F\\to \\GL_n(\\C)\\] of $\\pi$ is that of its Jacquet-Langlands transfer $\\JL(\\pi)$ (\\cite{HT},\\cite{Hen},\\cite{DKV}), and we say that $\\pi$ is symplectic if $\\phi_\\pi(\\WD_F)\\subseteq \\Sp_n(\\C)$. We denote by $\\symp(G)$, resp. $\\sdg(G)$, the subset of $\\gen(G)$, the elements of which are symplectic, resp. self-dual; $\\pi\\in \\sdg(G)$ if $\\pi^\\vee= \\pi$. In particular $\\symp(G)\\subseteq \\sdg(G)$. When $F$ is $p$-adic and only in this case, we set $\\sympc(G)$ to be the cuspidal part of $\\symp(G)$. In particular in many statements below, if we mention cuspidal representations, it will be implicit that for this part of the statement, the field $F$ has to be $p$-adic.\n\n\\begin{lemma}\\label{lm:globlz}\nLet $\\pi\\in \\sympc(G)$. There exist:\n\\begin{enumerate}\n\\item a number fields $\\mathfrak{F}$ and a place $v_0$ such that $\\mathfrak{F}_{v_0}\\simeq F$,\n\\item an $\\mathfrak{F}$-central division algebra $\\mathfrak{D}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{D}_{v_0}/\\mathfrak{F}_{v_0}\\simeq D/F$, \n\\item $\\mathfrak{D}_{w_0}$ is a division algebra of Hasse invariant opposite to that of $D$, for some finite place $w_0\\neq v_0$ of $\\mathfrak{F}$ lying over an odd prime number,\n\\item $\\mathfrak{D}_v$ is a split matrix algebra for any place $v$ of $\\mathfrak{F}$ different from $v_0$ and $w_0$, \n\\end{enumerate}\n\\item a cuspidal automorphic representation $\\Pi$ of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that:\n\\begin{enumerate}\n\\item $\\JL(\\Pi)$ is cuspidal and symplectic,\n\\item $\\Pi_{v_0}\\simeq \\pi$,\n\\item $\\Pi_{w_0}$ is cuspidal and $E'$-distinguished for some quadratic extension $E'/\\mathfrak{F}_{w_0}$, \n\\end{enumerate} \n\\item a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ such that:\n\\begin{enumerate}\n\\item $\\mathfrak{E}_{v_0}/\\mathfrak{F}_{v_0}\\simeq E/F$ and $\\mathfrak{E}_{w_0}/\\mathfrak{F}_{w_0} \\simeq E'/\\mathfrak{F}_{w_0}$,\n\\item $\\mathfrak{E}_v/\\mathfrak{F}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$ splits at the finite number of places $v\\notin \\{v_0,w_0\\}$ such that $\\pi_v$ is not isomorphic to a principal series induced from a Borel subgroup of $\\GL_m(\\mathfrak{D}_v)\\simeq \\GL_n(\\mathfrak{F}_v)$,\n\\item $\\mathfrak{E}/\\mathfrak{F}$ embeds into $\\CM_m(\\mathfrak{D})/\\mathfrak{F}$. \n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(1) The existence of $\\mathfrak{F}$ is well-known.\n\n\\begin{fact}\\label{fct}\nBy construction, the cuspidal automorphic representation $\\Pi$ in Lemma \\ref{lm:globlz} is such that $\\Pi_v$ is $\\mathfrak{E}_v$-distinguished \nfor all $v\\neq v_0$. \n\\end{fact}\n\\begin{proof}\nIt suffices to consider a place $v\\neq v_0,w_0$. Note that $\\mathfrak{D}_v$ is split by our choice of $\\mathfrak{D}$, and that $\\Pi_v=\\JL(\\Pi_v)=\\JL(\\Pi)_v$ is symplectic by the discussion at the beginning of Section \\ref{sec:conv}. Hence if $\\mathfrak{E}_v\\simeq \\mathfrak{F}_v\\times \\mathfrak{F}_v$, the result follows from the equality $\\gadist(\\GL_n(\\mathfrak{F}_v))=\\symp(\\GL_n(\\mathfrak{F}_v))$, with $A=\\mathfrak{F}_v\\times \\mathfrak{F}_v$, recalled at the beginning of Section \\ref{sec:not}. If not, i.e. if $\\mathfrak{E}_v/\\mathfrak{F}_v$ is quadratic, then by assumption $\\Pi_v$ is a principal series induced from a Borel subgroup. The result then follows from \\cite[Theorem 10.4]{MOY}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nWe only need to prove that (ii) implies (i). We assume that $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$. Then Theorem \\ref{thm:key} asserts that $\\beta_{\\pi,E,\\psi}(0)=1$, further Theorem \\ref{thm:key1} implies that $J_{\\pi,E}(s)$ has a pole at $s=0$, and we conclude thanks to Proposition \\ref{prop:key} that $\\pi$ is $E$-distinguished. \n\\end{proof}",
    "post_theorem_intro_text_len": 7102,
    "post_theorem_intro_text": "The above statement was predicted by Prasad and Takloo-Bighash in \\cite[Conjecture 1]{PTB}, in a slightly more general form involving a character twist. The heart of the conjecture is its cuspidal case although the reduction from the discrete to the cuspidal case, done in \\cite{SX24}, uses local intertwining periods in a non trivial way. As written above, the general case follows from the cuspidal case by \\cite{Sjnt}, \\cite{SX24}, and \\cite{Ch}. \n\nWe now review the techniques involved in the previous proof of the conjecture in the cuspidal case. The proof of the direct implication is based on establishing special cases of the direct implication of the Guo--Jacquet conjecture, via the global trace formula approach proposed by Guo in \\cite{Guo96}. Once this global conjecture established, the direct implication of the Prasad and Takloo-Bighash conjecture follows from an argument of Prasad in \\cite{Pr07}, as we shall recall in Section \\ref{sec:direct}. We observe in passing that the direct implication of the Guo-Jacquet conjecture is now a consequence of the recent \\cite[Theorem 1.4]{MOY}, established by a totally new method, which can be thought of as a global analogue of the approach developed in this paper. In \\cite{FMW}, Feigon, Martin and Whitehouse prove the direct implication of the split cuspidal case of Theorem \\ref{conj:PTB}, by establishing a special case of the Guo-Jacquet conjecture. More on the Guo-Jacquet conjecture is further established by Xue in \\cite{Xue21}, which provides two proofs of the direct implication in the cuspidal case. Xue also proves the converse implication in the cuspidal case, under the assumption $d\\leq 2$, by the global trace formula approach as well. Then, by type theoretic methods, S\\'echerre \\cite{Sec24} provides a full proof of the converse implication, under the assumption that $p\\neq 2$, which we remove here with our new proof. One common point of our proof of the converse implication and that of S\\'echerre, probably the only one, is that they rely on the direct implication.  \n\nTheorem \\ref{conj:PTB} should also follow from a local trace formula approach proving a special case of the conjecture in \\cite{Wan}, as explained and started by Suzuki in \\cite{Strace}. The advantage of this approach is that the dichotomy conjecture is put into a very general framework. On the other hand it seems to be expected that the remaining open problems there are technically highly challenging. \n\nOur method here looks much simpler, and also very natural. However, it uses two local statements from \\cite{MOY}, the proof of which is quite technical. \n\nThe statement of Theorem \\ref{conj:PTB} also makes sense over Archimedean local fields, and its more general version involving the character twist was recently settled by Tamori and Suzuki \\cite{ST23}.  \n\nAs a next step, following a strategy explained to us by Miyu Suzuki, we intend to deduce the conjecture for non Archimedean local fields of odd positive characteristic, via the theory of close local fields.\n\nLet's discuss and summarize the proof of the converse implication of Theorem \\ref{conj:PTB} in the cuspidal case. It was observed in \\cite{MatJFA} that the functional equation of local intertwining periods contains much information about distinction of quotients of induced representations, and this was used in \\cite{SX24} to reduce the discrete case of Theorem \\ref{conj:PTB} to its cuspidal case. Our main and new discovery, is that this functional equation actually also encodes most of the information about the distinction of cuspidal representations. Namely, let $\\pi$ be a cuspidal representation of $G$. It is well-known to experts that $\\pi$ is $E$-distinguished if and only if the open intertwining period \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]  has a pole at $ s=0$  (see Section \\ref{sec:signFE} for this fact, and Section \\ref{sec:OIP} for unexplained notations). \nNow assume that $\\pi$ has a symplectic Langlands parameter, so that in particular $\\pi=\\pi^\\vee$. Then by multiplicity at most one, we have a functional equation \n\\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s) J_{\\pi,E}(s),\\] where \n\\[M_{\\pi}(s):\\pi[s/2]\\times \\pi[-s/2]\\to \\pi[-s/2]\\times \\pi[s/2]\\] is the standard intertwining operator. One first main observation, proved in Theorem \\ref{thm:key1}, is that if we normalize $\\alpha_{\\pi,E}(s)$ by the Shahidi exterior and symmetric square gamma factors as \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\alpha_{\\pi,E}(s),\\] then $J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$, in other words:\n\\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation} It remains to compute $\\beta_{\\pi,E,\\psi}(0)$, which we do by a local to global argument in Section \\ref{sec:conv}. In order to achieve this, in Lemma \\ref{lm:globlz}, we globalize $E/F$ as the completion at an inert place $v_0$ of a quadratic extension $\\mathfrak{E}/\\mathfrak{F}$ of number fields, and with the help of new results of Takanashi and Wakatsuki \\cite{TW}, we globalize $\\pi$ as the local component $\\pi=\\Pi_{v_0}$ of a cuspidal automorphic representation $\\Pi$ with symplectic Jacquet-Langlands transfer, in such a way that $\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$ of $\\mathfrak{F}$. Then, in order to put the globalization argument to good use, we prove in Proposition \\ref{prop:sign dist} that the direct implication in \\eqref{eq:iff} holds as soon as $\\pi$ is generic and $G$ is $F$-split, even if we  allow $E=F\\times F$ (see Sections \\ref{sec:OIP} and \\ref{sec:signFE} for the precise meaning of this latter assertion). From this we deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$ of the number field globalizing $F$ in Corollary \\ref{cor:key}. The combination of Proposition \\ref{prop:sign dist} and Corollary \\ref{cor:key}, i.e. the formula for $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$, is our second main observation, and we appeal to the main local result of \\cite{MOY} in its proof, as well as to the direct implication of Theorem \\ref{conj:PTB}. Finally, the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$ being equal to one, as we prove in Lemma \\ref{lm:product of c is one} thanks to another result of \\cite{MOY}, we deduce in Theorem \\ref{thm:key} the expected formula for $\\beta_{\\pi,E,\\psi}(0)$. The final formula that we obtain is that if $\\pi$ is cuspidal with symplectic L-parameter:\n\\[\\beta_{\\pi,E,\\psi}(0)=(-1)^m\\e_E(\\phi_\\pi).\\] The converse implication of the cuspidal case of Theorem \\ref{conj:PTB} follows; it is Corollary \\ref{cor:main}.\n\n\\begin{Ac}\nWe thank Miyu Suzuki for sharing her insights and notes on the positive characteristic case, which served as an impetus for completing the proof in characteristic zero. The idea of the proof occurred to the author when he was visiting Huajie Li at the YMSC of Tsinghua university. We thank him and the YMSC for the hospitality. \n\\end{Ac}",
    "sketch": "For Theorem~\\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.\n\n- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \\cite{Sjnt}, \\cite{SX24}, \\cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.\n\n- **Key characterization via poles of open intertwining periods.** For cuspidal $\\pi$, it is “well-known” that $\\pi$ is $E$-distinguished **iff** the open intertwining period\n  \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]\n  “has a pole at $s=0$.”\n\n- **Functional equation and normalization.** Assuming $\\pi$ has symplectic parameter (so “in particular $\\pi=\\pi^\\vee$”), multiplicity-one gives a functional equation\n  \\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s)\\,J_{\\pi,E}(s),\\]\n  with $M_{\\pi}(s)$ the standard intertwining operator. The first main observation (Theorem~\\ref{thm:key1}) is that after normalizing\n  \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\,\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\,\\alpha_{\\pi,E}(s),\\]\n  one has\n  \\[\\pi\\ \\text{distinguished}\\iff \\beta_{\\pi,E,\\psi}(0)=1\\tag{\\ref{eq:iff}}\\]\n  i.e. “$J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$.”\n\n- **Compute $\\beta_{\\pi,E,\\psi}(0)$ by local-to-global.** “It remains to compute $\\beta_{\\pi,E,\\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\\mathfrak{E}/\\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \\cite{TW}) globalize $\\pi$ as $\\pi=\\Pi_{v_0}$ for a cuspidal automorphic $\\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$.”\n\n- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove (Proposition~\\ref{prop:sign dist}) that the direct implication in \\eqref{eq:iff} holds when “$\\pi$ is generic and $G$ is $F$-split,” even allowing $E=F\\times F$. From this they “deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$” (Corollary~\\ref{cor:key}). This combination is called the “second main observation” and it uses “the main local result of \\cite{MOY} … as well as … the direct implication of Theorem~\\ref{conj:PTB}.”\n\n- **Global product formula forces the local value at $v_0$.** They show “the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$” equals $1$ (Lemma~\\ref{lm:product of c is one}, using another result of \\cite{MOY}); hence one solves for $\\beta_{\\pi,E,\\psi}(0)$ and obtains in Theorem~\\ref{thm:key} the formula (for cuspidal $\\pi$ with symplectic parameter)\n  \\[\\beta_{\\pi,E,\\psi}(0)=(-1)^m\\,\\e_E(\\phi_\\pi).\\]\n  Combined with \\eqref{eq:iff}, this yields the “converse implication of the cuspidal case of Theorem~\\ref{conj:PTB}” (Corollary~\\ref{cor:main}).",
    "expanded_sketch": "For Theorem~\\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.\n\n- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \\cite{Sjnt}, \\cite{SX24}, \\cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.\n\n- **Key characterization via poles of open intertwining periods.** For cuspidal $\\pi$, it is “well-known” that $\\pi$ is $E$-distinguished **iff** the open intertwining period\n  \\[J_{\\pi,E}(s):\\pi[s/2]\\times \\pi^\\vee[-s/2]\\to \\mathbb{C}\\]\n  “has a pole at $s=0$.”\n\n- **Functional equation and normalization.** Assuming $\\pi$ has symplectic parameter (so “in particular $\\pi=\\pi^\\vee$”), multiplicity-one gives a functional equation\n  \\[J_{\\pi,E}(-s)\\circ M_{\\pi}(s)=\\alpha_{\\pi,E}(s)\\,J_{\\pi,E}(s),\\]\n  with $M_{\\pi}(s)$ the standard intertwining operator. We first use the following theorem.\n\n  \\begin{theorem}\\label{thm:key1}\nLet $\\pi\\in\\cusp(G)$ be symplectic. Assume that $A/F=E/F$ is quadratic. Then:\n\\begin{enumerate}\n    \\item if $J_{\\pi,E}(s)$ has a pole at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=1$,\n    \\item if $J_{\\pi,E}(s)$ is holomorphic at $s=0$, then $\\beta_{\\pi,E,\\psi}(0)=-1$. \n    \\end{enumerate} \n\\end{theorem}\n\n  More precisely, after normalizing\n  \\[\\beta_{\\pi,E,\\psi}(s)=\\gamma(-s,\\mathrm{JL}(\\pi),\\wedge^2,\\psi)\\,\\gamma(s,\\mathrm{JL}(\\pi),\\mathrm{sym}^2,\\psi)\\,\\alpha_{\\pi,E}(s),\\]\n  one has the key equivalence\n  \\begin{equation}\\label{eq:iff} \\pi \\mbox{ is distinguished} \\iff \\beta_{\\pi,E,\\psi}(0)=1 .\\end{equation}\n  i.e. “$J_{\\pi,E}(s)$ has a pole at $s=0$ if and only if $\\beta_{\\pi,E,\\psi}(0)=1$.”\n\n- **Compute $\\beta_{\\pi,E,\\psi}(0)$ by local-to-global.** “It remains to compute $\\beta_{\\pi,E,\\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\\mathfrak{E}/\\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \\cite{TW}) globalize $\\pi$ as $\\pi=\\Pi_{v_0}$ for a cuspidal automorphic $\\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\\Pi_v$ is moreover $\\mathfrak{E}_v$-distinguished for all places $v\\neq v_0$.”\n\n- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove the following proposition.\n\n  \\begin{proposition}\\label{prop:sign dist}\nAssume that either $G=\\GL_n(F)$ is split and $\\pi\\in \\gadist(G)$, or that $\\pi\\in \\cadist(G)$, then $\\beta_{\\pi,A,\\psi}(0)=1$. \n\\end{proposition}\n\n  From this they “deduce a formula for $\\beta_{\\Pi_v,\\mathfrak{E}_v,\\Psi_v}(0)$ for all places $v\\neq v_0$” via the following corollary.\n\n  \\begin{corollary}\\label{cor:key}\nLet $\\pi$ be as in Proposition \\ref{prop:sign dist}, but assume moreover that it is unitary, and let $c(\\pi,\\psi)$ be as in Theorem \\ref{thm:almost explicit constant}. Then $c(\\pi,\\psi)=(-1)^m\\eta_{A/F}(-1)^{n/2}$.\n\\end{corollary}\n\n  This combination is called the “second main observation” and it uses “the main local result of \\cite{MOY} … as well as … the direct implication in establishing the main theorem.”\n\n- **Global product formula forces the local value at $v_0$.** They show “the product of all $\\beta_{\\Pi_v,E_v,\\Psi_v}(0)$ including $\\beta_{\\pi,E,\\psi}(0)$” equals $1$ by the following lemma.\n\n  \\begin{lemma}\\label{lm:product of c is one}\nLet $\\mathfrak{E}/\\mathfrak{F}$ be a quadratic extension of number fields, and $\\Psi:\\mathfrak{F}\\backslash \\A_\\mathfrak{F}\\to \\C^\\times$ be a non trivial character. Let $\\mathfrak{D}$ be an $\\mathfrak{F}$-central division algebra. Assume moreover that \n$\\mathfrak{E}$ is contained inside $\\CM_m(\\mathfrak{D})$. \n If $\\Pi$ is a self-dual cuspidal automorphic representation of $\\GL_m(\\mathfrak{D}_{\\A_\\mathfrak{F}})$ such that $\\JL(\\Pi)$ is cuspidal, then \\[\\prod_v c(\\Pi_v,\\Psi_v)=1,\\] where the product is over all places of $F$. More specifically \\[\\prod_{v\\in S} c(\\Pi_v,\\Psi_v)=1,\\] where $S$ is the set of places such that of $\\mathfrak{F}$ outside of which $\\mathfrak{D}_v$ is split, and $\\mathfrak{E}_v/\\mathfrak{F}_v$, $\\pi_v$ and $\\Psi_v$ are all unramified.\n\\end{lemma}\n\n  Hence one solves for $\\beta_{\\pi,E,\\psi}(0)$ and obtains the value asserted in the following theorem:\n  \\begin{theorem}\\label{thm:key}\nLet $\\pi\\in \\sympc(G)$. Then \n\\[c(\\pi,\\psi)=(-1)^{m}\\eta_{E/F}(-1)^{n/2}.\\]\n\\end{theorem}\n  Together with the equation above \\eqref{eq:iff}, this yields the “converse implication of the cuspidal case in establishing the main theorem,” formulated as:\n\n  \\begin{corollary}\\label{cor:main}\nLet $\\pi\\in\\cusp(G)$ and let $\\phi=\\phi_\\pi$ the corresponding $L$-parameter.\nThe following two conditions are equivalent.\n\\begin{itemize}\n\\item[(i)] $\\pi$ is $H$-distinguished.\n\\item[(ii)] $\\phi$ takes values in $\\Sp_n(\\C)$ and satisfies $\\e_E(\\phi)=(-1)^m$.\n\\end{itemize}\n\\end{corollary}\n",
    "expanded_theorem": "\\label{conj:PTB}\nLet $\\pi$ be a generic representation of $G$, and $\\phi=\\phi_\\pi$.\n\\begin{itemize}\n\\item[(1)] If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$.\n\\item[(2)] Suppose $\\phi$ takes values in $\\Sp_n(\\mathbb{C})$ and we write it in the following form:\n    \\[\n    \\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n    \\]\nwhere $I_\\phi,  J_\\phi$ are finite sets and $\\phi_k\\,\\colon WD_F\\to\\GL_{n_k}(\\mathbb{C})$ is a discrete $L$-parameter such that\n\\begin{itemize}\n\\item for each $i\\in I_\\phi$,  $n_i$ is even and $\\phi_i$ takes values in $\\Sp_{n_i}(\\mathbb{C})$,\n\\item for any $i,  i'\\in I_\\phi$,  we have $\\phi_i\\not\\simeq\\phi_{i'}$ if $i\\neq i'$. \n\\end{itemize}\nThen $\\pi$ is $E$-distinguished if and only if we have \n    \\[\n    \\e_E(\\phi_i) = (-1)^{n_i/d},   \\qquad  i\\in I_\\phi.\n    \\]\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Implication"
    ],
    "mcq": {
      "question": "Let $F$ be a finite extension of $\\mathbb{Q}_p$, let $D$ be a central division $F$-algebra of dimension $d^2$, let $m\\ge 1$, set $G=\\mathrm{GL}_m(D)$, and write $n=md$. Assume $n$ is even, so that a quadratic extension $E/F$ embeds in $\\mathrm{M}_m(D)$. Let $H$ be the centralizer of $E^\\times$ in $G$, and say that a smooth representation $\\pi$ of $G$ is $E$-distinguished if $\\operatorname{Hom}_H(\\pi,\\mathbb{C})\\ne 0$. Let $\\pi$ be a generic representation of $G$, and let $\\phi=\\phi_\\pi:\\mathrm{WD}_F\\to \\mathrm{GL}_n(\\mathbb{C})$ be its $L$-parameter, defined via the Jacquet--Langlands transfer. For any symplectic $L$-parameter $\\theta:\\mathrm{WD}_F\\to \\mathrm{Sp}_r(\\mathbb{C})$, define\n\\[\n\\varepsilon_E(\\theta):=\\eta_{E/F}(-1)^{r/2}\\,\\epsilon\\!\\left(\\tfrac12,\\theta_{|\\mathrm{WD}_E},\\psi_E\\right)\n=\\eta_{E/F}(-1)^{r/2}\\,\\epsilon\\!\\left(\\tfrac12,\\theta,\\psi\\right)\\epsilon\\!\\left(\\tfrac12,\\eta_{E/F}\\otimes\\theta,\\psi\\right),\n\\]\nwhere $\\eta_{E/F}$ is the quadratic character of $F^\\times$ with kernel $N_{E/F}(E^\\times)$, $\\psi:F\\to\\mathbb{C}^\\times$ is nontrivial, and $\\psi_E=\\psi\\circ\\operatorname{tr}_{E/F}$. Suppose further, when applicable, that $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and decomposes as\n\\[\n\\phi = \\bigoplus_{i\\in I_\\phi} \\phi_i \\oplus \\bigoplus_{j\\in J_\\phi}(\\phi_j\\oplus\\phi_j^\\vee),\n\\]\nwith $I_\\phi,J_\\phi$ finite, each $\\phi_k:\\mathrm{WD}_F\\to\\mathrm{GL}_{n_k}(\\mathbb{C})$ a discrete $L$-parameter, each $n_i$ even and each $\\phi_i$ taking values in $\\mathrm{Sp}_{n_i}(\\mathbb{C})$ for $i\\in I_\\phi$, and $\\phi_i\\not\\simeq \\phi_{i'}$ whenever $i\\ne i'$. Which statement about $E$-distinction holds?",
      "correct_choice": {
        "label": "A",
        "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "C",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished whenever\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "D",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if there exists a choice of nontrivial character $\\psi:F\\to\\mathbb{C}^\\times$ such that\n\\[\n\\varepsilon_E(\\phi_i)=(-1)^{n_i/d}\\qquad\\text{for every }i\\in I_\\phi.\n\\]"
        },
        {
          "label": "E",
          "text": "If $\\pi$ is $E$-distinguished, then $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$. Moreover, if $\\phi$ takes values in $\\mathrm{Sp}_n(\\mathbb{C})$ and has the displayed decomposition, then $\\pi$ is $E$-distinguished if and only if\n\\[\n\\varepsilon_E(\\phi_k)=(-1)^{n_k/d}\\qquad\\text{for every }k\\in I_\\phi\\sqcup J_\\phi.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "sign exponent depends on $n_i/d$, not $n_i$",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the converse direction of the criterion",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "$\\varepsilon_E(\\theta)$ is independent of $\\psi$ for symplectic parameters; no existential choice of $\\psi$ is needed",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "criterion only constrains the symplectic self-dual summands indexed by $I_\\phi$, not the paired summands in $J_\\phi$",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It sets up notation and hypotheses, but the decisive criterion distinguishing the correct answer from the near-miss alternatives is not stated in the prompt itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct choice is the full statement of a specific distinction criterion, while the other options are slight perturbations of that statement. It functions mainly as recognition of the theorem rather than application to a new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the distractors differ in subtle ways: the exponent n_i/d versus n_i, 'if and only if' versus a one-way implication, unnecessary dependence on ψ, and whether the condition applies to I_φ or all summands. However, the item still leans much more toward precise recall than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: each alters a mathematically meaningful component of the criterion in a plausible way, reflecting realistic failure modes such as sign mistakes, quantifier weakening, spurious auxiliary-data dependence, and overextending the condition to the wrong summands."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-crafted theorem-recognition MCQ with high-quality distractors, but it is largely a restatement/recall task rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.20055v1",
    "paper_link": "http://arxiv.org/abs/2512.20055v1",
    "theorems_cnt": 8,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:lower}\nFix an integer $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k-o(1)},\n\\qquad\nc_k:=\\frac{k-2}{e ((k-2)!)^{1/(k-2)}}.\n\\]",
      "start_pos": 7643,
      "end_pos": 7812,
      "label": "thm:lower"
    },
    "ref_dict": {
      "prob:856": "\\begin{prob}\\label{prob:856}\nLet $k\\ge 3$. Estimate $f_k(N)$.\n\\end{prob}",
      "thm:lower": "\\begin{thm}\\label{thm:lower}\nFix an integer $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k-o(1)},\n\\qquad\nc_k:=\\frac{k-2}{e ((k-2)!)^{1/(k-2)}}.\n\\]\n\\end{thm}",
      "eq:Erdos-threshold": "\\begin{equation}\\label{eq:Erdos-threshold}\n\\sum_{i=1}^s \\frac{1}{a_i} > b_k \\log x,\n\\end{equation}",
      "thm:equivalence": "\\begin{thm}\\label{thm:equivalence}\nFor each $k\\ge 3$, we have $\\mu_k^{\\mathrm S} = 2$ if and only if $f_k(N) = (\\log N)^{1 - o(1)}$.\n\\end{thm}",
      "prob:857": "\\begin{prob}\\label{prob:857}\nLet $m=m(n,k)$ be minimal such that any distinct collection of sets\n$A_1,\\dots,A_m\\subseteq [n]$ contains a $k$-sunflower. Estimate $m(n,k)$, or ideally determine\nits asymptotic behavior.\n\\end{prob}",
      "lem:harmonic-SS": "\\begin{lem}\\label{lem:harmonic-SS}\nLet\n\\[\nH_\\ell(N) := \\sum_{\\substack{n\\le N\\\\ \\omega(n)=\\Omega(n)=\\ell}} \\frac1n.\n\\]Then for every $0<\\eta<1$ we have\n\\begin{equation}\\label{eq:H-ell-lower}\n  H_\\ell(N) \\gg_\\eta \\frac{(\\log\\log N)^\\ell}{\\ell!}\n  \\qquad (N\\to\\infty,\\ 1\\le \\ell\\le (1-\\eta)\\log\\log N).\n\\end{equation}\n\\end{lem}",
      "thm:B-implies-A": "\\begin{thm}\\label{thm:B-implies-A}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ll (\\log N)^{\\mu_k^{\\mathrm S}-1+o(1)}.\n\\]\n\\end{thm}",
      "thm:mu-lb-implies-f-lb": "\\begin{thm}\\label{thm:mu-lb-implies-f-lb}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[f_k(N) \\ge (\\log N)^{\\log\\mu_k^{\\mathrm S} - o(1)}.\\]\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 2671,
    "pre_theorem_intro_text": "\\subsection{The problem and background}Fix integers $k\\ge 3$ and $N\\ge 1$. Following Erd\\H{o}s~\\cite{Er70}, we study subsets\n$A\\subseteq [N]:=\\{1,2,\\dots,N\\}$ that avoid certain ``equal--LCM'' configurations.\nWe say that $A$ \\emph{contains an LCM-$k$-tuple} if there exist distinct\n$a_1,\\dots,a_k\\in A$ such that all pairwise least common multiples coincide, i.e.\n\\[\n\\operatorname{lcm}(a_i,a_j)=\\operatorname{lcm}(a_1,a_2)\\qquad (1\\le i<j\\le k).\n\\]\nWe say $A$ is \\emph{LCM-$k$-free} if it contains no LCM-$k$-tuple.\n\nIn~\\cite[p.~124]{Er70}, Erd\\H{o}s introduced the counting function $g(k,x)$, defined as\nthe largest integer $s$ for which one can find\n\\[\n1\\le a_1<\\cdots<a_s\\le x\n\\]\nwith the property that no $k$ of the $a_i$ form an LCM-$k$-tuple. Erd\\H{o}s\noriginally conjectured that $g(k,x)=o(x)$ for every $k\\ge 3$, but later proved that\nthis fails for all $k\\ge 4$; see~\\cite[Theorem~1]{Er70}. In particular, for each\n$k\\ge 4$ there exists $\\delta_k>0$ such that\n\\[\ng(k,x)\\ge \\delta_k x\n\\]\nfor all sufficiently large $x$.\n\nErd\\H{o}s also considered a harmonic analogue. He showed~\\cite[Eq.~(19)]{Er70} that for each\n$k\\ge 3$ there exists $b_k>0$ such that whenever $x$ is sufficiently large and\n$1\\le a_1<\\cdots<a_s\\le x$ satisfy\n\\begin{equation}\\label{eq:Erdos-threshold}\n\\sum_{i=1}^s \\frac{1}{a_i} > b_k \\log x,\n\\end{equation}\nthen some $k$ of the $a_i$ form an LCM-$k$-tuple. In~\\cite[p.~127]{Er70}, Erd\\H{o}s wrote:\n\\begin{quote}\n\\emph{I do not know how much \\eqref{eq:Erdos-threshold} can be weakened so that there should always\nbe $k$ $a$'s every two of which have the same least common multiple.}\n\\end{quote}\nIf we define\n\\[\nf_k(N)\\;:=\\;\\max\\left\\{\\sum_{a\\in A}\\frac1a:\\ A\\subseteq [N]\\ \\text{is LCM-$k$-free}\\right\\}.\n\\]\nthen Erd\\H{o}s is asking the following question.\n\\begin{prob}\\label{prob:856}\nLet $k\\ge 3$. Estimate $f_k(N)$.\n\\end{prob}\n\nProblem~\\ref{prob:856} appears as Problem~\\#856 on Bloom's Erd\\H{o}s Problems website~\\cite{EP856}\nand may be viewed as the harmonic counterpart of the counting problem introduced by Erd\\H{o}s\nin~\\cite[p.~124]{Er70}.\n\nA trivial bound is\n\\[\nf_k(N)\\le \\sum_{n\\le N}\\frac1n = \\log N + O(1).\n\\]\n\nErd\\H{o}s~\\cite{Er70} provided a short proof of the improved upper bound $f_k(N)\\ll_k \\log N/\\log\\log N$. It is observed in the comment section on the Erd\\H{o}s Problems website~\\cite{EP856}\nthat one can further improve this to $f_k(N)\\leq \\log N \\cdot \\exp\\left(-\\Omega_k\\left(\\frac{\\log\\log N}{\\log\\log\\log N}\\right)\\right)$. To our knowledge, no lower bounds of comparable strength were previously available.\n\n\\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.",
    "context": "\\subsection{The problem and background}Fix integers $k\\ge 3$ and $N\\ge 1$. Following Erd\\H{o}s~\\cite{Er70}, we study subsets\n$A\\subseteq [N]:=\\{1,2,\\dots,N\\}$ that avoid certain ``equal--LCM'' configurations.\nWe say that $A$ \\emph{contains an LCM-$k$-tuple} if there exist distinct\n$a_1,\\dots,a_k\\in A$ such that all pairwise least common multiples coincide, i.e.\n\\[\n\\operatorname{lcm}(a_i,a_j)=\\operatorname{lcm}(a_1,a_2)\\qquad (1\\le i<j\\le k).\n\\]\nWe say $A$ is \\emph{LCM-$k$-free} if it contains no LCM-$k$-tuple.\n\nIn~\\cite[p.~124]{Er70}, Erd\\H{o}s introduced the counting function $g(k,x)$, defined as\nthe largest integer $s$ for which one can find\n\\[\n1\\le a_1<\\cdots<a_s\\le x\n\\]\nwith the property that no $k$ of the $a_i$ form an LCM-$k$-tuple. Erd\\H{o}s\noriginally conjectured that $g(k,x)=o(x)$ for every $k\\ge 3$, but later proved that\nthis fails for all $k\\ge 4$; see~\\cite[Theorem~1]{Er70}. In particular, for each\n$k\\ge 4$ there exists $\\delta_k>0$ such that\n\\[\ng(k,x)\\ge \\delta_k x\n\\]\nfor all sufficiently large $x$.\n\nErd\\H{o}s also considered a harmonic analogue. He showed~\\cite[Eq.~(19)]{Er70} that for each\n$k\\ge 3$ there exists $b_k>0$ such that whenever $x$ is sufficiently large and\n$1\\le a_1<\\cdots<a_s\\le x$ satisfy\n\\begin{equation}\\label{eq:Erdos-threshold}\n\\sum_{i=1}^s \\frac{1}{a_i} > b_k \\log x,\n\\end{equation}\nthen some $k$ of the $a_i$ form an LCM-$k$-tuple. In~\\cite[p.~127]{Er70}, Erd\\H{o}s wrote:\n\\begin{quote}\n\\emph{I do not know how much \\eqref{eq:Erdos-threshold} can be weakened so that there should always\nbe $k$ $a$'s every two of which have the same least common multiple.}\n\\end{quote}\nIf we define\n\\[\nf_k(N)\\;:=\\;\\max\\left\\{\\sum_{a\\in A}\\frac1a:\\ A\\subseteq [N]\\ \\text{is LCM-$k$-free}\\right\\}.\n\\]\nthen Erd\\H{o}s is asking the following question.\n\\begin{prob}\\label{prob:856}\nLet $k\\ge 3$. Estimate $f_k(N)$.\n\\end{prob}\n\nA trivial bound is\n\\[\nf_k(N)\\le \\sum_{n\\le N}\\frac1n = \\log N + O(1).\n\\]\n\nErd\\H{o}s~\\cite{Er70} provided a short proof of the improved upper bound $f_k(N)\\ll_k \\log N/\\log\\log N$. It is observed in the comment section on the Erd\\H{o}s Problems website~\\cite{EP856}\nthat one can further improve this to $f_k(N)\\leq \\log N \\cdot \\exp\\left(-\\Omega_k\\left(\\frac{\\log\\log N}{\\log\\log\\log N}\\right)\\right)$. To our knowledge, no lower bounds of comparable strength were previously available.\n\n\\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.",
    "full_context": "\\subsection{The problem and background}Fix integers $k\\ge 3$ and $N\\ge 1$. Following Erd\\H{o}s~\\cite{Er70}, we study subsets\n$A\\subseteq [N]:=\\{1,2,\\dots,N\\}$ that avoid certain ``equal--LCM'' configurations.\nWe say that $A$ \\emph{contains an LCM-$k$-tuple} if there exist distinct\n$a_1,\\dots,a_k\\in A$ such that all pairwise least common multiples coincide, i.e.\n\\[\n\\operatorname{lcm}(a_i,a_j)=\\operatorname{lcm}(a_1,a_2)\\qquad (1\\le i<j\\le k).\n\\]\nWe say $A$ is \\emph{LCM-$k$-free} if it contains no LCM-$k$-tuple.\n\nIn~\\cite[p.~124]{Er70}, Erd\\H{o}s introduced the counting function $g(k,x)$, defined as\nthe largest integer $s$ for which one can find\n\\[\n1\\le a_1<\\cdots<a_s\\le x\n\\]\nwith the property that no $k$ of the $a_i$ form an LCM-$k$-tuple. Erd\\H{o}s\noriginally conjectured that $g(k,x)=o(x)$ for every $k\\ge 3$, but later proved that\nthis fails for all $k\\ge 4$; see~\\cite[Theorem~1]{Er70}. In particular, for each\n$k\\ge 4$ there exists $\\delta_k>0$ such that\n\\[\ng(k,x)\\ge \\delta_k x\n\\]\nfor all sufficiently large $x$.\n\nErd\\H{o}s also considered a harmonic analogue. He showed~\\cite[Eq.~(19)]{Er70} that for each\n$k\\ge 3$ there exists $b_k>0$ such that whenever $x$ is sufficiently large and\n$1\\le a_1<\\cdots<a_s\\le x$ satisfy\n\\begin{equation}\\label{eq:Erdos-threshold}\n\\sum_{i=1}^s \\frac{1}{a_i} > b_k \\log x,\n\\end{equation}\nthen some $k$ of the $a_i$ form an LCM-$k$-tuple. In~\\cite[p.~127]{Er70}, Erd\\H{o}s wrote:\n\\begin{quote}\n\\emph{I do not know how much \\eqref{eq:Erdos-threshold} can be weakened so that there should always\nbe $k$ $a$'s every two of which have the same least common multiple.}\n\\end{quote}\nIf we define\n\\[\nf_k(N)\\;:=\\;\\max\\left\\{\\sum_{a\\in A}\\frac1a:\\ A\\subseteq [N]\\ \\text{is LCM-$k$-free}\\right\\}.\n\\]\nthen Erd\\H{o}s is asking the following question.\n\\begin{prob}\\label{prob:856}\nLet $k\\ge 3$. Estimate $f_k(N)$.\n\\end{prob}\n\nA trivial bound is\n\\[\nf_k(N)\\le \\sum_{n\\le N}\\frac1n = \\log N + O(1).\n\\]\n\nErd\\H{o}s~\\cite{Er70} provided a short proof of the improved upper bound $f_k(N)\\ll_k \\log N/\\log\\log N$. It is observed in the comment section on the Erd\\H{o}s Problems website~\\cite{EP856}\nthat one can further improve this to $f_k(N)\\leq \\log N \\cdot \\exp\\left(-\\Omega_k\\left(\\frac{\\log\\log N}{\\log\\log\\log N}\\right)\\right)$. To our knowledge, no lower bounds of comparable strength were previously available.\n\n\\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.\n\n\\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.\n\nProblem~\\ref{prob:857} is widely known as the Erd\\H{o}s--Szemer\\'edi sunflower conjecture~\\cite{ErSz78},\nand it appears as Problem~\\#857 on Bloom's Erd\\H{o}s Problems website~\\cite{EP857}. Following the\nnotation of~\\cite{NaSa17}, let $F_k(n)$ be the maximum size of a $k$-sunflower-free family\n$\\mathcal{F}\\subseteq 2^{[n]}$, and define the \\emph{Erd\\H{o}s--Szemer\\'edi $k$-sunflower-free capacity}\n\\[\n\\mu_k^{\\mathrm S}:=\\limsup_{n\\to\\infty} F_k(n)^{1/n}.\n\\]Trivially $\\mu_k^{\\mathrm S}\\le 2$, and the celebrated Erd\\H{o}s--Szemer\\'edi sunflower conjecture~\\cite{ErSz78} asserts that\n$\\mu_k^{\\mathrm S}<2$ for every $k\\ge 3$. A standard tensor power argument\\footnote{Here is a concise description of the argument. Let $\\mathcal{F}_n\\subseteq 2^{[n]}$ be a $k$-sunflower-free family. Choose $r\\in\\{0,1,\\dots,n\\}$ maximizing $|\\mathcal{F}_n\\cap\\binom{[n]}{r}|$, and set $\\mathcal{F}_n':=\\mathcal{F}_n\\cap\\binom{[n]}{r}$, so that $|\\mathcal{F}_n'|\\ge \\frac{1}{n+1}|\\mathcal{F}_n|$. For each integer $t\\ge 2$, partition $[tn]$ into $t$ subsets $S_1\\sqcup\\cdots\\sqcup S_t$ with $|S_i|=n$. For each $i$ fix a bijection $\\pi_i:[n]\\to S_i$ and define a copy of $\\mathcal{F}_n'$ on $S_i$ by $\\mathcal{F}_n'^{(i)}:=\\{\\pi_i(F):F\\in\\mathcal{F}_n'\\}\\subseteq 2^{S_i}$. Then one checks that $\\mathcal{F}_{n}^{\\otimes t}:=\\{F_1\\sqcup\\cdots\\sqcup F_t:\\ F_i\\in\\mathcal{F}_n'^{(i)}\\}\\subseteq 2^{[tn]}$ is $k$-sunflower-free and satisfies $|\\mathcal{F}_{n}^{\\otimes t}|=|\\mathcal{F}_n'|^t\\ge (|\\mathcal{F}_n|/(n+1))^t$.} shows that the limsup can be replaced with a limit:\n\\begin{equation}\n \\label{eq:lim}\n \\mu_k^{\\mathrm S} =\\lim_{n\\to\\infty} F_k(n)^{1/n}. \n\\end{equation}\n\n\\begin{thm}\\label{thm:equivalence}\nFor each $k\\ge 3$, we have $\\mu_k^{\\mathrm S} = 2$ if and only if $f_k(N) = (\\log N)^{1 - o(1)}$.\n\\end{thm}\nThe implication ``$\\Rightarrow$'' is obtained by constructing a LCM-$k$-free set of squarefree integers\nwhose prime-support sets form a large cosunflower-free family in an appropriate weighted\nproduct measure. The implication ``$\\Leftarrow$'' follows from Theorem~\\ref{thm:B-implies-A} below. \n\\begin{thm}\\label{thm:B-implies-A}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ll (\\log N)^{\\mu_k^{\\mathrm S}-1+o(1)}.\n\\]\n\\end{thm}\nIts proof is a double-counting argument similar to Erd\\H{o}s's original argument in~\\cite{Er70}. To each integer $m$ we\nassociate a set system on the prime divisors of $m$ and show that LCM-$k$-freeness forces these set\nsystems to be $k$-sunflower-free. We then combine the resulting estimate with a Sathe--Selberg type\nlower bound for the harmonic sum of squarefree almost primes.\n\nFinally, we wonder how good a lower bound for $f_k(N)$ we can establish given knowledge of $\\mu_k^{\\mathrm S}$. While we can obtain a lower bound by following the proof of Theorem~\\ref{thm:equivalence}, it is not good when $\\mu_k^{\\mathrm S}$ is smaller than $2$. Towards an improvement on Theorem~\\ref{thm:lower}, we show that\n\\begin{thm}\\label{thm:mu-lb-implies-f-lb}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[f_k(N) \\ge (\\log N)^{\\log\\mu_k^{\\mathrm S} - o(1)}.\\]\n\\end{thm}\nFor $k=3$, the best current upper bound due to Naslund and Sawin~\\cite[Theorem~1]{NaSa17} states that\n\\[\n\\mu_3^{\\mathrm S}\\le \\frac{3}{2^{2/3}}=1.889881574\\dots\\, .\n\\]\nIn the opposite direction, a construction of Deuber, Erd\\H{o}s, Gunderson, Kostochka and\nMeyer~\\cite{DEGKM97} yields \\(\\mu_3^{\\mathrm S}>1.551\\). Hence, by Theorems~\\ref{thm:B-implies-A} and~\\ref{thm:mu-lb-implies-f-lb}, we obtain the following consequence.\n\\begin{cor}\\label{cor:f3-upper}\nAs $N\\to\\infty$,\n\\[\n(\\log N)^{\\log(1.551) - o(1)} \\leq f_3(N)\\ll (\\log N)^{\\frac{3}{2^{2/3}}-1+o(1)}.\n\\]\n\\end{cor}\nUnfortunately, Theorem~\\ref{thm:mu-lb-implies-f-lb} never gives an exponent greater than $\\log 2 < 0.7$, so it is strictly worse than Theorem~\\ref{thm:lower} for $k \\geq 7$. We leave it as an open question to obtain the correct exponent.\n\\begin{prob}\n In terms of $\\mu_k^{\\mathrm S}$, determine the smallest $c > 0$ such that for all sufficiently large $N$, we have\n $$f_k(N) \\leq (\\log N)^{c + o(1)}.$$\n\\end{prob}\n\n\\medskip\\noindent\n\\textbf{2. Prime bucketing into $t$ disjoint blocks of harmonic sum $\\ge 1$.}\nNow fix $N$ large (how large will be specified later) and set\n\\[\nL:=\\log\\log N,\\qquad t:=\\left\\lfloor L-2\\log L\\right\\rfloor,\\qquad\n\\delta:=\\frac{1}{L},\\qquad y:=L^2,\\qquad x:=N^{1/t}.\n\\]\nFor $N$ sufficiently large we have $t\\ge t_0$, $y<x$, and for every prime $p>y$ we have $1/p\\le 1/y\\le\\delta$. By Mertens' theorem,\n\\[\n\\sum_{p\\le u}\\frac1p=\\log\\log u+O(1)\\qquad(u\\to\\infty),\n\\]\nso\n\\begin{equation}\\label{eq:enough-harmonic-mass}\n\\sum_{y<p\\le x}\\frac1p= \\left(L- \\log t\\right) -\\log\\log y + O(1) \\geq t(1+\\delta)\n\\end{equation}for $N$ sufficiently large.\n\nWe next apply \\eqref{eq:sathe-selberg-squarefree} with $x=t$. To ensure that the uniformity range for $\\ell$ is satisfied, we restrict the\nintegral to a suitable subinterval where $\\log\\log t$ is comparable to $L$.\nSet \\(I := \\left\\{t\\in[2,N] : \\tfrac12 L \\le \\log\\log t \\le L \\right\\}\\). Thus, for $t\\in I$,\n\\[\n\\ell \\le (1-\\eta)L \\le 2(1-\\eta)\\log\\log t.\n\\]\nChoose $\\varepsilon=2\\eta$, so that $2(1-\\eta) = 2-\\varepsilon$. Then for\nall $t\\in I$ and all $2\\le \\ell\\le (1-\\eta)L$ we have\n\\[\n1\\le \\ell\\le (2-\\varepsilon)\\log\\log t.\n\\]\nHence \\eqref{eq:sathe-selberg-squarefree} applies to $A_\\ell(t)$ with $x=t$ for all $t\\in I$, giving\n\\begin{equation}\\label{eq:A-ell-asymp}\nA_\\ell(t)\n= G\\left(\\frac{\\ell-1}{\\log\\log t}\\right)\n \\frac{t}{\\log t}\n \\frac{(\\log\\log t)^{\\ell-1}}{(\\ell-1)!}\n \\left(1 + O_{\\eta}\\left(\\frac{\\ell}{(\\log\\log t)^2}\\right)\\right)\n\\end{equation}\nfor all $t\\in I$. Substituting \\eqref{eq:A-ell-asymp} into \\eqref{eq:H-ell-int} and restricting the integral to $t\\in I$, we obtain\n\\begin{align*}\nH_\\ell(N)\n &\\ge \\int_{t\\in I} \\frac{A_\\ell(t)}{t^2} dt\\\\\n&= \\frac1{(\\ell-1)!}\\int_{t\\in I}\n G\\left(\\frac{\\ell-1}{\\log\\log t}\\right)\n (\\log\\log t)^{\\ell-1}\n \\frac{dt}{t\\log t}\n \\left(1 + O_{\\eta}\\left(\\frac{\\ell}{(\\log\\log t)^2}\\right)\\right).\n\\end{align*}For $t\\in I$ we have $\\log\\log t\\asymp L$, and since\n$\\ell\\le (1-\\eta)L$, we have\n\\[\n\\frac{\\ell}{(\\log\\log t)^2}\n\\ll_\\eta \\frac1L.\n\\]\nThus, for $N$ sufficiently large, the factor\n$1 + O_{\\eta}\\bigl(\\ell/(\\log\\log t)^2\\bigr)$ is bounded below by, say, $1/2$ on~$I$.\nMoreover, for $t\\in I$ we have\n\\[\n0 \\le \\frac{\\ell-1}{\\log\\log t}\n\\le \\frac{(1-\\eta)L}{L/2}\n= 2(1-\\eta) < 2,\n\\]\nso that the argument of $G$ stays in the compact interval\n$[0,2(1-\\eta)]\\subseteq[0,2)$. By continuity and positivity of $G$ on $[0,2)$,\nthere exists a constant $c_\\eta>0$ such that\n\\[\nG(z) \\ge c_\\eta \\qquad\\text{for all } 0\\le z\\le 2(1-\\eta).\n\\]Therefore, using the substitution $u=\\log\\log t$ (so $du = dt/(t\\log t)$), we\nobtain\n\\begin{align*}\nH_\\ell(N)\n&\\ge \\frac{1}{2(\\ell-1)!} \\int_{t\\in I}\n G\\left(\\frac{\\ell-1}{\\log\\log t}\\right)\n (\\log\\log t)^{\\ell-1}\n \\frac{dt}{t\\log t}\n\\\\&\\ge\n\\frac{c_\\eta}{2(\\ell-1)!}\n\\int_{\\frac12 L}^{L} u^{\\ell-1} du.\n\\end{align*}\nThe last integral can be evaluated explicitly:\n\\[\n\\int_{\\frac12 L}^{L} u^{\\ell-1} du\n= \\frac{L^\\ell}{\\ell}\\left(1 - 2^{-\\ell}\\right)\n\\ge \\frac{L^\\ell}{2\\ell}.\n\\]Hence\n\\[\nH_\\ell(N)\n\\ge\n\\frac{c_\\eta}{2(\\ell-1)!}\\cdot \\frac{L^\\ell}{2\\ell}\n= \\frac{c_\\eta}{4} \\frac{L^\\ell}{\\ell!}.\n\\]This shows that for all sufficiently large $N$ and all integers\n$2\\le \\ell\\le (1-\\eta)L$ we have\n\\[\nH_\\ell(N) \\ge C_\\eta\\,\\frac{L^\\ell}{\\ell!}\n\\]\nwith $C_\\eta = c_\\eta/4>0$ depending only on~$\\eta$. Together with the case\n$\\ell=1$ treated above, this proves that\n\\[\nH_\\ell(N) \\gg_\\eta \\frac{L^\\ell}{\\ell!}\n\\]\nuniformly for $1\\le\\ell\\le(1-\\eta)L$ as $N\\to\\infty$. This is~\\eqref{eq:H-ell-lower}, and the lemma is proved.\\qed",
    "post_theorem_intro_text_len": 6602,
    "post_theorem_intro_text": "Our construction uses a ``prime bucketing'' scheme, which will be used for all lower bound constructions. We partition primes in a suitable interval\ninto many disjoint blocks of comparable harmonic sum and form squarefree integers by choosing\nexactly $k-2$ primes from each block. A short combinatorial lemma forces LCM-$k$-freeness, while the\nharmonic sum factorizes and can be optimized over a free parameter.\n\n\\subsection{Connections with the sunflower problem}\nWe next investigate the optimal exponent in the polylogarithmic bound.  Erd\\H{o}s suggested that Problem~\\ref{prob:856} is related to the sunflower problem. Recall that a collection of $k$ distinct sets $S_1,\\dots,S_k$ is called a \\emph{$k$-sunflower} if\n\\[\nS_i\\cap S_j = S_1\\cap S_2 \\qquad \\text{for all }1\\le i<j\\le k.\n\\]\nThis common intersection is called the \\emph{kernel}. A family of sets $\\mathcal F$ is said to be\n\\emph{$k$-sunflower-free} if it contains no $k$ distinct members forming a $k$-sunflower. Dually, a collection of $k$ distinct sets $S_1,\\dots,S_k$ is called a \\emph{$k$-cosunflower} if \\[\nS_i\\cup S_j = S_1\\cup S_2 \\qquad \\text{for all }1\\le i<j\\le k.\n\\] A family $\\mathcal F$ is \\emph{$k$-cosunflower-free} if it\ncontains no $k$ distinct members forming a $k$-cosunflower.\n\nIn~\\cite[p.~127]{Er70}, Erd\\H{o}s wrote:\n\\begin{quote}\n\\emph{Problem~\\ref{prob:856} seems connected with the following combinatorial problem:}\n\\end{quote}\n\n\\begin{prob}\\label{prob:857}\nLet $m=m(n,k)$ be minimal such that any distinct collection of sets\n$A_1,\\dots,A_m\\subseteq [n]$ contains a $k$-sunflower. Estimate $m(n,k)$, or ideally determine\nits asymptotic behavior.\n\\end{prob}\n\nProblem~\\ref{prob:857} is widely known as the Erd\\H{o}s--Szemer\\'edi sunflower conjecture~\\cite{ErSz78},\nand it appears as Problem~\\#857 on Bloom's Erd\\H{o}s Problems website~\\cite{EP857}. Following the\nnotation of~\\cite{NaSa17}, let $F_k(n)$ be the maximum size of a $k$-sunflower-free family\n$\\mathcal{F}\\subseteq 2^{[n]}$, and define the \\emph{Erd\\H{o}s--Szemer\\'edi $k$-sunflower-free capacity}\n\\[\n\\mu_k^{\\mathrm S}:=\\limsup_{n\\to\\infty} F_k(n)^{1/n}.\n\\]Trivially $\\mu_k^{\\mathrm S}\\le 2$, and the celebrated Erd\\H{o}s--Szemer\\'edi sunflower conjecture~\\cite{ErSz78} asserts that\n$\\mu_k^{\\mathrm S}<2$ for every $k\\ge 3$. A standard tensor power argument\\footnote{Here is a concise description of the argument. Let $\\mathcal{F}_n\\subseteq 2^{[n]}$ be a $k$-sunflower-free family. Choose $r\\in\\{0,1,\\dots,n\\}$ maximizing $|\\mathcal{F}_n\\cap\\binom{[n]}{r}|$, and set $\\mathcal{F}_n':=\\mathcal{F}_n\\cap\\binom{[n]}{r}$, so that $|\\mathcal{F}_n'|\\ge \\frac{1}{n+1}|\\mathcal{F}_n|$. For each integer $t\\ge 2$, partition $[tn]$ into $t$ subsets $S_1\\sqcup\\cdots\\sqcup S_t$ with $|S_i|=n$. For each $i$ fix a bijection $\\pi_i:[n]\\to S_i$ and define a copy of $\\mathcal{F}_n'$ on $S_i$ by $\\mathcal{F}_n'^{(i)}:=\\{\\pi_i(F):F\\in\\mathcal{F}_n'\\}\\subseteq 2^{S_i}$. Then one checks that $\\mathcal{F}_{n}^{\\otimes t}:=\\{F_1\\sqcup\\cdots\\sqcup F_t:\\ F_i\\in\\mathcal{F}_n'^{(i)}\\}\\subseteq 2^{[tn]}$ is $k$-sunflower-free and satisfies $|\\mathcal{F}_{n}^{\\otimes t}|=|\\mathcal{F}_n'|^t\\ge (|\\mathcal{F}_n|/(n+1))^t$.} shows that the limsup can be replaced with a limit:\n\\begin{equation}\n    \\label{eq:lim}\n  \\mu_k^{\\mathrm S} =\\lim_{n\\to\\infty} F_k(n)^{1/n}.  \n\\end{equation}\n\nIn this paper, we make Erd\\H{o}s's suggested connection between Problem~\\ref{prob:856} and Problem~\\ref{prob:857} precise by showing the Erd\\H{o}s--Szemer\\'edi sunflower conjecture is equivalent to a tight bound on $f_k(N)$.\n\n\\begin{thm}\\label{thm:equivalence}\nFor each $k\\ge 3$, we have $\\mu_k^{\\mathrm S} = 2$ if and only if $f_k(N) = (\\log N)^{1 - o(1)}$.\n\\end{thm}\nThe implication ``$\\Rightarrow$'' is obtained by constructing a LCM-$k$-free set of squarefree integers\nwhose prime-support sets form a large cosunflower-free family in an appropriate weighted\nproduct measure. The implication ``$\\Leftarrow$'' follows from Theorem~\\ref{thm:B-implies-A} below. \n\\begin{thm}\\label{thm:B-implies-A}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ll (\\log N)^{\\mu_k^{\\mathrm S}-1+o(1)}.\n\\]\n\\end{thm}\nIts proof is a double-counting argument similar to Erd\\H{o}s's original argument in~\\cite{Er70}. To each integer $m$ we\nassociate a set system on the prime divisors of $m$ and show that LCM-$k$-freeness forces these set\nsystems to be $k$-sunflower-free. We then combine the resulting estimate with a Sathe--Selberg type\nlower bound for the harmonic sum of squarefree almost primes. \n\nFinally, we wonder how good a lower bound for $f_k(N)$ we can establish given knowledge of $\\mu_k^{\\mathrm S}$. While we can obtain a lower bound by following the proof of Theorem~\\ref{thm:equivalence}, it is not good when $\\mu_k^{\\mathrm S}$ is smaller than $2$. Towards an improvement on Theorem~\\ref{thm:lower}, we show that\n\\begin{thm}\\label{thm:mu-lb-implies-f-lb}\nFix $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[f_k(N) \\ge (\\log N)^{\\log\\mu_k^{\\mathrm S} - o(1)}.\\]\n\\end{thm}\nFor $k=3$, the best current upper bound due to Naslund and Sawin~\\cite[Theorem~1]{NaSa17} states that\n\\[\n\\mu_3^{\\mathrm S}\\le \\frac{3}{2^{2/3}}=1.889881574\\dots\\, .\n\\]\nIn the opposite direction, a construction of Deuber, Erd\\H{o}s, Gunderson, Kostochka and\nMeyer~\\cite{DEGKM97} yields \\(\\mu_3^{\\mathrm S}>1.551\\). Hence, by Theorems~\\ref{thm:B-implies-A} and~\\ref{thm:mu-lb-implies-f-lb}, we obtain the following consequence.\n\\begin{cor}\\label{cor:f3-upper}\nAs $N\\to\\infty$,\n\\[\n(\\log N)^{\\log(1.551) - o(1)} \\leq f_3(N)\\ll (\\log N)^{\\frac{3}{2^{2/3}}-1+o(1)}.\n\\]\n\\end{cor}\nUnfortunately, Theorem~\\ref{thm:mu-lb-implies-f-lb} never gives an exponent greater than $\\log 2 < 0.7$, so it is strictly worse than Theorem~\\ref{thm:lower} for $k \\geq 7$. We leave it as an open question to obtain the correct exponent.\n\\begin{prob}\n    In terms of $\\mu_k^{\\mathrm S}$, determine the smallest $c > 0$ such that for all sufficiently large $N$, we have\n    $$f_k(N) \\leq (\\log N)^{c + o(1)}.$$\n\\end{prob}\n\nThe rest of the paper is organized as follows.\nIn Section~\\ref{sec:preliminaries} we fix notation and record two analytic inputs.\nIn Section~\\ref{sec:lower} we prove our unconditional lower bound\n(Theorem~\\ref{thm:lower}). In Section~\\ref{sec:upper} we prove the capacity-based upper bound (Theorem~\\ref{thm:B-implies-A}).\nIn Section~\\ref{sec:equivalence} we prove the full-density equivalence in\nTheorem~\\ref{thm:equivalence}. In Section~\\ref{sec:lower-bound-via-capacity} we establish the capacity-based lower bound Theorem~\\ref{thm:mu-lb-implies-f-lb}.\nFinally, the appendix contains the proof of the technical Lemma~\\ref{lem:harmonic-SS}.",
    "sketch": "To prove Theorem~\\ref{thm:lower}, they use a ``prime bucketing'' construction: partition primes in a suitable interval into many disjoint blocks of comparable harmonic sum, and form squarefree integers by choosing exactly $k-2$ primes from each block. A short combinatorial lemma then forces LCM-$k$-freeness, while the harmonic sum factorizes and can be optimized over a free parameter.",
    "expanded_sketch": "To prove the main theorem, they use a ``prime bucketing'' construction: partition primes in a suitable interval into many disjoint blocks of comparable harmonic sum, and form squarefree integers by choosing exactly $k-2$ primes from each block. A short combinatorial lemma then forces LCM-$k$-freeness, while the harmonic sum factorizes and can be optimized over a free parameter.,",
    "expanded_theorem": "\\label{thm:lower}\nFix an integer $k\\ge 3$. Then, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k-o(1)},\n\\qquad\nc_k:=\\frac{k-2}{e ((k-2)!)^{1/(k-2)}}.\n\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Fix an integer $k\\ge 3$. For $N\\ge 1$, let $[N]=\\{1,2,\\dots,N\\}$. A set $A\\subseteq [N]$ is called LCM-$k$-free if it contains no $k$ distinct elements $a_1,\\dots,a_k\\in A$ such that all pairwise least common multiples are equal, i.e. $\\operatorname{lcm}(a_i,a_j)$ has the same value for every $1\\le i<j\\le k$. Define\n\\[\nf_k(N):=\\max\\left\\{\\sum_{a\\in A}\\frac1a:\\ A\\subseteq [N]\\text{ is LCM-}k\\text{-free}\\right\\}.\n\\]\nWhich quantitative lower bound holds for $f_k(N)$ as $N\\to\\infty$?",
      "correct_choice": {
        "label": "A",
        "text": "For every fixed integer $k\\ge 3$, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k-o(1)},\n\\qquad c_k:=\\frac{k-2}{e\\,((k-2)!)^{1/(k-2)}}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every fixed integer $k\\ge 3$, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{1-o(1)}.\n\\]"
        },
        {
          "label": "C",
          "text": "For every fixed integer $k\\ge 3$, there exists a constant $c_k'>0$ such that as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k'-o(1)}.\n\\]"
        },
        {
          "label": "D",
          "text": "For every fixed integer $k\\ge 3$, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{c_k+o(1)},\n\\qquad c_k:=\\frac{k-2}{e\\,((k-2)!)^{1/(k-2)}}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every fixed integer $k\\ge 3$, as $N\\to\\infty$,\n\\[\nf_k(N)\\ge (\\log N)^{d_k-o(1)},\n\\qquad d_k:=\\frac{k-1}{e\\,((k-1)!)^{1/(k-1)}}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "optimization-over-free-parameter only yields a specific exponent strictly below 1",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "explicit formula for the exponent constant",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "sign of the little-o term in the exponent",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "block size is exactly $k-2$, so replacing it by $k-1$ alters the optimized exponent",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and asks for an asymptotic lower bound, but it does not state or strongly hint at the exact exponent or the correct formula for c_k."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially asking the test-taker to recognize the theorem’s stated conclusion among nearby variants. It is not a pure restatement in the stem, but it is still very close to theorem recall rather than an independently derived consequence."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle ways (missing factor e, sign of o(1), shifted parameter, weaker existential form). However, the problem mainly tests recognition/recall of the exact bound rather than genuine generative mathematical reasoning from the definitions."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors are mathematically plausible and target common errors, but choice C is also true as stated, just weaker than A. Since the stem asks which lower bound holds rather than which is strongest/sharpest, this creates ambiguity and weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "Good on avoiding answer leakage, but only moderate as a reasoning question. Its biggest flaw is ambiguity: at least one distractor is also true, so the item does not cleanly function as a single-correct MCQ."
    }
  },
  {
    "id": "2512.20373v1",
    "paper_link": "http://arxiv.org/abs/2512.20373v1",
    "theorems_cnt": 7,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:sup_sup}\n  Assume \\eqref{eq:degen} and \\eqref{eq:powerlike}. Let $u$ be a strong solution to \\eqref{eq:pde}--\\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\widetilde u$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a supersolution and\n  \\begin{equation}\n    \\label{eq:sup_sup_n}\n    u(x,t)\n    \\le\n    \\widetilde u(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sup_sup_nn}\n    \\supp u(t)\n    \\subset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    g^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\sup_{\\R^{\\SpDim}} u_{0}$, $R(0)$.",
      "start_pos": 18520,
      "end_pos": 19424,
      "label": "t:sup_sup"
    },
    "ref_dict": {
      "eq:degen": "\\begin{equation}\n  \\label{eq:degen}\n  1<p<N\n  \\,,\n  \\qquad\n  p+m-3>0\n  \\,.\n\\end{equation}",
      "p:main_sup_sub": "\\begin{proposition}\n  \\label{p:main_sup_sub}\n  The constants $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and\n  $\\nu_{0}$ can be selected so that $\\ups$ as defined in\n  \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a strong supersolution to\n  \\eqref{eq:pde}. Alternatively, we may select such constants so that\n  $\\ups$ is a strong subsolution to \\eqref{eq:pde}.\n\\end{proposition}",
      "r:rad_N": "\\begin{remark}\n  \\label{r:rad_N}\n  Clearly the sup estimate valid for $v$ as in Theorems\n  \\ref{t:sup_sup_dens} and \\ref{t:sub_sub_dens} is the same as in\n  \\eqref{eq:sup_decay}, which should be compared with the result valid for $\\rw(s)\\sim s^{-\\alpha}$, $\\alpha>p$, that is the universal bound $t^{-1/(p+m-3)}$ (see \\cite{Andreucci:Tedeev:2021b}). Our present bound however depends on the initial data, as the known bound valid in the subcritical case $\\alpha<p$, whose decay rate is $t^{-(N-\\alpha)/K}$, $K=(N-\\alpha)(p+m-3)+p-\\alpha$.\n\\end{remark}",
      "s:ssb": "\\begin{split}\n      \\ups(x,0)\n      &\\ge\n        C_{*}\n        t_{0}^{-\\frac{1}{p+m-3}}\n        \\Big[\n        \\tar(\\varGamma \\log t_{0})^{\\frac{1}{p-1}}\n        -\n        \\rat(\\rpol_{0})^{\\frac{1}{p-1}}\n        \\Big]^{\\frac{p-1}{p+m-3}}\n        \\\\\n      &\\ge\n        C_{*}\n        t_{0}^{-\\frac{1}{p+m-3}}\n        \\Big[\n        (\\varGamma \\log t_{0})^{\\frac{p-\\alpha_{2}}{\\alpha_{2}(p-1)}}\n        -\n        \\rat(\\rpol_{0})^{\\frac{1}{p-1}}\n        \\Big]^{\\frac{p-1}{p+m-3}}\n        \\ge\n        M\n        \\,,\n    \\end{split}\n  \\end{equation}\n  where last inequality is guaranteed by a suitable choice of $\\varGamma$, of course preserving \\eqref{eq:sss_ups_Gamma0}.\n\\end{proof}\n\n\\section{Self similar subsolutions}\n\\label{s:ssb}\n\nWe look for subsolutions in the radial form\n\\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2}. We use here the same\nnotation introduced in Section~\\ref{s:sss}, and we look at the set\nwhere $\\ups>0$. In this Section, the choice of the constants is more\ndelicate; we find it convenient to establish it from the beginning. In\nthis connection, let us note that the constant $\\tilde\\mu_{1}>0$ is\ndefined in \\eqref{eq:sss_ubs_mu1}, $\\tilde\\mu_{2}>0$ in\n\\eqref{eq:sss_ubs_mu2}, and $\\tilde\\mu_{3}\\ge 0$ in\n\\eqref{eq:sss_ubs_mu3}; the $\\tilde\\mu_{i}$ depend on $p$, $m$,\n$\\alpha_{1}$, $\\alpha_{2}$ only. The constant $\\nu_{0}$ is chosen as\nin \\eqref{eq:sss_ups_c1}. We also introduce for future reference a\ngiven constant $\\lambda>0$.\n\\\\\nWe fix $t_{0}>1$ such that\n\\begin{equation}\n  \\label{eq:sss_ubs_t0}\n  \\begin{split}\n    \\log t_{0}\n    \\ge\n      \\max&\\Big\\{\n      4\n      \\frac{p-\\alpha_{1}}{\\alpha_{1}}\n      \\,,\n      \\tilde\\mu_{4}^{-1}\n      \\,,\n      \\frac{2\\tilde\\mu_{1}(N-1)+\\tilde\\mu_{3}\n      +\n      2\\tilde\\mu_{1}\n      \\alpha_{2}^{2}}\n      {\\tilde\\mu_{4}}\n      \\,,\n    \\\\\n    &\\quad\n      \\frac{2(\\nu_{0}p)^{p-1}}{\\tilde\\mu_{4}(p+m-3)^{p-2}}\n      (\n      N\n      +\n      \\alpha_{2}^{2}\n      )\n      \\Big\\}\n      \\,,\n  \\end{split}",
      "eq:sup_decay": "\\begin{equation}\n  \\label{eq:sup_decay}\n  u(x,t)\n  \\le\n  \\gamma\n  \\Big[\n  \\frac{\\exw^{(-1)}(\\log t)^{p}}{t\\log t}\n  \\Big]^{\\frac{1}{p+m-3}}\n  \\,,\n  \\quad\n  x\\in\\RN\n  \\,,\n  t\\gg 1\n  \\,,\n\\end{equation}",
      "eq:fsp": "\\begin{equation}\n  \\label{eq:fsp}\n  \\bar R(t)\n  \\le\n  \\gamma\n  \\exw^{(-1)}(\\log t)\n  \\,,\n  \\quad\n  t\\gg 1\n  \\,,\n\\end{equation}",
      "t:sub_sub": "\\begin{theorem}\n  \\label{t:sub_sub}\n  Assume \\eqref{eq:degen} and \\eqref{eq:powerlike}. Let $u$ be a strong solution to \\eqref{eq:pde}--\\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$, such that $u_{0}(x)\\ge \\eps>0$ for $\\abs{x}\\le\\ell$, $\\ell>0$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution and\n  \\begin{equation}\n    \\label{eq:sub_sub_n}\n    u(x,t)\n    \\ge\n    \\ups(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sub_sub_nn}\n    \\supp u(t)\n    \\supset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    \\exw^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\eps$, $\\ell$.\n\\end{theorem}",
      "l:sss_comp": "\\begin{lemma}\n  \\label{l:sss_comp}\n  Let $L>0$, $M>0$ be given positive numbers. Then it is possible to select $\\rpol_{0}$, $C_{*}$, $t_{0}$, $\\varGamma$ so that $\\ups$ is a supersolution and \n  \\begin{equation}\n    \\label{eq:sss_comp_n}\n    \\ups(x,0)\n    \\ge\n    M\n    \\,,\n    \\qquad\n    \\abs{x}\\le L\n    \\,.\n  \\end{equation}\n\\end{lemma}",
      "eq:sss_sup": "\\begin{alignat}2\n  \\label{eq:sss_sup}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\rat(\\rpol)^{\\frac{1}{p-1}}\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol\\ge \\rpol_{0}\n            \\,,\n  \\\\\n  \\label{eq:sss_ups_2}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\ram(\\rpol)\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol<\\rpol_{0}\n            \\,,\n\\end{alignat}",
      "eq:powerlike": "\\begin{equation}\n  \\label{eq:powerlike}\n  \\alpha_{1}\n  \\frac{\\exw(s)}{s}\n  \\le\n  \\exw'(s)\n  \\le\n  \\alpha_{2}\n  \\frac{\\exw(s)}{s}\n  \\,,\n  \\qquad\n  s>0\n  \\,.\n\\end{equation}",
      "l:sss_excf": "\\begin{lemma}\n  \\label{l:sss_excf}\n  We have\n  \\begin{equation}\n    \\label{eq:sss_excf_n}\n    \\frac{\\exw(\\rpol)}{\\alpha_{2}}\n    \\le\n    \\excf(\\rpol)\n    \\le\n    \\frac{\\exw(\\rpol)}{\\alpha_{1}}\n    \\,,\n    \\qquad\n    \\rpol>0\n    \\,,\n  \\end{equation}\n  and its obvious consequence\n  \\begin{equation}\n    \\label{eq:sss_excf_nn}\n    \\alpha_{1}\n    \\frac{\\excf(\\rpol)}{\\rpol}\n    \\le\n    \\excf'(\\rpol)\n    =\n    \\frac{\\exw(\\rpol)}{\\rpol}\n    \\le\n    \\alpha_{2}\n    \\frac{\\excf(\\rpol)}{\\rpol}\n    \\,,\n    \\qquad\n    \\rpol>0\n    \\,.\n  \\end{equation}\n\\end{lemma}",
      "t:transform": "\\begin{theorem}\n  \\label{t:transform}\nRadial solutions $U(\\rpol,t)$ [respectively, supersolutions and subsolutions] of \\eqref{eq:pde} correspond, thru a suitable transformation of the space variable, to radial solutions $v(s,t)$ [respectively, supersolutions and subsolutions] of \\eqref{eq:rad_N},\nfor a suitable positive function $\\rw\\in C([0,+\\infty))$ such that $\\rw(0)=1$ and\n\\begin{equation}\n  \\label{eq:rad_rw_est}\n  \\rw(s)\n  \\sim\n  \\frac{\\exw^{(-1)}(\\log s)^{p}}{(\\log s)^{p}}\n  \\frac{1}{s^{p}}\n  \\,.\n\\end{equation}\nIn fact, the transformation $r=\\rpolf(s)$, $s>0$, is such that\n\\begin{equation}\n  \\label{eq:rad_r_asy_n}\n  \\rpolf(s)\n  \\sim\n  \\exw^{(-1)}(\\log s)\n  \\,.\n\\end{equation}\n\\end{theorem}",
      "t:sub_sub_dens": "\\begin{theorem}\n  \\label{t:sub_sub_dens}\n  Let $v$ be a strong solution to \\eqref{eq:rad_N}, for the specific $\\rw$ found in Theorem~\\ref{t:transform}, under assumptions \\eqref{eq:degen} and \\eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported, bounded and\n$v(x,0)\\ge \\eps>0$ for $\\abs{x}\\le\\ell$, $\\ell>0$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups(\\rpolf(\\abs{x}),t)$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution \\eqref{eq:rad_N}, and\n  \\begin{equation}\n    \\label{eq:sub_sub_dens_n}\n    v(x,t)\n    \\ge\n    \\ups(\\rpolf(\\abs{x}),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sub_sub_dens_nn}\n    \\supp v(t)\n    \\supset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\quad\n    \\log\\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  The constants involved in \\eqref{eq:sub_sub_dens_n}--\\eqref{eq:sub_sub_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.\n\\end{theorem}",
      "eq:rad_rw_est": "\\begin{equation}\n  \\label{eq:rad_rw_est}\n  \\rw(s)\n  \\sim\n  \\frac{\\exw^{(-1)}(\\log s)^{p}}{(\\log s)^{p}}\n  \\frac{1}{s^{p}}\n  \\,.\n\\end{equation}",
      "s:pre": "\\label{s:pre}\n\nNote that as a consequence of \\eqref{eq:powerlike} we have\n\\begin{equation}\n  \\label{eq:powerlike_inv}\n  \\alpha_{2}^{-1}\n  \\frac{\\exw^{(-1)}(t)}{t}\n  \\le\n  \\der{\\exw^{(-1)}}{t}(t)\n  \\le",
      "eq:weaksol_a": "\\begin{equation}\n    \\label{eq:weaksol_a}\n    \\int_{0}^{+\\infty}\n    \\int_{\\RN}\n    \\{\n    -\n    u\n    \\eta_{t}\n    +\n    u^{m-1}\n    \\abs{\\grad u}^{p-2}\n    \\grad u\n    \\grad \\eta\n    \\}\n    \\ew(x)\n    \\di x\n    \\di t\n    =\n    0\n    \\,.\n  \\end{equation}",
      "eq:sup_sup_dens_nn": "\\begin{equation}\n    \\label{eq:sup_sup_dens_nn}\n    \\supp v(t)\n    \\subset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\qquad\n    \\log \\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,.\n  \\end{equation}",
      "eq:sub_sub_dens_n": "\\begin{equation}\n    \\label{eq:sub_sub_dens_n}\n    v(x,t)\n    \\ge\n    \\ups(\\rpolf(\\abs{x}),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}",
      "eq:initd": "\\begin{alignat}{2}\n  \\label{eq:pde}\n  \\ew(x)\n  \\pder{u}{t}\n  -\n  \\Div\\big(\n  \\ew(x)\n  u^{m-1}\n  \\abs{\\grad u}^{p-2}\n  \\grad u\n  \\big)\n  &=\n    0\n    \\,,\n  &\\qquad&\n           \\text{in $S_{T}$,}\n  \\\\\n  \\label{eq:initd}\n  u(x,0)\n  &=\n    u_{0}(x)\n    \\,,\n  &\\qquad&\n           x\\in\\RN\n           \\,.\n\\end{alignat}",
      "eq:powerlike_lt": "\\begin{gather}\n  \\label{eq:powerlike_gt}\n  \\lambda^{\\alpha_{1}}\n  \\exw(\\rpol)\n  \\le\n  \\exw(\\lambda\\rpol)\n  \\le\n  \\lambda^{\\alpha_{2}}\n  \\exw(\\rpol)\n  \\,,\n  \\qquad\n  \\lambda\\ge 1\n  \\,,\n  \\\\\n  \\label{eq:powerlike_lt}\n  \\lambda^{\\alpha_{2}}\n  \\exw(\\rpol)\n  \\le\n  \\exw(\\lambda\\rpol)\n  \\le\n  \\lambda^{\\alpha_{1}}\n  \\exw(\\rpol)\n  \\,,\n  \\qquad\n  \\lambda\\le 1\n  \\,.\n\\end{gather}",
      "eq:sss_powerlike_gt_excf": "\\begin{equation}\n  \\label{eq:sss_powerlike_gt_excf}\n  \\lambda^{\\frac{1}{\\alpha_{2}}}\n  \\excf^{(-1)}(\\rpol)\n  \\le\n  \\excf^{(-1)}(\\lambda\\rpol)\n  \\le\n  \\lambda^{\\frac{1}{\\alpha_{1}}}\n  \\excf^{(-1)}(\\rpol)\n  \\,,\n  \\qquad\n  \\lambda\\ge 1\n  \\,.\n\\end{equation}",
      "l:ssb_comp": "\\begin{lemma}\n  \\label{l:ssb_comp}\n  Let $\\ell>0$, $\\eps>0$ be given positive numbers. Then it is possible to select $\\rpol_{0}$, $C_{*}$, $t_{0}$ and $\\varGamma$ so that $\\ups$ is a subsolution and\n  \\begin{gather}\n    \\label{eq:ssb_comp_n}\n    \\supp \\ups(0)\n    \\subset\n    \\{\\abs{x}\\le \\ell\\}\n    \\,,\n    \\\\\n    \\label{eq:ssb_comp_nn}\n    \\ups(x,0)\n    \\le\n    \\eps\n    \\,,\n    \\quad\n    x\\in\\RN\n    \\,.\n  \\end{gather}\n\\end{lemma}",
      "t:sup_sup": "\\begin{theorem}\n  \\label{t:sup_sup}\n  Assume \\eqref{eq:degen} and \\eqref{eq:powerlike}. Let $u$ be a strong solution to \\eqref{eq:pde}--\\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a supersolution and\n  \\begin{equation}\n    \\label{eq:sup_sup_n}\n    u(x,t)\n    \\le\n    \\ups(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sup_sup_nn}\n    \\supp u(t)\n    \\subset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    \\exw^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\sup_{\\RN} u_{0}$, $R(0)$.\n\\end{theorem}",
      "eq:sss_ups_2": "\\begin{alignat}2\n  \\label{eq:sss_sup}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\rat(\\rpol)^{\\frac{1}{p-1}}\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol\\ge \\rpol_{0}\n            \\,,\n  \\\\\n  \\label{eq:sss_ups_2}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\ram(\\rpol)\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol<\\rpol_{0}\n            \\,,\n\\end{alignat}",
      "s:sss": "\\begin{split}\n      \\auxf'(\\rpol)\n      &=\n        \\rat(\\rpol)^{\\frac{1}{p-1}-1}\n        \\frac{\\rat'(\\rpol)}{p-1}\n        \\Big(\n        \\delta_{1}\n        +\n        \\frac{\\delta_{2}}{\\excf(\\rpol)}\n        \\Big)\n        -\n        \\delta_{2}\n        \\rat(\\rpol)^{\\frac{1}{p-1}}\n        \\frac{\\excf'(\\rpol)}{\\excf(\\rpol)^{2}}\n        \\\\\n      &=\n        \\delta_{1}\n        \\rat(\\rpol)^{\\frac{1}{p-1}-1}\n        \\frac{\\rat'(\\rpol)}{p-1}\n        +\n        \\delta_{2}\n        \\frac{\\rat(\\rpol)^{\\frac{1}{p-1}}}{\\rpol\\excf(\\rpol)}\n        \\Big[\n        \\frac{\\rpol}{p-1}\n        \\frac{\\rat'(\\rpol)}{\\rat(\\rpol)}\n        -\n        \\frac{\\excf'(\\rpol)\\rpol}{\\excf(\\rpol)}\n        \\Big]\n        \\\\\n      &=\n        \\delta_{1}\n        \\rat(\\rpol)^{\\frac{1}{p-1}-1}\n        \\frac{\\rat'(\\rpol)}{p-1}\n        +\n        \\frac{p}{p-1}\n        \\delta_{2}\n        \\frac{\\rat(\\rpol)^{\\frac{1}{p-1}}}{\\rpol\\excf(\\rpol)}\n        \\Big[\n        1\n        -\n        \\frac{\\excf'(\\rpol)\\rpol}{\\excf(\\rpol)}\n        \\Big]\n        \\\\\n      &\\ge\n        \\frac{\\rat(\\rpol)^{\\frac{1}{p-1}}}{\\rpol}\n        \\Big\\{\n        \\delta_{1}\n        \\frac{p-\\alpha_{2}}{p-1}\n        +\n        \\frac{p}{p-1}\n        \\frac{\\delta_{2}}{\\excf(\\rpol)}\n        \\Big[\n        1\n        -\n        \\frac{\\excf'(\\rpol)\\rpol}{\\excf(\\rpol)}\n        \\Big]\n        \\Big\\}\n        \\,.\n    \\end{split}\n  \\end{equation}\n  The claim follows from the inequality $\\auxf'(\\rpol)\\ge 0$, which can be proved, for $\\rpol$ as in \\eqref{eq:sss_comb_n}, on applying next \\eqref{eq:sss_excf_n}.\n\\end{proof}\n\n\\section{Self similar supersolutions}\n\\label{s:sss}\n\nWe assume here $\\varGamma\\ge 1$ and\n\\begin{equation}\n  \\label{eq:sss_ups_t0r0}\n  \\excf(\\rpol_{0})\n  <\n  \\log t_{0}\n  \\le\n  \\varGamma\n  \\log t_{0}\n  \\,,\n\\end{equation}\nso that the support of $\\ups(t)$ contains the ball $\\rpol\\le\\rpol_{0}$ for all $t\\ge 0$; this follows from the fact that $\\rat$, and therefore $\\tar$, is an increasing function as proved in Lemma~\\ref{l:sss_rat}. By the same token, the support of $\\ups(t)$ is the ball of radius $\\excf^{(-1)}(\\tau)$.\n\nFor the reader's convenience we recall that we assume $\\alpha_{2}<p$, and note that $\\rpol_{0}$ will be chosen as in \\eqref{eq:sss_ups_r0}, $C_{*}$ as in \\eqref{eq:sss_ups_C}, $\\nu_{0}$ as in \\eqref{eq:sss_ups_c1}, $\\varGamma$ as in \\eqref{eq:sss_ups_Gamma0}, while $t_{0}$ is required to satisfy \\eqref{eq:sss_ups_t0r0} and \\eqref{eq:sss_ups_ineqs5}.\n\n1) Case $\\rpol\\ge \\rpol_{0}$.\nWith the notation above, we write\n\\begin{equation}\n  \\label{eq:sss_sup_2}\n  \\ups(x,t)\n  =\n  \\frac{C_{*}\\brt(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n  \\,,\n  \\quad\n  \\brt(\\rpol,\\tau)\n  :=\n  \\big[\n  \\tar(\\tau)^{\\frac{1}{p-1}}\n  -\n  \\rat(\\rpol)^{\\frac{1}{p-1}}\n  \\big]_{+}\n  \\,.\n\\end{equation}\nClearly we may restrict in our calculations to the open set where $\\brt>0$. Thus we compute\n\\begin{equation}\n  \\label{eq:sss_ups_pow}\n  \\ups(x,t)^{m-1}\n  =\n  \\frac{C_{*}^{m-1}\\brt(\\rpol,\\tau)^{\\frac{(m-1)(p-1)}{p+m-3}}}{(t+t_{0})^{\\frac{m-1}{p+m-3}}}\n  \\,,\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:sss_ups_rder}\n  \\ups_{\\rpol}(x,t)\n  =\n  -\n  \\frac{1}{p+m-3}\n  \\frac{C_{*}\\brt(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}-1}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n  \\rat(\\rpol)^{\\frac{1}{p-1}-1}\n  \\rat'(\\rpol)\n  \\,.\n\\end{equation}\nThus, on recalling $\\rat'(\\rpol)\\ge 0$, we obtain\n\\begin{equation}\n  \\label{eq:sss_ups_diff}\n  \\begin{split}\n    I_{1}\n    &:=\n      \\ups^{m-1}\n      \\abs{\\ups_{\\rpol}}^{p-2}\n      \\ups_{\\rpol}\n    \\\\\n    &=\n      -\n      \\frac{\n      C_{*}^{p+m-2}\n      }{\n      (p+m-3)^{p-1}\n      }\n      \\,\n      \\frac{\n      \\brt(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}}\n      }{\n      (t+t_{0})^{\\frac{p+m-2}{p+m-3}}\n      }\n      \\rat(\\rpol)^{2-p}\n      \\rat'(\\rpol)^{p-1}\n      \\,.\n  \\end{split}",
      "eq:sub_sub_dens_nn": "\\begin{equation}\n    \\label{eq:sub_sub_dens_nn}\n    \\supp v(t)\n    \\supset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\quad\n    \\log\\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}",
      "t:sup_sup_dens": "\\begin{theorem}\n  \\label{t:sup_sup_dens}\n  Let $v$ be a strong solution to \\eqref{eq:rad_N}, for the specific $\\rw$ found in Theorem~\\ref{t:transform}, under assumptions \\eqref{eq:degen} and \\eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported and bounded.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups(\\rpolf(\\abs{x}),t)$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a supersolution to \\eqref{eq:rad_N}, and\n  \\begin{equation}\n    \\label{eq:sup_sup_dens_n}\n    v(x,t)\n    \\le\n    \\ups(\\rpolf(\\abs{x}),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sup_sup_dens_nn}\n    \\supp v(t)\n    \\subset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\qquad\n    \\log \\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,.\n  \\end{equation}\n  The constants involved in \\eqref{eq:sup_sup_dens_n}--\\eqref{eq:sup_sup_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.\n\\end{theorem}",
      "eq:rad_N": "\\begin{equation}\n  \\label{eq:rad_N}\n  \\rw(\\abs{x})\n  \\pder{v}{t}\n  =\n  \\Div(\n  v^{m-1}\n  \\abs{\\grad v}^{p-2}\n  \\grad v\n  )\n  \\,,\n  \\qquad\n  x\\in\\R^{N}\n  \\,,\n  t>0\n  \\,.\n\\end{equation}",
      "s:rad": "\\begin{split}\n      \\max_{x\\in\\RN}\n      \\ups(x,0)\n      &=\n        \\ups(0,0)\n        =\n        \\frac{C_{*}}{t_{0}^{\\frac{1}{p+m-3}}}\n        \\Big[\n        \\tar(\\varGamma \\log t_{0})^{\\frac{1}{p-1}}\n        -\n        (1-\\nu_{0})\n        \\rat(\\rpol_{0})^{\\frac{1}{p-1}}\n        \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n      \\\\\n      &\\le\n        C_{*}\n        \\tar(\\varGamma \\log t_{0})^{p+m-3}\n        =\n        C_{*}\n        \\tar\\big(2^{\\frac{\\alpha_{2}(p-1)}{p-\\alpha_{2}}}\\excf(\\rpol_{0})\\big)^{p+m-3}\n      \\\\\n      &\\le\n        \\lambda\\tar\\big(2^{\\frac{\\alpha_{2}(p-1)}{p-\\alpha_{2}}}\\lambda^{p+m-3}\\big)^{p+m-3}\n        \\,.\n    \\end{split}\n  \\end{equation}\n  Clearly, by selecting a suitable $\\lambda>0$, both \\eqref{eq:ssb_comp_n} and \\eqref{eq:ssb_comp_nn} are satisfied.\n\\end{proof}\n\n\\section{Transformation to an inhomogeneous density equation}\n\\label{s:rad}\n\nIn this Section we prove Theorem~\\ref{t:transform}.\n\\\\\nEssentially, we want to move the weight out of the right hand side of \\eqref{eq:pde_rad}.\nTo this end, we perform a change of the independent variable $\\rpol$, introducing the function $\\rpolf(s)$, $s>0$, and the new unknown $v(s,t)=U(\\rpolf(s),t)$. Then clearly\n\\begin{equation*}\n  U_{r}\n  =\n  v_{s}\n  \\rpolf_{s}^{-1}\n  \\,,\n  \\quad\n  U^{m-1}\n  \\abs{U_{\\rpol}}^{p-2}\n  U_{\\rpol}\n  =\n  v^{m-1}\n  \\abs{v_{s}}^{p-2}\n  v_{s}\n  \\abs{\\rpolf_{s}}^{-(p-2)}\n  \\rpolf_{s}^{-1}\n  \\,.\n\\end{equation*}\nOn recalling also\n$\\partial/\\partial\\rpol=\\rpolf_{s}^{-1}\\partial/\\partial s$, and on\nassuming $\\rpolf_{s}>0$, we get that the right hand side of\n\\eqref{eq:pde_rad} equals\n\\begin{equation}\n  \\label{eq:div_new}\n  \\pder{}{s}\n  \\big[\n  s^{N-1}\n  v^{m-1}\n  \\abs{v_{s}}^{p-2}\n  v_{s}\n  Y(s)\n  \\big]\n  \\rpolf_{s}^{-1}\n  \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n  \\label{eq:div_coeff}\n  Y(s)\n  :=\n  \\ew(\\rpolf(s))\n  \\frac{\\rpolf(s)^{N-1}}{s^{N-1}}\n  \\rpolf_{s}(s)^{-(p-2)-1}\n  \\,,\n  \\qquad\n  s>0\n  \\,.\n\\end{equation}\nWe choose here $\\rpolf$ as a solution to $Y(s)=1$, that is more specifically to the problem\n\\begin{equation}\n  \\label{eq:rad_rpolf}\n  \\rpolf_{s}(s)\n  =\n  \\Big(\n  \\ew(\\rpolf(s))\n  \\frac{\\rpolf(s)^{N-1}}{s^{N-1}}\n  \\Big)^{\\frac{1}{p-1}}\n  \\,,\n  \\qquad\n  \\rpolf(1)\n  =\n  \\rpol_{*}\n  \\,.\n\\end{equation}\nWe prove in Section~\\ref{s:app} that for a suitable $\\rpol_{*}>0$, depending only on $N$, $p$ and $\\exw$, the solution $\\rpolf(s)$ is defined and increasing over $(0,+\\infty)$ and satisfies\n\\begin{equation*}\n  \\lim_{s\\to 0+}\n  \\rpolf(s)\n  =\n  0\n  \\,,\n  \\qquad\n  \\lim_{s\\to +\\infty}\n  \\rpolf(s)\n  =\n  +\\infty\n  \\,,\n\\end{equation*}\nso that it can be extended to a continuous function over $[0,+\\infty)$. Even more, we have $\\rpolf\\in C^{1}([0,+\\infty))$, with $\\rpolf_{s}(0)=1$. \n\\\\\nThen \\eqref{eq:pde_rad} becomes\n\\begin{equation*}\n  \\rpolf^{N-1}\n  \\ew(\\rpolf)\n  \\rpolf_{s}\n  \\pder{v}{t}\n  =\n  \\pder{}{s}\n  \\big[\n  s^{N-1}\n  v^{m-1}\n  \\abs{v_{s}}^{p-2}\n  v_{s}\n  \\big]\n  \\,,\n\\end{equation*}\nthat is in view of our choice $Y(s)=1$,\n\\begin{equation}\n  \\label{eq:rad_new}\n  \\rpolf_{s}^{p}\n  \\pder{v}{t}\n  =\n  \\frac{1}{s^{N-1}}\n  \\pder{}{s}\n  \\big[\n  s^{N-1}\n  v^{m-1}\n  \\abs{v_{s}}^{p-2}\n  v_{s}\n  \\big]\n  \\,.\n\\end{equation}\nThe right hand side of \\eqref{eq:rad_new} can of course be understood as the divergence in the space variable $x$ such that $\\abs{x}=s$; we use for the sake of simplicity the old variable names. Then if we define\n\\begin{equation}\n  \\label{eq:rad_rw}\n  \\rw(s)=\\rpolf_{s}(s)^{p}\n  \\,,\n\\end{equation}\nwe have the equation \\eqref{eq:rad_N}\nfor the radial function $v$. According to the results on $\\rpolf$ outlined above, $\\rw$ is a continuos positive function on $[0,+\\infty)$.\n\nNext we estimate the asymptotic behavior of $\\rw(s)$. We denote by $C_{i}$ constants depending only on $N$, $p$, $\\exw$, possibly varying from line to line. We get from \\eqref{eq:rad_rpolf} that\n\\begin{equation*}\n  \\frac{1}{(\\ew(\\rpol)\\rpol^{N-1})^{\\frac{1}{p-1}}}\n  =\n  \\frac{1}{\\flopr(\\rpol)^{\\frac{N-1}{p-1}}}\\flopr_{\\rpol}(\\rpol)\n  \\,,\n\\end{equation*}\nwhere $\\flopr=\\rpolf^{(-1)}$.\nThus, on defining\n\\begin{equation*}\n  \\varphi(\\rpol)\n  :=\n  \\int_{\\rpol}^{+\\infty}\n  \\frac{\\di z}{(\\ew(z)z^{N-1})^{\\frac{1}{p-1}}}\n  \\,,  \n\\end{equation*}\nwe get\n\\begin{equation}\n  \\label{eq:rad_rs}\n  \\varphi(\\rpol)\n  =\n  \\int_{\\rpol}^{+\\infty}\n  \\frac{1}{\\flopr(z)^{\\frac{N-1}{p-1}}}\\flopr_{z}(z)\n  \\di z\n  =\n  \\frac{p-1}{N-p}\n  \\flopr(\\rpol)^{-\\frac{N-p}{p-1}}\n  \\,.\n\\end{equation}\nDefine the function\n\\begin{equation}\n  \\label{eq:rad_H}\n  \\Hf(r)\n  =\n  \\Big[\n  \\frac{\\exw(\\rpol)}{\\rpol}\n  \\big(\n  \\ew(\\rpol)\n  \\rpol^{N-1}\n  \\big)^{\\frac{1}{p-1}}\n  \\Big]^{-1}\n  \\,,\n  \\qquad\n  \\rpol>0\n  \\,.\n\\end{equation}\nLet us calculate, recalling that $\\ew(\\rpol)=e^{\\exw(\\rpol)}$,\n\\begin{equation}\n  \\label{eq:rad_H_der}\n  \\begin{split}\n    \\Hf'(\\rpol)\n    &=\n      -\n      \\Big[\n      \\frac{\\exw(\\rpol)}{\\rpol}\n      \\big(\n      \\ew(\\rpol)\n      \\rpol^{N-1}\n      \\big)^{\\frac{1}{p-1}}\n      \\Big]^{-2}\n      \\big(\n      \\ew(\\rpol)\n      \\rpol^{N-1}\n      \\big)^{\\frac{1}{p-1}}\n      \\\\\n    &\\quad\n      \\times\\Big\\{\n      \\Big[\n      \\frac{\\exw'(\\rpol)}{\\rpol}\n      -\n      \\frac{\\exw(\\rpol)}{\\rpol^{2}}\n      \\Big]\n      +\n      \\frac{\\exw(\\rpol)}{(p-1)\\rpol}\n      \\big(\n      \\ew(\\rpol)\n      \\rpol^{N-1}\n      \\big)^{-1}\n  \\\\\n    &\\qquad\n      \\times\\big[\n      \\exw'(\\rpol)\n      \\ew(\\rpol)\n      \\rpol^{N-1}\n      +\n      (N-1)\n      \\ew(\\rpol)\n      \\rpol^{N-2}\n      \\big]\n      \\Big\\}\n      \\\\\n    &=\n      -\n      \\Big[\n      \\frac{\\exw(\\rpol)}{\\rpol}\n      \\big(\n      \\ew(\\rpol)\n      \\rpol^{N-1}\n      \\big)^{\\frac{1}{p-1}}\n      \\Big]^{-1}\n      \\Big\\{\n      \\Big[\n      \\frac{\\exw'(\\rpol)}{\\exw(\\rpol)}\n      -\n      \\frac{1}{\\rpol}\n      \\Big]\n      \\\\\n    &\\quad\n      +\n      \\frac{1}{p-1}\n      \\Big[\n      \\exw'(\\rpol)\n      +\n      \\frac{N-1}{\\rpol}\n      \\Big]\n      \\Big\\}\n      \\,.\n  \\end{split}",
      "eq:sup_sup_dens_n": "\\begin{equation}\n    \\label{eq:sup_sup_dens_n}\n    v(x,t)\n    \\le\n    \\ups(\\rpolf(\\abs{x}),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}",
      "eq:pde": "\\begin{alignat}{2}\n  \\label{eq:pde}\n  \\ew(x)\n  \\pder{u}{t}\n  -\n  \\Div\\big(\n  \\ew(x)\n  u^{m-1}\n  \\abs{\\grad u}^{p-2}\n  \\grad u\n  \\big)\n  &=\n    0\n    \\,,\n  &\\qquad&\n           \\text{in $S_{T}$,}\n  \\\\\n  \\label{eq:initd}\n  u(x,0)\n  &=\n    u_{0}(x)\n    \\,,\n  &\\qquad&\n           x\\in\\RN\n           \\,.\n\\end{alignat}"
    },
    "pre_theorem_intro_text_len": 11567,
    "pre_theorem_intro_text": "\\label{s:intro}\n\nWe look at the Cauchy problem for the doubly degenerate weighted parabolic\nequation\n\\begin{alignat}{2}\n  \\label{eq:pde}\n  f(x)\n  \\pder{u}{t}\n  -\n  \\Div\\big(\n  f(x)\n  u^{m-1}\n  \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n  \\operatorname{\\nabla} u\n  \\big)\n  &=\n    0\n    \\,,\n  &\\qquad&\n           \\text{in $S_{T}$,}\n  \\\\\n  \\label{eq:initd}\n  u(x,0)\n  &=\n    u_{0}(x)\n    \\,,\n  &\\qquad&\n           x\\in\\R^{\\SpDim}\n           \\,.\n\\end{alignat}\nHere $S_{T}=\\R^{\\SpDim}\\times (0,T)$, $0<T\\le +\\infty$, $x=(x_{1},\\dots,x_{N})$, $\\operatorname{\\nabla} u$ [respectively, $\\Div$] is the gradient [respectively, the divergence] with respect to $x$, and we denote\n\\begin{equation}\n  \\label{eq:expweight}\n  f(x)\n  =\n  e^{g(\\lvertx\\rvert)}\n  \\,,\n  \\qquad\n  x\\in\\R^{\\SpDim}\n  \\,.\n\\end{equation}\nWe assume in this paper, even without further reference, that $u\\ge 0$, $u_{0}\\ge 0$, $u_{0}\\in L^{\\infty}(\\R^{\\SpDim})$ is compactly supported, and that\n\\begin{equation}\n  \\label{eq:degen}\n  1<p<N\n  \\,,\n  \\qquad\n  p+m-3>0\n  \\,.\n\\end{equation}\nOn $g$ we assume that $g\\in C([0,+\\infty])\\cap C^{1}((0,+\\infty))$ is such that $g(0)=0$, $g(s)>0$ for $s>0$, and for given $0<\\alpha_{1}\\le \\alpha_{2}<p$\n\\begin{equation}\n  \\label{eq:powerlike}\n  \\alpha_{1}\n  \\frac{g(s)}{s}\n  \\le\n  g'(s)\n  \\le\n  \\alpha_{2}\n  \\frac{g(s)}{s}\n  \\,,\n  \\qquad\n  s>0\n  \\,.\n\\end{equation}\n\nIn \\cite{Andreucci:Tedeev:2025} we obtained sup bounds for solutions\nto \\eqref{eq:pde}--\\eqref{eq:initd} in the range $\\alpha_{2}<1$, and\nestimates of the support for all $\\alpha_{2}<Np/(p-1)$. The approach\nof \\cite{Andreucci:Tedeev:2025} relies on suitable embedding\ninequalities and iterative estimates, and is valid for radial or\ncompactly supported solutions; see also in this connection\n\\cite{Andreucci:Tedeev:2015} and \\cite{Andreucci:Tedeev:2022b}. Here\nwe present instead new results both for the decay of the solutions in\nthe $L^{\\infty}$ norm and for the finite speed of propagation, by\nmeans of comparison with explicit supersolutions and subsolutions, in\nthe range $\\alpha_{1}$, $\\alpha_{2}\\in (0,p)$. Our approach is\ninspired by the paper \\cite{Grillo:Muratori:Vazquez:2017}, where the\nauthors considered the porous media equation on manifolds. We treat\nhere the case of the doubly nonlinear equation, and, as a further\nnovelty even in the case of the porous media equation, we deal\nwith the case of non-power nonlinearities (satisfying\n\\eqref{eq:powerlike}). For example the Zygmund function\n\\begin{equation*}\n  g(s)\n  =\n  s^{\\alpha}\n  [\\log(s+c)]^{\\beta}\n  \\,,\n  \\quad\n  s\\ge 0\n  \\,,\n  \\alpha>0\n  \\,,\n  \\beta>0\n  \\,,\n  c>1\n  \\,,\n\\end{equation*}\nfalls in this class with $\\alpha_{1}=\\alpha$, $\\alpha_{2}=\\alpha+\\beta$.\n\\\\\nWe show here that the asymptotic rates found in \\cite{Andreucci:Tedeev:2025} are indeed optimal. These bounds are given by\n\\begin{equation}\n  \\label{eq:sup_decay}\n  u(x,t)\n  \\le\n  \\gamma\n  \\Big[\n  \\frac{g^{(-1)}(\\log t)^{p}}{t\\log t}\n  \\Big]^{\\frac{1}{p+m-3}}\n  \\,,\n  \\quad\n  x\\in\\R^{\\SpDim}\n  \\,,\n  t\\gg 1\n  \\,,\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:fsp}\n  \\bar R(t)\n  \\le\n  \\gamma\n  g^{(-1)}(\\log t)\n  \\,,\n  \\quad\n  t\\gg 1\n  \\,,\n\\end{equation}\nwhere, for solutions whose support is bounded for $t\\ge 0$ we define\n\\begin{equation}\n  \\label{eq:main_R}\n  \\bar R(t)\n  =\n  \\inf\\{\n  r>0\n  \\mid\n  \\supp u(t) \\subset B_{r}(0)\n  \\}\n  \\,.\n\\end{equation}\nHere $\\gamma>0$ is a constant depending on $u_{0}$.\nAccording to our present results, the estimates in \\eqref{eq:sup_decay}, \\eqref{eq:fsp} are indeed sharp, at least for $\\alpha_{2}<p$, and, even more, they are extended to such a range for bounded and compactly supported initial data. In addition the comparison results obtained here yield more precise pointwise estimates, tracking the dependence on $\\lvertx\\rvert$, in comparison to the bounds for the $L^{\\infty}(\\R^{\\SpDim})$ obtained in the quoted papers of the present authors. This is an obvious effect of the barrier function method, which however is in principle less general than the approach through integral inequalities. \n\nAs in \\cite{Grillo:Muratori:Vazquez:2017}, we may employ our barrier functions to investigate an equation with inhomogeneous density, i.e.,\n\\begin{equation}\n  \\label{eq:rad_N}\n  \\rho(\\lvertx\\rvert)\n  \\pder{v}{t}\n  =\n  \\Div(\n  v^{m-1}\n  \\lvert\\operatorname{\\nabla} v\\rvert^{p-2}\n  \\operatorname{\\nabla} v\n  )\n  \\,,\n  \\qquad\n  x\\in\\numberspacefont{R}^{N}\n  \\,,\n  t>0\n  \\,.\n\\end{equation}\nFor the concept of solutions of \\eqref{eq:rad_N}, and other related\ninformation, we refer the reader to \\cite{Andreucci:Tedeev:2021b}. In that paper, in fact in a more general setting, we considered equations modeled after \\eqref{eq:rad_N}, with, roughly speaking, $\\rho(s)$ behaving like $s^{-\\alpha}$, $\\alpha<p$ for large $s>0$. Here we deal with the new case \\eqref{eq:rad_rw_est}, in which essentially $\\rho(s)$ behaves like the power $s^{-p}$ multiplied by a powerlike function of $\\log s$; this factor may be increasing or decreasing in $s$ according to the values of $\\alpha_{1}$, $\\alpha_{2}$. We refer to Remark~\\ref{r:rad_N} for further comments on this point.\n\nThe Cauchy problem for nonlinear, and linear, parabolic equations\ninvolving coefficients strongly depending on the space variable has\nreceived great attention in the literature. We quote\n\\cite{Grigoryan:2006} which analyzes stochastical completeness and\nother qualitative properties of weighted manifolds. In\n\\cite{Grillo:Muratori:2016} the authors study the smoothness and\ntemporal decay estimates of the solution to the Cauchy problem for the\nporous media equation on Cartan-Hadamard manifolds supporting\nPoincar\\'{e} inequality. See also \\cite{Grillo:Muratori:Vazquez:2017}\nfor precise two-sides space-time estimates of the solution to the  same problem,\nin the setting of\nvarious classes of Cartan-Hadamard manifolds, \n\\cite{Muratori:2021} for a survey of results, and again\n\\cite{Muratori:Roncoroni:2022}\nfor new Sobolev type inequalities as well as sharp bounds of solutions.\nWe also quote \\cite{Tedeev:2007}, dealing with the influence of an inhomogeneous density on\nthe interface blow up phenomenon, on finite speed of\npropagation, and in general on the \nbehavior of  the solution to the Cauchy problem for doubly degenerate\nparabolic equations. \nFinally\n\\cite{Tedeev:2025} investigates the blow up of the solution itself in the same setting.\n\n\\subsection{Main results}\n\\label{s:main}\n\nWe define here the spaces $L^{p}_{f}(\\R^{\\SpDim})$ in a standard way, as the $L^{p}$ spaces relative to the measure $f(x)\\di x$.\n\n\\begin{definition}\n  \\label{d:weaksol}\n  A function $u\\ge 0$ is a weak solution to \\eqref{eq:pde} if\n  \\begin{equation*}\n    u\\in C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim})) \\cap L^{\\infty}(\\R^{\\SpDim}\\times(\\bar t,+\\infty))\n  \\end{equation*}\n  for all $\\bar t>0$, and\n  \\begin{equation*}\n    \\operatorname{\\nabla} u^{\\frac{p+m-2}{p-1}}\n    \\in\n    L^{p}_{\\textup{loc}}\n    (S_{\\infty})\n    \\,.\n  \\end{equation*}\n  In addition we require the standard integral formulation, i.e., for all $\\eta\\in C_{0}^{\\infty}(S_{\\infty})$,\n  \\begin{equation}\n    \\label{eq:weaksol_a}\n    \\int_{0}^{+\\infty}\n    \\int_{\\R^{\\SpDim}}\n    \\{\n    -\n    u\n    \\eta_{t}\n    +\n    u^{m-1}\n    \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n    \\operatorname{\\nabla} u\n    \\operatorname{\\nabla} \\eta\n    \\}\n    f(x)\n    \\di x\n    \\di t\n    =\n    0\n    \\,.\n  \\end{equation}\n  \\\\\n  If we also have $u_{t}\\in L^{\\infty}_{\\textup{loc}}(0,+\\infty); L^{1}_{f\\,\\textup{loc}}(\\R^{\\SpDim}))$, then $u$ is a strong solution.\n  \\\\\n  Supersolutions [respectively, subsolutions] to \\eqref{eq:pde} are defined similarly, with the difference that, for $\\eta\\ge0$, we require in \\eqref{eq:weaksol_a} an inequality sign $\\ge $ [respectively, $\\le$].\n  \\\\\n  Solutions to the Cauchy problem \\eqref{eq:pde}--\\eqref{eq:initd} are solutions to \\eqref{eq:pde} satisfying $u(0)=u_{0}$ in the sense of $C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim}))$.\n\\end{definition}\n\nExistence and comparison results for strong solutions to the problems which we consider here can be proved, at least for sufficiently regular initial data, according for example to the methods of \\cite{Degtyarev:Tedeev:2012}, \\cite{Tsutsumi:1988}; here we focus rather on estimates of such solutions.\n\nWhen we consider radial solutions of the form $u(x,t)=U(r,t)$,\n$r=\\lvertx\\rvert$, the equation \\eqref{eq:pde} can be written as\n\\begin{equation}\n  \\label{eq:pde_rad}\n  r^{N-1}\n  f(r)\n  \\pder{U}{t}\n  =\n  \\pder{}{r}\n  \\big[\n  r^{N-1}\n  f(r)\n  U^{m-1}\n  \\abs{U_{r}}^{p-2}\n  U_{r}\n  \\big]\n  \\,,\n  \\qquad\n  r>0\n  \\,,\n  t>0\n  \\,,\n\\end{equation}\nwhich in turn can be rephrased as\n\\begin{equation}\n  \\label{eq:sss_rad}\n  \\begin{split}\n    \\pder{U}{t}\n    &=\n      \\pder{}{r}\n      \\big[\n      U^{m-1}\n      \\abs{U_{r}}^{p-2}\n      U_{r}\n      \\big]\n    \\\\\n    &\\quad+\n      \\Big(\n      \\frac{N-1}{r}\n      +\n      g'\n      \\Big)\n      \\big[\n      U^{m-1}\n      \\abs{U_{r}}^{p-2}\n      U_{r}\n      \\big]\n      \\,,\n      \\qquad\n      r>0\n      \\,,\n      t>0\n      \\,.\n  \\end{split}\n\\end{equation}\nOf course the differential inequalities characterizing radial supersolutions and subsolutions can be rephrased similarly.\nSpecifically, we consider supersolutions and subsolutions of the form\n\\begin{alignat}2\n  \\label{eq:sss_sup}\n  \\widetilde u(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    E(\\tau)^{\\frac{1}{p-1}}\n    -\n    J(r)^{\\frac{1}{p-1}}\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            r\\ge \\rpol_{0}\n            \\,,\n  \\\\\n  \\label{eq:sss_ups_2}\n  \\widetilde u(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    E(\\tau)^{\\frac{1}{p-1}}\n    -\n    I(r)\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            r<\\rpol_{0}\n            \\,,\n\\end{alignat}\nwhere $r=\\lvertx\\rvert$, and\n\\begin{gather*}\n  G(r)\n  =\n  \\int_{0}^{r}\n  \\frac{g(s)}{s}\n  \\di s\n  \\,,\n  \\quad\n  r\\ge 0\n  \\,;\n  \\qquad\n  J(r)\n  =\n  \\frac{r^{p}}{G(r)}\n  \\,,\n  \\quad\n  r>0\n  \\,.\n  \\\\\n  \\tau\n  =\n  \\varGamma\n  \\log(t+t_{0})\n  \\,,\n  \\quad\n  t\\ge 0\n  \\,;\n  \\\\\n  E(\\tau)\n  =\n  J(G^{(-1)}(\\tau))\n  =\n  \\frac{G^{(-1)}(\\tau)^{p}}{\\tau}\n  \\,,\n  \\quad\n  \\tau>0\n  \\,;\n  \\\\\n  I(r)\n  =\n  \\nu_{0}\n  \\Big(\n  \\frac{r^{p}}{G(\\rpol_{0})}\n  \\Big)^{\\frac{1}{p-1}}\n  +\n  (1-\\nu_{0})\n  J(\\rpol_{0})^{\\frac{1}{p-1}}\n  \\,,\n  \\quad\n  r>0\n  \\,.\n\\end{gather*}\nNote that the integral defining $G$ converges according to \\eqref{eq:powerlike_lt}.\n\nHere $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and $\\nu_{0}$ are positive constants to be chosen. In particular, $\\nu_{0}\\in(0,1)$ will be selected so that $\\widetilde u$ and $\\ups_{r}$ are continuous even at $r=\\rpol_{0}$; in fact, clearly, the continuity of $\\widetilde u$ holds true for any $\\nu_{0}\\in(0,1)$.\n\nWe also remark that the radius $\\tilde r(t)$ of the support of $\\widetilde u(t)$ is characterized by $E(\\tau)=J(\\tilder(t))$, i.e., by $\\tilder(t)=G^{(-1)}(\\tau)$; according to Lemma~\\ref{l:sss_excf} and \\eqref{eq:sss_powerlike_gt_excf} this yields\n\\begin{equation}\n  \\label{eq:sup_sub_radius}\n  \\gamma_{1}\n  g^{(-1)}(\\log t)\n  \\le\n  \\tilde r(t)\n  \\le\n  \\gamma_{2}\n  g^{(-1)}(\\log t)\n  \\,,\n  \\qquad\n  t>t_{0}\n  \\,,\n\\end{equation}\nfor two suitable constants $\\gamma_{1}$, $\\gamma_{2}>0$.\n\n\\begin{proposition}\n  \\label{p:main_sup_sub}\n  The constants $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and\n  $\\nu_{0}$ can be selected so that $\\widetilde u$ as defined in\n  \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a strong supersolution to\n  \\eqref{eq:pde}. Alternatively, we may select such constants so that\n  $\\widetilde u$ is a strong subsolution to \\eqref{eq:pde}.\n\\end{proposition}",
    "context": "We look at the Cauchy problem for the doubly degenerate weighted parabolic\nequation\n\\begin{alignat}{2}\n \\label{eq:pde}\n f(x)\n \\pder{u}{t}\n -\n \\Div\\big(\n f(x)\n u^{m-1}\n \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n \\operatorname{\\nabla} u\n \\big)\n &=\n 0\n \\,,\n &\\qquad&\n \\text{in $S_{T}$,}\n \\\\\n \\label{eq:initd}\n u(x,0)\n &=\n u_{0}(x)\n \\,,\n &\\qquad&\n x\\in\\R^{\\SpDim}\n \\,.\n\\end{alignat}\nHere $S_{T}=\\R^{\\SpDim}\\times (0,T)$, $0<T\\le +\\infty$, $x=(x_{1},\\dots,x_{N})$, $\\operatorname{\\nabla} u$ [respectively, $\\Div$] is the gradient [respectively, the divergence] with respect to $x$, and we denote\n\\begin{equation}\n \\label{eq:expweight}\n f(x)\n =\n e^{g(\\lvertx\\rvert)}\n \\,,\n \\qquad\n x\\in\\R^{\\SpDim}\n \\,.\n\\end{equation}\nWe assume in this paper, even without further reference, that $u\\ge 0$, $u_{0}\\ge 0$, $u_{0}\\in L^{\\infty}(\\R^{\\SpDim})$ is compactly supported, and that\n\\begin{equation}\n \\label{eq:degen}\n 1<p<N\n \\,,\n \\qquad\n p+m-3>0\n \\,.\n\\end{equation}\nOn $g$ we assume that $g\\in C([0,+\\infty])\\cap C^{1}((0,+\\infty))$ is such that $g(0)=0$, $g(s)>0$ for $s>0$, and for given $0<\\alpha_{1}\\le \\alpha_{2}<p$\n\\begin{equation}\n \\label{eq:powerlike}\n \\alpha_{1}\n \\frac{g(s)}{s}\n \\le\n g'(s)\n \\le\n \\alpha_{2}\n \\frac{g(s)}{s}\n \\,,\n \\qquad\n s>0\n \\,.\n\\end{equation}\n\n\\begin{definition}\n \\label{d:weaksol}\n A function $u\\ge 0$ is a weak solution to \\eqref{eq:pde} if\n \\begin{equation*}\n u\\in C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim})) \\cap L^{\\infty}(\\R^{\\SpDim}\\times(\\bar t,+\\infty))\n \\end{equation*}\n for all $\\bar t>0$, and\n \\begin{equation*}\n \\operatorname{\\nabla} u^{\\frac{p+m-2}{p-1}}\n \\in\n L^{p}_{\\textup{loc}}\n (S_{\\infty})\n \\,.\n \\end{equation*}\n In addition we require the standard integral formulation, i.e., for all $\\eta\\in C_{0}^{\\infty}(S_{\\infty})$,\n \\begin{equation}\n \\label{eq:weaksol_a}\n \\int_{0}^{+\\infty}\n \\int_{\\R^{\\SpDim}}\n \\{\n -\n u\n \\eta_{t}\n +\n u^{m-1}\n \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n \\operatorname{\\nabla} u\n \\operatorname{\\nabla} \\eta\n \\}\n f(x)\n \\di x\n \\di t\n =\n 0\n \\,.\n \\end{equation}\n \\\\\n If we also have $u_{t}\\in L^{\\infty}_{\\textup{loc}}(0,+\\infty); L^{1}_{f\\,\\textup{loc}}(\\R^{\\SpDim}))$, then $u$ is a strong solution.\n \\\\\n Supersolutions [respectively, subsolutions] to \\eqref{eq:pde} are defined similarly, with the difference that, for $\\eta\\ge0$, we require in \\eqref{eq:weaksol_a} an inequality sign $\\ge $ [respectively, $\\le$].\n \\\\\n Solutions to the Cauchy problem \\eqref{eq:pde}--\\eqref{eq:initd} are solutions to \\eqref{eq:pde} satisfying $u(0)=u_{0}$ in the sense of $C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim}))$.\n\\end{definition}\n\nWhen we consider radial solutions of the form $u(x,t)=U(r,t)$,\n$r=\\lvertx\\rvert$, the equation \\eqref{eq:pde} can be written as\n\\begin{equation}\n \\label{eq:pde_rad}\n r^{N-1}\n f(r)\n \\pder{U}{t}\n =\n \\pder{}{r}\n \\big[\n r^{N-1}\n f(r)\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\,,\n \\qquad\n r>0\n \\,,\n t>0\n \\,,\n\\end{equation}\nwhich in turn can be rephrased as\n\\begin{equation}\n \\label{eq:sss_rad}\n \\begin{split}\n \\pder{U}{t}\n &=\n \\pder{}{r}\n \\big[\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\\\\n &\\quad+\n \\Big(\n \\frac{N-1}{r}\n +\n g'\n \\Big)\n \\big[\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\,,\n \\qquad\n r>0\n \\,,\n t>0\n \\,.\n \\end{split}\n\\end{equation}\nOf course the differential inequalities characterizing radial supersolutions and subsolutions can be rephrased similarly.\nSpecifically, we consider supersolutions and subsolutions of the form\n\\begin{alignat}2\n \\label{eq:sss_sup}\n \\widetilde u(x,t)\n &=\n \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n \\Big[\n E(\\tau)^{\\frac{1}{p-1}}\n -\n J(r)^{\\frac{1}{p-1}}\n \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n \\,,\n &\\quad&\n r\\ge \\rpol_{0}\n \\,,\n \\\\\n \\label{eq:sss_ups_2}\n \\widetilde u(x,t)\n &=\n \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n \\Big[\n E(\\tau)^{\\frac{1}{p-1}}\n -\n I(r)\n \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n \\,,\n &\\quad&\n r<\\rpol_{0}\n \\,,\n\\end{alignat}\nwhere $r=\\lvertx\\rvert$, and\n\\begin{gather*}\n G(r)\n =\n \\int_{0}^{r}\n \\frac{g(s)}{s}\n \\di s\n \\,,\n \\quad\n r\\ge 0\n \\,;\n \\qquad\n J(r)\n =\n \\frac{r^{p}}{G(r)}\n \\,,\n \\quad\n r>0\n \\,.\n \\\\\n \\tau\n =\n \\varGamma\n \\log(t+t_{0})\n \\,,\n \\quad\n t\\ge 0\n \\,;\n \\\\\n E(\\tau)\n =\n J(G^{(-1)}(\\tau))\n =\n \\frac{G^{(-1)}(\\tau)^{p}}{\\tau}\n \\,,\n \\quad\n \\tau>0\n \\,;\n \\\\\n I(r)\n =\n \\nu_{0}\n \\Big(\n \\frac{r^{p}}{G(\\rpol_{0})}\n \\Big)^{\\frac{1}{p-1}}\n +\n (1-\\nu_{0})\n J(\\rpol_{0})^{\\frac{1}{p-1}}\n \\,,\n \\quad\n r>0\n \\,.\n\\end{gather*}\nNote that the integral defining $G$ converges according to \\eqref{eq:powerlike_lt}.\n\nWe also remark that the radius $\\tilde r(t)$ of the support of $\\widetilde u(t)$ is characterized by $E(\\tau)=J(\\tilder(t))$, i.e., by $\\tilder(t)=G^{(-1)}(\\tau)$; according to Lemma~\\ref{l:sss_excf} and \\eqref{eq:sss_powerlike_gt_excf} this yields\n\\begin{equation}\n \\label{eq:sup_sub_radius}\n \\gamma_{1}\n g^{(-1)}(\\log t)\n \\le\n \\tilde r(t)\n \\le\n \\gamma_{2}\n g^{(-1)}(\\log t)\n \\,,\n \\qquad\n t>t_{0}\n \\,,\n\\end{equation}\nfor two suitable constants $\\gamma_{1}$, $\\gamma_{2}>0$.\n\n\\begin{proposition}\n \\label{p:main_sup_sub}\n The constants $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and\n $\\nu_{0}$ can be selected so that $\\widetilde u$ as defined in\n \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a strong supersolution to\n \\eqref{eq:pde}. Alternatively, we may select such constants so that\n $\\widetilde u$ is a strong subsolution to \\eqref{eq:pde}.\n\\end{proposition}",
    "full_context": "We look at the Cauchy problem for the doubly degenerate weighted parabolic\nequation\n\\begin{alignat}{2}\n \\label{eq:pde}\n f(x)\n \\pder{u}{t}\n -\n \\Div\\big(\n f(x)\n u^{m-1}\n \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n \\operatorname{\\nabla} u\n \\big)\n &=\n 0\n \\,,\n &\\qquad&\n \\text{in $S_{T}$,}\n \\\\\n \\label{eq:initd}\n u(x,0)\n &=\n u_{0}(x)\n \\,,\n &\\qquad&\n x\\in\\R^{\\SpDim}\n \\,.\n\\end{alignat}\nHere $S_{T}=\\R^{\\SpDim}\\times (0,T)$, $0<T\\le +\\infty$, $x=(x_{1},\\dots,x_{N})$, $\\operatorname{\\nabla} u$ [respectively, $\\Div$] is the gradient [respectively, the divergence] with respect to $x$, and we denote\n\\begin{equation}\n \\label{eq:expweight}\n f(x)\n =\n e^{g(\\lvertx\\rvert)}\n \\,,\n \\qquad\n x\\in\\R^{\\SpDim}\n \\,.\n\\end{equation}\nWe assume in this paper, even without further reference, that $u\\ge 0$, $u_{0}\\ge 0$, $u_{0}\\in L^{\\infty}(\\R^{\\SpDim})$ is compactly supported, and that\n\\begin{equation}\n \\label{eq:degen}\n 1<p<N\n \\,,\n \\qquad\n p+m-3>0\n \\,.\n\\end{equation}\nOn $g$ we assume that $g\\in C([0,+\\infty])\\cap C^{1}((0,+\\infty))$ is such that $g(0)=0$, $g(s)>0$ for $s>0$, and for given $0<\\alpha_{1}\\le \\alpha_{2}<p$\n\\begin{equation}\n \\label{eq:powerlike}\n \\alpha_{1}\n \\frac{g(s)}{s}\n \\le\n g'(s)\n \\le\n \\alpha_{2}\n \\frac{g(s)}{s}\n \\,,\n \\qquad\n s>0\n \\,.\n\\end{equation}\n\n\\begin{definition}\n \\label{d:weaksol}\n A function $u\\ge 0$ is a weak solution to \\eqref{eq:pde} if\n \\begin{equation*}\n u\\in C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim})) \\cap L^{\\infty}(\\R^{\\SpDim}\\times(\\bar t,+\\infty))\n \\end{equation*}\n for all $\\bar t>0$, and\n \\begin{equation*}\n \\operatorname{\\nabla} u^{\\frac{p+m-2}{p-1}}\n \\in\n L^{p}_{\\textup{loc}}\n (S_{\\infty})\n \\,.\n \\end{equation*}\n In addition we require the standard integral formulation, i.e., for all $\\eta\\in C_{0}^{\\infty}(S_{\\infty})$,\n \\begin{equation}\n \\label{eq:weaksol_a}\n \\int_{0}^{+\\infty}\n \\int_{\\R^{\\SpDim}}\n \\{\n -\n u\n \\eta_{t}\n +\n u^{m-1}\n \\lvert\\operatorname{\\nabla} u\\rvert^{p-2}\n \\operatorname{\\nabla} u\n \\operatorname{\\nabla} \\eta\n \\}\n f(x)\n \\di x\n \\di t\n =\n 0\n \\,.\n \\end{equation}\n \\\\\n If we also have $u_{t}\\in L^{\\infty}_{\\textup{loc}}(0,+\\infty); L^{1}_{f\\,\\textup{loc}}(\\R^{\\SpDim}))$, then $u$ is a strong solution.\n \\\\\n Supersolutions [respectively, subsolutions] to \\eqref{eq:pde} are defined similarly, with the difference that, for $\\eta\\ge0$, we require in \\eqref{eq:weaksol_a} an inequality sign $\\ge $ [respectively, $\\le$].\n \\\\\n Solutions to the Cauchy problem \\eqref{eq:pde}--\\eqref{eq:initd} are solutions to \\eqref{eq:pde} satisfying $u(0)=u_{0}$ in the sense of $C([0,+\\infty); L^{1}_{f}(\\R^{\\SpDim}))$.\n\\end{definition}\n\nWhen we consider radial solutions of the form $u(x,t)=U(r,t)$,\n$r=\\lvertx\\rvert$, the equation \\eqref{eq:pde} can be written as\n\\begin{equation}\n \\label{eq:pde_rad}\n r^{N-1}\n f(r)\n \\pder{U}{t}\n =\n \\pder{}{r}\n \\big[\n r^{N-1}\n f(r)\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\,,\n \\qquad\n r>0\n \\,,\n t>0\n \\,,\n\\end{equation}\nwhich in turn can be rephrased as\n\\begin{equation}\n \\label{eq:sss_rad}\n \\begin{split}\n \\pder{U}{t}\n &=\n \\pder{}{r}\n \\big[\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\\\\n &\\quad+\n \\Big(\n \\frac{N-1}{r}\n +\n g'\n \\Big)\n \\big[\n U^{m-1}\n \\abs{U_{r}}^{p-2}\n U_{r}\n \\big]\n \\,,\n \\qquad\n r>0\n \\,,\n t>0\n \\,.\n \\end{split}\n\\end{equation}\nOf course the differential inequalities characterizing radial supersolutions and subsolutions can be rephrased similarly.\nSpecifically, we consider supersolutions and subsolutions of the form\n\\begin{alignat}2\n \\label{eq:sss_sup}\n \\widetilde u(x,t)\n &=\n \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n \\Big[\n E(\\tau)^{\\frac{1}{p-1}}\n -\n J(r)^{\\frac{1}{p-1}}\n \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n \\,,\n &\\quad&\n r\\ge \\rpol_{0}\n \\,,\n \\\\\n \\label{eq:sss_ups_2}\n \\widetilde u(x,t)\n &=\n \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n \\Big[\n E(\\tau)^{\\frac{1}{p-1}}\n -\n I(r)\n \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n \\,,\n &\\quad&\n r<\\rpol_{0}\n \\,,\n\\end{alignat}\nwhere $r=\\lvertx\\rvert$, and\n\\begin{gather*}\n G(r)\n =\n \\int_{0}^{r}\n \\frac{g(s)}{s}\n \\di s\n \\,,\n \\quad\n r\\ge 0\n \\,;\n \\qquad\n J(r)\n =\n \\frac{r^{p}}{G(r)}\n \\,,\n \\quad\n r>0\n \\,.\n \\\\\n \\tau\n =\n \\varGamma\n \\log(t+t_{0})\n \\,,\n \\quad\n t\\ge 0\n \\,;\n \\\\\n E(\\tau)\n =\n J(G^{(-1)}(\\tau))\n =\n \\frac{G^{(-1)}(\\tau)^{p}}{\\tau}\n \\,,\n \\quad\n \\tau>0\n \\,;\n \\\\\n I(r)\n =\n \\nu_{0}\n \\Big(\n \\frac{r^{p}}{G(\\rpol_{0})}\n \\Big)^{\\frac{1}{p-1}}\n +\n (1-\\nu_{0})\n J(\\rpol_{0})^{\\frac{1}{p-1}}\n \\,,\n \\quad\n r>0\n \\,.\n\\end{gather*}\nNote that the integral defining $G$ converges according to \\eqref{eq:powerlike_lt}.\n\nWe also remark that the radius $\\tilde r(t)$ of the support of $\\widetilde u(t)$ is characterized by $E(\\tau)=J(\\tilder(t))$, i.e., by $\\tilder(t)=G^{(-1)}(\\tau)$; according to Lemma~\\ref{l:sss_excf} and \\eqref{eq:sss_powerlike_gt_excf} this yields\n\\begin{equation}\n \\label{eq:sup_sub_radius}\n \\gamma_{1}\n g^{(-1)}(\\log t)\n \\le\n \\tilde r(t)\n \\le\n \\gamma_{2}\n g^{(-1)}(\\log t)\n \\,,\n \\qquad\n t>t_{0}\n \\,,\n\\end{equation}\nfor two suitable constants $\\gamma_{1}$, $\\gamma_{2}>0$.\n\n\\begin{proposition}\n \\label{p:main_sup_sub}\n The constants $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and\n $\\nu_{0}$ can be selected so that $\\widetilde u$ as defined in\n \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a strong supersolution to\n \\eqref{eq:pde}. Alternatively, we may select such constants so that\n $\\widetilde u$ is a strong subsolution to \\eqref{eq:pde}.\n\\end{proposition}\n\n\\begin{theorem}\n \\label{t:sub_sub_dens}\n Let $v$ be a strong solution to \\eqref{eq:rad_N}, for the specific $\\rw$ found in Theorem~\\ref{t:transform}, under assumptions \\eqref{eq:degen} and \\eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported, bounded and\n$v(x,0)\\ge \\eps>0$ for $\\abs{x}\\le\\ell$, $\\ell>0$.\n \\\\\n Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups(\\rpolf(\\abs{x}),t)$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution \\eqref{eq:rad_N}, and\n \\begin{equation}\n \\label{eq:sub_sub_dens_n}\n v(x,t)\n \\ge\n \\ups(\\rpolf(\\abs{x}),t)\n \\,,\n \\qquad\n (x,t)\\in S_{\\infty}\n \\,.\n \\end{equation}\n We have also\n \\begin{equation}\n \\label{eq:sub_sub_dens_nn}\n \\supp v(t)\n \\supset\n B_{\\tilde R(t)}(0)\n \\,,\n \\quad\n \\log\\tilde R(t)\n \\sim\n \\log t\n \\,,\n \\qquad\n t>t_{0}\n \\,,\n \\end{equation}\n The constants involved in \\eqref{eq:sub_sub_dens_n}--\\eqref{eq:sub_sub_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.\n\\end{theorem}\n\nWe work in the set $\\trb>0$. Then we compute\n\\begin{equation}\n \\label{eq:sss_ups_diffs}\n \\begin{split}\n L_{1}\n &:=\n \\ups^{m-1}\n \\abs{\\ups_{\\rpol}}^{p-2}\n \\ups_{\\rpol}\n \\\\\n &=\n -\n \\frac{\n C_{*}^{p+m-2}\n (\\nu_{0}p)^{p-1}\n }{\n (p+m-3)^{p-1}\n }\n \\,\n \\frac{\n \\trb(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}}\n }{\n (t+t_{0})^{\\frac{p+m-2}{p+m-3}}\n }\n \\frac{\\rpol}{\\excf(\\rpol_{0})}\n \\,.\n \\end{split}\n\\end{equation}\nWe recall that, in order for $\\ups$ to be a supersolution, we need to check (see also \\eqref{eq:sss_ups_ineq})\n\\begin{equation}\n \\label{eq:sss_ups_ineqs}\n \\pder{\\ups}{t}\n \\ge\n \\pder{L_{1}}{\\rpol}\n +\n L_{1}\n \\Big(\n \\frac{N-1}{\\rpol}\n +\n \\exw'(\\rpol)\n \\Big)\n \\,,\n \\qquad\n \\rpol<\\rpol_{0}\n \\,.\n\\end{equation}\nThe time derivative $\\ups_{t}$ is given by \\eqref{eq:sss_ups_tder},\nwhen we formally substitute $\\brt(\\rpol,\\tau)$ with $\\trb(\\rpol,\\tau)$.\nNext we calculate\n\\begin{equation}\n \\label{eq:sss_ups_diffs2}\n \\begin{split}\n \\pder{L_{1}}{\\rpol}\n &=\n \\frac{\n C_{*}^{p+m-2}\n (\\nu_{0}p)^{p-1}\n }{\n (p+m-3)^{p-1}\n }\n \\,\n \\frac{\n \\trb(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}-1}\n }{\n (t+t_{0})^{\\frac{p+m-2}{p+m-3}}\n \\excf(\\rpol_{0})\n }\n \\\\\n &\\quad\\times\n \\Big[\n \\frac{p\\nu_{0}}{p+m-3}\n \\frac{\\rpol^{\\frac{p}{p-1}}}{\\excf(\\rpol_{0})^{\\frac{1}{p-1}}}\n -\n \\trb(\\rpol,\\tau)\n \\Big]\n \\,.\n \\end{split}\n\\end{equation}\nUpon dividing both sides of \\eqref{eq:sss_ups_ineqs} by\n\\begin{equation*}\n \\frac{C_{*}}{p+m-3}\n \\,\n \\frac{\\trb(\\rpol,\\tau)^{\\frac{p-1}{p+m-3}-1}}{(t+t_{0})^{\\frac{p+m-2}{p+m-3}}}\n \\,,\n\\end{equation*}\nwe obtain the equivalent version of \\eqref{eq:sss_ups_ineqs} (here $K_{0}$ is as in \\eqref{eq:sss_ups_ineq2})\n\\begin{equation}\n \\label{eq:sss_ups_ineqs2}\n \\begin{split}\n K_{0}&=\n (p-1)\n (t+t_{0})\n \\der{}{t}\n \\big(\n \\tar(\\tau)^{\\frac{1}{p-1}}\n \\big)\n \\ge\n \\trb(\\rpol,\\tau)\n \\\\\n &\\quad+\n \\frac{C_{*}^{p+m-3}(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}\n \\frac{1}{\\excf(\\rpol_{0})}\n \\Big[\n \\frac{p\\nu_{0}}{p+m-3}\n \\frac{\\rpol^{\\frac{p}{p-1}}}{\\excf(\\rpol_{0})^{\\frac{1}{p-1}}}\n -\n \\trb(\\rpol,\\tau)\n \\Big]\n \\\\\n &\\quad\n -\n \\Big(\n \\frac{N-1}{\\rpol}\n +\n \\exw'(\\rpol)\n \\Big)\n \\frac{C_{*}^{p+m-3}(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}\n \\frac{\\trb(\\rpol,\\tau)\\rpol}{\\excf(\\rpol_{0})}\n =:\n K_{3}\n \\,.\n \\end{split}\n\\end{equation}\nOn recalling \\eqref{eq:sss_ups_k0}, we see that $K_{0}>0$, so that we need only show\n\\begin{equation}\n \\label{eq:sss_ups_ineqs3}\n 0\n \\ge\n K_{3}\n \\,.\n\\end{equation}\nClearly, this is implied by\n\\begin{equation}\n \\label{eq:sss_ups_ineqs4}\n \\begin{split}\n & \\Big(\n \\frac{(N-1)C_{*}^{p+m-3}(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}\\excf(\\rpol_{0})}\n -\n 1\n \\Big)\n \\trb(\\rpol,\\tau)\n \\ge\n \\\\\n &\\quad\n \\frac{C_{*}^{p+m-3}(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}\n \\frac{1}{\\excf(\\rpol_{0})}\n \\Big[\n \\frac{p\\nu_{0}}{p+m-3}\n \\frac{\\rpol^{\\frac{p}{p-1}}}{\\excf(\\rpol_{0})^{\\frac{1}{p-1}}}\n -\n \\trb(\\rpol,\\tau)\n \\Big]\n \\,.\n \\end{split}\n\\end{equation}\nThe left hand side of \\eqref{eq:sss_ups_ineqs4} is nonnegative, owing to our choice of $C_{*}$ and to \\eqref{eq:sss_ups_c1} (we use the second term in the $\\max$ function in \\eqref{eq:sss_ups_C} here). Thus we only need to show that the right hand side there is nonpositive for $\\rpol<\\rpol_{0}$. According to the definitions of $\\tau$ and of $\\trb(\\rpol,\\tau)$ this in turn follows from\n\\begin{equation}\n \\label{eq:sss_ups_ineqs5}\n \\begin{split}\n &\\frac{p\\nu_{0}}{p+m-3}\n \\frac{\\rpol_{0}^{\\frac{p}{p-1}}}{\\excf(\\rpol_{0})^{\\frac{1}{p-1}}}\n +\n \\ram(\\rpol_{0})\n \\le\n \\tar(\\log t_{0})^{\\frac{1}{p-1}}\n \\\\\n &\\quad\\le\n \\tar(\\varGamma \\log t_{0})^{\\frac{1}{p-1}}\n \\le\n \\tar(\\tau)^{\\frac{1}{p-1}}\n \\,,\n \\end{split}\n\\end{equation}\nwhere the last two inequalities follow trivially from the increasing character of $\\tar(\\tau)$ and $\\varGamma\\ge 1$. But since $\\rpol_{0}$ has been fixed, this amounts simply to choosing $t_{0}=t_{0}(\\rpol_{0})$ so that both \\eqref{eq:sss_ups_t0r0} and \\eqref{eq:sss_ups_ineqs5} are satisfied; actually it is easily seen that \\eqref{eq:sss_ups_t0r0} follows from \\eqref{eq:sss_ups_ineqs5}. Note that $r_{0}$, $C_{*}$ and $t_{0}$ are independent of $\\varGamma$.\n\nWe look for subsolutions in the radial form\n\\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2}. We use here the same\nnotation introduced in Section~\\ref{s:sss}, and we look at the set\nwhere $\\ups>0$. In this Section, the choice of the constants is more\ndelicate; we find it convenient to establish it from the beginning. In\nthis connection, let us note that the constant $\\tilde\\mu_{1}>0$ is\ndefined in \\eqref{eq:sss_ubs_mu1}, $\\tilde\\mu_{2}>0$ in\n\\eqref{eq:sss_ubs_mu2}, and $\\tilde\\mu_{3}\\ge 0$ in\n\\eqref{eq:sss_ubs_mu3}; the $\\tilde\\mu_{i}$ depend on $p$, $m$,\n$\\alpha_{1}$, $\\alpha_{2}$ only. The constant $\\nu_{0}$ is chosen as\nin \\eqref{eq:sss_ups_c1}. We also introduce for future reference a\ngiven constant $\\lambda>0$.\n\\\\\nWe fix $t_{0}>1$ such that\n\\begin{equation}\n \\label{eq:sss_ubs_t0}\n \\begin{split}\n \\log t_{0}\n \\ge\n \\max&\\Big\\{\n 4\n \\frac{p-\\alpha_{1}}{\\alpha_{1}}\n \\,,\n \\tilde\\mu_{4}^{-1}\n \\,,\n \\frac{2\\tilde\\mu_{1}(N-1)+\\tilde\\mu_{3}\n +\n 2\\tilde\\mu_{1}\n \\alpha_{2}^{2}}\n {\\tilde\\mu_{4}}\n \\,,\n \\\\\n &\\quad\n \\frac{2(\\nu_{0}p)^{p-1}}{\\tilde\\mu_{4}(p+m-3)^{p-2}}\n (\n N\n +\n \\alpha_{2}^{2}\n )\n \\Big\\}\n \\,,\n \\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n \\label{eq:ssb_ups_mu4}\n \\tilde\\mu_{4}\n =\n 2^{-\\frac{\\alpha_{2}(p-1)}{p-\\alpha_{2}}}\n \\frac{\\tilde\\mu_{2}\\alpha_{1}}{p-\\alpha_{1}}\n >0\n \\,. \n\\end{equation}\nNext we select $\\rpol_{0}>0$ such that\n\\begin{equation}\n \\label{eq:ssb_ups_r0}\n \\excf(r_{0})\n <\n \\min\n \\Big\\{\n 1\n \\,,\n \\lambda^{p+m-3}\n \\,,\n \\tilde\\mu_{4}\n C_{*}^{p+m-3}\n \\log t_{0}\n \\Big\\}\n \\,,\n\\end{equation}\nwhere we set\n\\begin{equation}\n \\label{eq:ssb_ups_C}\n \\begin{split}\n C_{*}^{-(p+m-3)}\n &=\n \\max\\Big\\{\n \\lambda^{-(p+m-3)}\n \\,,\n 2\\frac{\\tilde\\mu_{1}(N-1)+\\tilde\\mu_{3}}{\\excf(\\rpol_{0})}\n +\n 2\\tilde\\mu_{1}\n \\alpha_{2}^{2}\n \\,,\n \\\\\n &\\quad\n \\frac{2(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}\n \\frac{1}{\\excf(\\rpol_{0})}\n (\n N\n +\n \\alpha_{2}^{2}\n \\excf(\\rpol_{0})\n )\n \\Big\\}\n \\,.\n \\end{split}\n\\end{equation}\nClearly we have to show that \\eqref{eq:ssb_ups_r0}, \\eqref{eq:ssb_ups_C} are compatible, that is that \n\\begin{equation}\n \\label{eq:ssb_ups_r0_i}\n \\excf(\\rpol_{0})C_{*}^{-(p+m-3)}\n <\n \\tilde\\mu_{4}\n \\log t_{0}\n\\end{equation}\ncan be fulfilled by choosing a small enough $\\rpol_{0}$. Indeed, also invoking $\\excf(\\rpol_{0})<\\min\\{1,\\lambda^{p+m-3}\\}$ which is certainly meaningful, we compute from \\eqref{eq:ssb_ups_C}\n\\begin{equation}\n \\label{eq:ssb_ups_r0_ii}\n \\begin{split}\n \\excf(\\rpol_{0})C_{*}^{-(p+m-3)}\n &<\n \\max\\Big\\{\n 1\n \\,,\n 2\\tilde\\mu_{1}(N-1)+\\tilde\\mu_{3}\n +\n 2\\tilde\\mu_{1}\n \\alpha_{2}^{2}\n \\,,\n \\\\\n &\\quad\n \\frac{2(\\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}\n (\n N\n +\n \\alpha_{2}^{2}\n )\n \\Big\\}\n \\le\n \\tilde\\mu_{4}\n \\log t_{0}\n \\,,\n \\end{split}\n\\end{equation}\naccording to our choice of $t_{0}$.\nFinally we select\n\\begin{equation}\n \\label{eq:ssb_ups_Gamma0}\n \\varGamma\n =\n 2^{\\frac{\\alpha_{2}(p-1)}{p-\\alpha_{2}}}\n \\frac{\\excf(\\rpol_{0})}{\\log t_{0}}\n \\,.\n\\end{equation}",
    "post_theorem_intro_text_len": 5925,
    "post_theorem_intro_text": "\\begin{theorem}\n  \\label{t:sub_sub}\n  Assume \\eqref{eq:degen} and \\eqref{eq:powerlike}. Let $u$ be a strong solution to \\eqref{eq:pde}--\\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$, such that $u_{0}(x)\\ge \\varepsilon>0$ for $\\lvertx\\rvert\\le\\ell$, $\\ell>0$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\widetilde u$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution and\n  \\begin{equation}\n    \\label{eq:sub_sub_n}\n    u(x,t)\n    \\ge\n    \\widetilde u(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sub_sub_nn}\n    \\supp u(t)\n    \\supset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    g^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\varepsilon$, $\\ell$.\n\\end{theorem}\n\n\\begin{proof}[Proofs of Theorems \\ref{t:sup_sup} and \\ref{t:sub_sub}]\n  The proof follows straightforwardly from an application of the method given in the proof of \\cite[Theorem~1]{Tsutsumi:1988}, when we take into account Proposition~\\ref{p:main_sup_sub} and Lemmas \\ref{l:sss_comp} and \\ref{l:ssb_comp}.\n\\end{proof}\n\nNext we apply our results to a different kind of equation, that is \\eqref{eq:rad_N}; see also \\cite{Vazquez:2015}, \\cite{Grillo:Muratori:Vazquez:2017} for the case of the porous media equation.\n\n\\begin{theorem}\n  \\label{t:transform}\nRadial solutions $U(r,t)$ [respectively, supersolutions and subsolutions] of \\eqref{eq:pde} correspond, thru a suitable transformation of the space variable, to radial solutions $v(s,t)$ [respectively, supersolutions and subsolutions] of \\eqref{eq:rad_N},\nfor a suitable positive function $\\rho\\in C([0,+\\infty))$ such that $\\rho(0)=1$ and\n\\begin{equation}\n  \\label{eq:rad_rw_est}\n  \\rho(s)\n  \\sim\n  \\frac{g^{(-1)}(\\log s)^{p}}{(\\log s)^{p}}\n  \\frac{1}{s^{p}}\n  \\,.\n\\end{equation}\nIn fact, the transformation $r=\\hat\\rpol(s)$, $s>0$, is such that\n\\begin{equation}\n  \\label{eq:rad_r_asy_n}\n  \\hat\\rpol(s)\n  \\sim\n  g^{(-1)}(\\log s)\n  \\,.\n\\end{equation}\n\\end{theorem}\n\nHere and in the following we use the notation, for positive functions $F_{1}$, $F_{2}$,\n\\begin{equation}\n  \\label{eq:rad_sim}\n  F_{1}(s)\\sim F_{2}(s)\n  \\quad\n  \\text{if and only if}\n  \\quad\n  C_{1}\n  \\le\n  \\frac{F_{2}(s)}{F_{1}(s)}\n  \\le\n  C_{2}\n  \\,,\n  \\quad\n  s\\ge \\bar s\n  \\,,\n\\end{equation}\nfor suitable $C_{1}$, $C_{2}$, $\\bar s>0$.\n\nOwing to Theorem~\\ref{t:transform}, supersolutions and subsolutions of the type \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} yield radial counterparts for equation \\eqref{eq:rad_N}. Thus, from our  Theorems \\ref{t:sup_sup}, \\ref{t:sub_sub} the following comparison results follow immediately.\n\n\\begin{theorem}\n  \\label{t:sup_sup_dens}\n  Let $v$ be a strong solution to \\eqref{eq:rad_N}, for the specific $\\rho$ found in Theorem~\\ref{t:transform}, under assumptions \\eqref{eq:degen} and \\eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported and bounded.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\widetilde u(\\hat\\rpol(\\lvertx\\rvert),t)$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a supersolution to \\eqref{eq:rad_N}, and\n  \\begin{equation}\n    \\label{eq:sup_sup_dens_n}\n    v(x,t)\n    \\le\n    \\widetilde u(\\hat\\rpol(\\lvertx\\rvert),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sup_sup_dens_nn}\n    \\supp v(t)\n    \\subset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\qquad\n    \\log \\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,.\n  \\end{equation}\n  The constants involved in \\eqref{eq:sup_sup_dens_n}--\\eqref{eq:sup_sup_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.\n\\end{theorem}\n\n\\begin{theorem}\n  \\label{t:sub_sub_dens}\n  Let $v$ be a strong solution to \\eqref{eq:rad_N}, for the specific $\\rho$ found in Theorem~\\ref{t:transform}, under assumptions \\eqref{eq:degen} and \\eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported, bounded and\n$v(x,0)\\ge \\varepsilon>0$ for $\\lvertx\\rvert\\le\\ell$, $\\ell>0$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\widetilde u(\\hat\\rpol(\\lvertx\\rvert),t)$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution \\eqref{eq:rad_N}, and\n  \\begin{equation}\n    \\label{eq:sub_sub_dens_n}\n    v(x,t)\n    \\ge\n    \\widetilde u(\\hat\\rpol(\\lvertx\\rvert),t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sub_sub_dens_nn}\n    \\supp v(t)\n    \\supset\n    B_{\\tilde R(t)}(0)\n    \\,,\n    \\quad\n    \\log\\tilde R(t)\n    \\sim\n    \\log t\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  The constants involved in \\eqref{eq:sub_sub_dens_n}--\\eqref{eq:sub_sub_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.\n\\end{theorem}\n\n\\begin{remark}\n  \\label{r:rad_N}\n  Clearly the sup estimate valid for $v$ as in Theorems\n  \\ref{t:sup_sup_dens} and \\ref{t:sub_sub_dens} is the same as in\n  \\eqref{eq:sup_decay}, which should be compared with the result valid for $\\rho(s)\\sim s^{-\\alpha}$, $\\alpha>p$, that is the universal bound $t^{-1/(p+m-3)}$ (see \\cite{Andreucci:Tedeev:2021b}). Our present bound however depends on the initial data, as the known bound valid in the subcritical case $\\alpha<p$, whose decay rate is $t^{-(N-\\alpha)/K}$, $K=(N-\\alpha)(p+m-3)+p-\\alpha$.\n\\end{remark}\n\n\\textbf{Plan of the paper.}\nSection~\\ref{s:pre} contains some elementary auxiliary results.\nThen Proposition~\\ref{p:main_sup_sub} is proved in Sections \\ref{s:sss} (supersolutions) and \\ref{s:ssb} (subsolutions).\nFinally in Section~\\ref{s:rad} we carry out the transformation of variables linking \\eqref{eq:pde} and \\eqref{eq:rad_N}.",
    "sketch": "The post-theorem text gives the following proof sketch for Theorems~\\ref{t:sup_sup} and \\ref{t:sub_sub}: “The proof follows straightforwardly from an application of the method given in the proof of \\cite[Theorem~1]{Tsutsumi:1988}, when we take into account Proposition~\\ref{p:main_sup_sub} and Lemmas \\ref{l:sss_comp} and \\ref{l:ssb_comp}.”",
    "expanded_sketch": "The post-theorem text gives the following proof sketch for the main theorem and \n\\begin{theorem}\n  \\label{t:sub_sub}\n  Assume \\eqref{eq:degen} and \\eqref{eq:powerlike}. Let $u$ be a strong solution to \\eqref{eq:pde}--\\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$, such that $u_{0}(x)\\ge \\eps>0$ for $\\abs{x}\\le\\ell$, $\\ell>0$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\ups$ as in \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a subsolution and\n  \\begin{equation}\n    \\label{eq:sub_sub_n}\n    u(x,t)\n    \\ge\n    \\ups(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,.\n  \\end{equation}\n  We have also\n  \\begin{equation}\n    \\label{eq:sub_sub_nn}\n    \\supp u(t)\n    \\supset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    \\exw^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\eps$, $\\ell$.\n\\end{theorem}\n: “The proof follows straightforwardly from an application of the method given in the proof of \\cite[Theorem~1]{Tsutsumi:1988}, when we take into account\n\\begin{proposition}\n  \\label{p:main_sup_sub}\n  The constants $C_{*}$, $\\varGamma$, $t_{0}> 1$, $\\rpol_0$ and\n  $\\nu_{0}$ can be selected so that $\\ups$ as defined in\n  \\eqref{eq:sss_sup}--\\eqref{eq:sss_ups_2} is a strong supersolution to\n  \\eqref{eq:pde}. Alternatively, we may select such constants so that\n  $\\ups$ is a strong subsolution to \\eqref{eq:pde}.\n\\end{proposition}\n and\n\\begin{lemma}\n  \\label{l:sss_comp}\n  Let $L>0$, $M>0$ be given positive numbers. Then it is possible to select $\\rpol_{0}$, $C_{*}$, $t_{0}$, $\\varGamma$ so that $\\ups$ is a supersolution and \n  \\begin{equation}\n    \\label{eq:sss_comp_n}\n    \\ups(x,0)\n    \\ge\n    M\n    \\,,\n    \\qquad\n    \\abs{x}\\le L\n    \\,.\n  \\end{equation}\n\\end{lemma}\n and\n\\begin{lemma}\n  \\label{l:ssb_comp}\n  Let $\\ell>0$, $\\eps>0$ be given positive numbers. Then it is possible to select $\\rpol_{0}$, $C_{*}$, $t_{0}$ and $\\varGamma$ so that $\\ups$ is a subsolution and\n  \\begin{gather}\n    \\label{eq:ssb_comp_n}\n    \\supp \\ups(0)\n    \\subset\n    \\{\\abs{x}\\le \\ell\\}\n    \\,,\n    \\\\\n    \\label{eq:ssb_comp_nn}\n    \\ups(x,0)\n    \\le\n    \\eps\n    \\,,\n    \\quad\n    x\\in\\RN\n    \\,.\n  \\end{gather}\n\\end{lemma}\n.”",
    "expanded_theorem": "\\label{t:sup_sup}\n  \\begin{equation}\n  \\label{eq:degen}\n  1<p<N\n  \\,,\n  \\qquad\n  p+m-3>0\n  \\,,\n\\end{equation}\n  and\n  \\begin{equation}\n  \\label{eq:powerlike}\n  \\alpha_{1}\n  \\frac{\\exw(s)}{s}\n  \\le\n  \\exw'(s)\n  \\le\n  \\alpha_{2}\n  \\frac{\\exw(s)}{s}\n  \\,,\n  \\qquad\n  s>0\n  \\,,\n\\end{equation}\n  hold. Let $u$ be a strong solution to\n  \\begin{alignat}{2}\n  \\label{eq:pde}\n  \\ew(x)\n  \\pder{u}{t}\n  -\n  \\Div\\big(\n  \\ew(x)\n  u^{m-1}\n  \\abs{\\grad u}^{p-2}\n  \\grad u\n  \\big)\n  &=\n    0\n    \\,,\n  &\\qquad&\n           \\text{in $S_{T}$,}\n  \\\\\n  \\label{eq:initd}\n  u(x,0)\n  &=\n    u_{0}(x)\n    \\,,\n  &\\qquad&\n           x\\in\\RN\n           \\,,\n\\end{alignat}\n  with a compactly supported and bounded data $u_{0}$.\n  \\\\\n  Then there exist constants $t_{0}$, $C_{*}$, $\\rpol_{0}$ and $\\varGamma$ such that $\\widetilde u$ defined by\n  \\begin{alignat}2\n  \\label{eq:sss_sup}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\rat(\\rpol)^{\\frac{1}{p-1}}\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol\\ge \\rpol_{0}\n            \\,,\n  \\\\\n  \\label{eq:sss_ups_2}\n  \\ups(x,t)\n  &=\n    \\frac{C_{*}}{(t+t_{0})^{\\frac{1}{p+m-3}}}\n    \\Big[\n    \\tar(\\tau)^{\\frac{1}{p-1}}\n    -\n    \\ram(\\rpol)\n    \\Big]^{\\frac{p-1}{p+m-3}}_{+}\n    \\,,\n    &\\quad&\n            \\rpol<\\rpol_{0}\n            \\,,\n\\end{alignat}\n  is a supersolution and\n  \\begin{equation}\n    \\label{eq:sup_sup_n}\n    u(x,t)\n    \\le\n    \\widetilde u(x,t)\n    \\,,\n    \\qquad\n    (x,t)\\in S_{\\infty}\n    \\,,\n  \\end{equation}\n  holds.\n  We have also\n  \\begin{equation}\n    \\label{eq:sup_sup_nn}\n    \\supp u(t)\n    \\subset\n    B_{R(t)}(0)\n    \\,,\n    \\quad\n    R(t)\n    =\n    \\gamma\n    g^{(-1)}(\\log t)\n    \\,,\n    \\qquad\n    t>t_{0}\n    \\,,\n  \\end{equation}\n  for a suitable constant $\\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\\sup_{\\R^{\\SpDim}} u_{0}$, $R(0)\\,.",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let \\(1<p<N\\) and \\(p+m-3>0\\). Let \\(g\\in C([0,\\infty))\\cap C^1((0,\\infty))\\) satisfy \\(g(0)=0\\), \\(g(s)>0\\) for \\(s>0\\), and for some \\(0<\\alpha_1\\le \\alpha_2<p\\),\n\\[\n\\alpha_1\\frac{g(s)}{s}\\le g'(s)\\le \\alpha_2\\frac{g(s)}{s},\\qquad s>0.\n\\]\nSet \\(f(x)=e^{g(|x|)}\\), and consider the Cauchy problem\n\\[\nf(x)u_t-\\operatorname{div}\\big(f(x)u^{m-1}|\\nabla u|^{p-2}\\nabla u\\big)=0\n\\quad\\text{in }\\mathbb R^N\\times(0,T),\n\\qquad\nu(x,0)=u_0(x),\n\\]\nwhere \\(u\\) is a nonnegative strong solution and \\(u_0\\) is bounded and compactly supported. For \\(r=|x|\\), define\n\\[\nG(r)=\\int_0^r \\frac{g(s)}{s}\\,ds,\n\\qquad\nJ(r)=\\frac{r^p}{G(r)},\n\\qquad\n\\tau=\\Gamma\\log(t+t_0),\n\\qquad\nE(\\tau)=J(G^{-1}(\\tau))=\\frac{(G^{-1}(\\tau))^p}{\\tau},\n\\]\nand\n\\[\nI(r)=\\nu_0\\Big(\\frac{r^p}{G(r_0)}\\Big)^{\\!1/(p-1)}+(1-\\nu_0)J(r_0)^{1/(p-1)},\n\\]\nwith the fixed constant \\(\\nu_0\\) from the barrier construction. Under these assumptions, which quantitative estimate holds?",
      "correct_choice": {
        "label": "A",
        "text": "There exist constants \\(t_0>0\\), \\(C_*>0\\), \\(r_0>0\\), \\(\\Gamma>0\\), and \\(\\gamma>0\\) such that the function \\(\\widetilde u\\) defined for \\(r=|x|\\) by\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-J(r)^{1/(p-1)}\\Big]_+^{(p-1)/(p+m-3)},\\qquad r\\ge r_0,\n\\]\nand\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-I(r)\\Big]_+^{(p-1)/(p+m-3)},\\qquad r<r_0,\n\\]\nis a supersolution, and\n\\[\nu(x,t)\\le \\widetilde u(x,t)\\qquad \\text{for all }(x,t)\\in \\mathbb R^N\\times(0,\\infty).\n\\]\nMoreover,\n\\[\n\\operatorname{supp} u(t)\\subset B_{R(t)}(0),\n\\qquad\nR(t)=\\gamma\\,g^{-1}(\\log t),\n\\qquad t>t_0.\n\\]\nThe constants depend only on the parameters in the assumptions and on \\(\\sup_{\\mathbb R^N}u_0\\) and the initial support radius \\(R(0)\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exist constants \\(t_0>0\\), \\(C_*>0\\), \\(r_0>0\\), \\(\\Gamma>0\\), and \\(\\gamma>0\\) such that the function \\(\\widetilde u\\) defined for \\(r=|x|\\) by\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-J(r)^{1/(p-1)}\\Big]_+^{(p-1)/(p+m-3)},\\qquad r\\ge r_0,\n\\]\nand\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-I(r)\\Big]_+^{(p-1)/(p+m-3)},\\qquad r<r_0,\n\\]\nis a supersolution, and\n\\[\nu(x,t)\\le \\widetilde u(x,t)\\qquad \\text{for all }(x,t)\\in \\mathbb R^N\\times(0,\\infty).\n\\]\nMoreover,\n\\[\n\\operatorname{supp} u(t)\\subset B_{R(t)}(0),\n\\qquad\nR(t)=\\gamma\\,g^{-1}(t),\n\\qquad t>t_0.\n\\]\nThe constants depend only on the parameters in the assumptions and on \\(\\sup_{\\mathbb R^N}u_0\\) and the initial support radius \\(R(0)\\)."
        },
        {
          "label": "C",
          "text": "There exist constants \\(t_0>0\\), \\(C_*>0\\), \\(r_0>0\\), and \\(\\Gamma>0\\) such that the function \\(\\widetilde u\\) defined for \\(r=|x|\\) by\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-J(r)^{1/(p-1)}\\Big]_+^{(p-1)/(p+m-3)},\\qquad r\\ge r_0,\n\\]\nand\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-I(r)\\Big]_+^{(p-1)/(p+m-3)},\\qquad r<r_0,\n\\]\nis a supersolution, and\n\\[\nu(x,t)\\le \\widetilde u(x,t)\\qquad \\text{for all }(x,t)\\in \\mathbb R^N\\times(0,\\infty).\n\\]"
        },
        {
          "label": "D",
          "text": "There exist constants \\(t_0>0\\), \\(C_*>0\\), \\(r_0>0\\), \\(\\Gamma>0\\), and \\(\\gamma>0\\) depending only on \\(p,m,N,\\alpha_1,\\alpha_2\\) such that the function \\(\\widetilde u\\) defined for \\(r=|x|\\) by\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-J(r)^{1/(p-1)}\\Big]_+^{(p-1)/(p+m-3)},\\qquad r\\ge r_0,\n\\]\nand\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-I(r)\\Big]_+^{(p-1)/(p+m-3)},\\qquad r<r_0,\n\\]\nis a supersolution, and\n\\[\nu(x,t)\\le \\widetilde u(x,t)\\qquad \\text{for all }(x,t)\\in \\mathbb R^N\\times(0,\\infty).\n\\]\nMoreover,\n\\[\n\\operatorname{supp} u(t)\\subset B_{R(t)}(0),\n\\qquad\nR(t)=\\gamma\\,g^{-1}(\\log t),\n\\qquad t>t_0.\n\\]"
        },
        {
          "label": "E",
          "text": "There exist constants \\(t_0>0\\), \\(C_*>0\\), \\(r_0>0\\), \\(\\Gamma>0\\), and \\(\\gamma>0\\) such that the function \\(\\widetilde u\\) defined for \\(r=|x|\\) by\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-J(r)^{1/(p-1)}\\Big]_+^{(p-1)/(p+m-3)},\\qquad r\\ge r_0,\n\\]\nand\n\\[\n\\widetilde u(x,t)=\\frac{C_*}{(t+t_0)^{1/(p+m-3)}}\\Big[E(\\tau)^{1/(p-1)}-I(r)\\Big]_+^{(p-1)/(p+m-3)},\\qquad r<r_0,\n\\]\nis a subsolution, and\n\\[\n\\widetilde u(x,t)\\le u(x,t)\\qquad \\text{for all }(x,t)\\in \\mathbb R^N\\times(0,\\infty).\n\\]\nMoreover,\n\\[\n\\operatorname{supp} u(t)\\supset B_{R(t)}(0),\n\\qquad\nR(t)=\\gamma\\,g^{-1}(\\log t),\n\\qquad t>t_0.\n\\]\nThe constants depend only on the parameters in the assumptions and on \\(\\sup_{\\mathbb R^N}u_0\\) and the initial support radius \\(R(0)\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "support-growth scale uses g^{-1}(\\log t), not g^{-1}(t)",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the explicit support-radius conclusion and parameter dependence statement",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "quantifier_dependence",
          "tampered_component": "constants must depend on the initial size/support data, not only structural parameters",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "replaced the supersolution comparison theorem by the subsolution/lower-bound theorem, but kept the wrong data dependence",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It provides hypotheses and definitions, then asks which estimate holds; the correct conclusion is not directly stated in the stem itself."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is very close to asking for the precise theorem statement under the given assumptions. The answer is selected by matching a known conclusion rather than synthesizing a new consequence, though the variants do introduce some comparative selection."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle alternatives such as g^{-1}(log t) versus g^{-1}(t), supersolution versus subsolution, and dependence of constants. However, this is mainly theorem recognition/verification rather than strong generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and target common errors, but choice C is a weaker true statement rather than a clearly false distractor. That weakens the single-best-answer structure and reduces distractor quality overall."
      },
      "total_score": 5,
      "overall_assessment": "A moderately good but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses mostly plausible distractors, but it is close to restating a theorem and is weakened by the presence of a weaker true option, which makes the item less clean as a single-answer question."
    }
  },
  {
    "id": "2512.20498v1",
    "paper_link": "http://arxiv.org/abs/2512.20498v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "teorema",
      "content": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.",
      "start_pos": 13260,
      "end_pos": 13630,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{teorema}\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.\n\\end{teorema}",
      "lmm:0-hom": "\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}",
      "eq:def_s_mu_k": "\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}",
      "eq:def-unique_blowup": "\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}",
      "cor:rectif_criterion": "\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}"
    },
    "pre_theorem_intro_text_len": 3739,
    "pre_theorem_intro_text": "The notion of blow-up of a geometric or analytic object, i.e.\\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale\".\nFor this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.\n\nFor a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.",
    "context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}",
    "full_context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\nThe main result of this note provides an answer to these questions.\n\nThe idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nThe statement is trivial for $k=n$, thus we assume $k \\in \\{0,1,\\dots,n-1\\}$ be fixed throughout the proof.\nLet us moreover fix $\\gamma \\in \\left (0,\\frac{1}{16}\\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\\{V_\\ell\\}_{\\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\\overline{C(0,V_\\ell,\\gamma)}$ for some $\\ell \\in \\{1,\\dots,m\\}$. For every $\\ell \\in \\{1,\\dots,m\\}$ and every $i \\in \\mathbb{N}$, we define the set\n\\begin{equation}\nE_{\\ell,i} \\coloneqq \\left \\{ x \\in \\mathcal{S}^k \\colon D_x \\subset \\overline{C(0,V_\\ell,\\gamma)},\\,\\,\\, \\alpha(x) \\leq i,\\,\\,\\, \\eta(x)\\geq \\frac{1}{i},\\,\\,\\, d(c\\mu_{x},\\omega)\\geq \\frac{1}{i} \\,\\,\\,\\forall |c| \\in \\left [\\frac{1}{i},1 \\right ],\\, \\forall\\omega \\in \\operatorname{Hom}_{k+1}^i \\right \\},\n\\end{equation}\nwhere $D_x$ and $\\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\\mu_x$ given by Lemma \\ref{lmm:0-hom}, $\\eta=\\eta(x)$ is given by Lemma \\ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\\alpha-n)$-homogeneous measures described in Section \\ref{sec:notations}.\nSince $\\mathcal{S}^k= \\bigcup_{i \\in \\mathbb{N}}\\bigcup_{\\ell=1}^m E_{i,\\ell}$, it is enough to prove that each $E_{i,\\ell}$ can be covered by a countable union of Lipschitz graphs.\n\nLet us fix $i \\in \\mathbb{N}$ and $\\ell \\in \\{1,\\dots,m\\}$. \nThen fix the value $\\varepsilon$ given by Lemma \\ref{lmm:too_close} for $\\beta=i$, $\\delta=\\frac{1}{i}$, $M=i \\cdot 2^\\alpha$ and, for every $\\rho>0$, define the set\n\\begin{equation}\nE_{i,\\ell,\\rho} \\coloneqq\n\\left \\{x \\in E_{i,\\ell} \\colon \\inf_{|c| \\in [\\eta,1]}d(\\mu_{x,r},c\\mu_x)< \\varepsilon \\,\\,\\,\\forall r \\in (0,\\rho) \\right \\}.\n\\end{equation}\nIf $x \\in E_{i,\\ell}$, then it belongs to $E_{i,\\ell,\\rho}$ for some $\\rho>0$, because the existence of a sequence $r_j \\to 0$ such that $d(\\mu_{x,r_j},c\\mu_x) \\geq \\varepsilon$ for every $j \\in \\mathbb{N}$ and every $|c| \\in [\\eta,1]$ would contradict $x \\in \\mathcal{S}$. Therefore\n\\begin{equation}\nE_{i,\\ell} = \\bigcup_{\\rho \\in \\mathbb{Q} \\cap (0,+\\infty)} E_{i,\\ell,\\rho}\n\\end{equation}\nand it suffice to prove that, for a fixed $\\rho>0$, the set $E_{i,\\ell,\\rho}$ can be covered with a countable union of Lipschitz graphs.\nIn order to do so, we claim that, for every $x \\in E_{i,\\ell,\\rho}$, there exists $r>0$ such that\n\\begin{equation}\\label{eq:outside_cones}\nE_{i,\\ell,\\rho} \\cap B_{r}(x) \\setminus C(x,D_x,3\\gamma)= \\{x\\}.\n\\end{equation}\nLet us fix $x\\in E_{i,\\ell,\\rho}$, let $\\mu_x$ be $(\\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \\in \\mathbb{N}$ there exists \n\\begin{equation}\\label{eq:y_j_final}\ny_j \\in E_{i,\\ell,\\rho} \\cap \\overline{B_{\\frac{r_j}{2}}(x)} \\setminus\\Big( B_{\\frac{r_j}{4}}(x) \\cup C(x,D_x,3\\gamma) \\Big),\n\\end{equation}\nfor some $r_j \\leq \\rho$ such that $r_j \\downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \\in \\left [\\frac{1}{i},1 \\right ]$ such that\n\\begin{equation}\\label{eq:distance_muj}\nd( c_j\\mu_{y_j}, \\mu_{y_j,r_j})< \\varepsilon\n\\qquad\n\\forall j \\in \\mathbb{N}.\n\\end{equation}\nUp to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\\mu_{x,r_j} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ for some $c \\in \\left [\\frac{1}{i},1 \\right ]$.\nWe now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \\eqref{eq:distance_muj}.",
    "post_theorem_intro_text_len": 5662,
    "post_theorem_intro_text": "The idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\medskip\n\nA first outcome of Theorem \\ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\nThe above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \\cite[Theorem 11.8]{simonlectures83}, where one usually sets $\\nu_x=\\theta_0 \\mathcal{H}^k \\llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\\tau_{x,r_j})_\\# \\mu \\overset{\\ast}{\\rightharpoonup} \\theta_0 \\mathcal{H}^k \\llcorner V_x$ as $r \\to 0^+$.\n\nWe point out that Corollary \\ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \\cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.\n\n\\medskip\n\nA further consequence of Theorem \\ref{thm:main} is the following extension of \\cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\\mathcal{S}_{\\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\\mu$ is more diffused than a $k$-dimensional measure is $|\\mu|$-negligible.\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\nWe conclude this introduction with a comment on the definition of unique blow-up at a point.\nAs already said above, we say that $\\mu_x$ is the unique blow-up of $\\mu$ at $x \\in \\operatorname{supp}\\mu$ if, for every sequence $r_j \\to 0$, \\emph{there exists} a subsequence for which \\eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \\cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\\{r_j\\}_j$: according to \\cite{mattila_unique05}, one says that  $\\mu$ has unique blow-up at $x \\in \\mathbb{R}$ if there exists $\\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\\tau_{x,r_j})_\\# \\mu$, where $c_j>0$ and $r_j \\to 0^+$, is of the form $c\\nu$ for some $c>0$. \n\n\\cite[Lemma 2.5]{mattila_unique05} shows, provided $\\mu$ is positive, that this condition actually implies $\\mu_{x,r} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ as $r \\to 0$ and, subsequently, our definition (they are actually equivalent).\nHowever, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \\rho_j$ such that\n\\begin{equation}\n\\lim_{j \\to +\\infty} \\frac{1}{a_j} \\sum_{i=j+1}^{+\\infty} a_i = 0,\n\\qquad\n\\lim_{j \\to +\\infty} \\frac{\\rho_{j+1}}{\\rho_j}=0.\n\\end{equation}\nThen the Radon measure on $\\mathbb{R}$ defined as\n\\begin{equation}\n\\mu \\coloneqq \\sum_{j \\in \\mathbb{N}} a_j\\left ( \\delta_{2\\rho_j + \\rho_j^2} - \\delta_{2\\rho_j - \\rho_j^2} \\right )\n\\end{equation}\nhas the null measure as unique blow-up at $x=0$ in the sense of \\cite{mattila_unique05}, but $\\mu_{0,\\rho_j} =\\frac{1}{|\\mu|(B_{\\rho_j}(0))} (\\tau_{0,\\rho_j})_\\# \\mu $ has no converging subsequences because $|\\mu_{x,\\rho_j}|\\big(B_2(0)\\big) \\to +\\infty$ as $j\\to +\\infty$.\n\n\\subsection*{Acknowledgments}\nThe author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations\" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems\" at ESI (Wien), which the author wishes to thank.",
    "sketch": "The post-theorem introduction gives the following proof idea for Theorem~\\ref{thm:main}.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.",
    "expanded_sketch": "The post-theorem introduction gives the following proof idea for the main theorem.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.",
    "expanded_theorem": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.,",
    "theorem_type": [
      "Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(\\mu\\) be a signed Radon measure on \\(\\mathbb{R}^n\\). For \\(x\\in \\operatorname{supp}\\mu\\), say that \\(\\mu\\) has a unique blow-up at \\(x\\) if there exists a Radon measure \\(\\mu_x\\), with either \\(|\\mu_x|(B_1(0))=1\\) or \\(\\mu_x=0\\), such that for every sequence \\(r_j\\to 0^+\\) there are a subsequence and a constant \\(c\\neq 0\\) for which\n\\[\n\\mu_{x,r_j}:=\\frac{1}{|\\mu|(B_{r_j}(x))}(\\tau_{x,r_j})_\\#\\mu \\overset{*}{\\rightharpoonup} c\\mu_x,\n\\qquad \\tau_{x,r}(y)=\\frac{y-x}{r}.\n\\]\nLet \\(\\mathcal S_\\mu\\) be the set of points where \\(\\mu\\) has a unique blow-up. For \\(x\\in \\mathcal S_\\mu\\), let\n\\[\nD_x:=\\{y\\in\\mathbb R^n:(\\tau_{y,1})_\\#\\mu_x=\\mu_x\\}\n\\]\nbe the invariant linear subspace of the blow-up, and for \\(k\\in\\{0,1,\\dots,n\\}\\) define\n\\[\n\\mathcal S_\\mu^k:=\\{x\\in\\mathcal S_\\mu:\\dim D_x=k\\}.\n\\]\nWhich statement holds for every signed Radon measure \\(\\mu\\) on \\(\\mathbb R^n\\) and every \\(k\\in\\{0,1,\\dots,n\\}\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is exactly \\(D_x\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\((k+1)\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^{k+1}\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) contains \\(D_x\\)."
        },
        {
          "label": "C",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs."
        },
        {
          "label": "D",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) exists and is exactly \\(D_x\\)."
        },
        {
          "label": "E",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a finite union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is a \\(k\\)-plane parallel to \\(D_x\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dimension_from_cone_containment",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped_a.e._tangent_plane_identification",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "a.e._pointwise_tangent_identification_upgraded_to_every_point",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "countable_cover_replaced_by_finite_cover_and_exact_plane_identification_relaxed_to_parallelism",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the objects and hypotheses but does not explicitly reveal the conclusion. There is no obvious lexical or structural cue that singles out choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-selection question: it asks for the correct structural conclusion under the stated assumptions. However, it is not a pure tautology because the options include weaker and stronger nearby variants that must be distinguished."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare rectifiability, graph structure, dimension, and the a.e. versus everywhere tangent-plane conclusion. Still, the problem primarily tests recognition or recall of the precise theorem rather than substantial generative derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: wrong dimension, weaker true statement, overstrong everywhere claim, and unjustified finiteness/exactness upgrades. They are distinct and well aligned with the theorem’s subtleties."
      },
      "total_score": 6,
      "overall_assessment": "A solid advanced-theory MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than genuine generative reasoning."
    }
  },
  {
    "id": "2512.20498v1",
    "paper_link": "http://arxiv.org/abs/2512.20498v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "teorema",
      "content": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.",
      "start_pos": 13260,
      "end_pos": 13630,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:main": "\\begin{teorema}\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.\n\\end{teorema}",
      "lmm:0-hom": "\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}",
      "eq:def_s_mu_k": "\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}",
      "eq:def-unique_blowup": "\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}",
      "cor:rectif_criterion": "\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}"
    },
    "pre_theorem_intro_text_len": 3739,
    "pre_theorem_intro_text": "The notion of blow-up of a geometric or analytic object, i.e.\\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale\".\nFor this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.\n\nFor a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.",
    "context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}",
    "full_context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\nThe main result of this note provides an answer to these questions.\n\nThe idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nThe statement is trivial for $k=n$, thus we assume $k \\in \\{0,1,\\dots,n-1\\}$ be fixed throughout the proof.\nLet us moreover fix $\\gamma \\in \\left (0,\\frac{1}{16}\\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\\{V_\\ell\\}_{\\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\\overline{C(0,V_\\ell,\\gamma)}$ for some $\\ell \\in \\{1,\\dots,m\\}$. For every $\\ell \\in \\{1,\\dots,m\\}$ and every $i \\in \\mathbb{N}$, we define the set\n\\begin{equation}\nE_{\\ell,i} \\coloneqq \\left \\{ x \\in \\mathcal{S}^k \\colon D_x \\subset \\overline{C(0,V_\\ell,\\gamma)},\\,\\,\\, \\alpha(x) \\leq i,\\,\\,\\, \\eta(x)\\geq \\frac{1}{i},\\,\\,\\, d(c\\mu_{x},\\omega)\\geq \\frac{1}{i} \\,\\,\\,\\forall |c| \\in \\left [\\frac{1}{i},1 \\right ],\\, \\forall\\omega \\in \\operatorname{Hom}_{k+1}^i \\right \\},\n\\end{equation}\nwhere $D_x$ and $\\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\\mu_x$ given by Lemma \\ref{lmm:0-hom}, $\\eta=\\eta(x)$ is given by Lemma \\ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\\alpha-n)$-homogeneous measures described in Section \\ref{sec:notations}.\nSince $\\mathcal{S}^k= \\bigcup_{i \\in \\mathbb{N}}\\bigcup_{\\ell=1}^m E_{i,\\ell}$, it is enough to prove that each $E_{i,\\ell}$ can be covered by a countable union of Lipschitz graphs.\n\nLet us fix $i \\in \\mathbb{N}$ and $\\ell \\in \\{1,\\dots,m\\}$. \nThen fix the value $\\varepsilon$ given by Lemma \\ref{lmm:too_close} for $\\beta=i$, $\\delta=\\frac{1}{i}$, $M=i \\cdot 2^\\alpha$ and, for every $\\rho>0$, define the set\n\\begin{equation}\nE_{i,\\ell,\\rho} \\coloneqq\n\\left \\{x \\in E_{i,\\ell} \\colon \\inf_{|c| \\in [\\eta,1]}d(\\mu_{x,r},c\\mu_x)< \\varepsilon \\,\\,\\,\\forall r \\in (0,\\rho) \\right \\}.\n\\end{equation}\nIf $x \\in E_{i,\\ell}$, then it belongs to $E_{i,\\ell,\\rho}$ for some $\\rho>0$, because the existence of a sequence $r_j \\to 0$ such that $d(\\mu_{x,r_j},c\\mu_x) \\geq \\varepsilon$ for every $j \\in \\mathbb{N}$ and every $|c| \\in [\\eta,1]$ would contradict $x \\in \\mathcal{S}$. Therefore\n\\begin{equation}\nE_{i,\\ell} = \\bigcup_{\\rho \\in \\mathbb{Q} \\cap (0,+\\infty)} E_{i,\\ell,\\rho}\n\\end{equation}\nand it suffice to prove that, for a fixed $\\rho>0$, the set $E_{i,\\ell,\\rho}$ can be covered with a countable union of Lipschitz graphs.\nIn order to do so, we claim that, for every $x \\in E_{i,\\ell,\\rho}$, there exists $r>0$ such that\n\\begin{equation}\\label{eq:outside_cones}\nE_{i,\\ell,\\rho} \\cap B_{r}(x) \\setminus C(x,D_x,3\\gamma)= \\{x\\}.\n\\end{equation}\nLet us fix $x\\in E_{i,\\ell,\\rho}$, let $\\mu_x$ be $(\\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \\in \\mathbb{N}$ there exists \n\\begin{equation}\\label{eq:y_j_final}\ny_j \\in E_{i,\\ell,\\rho} \\cap \\overline{B_{\\frac{r_j}{2}}(x)} \\setminus\\Big( B_{\\frac{r_j}{4}}(x) \\cup C(x,D_x,3\\gamma) \\Big),\n\\end{equation}\nfor some $r_j \\leq \\rho$ such that $r_j \\downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \\in \\left [\\frac{1}{i},1 \\right ]$ such that\n\\begin{equation}\\label{eq:distance_muj}\nd( c_j\\mu_{y_j}, \\mu_{y_j,r_j})< \\varepsilon\n\\qquad\n\\forall j \\in \\mathbb{N}.\n\\end{equation}\nUp to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\\mu_{x,r_j} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ for some $c \\in \\left [\\frac{1}{i},1 \\right ]$.\nWe now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \\eqref{eq:distance_muj}.",
    "post_theorem_intro_text_len": 5662,
    "post_theorem_intro_text": "The idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\medskip\n\nA first outcome of Theorem \\ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\nThe above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \\cite[Theorem 11.8]{simonlectures83}, where one usually sets $\\nu_x=\\theta_0 \\mathcal{H}^k \\llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\\tau_{x,r_j})_\\# \\mu \\overset{\\ast}{\\rightharpoonup} \\theta_0 \\mathcal{H}^k \\llcorner V_x$ as $r \\to 0^+$.\n\nWe point out that Corollary \\ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \\cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.\n\n\\medskip\n\nA further consequence of Theorem \\ref{thm:main} is the following extension of \\cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\\mathcal{S}_{\\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\\mu$ is more diffused than a $k$-dimensional measure is $|\\mu|$-negligible.\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\nWe conclude this introduction with a comment on the definition of unique blow-up at a point.\nAs already said above, we say that $\\mu_x$ is the unique blow-up of $\\mu$ at $x \\in \\operatorname{supp}\\mu$ if, for every sequence $r_j \\to 0$, \\emph{there exists} a subsequence for which \\eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \\cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\\{r_j\\}_j$: according to \\cite{mattila_unique05}, one says that  $\\mu$ has unique blow-up at $x \\in \\mathbb{R}$ if there exists $\\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\\tau_{x,r_j})_\\# \\mu$, where $c_j>0$ and $r_j \\to 0^+$, is of the form $c\\nu$ for some $c>0$. \n\n\\cite[Lemma 2.5]{mattila_unique05} shows, provided $\\mu$ is positive, that this condition actually implies $\\mu_{x,r} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ as $r \\to 0$ and, subsequently, our definition (they are actually equivalent).\nHowever, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \\rho_j$ such that\n\\begin{equation}\n\\lim_{j \\to +\\infty} \\frac{1}{a_j} \\sum_{i=j+1}^{+\\infty} a_i = 0,\n\\qquad\n\\lim_{j \\to +\\infty} \\frac{\\rho_{j+1}}{\\rho_j}=0.\n\\end{equation}\nThen the Radon measure on $\\mathbb{R}$ defined as\n\\begin{equation}\n\\mu \\coloneqq \\sum_{j \\in \\mathbb{N}} a_j\\left ( \\delta_{2\\rho_j + \\rho_j^2} - \\delta_{2\\rho_j - \\rho_j^2} \\right )\n\\end{equation}\nhas the null measure as unique blow-up at $x=0$ in the sense of \\cite{mattila_unique05}, but $\\mu_{0,\\rho_j} =\\frac{1}{|\\mu|(B_{\\rho_j}(0))} (\\tau_{0,\\rho_j})_\\# \\mu $ has no converging subsequences because $|\\mu_{x,\\rho_j}|\\big(B_2(0)\\big) \\to +\\infty$ as $j\\to +\\infty$.\n\n\\subsection*{Acknowledgments}\nThe author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations\" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems\" at ESI (Wien), which the author wishes to thank.",
    "sketch": "The post-theorem introduction gives the following proof idea for Theorem~\\ref{thm:main}.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.",
    "expanded_sketch": "The post-theorem introduction gives the following proof idea for the main theorem.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.",
    "expanded_theorem": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.,",
    "theorem_type": [
      "Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(\\mu\\) be a signed Radon measure on \\(\\mathbb{R}^n\\). For \\(x\\in \\operatorname{supp}\\mu\\), say that \\(\\mu\\) has a unique blow-up at \\(x\\) if there exists a Radon measure \\(\\mu_x\\), with either \\(|\\mu_x|(B_1(0))=1\\) or \\(\\mu_x=0\\), such that for every sequence \\(r_j\\to 0^+\\) there are a subsequence and a constant \\(c\\neq 0\\) for which\n\\[\n\\mu_{x,r_j}:=\\frac{1}{|\\mu|(B_{r_j}(x))}(\\tau_{x,r_j})_\\#\\mu \\overset{*}{\\rightharpoonup} c\\mu_x,\n\\qquad \\tau_{x,r}(y)=\\frac{y-x}{r}.\n\\]\nLet \\(\\mathcal S_\\mu\\) be the set of points where \\(\\mu\\) has a unique blow-up. For \\(x\\in \\mathcal S_\\mu\\), let\n\\[\nD_x:=\\{y\\in\\mathbb R^n:(\\tau_{y,1})_\\#\\mu_x=\\mu_x\\}\n\\]\nbe the invariant linear subspace of the blow-up, and for \\(k\\in\\{0,1,\\dots,n\\}\\) define\n\\[\n\\mathcal S_\\mu^k:=\\{x\\in\\mathcal S_\\mu:\\dim D_x=k\\}.\n\\]\nWhich statement holds for every signed Radon measure \\(\\mu\\) on \\(\\mathbb R^n\\) and every \\(k\\in\\{0,1,\\dots,n\\}\\)?",
      "correct_choice": {
        "label": "A",
        "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is exactly \\(D_x\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\((k+1)\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^{k+1}\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) contains \\(D_x\\)."
        },
        {
          "label": "C",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs."
        },
        {
          "label": "D",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) exists and is exactly \\(D_x\\)."
        },
        {
          "label": "E",
          "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a finite union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is a \\(k\\)-plane parallel to \\(D_x\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dimension_from_cone_containment",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "dropped_a.e._tangent_plane_identification",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "a.e._pointwise_tangent_identification_upgraded_to_every_point",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "countable_cover_replaced_by_finite_cover_and_exact_plane_identification_relaxed_to_parallelism",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives only definitions and setup; it does not state or strongly hint at the full conclusion in choice A. The correct answer is not leaked explicitly or trivially."
      },
      "TAS": {
        "score": 1,
        "justification": "This is not tautological from the definitions, but it is essentially a theorem-identification item: the correct option states a precise structural result, while the others are nearby variants. So it is more than a restatement of the stem, but still close to recall of a known theorem."
      },
      "GPS": {
        "score": 1,
        "justification": "Selecting A requires some care with subtle distinctions such as countable vs finite cover, k-dimensional vs (k+1)-dimensional graphs, and almost-everywhere vs every-point tangent identification. However, the task is still primarily theorem recognition rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one is a weaker true statement, others are plausible overstatements or dimension/quantifier mistakes that reflect common mathematical failure modes. They are distinct and discriminating."
      },
      "total_score": 6,
      "overall_assessment": "A solid high-level theorem-recognition MCQ with no answer leakage and strong distractors, though it tests recall/comparison more than genuinely generative reasoning."
    }
  },
  {
    "id": "2512.20751v1",
    "paper_link": "http://arxiv.org/abs/2512.20751v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}",
      "start_pos": 1552652,
      "end_pos": 1554414,
      "label": "Teo1"
    },
    "ref_dict": {
      "Equation": "\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}",
      "Teo1": "\\begin{theorem}\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6996,
    "pre_theorem_intro_text": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}.  \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nSecond-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \\eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\\varphi(t) \\ge c\\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \\cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \\cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \\cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \\eqref{Equation} and on models where the damping parameter depends on time, such as \\cite{1,2,3,4,6,7}.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n\t\\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\t\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n\t$$\n\tW(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W3}{($W_3$)}] There exists $\\lambda > 0$ such that\n\t$$\n\t\\langle \\nabla W(u), u - u_* \\rangle > 0 \\quad \\text{for all} \\quad u \\in B_\\lambda(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,\n\t$$\n\t\\lim_{\\|u\\| \\to +\\infty} W(u) = +\\infty.\n\t$$\n\n\t\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n\t$$\n\t\\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n\t$$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking  \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential  \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential  \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential  \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.",
    "context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}",
    "full_context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\nIn this section we discuss the dynamics of the conservative system\n\\begin{equation}\\label{EqConservative}\n \\ddot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwhich arises as the limit of \\eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative\ncase, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \\eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\\mathbb{R}^{2N}$ by setting $x=u$ and $y=\\dot{u}$:\n\\begin{equation}\\label{S0}\n \\begin{cases}\n \\dot{x} = y,\\\\[2mm]\n \\dot{y} = -\\nabla W(x).\n \\end{cases}\n\\end{equation}\nThe natural energy associated with \\eqref{EqConservative},\n$$\nE(x,y) := \\frac12\\|y\\|^2 + W(x), \\quad (x,y)\\in B_\\eta(0)\\times \\mathbb{R}^N,\n$$\nwhere $\\eta=\\min\\{\\delta,\\lambda\\}$ with $\\delta$ and $\\lambda$ given in \\hyperlink{W2}{$(W_2)$}-\\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \\eqref{S0}, then deriving \n$E(x(t),y(t))$ along the trajectory yields\n$$\n\\frac{d}{dt}E(x(t),y(t))= \\langle y(t),\\dot{y}(t)\\rangle + \\langle \\nabla W(x(t)),\\dot{x}(t)\\rangle.\n$$\nUsing $\\dot{x}=y$ and $\\dot{y}=-\\nabla W(x)$, we immediately obtain\n$$\n\\frac{d}{dt}E(x(t),y(t))= -\\langle y(t),\\nabla W(x(t))\\rangle+ \\langle \\nabla W(x(t)),y(t)\\rangle= 0.\n$$\nThus, if $u(t)$ is any solution of \\eqref{EqConservative}, then the energy is conserved in time, that is,\n\\begin{equation}\\label{EnergyConservation}\n \\frac12\\|\\dot{u}(t)\\|^2 + W(u(t)) = \\tau,\\quad \\forall\\, t\\in\\mathbb{R},\n\\end{equation}\nfor some $\\tau\\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that \n$(0,0)$ is stable: for every $\\varepsilon>0$ there exists \n$\\delta>0$ such that\n$$\n\\|u(0)\\| + \\|\\dot{u}(0)\\| < \\delta\\quad\\Longrightarrow\\quad\\sup_{t\\ge 0} \\big( \\|u(t)\\| + \\|\\dot{u}(t)\\| \\big) \\le \\varepsilon.\n$$\nHowever, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\\dot{u}(t))$ were to converge to $(0,0)$ as $t\\to+\\infty$, then \\eqref{EnergyConservation} would force $\\tau=0$, which implies that $u(t)\\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.\n\nWe now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \\ref{Sec2}. Assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W5}{$(W_5)$}. The conservative system \\eqref{EqConservative} arises as the limit of the dissipative equation \\eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \\ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \\to 0$, and let $u_n$ be the solution of the perturbed problem\n\\begin{equation}\\label{EqDamped-n}\n \\ddot u_n(t) + a_n \\dot u_n(t) + \\nabla W(u_n(t)) = 0,\n\\end{equation}\nwith fixed initial data $(u_n(0),\\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\\neq(0,0)$ and $\\|(z_0,z_1)\\|<\\sigma$. Here $\\sigma>0$ depends on a prescribed $\\varepsilon\\in(0,\\delta)$, with $\\delta>0$ given in \\hyperlink{W2}{\\((W_2)\\)}, and is provided by the uniform stability result of Lemma \\ref{L5}. In particular, Lemma \\ref{L5} ensures that\n$$\n\\lim_{t\\to+\\infty} (u_n(t),\\dot u_n(t)) = (0,0)\\quad\\text{and}\\quad\\|u_n(t)\\| + \\|\\dot u_n(t)\\| \\le \\varepsilon \\quad\\text{for all } t\\ge 0\\text{ and }n\\in\\mathbb{N}.\n$$\nMoreover, by Lemma \\ref{L7}, each $u_n$ satisfies the exponential decay estimate\n\\begin{equation}\\label{Exp}\n \\|u_n(t)\\| + \\|\\dot u_n(t)\\|\\le C e^{-\\gamma(a_n) t}, \\qquad t\\ge 0,\n\\end{equation}\nwhere $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\\mathrm{loc}}([0,+\\infty))$ to a limit\nfunction $u_0$, which solves the conservative system\n$$\n\\ddot u_0 + \\nabla W(u_0)=0\n$$\nwith initial data $u_0(0)=z_0$ and $\\dot u_0(0)=z_1$, and satisfying\n\\begin{equation*}\n \\sup_{t\\ge 0}\\left(\\|u_0(t)\\|+\\|\\dot{u}_0(t)\\|\\right) \\leq \\epsilon.\n\\end{equation*}\nAlthough each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that\n$$\n\\lim_{t\\to+\\infty} u_0(t)=0\\quad\\text{and}\\quad\n\\lim_{t\\to+\\infty} \\dot u_0(t)=0.\n$$\nSince $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\\to 0$ and $\\|\\dot u_0(t)\\|\\to 0$. Inserting these limits into the conservation law \\eqref{EnergyConservation}, we obtain $\\tau=0$. But $\\tau=0$ implies\n$$\n\\dot u_0(t)\\equiv 0\\quad\\text{and}\\quad W(u_0(t))\\equiv 0,\n$$\nso that $u_0(t)= 0$ for all $t\\in\\mathbb{R}$. This contradicts the fact that $(u_0(0),\\dot u_0(0))\\neq(0,0)$. Therefore,\n$$\n\\lim_{t \\to +\\infty} (u_0(t),\\dot{u}_0(t)) \\neq (0,0).\n$$\nThis argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\\gamma(a)\\to 0$ as $a\\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.\n\n\\begin{lemma}\\label{asd}\n Let $B\\subset \\mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.\n Then there exists a bounded set $K\\subset \\mathbb{R}^{2N}$ such that \n $$\n T(t) B \\subset K,\\quad \\forall\\, t\\ge 0.\n $$\n In other words, $K$ serves as a absorbing set for the family of semigroup $\\{T(t)\\}_{t\\ge 0}$.\n\\end{lemma}\n\\begin{proof}\nFix $a>0$ and let $z_0=(x_0,y_0)\\in B$. Since $B$ is bounded in $\\mathbb{R}^{2N}$ and $W$ is continuous by \\hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that\n$$\n\\sup_{z_0\\in B} E(z_0)\\le R_B.\n$$\nTherefore, by the monotonicity property of the energy in \\eqref{decreasing-property}, we obtain\n$$\nW(x_a(t))\\leq R_B\\quad\\text{and}\\quad\\|y_a(t)\\|^2\\leq 2R_B\\quad\\forall \\, t\\geq 0,\n$$\nwhere $x_a$ and $y_a$ denote the components of the trajectory $\\varphi_a(t;z_0)$. Next, by the coercivity assumption \\hyperlink{W4}{$(W_4)$}, the sublevel set\n$$\n\\mathcal{S}_{B} \\coloneqq \\{ x \\in \\mathbb{R}^{N}:W(x)\\le R_B\\}\n$$\nis bounded. Consequently, we may define the positive real number\n$$\nC_B \\coloneqq \\sup \\left\\{\\|x\\|^{2} : x \\in \\mathcal{S}_{B}\\right\\} < \\infty.\n$$\nIn this way, we have \n$$\n\\Vert T(t)z_0 \\Vert_{\\ast}^2 = \\Vert (x_a(t), y_a(t))\\Vert_{\\ast}^2 = \\Vert x_a(t) \\Vert^2 + \\Vert y_a(t) \\Vert^2 \\leq C_B + 2R_B,\\quad \\forall\\, t\\ge 0, \n$$\nNow, defining\n$$\nK\\coloneqq\\Big\\{(x,y)\\in\\mathbb{R}^{2N}:\\|(x,y)\\|_{\\ast}^{2}\\le \\sqrt{C_B + 2R_B}\\Big\\}\n$$\nit follows the inclusion $T(t)B \\subset K$ for all $t\\geq 0$, which finishes our proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 6953,
    "post_theorem_intro_text": "The Theorem \\ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \\eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\\to +\\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system\n\\begin{equation}\\label{S-intro}\n\t\\begin{cases}\n\t\t\\dot{x}=y,\\\\[1mm]\n\t\t\\dot{y}=-a\\,y-\\nabla W(x),\n\t\\end{cases}\n\\end{equation}\nwhich encodes the full dynamics of \\eqref{Equation} in the phase space $\\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional\n\\begin{equation}\n\tV_a(x,y):=\\frac{1}{2}\\|y\\|^2+2W(x)+\\frac{1}{2}\\|y+a(x-u_*)\\|^2,\n\\end{equation}\nwhich can be interpreted as a smooth perturbation of the mechanical energy\n$$\nE(x,y):=\\frac12\\|y\\|^2+W(x).\n$$\nThe functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \\hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\\to 0^{+}$, since the corresponding rate $\\gamma(a)$ necessarily satisfies $\\gamma(a)\\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \\eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.\n\nWe emphasize that the conclusions of Theorem \\ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \\hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \\ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.\n\nWhen $a=0$, the equation \\eqref{Equation} reduces to the conservative system\n$$\n\\ddot u(t) + \\nabla W(u(t))=0,\n$$\ncorresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\\ddot{u}(t)$ its acceleration, and $\\nabla W(u)$ the force derived from $W$. Under assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\\mathbb{R}^{N}$, that is, \n$$\n\\sup_{t\\in\\mathbb{R}}\\left(\\|u(t)\\|+\\|\\dot u(t)\\|\\right)<\\infty.\n$$\nThus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.\n\nIn addition to the study of the asymptotic stability of \\eqref{Equation}, two additional components complement our study. In Section \\ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \\eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\\mathcal{A}$ contains the entire set of equilibria of the system, that is, \n$$\\{(x,0)\\in\\mathbb{R}^{2N}:\\nabla W (x)=0\\}\\subset \\mathcal{A}.\n$$ \nFor readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \\cite{Livro-Alexandre,Hale,Temam}. Finally, Section \\ref{Sec5} presents numerical simulations for some cases of \\eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \\emph{R}. The simulations rely on a functional-programming framework supported by \\texttt{tidyverse} (data manipulation), \\texttt{deSolve} (time-integration of differential equations), and \\texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \\eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \\emph{R} scripts at the end of Section \\ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.\n\nThe plan of the paper is as follows. Section \\ref{Sec2} contains the proof of Theorem \\ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \\ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\\to 0^{+}$. Section \\ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \\ref{Sec2}. Finally, Section \\ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.",
    "sketch": "To prove Theorem~\\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \\eqref{Equation} written as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).",
    "expanded_sketch": "To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwritten as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).",
    "expanded_theorem": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of\n\t\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\twith initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of the equation above,\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}",
    "theorem_type": [
      "Universal–Existential",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $u_a:\\mathbb{R}\\to\\mathbb{R}^N$ solve the damped second-order system\n$$\n\\ddot u(t)+a\\dot u(t)+\\nabla W(u(t))=0,\n$$\nwhere $a>0$ and $W:\\mathbb{R}^N\\to\\mathbb{R}$ is a potential. Assume $W\\in C^2(\\mathbb{R}^N)$, there exist $u_*\\in\\mathbb{R}^N$ and radii $\\delta,\\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\\in B_\\delta(u_*)\\setminus\\{u_*\\}$, $\\langle \\nabla W(x),x-u_*\\rangle>0$ for every $x\\in B_\\lambda(u_*)\\setminus\\{u_*\\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate\n$$\n\\alpha\\|x-u_*\\|^2\\le W(x)\\le \\beta\\|x-u_*\\|^2,\n\\qquad\n\\langle \\nabla W(x),x-u_*\\rangle\\ge \\mu\\|x-u_*\\|^2\n$$\nfor all $x\\in B_\\lambda(u_*)$, for some $\\alpha,\\beta,\\mu>0$. Which statement holds about the stability and decay of solutions?",
      "correct_choice": {
        "label": "A",
        "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally (that is, with $\\delta=\\lambda=+\\infty$), then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand again\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$\nIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist constants $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a\\in[a_1,a_2].\n$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exist constants $R>0$, $\\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0.\n$$"
        },
        {
          "label": "C",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"
        },
        {
          "label": "D",
          "text": "For every $\\varepsilon>0$ there exists $\\rho>0$ such that for every $a>0$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, there exist constants $\\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a>0.\n$$"
        },
        {
          "label": "E",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand every bounded trajectory converges to some equilibrium point of $W$ as $t\\to+\\infty$.\n$$\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "extra global exponential estimate without (W5)",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the global-positivity/global-monotonicity conclusion and the uniform-in-$[a_1,a_2]$ exponential constants",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniformity in $a$ down to $a=0$ for stability radius and exponential rate",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "limit identified only as some equilibrium rather than the distinguished $u_*$ under global assumptions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion; it gives hypotheses and asks which asymptotic claim follows. Although the assumptions are detailed, they do not directly reveal the exact correct choice, especially because the options differ in subtle quantifier and uniformity claims."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to a theorem-recall question: the correct option is essentially a full theorem statement under the listed assumptions. Still, it is not a pure restatement, since the distractors alter important logical features such as convergence clauses, global hypotheses, and uniformity in the damping parameter."
      },
      "GPS": {
        "score": 2,
        "justification": "Selecting the correct option requires meaningful reasoning about which conclusions are justified: local vs global attraction, dependence on bounds for a, and when exponential decay can be uniform. The solver must distinguish valid conclusions from stronger but false variants and a weaker true variant."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and reflect common failure modes: overclaiming uniformity in a, dropping necessary radial monotonicity for global attraction, omitting convergence, or extending exponential estimates to a→0. They are distinct and well-targeted."
      },
      "total_score": 7,
      "overall_assessment": "A strong, technically sophisticated MCQ with high-quality distractors and genuine logical discrimination, though it remains somewhat theorem-statement-driven rather than fully generative."
    }
  },
  {
    "id": "2512.20751v1",
    "paper_link": "http://arxiv.org/abs/2512.20751v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}",
      "start_pos": 1552652,
      "end_pos": 1554414,
      "label": "Teo1"
    },
    "ref_dict": {
      "Equation": "\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}",
      "Teo1": "\\begin{theorem}\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 6996,
    "pre_theorem_intro_text": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}.  \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nSecond-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \\eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\\varphi(t) \\ge c\\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \\cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \\cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \\cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \\eqref{Equation} and on models where the damping parameter depends on time, such as \\cite{1,2,3,4,6,7}.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n\t\\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\t\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n\t$$\n\tW(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W3}{($W_3$)}] There exists $\\lambda > 0$ such that\n\t$$\n\t\\langle \\nabla W(u), u - u_* \\rangle > 0 \\quad \\text{for all} \\quad u \\in B_\\lambda(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,\n\t$$\n\t\\lim_{\\|u\\| \\to +\\infty} W(u) = +\\infty.\n\t$$\n\n\t\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n\t$$\n\t\\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n\t$$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking  \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential  \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential  \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential  \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.",
    "context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}",
    "full_context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\nIn this section we discuss the dynamics of the conservative system\n\\begin{equation}\\label{EqConservative}\n \\ddot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwhich arises as the limit of \\eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative\ncase, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \\eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\\mathbb{R}^{2N}$ by setting $x=u$ and $y=\\dot{u}$:\n\\begin{equation}\\label{S0}\n \\begin{cases}\n \\dot{x} = y,\\\\[2mm]\n \\dot{y} = -\\nabla W(x).\n \\end{cases}\n\\end{equation}\nThe natural energy associated with \\eqref{EqConservative},\n$$\nE(x,y) := \\frac12\\|y\\|^2 + W(x), \\quad (x,y)\\in B_\\eta(0)\\times \\mathbb{R}^N,\n$$\nwhere $\\eta=\\min\\{\\delta,\\lambda\\}$ with $\\delta$ and $\\lambda$ given in \\hyperlink{W2}{$(W_2)$}-\\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \\eqref{S0}, then deriving \n$E(x(t),y(t))$ along the trajectory yields\n$$\n\\frac{d}{dt}E(x(t),y(t))= \\langle y(t),\\dot{y}(t)\\rangle + \\langle \\nabla W(x(t)),\\dot{x}(t)\\rangle.\n$$\nUsing $\\dot{x}=y$ and $\\dot{y}=-\\nabla W(x)$, we immediately obtain\n$$\n\\frac{d}{dt}E(x(t),y(t))= -\\langle y(t),\\nabla W(x(t))\\rangle+ \\langle \\nabla W(x(t)),y(t)\\rangle= 0.\n$$\nThus, if $u(t)$ is any solution of \\eqref{EqConservative}, then the energy is conserved in time, that is,\n\\begin{equation}\\label{EnergyConservation}\n \\frac12\\|\\dot{u}(t)\\|^2 + W(u(t)) = \\tau,\\quad \\forall\\, t\\in\\mathbb{R},\n\\end{equation}\nfor some $\\tau\\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that \n$(0,0)$ is stable: for every $\\varepsilon>0$ there exists \n$\\delta>0$ such that\n$$\n\\|u(0)\\| + \\|\\dot{u}(0)\\| < \\delta\\quad\\Longrightarrow\\quad\\sup_{t\\ge 0} \\big( \\|u(t)\\| + \\|\\dot{u}(t)\\| \\big) \\le \\varepsilon.\n$$\nHowever, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\\dot{u}(t))$ were to converge to $(0,0)$ as $t\\to+\\infty$, then \\eqref{EnergyConservation} would force $\\tau=0$, which implies that $u(t)\\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.\n\nWe now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \\ref{Sec2}. Assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W5}{$(W_5)$}. The conservative system \\eqref{EqConservative} arises as the limit of the dissipative equation \\eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \\ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \\to 0$, and let $u_n$ be the solution of the perturbed problem\n\\begin{equation}\\label{EqDamped-n}\n \\ddot u_n(t) + a_n \\dot u_n(t) + \\nabla W(u_n(t)) = 0,\n\\end{equation}\nwith fixed initial data $(u_n(0),\\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\\neq(0,0)$ and $\\|(z_0,z_1)\\|<\\sigma$. Here $\\sigma>0$ depends on a prescribed $\\varepsilon\\in(0,\\delta)$, with $\\delta>0$ given in \\hyperlink{W2}{\\((W_2)\\)}, and is provided by the uniform stability result of Lemma \\ref{L5}. In particular, Lemma \\ref{L5} ensures that\n$$\n\\lim_{t\\to+\\infty} (u_n(t),\\dot u_n(t)) = (0,0)\\quad\\text{and}\\quad\\|u_n(t)\\| + \\|\\dot u_n(t)\\| \\le \\varepsilon \\quad\\text{for all } t\\ge 0\\text{ and }n\\in\\mathbb{N}.\n$$\nMoreover, by Lemma \\ref{L7}, each $u_n$ satisfies the exponential decay estimate\n\\begin{equation}\\label{Exp}\n \\|u_n(t)\\| + \\|\\dot u_n(t)\\|\\le C e^{-\\gamma(a_n) t}, \\qquad t\\ge 0,\n\\end{equation}\nwhere $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\\mathrm{loc}}([0,+\\infty))$ to a limit\nfunction $u_0$, which solves the conservative system\n$$\n\\ddot u_0 + \\nabla W(u_0)=0\n$$\nwith initial data $u_0(0)=z_0$ and $\\dot u_0(0)=z_1$, and satisfying\n\\begin{equation*}\n \\sup_{t\\ge 0}\\left(\\|u_0(t)\\|+\\|\\dot{u}_0(t)\\|\\right) \\leq \\epsilon.\n\\end{equation*}\nAlthough each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that\n$$\n\\lim_{t\\to+\\infty} u_0(t)=0\\quad\\text{and}\\quad\n\\lim_{t\\to+\\infty} \\dot u_0(t)=0.\n$$\nSince $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\\to 0$ and $\\|\\dot u_0(t)\\|\\to 0$. Inserting these limits into the conservation law \\eqref{EnergyConservation}, we obtain $\\tau=0$. But $\\tau=0$ implies\n$$\n\\dot u_0(t)\\equiv 0\\quad\\text{and}\\quad W(u_0(t))\\equiv 0,\n$$\nso that $u_0(t)= 0$ for all $t\\in\\mathbb{R}$. This contradicts the fact that $(u_0(0),\\dot u_0(0))\\neq(0,0)$. Therefore,\n$$\n\\lim_{t \\to +\\infty} (u_0(t),\\dot{u}_0(t)) \\neq (0,0).\n$$\nThis argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\\gamma(a)\\to 0$ as $a\\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.\n\n\\begin{lemma}\\label{asd}\n Let $B\\subset \\mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.\n Then there exists a bounded set $K\\subset \\mathbb{R}^{2N}$ such that \n $$\n T(t) B \\subset K,\\quad \\forall\\, t\\ge 0.\n $$\n In other words, $K$ serves as a absorbing set for the family of semigroup $\\{T(t)\\}_{t\\ge 0}$.\n\\end{lemma}\n\\begin{proof}\nFix $a>0$ and let $z_0=(x_0,y_0)\\in B$. Since $B$ is bounded in $\\mathbb{R}^{2N}$ and $W$ is continuous by \\hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that\n$$\n\\sup_{z_0\\in B} E(z_0)\\le R_B.\n$$\nTherefore, by the monotonicity property of the energy in \\eqref{decreasing-property}, we obtain\n$$\nW(x_a(t))\\leq R_B\\quad\\text{and}\\quad\\|y_a(t)\\|^2\\leq 2R_B\\quad\\forall \\, t\\geq 0,\n$$\nwhere $x_a$ and $y_a$ denote the components of the trajectory $\\varphi_a(t;z_0)$. Next, by the coercivity assumption \\hyperlink{W4}{$(W_4)$}, the sublevel set\n$$\n\\mathcal{S}_{B} \\coloneqq \\{ x \\in \\mathbb{R}^{N}:W(x)\\le R_B\\}\n$$\nis bounded. Consequently, we may define the positive real number\n$$\nC_B \\coloneqq \\sup \\left\\{\\|x\\|^{2} : x \\in \\mathcal{S}_{B}\\right\\} < \\infty.\n$$\nIn this way, we have \n$$\n\\Vert T(t)z_0 \\Vert_{\\ast}^2 = \\Vert (x_a(t), y_a(t))\\Vert_{\\ast}^2 = \\Vert x_a(t) \\Vert^2 + \\Vert y_a(t) \\Vert^2 \\leq C_B + 2R_B,\\quad \\forall\\, t\\ge 0, \n$$\nNow, defining\n$$\nK\\coloneqq\\Big\\{(x,y)\\in\\mathbb{R}^{2N}:\\|(x,y)\\|_{\\ast}^{2}\\le \\sqrt{C_B + 2R_B}\\Big\\}\n$$\nit follows the inclusion $T(t)B \\subset K$ for all $t\\geq 0$, which finishes our proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 6953,
    "post_theorem_intro_text": "The Theorem \\ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \\eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\\to +\\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system\n\\begin{equation}\\label{S-intro}\n\t\\begin{cases}\n\t\t\\dot{x}=y,\\\\[1mm]\n\t\t\\dot{y}=-a\\,y-\\nabla W(x),\n\t\\end{cases}\n\\end{equation}\nwhich encodes the full dynamics of \\eqref{Equation} in the phase space $\\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional\n\\begin{equation}\n\tV_a(x,y):=\\frac{1}{2}\\|y\\|^2+2W(x)+\\frac{1}{2}\\|y+a(x-u_*)\\|^2,\n\\end{equation}\nwhich can be interpreted as a smooth perturbation of the mechanical energy\n$$\nE(x,y):=\\frac12\\|y\\|^2+W(x).\n$$\nThe functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \\hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\\to 0^{+}$, since the corresponding rate $\\gamma(a)$ necessarily satisfies $\\gamma(a)\\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \\eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.\n\nWe emphasize that the conclusions of Theorem \\ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \\hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \\ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.\n\nWhen $a=0$, the equation \\eqref{Equation} reduces to the conservative system\n$$\n\\ddot u(t) + \\nabla W(u(t))=0,\n$$\ncorresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\\ddot{u}(t)$ its acceleration, and $\\nabla W(u)$ the force derived from $W$. Under assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\\mathbb{R}^{N}$, that is, \n$$\n\\sup_{t\\in\\mathbb{R}}\\left(\\|u(t)\\|+\\|\\dot u(t)\\|\\right)<\\infty.\n$$\nThus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.\n\nIn addition to the study of the asymptotic stability of \\eqref{Equation}, two additional components complement our study. In Section \\ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \\eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\\mathcal{A}$ contains the entire set of equilibria of the system, that is, \n$$\\{(x,0)\\in\\mathbb{R}^{2N}:\\nabla W (x)=0\\}\\subset \\mathcal{A}.\n$$ \nFor readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \\cite{Livro-Alexandre,Hale,Temam}. Finally, Section \\ref{Sec5} presents numerical simulations for some cases of \\eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \\emph{R}. The simulations rely on a functional-programming framework supported by \\texttt{tidyverse} (data manipulation), \\texttt{deSolve} (time-integration of differential equations), and \\texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \\eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \\emph{R} scripts at the end of Section \\ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.\n\nThe plan of the paper is as follows. Section \\ref{Sec2} contains the proof of Theorem \\ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \\ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\\to 0^{+}$. Section \\ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \\ref{Sec2}. Finally, Section \\ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.",
    "sketch": "To prove Theorem~\\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \\eqref{Equation} written as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).",
    "expanded_sketch": "To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwritten as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).",
    "expanded_theorem": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of\n\t\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\twith initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of the equation above,\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}",
    "theorem_type": [
      "Universal–Existential",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Let $u_a:\\mathbb{R}\\to\\mathbb{R}^N$ solve the damped second-order system\n$$\n\\ddot u(t)+a\\dot u(t)+\\nabla W(u(t))=0,\n$$\nwhere $a>0$ and $W:\\mathbb{R}^N\\to\\mathbb{R}$ is a potential. Assume $W\\in C^2(\\mathbb{R}^N)$, there exist $u_*\\in\\mathbb{R}^N$ and radii $\\delta,\\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\\in B_\\delta(u_*)\\setminus\\{u_*\\}$, $\\langle \\nabla W(x),x-u_*\\rangle>0$ for every $x\\in B_\\lambda(u_*)\\setminus\\{u_*\\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate\n$$\n\\alpha\\|x-u_*\\|^2\\le W(x)\\le \\beta\\|x-u_*\\|^2,\n\\qquad\n\\langle \\nabla W(x),x-u_*\\rangle\\ge \\mu\\|x-u_*\\|^2\n$$\nfor all $x\\in B_\\lambda(u_*)$, for some $\\alpha,\\beta,\\mu>0$. Which statement holds about the stability and decay of solutions?",
      "correct_choice": {
        "label": "A",
        "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally (that is, with $\\delta=\\lambda=+\\infty$), then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand again\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$\nIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist constants $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a\\in[a_1,a_2].\n$$"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exist constants $R>0$, $\\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0.\n$$"
        },
        {
          "label": "C",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"
        },
        {
          "label": "D",
          "text": "For every $\\varepsilon>0$ there exists $\\rho>0$ such that for every $a>0$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, there exist constants $\\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a>0.\n$$"
        },
        {
          "label": "E",
          "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand every bounded trajectory converges to some equilibrium point of $W$ as $t\\to+\\infty$.\n$$\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "extra global exponential estimate without (W5)",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped the global-positivity/global-monotonicity conclusion and the uniform-in-$[a_1,a_2]$ exponential constants",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "uniformity in $a$ down to $a=0$ for stability radius and exponential rate",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "limit identified only as some equilibrium rather than the distinguished $u_*$ under global assumptions",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the conclusion outright. It gives hypotheses and asks which global/local stability and decay statement is valid; the correct answer is not leaked explicitly, though the extra quadratic assumptions do hint that some exponential decay statement may be relevant."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to theorem recognition: the correct option is essentially a full theorem statement assembled from the listed assumptions. However, it is not a pure verbatim restatement, since the alternatives differ in subtle quantifiers, global-vs-local assumptions, and uniformity claims."
      },
      "GPS": {
        "score": 1,
        "justification": "Moderate reasoning is required to compare the strength of the claims, especially the dependence on a, the need for global positivity/monotonicity, and when exponential rates can be uniform. Still, the task is mainly discriminating among theorem variants rather than generating a conclusion from scratch."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: unjustified uniformity as a -> 0, overly strong global exponential decay, omission of global conclusions, and convergence only to some equilibrium. They are distinct and well aligned with common misunderstandings."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it remains largely a theorem-identification item rather than a deeply generative reasoning question."
    }
  },
  {
    "id": "2512.21134v2",
    "paper_link": "http://arxiv.org/abs/2512.21134v2",
    "theorems_cnt": 6,
    "theorem": {
      "env_name": "theorem",
      "content": "The number of elements in ${DRP}_n$ of a fixed width $r$ and height $p$ is $F(n,r,p)= {r-1 \\choose p-1} {n+1 \\choose r+p} $",
      "start_pos": 14973,
      "end_pos": 15126,
      "label": null
    },
    "ref_dict": {
      "rev": "\\begin{lemma}\\label{rev}\nAn element $\\rho \\in \\mathcal{LS}_{n}$, as expressed in \\eqref{1}, is reversible; that is, $\\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix} \\in \\mathcal{DORP}_{n}$ if and only if $a_{p} \\leq \\min A_{1}$ and $1 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$.\n\\end{lemma}",
      "1": "\\begin{equation}\n\\label{1}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_1 & \\dots & a_p \\end{pmatrix} \\quad (1 \\leq p \\leq n),\n\\end{equation}",
      "coo": "\\begin{equation}\\label{coo} F(n,p)= \\sum_{r=p}^{n-p+1}{r-1 \\choose p-1} {n+1 \\choose r+p}.\\end{equation}",
      "id": "\\begin{lemma}[\\cite{44}, (3b), p.8]\\label{id}\nFor all n \\( m, n, k\\in \\mathbb{N} \\), it follows that\n\\[\n\\sum_{j=k}^{m} \\binom{j}{k} \\binom{m+n-j}{n} = \\binom{m+n+1}{n+k+1}.\n\\]\n\\end{lemma}",
      "2": "\\begin{equation}\\label{2}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix},\n\\end{equation}",
      "nums": "\\begin{equation}\\label{nums}\ns_{0}=1, \\quad s_{n}= \\frac{1}{n+1} \\sum\\limits_{r=0}^{n}\\binom{n+1}{n-r}\\binom{n+r}{r}, \\quad n\\geq 1.\n\\end{equation}",
      "qn2": "\\begin{equation}\\label{qn2}\n\t\\mathcal{DORP}_{n} = \\mathcal{LS}_{n} \\cup DRP_{n}.\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 10711,
    "pre_theorem_intro_text": "Let $[n]$ denote the finite $n-$chain $\\{1, 2, \\dots, n\\}$. A function $\\rho$ with  domain and range both being subsets of $[n]$ is termed a \\emph{partial transformation} of $[n]$, and it is said to be \\emph{full} (or \\emph{total}) if its domain is the whole of $[n]$. The set consisting of all partial transformations on $[n]$ is usually denoted by $\\mathcal{P}_{n}$ and is known as \\emph{the semigroup of all partial transformations} on $[n]$, more commonly referred to as \\emph{the partial symmetric monoid}. A transformation $\\rho \\in \\mathcal{P}_{n}$ is called an \\emph{isotone} (or order preserving) function (resp., an \\emph{antitone} (order reversing) function) if (for all $x,y \\in \\mathop{\\rm Dom}\\nolimits\\,\\rho$) $x \\leq y$ implies $x\\rho \\leq y\\rho$ (resp., $x\\rho \\geq y\\rho$); it is called \\emph{monotone} if it is either isotone or antitone or both; and it is called \\emph{order decreasing} if (for all $a \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho$) $a\\rho \\leq a$. Let $\\mathcal{POD}_{n}$ be \\emph{the monoid of all monotone partial transformations} on $[n]$. This monoid first appeared in Fernandes \\cite{vm} and subsequently in Umar \\cite{auc} and East \\emph{et. al.}, \\cite{east}, where its congruence/rank properties and combinatorial properties, and its maximal subsemigroups were studied, respectively. The algebraic properties of various subsemigroups of monotone maps in various classes have been investigated over the years, for example see \\cite{bugay, bugay1, il}.\n\nIn line with \\cite{auc},  $\\mathcal{LS}_{n}$ (\\emph{the semigroup of all isotone and order-decreasing partial transformations} on $[n]$) is  refer to as the \\emph{large} \\emph{Schr\\\"{o}der} monoid, which is defined as:\n\n\\begin{equation}\\label{qn111}\n\\mathcal{LS}_{n} = \\mathcal{OP}_n \\cap \\mathcal{DP}_n,\n\\end{equation}\n\n\\noindent where  $\\mathcal{OP}_n$ and $\\mathcal{DP}_n$ denote  the \\emph{semigroup of all isotone partial transformations} on $[n]$  and \\emph{the semigroup of all order-decreasing partial transformations} on $[n]$, respectively. The order of this monoid, obtained in \\cite{al3}, corresponds to the \\emph{double} (or \\emph{large}) \\emph{Schr\\\"{o}der} number:\n\n\\begin{equation}\\label{nums}\ns_{0}=1, \\quad s_{n}= \\frac{1}{n+1} \\sum\\limits_{r=0}^{n}\\binom{n+1}{n-r}\\binom{n+r}{r}, \\quad n\\geq 1.\n\\end{equation}\n\n Now, let\n\\begin{equation}\\label{qn1}\n \t\\mathcal{DORP}_{n} = \\{\\rho \\in \\mathcal{POD}_{n} : \\,   \\rho \\text{ is \\it decreasing} \\}\n\\end{equation}\n\\noindent be the set of all decreasing maps in $\\mathcal{POD}_{n}$. This set can equivalently be expressed as:\n\\begin{align*}\n\\mathcal{DORP}_{n} &= \\{\\rho \\in \\mathcal{P}_{n} : \\,   \\rho \\text{ is {\\it monotone} and \\it decreasing} \\} \\\\\n&= \\{\\rho \\in \\mathcal{P}_{n} : \\,   \\rho \\text{ is {\\it isotone} and \\it decreasing} \\} \\cup \\{\\rho \\in \\mathcal{P}_{n} : \\,   \\rho \\text{ is {\\it antitone} and \\it decreasing} \\}.\n\\end{align*}\n\\noindent Therefore, if we let $DRP_{n} = \\{\\rho \\in \\mathcal{P}_{n} : \\,   \\rho  \\text{ is {\\it antitone} and \\it decreasing}\\},$ then\n\\begin{equation}\\label{qn2}\n\t\\mathcal{DORP}_{n} = \\mathcal{LS}_{n} \\cup DRP_{n}.\n\\end{equation}\nMoreover, the set $\\mathcal{DORP}_{n}$ can be expressed as:\n\n$$\n\\mathcal{DORP}_{n} = \\mathcal{POD}_{n} \\cap \\mathcal{DP}_{n}.\n$$\n\\noindent It is a routine matter to show that $\\mathcal{DORP}_{n}$ is a monoid. It shall be refer to  as the \\emph{monoid of all monotone and decreasing partial transformations} on $[n]$. There seems to be no prior discussion of the monoid \\(\\mathcal{DORP}_{n}\\) in the literature. This paper investigates certain algebraic features of the monoid \\(\\mathcal{DORP}_{n}\\) and its rank properties.\n\nFor  map $\\rho\\in \\mathcal{DORP}_{n}$, we shall adopt the notations $\\mathop{\\rm Dom}\\nolimits \\rho$, $1_{[n]}$, $b(\\rho) = |\\mathop{\\rm Dom}\\nolimits \\, \\rho|$, $\\mathop{\\rm Im}\\nolimits \\rho$,  $h(\\rho) = |\\mathop{\\rm Im}\\nolimits \\, \\rho|$, and $F(\\rho) = \\{x \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho : x\\rho = x\\}$ to denote the domain set of  $\\rho$, the identity mapping on $[n]$, the width of $\\rho$, the image set of $\\rho$, the height of $\\rho$,  and the set of fixed points of $\\rho$, respectively. Additionally, we will let $f(\\rho) = |F(\\rho)|$ denote the number of fixed points of $\\rho$.  A subset $X$ of $[n]$ is said to be \\emph{convex} if, for any $ x, y\\in X $ such that $x \\leq y$, and for any $c \\in [n]$, if $x < c < y$, then $c \\in X$. We shall be using the notations $E(S)$ and $\\textbf{0}$ to denote the set of idempotents  and the zero element of a semigroup $S$, respectively.\n\nFurthermore, we shall adopt the right-hand composition of two transformations, say $\\rho$ and $\\sigma$ in $\\mathcal{P}_{n}$, defined as\n$$\nx(\\rho \\circ \\sigma) = ((x)\\rho) \\sigma\n$$\n\\noindent for all $x \\in \\mathop{\\rm Dom}\\nolimits\\, \\rho$. To be concise, we will denote \\(\\rho \\sigma\\) as \\(\\rho \\circ \\sigma\\).\n\n  Moreover, for $0\\le p\\le n-1$,  let \\begin{equation} \\label{kn} I(n, \\,p)=\\{\\rho\\in  \\mathcal{DORP}_{n}: \\, |\\mathop{\\rm Im}\\nolimits \\, \\rho|\\le p\\}\\end{equation}\n  \\noindent be the two-sided ideal of $\\mathcal{DORP}_{n}$, which consist all decreasing and monotone transformations in  $\\mathcal{DORP}_{n}$, each with a height not more than $p$.\n\n  Additionally, given that $p\\geq 1$,  denote \\begin{equation}\\label{knn} {RQ}_{p}(n)= I(n, \\,p)/ I(n, \\, p-1)  \\end{equation}\n  \\noindent to be the Rees quotient semigroup of $I(n, \\,p)$, where the elements in ${RQ}_{p}(n)$ can be regarded as the elements of $\\mathcal{DORP}_{n}$ that possess a height of precisely $p$. If the product of two elements from ${RQ}_{p}(n)$ has a height that does not reach $p$, then the result is $\\textbf{0}$; otherwise, it is an element of ${RQ}_{p}(n) \\setminus \\{\\textbf{0}\\}$.\n\n\\indent According to \\cite{HRS}, every $\\rho \\in \\mathcal{LS}_{n}$ can be represented in two-line notation as\n\n\\begin{equation}\n\\label{1}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_1 & \\dots & a_p \\end{pmatrix} \\quad (1 \\leq p \\leq n),\n\\end{equation}\n\n\\noindent where $a_{i} \\leq \\min A_{i}$ for all $i\\in\\{1,\\ldots, p\\}$ since $\\rho$ is a decreasing transformation, and each of the sets $A_i$ $(1 \\leq i \\leq p)$ denotes an equivalence class determined by the following relation:\n\n$$\\textnormal{ker } \\rho = \\{(x, y) \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho \\times \\mathop{\\rm Dom}\\nolimits \\, \\rho : x\\rho = y\\rho\\}.$$\n\n\\noindent The collection of these equivalence classes shall be denoted as $\\textnormal{\\bf Ker } \\rho = \\{A_1,  \\dots, A_p\\}$. In addition, the kernel $\\textnormal{\\bf Ker } \\rho$ is linearly ordered (meaning that for any indices $i < j$, the relation $A_{i} < A_{j}$ is equivalent to $x < y$ for all $x \\in A_{i}$ and $y \\in A_{j}$). Furthermore, we can assume without loss of generality that $1 \\leq a_{1} < \\dots < a_{p} \\leq n$. For a comprehensive introduction to the basic ideas of semigroup theory, we recommend the texts by Howie \\cite{howi} and Higgins \\cite{ph}.\n\nThis paragraph offers a concise explanation of the framework of the paper. Section 1 provide definitions of some basic terms and compute the order of the monoid \\( \\mathcal{DORP}_{n} \\). In Section 2, we characterize of all the Green's relations along with their starred counterparts in the monoid \\( \\mathcal{DORP}_{n} \\) and its two-sided ideal \\( I(n, \\,p) \\). Moreover, we demonstrate that the monoid \\( \\mathcal{DORP}_{n} \\) and its two-sided ideals are abundant semigroups. Finally, in Section 3, we calculate the rank of the Rees quotient semigroup \\( RQ_{n}(p) \\), the two-sided ideal \\( I(n, \\,p) \\), and the monoid \\( \\mathcal{DORP}_{n} \\). The computation of the rank of this monoid and its two-sided ideals differs from the conventional understanding associated with certain known monoids, where most arguments or generating sets are derive from elements of height \\(n-1\\). However, in the monoid \\( \\mathcal{DORP}_{n} \\), elements of height \\(n-1\\) are insufficient to generate the structure; additional elements below this height are necessary, as will be discussed in Section 3.\n\nNow let us briefly examine the elements in $DRP_{n}$. First, notice that every element of height 1 in $\\mathcal{DORP}_{n}$ is both isotone and antitone; that is, it belongs to both $\\mathcal{LS}_{n}$ and $DRP_{n}$. However, antitones of height greater than or equal to 2 cannot be isotones. Therefore, we have the following remark.\n\n\\begin{remark}\\label{rmk1}\nEvery element in $DRP_{n}$ of height $1 < p < n$ of the form\n\\begin{equation}\\label{2}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix},\n\\end{equation}\n has the following properties:\n\\begin{itemize}\n  \\item[(i)] $1 \\leq a_{1} < \\cdots < a_{p} \\leq \\min A_{1}$;\n  \\item[(ii)] Note that when multiplying two isotone maps or two antitone maps, the result remains isotone. In contrast, the product of an isotone map with an antitone map yields an antitone map. Similarly, when an antitone map is multiplied by an isotone map, the outcome is antitone as well.\n\n  \\item[(iii)] It should also be emphasized that all elements in $\\mathcal{DORP}_{n}$ of height greater than $\\lceil \\frac{n}{2} \\rceil$ is necessarily isotone.\n\\end{itemize}\n\\end{remark}\n\nConsequently, every element in $DRP_{n}$ has height $p$ that ranges from $1 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$; thus, we have the following lemma.\n\n\\begin{lemma}\\label{rev}\nAn element $\\rho \\in \\mathcal{LS}_{n}$, as expressed in \\eqref{1}, is reversible; that is, $\\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix} \\in \\mathcal{DORP}_{n}$ if and only if $a_{p} \\leq \\min A_{1}$ and $1 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$.\n\\end{lemma}\n\\begin{proof} Let $w=\\min (\\mathop{\\rm Dom}\\nolimits \\, \\rho) =\\min A_{1}\\geq a_{p}$, by the order decreasing property. However, notice that $n-p+1\\geq w $ and $a_{p}\\geq p$,  and so,  $n-p+1\\geq w\\geq a_{p}\\geq p$, which implies $n+1\\geq 2p$. Hence $p\\leq \\lfloor \\frac{n+1}{2}\\rfloor=\\lceil\\frac{n}{2}\\rceil$. The converse is trivial.\n\n\\end{proof}\nWe note the following well known combinatorial identity, which is useful in our subsequent discussions.\n\n \\begin{lemma}[\\cite{44}, (3b), p.8]\\label{id}\nFor all n \\( m, n, k\\in \\mathbb{N} \\), it follows that\n\\[\n\\sum_{j=k}^{m} \\binom{j}{k} \\binom{m+n-j}{n} = \\binom{m+n+1}{n+k+1}.\n\\]\n\\end{lemma}\n\n It is now clear that to obtain the size of $\\mathcal{DORP}_{n}$, it is sufficient to compute the size of the set $DRP_{n}$. For a transformation $\\rho$ on a finite chain, let $b(\\rho) = r$ and $h(\\rho) = p$. Then, $p \\leq r \\leq n$. Define the combinatorial functions:\n\n\\[F(n,r,p)=|\\{\\rho\\in {DRP}_n : \\, b(\\rho)=r \\text{ and } h( \\rho)=p\\}|\\]\n\\noindent and\n\\[F(n,p)=|\\{\\rho\\in {DRP}_n : \\,  h(\\rho)=p\\}|.\\]\n\nThen we present the following theorem.",
    "context": "For map $\\rho\\in \\mathcal{DORP}_{n}$, we shall adopt the notations $\\mathop{\\rm Dom}\\nolimits \\rho$, $1_{[n]}$, $b(\\rho) = |\\mathop{\\rm Dom}\\nolimits \\, \\rho|$, $\\mathop{\\rm Im}\\nolimits \\rho$, $h(\\rho) = |\\mathop{\\rm Im}\\nolimits \\, \\rho|$, and $F(\\rho) = \\{x \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho : x\\rho = x\\}$ to denote the domain set of $\\rho$, the identity mapping on $[n]$, the width of $\\rho$, the image set of $\\rho$, the height of $\\rho$, and the set of fixed points of $\\rho$, respectively. Additionally, we will let $f(\\rho) = |F(\\rho)|$ denote the number of fixed points of $\\rho$. A subset $X$ of $[n]$ is said to be \\emph{convex} if, for any $ x, y\\in X $ such that $x \\leq y$, and for any $c \\in [n]$, if $x < c < y$, then $c \\in X$. We shall be using the notations $E(S)$ and $\\textbf{0}$ to denote the set of idempotents and the zero element of a semigroup $S$, respectively.\n\nMoreover, for $0\\le p\\le n-1$, let \\begin{equation} \\label{kn} I(n, \\,p)=\\{\\rho\\in \\mathcal{DORP}_{n}: \\, |\\mathop{\\rm Im}\\nolimits \\, \\rho|\\le p\\}\\end{equation}\n \\noindent be the two-sided ideal of $\\mathcal{DORP}_{n}$, which consist all decreasing and monotone transformations in $\\mathcal{DORP}_{n}$, each with a height not more than $p$.\n\nAdditionally, given that $p\\geq 1$, denote \\begin{equation}\\label{knn} {RQ}_{p}(n)= I(n, \\,p)/ I(n, \\, p-1) \\end{equation}\n \\noindent to be the Rees quotient semigroup of $I(n, \\,p)$, where the elements in ${RQ}_{p}(n)$ can be regarded as the elements of $\\mathcal{DORP}_{n}$ that possess a height of precisely $p$. If the product of two elements from ${RQ}_{p}(n)$ has a height that does not reach $p$, then the result is $\\textbf{0}$; otherwise, it is an element of ${RQ}_{p}(n) \\setminus \\{\\textbf{0}\\}$.\n\nNow let us briefly examine the elements in $DRP_{n}$. First, notice that every element of height 1 in $\\mathcal{DORP}_{n}$ is both isotone and antitone; that is, it belongs to both $\\mathcal{LS}_{n}$ and $DRP_{n}$. However, antitones of height greater than or equal to 2 cannot be isotones. Therefore, we have the following remark.\n\n\\[F(n,r,p)=|\\{\\rho\\in {DRP}_n : \\, b(\\rho)=r \\text{ and } h( \\rho)=p\\}|\\]\n\\noindent and\n\\[F(n,p)=|\\{\\rho\\in {DRP}_n : \\, h(\\rho)=p\\}|.\\]\n\nThen we present the following theorem.",
    "full_context": "For map $\\rho\\in \\mathcal{DORP}_{n}$, we shall adopt the notations $\\mathop{\\rm Dom}\\nolimits \\rho$, $1_{[n]}$, $b(\\rho) = |\\mathop{\\rm Dom}\\nolimits \\, \\rho|$, $\\mathop{\\rm Im}\\nolimits \\rho$, $h(\\rho) = |\\mathop{\\rm Im}\\nolimits \\, \\rho|$, and $F(\\rho) = \\{x \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho : x\\rho = x\\}$ to denote the domain set of $\\rho$, the identity mapping on $[n]$, the width of $\\rho$, the image set of $\\rho$, the height of $\\rho$, and the set of fixed points of $\\rho$, respectively. Additionally, we will let $f(\\rho) = |F(\\rho)|$ denote the number of fixed points of $\\rho$. A subset $X$ of $[n]$ is said to be \\emph{convex} if, for any $ x, y\\in X $ such that $x \\leq y$, and for any $c \\in [n]$, if $x < c < y$, then $c \\in X$. We shall be using the notations $E(S)$ and $\\textbf{0}$ to denote the set of idempotents and the zero element of a semigroup $S$, respectively.\n\nMoreover, for $0\\le p\\le n-1$, let \\begin{equation} \\label{kn} I(n, \\,p)=\\{\\rho\\in \\mathcal{DORP}_{n}: \\, |\\mathop{\\rm Im}\\nolimits \\, \\rho|\\le p\\}\\end{equation}\n \\noindent be the two-sided ideal of $\\mathcal{DORP}_{n}$, which consist all decreasing and monotone transformations in $\\mathcal{DORP}_{n}$, each with a height not more than $p$.\n\nAdditionally, given that $p\\geq 1$, denote \\begin{equation}\\label{knn} {RQ}_{p}(n)= I(n, \\,p)/ I(n, \\, p-1) \\end{equation}\n \\noindent to be the Rees quotient semigroup of $I(n, \\,p)$, where the elements in ${RQ}_{p}(n)$ can be regarded as the elements of $\\mathcal{DORP}_{n}$ that possess a height of precisely $p$. If the product of two elements from ${RQ}_{p}(n)$ has a height that does not reach $p$, then the result is $\\textbf{0}$; otherwise, it is an element of ${RQ}_{p}(n) \\setminus \\{\\textbf{0}\\}$.\n\nNow let us briefly examine the elements in $DRP_{n}$. First, notice that every element of height 1 in $\\mathcal{DORP}_{n}$ is both isotone and antitone; that is, it belongs to both $\\mathcal{LS}_{n}$ and $DRP_{n}$. However, antitones of height greater than or equal to 2 cannot be isotones. Therefore, we have the following remark.\n\n\\[F(n,r,p)=|\\{\\rho\\in {DRP}_n : \\, b(\\rho)=r \\text{ and } h( \\rho)=p\\}|\\]\n\\noindent and\n\\[F(n,p)=|\\{\\rho\\in {DRP}_n : \\, h(\\rho)=p\\}|.\\]\n\nThen we present the following theorem.\n\nFor map $\\rho\\in \\mathcal{DORP}_{n}$, we shall adopt the notations $\\dom \\rho$, $1_{[n]}$, $b(\\rho) = |\\dom \\, \\rho|$, $\\im \\rho$, $h(\\rho) = |\\im \\, \\rho|$, and $F(\\rho) = \\{x \\in \\dom \\, \\rho : x\\rho = x\\}$ to denote the domain set of $\\rho$, the identity mapping on $[n]$, the width of $\\rho$, the image set of $\\rho$, the height of $\\rho$, and the set of fixed points of $\\rho$, respectively. Additionally, we will let $f(\\rho) = |F(\\rho)|$ denote the number of fixed points of $\\rho$. A subset $X$ of $[n]$ is said to be \\emph{convex} if, for any $ x, y\\in X $ such that $x \\leq y$, and for any $c \\in [n]$, if $x < c < y$, then $c \\in X$. We shall be using the notations $E(S)$ and $\\textbf{0}$ to denote the set of idempotents and the zero element of a semigroup $S$, respectively.\n\nAdditionally, given that $p\\geq 1$, denote \\begin{equation}\\label{knn} {RQ}_{p}(n)= I(n, \\,p)/ I(n, \\, p-1) \\end{equation}\n \\noindent to be the Rees quotient semigroup of $I(n, \\,p)$, where the elements in ${RQ}_{p}(n)$ can be regarded as the elements of $\\mathcal{DORP}_{n}$ that possess a height of precisely $p$. If the product of two elements from ${RQ}_{p}(n)$ has a height that does not reach $p$, then the result is $\\textbf{0}$; otherwise, it is an element of ${RQ}_{p}(n) \\setminus \\{\\textbf{0}\\}$.\n\n\\[F(n,r,p)=|\\{\\rho\\in {DRP}_n : \\, b(\\rho)=r \\text{ and } h( \\rho)=p\\}|\\]\n\\noindent and\n\\[F(n,p)=|\\{\\rho\\in {DRP}_n : \\, h(\\rho)=p\\}|.\\]\n\nLet $\\rho\\in{DRP}_n$ be as expressed in \\eqref{2}, where $b(\\rho)=r$ and $h(\\rho)=p$. Moreover, let $w=\\min(\\dom \\, \\rho)$. Note that $p\\leq w\\leq n - p + 1$, from the proof of Lemma \\ref{rev}, and for all $x_i \\in \\dom \\, \\rho$ and $a_j \\in \\im \\, \\rho$, we have $ a_j\\leq x_i$. To count the number of $\\rho\\in{DRP}_n$, we first choose the domain elements. Now, since $w\\in \\dom \\, \\rho$, we can choose the remaining $r-1$ elements from $[n]\\setminus\\{1,\\ldots,w\\}$, i.e., in ${n - w \\choose r - 1}$ ways. Next, we partition the $r$ elements in the domain into $p$ convex (modulo $\\dom \\, \\rho$) blocks in ${r - 1 \\choose p - 1}$ ways. Then, we can choose the $p$ images from the set $\\{1,\\ldots,w\\}$ in ${w \\choose p}$ ways. Finally, taking the sum of the product: ${n-w\\choose r-1} {r-1\\choose p-1} {w\\choose p}$, from $w=p$ to $w=n-p+1$ produces\n\\begin{align*} F(n,r,p)&=\\sum_{w=p}^{n-p+1} {n-w\\choose r-1} {r-1\\choose p-1} {w\\choose p}\\\\&=\n{r-1\\choose p-1}\\sum_{w=p}^{n-p+1} {n-w\\choose r-1} {w\\choose p}\n\\\\&=\n{r-1 \\choose p-1} {n+1 \\choose r+p}, \\quad (\\text{by Lemma \\ref{id}})\n\\end{align*}\n\\noindent as postulated.\n\nThe number of elements in ${DRP}_n$ of a fixed height $1\\le p\\le \\lceil\\frac{n}{2}\\rceil$ is\n\\begin{equation}\\label{coo} F(n,p)= \\sum_{r=p}^{n-p+1}{r-1 \\choose p-1} {n+1 \\choose r+p}.\\end{equation}\n\\end{corollary}\n\\begin{proof}\n We get the number by varying the width within the possible range which is $p\\leq r\\leq n-p+1$.\n\\end{proof}\n\nWe will now state the following lemma.\n\\begin{lemma} Let $F(n,p)$ be as in \\eqref{coo} and let $a_{n} = \\sum\\limits_{p=2}^{\\lceil\\frac{n}{2} \\rceil} F(n, p)$. Then\n\\[\na_{n} = \\sum\\limits_{p=2}^{\\lceil\\frac{n}{2} \\rceil} \\sum_{r=p}^{n-p+1} {r-1 \\choose p-1} {n+1 \\choose r+p}.\n\\]\n\\end{lemma}\n\n\\noindent\\textbf{(ii.)} The result follows from Remark \\ref{remm}.\n\\end{proof}\nNext, we have the following theorem.\n\\begin{theorem} Let $\\mathcal{DORP}_{n}$ be as defined in \\eqref{qn1}. Then the number of convex vital elements of height $2 \\leq p\\leq \\lceil \\frac{n}{2} \\rceil$ in $\\mathcal{DORP}_{n}$ is \\[(n+2)\\left(\\left\\lceil \\frac{n}{2} \\right\\rceil-1\\right)-\\left(\\left\\lceil \\frac{n}{2} \\right\\rceil\\left(\\left\\lceil \\frac{n}{2} \\right\\rceil+1\\right)-2\\right).\\]\n\\end{theorem}\n\\begin{proof} By summing the convex vital elements in Lemma \\ref{nid}(i) over $2 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil,$ together with some algebraic manipulations, we obtain \\[\\sum_{p=2}^{\\left\\lceil \\frac{n}{2} \\right\\rceil} (n - 2p + 2) = (n + 2)\\left(\\left\\lceil \\frac{n}{2} \\right\\rceil - 1\\right) - \\left(\\left\\lceil \\frac{n}{2} \\right\\rceil\\left(\\left\\lceil \\frac{n}{2} \\right\\rceil + 1\\right) - 2\\right).\\]\n\\end{proof}\n\n\\begin{lemma}\\label{mminknp} For $0\\leq p\\le n-1$,\n$W(p)$ is the minimum generating set of $I(n, \\,p)$.\n\\end{lemma}\n\\begin{proof} The result follows from the fact that $G(p)$ is the minimum generating set of $\\langle J^{*}_{p} \\rangle$ by Lemma \\ref{minnn}, and also from the fact that each minimum generating set of the ideal $I(n, \\,p)$ must contain all the extreme convex vital elements below height $p$, as stated in Remark \\ref{extr}.\n\\end{proof}\n\\begin{lemma}\\label{ooknp} Let $W(p)$ be as defined in \\eqref{w}. Then \\[|W(p)|=\\left\\{\n \\begin{array}{lll}\n 1, & \\hbox{if $p=0$;} \\\\\n 2^{n}-1, & \\hbox{if $p=1$;} \\\\\n (n-2)+\\sum\\limits_{r=p}^{n}{\\binom{n}{r}}{\\binom{r-1}{p-1}}, & \\hbox{if $p \\in \\{2, \\ldots, n-1\\}$.}\n \\end{array}\n \\right.\\]\n\\end{lemma}\n\\begin{proof} For the cases $p=0$ and $p=1$, the result is trivial. Next, if $2 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$, then $|W(p)|=|G(p)|+|\\{\\delta_{p, \\, i}, \\, \\, \\delta_{p, \\, n-i+1}: \\, 2\\leq i\\le p-1\\}|$. Thus, by Lemma \\ref{nid2}, we see that $$|W(p)|=(n-2p+2)+\\sum\\limits_{r=p}^{n}{\\binom{n}{r}}{\\binom{r-1}{p-1}}+2(p-2)=(n-2)+\\sum\\limits_{r=p}^{n}{\\binom{n}{r}}{\\binom{r-1}{p-1}}.$$\n\n\\begin{equation}\\label{2}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix},\n\\end{equation}\n\n\\begin{lemma}[\\cite{44}, (3b), p.8]\\label{id}\nFor all n \\( m, n, k\\in \\mathbb{N} \\), it follows that\n\\[\n\\sum_{j=k}^{m} \\binom{j}{k} \\binom{m+n-j}{n} = \\binom{m+n+1}{n+k+1}.\n\\]\n\\end{lemma}\n\n\\begin{lemma}\\label{rev}\nAn element $\\rho \\in \\mathcal{LS}_{n}$, as expressed in \\eqref{1}, is reversible; that is, $\\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix} \\in \\mathcal{DORP}_{n}$ if and only if $a_{p} \\leq \\min A_{1}$ and $1 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$.\n\\end{lemma}",
    "post_theorem_intro_text_len": 2923,
    "post_theorem_intro_text": "\\begin{proof}\n\n Let $\\rho\\in{DRP}_n$ be as expressed in \\eqref{2}, where $b(\\rho)=r$ and $h(\\rho)=p$. Moreover, let $w=\\min(\\mathop{\\rm Dom}\\nolimits \\, \\rho)$. Note that $p\\leq w\\leq n - p + 1$, from the proof of Lemma \\ref{rev},   and for all $x_i \\in \\mathop{\\rm Dom}\\nolimits \\, \\rho$ and $a_j \\in \\mathop{\\rm Im}\\nolimits \\, \\rho$, we have $ a_j\\leq x_i$. To count the number of $\\rho\\in{DRP}_n$, we first choose the domain elements. Now, since $w\\in \\mathop{\\rm Dom}\\nolimits \\, \\rho$, we can choose the remaining $r-1$ elements from $[n]\\setminus\\{1,\\ldots,w\\}$, i.e., in ${n - w \\choose r - 1}$ ways. Next, we partition the $r$ elements in the domain into $p$  convex (modulo $\\mathop{\\rm Dom}\\nolimits \\, \\rho$) blocks in ${r - 1 \\choose p - 1}$ ways. Then, we can choose the $p$ images from the set $\\{1,\\ldots,w\\}$  in ${w \\choose p}$ ways. Finally, taking the sum of the product: ${n-w\\choose r-1} {r-1\\choose p-1} {w\\choose p}$, from $w=p$ to $w=n-p+1$ produces\n\\begin{align*}  F(n,r,p)&=\\sum_{w=p}^{n-p+1} {n-w\\choose r-1} {r-1\\choose p-1} {w\\choose p}\\\\&=\n{r-1\\choose p-1}\\sum_{w=p}^{n-p+1} {n-w\\choose r-1}  {w\\choose p}\n\\\\&=\n{r-1 \\choose p-1} {n+1 \\choose r+p}, \\quad (\\text{by Lemma \\ref{id}})\n\\end{align*}\n\\noindent as postulated.\n\n\\end{proof}\n\\bigskip\n\n\\begin{corollary}\\label{co}\n\nThe number of elements in ${DRP}_n$ of a fixed height $1\\le p\\le \\lceil\\frac{n}{2}\\rceil$ is\n\\begin{equation}\\label{coo} F(n,p)= \\sum_{r=p}^{n-p+1}{r-1 \\choose p-1} {n+1 \\choose r+p}.\\end{equation}\n\\end{corollary}\n\\begin{proof}\n We get the number by varying the width within the possible range which is $p\\leq r\\leq n-p+1$.\n\\end{proof}\n\nWe will now state the following lemma.\n\\begin{lemma} Let $F(n,p)$ be as in  \\eqref{coo} and let $a_{n} = \\sum\\limits_{p=2}^{\\lceil\\frac{n}{2} \\rceil} F(n, p)$. Then\n\\[\na_{n} =  \\sum\\limits_{p=2}^{\\lceil\\frac{n}{2} \\rceil}  \\sum_{r=p}^{n-p+1}  {r-1 \\choose p-1} {n+1 \\choose r+p}.\n\\]\n\\end{lemma}\n\nAt this point, we present the following result.\n\n\\begin{theorem} Let $s_{n}$ be as defined in \\eqref{nums}. Then $|\\mathcal{DORP}_{n}|=s_{n}+a_{n}$.\n\\end{theorem}\n\\begin{proof} Notice that, as in \\eqref{qn2}, $\\mathcal{DORP}_{n} = \\mathcal{LS}_{n} \\cup DRP_{n}$. However, elements of exactly height $1$ are both isotone and antitone maps, so  $$|\\mathcal{LS}_{n} \\cap DRP_{n}|=F(n,1).$$ \\noindent Thus,  $$|\\mathcal{DORP}_{n}| =|\\mathcal{LS}_{n} \\cup DRP_{n}|= |\\mathcal{LS}_{n}|+| DRP_{n}|-| \\mathcal{LS}_{n} \\cap DRP_{n}|,$$ \\noindent and therefore\n\\[|\\mathcal{DORP}_{n}| =s_{n}+\\sum\\limits_{p=1}^{\\lceil\\frac{n}{2} \\rceil} {F(n, p)}-F(n,1)=s_{n}+ F(n,1)+\\sum\\limits_{p=2}^{\\lceil\\frac{n}{2} \\rceil} {F(n, p)}-F(n,1)=s_{n}+a_{n},\\] as required.\n\\end{proof}\n\n\\begin{remark}\n It is  significant to highlight that the triangle of numbers $F(n,\\, p)$ and the sequences $a_{n}$ and $s_{n}+a_{n}$ are not yet recorded in The Online Encyclopedia of Integer Sequences \\cite{slone}.\n\\end{remark}",
    "sketch": "Let $\\rho\\in{DRP}_n$ be as expressed in \\eqref{2}, with $b(\\rho)=r$, $h(\\rho)=p$, and let $w=\\min(\\mathop{\\rm Dom}\\nolimits \\, \\rho)$. From the proof of Lemma \\ref{rev}, $p\\leq w\\leq n-p+1$, and for all $x_i\\in\\mathop{\\rm Dom}\\nolimits\\,\\rho$ and $a_j\\in\\mathop{\\rm Im}\\nolimits\\,\\rho$ we have $a_j\\le x_i$. To count such $\\rho$, first choose the domain: since $w\\in\\mathop{\\rm Dom}\\nolimits\\,\\rho$, choose the remaining $r-1$ domain elements from $[n]\\setminus\\{1,\\ldots,w\\}$ in ${n-w\\choose r-1}$ ways. Next, partition the $r$ domain elements into $p$ convex (modulo $\\mathop{\\rm Dom}\\nolimits\\,\\rho$) blocks in ${r-1\\choose p-1}$ ways. Then choose the $p$ images from $\\{1,\\ldots,w\\}$ in ${w\\choose p}$ ways. Summing over $w$ gives\n\\[\nF(n,r,p)=\\sum_{w=p}^{n-p+1}{n-w\\choose r-1}{r-1\\choose p-1}{w\\choose p}={r-1\\choose p-1}\\sum_{w=p}^{n-p+1}{n-w\\choose r-1}{w\\choose p}={r-1\\choose p-1}{n+1\\choose r+p}\\quad(\\text{by Lemma \\ref{id}}),\n\\]\nas claimed.",
    "expanded_sketch": "Let $\\rho\\in{DRP}_n$ be as expressed in\n\\begin{equation}\\label{2}\n\\rho = \\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix},\n\\end{equation}\nwith $b(\\rho)=r$, $h(\\rho)=p$, and let $w=\\min(\\mathop{\\rm Dom}\\nolimits \\, \\rho)$. From the proof of the following lemma,\n\\begin{lemma}\\label{rev}\nAn element $\\rho \\in \\mathcal{LS}_{n}$, as expressed in \\eqref{1}, is reversible; that is, $\\begin{pmatrix} A_1 & \\dots & A_p \\\\ a_p & \\dots & a_1 \\end{pmatrix} \\in \\mathcal{DORP}_{n}$ if and only if $a_{p} \\leq \\min A_{1}$ and $1 \\leq p \\leq \\lceil \\frac{n}{2} \\rceil$.\n\\end{lemma}\nwe have $p\\leq w\\leq n-p+1$, and for all $x_i\\in\\mathop{\\rm Dom}\\nolimits\\,\\rho$ and $a_j\\in\\mathop{\\rm Im}\\nolimits\\,\\rho$ we have $a_j\\le x_i$. To count such $\\rho$, first choose the domain: since $w\\in\\mathop{\\rm Dom}\\nolimits\\,\\rho$, choose the remaining $r-1$ domain elements from $[n]\\setminus\\{1,\\ldots,w\\}$ in ${n-w\\choose r-1}$ ways. Next, partition the $r$ domain elements into $p$ convex (modulo $\\mathop{\\rm Dom}\\nolimits\\,\\rho$) blocks in ${r-1\\choose p-1}$ ways. Then choose the $p$ images from $\\{1,\\ldots,w\\}$ in ${w\\choose p}$ ways. Summing over $w$ gives\n\\[\nF(n,r,p)=\\sum_{w=p}^{n-p+1}{n-w\\choose r-1}{r-1\\choose p-1}{w\\choose p}={r-1\\choose p-1}\\sum_{w=p}^{n-p+1}{n-w\\choose r-1}{w\\choose p}={r-1\\choose p-1}{n+1\\choose r+p}\\quad(\\text{by the following lemma}),\n\\]\n\\begin{lemma}[\\cite{44}, (3b), p.8]\\label{id}\nFor all n \\( m, n, k\\in \\mathbb{N} \\), it follows that\n\\[\n\\sum_{j=k}^{m} \\binom{j}{k} \\binom{m+n-j}{n} = \\binom{m+n+1}{n+k+1}.\n\\]\n\\end{lemma}\nas claimed.",
    "expanded_theorem": "The number of elements in ${DRP}_n$ of a fixed width $r$ and height $p$ is $F(n,r,p)= {r-1 \\choose p-1} {n+1 \\choose r+p} $,",
    "theorem_type": [
      "Classification or Bijection",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $[n]=\\{1,2,\\dots,n\\}$, and let ${DRP}_n$ denote the set of order-reversing partial transformations of $[n]$. For $\\rho\\in {DRP}_n$, define its width by $b(\\rho)=|\\operatorname{dom}\\rho|$ and its height by $h(\\rho)=|\\operatorname{im}\\rho|$. Also define\n\\[\nF(n,r,p)=\\left|\\{\\rho\\in {DRP}_n: b(\\rho)=r \\text{ and } h(\\rho)=p\\}\\right|.\n\\]\nWhich explicit formula gives the number of elements of ${DRP}_n$ with fixed width $r$ and height $p$?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\nF(n,r,p)=\\binom{r-1}{p-1}\\binom{n+1}{r+p}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\nF(n,r,p)=\\binom{r-1}{p-1}\\binom{n}{r+p-1}.\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\nF(n,r,p)=\\binom{r-1}{p-1}\\sum_{w=p}^{\\,n-p+1}\\binom{n-w}{r-1}\\binom{w}{p}.\n\\]"
        },
        {
          "label": "D",
          "text": "\\[\nF(n,r,p)=\\binom{r}{p}\\binom{n+1}{r+p}.\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\nF(n,r,p)=\\binom{r-1}{p-1}\\binom{n+1}{r+p-1}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "Vandermonde-index shift after summing over w",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "closed-form binomial evaluation by Lemma \\ref{id}",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "number of convex block partitions of the domain",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "final upper binomial parameter r+p from convolution identity",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the objects and asks for the counting formula, but it does not reveal or strongly hint at the specific closed form. There is no explicit answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially a direct request for the known formula for F(n,r,p). It tests recall/recognition of the theorem rather than applying it in a new context."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact closed form from nearby off-by-one variants and from the weaker true summation formula, but the task is still mainly formula recognition rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and well targeted: several encode common indexing/convolution mistakes, and one option is a weaker true expression that increases subtlety."
      },
      "total_score": 5,
      "overall_assessment": "A solid multiple-choice item with no answer leakage and strong distractors, but it is largely theorem-recall and therefore weak on tautology avoidance and deep generative reasoning."
    }
  },
  {
    "id": "2512.21444v1",
    "paper_link": "http://arxiv.org/abs/2512.21444v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\]",
      "start_pos": 6586,
      "end_pos": 6763,
      "label": "thm.Pen"
    },
    "ref_dict": {
      "fig.PenTiles": "\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}",
      "fig.PenPatch": "\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}",
      "eqn.PenVertTrivEst": "\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3356,
    "pre_theorem_intro_text": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings  \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.",
    "context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}",
    "full_context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\n\\begin{figure}[h]\n \\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nThe work of de Bruijn, et al., mentioned at the beginning of this section (see references listed above), implies the following result.\n\\begin{theorem}\\label{thm.PenVerts}\nEvery non-singular point set $\\Lambda_\\omega$, constructed as above, is the set of vertices of a unit length rhombic Penrose tiling. Furthermore, the vertex set of any unit length rhombic Penrose tiling can be obtained, after rotation, as the limit in an appropriate topology (e.g. the Chabauty-Fell topology) of a sequence of non-singular point sets $\\Lambda_\\omega$.\n\\end{theorem}\nAn example of a RPT obtained using the cut and project method described above is shown in Figure \\ref{fig.PenPatch}. Theorem \\ref{thm.PenVerts} will allow us to derive Theorem \\ref{thm.Pen} from the following result.\n\\begin{theorem}\\label{thm.PenCPVersion}\nFor the cut and project sets $\\Lambda_{m,\\omega}$ defined above, there is a constant $C>0$ with the property that, for any $\\omega\\in\\C$, any integer $1\\le m\\le 4$, and any $R\\ge 2$,\n\\[\\left|\\#\\left(\\Lambda_{m,\\omega}\\cap B_R\\right)-\\pi R^2\\mr{dens}(\\cL)\\left|W_{m,\\omega}\\right|\\right|\\le CR^{2/3}(\\log R)^{2/3},\\]\nwhere $|W_{m,\\omega}|$ denotes the area of $W_{m,\\omega}$, and $\\mr{dens}(\\cL)$ is the reciprocal of the volume of a fundamental domain for $\\R^4/\\cL$.\n\\end{theorem}\nAn important point is that the error term in this theorem is uniform in $\\omega$, which allows us to take limits of non-singular point sets, to account for all possible vertex sets of RPTs, in order to establish Theorem \\ref{thm.Pen}. There are a few other small details that have to be ironed out, but they are not difficult to justify, so we postpone their discussion until the end of Section \\ref{sec.PenProof}. The majority of the remainder of the paper will be focused on proving Theorem \\ref{thm.PenCPVersion}.\n\nIf $W\\subseteq\\R^2$ then we write $\\chi_W$ for the indicator function of $W$. The similarity in notation to $\\chi_R$, as defined above, will not cause confusion in what follows. Observe that\n\\[\\widehat{\\chi}_W(y)=\\int_We(-x\\cdot y)\\mr{d}x=\\frac{-1}{(2\\pi|y|)^2}\\int_W\\nabla_x^2\\left( e(-x\\cdot y)\\right)\\mr{d}x,\\]\nwhere $\\nabla_x^2=\\nabla_x\\cdot\\nabla_x$ is the Laplacian. By the Divergence Theorem, under reasonable hypotheses on $W$, this allows us to express the Fourier transform as a line integral over the boundary of $W$. Applying this argument in the case when $W$ is a polygon leads to a particularly nice formula for the Fourier transform, which we reproduce here in the form given in \\cite{BranColzTrav1997}.\n\\begin{proposition}\\label{prop.FourTranPoly}\n Let $P\\subseteq\\R^2$ be a polygon with counterclockwise oriented vertices $\\{a_j\\}_{j=1}^m$. Denote by $\\sigma_j$ the unit vector parallel to the side $[a_j,a_{j+1}]$ and by $v_j$ the outward unit normal to this side. Then, with $a_{m+1}:=a_1$, we have\n \\[\\widehat{\\chi}_P(y)=\\frac{-1}{(2\\pi|y|)^2}\\sum_{j=1}^m\\left(\\frac{e(-y\\cdot a_{j+1})-e(-y\\cdot a_{j})}{y\\cdot \\sigma_j}\\right)y\\cdot v_j.\\]\n\\end{proposition}\nThis proposition is proved as \\cite[Lemma 2.2]{BranColzTrav1997}. The quantity in parenthesis in the sum is interpreted to equal\n\\[-\\left(2\\pi i |a_{j+1}-a_j|\\right)e\\left(\\frac{-y\\cdot(a_j+a_{j+1})}{2}\\right),\\]\nwhen $y\\cdot\\sigma_j=0$. Proposition \\ref{prop.FourTranPoly} implies that\n\\begin{equation}\\label{eqn.PolyFourEst}\n \\widehat{\\chi}_P(y)\\ll_P \\frac{1}{|y|}\\sum_{j=1}^m\\min\\left\\{1,\\frac{1}{|y\\cdot\\sigma_j|}\\right\\}.\n\\end{equation}\nOne thing to note is that these Fourier coefficients decay in most directions like $|y|^{-2}$. However, in directions perpendicular to the sides of $P$, they decay only like $|y|^{-1}$. This turns out to be a source of some difficulty, which is given special attention in the proofs below.\n\n\\noindent\\emph{Bound for $E_2$:} To estimate $E_2$, first write\n\\[\\mc{S}_{m,n}=\\left\\{\\theta\\in L^\\circledast : \\frac{2^{m-1}}{R}<|\\theta|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\theta^\\star|<\\frac{(n+1)R}{M2^m}\\right\\},\\]\nwith $M$ chosen as in \\eqref{eqn.MChoice}. Then we have that\n\\begin{align}\n E_2&\\ll R^{1/2}\\sum_{m=1}^{\\lceil \\log_2R\\rceil}\\sum_{n=1}^\\infty\\sum_{\\theta\\in\\mc{S}_{m,n}}\\frac{1}{|\\theta|^{3/2}}|\\widehat{\\chi}_W(\\theta^\\star)|\\left(1+\\delta^2|\\theta^\\star|^2\\right)^{-N}\\nonumber\\\\\n &\\ll R^2 \\sum_{m=1}^{\\lceil\\log_2R\\rceil}2^{-3m/2}\\sum_{n=1}^\\infty\\left(1+\\frac{\\delta^2n^2R^2}{2^{2m}}\\right)^{-N}\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|.\\label{eqn.E2Bound1}\n\\end{align}\nHere we will use the following lemma.\n\\begin{lemma}\\label{lem.ChiWEst2}\n With notation as above, for $m,n\\in\\N$ with $m\\le\\lceil \\log_2R\\rceil$, we have that\n \\[\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|\\ll\\frac{2^{2m}\\log n}{nR^2}.\\]\n\\end{lemma}\n\\begin{proof}\n Let us write\n \\[\\tilde{\\mc{S}}=\\left\\{\\lambda\\in \\frac{1}{25}L : \\frac{2^{m-1}}{R}<|\\lambda|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\lambda^\\star|<\\frac{(n+1)R}{M2^m},~|\\mr{arg}(\\lambda^\\star)|\\le \\frac{\\pi}{5}\\right\\}.\\]\n By the argument used in the proof of Lemma \\ref{lem.ChiWEst1} leading up to equation \\eqref{eqn.ChiWEst1}, it is sufficient for this lemma to prove that\n \\begin{equation*}\n \\sum_{\\lambda\\in\\tilde{\\mc{S}}}\\frac{1}{|\\lambda^\\star|^2|\\sin(\\mathrm{arg}(\\lambda^\\star))|}\\ll\\frac{2^{2m}\\log n}{nR^2}.\n \\end{equation*}\n Label the elements of $\\star(\\tilde{\\mc{S}})$ as\n \\[\\lambda_j^\\star=\\rho_je^{it_j},\\]\n for $1\\le j\\le J:=|\\tilde{\\mc{S}}|$, with each $\\rho_j\\in\\R^+$ and with $0< |t_1|\\le |t_2|\\le\\cdots \\le |t_J|\\le\\pi/5$. Also, label the element of $\\tilde{\\mc{S}}$ corresponding to each $\\lambda_j^\\star$ as $\\lambda_j$.\n\n\\noindent\\emph{Proof of Theorem \\ref{thm.Pen}:} Let $V$ be the vertex set of a unit length RPT. If $V$ can be constructed as a cut and project set, as $V=\\Lambda_\\omega$ for some $\\omega\\in\\C$, then the result of Theorem \\ref{thm.Pen} follows immediately from Theorem \\ref{thm.PenCPVersion}. Otherwise, by Theorem \\ref{thm.PenVerts}, there is a sequence of complex numbers $\\{\\omega_j\\}_{j=1}^\\infty$ with the property that, after a possible rotation,\n\\[V=\\lim_{j\\rar\\infty}\\Lambda_{\\omega_j},\\]\nwhere the limit is taken in the Chabauty-Fell topology. For $R\\ge 2$, the number of points in $\\R^2$ which lie on the boundary of $B_R$, and which are the limits of sequences of points $\\lambda_j\\in\\Lambda_{\\omega_j}$, as $j\\rar\\infty,$ must be $\\ll R^{2/3}(\\log R)^{2/3}$. This follows from the uniformity over $\\omega\\in \\C$ of the error term in Theorem \\ref{thm.PenCPVersion}. It then follows, again by uniformity, that\n\\begin{align}\n \\#(V\\cap B_R)&=\\lim_{j\\rar\\infty}\\#\\left(\\Lambda_{\\omega_j}\\cap B_R\\right)+O(R^{2/3}(\\log R)^{2/3})\\nonumber\\\\\n &=\\pi R^2\\mr{dens}(\\cL)\\sum_{m=1}^4|W_{m,\\omega+m}|+O( R^{2/3}(\\log R)^{2/3}).\\label{eqn.ChabLim1}\n\\end{align}\nFinally, we compute that\n\\[\\mr{dens}(\\cL)=\\mr{disc}(L)^{-1/2}=\\frac{4}{25\\sqrt{5}}.\\]\nSubstituting this in \\eqref{eqn.ChabLim1}, together with the areas of the pentagonal windows, completes the proof of Theorem \\ref{thm.Pen}.",
    "post_theorem_intro_text_len": 4426,
    "post_theorem_intro_text": "Readers who are familiar with the analogous counting problem for integer lattice points in the plane, called the Gauss circle problem, may at this point recognize some similarities. Let us write\n\\[\\#(\\mathbb{Z}^2\\cap B_R)=\\pi R^2 + E(R).\\]\nGauss himself observed that $|E(R)|\\le 2\\sqrt{2}\\pi R$, and he conjectured that it should be possible to improve the exponent on $R$  to obtain an error term of $\\ll R^{1/2}$. The next substantial result was published in 1906 by Sierpi\\'nski \\cite{Sier1906}, who showed that $E(R)\\ll R^{2/3}$ (see also \\cite{Schi1972}). Subsequent improvements have been made by many authors (the introduction of \\cite{Huxl2003} contains a list), and currently the best known upper bound, due to Martin Huxley \\cite{Huxl2003}, is $E(R)\\ll R^{131/208}(\\log R)^\\tau$, with $\\tau=18627/8320$.\n\nThe important link which allows us to employ techniques used in the Gauss circle problem in order to study vertices of Penrose tilings is a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace. We describe this connection in detail in Section \\ref{sec.ModelSets}. With this as a general framework, we are able in Sections \\ref{sec.Fourier}-\\ref{sec.PenProof} to bring in tools from Fourier analysis and to approach our point counting problem through the lens of the Poisson summation formula (PSF). Results of a similar flavor, estimating discrepancies of points in growing patches of cut and projects sets, have recently been explored by Koivusalo and Lagac{\\'e} in \\cite{KoivLagacBjorHart2025}, and by R\\\"uhr, Smilansky, and Weiss in \\cite{RuhrSmilWeis2024}.\n\nWe remark that our proof, although structurally similar, is not a na\\\"ive application of known methods (e.g. Huxley's Fourier analytic approach to lattice point counting in regions with smooth boundary, using Van der Corput's method). Here, there are two new difficulties which are not encountered in the classical versions of the Gauss circle problem. The first is that, for the cut and project sets that we consider, the problem amounts to counting points in a lattice in a $4$-dimensional cylinder which is only growing radially. Since the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction. Secondly, the lattices from which we derive RPTs are irrational. This has the effect that, depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries. This is a real phenomenon, which can potentially cause a blow up in the Fourier coefficients after using the PSF. It is one of the main issues that we must confront using the specific geometric and algebraic data which define RPTs.\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nFinally, there is the question of how these results could be generalized. Several parts of the proofs below can be adapted quite well to deal with much wider classes of cut and project sets. However, there are some crucial details in the proof, in particular those contained in Section \\ref{sec.PenProof}, which form a narrow gate that any more general result would need to pass through. The cut and project sets used to define RPTs are extremely special, both in the algebraic properties of the lattice involved and in the exact shape, symmetry, and orientation of the windows involved, with respect to the specific underlying lattice. Any attempt to modify the lattice, or change the windows (even by a small rotation) will destroy important features of the argument and make it difficult or even impossible to control the sizes of the Fourier coefficients involved.\n\nOn the other hand, it seems likely that the arguments given here should apply equally well, with the appropriate modifications, to vertex sets of Ammann-Beenker tilings (see \\cite[Section 10.4]{GrunShep1987} and \\cite{Been1982}). We leave it to the interested reader to work out the details.",
    "sketch": "To prove Theorem~\\ref{thm.Pen}, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed in Section~\\ref{sec.ModelSets}). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (Sections~\\ref{sec.Fourier}--\\ref{sec.PenProof}).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in Section~\\ref{sec.PenProof}, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.",
    "expanded_sketch": "To prove the main theorem, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed later). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (proved later).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in the later argument, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.",
    "expanded_theorem": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $V$ be the vertex set of any unit length rhombic Penrose tiling, meaning a tiling of the plane by copies of the two Penrose rhombi of side length $1$ with angles $72^\\circ$ and $36^\\circ$, placed edge-to-edge and satisfying the usual matching rules. Let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin, and define\n\\[\nC_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}, \\qquad \\phi=\\frac{1+\\sqrt{5}}{2}.\n\\]\nWhich quantitative estimate holds for the number of vertices of the tiling in $B_R$ when $R\\ge 2$?",
      "correct_choice": {
        "label": "A",
        "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{2/3}(\\log R)^{2/3}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{1/2}(\\log R)^{2/3}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + o(R).\n\\]"
        },
        {
          "label": "D",
          "text": "For every such $V$ there exists a constant $K_V>0$ such that for all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(K_V R^{2/3}(\\log R)^{2/3}\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{2/3}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "error_exponent_from_anisotropic_smoothing",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "explicit_sharp_error_rate",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity_over_all_RPT_vertex_sets",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "log_factor_needed_to_control_fourier_coefficients",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct error term or identify choice A. It only defines the setting and the main-density constant, so the answer is not leaked."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: it asks which asymptotic statement holds, and the correct option is the theorem statement itself with its precise error term."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ by subtle issues of strength, quantifiers, and logarithmic losses. However, the item mainly tests recognition of the exact known result rather than generating a conclusion from supplied mathematical structure."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: one is too strong, one is weaker-but-true, one alters uniformity/quantifiers, and one removes the log factor. These align well with common failure modes in asymptotic reasoning."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is mostly a direct restatement of a known result rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.21444v1",
    "paper_link": "http://arxiv.org/abs/2512.21444v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\]",
      "start_pos": 6586,
      "end_pos": 6763,
      "label": "thm.Pen"
    },
    "ref_dict": {
      "fig.PenTiles": "\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}",
      "fig.PenPatch": "\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}",
      "eqn.PenVertTrivEst": "\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3356,
    "pre_theorem_intro_text": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings  \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.",
    "context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}",
    "full_context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\n\\begin{figure}[h]\n \\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nThe work of de Bruijn, et al., mentioned at the beginning of this section (see references listed above), implies the following result.\n\\begin{theorem}\\label{thm.PenVerts}\nEvery non-singular point set $\\Lambda_\\omega$, constructed as above, is the set of vertices of a unit length rhombic Penrose tiling. Furthermore, the vertex set of any unit length rhombic Penrose tiling can be obtained, after rotation, as the limit in an appropriate topology (e.g. the Chabauty-Fell topology) of a sequence of non-singular point sets $\\Lambda_\\omega$.\n\\end{theorem}\nAn example of a RPT obtained using the cut and project method described above is shown in Figure \\ref{fig.PenPatch}. Theorem \\ref{thm.PenVerts} will allow us to derive Theorem \\ref{thm.Pen} from the following result.\n\\begin{theorem}\\label{thm.PenCPVersion}\nFor the cut and project sets $\\Lambda_{m,\\omega}$ defined above, there is a constant $C>0$ with the property that, for any $\\omega\\in\\C$, any integer $1\\le m\\le 4$, and any $R\\ge 2$,\n\\[\\left|\\#\\left(\\Lambda_{m,\\omega}\\cap B_R\\right)-\\pi R^2\\mr{dens}(\\cL)\\left|W_{m,\\omega}\\right|\\right|\\le CR^{2/3}(\\log R)^{2/3},\\]\nwhere $|W_{m,\\omega}|$ denotes the area of $W_{m,\\omega}$, and $\\mr{dens}(\\cL)$ is the reciprocal of the volume of a fundamental domain for $\\R^4/\\cL$.\n\\end{theorem}\nAn important point is that the error term in this theorem is uniform in $\\omega$, which allows us to take limits of non-singular point sets, to account for all possible vertex sets of RPTs, in order to establish Theorem \\ref{thm.Pen}. There are a few other small details that have to be ironed out, but they are not difficult to justify, so we postpone their discussion until the end of Section \\ref{sec.PenProof}. The majority of the remainder of the paper will be focused on proving Theorem \\ref{thm.PenCPVersion}.\n\nIf $W\\subseteq\\R^2$ then we write $\\chi_W$ for the indicator function of $W$. The similarity in notation to $\\chi_R$, as defined above, will not cause confusion in what follows. Observe that\n\\[\\widehat{\\chi}_W(y)=\\int_We(-x\\cdot y)\\mr{d}x=\\frac{-1}{(2\\pi|y|)^2}\\int_W\\nabla_x^2\\left( e(-x\\cdot y)\\right)\\mr{d}x,\\]\nwhere $\\nabla_x^2=\\nabla_x\\cdot\\nabla_x$ is the Laplacian. By the Divergence Theorem, under reasonable hypotheses on $W$, this allows us to express the Fourier transform as a line integral over the boundary of $W$. Applying this argument in the case when $W$ is a polygon leads to a particularly nice formula for the Fourier transform, which we reproduce here in the form given in \\cite{BranColzTrav1997}.\n\\begin{proposition}\\label{prop.FourTranPoly}\n Let $P\\subseteq\\R^2$ be a polygon with counterclockwise oriented vertices $\\{a_j\\}_{j=1}^m$. Denote by $\\sigma_j$ the unit vector parallel to the side $[a_j,a_{j+1}]$ and by $v_j$ the outward unit normal to this side. Then, with $a_{m+1}:=a_1$, we have\n \\[\\widehat{\\chi}_P(y)=\\frac{-1}{(2\\pi|y|)^2}\\sum_{j=1}^m\\left(\\frac{e(-y\\cdot a_{j+1})-e(-y\\cdot a_{j})}{y\\cdot \\sigma_j}\\right)y\\cdot v_j.\\]\n\\end{proposition}\nThis proposition is proved as \\cite[Lemma 2.2]{BranColzTrav1997}. The quantity in parenthesis in the sum is interpreted to equal\n\\[-\\left(2\\pi i |a_{j+1}-a_j|\\right)e\\left(\\frac{-y\\cdot(a_j+a_{j+1})}{2}\\right),\\]\nwhen $y\\cdot\\sigma_j=0$. Proposition \\ref{prop.FourTranPoly} implies that\n\\begin{equation}\\label{eqn.PolyFourEst}\n \\widehat{\\chi}_P(y)\\ll_P \\frac{1}{|y|}\\sum_{j=1}^m\\min\\left\\{1,\\frac{1}{|y\\cdot\\sigma_j|}\\right\\}.\n\\end{equation}\nOne thing to note is that these Fourier coefficients decay in most directions like $|y|^{-2}$. However, in directions perpendicular to the sides of $P$, they decay only like $|y|^{-1}$. This turns out to be a source of some difficulty, which is given special attention in the proofs below.\n\n\\noindent\\emph{Bound for $E_2$:} To estimate $E_2$, first write\n\\[\\mc{S}_{m,n}=\\left\\{\\theta\\in L^\\circledast : \\frac{2^{m-1}}{R}<|\\theta|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\theta^\\star|<\\frac{(n+1)R}{M2^m}\\right\\},\\]\nwith $M$ chosen as in \\eqref{eqn.MChoice}. Then we have that\n\\begin{align}\n E_2&\\ll R^{1/2}\\sum_{m=1}^{\\lceil \\log_2R\\rceil}\\sum_{n=1}^\\infty\\sum_{\\theta\\in\\mc{S}_{m,n}}\\frac{1}{|\\theta|^{3/2}}|\\widehat{\\chi}_W(\\theta^\\star)|\\left(1+\\delta^2|\\theta^\\star|^2\\right)^{-N}\\nonumber\\\\\n &\\ll R^2 \\sum_{m=1}^{\\lceil\\log_2R\\rceil}2^{-3m/2}\\sum_{n=1}^\\infty\\left(1+\\frac{\\delta^2n^2R^2}{2^{2m}}\\right)^{-N}\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|.\\label{eqn.E2Bound1}\n\\end{align}\nHere we will use the following lemma.\n\\begin{lemma}\\label{lem.ChiWEst2}\n With notation as above, for $m,n\\in\\N$ with $m\\le\\lceil \\log_2R\\rceil$, we have that\n \\[\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|\\ll\\frac{2^{2m}\\log n}{nR^2}.\\]\n\\end{lemma}\n\\begin{proof}\n Let us write\n \\[\\tilde{\\mc{S}}=\\left\\{\\lambda\\in \\frac{1}{25}L : \\frac{2^{m-1}}{R}<|\\lambda|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\lambda^\\star|<\\frac{(n+1)R}{M2^m},~|\\mr{arg}(\\lambda^\\star)|\\le \\frac{\\pi}{5}\\right\\}.\\]\n By the argument used in the proof of Lemma \\ref{lem.ChiWEst1} leading up to equation \\eqref{eqn.ChiWEst1}, it is sufficient for this lemma to prove that\n \\begin{equation*}\n \\sum_{\\lambda\\in\\tilde{\\mc{S}}}\\frac{1}{|\\lambda^\\star|^2|\\sin(\\mathrm{arg}(\\lambda^\\star))|}\\ll\\frac{2^{2m}\\log n}{nR^2}.\n \\end{equation*}\n Label the elements of $\\star(\\tilde{\\mc{S}})$ as\n \\[\\lambda_j^\\star=\\rho_je^{it_j},\\]\n for $1\\le j\\le J:=|\\tilde{\\mc{S}}|$, with each $\\rho_j\\in\\R^+$ and with $0< |t_1|\\le |t_2|\\le\\cdots \\le |t_J|\\le\\pi/5$. Also, label the element of $\\tilde{\\mc{S}}$ corresponding to each $\\lambda_j^\\star$ as $\\lambda_j$.\n\n\\noindent\\emph{Proof of Theorem \\ref{thm.Pen}:} Let $V$ be the vertex set of a unit length RPT. If $V$ can be constructed as a cut and project set, as $V=\\Lambda_\\omega$ for some $\\omega\\in\\C$, then the result of Theorem \\ref{thm.Pen} follows immediately from Theorem \\ref{thm.PenCPVersion}. Otherwise, by Theorem \\ref{thm.PenVerts}, there is a sequence of complex numbers $\\{\\omega_j\\}_{j=1}^\\infty$ with the property that, after a possible rotation,\n\\[V=\\lim_{j\\rar\\infty}\\Lambda_{\\omega_j},\\]\nwhere the limit is taken in the Chabauty-Fell topology. For $R\\ge 2$, the number of points in $\\R^2$ which lie on the boundary of $B_R$, and which are the limits of sequences of points $\\lambda_j\\in\\Lambda_{\\omega_j}$, as $j\\rar\\infty,$ must be $\\ll R^{2/3}(\\log R)^{2/3}$. This follows from the uniformity over $\\omega\\in \\C$ of the error term in Theorem \\ref{thm.PenCPVersion}. It then follows, again by uniformity, that\n\\begin{align}\n \\#(V\\cap B_R)&=\\lim_{j\\rar\\infty}\\#\\left(\\Lambda_{\\omega_j}\\cap B_R\\right)+O(R^{2/3}(\\log R)^{2/3})\\nonumber\\\\\n &=\\pi R^2\\mr{dens}(\\cL)\\sum_{m=1}^4|W_{m,\\omega+m}|+O( R^{2/3}(\\log R)^{2/3}).\\label{eqn.ChabLim1}\n\\end{align}\nFinally, we compute that\n\\[\\mr{dens}(\\cL)=\\mr{disc}(L)^{-1/2}=\\frac{4}{25\\sqrt{5}}.\\]\nSubstituting this in \\eqref{eqn.ChabLim1}, together with the areas of the pentagonal windows, completes the proof of Theorem \\ref{thm.Pen}.",
    "post_theorem_intro_text_len": 4426,
    "post_theorem_intro_text": "Readers who are familiar with the analogous counting problem for integer lattice points in the plane, called the Gauss circle problem, may at this point recognize some similarities. Let us write\n\\[\\#(\\mathbb{Z}^2\\cap B_R)=\\pi R^2 + E(R).\\]\nGauss himself observed that $|E(R)|\\le 2\\sqrt{2}\\pi R$, and he conjectured that it should be possible to improve the exponent on $R$  to obtain an error term of $\\ll R^{1/2}$. The next substantial result was published in 1906 by Sierpi\\'nski \\cite{Sier1906}, who showed that $E(R)\\ll R^{2/3}$ (see also \\cite{Schi1972}). Subsequent improvements have been made by many authors (the introduction of \\cite{Huxl2003} contains a list), and currently the best known upper bound, due to Martin Huxley \\cite{Huxl2003}, is $E(R)\\ll R^{131/208}(\\log R)^\\tau$, with $\\tau=18627/8320$.\n\nThe important link which allows us to employ techniques used in the Gauss circle problem in order to study vertices of Penrose tilings is a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace. We describe this connection in detail in Section \\ref{sec.ModelSets}. With this as a general framework, we are able in Sections \\ref{sec.Fourier}-\\ref{sec.PenProof} to bring in tools from Fourier analysis and to approach our point counting problem through the lens of the Poisson summation formula (PSF). Results of a similar flavor, estimating discrepancies of points in growing patches of cut and projects sets, have recently been explored by Koivusalo and Lagac{\\'e} in \\cite{KoivLagacBjorHart2025}, and by R\\\"uhr, Smilansky, and Weiss in \\cite{RuhrSmilWeis2024}.\n\nWe remark that our proof, although structurally similar, is not a na\\\"ive application of known methods (e.g. Huxley's Fourier analytic approach to lattice point counting in regions with smooth boundary, using Van der Corput's method). Here, there are two new difficulties which are not encountered in the classical versions of the Gauss circle problem. The first is that, for the cut and project sets that we consider, the problem amounts to counting points in a lattice in a $4$-dimensional cylinder which is only growing radially. Since the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction. Secondly, the lattices from which we derive RPTs are irrational. This has the effect that, depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries. This is a real phenomenon, which can potentially cause a blow up in the Fourier coefficients after using the PSF. It is one of the main issues that we must confront using the specific geometric and algebraic data which define RPTs.\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nFinally, there is the question of how these results could be generalized. Several parts of the proofs below can be adapted quite well to deal with much wider classes of cut and project sets. However, there are some crucial details in the proof, in particular those contained in Section \\ref{sec.PenProof}, which form a narrow gate that any more general result would need to pass through. The cut and project sets used to define RPTs are extremely special, both in the algebraic properties of the lattice involved and in the exact shape, symmetry, and orientation of the windows involved, with respect to the specific underlying lattice. Any attempt to modify the lattice, or change the windows (even by a small rotation) will destroy important features of the argument and make it difficult or even impossible to control the sizes of the Fourier coefficients involved.\n\nOn the other hand, it seems likely that the arguments given here should apply equally well, with the appropriate modifications, to vertex sets of Ammann-Beenker tilings (see \\cite[Section 10.4]{GrunShep1987} and \\cite{Been1982}). We leave it to the interested reader to work out the details.",
    "sketch": "To prove Theorem~\\ref{thm.Pen}, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed in Section~\\ref{sec.ModelSets}). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (Sections~\\ref{sec.Fourier}--\\ref{sec.PenProof}).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in Section~\\ref{sec.PenProof}, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.",
    "expanded_sketch": "To prove the main theorem, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed later). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (proved later).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in the later argument, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.",
    "expanded_theorem": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\],",
    "theorem_type": [
      "Asymptotic or Limit",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $V$ be the vertex set of any unit length rhombic Penrose tiling, meaning a tiling of the plane by copies of the two Penrose rhombi of side length $1$ with angles $72^\\circ$ and $36^\\circ$, placed edge-to-edge and satisfying the usual matching rules. Let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin, and define\n\\[\nC_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}, \\qquad \\phi=\\frac{1+\\sqrt{5}}{2}.\n\\]\nWhich quantitative estimate holds for the number of vertices of the tiling in $B_R$ when $R\\ge 2$?",
      "correct_choice": {
        "label": "A",
        "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{2/3}(\\log R)^{2/3}\\right).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{1/2}(\\log R)^{2/3}\\right).\n\\]"
        },
        {
          "label": "C",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + o(R).\n\\]"
        },
        {
          "label": "D",
          "text": "For every such $V$ there exists a constant $K_V>0$ such that for all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(K_V R^{2/3}(\\log R)^{2/3}\\right).\n\\]"
        },
        {
          "label": "E",
          "text": "For every such $V$ and all $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2 + O\\!\\left(R^{2/3}\\right).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "error_exponent_from_anisotropic_smoothing",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "explicit_sharp_error_rate",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity_over_all_RPT_vertex_sets",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "log_factor_needed_to_control_fourier_coefficients",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the setting and the constant C_P but does not reveal the asymptotic error term, the logarithmic factor, or the quantifier structure. There is no explicit or trivial cue pointing to choice A."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is largely a theorem-identification question: one option states the precise target estimate, and the task is to pick the correct formulation among close variants. This is better than a pure restatement, but still only a mild reformulation rather than a genuinely new conclusion derived from the setup."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways: stronger false bounds, a weaker true statement, a quantifier-dependent variant, and omission of a log factor. However, the item primarily tests precise theorem recall/recognition rather than substantial generative mathematical reasoning from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: B is an over-strong error term, C is a weaker true statement, D probes uniformity versus tiling-dependent constants, and E tests whether the log factor is necessary. These are plausible and mathematically aligned with common failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as precise theorem recognition rather than a deeply generative reasoning task."
    }
  },
  {
    "id": "2512.21565v1",
    "paper_link": "http://arxiv.org/abs/2512.21565v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.",
      "start_pos": 7785,
      "end_pos": 7994,
      "label": "main1"
    },
    "ref_dict": {
      "main2": "\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}",
      "main1": "\\begin{thm}\n  \\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 3326,
    "pre_theorem_intro_text": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.",
    "context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.",
    "full_context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nThus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n \\label{main}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Then $\\mathbf E(X)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}\n\nHence, for a subset $Z \\subset W$, in order to show that $\\mathbf E(Z)$ is not finitely generated, it is enough to show that $\\mathbf E(\\iota^{-1}(Z))$ is not finitely generated.\nThus we may assume that $Z$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$.\nNote that if $Z$ is the support of a tropical variety in $\\mathbb R^n$, then so is $\\iota^{-1}(Z)$ by Lemma \\ref{trop var reduction}.\n\nLet $X$ be a tropical variety in $\\mathbb R^n$.\nAssume that $W := |X|$ is a $d$-dimensional rational affine linear subspace of $\\mathbb R^n$.\nBy Lemma \\ref{translation}, we may assume that $\\mathbf 0 \\in W$, i.e., $W$ is a linear subspace of $\\mathbb R^n$.\nSince $W$ is rational, the orthogonal complement $W'$ (with respect to the standard inner product) of $W$ is also rational.\nWe use the notations $\\Lambda(W) = W \\cap \\mathbb Z^n$ and $\\Lambda(W') = W' \\cap \\mathbb Z^n$.\nLet $\\{ \\bm q_1, \\ldots, \\bm q_{n-d} \\}$ be a basis of $\\Lambda(W')$.\nWe see that the congruence $\\mathbf E(W)$ is generated by the finite set $S := \\left\\{ \\left(\\overline{\\bm x^{\\bm q_i}}, 0 \\right) \\ | \\ i=1, \\ldots, n-d \\right\\}$.\n\n\\begin{prop}\n \\label{trop unbounded}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Assume that $|X|$ is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Then $|X|$ is unbounded in all rational directions.\n\\end{prop}\n\n\\begin{proof}[Proof of \\textquotedblleft Proposition \\ref{trop unbounded in f} $\\Longrightarrow$ Proposition \\ref{trop unbounded}\\textquotedblright]\n Assume that Proposition \\ref{trop unbounded in f} holds.\n Let $X$ be a tropical variety in $\\mathbb R^n$ whose support is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Take any $\\bm p \\in \\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$.\n If the function $f: |X| \\to \\mathbb R, \\bm a \\mapsto \\bm a \\cdot \\bm p$ is constant with the value $k$, then $|X|$ is included in the set $\\{ \\bm a \\in \\mathbb R^n \\ | \\ \\bm a \\cdot \\bm p = k \\}$, which is a proper rational affine linear subspace of $\\mathbb R^n$.\n This contradicts to the assumption, hence $f$ is not constant.\n Thus $f$ is upper unbounded.\n Since $\\bm p$ is an arbitrary element of $\\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$, $|X|$ is unbounded in all rational directions.\n\\end{proof}\n\n\\begin{proof}[Proof of the only if part of Theorem \\ref{main}]\n Let $X$ be a tropical variety such that $|X|$ is not a rational affine linear subspace of $\\mathbb R^n$.\n We may assume that $|X|$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$ by Lemma \\ref{fg reduction} and the observation after it.\n Hence, by Proposition \\ref{trop unbounded}, $|X|$ is unbounded in all rational directions.\n In order to use Theorem \\ref{not fg}, we check that $n \\geq 2$.\n If $n =0$, $|X| = \\{ \\mathbf 0 \\}$.\n It contradicts to the assumption that $|X|$ is not an affine linear subspace.\n If $n =1$, $|X|$ is a singleton or the whole space $\\mathbb R$ (by Lemma \\ref{n in n}).\n It also contradicts to the assumption.\n Thus $n \\geq 2$, and hence by Theorem \\ref{not fg}, $\\mathbf E(X)$ is not finitely generated. \\qedhere\n\\end{proof}",
    "post_theorem_intro_text_len": 2612,
    "post_theorem_intro_text": "Thus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}\n\nThe strategy of the proof of Theorem \\ref{main2} is as follows.\nFirst we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set.\nThen we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large.\nThe existence of such pair causes a contradiction.\nSee Section 3 for detail.\n\nNote that we cannot apply Theorem \\ref{main2} for showing the only if part of Theorem \\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.\nIn particular, if $|X|$ is included in a proper rational affine linear subspace of $\\mathbb R^n$, then $|X|$ is not unbounded in all rational directions.\nTo overcome this issue, we show that we may replace the ambient space with a lower dimensional one.\nSee Section 4 for detail.\n\nFinally, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L \\subset \\mathbb R^2$.\nSee Section \\ref{section min gen} for detail.\n\n\\begin{thm}\n  \\label{main3}\n  The set\n  $$\\mathcal S = \\left\\{ \\left( \\overline{0 \\oplus x^u y^v}, f_{\\Delta(\\{ (0,0), (u,v) \\})} \\right) \\ \\middle| \\ \\begin{aligned}\n  &\\text{$u$ and $v$ are integers with $\\gcd(u,v) = 1$,}\\\\\n  &\\text{ $u > 0$ or $(u,v) = (0,1)$}\n  \\end{aligned} \\right\\}$$\n  is a minimal generating set of $\\mathbf E(L)$.\n\\end{thm}\n\nThis paper is organized as follows.\nIn Section 2, we recall some basic definitions and facts about semirings, tropical algebra, tropical varieties, and Newton polytopes.\nAlso we show some lemmas we need later.\nIn Section 3, we consider a way to prove that a given congruence on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ is not finitely generated, and use it to show Theorem \\ref{main2}.\nIn Section 4, we discuss about the reduction of the ambient spaces.\nThe result of this section is necessary for showing Theorem \\ref{main1}.\nIn Section 5, we prove Theorem \\ref{main1}.\nIn Section 6, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L$.\nThis section is almost independent of Section 3-5.\nIn Appendix, we give proofs of some facts about free abelian groups and polytopes, which are used in this paper.",
    "sketch": "For Theorem~\\ref{main1}, the text says: “The only if part of this theorem is a consequence of the following more general result” (Theorem~\\ref{main2}), but “we cannot apply Theorem~\\ref{main2} for showing the only if part of Theorem~\\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.” “To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.”\n\nFor Theorem~\\ref{main2}, the stated proof strategy is: “First we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set. Then we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large. The existence of such pair causes a contradiction.”",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Universal"
    ],
    "mcq": {
      "question": "Let $X$ be a tropical variety in $\\mathbb R^n$, and let $|X|$ denote its support. For a subset $Z\\subset \\mathbb R^n$, define the congruence\n$$\\mathbf E(Z)=\\{(f,g)\\in \\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]\\mid f(\\mathbf a)=g(\\mathbf a)\\text{ for all }\\mathbf a\\in Z\\},$$\nwhere $\\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]$ is the tropical Laurent polynomial semiring. A rational affine linear subspace of $\\mathbb R^n$ means an affine translate of a linear subspace defined over $\\mathbb Q$ (equivalently, whose direction is spanned by rational vectors). Which statement holds for every such tropical variety $X$?",
      "correct_choice": {
        "label": "A",
        "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is an affine linear subspace of $\\mathbb R^n$."
        },
        {
          "label": "C",
          "text": "If $|X|$ is a rational affine linear subspace of $\\mathbb R^n$, then $\\mathbf E(|X|)$ is finitely generated."
        },
        {
          "label": "D",
          "text": "$\\mathbf E(|X|)$ is finitely generated whenever $|X|$ is contained in a rational affine linear subspace of $\\mathbb R^n$."
        },
        {
          "label": "E",
          "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a finite union of rational affine linear subspaces of $\\mathbb R^n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "rationality of affine linear subspace",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "only-if direction",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "equality with a rational affine linear subspace replaced by mere containment",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "single rational affine linear subspace replaced by finite union",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct choice explicitly or by a near-verbatim hint. It asks for an equivalent characterization, and the correct option must be identified from several closely related geometric statements."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially a theorem-recall question about an equivalence, so it is somewhat close to restating a known result. However, it is not fully tautological because the choices introduce subtly different conditions such as rationality, containment, and finite unions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact equivalence from weaker or altered variants, especially between equality, containment, and rationality. Still, the question mainly tests precise recall/recognition of a theorem rather than deeper generative problem solving."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target common failure modes: dropping rationality, weakening equality to containment, replacing a single subspace by a finite union, or reformulating via congruence equality with a larger subspace."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall than genuine generative reasoning."
    }
  },
  {
    "id": "2512.21565v1",
    "paper_link": "http://arxiv.org/abs/2512.21565v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.",
      "start_pos": 7785,
      "end_pos": 7994,
      "label": "main1"
    },
    "ref_dict": {
      "main2": "\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}",
      "main1": "\\begin{thm}\n  \\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}"
    },
    "pre_theorem_intro_text_len": 3326,
    "pre_theorem_intro_text": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.",
    "context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.",
    "full_context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nThus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n \\label{main}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Then $\\mathbf E(X)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}\n\nHence, for a subset $Z \\subset W$, in order to show that $\\mathbf E(Z)$ is not finitely generated, it is enough to show that $\\mathbf E(\\iota^{-1}(Z))$ is not finitely generated.\nThus we may assume that $Z$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$.\nNote that if $Z$ is the support of a tropical variety in $\\mathbb R^n$, then so is $\\iota^{-1}(Z)$ by Lemma \\ref{trop var reduction}.\n\nLet $X$ be a tropical variety in $\\mathbb R^n$.\nAssume that $W := |X|$ is a $d$-dimensional rational affine linear subspace of $\\mathbb R^n$.\nBy Lemma \\ref{translation}, we may assume that $\\mathbf 0 \\in W$, i.e., $W$ is a linear subspace of $\\mathbb R^n$.\nSince $W$ is rational, the orthogonal complement $W'$ (with respect to the standard inner product) of $W$ is also rational.\nWe use the notations $\\Lambda(W) = W \\cap \\mathbb Z^n$ and $\\Lambda(W') = W' \\cap \\mathbb Z^n$.\nLet $\\{ \\bm q_1, \\ldots, \\bm q_{n-d} \\}$ be a basis of $\\Lambda(W')$.\nWe see that the congruence $\\mathbf E(W)$ is generated by the finite set $S := \\left\\{ \\left(\\overline{\\bm x^{\\bm q_i}}, 0 \\right) \\ | \\ i=1, \\ldots, n-d \\right\\}$.\n\n\\begin{prop}\n \\label{trop unbounded}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Assume that $|X|$ is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Then $|X|$ is unbounded in all rational directions.\n\\end{prop}\n\n\\begin{proof}[Proof of \\textquotedblleft Proposition \\ref{trop unbounded in f} $\\Longrightarrow$ Proposition \\ref{trop unbounded}\\textquotedblright]\n Assume that Proposition \\ref{trop unbounded in f} holds.\n Let $X$ be a tropical variety in $\\mathbb R^n$ whose support is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Take any $\\bm p \\in \\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$.\n If the function $f: |X| \\to \\mathbb R, \\bm a \\mapsto \\bm a \\cdot \\bm p$ is constant with the value $k$, then $|X|$ is included in the set $\\{ \\bm a \\in \\mathbb R^n \\ | \\ \\bm a \\cdot \\bm p = k \\}$, which is a proper rational affine linear subspace of $\\mathbb R^n$.\n This contradicts to the assumption, hence $f$ is not constant.\n Thus $f$ is upper unbounded.\n Since $\\bm p$ is an arbitrary element of $\\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$, $|X|$ is unbounded in all rational directions.\n\\end{proof}\n\n\\begin{proof}[Proof of the only if part of Theorem \\ref{main}]\n Let $X$ be a tropical variety such that $|X|$ is not a rational affine linear subspace of $\\mathbb R^n$.\n We may assume that $|X|$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$ by Lemma \\ref{fg reduction} and the observation after it.\n Hence, by Proposition \\ref{trop unbounded}, $|X|$ is unbounded in all rational directions.\n In order to use Theorem \\ref{not fg}, we check that $n \\geq 2$.\n If $n =0$, $|X| = \\{ \\mathbf 0 \\}$.\n It contradicts to the assumption that $|X|$ is not an affine linear subspace.\n If $n =1$, $|X|$ is a singleton or the whole space $\\mathbb R$ (by Lemma \\ref{n in n}).\n It also contradicts to the assumption.\n Thus $n \\geq 2$, and hence by Theorem \\ref{not fg}, $\\mathbf E(X)$ is not finitely generated. \\qedhere\n\\end{proof}",
    "post_theorem_intro_text_len": 2612,
    "post_theorem_intro_text": "Thus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}\n\nThe strategy of the proof of Theorem \\ref{main2} is as follows.\nFirst we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set.\nThen we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large.\nThe existence of such pair causes a contradiction.\nSee Section 3 for detail.\n\nNote that we cannot apply Theorem \\ref{main2} for showing the only if part of Theorem \\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.\nIn particular, if $|X|$ is included in a proper rational affine linear subspace of $\\mathbb R^n$, then $|X|$ is not unbounded in all rational directions.\nTo overcome this issue, we show that we may replace the ambient space with a lower dimensional one.\nSee Section 4 for detail.\n\nFinally, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L \\subset \\mathbb R^2$.\nSee Section \\ref{section min gen} for detail.\n\n\\begin{thm}\n  \\label{main3}\n  The set\n  $$\\mathcal S = \\left\\{ \\left( \\overline{0 \\oplus x^u y^v}, f_{\\Delta(\\{ (0,0), (u,v) \\})} \\right) \\ \\middle| \\ \\begin{aligned}\n  &\\text{$u$ and $v$ are integers with $\\gcd(u,v) = 1$,}\\\\\n  &\\text{ $u > 0$ or $(u,v) = (0,1)$}\n  \\end{aligned} \\right\\}$$\n  is a minimal generating set of $\\mathbf E(L)$.\n\\end{thm}\n\nThis paper is organized as follows.\nIn Section 2, we recall some basic definitions and facts about semirings, tropical algebra, tropical varieties, and Newton polytopes.\nAlso we show some lemmas we need later.\nIn Section 3, we consider a way to prove that a given congruence on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ is not finitely generated, and use it to show Theorem \\ref{main2}.\nIn Section 4, we discuss about the reduction of the ambient spaces.\nThe result of this section is necessary for showing Theorem \\ref{main1}.\nIn Section 5, we prove Theorem \\ref{main1}.\nIn Section 6, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L$.\nThis section is almost independent of Section 3-5.\nIn Appendix, we give proofs of some facts about free abelian groups and polytopes, which are used in this paper.",
    "sketch": "For Theorem~\\ref{main1}, the text says: “The only if part of this theorem is a consequence of the following more general result” (Theorem~\\ref{main2}), but “we cannot apply Theorem~\\ref{main2} for showing the only if part of Theorem~\\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.” “To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.”\n\nFor Theorem~\\ref{main2}, the stated proof strategy is: “First we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set. Then we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large. The existence of such pair causes a contradiction.”",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Universal"
    ],
    "mcq": {
      "question": "Let $X$ be a tropical variety in $\\mathbb R^n$, and let $|X|$ denote its support. For a subset $Z\\subset \\mathbb R^n$, define the congruence\n$$\\mathbf E(Z)=\\{(f,g)\\in \\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]\\mid f(\\mathbf a)=g(\\mathbf a)\\text{ for all }\\mathbf a\\in Z\\},$$\nwhere $\\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]$ is the tropical Laurent polynomial semiring. A rational affine linear subspace of $\\mathbb R^n$ means an affine translate of a linear subspace defined over $\\mathbb Q$ (equivalently, whose direction is spanned by rational vectors). Which statement holds for every such tropical variety $X$?",
      "correct_choice": {
        "label": "A",
        "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is an affine linear subspace of $\\mathbb R^n$."
        },
        {
          "label": "C",
          "text": "If $|X|$ is a rational affine linear subspace of $\\mathbb R^n$, then $\\mathbf E(|X|)$ is finitely generated."
        },
        {
          "label": "D",
          "text": "$\\mathbf E(|X|)$ is finitely generated whenever $|X|$ is contained in a rational affine linear subspace of $\\mathbb R^n$."
        },
        {
          "label": "E",
          "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a finite union of rational affine linear subspaces of $\\mathbb R^n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "rationality of affine linear subspace",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "only-if direction",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "equality with a rational affine linear subspace replaced by mere containment",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "single rational affine linear subspace replaced by finite union",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the objects involved but does not reveal the finite-generation criterion or otherwise point directly to choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially an exact theorem-identification question: the correct option states the full characterization directly, so the task is largely to recognize the theorem rather than apply it."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning in comparing 'if and only if' versus weaker, stronger, or tampered variants, but the problem mainly tests recall of the precise statement rather than genuine derivation or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: they vary rationality, weaken to one direction, replace equality by containment, or broaden to finite unions, all of which reflect realistic failure modes."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed in terms of non-leaky wording and strong distractors, but it is primarily a theorem-recall/restatement item rather than a high-generative-reasoning question."
    }
  },
  {
    "id": "2512.21600v1",
    "paper_link": "http://arxiv.org/abs/2512.21600v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{th1}\nLet $p>3$.\nAssume that $\\Gamma$ is a simple closed smooth curve with unit length in $\\Omega$, and it is also a non-degenerate critical point of the functional\n\\begin{equation}\\label{e91}\n\\mathcal{K}(\\Gamma)=\\int_\\Gamma \\mathbf{\\Psi}^{\\frac{p+3}{2p}}dvol_{\\mathfrak{g}},\n\\end{equation}\nwhere $\\mathfrak{g}(X,Y)=\\langle A^*X,Y\\rangle$ is a Riemannian metric on $\\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\\Gamma$:\n\\begin{eqnarray*}\n\\Upsilon_0&=&-\\alpha^{1-p}\\left[ 2\\alpha^{-1}\\alpha'a_{22}+\\beta^{-1}\\beta'b_{11}+b_{22}-a_{33}+\\frac{p+3}2\\alpha^{p-2}\\beta^{-2}\\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p} \\beta^2 (a_{32})^2\\right. \\\\\n&&\\left.-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac{2}{p+3}\\alpha^{1-p}\\beta^2(b_{21})^2 +a_{11}\\left(\\beta^{-1}\\beta''+2\\alpha^{-1}\\beta^{-1}\\alpha'\\beta'-\\beta^{-2}(\\beta')^2\\right)\\right]>0,\n\\end{eqnarray*}\nwhere $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \\ref{lm8}, $\\alpha$ and $\\beta$ are defined in \\eqref{80}, and $\\mathbf{q}$ is defined in \\eqref{e88}.\nThen for each  integer $N>0$, there exists a sequence of $\\varepsilon$, \\textit{i.e.}, $\\{\\varepsilon_l\\}$ converging to $0$ such that \\eqref{1} has a positive solutions $u_{\\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\\varepsilon_l|\\log\\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\\Gamma$ at speed $\\varepsilon_l^{1+\\mu}$ for small positive constant $\\mu\\in(0,1/2)$. More precisely $u_{\\varepsilon_l}$ has the form\n\\begin{eqnarray*}\nu_{\\varepsilon_l}(y_1,y_2)\\approx -\\mathbf{\\Psi}^{\\frac1p}(y_1,y_2)+ \\mathbf{\\Psi}^{\\frac1p}(\\gamma(\\theta))\\sum_{k=1}^N w\\left(\\left[\\frac{\\mathbf{\\Psi}^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}{\\langle A^*n,n\\rangle}\\right]^{\\frac12}\\frac{t-\\varepsilon_l f_k}{\\varepsilon_l}\\right),\n\\end{eqnarray*}\nwhere $\\gamma$ is a natural parametrization of $\\Gamma$, $n$ is the unit vector defined in \\eqref{e102} and $w$ is  the unique solution of the following problem\n\\begin{equation}\\label{29}\n-w''=|1-w|^p -1, \\qquad w>0 \\quad \\mathrm{in} \\quad \\mathbb R, \\qquad w'(0)=w(\\pm \\infty).\n\\end{equation}\nIn the expression above, the functions $f_j$'s satisfy\n\\[\\|f_j\\|_{L^\\infty(0,1)}\\le C|\\log\\varepsilon_l|^2,\\qquad \\sum_{j=1}^N f_j =O(\\frac1{|\\log\\varepsilon_l|^{\\frac32}}),\\]\n\\[\\min_{1\\le j\\le N}(f_{j+1}-f_j)\\approx\\frac2{\\sqrt{p}}|\\log\\varepsilon_l|\\left[\\frac{\\langle A^*n,n\\rangle}{\\Psi^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}\\right]^{\\frac12}\\]\nand solves the Jacobi-Toda system for $j=1,2,\\cdots,N$\n\\begin{eqnarray*}\n\\nonumber&&\\varepsilon^2\\alpha^{1-p}\\beta\\left\\{-a_{11}f_j''+\\left[a_{22}-b_{11}-a_{11}\\left(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha'\\right)\\right]f_j'+\\left[a_{22}(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha') +b_{22}-a_{33}\\right]f_j\\right. \\\\\n\\nonumber&&+\\left[\\frac{p+3}2\\alpha^{p-2}\\beta^{-2} \\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p}\\beta^2(a_{32})^2-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac2{p+3}\\alpha^{1-p}\\beta^2 (b_{21})^2\\right]f_j \\\\\n&&C_0 p\\alpha_p\\left[e^{-\\sqrt{p}\\beta(f_j-f_{j-1})}-e^{-\\sqrt{p}\\beta (f_{j+1}-f_j)}\\right]\\approx0.\n\\end{eqnarray*}",
      "start_pos": 15227,
      "end_pos": 18420,
      "label": "th1"
    },
    "ref_dict": {
      "e52": "\\begin{equation}\\label{e52}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\mathbf{\\Psi}_0 (x), &\\mbox{in $\\Omega$,} \\\\\nu=0,&\\mbox{on $\\partial \\Omega$}.\n\\end{array}\\right.\n\\end{equation}",
      "lm1": "\\begin{lemma}\\label{lm1}\nIf $a\\ge c$ and $u_\\varepsilon$ is the minimizer of problem \\eqref{6},  $u_\\varepsilon(x)\\to a$ on compact sets in $B_1(0)$, as $\\varepsilon \\to 0$.\n\\end{lemma}",
      "lm8": "\\begin{lemma}\\label{lm8}\nUnder the stretched modified Fermi coordinate defined by $\\Phi_\\varepsilon(z,s)$, we get\n\\begin{eqnarray*}\n\\nonumber\\Div(A(\\varepsilon y)\\nabla v)&=& a_{11}(\\varepsilon z)v_{zz}+2\\varepsilon s a_{22}(\\varepsilon z)v_{sz} +\\left(a_{31}(\\varepsilon z)+\\varepsilon s a_{32}(\\varepsilon z) +\\varepsilon^2 s^2 a_{33}(\\varepsilon z)\\right)v_{ss} \\\\\n&& +\\varepsilon b_{11}(\\varepsilon z)v_z +\\left(\\varepsilon b_{21}(\\varepsilon z)+\\varepsilon^2 s b_{22}(\\varepsilon z)\\right) v_s +B_0(v),\n\\end{eqnarray*}\nwhere\n\\[a_{11}(\\theta)=\\frac{\\langle{A^*}n,n\\rangle}{1-\\langle\\gamma',n\\rangle^2},\\qquad a_{22}(\\theta)=-\\frac{\\langle{A^*}n,n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{\\langle{A^*_t}n,\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}, \\]\n\\[a_{31}(\\theta)=\\frac{\\langle{A^*}\\gamma',\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}, \\qquad a_{32}=\\frac{\\langle{A^*_t}\\gamma',\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}+ \\frac{2\\langle{A^*}\\gamma',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{2\\langle\\gamma',n'\\rangle\\langle{A^*}\\gamma',\\gamma'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}, \\]\n\\begin{eqnarray*}\na_{33}(\\theta)&=& \\frac1{1-\\langle\\gamma',n\\rangle^2}\\left(\\langle{A^*}n',n'\\rangle +2\\langle{A^*_t}\\gamma',n'\\rangle+\\frac12\\langle{A^*_{tt}}\\gamma',\\gamma'\\rangle\\right)  \\\\\n&&-\\frac{2\\langle\\gamma',n'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}\\left(\\langle{A^*_t} \\gamma',\\gamma'\\rangle +2\\langle{A^*}\\gamma',n'\\rangle\\right)  \\\\\n&&+\\left(\\frac{4\\langle\\gamma',n'\\rangle^2}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^3}- \\frac{\\langle n',n'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}\\right)\\langle{A^*}\\gamma',\\gamma'\\rangle,\n\\end{eqnarray*}\n\\[b_{11}(\\theta)=\\left[\\frac{\\langle{A^*}n,n\\rangle}{1-\\langle\\gamma',n\\rangle^2}\\right]_{\\theta}-\\frac12\\left(\\frac1{1-\\langle\\gamma',n\\rangle^2}\\right)_\\theta \\langle{A^*}n,n\\rangle -\\frac{\\langle{A^*}n,n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{\\langle{A^*_t}n,\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2},\\]\n\\[b_{21}(\\theta)=\\frac{\\langle\\gamma',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}a_{31}+a_{32},\\]\n\\[b_{22}(\\theta)=\\frac12\\frac{(1-\\langle\\gamma',n\\rangle^2)_\\theta}{1-\\langle\\gamma',n\\rangle^2}a_{22}+\\left[\\frac{\\langle n',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{2\\langle\\gamma',n'\\rangle^2}{(1-\\langle\\gamma',n\\rangle^2)^2}\\right]a_{31}+\\frac{\\langle\\gamma',n'\\rangle}{1-\\langle\\gamma', n\\rangle^2}a_{32} +\\partial_\\theta a_{22}+2a_{33}\\]\nand\n\\[B_0(v)=\\varepsilon a_{12}(\\varepsilon z,s)v_{zz}+\\varepsilon^2 a_{23}(\\varepsilon z.s)v_{zs}+\\varepsilon^3a_{33}(\\varepsilon z,s)v_{ss} +\\varepsilon^2 b_{12}(\\varepsilon z,s) v_z +\\varepsilon^3 b_{23}(\\varepsilon z,s)v_s.\\]\nIn the expression above, funtions $a_{12}$, $a_{23}$, $a_{33}$, $b_{12}$ and $b_{23}$ are smooth functions satisfying the following estimate\n\\[|a_{12}(\\varepsilon z, s)|\\le C(1+|s|), \\quad |a_{23}(\\varepsilon z, s)|\\le C(1+|s|^2), \\quad |a_{33}(\\varepsilon z, s)|\\le C(1+|s|^3),\\]\nand\n\\[|b_{12}(\\varepsilon z, s)|\\le C(1+|s|), \\quad |b_{23}(\\varepsilon z, s)|\\le C(1+|s|^2).\\]\n\\end{lemma}",
      "e102": "\\begin{equation}\\label{e102}\nn(\\theta)=\\frac{A(\\gamma(\\theta))\\nu(\\theta)}{|A(\\gamma(\\theta))\\nu(\\theta)|}.\n\\end{equation}",
      "1": "\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}",
      "th1": "\\begin{theorem}\\label{th1}\nLet $p>3$.\nAssume that $\\Gamma$ is a simple closed smooth curve with unit length in $\\Omega$, and it is also a non-degenerate critical point of the functional\n\\begin{equation}\\label{e91}\n\\mathcal{K}(\\Gamma)=\\int_\\Gamma \\mathbf{\\Psi}^{\\frac{p+3}{2p}}dvol_{\\mathfrak{g}},\n\\end{equation}\nwhere $\\mathfrak{g}(X,Y)=\\langle A^*X,Y\\rangle$ is a Riemannian metric on $\\R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\\Gamma$:\n\\begin{eqnarray*}\n\\Upsilon_0&=&-\\alpha^{1-p}\\left[ 2\\alpha^{-1}\\alpha'a_{22}+\\beta^{-1}\\beta'b_{11}+b_{22}-a_{33}+\\frac{p+3}2\\alpha^{p-2}\\beta^{-2}\\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p} \\beta^2 (a_{32})^2\\right. \\\\\n&&\\left.-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac{2}{p+3}\\alpha^{1-p}\\beta^2(b_{21})^2 +a_{11}\\left(\\beta^{-1}\\beta''+2\\alpha^{-1}\\beta^{-1}\\alpha'\\beta'-\\beta^{-2}(\\beta')^2\\right)\\right]>0,\n\\end{eqnarray*}\nwhere $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \\ref{lm8}, $\\alpha$ and $\\beta$ are defined in \\eqref{80}, and $\\mathbf{q}$ is defined in \\eqref{e88}.\nThen for each  integer $N>0$, there exists a sequence of $\\varepsilon$, \\textit{i.e.}, $\\{\\varepsilon_l\\}$ converging to $0$ such that \\eqref{1} has a positive solutions $u_{\\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\\varepsilon_l|\\log\\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\\Gamma$ at speed $\\varepsilon_l^{1+\\mu}$ for small positive constant $\\mu\\in(0,1/2)$. More precisely $u_{\\varepsilon_l}$ has the form\n\\begin{eqnarray*}\nu_{\\varepsilon_l}(y_1,y_2)\\approx -\\mathbf{\\Psi}^{\\frac1p}(y_1,y_2)+ \\mathbf{\\Psi}^{\\frac1p}(\\gamma(\\theta))\\sum_{k=1}^N w\\left(\\left[\\frac{\\mathbf{\\Psi}^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}{\\langle A^*n,n\\rangle}\\right]^{\\frac12}\\frac{t-\\varepsilon_l f_k}{\\varepsilon_l}\\right),\n\\end{eqnarray*}\nwhere $\\gamma$ is a natural parametrization of $\\Gamma$, $n$ is the unit vector defined in \\eqref{e102} and $w$ is  the unique solution of the following problem\n\\begin{equation}\\label{29}\n-w''=|1-w|^p -1, \\qquad w>0 \\quad \\mathrm{in} \\quad \\R, \\qquad w'(0)=w(\\pm \\infty).\n\\end{equation}\nIn the expression above, the functions $f_j$'s satisfy\n\\[\\|f_j\\|_{L^\\infty(0,1)}\\le C|\\log\\varepsilon_l|^2,\\qquad \\sum_{j=1}^N f_j =O(\\frac1{|\\log\\varepsilon_l|^{\\frac32}}),\\]\n\\[\\min_{1\\le j\\le N}(f_{j+1}-f_j)\\approx\\frac2{\\sqrt{p}}|\\log\\varepsilon_l|\\left[\\frac{\\langle A^*n,n\\rangle}{\\Psi^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}\\right]^{\\frac12}\\]\nand solves the Jacobi-Toda system for $j=1,2,\\cdots,N$\n\\begin{eqnarray*}\n\\nonumber&&\\varepsilon^2\\alpha^{1-p}\\beta\\left\\{-a_{11}f_j''+\\left[a_{22}-b_{11}-a_{11}\\left(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha'\\right)\\right]f_j'+\\left[a_{22}(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha') +b_{22}-a_{33}\\right]f_j\\right. \\\\\n\\nonumber&&+\\left[\\frac{p+3}2\\alpha^{p-2}\\beta^{-2} \\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p}\\beta^2(a_{32})^2-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac2{p+3}\\alpha^{1-p}\\beta^2 (b_{21})^2\\right]f_j \\\\\n&&C_0 p\\alpha_p\\left[e^{-\\sqrt{p}\\beta(f_j-f_{j-1})}-e^{-\\sqrt{p}\\beta (f_{j+1}-f_j)}\\right]\\approx0.\n\\end{eqnarray*}\n\\end{theorem}",
      "80": "\\begin{equation}\\label{80}\n\\alpha(\\theta)=\\mathbf{q}(\\theta,0), \\qquad \\beta(\\theta)=\\frac{\\left[\\mathbf{q}(\\theta,0)\\right]^{\\frac{p-1}2}}{\\left[a_{31}(\\theta)\\right]^{\\frac12}}.\n\\end{equation}",
      "e53": "\\begin{equation}\\label{e53}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\psi (x), &\\mbox{in $\\Omega$,} \\\\\n\\frac{\\partial u}{\\partial \\mathbf{n}}=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}",
      "e51": "\\begin{equation}\\label{e51}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=\\zeta(u)-t\\mathbf{\\Psi}_0 (x), &\\mbox{in $\\Omega$,} \\\\\nu=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}",
      "e90": "\\begin{equation}\\label{e90}\n\\left\\{\\begin{array}{ll}\n\\varepsilon^2 \\Div(\\nabla_{\\mathfrak{a}(y)}u)-V(y)u+u^p=0, \\qquad u>0, &\\mbox{in $\\Omega$,}\\\\\n\\nabla_{\\mathfrak{a}(y)} u\\cdot \\mathbf{n}=0,  &\\mbox{on $\\partial\\Omega$,}\n\\end{array}\\right.\n\\end{equation}",
      "2": "\\begin{equation}\\label{2}\n\\left\\{\\begin{array}{ll}\n-\\Div (A(x)\\nabla u)=|u|^p-t\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}",
      "e55": "\\begin{equation}\\label{e55}\n-\\varepsilon^2 \\Delta u+V(y)u=u^p, \\qquad \\mathrm{in} \\qquad \\R^\\mathcal{N}.\n\\end{equation}",
      "e88": "\\begin{equation}\\label{e88}\n\\mathbf{q}(\\theta, t)=\\mathbf{\\Psi}^{\\frac1p}(\\gamma(\\theta)+tn(\\theta)).\n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 10686,
    "pre_theorem_intro_text": "The Ambrosetti-Prodi problem is of the following form:\n\\begin{equation}\\label{e51}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=\\zeta(u)-t\\mathbf{\\Psi}_0 (x), &\\mbox{in $\\Omega$,} \\\\\nu=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}\nwhere $t>0$, $\\Omega\\subset \\mathbb R^\\mathcal{N}$ is a smooth bounded domain,  $\\mathbf{\\Psi}_0$ is the an eigenfunction of $-\\Delta$ subject to Dirichlet boundary condition corresponding to the first eigenvalue, the function $\\zeta(t)$ satisfies\n\\[-\\infty\\le\\mu=\\lim_{t\\to +\\infty} \\frac{\\zeta(t)}{t}>\\lim_{t\\to +\\infty} \\frac{\\zeta(t)}{t}=\\nu\\le +\\infty,\\]\nand the interval $(\\mu, \\nu)$ contains some eigenvalues of $-\\Delta$ subject to Dirichlet boundary condition.\n\nProblem \\eqref{e51} was first studied by Ambrosetti and Prodi \\cite{Ambrosetti_Prodi1972}. It was widely discussed in 1980's( see \\cite{deFigueiredo1984,deFigueiredo_Solimini1984,Lazer_McKenna1981,Lazer_McKenna1984,Lazer_McKenna1988,Ruf_Srikanth986a,Ruf_Srikanth1986,Solimini1985,Ruf_Solimini1986} for example). The main results in these literature are that if $g(t)$ grows subcritical at infinity, \\eqref{e51} has at least two solutions: one is local minimizer of the Euler-Lagrange functional, the other is the mountain-pass solution. Breuer, McKenna and Plum \\cite{Breuer_McKenna_Plum2003} considered the case that $\\zeta(t)=t^2$ and $\\Omega$ is a unit square in $\\mathbb R^2$. Using a computer assisted proof, they showed  that \\eqref{e51} has at least 4 solutions.  By comparing Morse index of mountain pass solution in different spaces, de Figueiredo, Srikanth and Santra \\cite{deFigueiredo_Srikanth_Santra2005}  found a non-radial solution of \\eqref{e51} under the condition $\\Omega$ is a unit ball and $\\zeta(t)=t^2$.\n\nHowever, in the case of $\\zeta(t)=|t|^p$, where $1<p<\\frac{\\mathcal{N}+2}{\\mathcal{N}-2}$ for $\\mathcal{N}\\ge3$ and $p>1$ for $\\mathcal{N}=2$, \\eqref{e51} becomes the following problem\n\\begin{equation}\\label{e52}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\mathbf{\\Psi}_0 (x), &\\mbox{in $\\Omega$,} \\\\\nu=0,&\\mbox{on $\\partial \\Omega$}.\n\\end{array}\\right.\n\\end{equation}\nDancer and Yan \\cite{Dancer_Yan2005} constructed arbitrary many peak solutions of \\eqref{e52} for $t>0$ large enough.  It shows that Lazer-McKenna conjecture \\cite{Lazer_McKenna1981} holds in this case. Dancer and Yan \\cite{Dancer_Yan2005} also proved that the mountain pass solution of \\eqref{e51} has a sharp peak near the boundary for $t$ large enough. As a conclusion,   \\eqref{e51} has solutions concentrating at some points in $\\Omega$ or the  ones on $\\partial \\Omega$ as $t\\to \\infty$. These results have been extended to different kind of nonlinearities( see \\cite{Dancer_Yan2005a, Dancer_Santra2007, delPino_Munoz2006, Li_Yan_Yang2007, Li_Yan_Yang2006, Molle_Passaseo2010, Wei_Yan2007} for instance and references therein).\n\nHowever, these results concern only point concentrating solutions. Based on some numerical evidence, Hollman and McKenna \\cite{Hollman_McKenna2011} asked \\eqref{e52} that  whether there exist other types of concentrations  for the solutions to \\eqref{e52} as $t\\to+\\infty$. For this problem, Manna and Santra \\cite{Manna_Santra2016} considered \\eqref{e52} under the condition $\\Omega\\subset \\mathbb R^2$ and $p>2$. They proved that \\eqref{e52} has a family of solutions clustering along a closed curve $\\Gamma\\subset \\Omega$, where $\\Gamma$ is a non-degenerate critical point of the functional\n\\begin{equation}\\label{e54}\n\\mathcal{K}_0(\\Gamma)=\\int_{\\Gamma} \\mathbf{\\Psi}_0^{\\frac{p+3}{2p}}(x)dvol.\n\\end{equation}\nLater, this result was extended to high dimensional case by Khemiri, Mahmoudi and Messaoudi  \\cite{Khemiri_Mahmoudi_Messaoudi2017}. Under the condition that $\\mathcal{N}\\ge 3$ and $\\Omega$ contain a $k$-dimensional compact submanifold $\\Gamma$, which is a non-degenerate critical point of the functional\n\\[\\mathcal{K}_1(\\Gamma)=\\int_{\\Gamma} \\mathbf{\\Psi}_0^{\\left(1-\\frac1p\\right)\\left(\\frac{p+1}{p-1}-\\frac{n-k}2\\right)}(x)dvol,\\]\nThey proved there exists a sequence $t=t_j\\to \\infty$, and  solutions  $u_{t_j}$ to \\eqref{e52}, which have  concentration layers concentrating near $\\Gamma$.\n\nBaraket \\textit{et. al.} \\cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:\n\\begin{equation}\\label{e53}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\psi (x), &\\mbox{in $\\Omega$,} \\\\\n\\frac{\\partial u}{\\partial \\mathbf{n}}=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}\nwhere  $\\Omega\\subset\\mathbb R^\\mathcal{N}$ is a smooth bound domain and $\\mathbf{n}$ is the unit outward normal vector of $\\partial\\Omega$.\nUnder the condition $\\mathcal{N}=2$, they proved that \\eqref{e53} has a solution $u_t$ concentrating along a curve $\\Gamma\\subset \\bar{\\Omega}$ with $t>0$ large enough. The curve $\\Gamma$ intersects $\\partial\\Omega$ with right angle and divides $\\Omega$ into two part. Moreover $\\Gamma$ is a non-degenerate critical point of the functional $\\mathcal{K}_2(\\Gamma)=\\int_{\\Gamma} \\psi^{\\frac{p+3}{2p}}(x)dvol$.  However, under the condition that  $\\psi\\equiv1$ and  $\\mathcal{N}\\ge 2$, Bendahou, Khemiri and Mahmoudi \\cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \\eqref{e53}, which has large number of spikes concentrating along an interior  straight line in $\\Omega$ for $t\\to+\\infty$. Using the similar method, Ao, Fu and Liu \\cite{Ao_Fu_Liu2022} constructed a similar type of  solutions  which concentrate along a segment of boundary $\\partial \\Omega$ in the two dimensional case.\n\nFrom these results, we recognize that the high dimensional concentration behaviours of Ambrosetti-Prodi type problem is similar to that of Nonlinear Schr\\\"{o}dinger equation:\n\\begin{equation}\\label{e55}\n-\\varepsilon^2 \\Delta u+V(y)u=u^p, \\qquad \\mathrm{in} \\qquad \\mathbb R^\\mathcal{N}.\n\\end{equation}\nIn 2003, Ambrosetti, Malchiodi and Ni \\cite{Ambrosetti_Malchiodi_Ni2003} raised the following conjecture:\n\\begin{conjecture}\nLet $\\Gamma$ be a $k$-dimensional submanifold in $\\mathbb R^\\mathcal{N}$ and a nondegenerate critical point of the following functional\n\\[\\mathcal{K}(\\Gamma)=\\int_\\Gamma V^{\\frac{p+1}{p-1}-\\frac12(\\mathcal{N}-k)}dvol,\\]\nwhere $1<p<\\frac{n+2-k}{n-2+k}$ for $\\mathcal{N}\\ge 3$ and $p>1$ for $\\mathcal{N}=2$.\nThen there exists a family of solutions to \\eqref{e55} concentrating along $\\Gamma$ at least for a subsequence $\\varepsilon=\\varepsilon_j\\to 0$.\n\\end{conjecture}\ndel Pino, Kowalczyk and Wei \\cite{delPino_Kowalczyk_Wei2007} first showed that this conjecture holds in the case of $\\mathcal{N}=2$ and $k=1$.  Wang, Wei and Yang \\cite{Wang_Wei_Yang2011} proved that this conjecture in the case of $\\mathcal{N}\\ge3$ and $k=\\mathcal{N}-1$.  And Mahmoudi, Sanchez and Yao \\cite{Mahmoudi_Sanchez_Yao2015} proved this conjecture is valid for all cases.\n\nHowever, the following more general Neumann version of \\eqref{e55} also has some solutions concentrating on high dimensional set:\n\\begin{equation}\\label{e90}\n\\left\\{\\begin{array}{ll}\n\\varepsilon^2 \\Div(\\nabla_{\\mathfrak{a}(y)}u)-V(y)u+u^p=0, \\qquad u>0, &\\mbox{in $\\Omega$,}\\\\\n\\nabla_{\\mathfrak{a}(y)} u\\cdot \\mathbf{n}=0,  &\\mbox{on $\\partial\\Omega$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb R^2$ is a smooth bounded domain, $\\mathbf{n}$ is the unit outward normal of $\\partial \\Omega$,\n\\[\\mathfrak{a}(y)=(\\mathfrak{a}_1(y),\\mathfrak{a}_2(y), \\cdots, \\mathfrak{a}_N(y))\\]\nand\n\\[\\nabla_{\\mathfrak{a}(y)}u=(\\mathfrak{a}_1(y)u_{y_1}, \\mathfrak{a}_2(y)u_{y_2}, \\cdots, \\mathfrak{a}_N(y)u_{y_N}).\\]\nIn the case of $\\mathfrak{a}\\equiv 1$ and $V(y)\\equiv 1$, Wei and Yang \\cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\\Gamma_2$ in $\\Omega$. The segment intersects $\\partial \\Omega$ with right angle and separates $\\Omega$ into two parts. Unlike the solutions constructed in \\cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \\cite{Wei_Yang2008}  have multiple  concentration layers.\nWei, Xu and Yang \\cite{Wei_Xu_Yang2018} considered the case that $\\mathfrak{a}(y)\\equiv 1$. Let $\\Gamma\\subset\\bar{\\Omega}$ be a curve intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Provided $\\Gamma$ is a nondegenerate critical point of $\\int_\\Gamma V^{\\frac{p+1}{p-1}-\\frac12}$, they constructed a  sequence of solutions to \\eqref{e90} clustering along $\\Gamma$. Recently, Wei and Yang \\cite{Wei_Yang2021} extended the result in \\cite{Wei_Yang2008} into the general case. They  constructed a sequence of solutions to \\eqref{e90} with multiple\nconcentration layers which concentrate near a closed curve $\\Gamma_0\\subset\\Omega$ or a curve $\\Gamma_1$ intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Moreover,  $\\Gamma_0$ and $\\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\\mathbb R^2$ with the metric $V^{\\frac{2(p+1)}{p-1}-1}[\\mathfrak{a}_2(y)dy_1^2+\\mathfrak{a}_1(y)dy_2^2]$.\n\nInspired by \\cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem\n\\begin{equation}\\label{2}\n\\left\\{\\begin{array}{ll}\n-\\Div (A(x)\\nabla u)=|u|^p-t\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and  $\\mathbf{\\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\\mathfrak{L}(u)=-\\Div (A(x)\\nabla u)$ on $\\Omega$.\nMoreover, $A(x)=\\{A_{ij}(x)\\}_{2\\times 2}$ is a symmetric positive defined matrix function, satisfying\n\\begin{equation}\\label{e98}\n\\lambda |\\alpha|^2 \\leq \\langle A(x)\\alpha, \\alpha \\rangle \\leq \\Lambda |\\alpha|^2,\\qquad \\mathrm{for} \\qquad x,\\alpha \\in \\mathbb R^n.\n\\end{equation}\nWe notice $\\Div(\\nabla_{\\mathfrak{a}(y)}u)$ is a special form of $\\Div (A(x)\\nabla u)$. Based on previous work, we suspect whether \\eqref{2} has a similar solution to that constructed in \\cite{Wei_Yang2021,Wei_Yang2008}.\n\nOur motivation for writing this paper is twofold. First, we plan to construct solutions to \\eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \\eqref{2}.\n\nLet $\\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \\eqref{2} if and only if $t^{-\\frac1p}u$ is the solution of the following problem\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nThen we get the following theorem.",
    "context": "Baraket \\textit{et. al.} \\cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:\n\\begin{equation}\\label{e53}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\psi (x), &\\mbox{in $\\Omega$,} \\\\\n\\frac{\\partial u}{\\partial \\mathbf{n}}=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}\nwhere $\\Omega\\subset\\mathbb R^\\mathcal{N}$ is a smooth bound domain and $\\mathbf{n}$ is the unit outward normal vector of $\\partial\\Omega$.\nUnder the condition $\\mathcal{N}=2$, they proved that \\eqref{e53} has a solution $u_t$ concentrating along a curve $\\Gamma\\subset \\bar{\\Omega}$ with $t>0$ large enough. The curve $\\Gamma$ intersects $\\partial\\Omega$ with right angle and divides $\\Omega$ into two part. Moreover $\\Gamma$ is a non-degenerate critical point of the functional $\\mathcal{K}_2(\\Gamma)=\\int_{\\Gamma} \\psi^{\\frac{p+3}{2p}}(x)dvol$. However, under the condition that $\\psi\\equiv1$ and $\\mathcal{N}\\ge 2$, Bendahou, Khemiri and Mahmoudi \\cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \\eqref{e53}, which has large number of spikes concentrating along an interior straight line in $\\Omega$ for $t\\to+\\infty$. Using the similar method, Ao, Fu and Liu \\cite{Ao_Fu_Liu2022} constructed a similar type of solutions which concentrate along a segment of boundary $\\partial \\Omega$ in the two dimensional case.\n\nHowever, the following more general Neumann version of \\eqref{e55} also has some solutions concentrating on high dimensional set:\n\\begin{equation}\\label{e90}\n\\left\\{\\begin{array}{ll}\n\\varepsilon^2 \\Div(\\nabla_{\\mathfrak{a}(y)}u)-V(y)u+u^p=0, \\qquad u>0, &\\mbox{in $\\Omega$,}\\\\\n\\nabla_{\\mathfrak{a}(y)} u\\cdot \\mathbf{n}=0, &\\mbox{on $\\partial\\Omega$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb R^2$ is a smooth bounded domain, $\\mathbf{n}$ is the unit outward normal of $\\partial \\Omega$,\n\\[\\mathfrak{a}(y)=(\\mathfrak{a}_1(y),\\mathfrak{a}_2(y), \\cdots, \\mathfrak{a}_N(y))\\]\nand\n\\[\\nabla_{\\mathfrak{a}(y)}u=(\\mathfrak{a}_1(y)u_{y_1}, \\mathfrak{a}_2(y)u_{y_2}, \\cdots, \\mathfrak{a}_N(y)u_{y_N}).\\]\nIn the case of $\\mathfrak{a}\\equiv 1$ and $V(y)\\equiv 1$, Wei and Yang \\cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\\Gamma_2$ in $\\Omega$. The segment intersects $\\partial \\Omega$ with right angle and separates $\\Omega$ into two parts. Unlike the solutions constructed in \\cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \\cite{Wei_Yang2008} have multiple concentration layers.\nWei, Xu and Yang \\cite{Wei_Xu_Yang2018} considered the case that $\\mathfrak{a}(y)\\equiv 1$. Let $\\Gamma\\subset\\bar{\\Omega}$ be a curve intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Provided $\\Gamma$ is a nondegenerate critical point of $\\int_\\Gamma V^{\\frac{p+1}{p-1}-\\frac12}$, they constructed a sequence of solutions to \\eqref{e90} clustering along $\\Gamma$. Recently, Wei and Yang \\cite{Wei_Yang2021} extended the result in \\cite{Wei_Yang2008} into the general case. They constructed a sequence of solutions to \\eqref{e90} with multiple\nconcentration layers which concentrate near a closed curve $\\Gamma_0\\subset\\Omega$ or a curve $\\Gamma_1$ intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Moreover, $\\Gamma_0$ and $\\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\\mathbb R^2$ with the metric $V^{\\frac{2(p+1)}{p-1}-1}[\\mathfrak{a}_2(y)dy_1^2+\\mathfrak{a}_1(y)dy_2^2]$.\n\nInspired by \\cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem\n\\begin{equation}\\label{2}\n\\left\\{\\begin{array}{ll}\n-\\Div (A(x)\\nabla u)=|u|^p-t\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and $\\mathbf{\\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\\mathfrak{L}(u)=-\\Div (A(x)\\nabla u)$ on $\\Omega$.\nMoreover, $A(x)=\\{A_{ij}(x)\\}_{2\\times 2}$ is a symmetric positive defined matrix function, satisfying\n\\begin{equation}\\label{e98}\n\\lambda |\\alpha|^2 \\leq \\langle A(x)\\alpha, \\alpha \\rangle \\leq \\Lambda |\\alpha|^2,\\qquad \\mathrm{for} \\qquad x,\\alpha \\in \\mathbb R^n.\n\\end{equation}\nWe notice $\\Div(\\nabla_{\\mathfrak{a}(y)}u)$ is a special form of $\\Div (A(x)\\nabla u)$. Based on previous work, we suspect whether \\eqref{2} has a similar solution to that constructed in \\cite{Wei_Yang2021,Wei_Yang2008}.\n\nOur motivation for writing this paper is twofold. First, we plan to construct solutions to \\eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \\eqref{2}.\n\nLet $\\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \\eqref{2} if and only if $t^{-\\frac1p}u$ is the solution of the following problem\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nThen we get the following theorem.",
    "full_context": "Baraket \\textit{et. al.} \\cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:\n\\begin{equation}\\label{e53}\n\\left\\{\\begin{array}{ll}\n-\\Delta u=|u|^p-t\\psi (x), &\\mbox{in $\\Omega$,} \\\\\n\\frac{\\partial u}{\\partial \\mathbf{n}}=0,&\\mbox{on $\\partial \\Omega$},\n\\end{array}\\right.\n\\end{equation}\nwhere $\\Omega\\subset\\mathbb R^\\mathcal{N}$ is a smooth bound domain and $\\mathbf{n}$ is the unit outward normal vector of $\\partial\\Omega$.\nUnder the condition $\\mathcal{N}=2$, they proved that \\eqref{e53} has a solution $u_t$ concentrating along a curve $\\Gamma\\subset \\bar{\\Omega}$ with $t>0$ large enough. The curve $\\Gamma$ intersects $\\partial\\Omega$ with right angle and divides $\\Omega$ into two part. Moreover $\\Gamma$ is a non-degenerate critical point of the functional $\\mathcal{K}_2(\\Gamma)=\\int_{\\Gamma} \\psi^{\\frac{p+3}{2p}}(x)dvol$. However, under the condition that $\\psi\\equiv1$ and $\\mathcal{N}\\ge 2$, Bendahou, Khemiri and Mahmoudi \\cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \\eqref{e53}, which has large number of spikes concentrating along an interior straight line in $\\Omega$ for $t\\to+\\infty$. Using the similar method, Ao, Fu and Liu \\cite{Ao_Fu_Liu2022} constructed a similar type of solutions which concentrate along a segment of boundary $\\partial \\Omega$ in the two dimensional case.\n\nHowever, the following more general Neumann version of \\eqref{e55} also has some solutions concentrating on high dimensional set:\n\\begin{equation}\\label{e90}\n\\left\\{\\begin{array}{ll}\n\\varepsilon^2 \\Div(\\nabla_{\\mathfrak{a}(y)}u)-V(y)u+u^p=0, \\qquad u>0, &\\mbox{in $\\Omega$,}\\\\\n\\nabla_{\\mathfrak{a}(y)} u\\cdot \\mathbf{n}=0, &\\mbox{on $\\partial\\Omega$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb R^2$ is a smooth bounded domain, $\\mathbf{n}$ is the unit outward normal of $\\partial \\Omega$,\n\\[\\mathfrak{a}(y)=(\\mathfrak{a}_1(y),\\mathfrak{a}_2(y), \\cdots, \\mathfrak{a}_N(y))\\]\nand\n\\[\\nabla_{\\mathfrak{a}(y)}u=(\\mathfrak{a}_1(y)u_{y_1}, \\mathfrak{a}_2(y)u_{y_2}, \\cdots, \\mathfrak{a}_N(y)u_{y_N}).\\]\nIn the case of $\\mathfrak{a}\\equiv 1$ and $V(y)\\equiv 1$, Wei and Yang \\cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\\Gamma_2$ in $\\Omega$. The segment intersects $\\partial \\Omega$ with right angle and separates $\\Omega$ into two parts. Unlike the solutions constructed in \\cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \\cite{Wei_Yang2008} have multiple concentration layers.\nWei, Xu and Yang \\cite{Wei_Xu_Yang2018} considered the case that $\\mathfrak{a}(y)\\equiv 1$. Let $\\Gamma\\subset\\bar{\\Omega}$ be a curve intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Provided $\\Gamma$ is a nondegenerate critical point of $\\int_\\Gamma V^{\\frac{p+1}{p-1}-\\frac12}$, they constructed a sequence of solutions to \\eqref{e90} clustering along $\\Gamma$. Recently, Wei and Yang \\cite{Wei_Yang2021} extended the result in \\cite{Wei_Yang2008} into the general case. They constructed a sequence of solutions to \\eqref{e90} with multiple\nconcentration layers which concentrate near a closed curve $\\Gamma_0\\subset\\Omega$ or a curve $\\Gamma_1$ intersecting $\\partial \\Omega$ with right angle and separating $\\Omega$ into two parts. Moreover, $\\Gamma_0$ and $\\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\\mathbb R^2$ with the metric $V^{\\frac{2(p+1)}{p-1}-1}[\\mathfrak{a}_2(y)dy_1^2+\\mathfrak{a}_1(y)dy_2^2]$.\n\nInspired by \\cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem\n\\begin{equation}\\label{2}\n\\left\\{\\begin{array}{ll}\n-\\Div (A(x)\\nabla u)=|u|^p-t\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and $\\mathbf{\\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\\mathfrak{L}(u)=-\\Div (A(x)\\nabla u)$ on $\\Omega$.\nMoreover, $A(x)=\\{A_{ij}(x)\\}_{2\\times 2}$ is a symmetric positive defined matrix function, satisfying\n\\begin{equation}\\label{e98}\n\\lambda |\\alpha|^2 \\leq \\langle A(x)\\alpha, \\alpha \\rangle \\leq \\Lambda |\\alpha|^2,\\qquad \\mathrm{for} \\qquad x,\\alpha \\in \\mathbb R^n.\n\\end{equation}\nWe notice $\\Div(\\nabla_{\\mathfrak{a}(y)}u)$ is a special form of $\\Div (A(x)\\nabla u)$. Based on previous work, we suspect whether \\eqref{2} has a similar solution to that constructed in \\cite{Wei_Yang2021,Wei_Yang2008}.\n\nOur motivation for writing this paper is twofold. First, we plan to construct solutions to \\eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \\eqref{2}.\n\nLet $\\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \\eqref{2} if and only if $t^{-\\frac1p}u$ is the solution of the following problem\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nThen we get the following theorem.\n\nOur motivation for writing this paper is twofold. First, we plan to construct solutions to \\eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \\eqref{2}.\n\nMultiplying the both sides of \\eqref{e71} by $w_{ji,x}$ and $Z_{ji}$, respectively and integrating, we get\n\\[a_{11}\\tilde{\\alpha}^{1-p}\\Lambda_1''=\\int_\\R hw_{ji,x}dx+c_{ji}\\int_\\R w_x^2dx\\]\nand\n\\[a_{11}\\tilde{\\alpha}^{1-p}\\Lambda_2''+\\lambda_0 \\Lambda_2=\\int_\\R hZ_{ji}dx+d_{ji}.\\]\nThen we get\n\\[\\|c_{ji}\\|_{L^2(0,1/\\varepsilon)}\\le C\\|\\Lambda_1\\|_{H^2(0,1/\\varepsilon)}+\\|h\\|_{L^2(\\mathfrak{S})}\\]\nand\n\\[\\|d_{ji}\\|_{L^2(0,1/\\varepsilon)}\\le C\\|\\Lambda_2\\|_{H^2(0,1/\\varepsilon)}+\\|h\\|_{L^2(\\mathfrak{S})}.\\]\nFrom this fact, we have\n\\begin{eqnarray*}\n&&\\|c_{j2}(w_{j1,x}-w_{j2,x})+d_{j2}(Z_{j1}-Z_{j2})\\|_{L^2(\\mathfrak{S})} \\\\\n&\\le& \\left[\\|c_{ji}\\|_{L^2(0,1/\\varepsilon)}+\\|d_{ji}\\|_{L^2(0,1/\\varepsilon)}\\right] \\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(0,1)} \\\\\n&\\le& C\\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(0,1)}\\left[\\|h\\|_{L^2(\\mathfrak{S})}+\\|\\Lambda_1\\|_{H^2(0, 1/\\varepsilon)}+\\|\\Lambda_2\\|_{H^2(0, 1/\\varepsilon)}\\right].\n\\end{eqnarray*}\nDenote\n\\[\\tilde{\\Lambda}_1(z)=-\\int_\\R \\phi_2(z,x)(w_{j1,x}-w_{j2,x})dx, \\quad \\mathrm{and} \\quad \\tilde{\\Lambda}_2(z)=-\\int_\\R \\phi_2(z,x)(Z_{j1}-Z_{j2})dx\\]\nFrom direct computation, we get\n\\[\\|\\tilde{\\Lambda}_1\\|_{H^2(0,1/\\varepsilon)}+\\|\\tilde{\\Lambda}_2\\|_{H^2(0,1/\\varepsilon)}\\le C\\|\\phi_2\\|_{H^2(\\mathfrak{S})}\\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(0,1)}\\]\nand\n\\[\\|p[|w_{j1}-1|^{p-2}(w_{j1}-1)-|w_{j2}-1|^{p-2}(w_{j2}-1)]\\phi_2\\|_{L^2(\\mathfrak{S})}\\le C\\|f_1-f_2\\|_{H^2(0,1)}\\|\\phi_2\\|_{H^2(\\mathfrak{S})}.\\]\nFrom the priori estimate in \\eqref{e21}, we arrive at\n\\[\\|\\phi^*\\|_{H^2(S)}\\le C\\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(0,1)}\\left[\\|h\\|_{L^2(\\mathfrak{S})}+\\|\\Lambda_1\\|_{H^2(0, 1/\\varepsilon)}+\\|\\Lambda_2\\|_{H^2(0, 1/\\varepsilon)}\\right].\\]\nHence \\eqref{e72} holds.\n\\end{proof}\n\\begin{proposition}\\label{pre1}\nProblem \\eqref{48} has a unique solution $\\phi=\\tilde{T}(h)$, where $\\tilde{T}$ is a linear operator and $\\phi$ satisfies the following estimate\n\\begin{equation}\\label{e74}\n\\|\\phi\\|_{H^2(\\mathfrak{S})}\\leq C\\|h\\|_{L^2(\\mathfrak{S})}.\n\\end{equation}\nMoreover, $\\tilde{T}$ is Lipschitz continuous on $\\mathbf{f}$ and $\\mathbf{e}$:\n\\begin{equation}\\label{e73}\n\\|\\tilde{T}_{\\mathbf{f}_1,\\mathbf{e}_1}-\\tilde{T}_{\\mathbf{f}_2,\\mathbf{e}_2}\\|\\le C\\left[ \\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(\\mathfrak{S})}+\\|\\mathbf{e}_1-\\mathbf{e}_2\\|_*\\right].\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nAccording to the argument above, we only need to solve the system \\eqref{50}-\\eqref{52} and the problem \\eqref{54}.\nWith Lemma \\ref{lme4}, we first consider problem \\eqref{50}-\\eqref{52} for fixed function $\\tilde{\\psi}$. It is written into the following problem\n\\begin{eqnarray}\\label{e22}\n\\nonumber \\tilde{\\phi}_j&=&T_j\\left[\\tilde{\\chi}_j h+p\\left[|w_j-1|^{p-2}(w_j-1)-|\\mathcal{V}-1|^{p-2}(\\mathcal{V}-1)\\right]\\tilde{\\chi}_j \\tilde{\\phi}_j \\right. \\\\ &&\\left.-p\\left[|\\mathcal{V}-1|^{p-2}(\\mathcal{V}-1)+1\\right]\\tilde{\\chi}_j\\tilde{\\psi}, \\Lambda_{j1}, \\Lambda_{j2}\\right], \\qquad j=1,2,\\cdots, N.\n\\end{eqnarray}\nFrom tedious computation and Sobolev imbbeding theorem in \\cite[Theorem 8.8]{Brezis2011}, we get\n\\[\\|\\Lambda_{j1}\\|_{H^2(0,1/\\varepsilon)}+\\|\\Lambda_{j2}\\|_{H^2(0,1/\\varepsilon)}\\leq C\\varepsilon^{1/2}\\sum_{k=1}^N\\|\\tilde{\\phi}_k\\|_{H^2(\\mathfrak{S})}+C\\|\\tilde{\\psi}\\|_{H^2(\\mathfrak{S})},\\]\n\\[\\|p[|\\mathcal{V}-1|^{p-2}(\\mathcal{V}-1)+1]\\tilde{\\chi}_j\\tilde{\\psi}\\|_{L^2(\\mathfrak{S})}\\le C\\|\\tilde{\\psi}\\|_{H^2(\\mathfrak{S})}\\]\nand\n\\[\\|p\\left[|w_j-1|^{p-2}(w_j-1)-|\\mathcal{V}-1|^{p-2}(\\mathcal{V}-1)\\right]\\tilde{\\chi}_j \\tilde{\\phi}_j\\|_{L^2(\\mathfrak{S})}\\leq C\\varepsilon^{1/2}\\|\\tilde{\\phi}_j\\|_{H^2(\\mathfrak{S})}.\\]\nUsing the fixed point theorem, we get \\eqref{e22} has a unique solution $\\tilde{\\phi}_j=\\tilde{\\phi}_j(\\tilde{\\psi})$, $j=1,2,\\cdots,N$. Moreover $\\tilde{\\phi}_j$ satisfies the following estimate\n\\begin{equation}\\label{e24}\n\\|\\tilde{\\phi}_j\\|_{H^2(\\mathfrak{S})}\\leq C\\left[\\|h\\|_{L^2(\\mathfrak{S})}+\\|\\tilde{\\psi}\\|_{H^2(\\mathfrak{S})}\\right].\n\\end{equation}\nIt is easy to get $\\phi_j$ is Lipschitz dependent on $\\psi$:\n\\begin{equation}\\label{e70}\n\\|\\tilde{\\phi}_j(\\tilde{\\psi}_1)-\\tilde{\\phi}_j(\\tilde{\\psi}_2)\\|_{H^2(\\mathfrak{S})}\\le C\\|\\tilde{\\psi}_1-\\tilde{\\psi}_2\\|_{H^2(\\mathfrak{S})}.\n\\end{equation}\n\nFor the other terms in \\eqref{77}, we define\n\\[\\Lambda_1=\\int_\\R p\\eta_\\delta^\\varepsilon \\left[|\\mathcal{V}-\\tilde{\\alpha}^{-1}\\bar{\\mathbf{q}}|^{p-2}(\\mathcal{V}-\\tilde{\\alpha}^{-1}\\bar{\\mathbf{q}})+|\\tilde{\\alpha}^{-1}\\bar{\\mathbf{q}}|^{p-2}(\\tilde{\\alpha}^{-1}\\bar{\\mathbf{q}}) \\right]\\hat{\\psi}( \\hat{\\varphi})w_{j,x}dx\\]\nand\n\\[\\Lambda_2=\\int_\\R \\eta_\\delta^\\varepsilon \\hat{N}(\\hat{\\varphi}+\\hat{\\psi}(\\hat{\\varphi}))w_{j,x}dx.\\]\nFrom direct computation, we get\n\\[\\|\\Lambda_1\\|_{L^2(0,1)}\\leq C\\|\\psi(\\hat{\\varphi})\\|_{L^\\infty }\\le C(\\varepsilon^{3/2-\\mu})^{\\min\\{p,2\\}}\\le C\\varepsilon^{\\frac52+\\mu_1}\\]\nand\n\\[\\|\\Lambda_2\\|_{L^2(0,1)}\\leq C\\varepsilon^{1/2}(\\varepsilon^{3/2-\\mu})^{\\min\\{p,2\\}}+(\\varepsilon^{3/2-\\mu})^{2\\min\\{p,2\\}}\\le C\\varepsilon^{\\frac52+\\mu_1}.\\]\\\nFrom \\eqref{e94}, we rewrite \\eqref{77} into the following form\n\\begin{eqnarray}\\label{73}\n\\nonumber&&\\varepsilon^2\\alpha^{1-p}\\beta\\left\\{-a_{11}f_j''+\\left[a_{22}-b_{11}-a_{11}\\left(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha'\\right)\\right]f_j'+\\left[a_{22}(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha') +b_{22}-a_{33}\\right]f_j\\right. \\\\\n\\nonumber&&+\\left[\\frac{p+3}2\\alpha^{p-2}\\beta^{-2} \\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p}\\beta^2(a_{32})^2-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac2{p+3}\\alpha^{1-p}\\beta^2 (b_{21})^2\\right]f_j \\\\\n\\nonumber&&\\left.+\\left[-2\\varepsilon a_{11}e_j' f_j'+2\\varepsilon a_{22}e_j' f_j\\right]\\left(\\int_\\R w_x^2 dx\\right)^{-1}\\int_\\R Z_x w_xdx\\right\\} \\\\\n&&C_0 p\\alpha_p\\left[e^{-\\sqrt{p}\\beta(f_j-f_{j-1})}-e^{-\\sqrt{p}\\beta (f_{j+1}-f_j)}\\right]=\\varepsilon^2 M_{1j\\varepsilon}.\n\\end{eqnarray}\nUsing the same procedure and \\eqref{e95}, we get \\eqref{82} is written into the following form\n\\begin{equation}\\label{e80}\n\\varepsilon(\\varepsilon^2a_{11}\\alpha^{1-p}e_j''+\\lambda_0 e_j) +p\\alpha_p C_1\\left[e^{-\\sqrt{p}\\beta(f_j-f_{j-1})}+e^{-\\sqrt{p}\\beta(f_{j+1}-f_j)}\\right]=\\varepsilon^2 M_{2j\\varepsilon}.\n\\end{equation}\nIn the expression \\eqref{73} and \\eqref{e80}, $M_{ij\\varepsilon}=A_{ij\\varepsilon}+K_{ij\\varepsilon}$, $i=1,2$, where $K_{ij\\varepsilon}$ is a compact operator and $A_{ij\\varepsilon}$ is a Lipschitz operator. It satisfies the following estimates\n\\[\\|A_{ij\\varepsilon}(\\mathbf{f}_1,\\mathbf{e}_1)-A_{ij\\varepsilon}(\\mathbf{f}_2,\\mathbf{e}_2)\\|_{L^2(0,1)}\\leq C\\varepsilon^{\\frac12+\\mu_1}\\left[\\|\\mathbf{f}_1-\\mathbf{f}_2\\|_{H^2(0,1)}+\\|\\mathbf{e}_1-\\mathbf{e}_2\\|_*\\right]\\]\nand\n\\begin{equation}\\label{e49}\n\\|A_{ij\\varepsilon}\\|_{L^2(0,1)}\\leq C\\varepsilon^{\\frac12+\\mu_1}, \\qquad \\|K_{ij\\varepsilon}\\|_{L^2(0,1)}\\leq C\\varepsilon^{\\frac12+\\mu_1}.\n\\end{equation}\n\\section{Proof of Theorem}\\label{sect8}\nTo prove our main theorem, we only need to find a solution to \\eqref{73}-\\eqref{e80}. To do job, we need some priori estimate.",
    "post_theorem_intro_text_len": 2419,
    "post_theorem_intro_text": "\\begin{remark}\n\nThe condition that  $\\Gamma$ is a non-degenerate critical point of the functional $\\mathcal{K}(\\Gamma)$ is equivalent to the condition that $\\Gamma$ is a non-degenerate geodesic embedded into the Riemannian manifold $(\\mathbb R^2,\\tilde{\\mathfrak{g}})$, where $\\tilde{\\mathfrak{g}}(y)=\\mathbf{\\Psi}^{\\frac{p+3}p}[A_{22}dy_1^2-2A_{12}(y)dy_1dy_2+A_{11}dy_2^2]$.\n\\end{remark}\nWe will use the infinite dimensional reduction method developed in \\cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to prove Theorem \\ref{th1}. Infinite dimensional reduction method is used to find solutions to elliptic partial differential equations concentrating on high dimensional sets. To construct a solution to \\eqref{1}, we first investigate the negative solution of \\eqref{1} and its property in Section \\ref{sectA}. However, we need to overcome the difficulty that $-\\Div (A(x)\\nabla u)$ is not a symmetric operator( see Lemma \\ref{lm1} for detail).  To get a local approximate solution to \\eqref{1}, we need to expand the operator $\\Div(A(\\varepsilon y)\\nabla v)$ in a proper way. Wei and Yang \\cite{Wei_Yang2021} developed a method to expand $\\Div(\\nabla_{\\mathfrak{a}}(y)u)$. However we find a easier way to expand this operator. Similar to \\cite{Wei_Yang2021}, to construct local approximate solution with $N$ concentration layers, we need a fine asymptotic estimate of the function $w$. But $w$ does not have the explicit expression(comparing \\cite[(3.1)]{Wei_Yang2021}). To overcome this difficulty, we use the method in \\cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$ in the Section \\ref{sectB}.\n\nThis paper is organized as follows. In Section \\ref{sect1}, we set up a new modified Fermi coordinate in some neighborhood of $\\Gamma$ and write down a local form of the operator $\\Div (A(\\varepsilon y)\\nabla u)$ in the stretched modified Fermi coordinate. In Section \\ref{sect2}, we find a local approximate solution to \\eqref{1}. Then we conduct the gluing procedure in Section \\ref{sect5} and the outer problem is solved. To solve the inner problem, we construct linear and nonlinear theory in Section \\ref{sect6} and Section \\ref{sect9}, respectively. From some involved calculation in Section \\ref{sect4}, we reduce the problem into some partial differential equations in Section \\ref{sect7}.  At last, Theorem \\ref{th1} is proven in Section \\ref{sect8}.",
    "sketch": "To prove Theorem~\\ref{th1}, the authors \"use the infinite dimensional reduction method\" from \\cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of \\eqref{1} concentrating on (high-dimensional) sets.\n\nKey ingredients/steps described in the introduction:\n\\begin{itemize}\n\\item \"First investigate the negative solution of \\eqref{1} and its property\" (Section~\\ref{sectA}), while overcoming that $-\\Div(A(x)\\nabla u)$ \"is not a symmetric operator\" (Lemma~\\ref{lm1}).\n\\item To build a local approximation, one must \"expand the operator $\\Div(A(\\varepsilon y)\\nabla v)$ in a proper way\"; although Wei--Yang \\cite{Wei_Yang2021} expand $\\Div(\\nabla_{\\mathfrak{a}}(y)u)$, the authors state they \"find a easier way to expand this operator.\"\n\\item As in \\cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires \"a fine asymptotic estimate of the function $w$\"; since $w$ has no explicit formula, they \"use the method in \\cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$\" (Section~\\ref{sectB}).\n\\item Paper roadmap implementing the reduction/gluing scheme:\n  \\begin{itemize}\n  \\item Section~\\ref{sect1}: set up a \"new modified Fermi coordinate\" near $\\Gamma$ and derive a local form of $\\Div(A(\\varepsilon y)\\nabla u)$ in stretched coordinates.\n  \\item Section~\\ref{sect2}: construct a local approximate solution to \\eqref{1}.\n  \\item Section~\\ref{sect5}: perform the \"gluing procedure\" and solve the outer problem.\n  \\item Sections~\\ref{sect6} and \\ref{sect9}: develop linear and nonlinear theory to solve the inner problem.\n  \\item Using \"involved calculation\" in Section~\\ref{sect4}, \"reduce the problem into some partial differential equations\" in Section~\\ref{sect7}.\n  \\item \"At last, Theorem~\\ref{th1} is proven in Section~\\ref{sect8}.\"\n  \\end{itemize}\n\\end{itemize}",
    "expanded_sketch": "To prove the main theorem, the authors “use the infinite dimensional reduction method” from \\cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nconcentrating on (high-dimensional) sets.\n\nKey ingredients/steps described in the introduction:\n\\begin{itemize}\n\\item First they “investigate the negative solution” of the boundary value problem\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nand its properties (later in the paper), while overcoming that $-\\Div(A(x)\\nabla u)$ “is not a symmetric operator.” In particular, they use the following auxiliary lemma.\n\n\\begin{lemma}\\label{lm1}\nIf $a\\ge c$ and $u_\\varepsilon$ is the minimizer of problem \\eqref{6},  $u_\\varepsilon(x)\\to a$ on compact sets in $B_1(0)$, as $\\varepsilon \\to 0$.\n\\end{lemma}\n\n\\item To build a local approximation, one must “expand the operator $\\Div(A(\\varepsilon y)\\nabla v)$ in a proper way”; although Wei--Yang \\cite{Wei_Yang2021} expand $\\Div(\\nabla_{\\mathfrak{a}}(y)u)$, the authors state they “find a easier way to expand this operator.”\n\n\\item As in \\cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires “a fine asymptotic estimate of the function $w$”; since $w$ has no explicit formula, they “use the method in \\cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$” (later in the paper).\n\n\\item Paper roadmap implementing the reduction/gluing scheme:\n  \\begin{itemize}\n  \\item Next they set up a “new modified Fermi coordinate” near $\\Gamma$ and derive a local form of $\\Div(A(\\varepsilon y)\\nabla u)$ in stretched coordinates.\n  \\item Next they construct a local approximate solution to the equation above.\n  \\item Next they perform the “gluing procedure” and solve the outer problem.\n  \\item Later they develop linear and nonlinear theory to solve the inner problem.\n  \\item Using “involved calculation” earlier, they “reduce the problem into some partial differential equations” later.\n  \\item This completes the proof of the main theorem.\n  \\end{itemize}\n\\end{itemize}",
    "expanded_theorem": "\\label{th1}\nLet $p>3$.\nAssume that $\\Gamma$ is a simple closed smooth curve with unit length in $\\Omega$, and it is also a non-degenerate critical point of the functional\n\\begin{equation}\\label{e91}\n\\mathcal{K}(\\Gamma)=\\int_\\Gamma \\mathbf{\\Psi}^{\\frac{p+3}{2p}}dvol_{\\mathfrak{g}},\n\\end{equation}\nwhere $\\mathfrak{g}(X,Y)=\\langle A^*X,Y\\rangle$ is a Riemannian metric on $\\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\\Gamma$:\n\\begin{eqnarray*}\n\\Upsilon_0&=&-\\alpha^{1-p}\\left[ 2\\alpha^{-1}\\alpha'a_{22}+\\beta^{-1}\\beta'b_{11}+b_{22}-a_{33}+\\frac{p+3}2\\alpha^{p-2}\\beta^{-2}\\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p} \\beta^2 (a_{32})^2\\right. \\\\\n&&\\left.-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac{2}{p+3}\\alpha^{1-p}\\beta^2(b_{21})^2 +a_{11}\\left(\\beta^{-1}\\beta''+2\\alpha^{-1}\\beta^{-1}\\alpha'\\beta'-\\beta^{-2}(\\beta')^2\\right)\\right]>0,\n\\end{eqnarray*}\nwhere the coefficients $a_{ij}$ and $b_{ij}$ are given by the following lemma.\n\n\\begin{lemma}\\label{lm8}\nUnder the stretched modified Fermi coordinate defined by $\\Phi_\\varepsilon(z,s)$, we get\n\\begin{eqnarray*}\n\\nonumber\\Div(A(\\varepsilon y)\\nabla v)&=& a_{11}(\\varepsilon z)v_{zz}+2\\varepsilon s a_{22}(\\varepsilon z)v_{sz} +\\left(a_{31}(\\varepsilon z)+\\varepsilon s a_{32}(\\varepsilon z) +\\varepsilon^2 s^2 a_{33}(\\varepsilon z)\\right)v_{ss} \\\\\n&& +\\varepsilon b_{11}(\\varepsilon z)v_z +\\left(\\varepsilon b_{21}(\\varepsilon z)+\\varepsilon^2 s b_{22}(\\varepsilon z)\\right) v_s +B_0(v),\n\\end{eqnarray*}\nwhere\n\\[a_{11}(\\theta)=\\frac{\\langle{A^*}n,n\\rangle}{1-\\langle\\gamma',n\\rangle^2},\\qquad a_{22}(\\theta)=-\\frac{\\langle{A^*}n,n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{\\langle{A^*_t}n,\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}, \\]\n\\[a_{31}(\\theta)=\\frac{\\langle{A^*}\\gamma',\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}, \\qquad a_{32}=\\frac{\\langle{A^*_t}\\gamma',\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2}+ \\frac{2\\langle{A^*}\\gamma',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{2\\langle\\gamma',n'\\rangle\\langle{A^*}\\gamma',\\gamma'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}, \\]\n\\begin{eqnarray*}\na_{33}(\\theta)&=& \\frac1{1-\\langle\\gamma',n\\rangle^2}\\left(\\langle{A^*}n',n'\\rangle +2\\langle{A^*_t}\\gamma',n'\\rangle+\\frac12\\langle{A^*_{tt}}\\gamma',\\gamma'\\rangle\\right)  \\\\\n&&-\\frac{2\\langle\\gamma',n'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}\\left(\\langle{A^*_t} \\gamma',\\gamma'\\rangle +2\\langle{A^*}\\gamma',n'\\rangle\\right)  \\\\\n&&+\\left(\\frac{4\\langle\\gamma',n'\\rangle^2}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^3}- \\frac{\\langle n',n'\\rangle}{\\left(1-\\langle\\gamma',n\\rangle^2\\right)^2}\\right)\\langle{A^*}\\gamma',\\gamma'\\rangle,\n\\end{eqnarray*}\n\\[b_{11}(\\theta)=\\left[\\frac{\\langle{A^*}n,n\\rangle}{1-\\langle\\gamma',n\\rangle^2}\\right]_{\\theta}-\\frac12\\left(\\frac1{1-\\langle\\gamma',n\\rangle^2}\\right)_\\theta \\langle{A^*}n,n\\rangle -\\frac{\\langle{A^*}n,n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{\\langle{A^*_t}n,\\gamma'\\rangle}{1-\\langle\\gamma',n\\rangle^2},\\]\n\\[b_{21}(\\theta)=\\frac{\\langle\\gamma',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}a_{31}+a_{32},\\]\n\\[b_{22}(\\theta)=\\frac12\\frac{(1-\\langle\\gamma',n\\rangle^2)_\\theta}{1-\\langle\\gamma',n\\rangle^2}a_{22}+\\left[\\frac{\\langle n',n'\\rangle}{1-\\langle\\gamma',n\\rangle^2}-\\frac{2\\langle\\gamma',n'\\rangle^2}{(1-\\langle\\gamma',n\\rangle^2)^2}\\right]a_{31}+\\frac{\\langle\\gamma',n'\\rangle}{1-\\langle\\gamma', n\\rangle^2}a_{32} +\\partial_\\theta a_{22}+2a_{33}\\]\nand\n\\[B_0(v)=\\varepsilon a_{12}(\\varepsilon z,s)v_{zz}+\\varepsilon^2 a_{23}(\\varepsilon z.s)v_{zs}+\\varepsilon^3a_{33}(\\varepsilon z,s)v_{ss} +\\varepsilon^2 b_{12}(\\varepsilon z,s) v_z +\\varepsilon^3 b_{23}(\\varepsilon z,s)v_s.\\]\nIn the expression above, funtions $a_{12}$, $a_{23}$, $a_{33}$, $b_{12}$ and $b_{23}$ are smooth functions satisfying the following estimate\n\\[|a_{12}(\\varepsilon z, s)|\\le C(1+|s|), \\quad |a_{23}(\\varepsilon z, s)|\\le C(1+|s|^2), \\quad |a_{33}(\\varepsilon z, s)|\\le C(1+|s|^3),\\]\nand\n\\[|b_{12}(\\varepsilon z, s)|\\le C(1+|s|), \\quad |b_{23}(\\varepsilon z, s)|\\le C(1+|s|^2).\\]\n\\end{lemma}\n\nMoreover,\n\\begin{equation}\\label{80}\n\\alpha(\\theta)=\\mathbf{q}(\\theta,0), \\qquad \\beta(\\theta)=\\frac{\\left[\\mathbf{q}(\\theta,0)\\right]^{\\frac{p-1}2}}{\\left[a_{31}(\\theta)\\right]^{\\frac12}}.\n\\end{equation}\nand\n\\begin{equation}\\label{e88}\n\\mathbf{q}(\\theta, t)=\\mathbf{\\Psi}^{\\frac1p}(\\gamma(\\theta)+tn(\\theta)).\n\\end{equation}\n\nThen for each  integer $N>0$, there exists a sequence of $\\varepsilon$, \\textit{i.e.}, $\\{\\varepsilon_l\\}$ converging to $0$ such that\n\\begin{equation}\\label{1}\n\\left\\{\\begin{array}{ll}\n-\\varepsilon^2\\Div (A(x)\\nabla u)=|u|^p-\\mathbf{\\Psi}(x), &\\mbox{in $\\Omega$,} \\\\\nu=0, & \\mbox{on $\\partial \\Omega$},\n\\end{array}\n\\right.\n\\end{equation}\nhas a positive solutions $u_{\\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\\varepsilon_l|\\log\\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\\Gamma$ at speed $\\varepsilon_l^{1+\\mu}$ for small positive constant $\\mu\\in(0,1/2)$. More precisely $u_{\\varepsilon_l}$ has the form\n\\begin{eqnarray*}\nu_{\\varepsilon_l}(y_1,y_2)\\approx -\\mathbf{\\Psi}^{\\frac1p}(y_1,y_2)+ \\mathbf{\\Psi}^{\\frac1p}(\\gamma(\\theta))\\sum_{k=1}^N w\\left(\\left[\\frac{\\mathbf{\\Psi}^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}{\\langle A^*n,n\\rangle}\\right]^{\\frac12}\\frac{t-\\varepsilon_l f_k}{\\varepsilon_l}\\right),\n\\end{eqnarray*}\nwhere $\\gamma$ is a natural parametrization of $\\Gamma$, $n$ is the unit vector defined by\n\\begin{equation}\\label{e102}\nn(\\theta)=\\frac{A(\\gamma(\\theta))\\nu(\\theta)}{|A(\\gamma(\\theta))\\nu(\\theta)|}.\n\\end{equation}\nand $w$ is  the unique solution of the following problem\n\\begin{equation}\\label{29}\n-w''=|1-w|^p -1, \\qquad w>0 \\quad \\mathrm{in} \\quad \\mathbb R, \\qquad w'(0)=w(\\pm \\infty).\n\\end{equation}\nIn the expression above, the functions $f_j$'s satisfy\n\\[\\|f_j\\|_{L^\\infty(0,1)}\\le C|\\log\\varepsilon_l|^2,\\qquad \\sum_{j=1}^N f_j =O(\\frac1{|\\log\\varepsilon_l|^{\\frac32}}),\\]\n\\[\\min_{1\\le j\\le N}(f_{j+1}-f_j)\\approx\\frac2{\\sqrt{p}}|\\log\\varepsilon_l|\\left[\\frac{\\langle A^*n,n\\rangle}{\\Psi^{\\frac{p-1}p}(1-\\langle\\gamma',n\\rangle)}\\right]^{\\frac12}\\]\nand solves the Jacobi-Toda system for $j=1,2,\\cdots,N$\n\\begin{eqnarray*}\n\\nonumber&&\\varepsilon^2\\alpha^{1-p}\\beta\\left\\{-a_{11}f_j''+\\left[a_{22}-b_{11}-a_{11}\\left(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha'\\right)\\right]f_j'+\\left[a_{22}(\\beta^{-1}\\beta'+2\\alpha^{-1}\\alpha') +b_{22}-a_{33}\\right]f_j\\right. \\\\\n\\nonumber&&+\\left[\\frac{p+3}2\\alpha^{p-2}\\beta^{-2} \\mathbf{q}_{tt}+\\frac{p+2}{2(p+3)}\\alpha^{1-p}\\beta^2(a_{32})^2-\\frac{p+1}{p+3}\\alpha^{1-p}\\beta^2 a_{32}b_{21}-\\frac2{p+3}\\alpha^{1-p}\\beta^2 (b_{21})^2\\right]f_j \\\\\n&&C_0 p\\alpha_p\\left[e^{-\\sqrt{p}\\beta(f_j-f_{j-1})}-e^{-\\sqrt{p}\\beta (f_{j+1}-f_j)}\\right]\\approx0.\n\\end{eqnarray*}",
    "theorem_type": [
      "Universal–Existential",
      "Existence"
    ],
    "mcq": {
      "question": "Let Ω⊂R2 be a smooth bounded domain, let A(x) be a smooth symmetric positive-definite 2×2 matrix field, and let Ψ be the first Dirichlet eigenfunction of L(u)=-Div(A(x)∇u) on Ω. Let Γ⊂Ω be a simple closed smooth curve of unit length with natural parametrization γ(θ), let ν(θ) be its Euclidean unit normal, and define n(θ)=A(γ(θ))ν(θ)/|A(γ(θ))ν(θ)|, q(θ,t)=Ψ(γ(θ)+tn(θ))^(1/p), α(θ)=q(θ,0), β(θ)=q(θ,0)^((p-1)/2)/a31(θ)^(1/2), and the Riemannian metric g(X,Y)=⟨A*X,Y⟩. Assume p>3, Γ is a nondegenerate critical point of K(Γ)=∫_Γ Ψ^((p+3)/(2p)) dvol_g, and on Γ one has Υ0>0, where Υ0=-α^(1-p)[2α^(-1)α'a22+β^(-1)β'b11+b22-a33+((p+3)/2)α^(p-2)β^(-2)qtt+((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2-((p+1)/(p+3))α^(1-p)β^2 a32 b21-(2/(p+3))α^(1-p)β^2(b21)^2+a11(β^(-1)β''+2α^(-1)β^(-1)α'β'-β^(-2)(β')^2)], and a11,a22,a31,a32,a33,b11,b21,b22 are the coefficients arising from the stretched modified Fermi-coordinate expansion of Div(A(εy)∇v) along Γ. Under these assumptions, which statement holds for the Dirichlet problem -ε2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω?",
      "correct_choice": {
        "label": "A",
        "text": "For each integer N>0, there exists a sequence {ε_l} with ε_l→0 such that the problem -ε_l2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω, has a positive solution u_{ε_l} with exactly N concentration layers at mutual distance O(ε_l|log ε_l|), and the center of mass of the N layers collapses to Γ at speed ε_l^(1+μ) for some small constant μ∈(0,1/2). More precisely, in the normal coordinates y=γ(θ)+tn(θ), one has u_{ε_l}(y1,y2)≈-Ψ(y1,y2)^(1/p)+Ψ(γ(θ))^(1/p)∑_{k=1}^N w(( [Ψ(γ(θ))^((p-1)/p)(1-⟨γ',n⟩)/⟨A*n,n⟩]^(1/2) (t-ε_l f_k)/ε_l )), where w is the unique solution of -w''=|1-w|^p-1, w>0 in R, w'(0)=w(±∞). In addition, the functions f_j satisfy ||f_j||_{L^∞(0,1)}≤C|log ε_l|^2, ∑_{j=1}^N f_j=O(1/|log ε_l|^(3/2)), and min_{1≤j≤N}(f_{j+1}-f_j)≈(2/√p)|log ε_l|[⟨A*n,n⟩/(Ψ^((p-1)/p)(1-⟨γ',n⟩))]^(1/2); moreover, for j=1,2,...,N they solve the Jacobi-Toda system ε_l2α^(1-p)β{ -a11 f_j'' + [a22-b11-a11(β^(-1)β'+2α^(-1)α')]f_j' + [a22(β^(-1)β'+2α^(-1)α')+b22-a33]f_j + [((p+3)/2)α^(p-2)β^(-2)qtt + ((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2 - ((p+1)/(p+3))α^(1-p)β^2 a32 b21 - (2/(p+3))α^(1-p)β^2(b21)^2]f_j } + C0 p α_p [e^(-√p β(f_j-f_{j-1})) - e^(-√p β(f_{j+1}-f_j))]≈0."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists varepsilon_0>0 such that for every integer N>0 and every varepsilon\na0,varepsilon_0), the problem -varepsilon^2Div(A(x)u)=|u|^p-si(x) in 9, u=0 on 99, has a positive solution u_varepsilon with exactly N concentration layers at mutual distance O(varepsilon|log varepsilon|), whose center of mass collapses to at speed varepsilon^(1+) for some small constant (0,1/2). More precisely, in the normal coordinates y=()+tn(), one has u_varepsilon(y_1,y_2)- si(y_1,y_2)^(1/p)+si(())^(1/p)_{k=1}^N w(([ si(())^((p-1)/p)(1-',n)/A*n,n]^(1/2)(t-varepsilon f_k)/varepsilon)), and the functions f_j satisfy the same bounds and JacobiToda system as in option A."
        },
        {
          "label": "C",
          "text": "For each integer N>0, there exists a sequence {\\varepsilon_l} with \\varepsilon_l\\to0 such that the problem -\\varepsilon_l^2\\Div(A(x)\\nabla u)=|u|^p-\\Psi(x) in \\Omega, u=0 on \\partial\\Omega, has a positive solution u_{\\varepsilon_l} with exactly N concentration layers whose center of mass collapses to \\Gamma. More precisely, in the normal coordinates y=\\gamma(\\theta)+tn(\\theta), one has u_{\\varepsilon_l}(y_1,y_2)\\approx-\\Psi(y_1,y_2)^{1/p}+\\Psi(\\gamma(\\theta))^{1/p}\\sum_{k=1}^N w\\!\\left(\\left[\\frac{\\Psi(\\gamma(\\theta))^{(p-1)/p}(1-\\langle\\gamma',n\\rangle)}{\\langle A^*n,n\\rangle}\\right]^{1/2}\\frac{t-\\varepsilon_l f_k}{\\varepsilon_l}\\right),"
        },
        {
          "label": "D",
          "text": "For each integer N>0, there exists a sequence {\\varepsilon_l} with \\varepsilon_l\\to0 such that the problem -\\varepsilon_l^2\\Div(A(x)\\nabla u)=|u|^p-\\Psi(x) in \\Omega, u=0 on \\partial\\Omega, has a positive solution u_{\\varepsilon_l} with exactly N concentration layers at mutual distance O(\\varepsilon_l), and the center of mass of the N layers collapses to \\Gamma at speed \\varepsilon_l^{1+\\mu} for some small constant \\mu\\in(0,1/2). More precisely, in the normal coordinates y=\\gamma(\\theta)+tn(\\theta), one has the same leading-order profile as in option A, while the functions f_j satisfy \\|f_j\\|_{L^\\infty(0,1)}\\le C|\\log\\varepsilon_l|^2, \\sum_{j=1}^N f_j=O(1/|\\log\\varepsilon_l|^{3/2}), and \\min_{1\\le j\\le N}(f_{j+1}-f_j)\\approx \\frac{2}{\\sqrt p}\\left[\\frac{\\langle A^*n,n\\rangle}{\\Psi^{(p-1)/p}(1-\\langle\\gamma',n\\rangle)}\\right]^{1/2}; moreover, for j=1,2,\\dots,N they solve the same Jacobi-Toda system as in option A."
        },
        {
          "label": "E",
          "text": "For each integer N>0, there exists a sequence {\\varepsilon_l} with \\varepsilon_l\\to0 such that the problem -\\varepsilon_l^2\\Div(A(x)\\nabla u)=|u|^p-\\Psi(x) in \\Omega, u=0 on \\partial\\Omega, has a positive solution u_{\\varepsilon_l} with exactly N concentration layers at mutual distance O(\\varepsilon_l|\\log \\varepsilon_l|), and the center of mass of the N layers collapses to \\Gamma at speed \\varepsilon_l^{1+\\mu} for some small constant \\mu\\in(0,1/2). More precisely, in the normal coordinates y=\\gamma(\\theta)+tn(\\theta), one has u_{\\varepsilon_l}(y_1,y_2)\\approx-\\Psi(y_1,y_2)^{1/p}+\\Psi(\\gamma(\\theta))^{1/p}\\sum_{k=1}^N w\\!\\left(\\left[\\frac{\\Psi(\\gamma(\\theta))^{(p-1)/p}(1-\\langle\\gamma',n\\rangle)}{\\langle A^*n,n\\rangle}\\right]^{1/2}\\frac{t-\\varepsilon_l f_k}{\\varepsilon_l}\\right), where w is the unique solution of -w''=|1-w|^p-1, w>0 in \\mathbb R, w'(0)=w(\\pm\\infty). In addition, the functions f_j satisfy the same bounds as in option A, and for j=1,2,\\dots,N they solve the Jacobi-Toda system obtained from option A after replacing the interaction term \\;C_0 p\\alpha_p\\big[e^{-\\sqrt p\\beta(f_j-f_{j-1})}-e^{-\\sqrt p\\beta(f_{j+1}-f_j)}\\big]\\; by \\;C_0 p\\alpha_p\\big[e^{-\\sqrt p\\beta(f_j-f_{j-1})}+e^{-\\sqrt p\\beta(f_{j+1}-f_j)}\\big]\\approx0."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "discrete_sequence_of_epsilons",
          "template_used": "uniformity_effectivity"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped quantitative layer-spacing speed and Jacobi-Toda details",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "logarithmic_layer_spacing",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "sign_structure_of_Toda_interaction",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives assumptions and asks for the resulting existence statement, but it does not itself reveal the conclusion. There is no explicit answer leakage beyond framing the theorem setup."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the stem lists the full hypotheses and asks which conclusion holds. The correct option is basically the theorem conclusion restated in full."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning/attention is needed to distinguish subtle variants such as sequence vs. all small ε, exact spacing scale, and anisotropic geometric factors. However, the task mainly rewards recognition of the exact theorem statement rather than genuine derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrong quantifiers, weakened conclusions, incorrect spacing scale, and incorrect geometric normalization/profile scaling."
      },
      "total_score": 5,
      "overall_assessment": "Technically well-constructed with strong distractors, but it is mostly a direct theorem restatement and therefore only weakly tests generative mathematical reasoning."
    }
  },
  {
    "id": "2512.21603v1",
    "paper_link": "http://arxiv.org/abs/2512.21603v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}",
      "start_pos": 9579,
      "end_pos": 9957,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:Reading": "\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1259,
    "pre_theorem_intro_text": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.",
    "context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.",
    "full_context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\n\\begin{abstract}\nWe study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.\n\\end{abstract}\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\nThe implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.\n\nAn $n\\times n$ skew-symmetrizable matrix $B$ is called\n\\begin{itemize}\n\\item \\emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;\n\\item \\emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\\le 3$ for all $i,j$.\n\\end{itemize}\n\n\\begin{definition}\nA \\emph{$g$-vector seed} is a pair $(C,(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n))$ consisting of the following data:\n\\begin{itemize}\n\\item[$(1)$] $C$ is a $2n\\times n$ integer matrix whose upper part is skew-symmetrizable.\n\\item[$(2)$] $\\mathbf{g}_1,\\ldots,\\mathbf{g}_n\\in\\mathbb Z^n$.\n\\end{itemize}\nThe tuple $(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n)$ is called the \\emph{$g$-vector tuple}.\n\\end{definition}\n\n\\begin{example}\\label{exam:rank 2 g-fan}\nAny nonzero $2 \\times 2$ skew-symmetrizable matrix is given by\n\\[\nB_{b,c}:=\\begin{bmatrix}0 & c \\\\ -b & 0\\end{bmatrix},\n\\]\nwhere $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\\mathcal{F}(B_{b,c})$ is well-known (see e.g. \\cite[Example~1.15]{GHKK18} or \\cite[Section~2]{N24}). In fact, for $bc \\ge 4$, $\\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\\pm}$ of slope $(-bc \\pm \\sqrt{bc(bc-4)})/2c$ (see Figure~\\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\\mathbb{R}^2\\setminus|\\mathcal{F}(B_{b,c})|)\\cup\\{0\\}$. In particular, the lattice point $(-2,b)$ is contained in $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B_{b,c})|$.\n\\end{example}\n\n\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nThe implication $(1)\\Rightarrow(2)$ follows from Theorem~\\ref{thm:Reading}, and $(2)\\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\\Rightarrow(1)$. By Theorem~\\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\\ge 4$ for some subset $I \\subset \\{1,\\ldots,n\\}$. By Proposition~\\ref{prop:rank 2 no lattice pt}, $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B')|\\neq\\emptyset$, which implies that $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B)|\\neq\\emptyset$ by Corollary~\\ref{cor:preserve lattice pt}. This completes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 485,
    "post_theorem_intro_text": "The implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.",
    "sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} \"via categorification using Jacobian algebras.\" A possible approach \"by methods similar to those in \\cite[Proposition~4.9]{A21}\" had been suggested in \\cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof \"using scattering diagrams and their pull-back constructions.\"",
    "expanded_sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known. \\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\nFor skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \\cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”",
    "expanded_theorem": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $B$ be a skew-symmetrizable $n\\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\\mathbb{R}^n$, forming the $g$-fan $\\mathcal{F}(B)$, and $|\\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?",
      "correct_choice": {
        "label": "A",
        "text": "$B$ is of finite type if and only if the $g$-fan $\\mathcal{F}(B)$ is complete, i.e. $|\\mathcal{F}(B)|=\\mathbb{R}^n$; equivalently, if and only if $|\\mathcal{F}(B)|$ contains every lattice point of $\\mathbb{Z}^n$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$B$ is of finite type if and only if the support $|\\mathcal{F}(B)|$ contains every nonzero lattice point of $\\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\\mathcal{F}(B)|$ is the origin."
        },
        {
          "label": "C",
          "text": "If $B$ is of finite type, then the $g$-fan $\\mathcal{F}(B)$ is complete, i.e. $|\\mathcal{F}(B)|=\\mathbb{R}^n$."
        },
        {
          "label": "D",
          "text": "$B$ is of finite type if and only if for every lattice point $\\mathbf{m}\\in\\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\\mathbf{m}\\in |\\mathcal{F}(B')|$."
        },
        {
          "label": "E",
          "text": "$B$ is of finite type if and only if the support $|\\mathcal{F}(B)|$ has nonempty interior in $\\mathbb{R}^n$; equivalently, if and only if $|\\mathcal{F}(B)|$ contains infinitely many lattice points of $\\mathbb{Z}^n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "all_lattice_points_exactness",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "iff_equivalence_dropped_to_forward_implication",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "same_matrix_vs_mutation_equivalent_matrix",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "completeness_replaced_by_nonempty_interior",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 1,
        "justification": "The stem partially leaks the correct direction by explicitly defining completeness as |F(B)| = R^n, which immediately suggests that any equivalent condition should at least imply containment of all lattice points. This makes choice A stand out, though it does not fully settle equivalence."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is not a verbatim restatement, but the correct answer is very close to a weakened reformulation of completeness ('all of R^n' to 'all lattice points in Z^n'). So it avoids full tautology, but only mildly."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is required to distinguish exact equivalence from nearby statements such as 'all but finitely many lattice points,' 'every nonzero lattice point,' or 'density.' However, the explicit completeness condition makes the intended answer comparatively easy to infer."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: B tests finite-exception weakening, C tests omission of the origin, D tests quantifier/mutation confusion, and E tests density versus equality/completeness. These align well with common failure modes."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-constructed MCQ with strong distractors, but the stem gives a substantial hint toward the correct answer and the correct option is close to a reformulation of the stated completeness condition rather than a fully independent conclusion."
    }
  },
  {
    "id": "2512.21603v1",
    "paper_link": "http://arxiv.org/abs/2512.21603v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}",
      "start_pos": 9579,
      "end_pos": 9957,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm:Reading": "\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 1259,
    "pre_theorem_intro_text": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.",
    "context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.",
    "full_context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\n\\begin{abstract}\nWe study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.\n\\end{abstract}\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\nThe implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.\n\nAn $n\\times n$ skew-symmetrizable matrix $B$ is called\n\\begin{itemize}\n\\item \\emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;\n\\item \\emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\\le 3$ for all $i,j$.\n\\end{itemize}\n\n\\begin{definition}\nA \\emph{$g$-vector seed} is a pair $(C,(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n))$ consisting of the following data:\n\\begin{itemize}\n\\item[$(1)$] $C$ is a $2n\\times n$ integer matrix whose upper part is skew-symmetrizable.\n\\item[$(2)$] $\\mathbf{g}_1,\\ldots,\\mathbf{g}_n\\in\\mathbb Z^n$.\n\\end{itemize}\nThe tuple $(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n)$ is called the \\emph{$g$-vector tuple}.\n\\end{definition}\n\n\\begin{example}\\label{exam:rank 2 g-fan}\nAny nonzero $2 \\times 2$ skew-symmetrizable matrix is given by\n\\[\nB_{b,c}:=\\begin{bmatrix}0 & c \\\\ -b & 0\\end{bmatrix},\n\\]\nwhere $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\\mathcal{F}(B_{b,c})$ is well-known (see e.g. \\cite[Example~1.15]{GHKK18} or \\cite[Section~2]{N24}). In fact, for $bc \\ge 4$, $\\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\\pm}$ of slope $(-bc \\pm \\sqrt{bc(bc-4)})/2c$ (see Figure~\\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\\mathbb{R}^2\\setminus|\\mathcal{F}(B_{b,c})|)\\cup\\{0\\}$. In particular, the lattice point $(-2,b)$ is contained in $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B_{b,c})|$.\n\\end{example}\n\n\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nThe implication $(1)\\Rightarrow(2)$ follows from Theorem~\\ref{thm:Reading}, and $(2)\\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\\Rightarrow(1)$. By Theorem~\\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\\ge 4$ for some subset $I \\subset \\{1,\\ldots,n\\}$. By Proposition~\\ref{prop:rank 2 no lattice pt}, $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B')|\\neq\\emptyset$, which implies that $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B)|\\neq\\emptyset$ by Corollary~\\ref{cor:preserve lattice pt}. This completes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 485,
    "post_theorem_intro_text": "The implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.",
    "sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} \"via categorification using Jacobian algebras.\" A possible approach \"by methods similar to those in \\cite[Proposition~4.9]{A21}\" had been suggested in \\cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof \"using scattering diagrams and their pull-back constructions.\"",
    "expanded_sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known. \\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\nFor skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \\cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”",
    "expanded_theorem": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $B$ be a skew-symmetrizable $n\\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\\mathbb{R}^n$, forming the $g$-fan $\\mathcal{F}(B)$, and $|\\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?",
      "correct_choice": {
        "label": "A",
        "text": "$B$ is of finite type if and only if the $g$-fan $\\mathcal{F}(B)$ is complete, i.e. $|\\mathcal{F}(B)|=\\mathbb{R}^n$; equivalently, if and only if $|\\mathcal{F}(B)|$ contains every lattice point of $\\mathbb{Z}^n$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$B$ is of finite type if and only if the support $|\\mathcal{F}(B)|$ contains every nonzero lattice point of $\\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\\mathcal{F}(B)|$ is the origin."
        },
        {
          "label": "C",
          "text": "If $B$ is of finite type, then the $g$-fan $\\mathcal{F}(B)$ is complete, i.e. $|\\mathcal{F}(B)|=\\mathbb{R}^n$."
        },
        {
          "label": "D",
          "text": "$B$ is of finite type if and only if for every lattice point $\\mathbf{m}\\in\\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\\mathbf{m}\\in |\\mathcal{F}(B')|$."
        },
        {
          "label": "E",
          "text": "$B$ is of finite type if and only if the support $|\\mathcal{F}(B)|$ has nonempty interior in $\\mathbb{R}^n$; equivalently, if and only if $|\\mathcal{F}(B)|$ contains infinitely many lattice points of $\\mathbb{Z}^n$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "all_lattice_points_exactness",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "iff_equivalence_dropped_to_forward_implication",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "same_matrix_vs_mutation_equivalent_matrix",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "completeness_replaced_by_nonempty_interior",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the characterization. It defines the objects involved and asks for the exact finite-type criterion, but the correct equivalence is not leaked."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is quite close to a direct theorem-recall question: it asks for the exact characterization of finite type in terms of the g-fan. However, it is not pure verbatim restatement because the options introduce nearby variants and weakened/altered formulations."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish the exact iff statement from a weaker true implication and from quantifier/property confusions. Still, the task is mainly recognition of a known theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: one is a weaker true statement, and the others reflect plausible mathematical errors involving lattice-point coverage, mutation-equivalence, and confusing completeness with nonempty interior. They are distinct and diagnostically useful."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it primarily tests precise recall/recognition rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.21611v1",
    "paper_link": "http://arxiv.org/abs/2512.21611v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Thm}\nLet $\\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\\frac{1}{2}, t)$-pair of $\\Ga$ with $t\\geq1$, and let $K=\\mathrm{Core}_H(M)$. Take $u\\in V(\\Ga)$. Then one of the followings holds.\n\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}\n\\end{enumerate}\nMoreover, examples for each of the above cases exist.",
      "start_pos": 14771,
      "end_pos": 16008,
      "label": "Thm"
    },
    "ref_dict": {
      "Thmcase:t=1": "\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}",
      "Thmcase:t=1_cycle": "\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}",
      "Thmcase:t=3": "\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}",
      "que:Sparl": "\\begin{question}[{\\cite[Question~5.8]{RSparl2019}}]\\label{que:Sparl}\nIs it true that, if $\\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\\leq\\Aut(\\Ga)$, the graphs $\\Alt_M(\\Ga)$ and $\\Ga_{\\mathcal{B}(M)}$ are both arc-transitive?\n\\end{question}",
      "Thmcase:t=2": "\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}",
      "prob": "\\begin{problem}\\label{prob}\nClassify maximal $(s,t)$-pairs for all $s\\in\\mathbb{N}\\cup\\{\\tfrac{1}{2}\\}$ and $t\\in\\mathbb{N}$ with $s<t$.\n\\end{problem}",
      "Thm": "\\begin{theorem}\\label{Thm}\nLet $\\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\\frac{1}{2}, t)$-pair of $\\Ga$ with $t\\geq1$, and let $K=\\mathrm{Core}_H(M)$. Take $u\\in V(\\Ga)$. Then one of the followings holds.\n\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}\n\\end{enumerate}\nMoreover, examples for each of the above cases exist.\n\\end{theorem}",
      "Thmcase:t=1_cover": "\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}"
    },
    "pre_theorem_intro_text_len": 7279,
    "pre_theorem_intro_text": "All graphs considered in this paper are finite, undirected, regular and simple. Let $\\Ga$ be a graph. For a positive integer $s$, a sequence $(u_0, u_1, \\ldots, u_s)$ of vertices of $\\Ga$ is said to be an \\emph{$s$-arc} if $u_i$ is adjacent to $u_{i+1}$ for $0\\leq i\\leq s-1$ and $u_{j-1}\\neq u_{j+1}$ for $1\\leq j\\leq s-1$. A $1$-arc is simply an \\emph{arc}. We use $V(\\Gamma)$, $E(\\Gamma)$, $A(\\Ga)$ and $\\Aut(\\Gamma)$ to denote its vertex set, edge set, arc set and (full) automorphism group, respectively.\n\nLet $G$ be a subgroup of $\\Aut(\\Ga)$. If $G$ is transitive on $V(\\Ga)$ and on $E(\\Ga)$ but not on $A(\\Ga)$, then we say that $\\Ga$ is \\emph{ $G$-half-arc-transitive}. In the literature, ``half-arc-transitive'' is also abbreviated as ``$\\frac{1}{2}$-arc-transitive'' or ``HAT''. When $\\Ga$ is $\\Aut(\\Ga)$-half-arc-transitive, we omit the prefix and simply say that $\\Ga$ is half-arc-transitive (or HAT).\nThe study of HAT graphs was initiated by Tutte in \\cite{Tutte1966}. Over the past half-century, substantial progress has been made, and we refer the reader to the survey articles \\cite{CPS2015,Marusic1998} for an overview. Within this broader development, considerable attention has been devoted to tetravalent HAT graphs; see, for example, \\cite{M-DM,MM1999,MN-JGT,MP1999,MX1997,PW-JCTB2007,RSparl2019,Spiga-Xia2021,Xia-2021,Zhou-JACO,Zhou-PAMS}.\n\nFor a positive integer $t$, we say that $\\Ga$ is \\emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \\emph{$t$-arc-transitive graph}.\nFor $s\\in\\mathbb{N}\\cup\\{\\frac{1}{2}\\}$, if $\\Ga$ is $(G,s)$-arc-transitive but not $(G,\\lfloor s+1\\rfloor)$-arc-transitive, then we say that $\\Ga$ is \\emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\\Aut(\\Ga)$ is called an \\emph{$(s,t)$-pair} of $\\Ga$ if $M\\leq H\\leq\\Aut(\\Ga)$ and $\\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.\nSuch a pair $(M, H)$ is called \\emph{maximal} if $M$ is a maximal subgroup of $H$.\nNote that, $\\Gamma$ admits a maximal $\\bigl(\\frac{1}{2},t\\bigr)$-pair for some positive integer $t$ if and only if $\\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.\nThe following problem is both natural and of considerable interest.\n\n\\begin{problem}\\label{prob}\nClassify maximal $(s,t)$-pairs for all $s\\in\\mathbb{N}\\cup\\{\\tfrac{1}{2}\\}$ and $t\\in\\mathbb{N}$ with $s<t$.\n\\end{problem}\n\nIn this paper we address Problem~\\ref{prob} for tetravalent graphs in the case $s=\\tfrac{1}{2}$.\nOur motivation is to better understand the automorphism groups of tetravalent $G$-HAT graphs $\\Gamma$.\nKnowing whether a given graph $\\Gamma$ is HAT is often difficult, largely because determining $\\Aut(\\Gamma)$ is hard in general.\nIf $\\Gamma$ is not half-arc-transitive, then $\\Aut(\\Gamma)$ is arc-transitive; consequently, $\\Gamma$ admits a maximal $\\bigl(\\frac{1}{2},t\\bigr)$-pair $(M,H)$ for some positive integer $t$ with $G\\le M$.\n\nAnother motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.\nLet $\\Ga$ be a tetravalent $M$-HAT graph for some $M\\leq\\Aut(\\Ga)$. It is easy to see that the action of\n$M$ on the arc set of $\\Ga$ has two orbits, denoted by $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$. For each $\\epsilon\\in \\{+,-\\}$,\nand each edge of $\\Ga$, the orbit $O_M^\\epsilon(\\Ga)$ contains exactly one of the two arcs corresponding to this edge.\nHence $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$ give two opposite orientations of the edges of $\\Ga$ and hence give two digraphs with the same vertex set as $\\Ga$, denoted by $D_M^+(\\Ga)$ and $D_M^-(\\Ga)$.\nA cycle of $\\Ga$ is called an \\emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\\Ga)$ (and hence also opposite orientations in $D_M^-(\\Ga)$). In a seminal paper~\\cite{Marusic1998} in 1998, Maru\\v si\\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\\Ga$ have the same length, half of which is called the \\emph{$M$-radius} of $\\Ga$, and any two alternating cycles of $\\Ga$ with nonempty intersection share the same number of vertices. This number is called the \\emph{$M$-attachment number} of $\\Ga$, denoted by $\\att_M(\\Ga)$.\n\nThe \\emph{graph of $M$-alternating cycles} of $\\Ga$, denoted $\\Alt_M(\\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\\cite{MP1999,PSparl2017}). In~\\cite{RSparl2019}, Rivera and \\v{S}parl initiated the study of the automorphism group of $\\Alt_M(\\Ga)$ for tetravalent $M$-HAT graphs $\\Ga$.\nThey proved that, if either $\\att_M(\\Ga)$ equals the $M$-radius or $\\Aut(\\Ga)$ contains an element interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, then $\\Alt_M(\\Ga)$ is arc-transitive (see~\\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\\Alt_M(\\Ga)$. Despite this, all computed examples in~\\cite{RSparl2019} still have $\\Alt_M(\\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\\Alt_M(\\Ga)$, and similarly for the related quotient graph $\\Ga_{\\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\\mathcal{B}(M)$ of $V(\\Ga)$; see~\\cite{MP1999} or~\\cite{RSparl2019} for definition). In particular, $\\Ga_{\\mathcal{B}(M)}=\\Ga$ whenever $\\att_M(\\Ga)=1$.\n\n\\begin{question}[{\\cite[Question~5.8]{RSparl2019}}]\\label{que:Sparl}\nIs it true that, if $\\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\\leq\\Aut(\\Ga)$, the graphs $\\Alt_M(\\Ga)$ and $\\Ga_{\\mathcal{B}(M)}$ are both arc-transitive?\n\\end{question}\n\nNote that any larger HAT group $N>M$ has the same orbits on $A(\\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\\Ga$ is $M$-HAT. Take $H\\leq\\Aut(\\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\\Ga$, and so $(M, H)$ is a maximal $(\\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\\{O_M^+(\\Ga),O_M^-(\\Ga)\\}$,  and so we deduce from~\\cite[Proposition~5.7]{RSparl2019} that $\\Alt_M(\\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.\n\nOur main theorem characterizes tetravalent graphs admitting a maximal $(\\frac{1}{2}, t)$-pair with $t\\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \\emph{core} of $X$ in $Y$, denoted by $\\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.\nLet $\\Ga$ be a graph. Assume that $G\\leq\\Aut(\\Ga)$ is such that $G$ is transitive on the vertex set of $\\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\\Ga)$. The \\emph{normal quotient graph\\/} $\\Ga_N$ of $\\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\\Ga)$ and with two different orbits adjacent if there exists an edge in $\\Ga$ between the vertices lying in those two orbits. If $\\Ga_N$ and $\\Ga$ have the same valency, then $\\Ga$ is a \\emph{cover} of $\\Ga_N$.",
    "context": "For a positive integer $t$, we say that $\\Ga$ is \\emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \\emph{$t$-arc-transitive graph}.\nFor $s\\in\\mathbb{N}\\cup\\{\\frac{1}{2}\\}$, if $\\Ga$ is $(G,s)$-arc-transitive but not $(G,\\lfloor s+1\\rfloor)$-arc-transitive, then we say that $\\Ga$ is \\emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\\Aut(\\Ga)$ is called an \\emph{$(s,t)$-pair} of $\\Ga$ if $M\\leq H\\leq\\Aut(\\Ga)$ and $\\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.\nSuch a pair $(M, H)$ is called \\emph{maximal} if $M$ is a maximal subgroup of $H$.\nNote that, $\\Gamma$ admits a maximal $\\bigl(\\frac{1}{2},t\\bigr)$-pair for some positive integer $t$ if and only if $\\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.\nThe following problem is both natural and of considerable interest.\n\nAnother motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.\nLet $\\Ga$ be a tetravalent $M$-HAT graph for some $M\\leq\\Aut(\\Ga)$. It is easy to see that the action of\n$M$ on the arc set of $\\Ga$ has two orbits, denoted by $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$. For each $\\epsilon\\in \\{+,-\\}$,\nand each edge of $\\Ga$, the orbit $O_M^\\epsilon(\\Ga)$ contains exactly one of the two arcs corresponding to this edge.\nHence $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$ give two opposite orientations of the edges of $\\Ga$ and hence give two digraphs with the same vertex set as $\\Ga$, denoted by $D_M^+(\\Ga)$ and $D_M^-(\\Ga)$.\nA cycle of $\\Ga$ is called an \\emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\\Ga)$ (and hence also opposite orientations in $D_M^-(\\Ga)$). In a seminal paper~\\cite{Marusic1998} in 1998, Maru\\v si\\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\\Ga$ have the same length, half of which is called the \\emph{$M$-radius} of $\\Ga$, and any two alternating cycles of $\\Ga$ with nonempty intersection share the same number of vertices. This number is called the \\emph{$M$-attachment number} of $\\Ga$, denoted by $\\att_M(\\Ga)$.\n\nThe \\emph{graph of $M$-alternating cycles} of $\\Ga$, denoted $\\Alt_M(\\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\\cite{MP1999,PSparl2017}). In~\\cite{RSparl2019}, Rivera and \\v{S}parl initiated the study of the automorphism group of $\\Alt_M(\\Ga)$ for tetravalent $M$-HAT graphs $\\Ga$.\nThey proved that, if either $\\att_M(\\Ga)$ equals the $M$-radius or $\\Aut(\\Ga)$ contains an element interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, then $\\Alt_M(\\Ga)$ is arc-transitive (see~\\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\\Alt_M(\\Ga)$. Despite this, all computed examples in~\\cite{RSparl2019} still have $\\Alt_M(\\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\\Alt_M(\\Ga)$, and similarly for the related quotient graph $\\Ga_{\\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\\mathcal{B}(M)$ of $V(\\Ga)$; see~\\cite{MP1999} or~\\cite{RSparl2019} for definition). In particular, $\\Ga_{\\mathcal{B}(M)}=\\Ga$ whenever $\\att_M(\\Ga)=1$.\n\n\\begin{question}[{\\cite[Question~5.8]{RSparl2019}}]\\label{que:Sparl}\nIs it true that, if $\\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\\leq\\Aut(\\Ga)$, the graphs $\\Alt_M(\\Ga)$ and $\\Ga_{\\mathcal{B}(M)}$ are both arc-transitive?\n\\end{question}\n\nNote that any larger HAT group $N>M$ has the same orbits on $A(\\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\\Ga$ is $M$-HAT. Take $H\\leq\\Aut(\\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\\Ga$, and so $(M, H)$ is a maximal $(\\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\\{O_M^+(\\Ga),O_M^-(\\Ga)\\}$, and so we deduce from~\\cite[Proposition~5.7]{RSparl2019} that $\\Alt_M(\\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.\n\nOur main theorem characterizes tetravalent graphs admitting a maximal $(\\frac{1}{2}, t)$-pair with $t\\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \\emph{core} of $X$ in $Y$, denoted by $\\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.\nLet $\\Ga$ be a graph. Assume that $G\\leq\\Aut(\\Ga)$ is such that $G$ is transitive on the vertex set of $\\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\\Ga)$. The \\emph{normal quotient graph\\/} $\\Ga_N$ of $\\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\\Ga)$ and with two different orbits adjacent if there exists an edge in $\\Ga$ between the vertices lying in those two orbits. If $\\Ga_N$ and $\\Ga$ have the same valency, then $\\Ga$ is a \\emph{cover} of $\\Ga_N$.",
    "full_context": "For a positive integer $t$, we say that $\\Ga$ is \\emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \\emph{$t$-arc-transitive graph}.\nFor $s\\in\\mathbb{N}\\cup\\{\\frac{1}{2}\\}$, if $\\Ga$ is $(G,s)$-arc-transitive but not $(G,\\lfloor s+1\\rfloor)$-arc-transitive, then we say that $\\Ga$ is \\emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\\Aut(\\Ga)$ is called an \\emph{$(s,t)$-pair} of $\\Ga$ if $M\\leq H\\leq\\Aut(\\Ga)$ and $\\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.\nSuch a pair $(M, H)$ is called \\emph{maximal} if $M$ is a maximal subgroup of $H$.\nNote that, $\\Gamma$ admits a maximal $\\bigl(\\frac{1}{2},t\\bigr)$-pair for some positive integer $t$ if and only if $\\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.\nThe following problem is both natural and of considerable interest.\n\nAnother motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.\nLet $\\Ga$ be a tetravalent $M$-HAT graph for some $M\\leq\\Aut(\\Ga)$. It is easy to see that the action of\n$M$ on the arc set of $\\Ga$ has two orbits, denoted by $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$. For each $\\epsilon\\in \\{+,-\\}$,\nand each edge of $\\Ga$, the orbit $O_M^\\epsilon(\\Ga)$ contains exactly one of the two arcs corresponding to this edge.\nHence $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$ give two opposite orientations of the edges of $\\Ga$ and hence give two digraphs with the same vertex set as $\\Ga$, denoted by $D_M^+(\\Ga)$ and $D_M^-(\\Ga)$.\nA cycle of $\\Ga$ is called an \\emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\\Ga)$ (and hence also opposite orientations in $D_M^-(\\Ga)$). In a seminal paper~\\cite{Marusic1998} in 1998, Maru\\v si\\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\\Ga$ have the same length, half of which is called the \\emph{$M$-radius} of $\\Ga$, and any two alternating cycles of $\\Ga$ with nonempty intersection share the same number of vertices. This number is called the \\emph{$M$-attachment number} of $\\Ga$, denoted by $\\att_M(\\Ga)$.\n\nThe \\emph{graph of $M$-alternating cycles} of $\\Ga$, denoted $\\Alt_M(\\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\\cite{MP1999,PSparl2017}). In~\\cite{RSparl2019}, Rivera and \\v{S}parl initiated the study of the automorphism group of $\\Alt_M(\\Ga)$ for tetravalent $M$-HAT graphs $\\Ga$.\nThey proved that, if either $\\att_M(\\Ga)$ equals the $M$-radius or $\\Aut(\\Ga)$ contains an element interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, then $\\Alt_M(\\Ga)$ is arc-transitive (see~\\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\\Ga)$ and $O_M^-(\\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\\Alt_M(\\Ga)$. Despite this, all computed examples in~\\cite{RSparl2019} still have $\\Alt_M(\\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\\Alt_M(\\Ga)$, and similarly for the related quotient graph $\\Ga_{\\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\\mathcal{B}(M)$ of $V(\\Ga)$; see~\\cite{MP1999} or~\\cite{RSparl2019} for definition). In particular, $\\Ga_{\\mathcal{B}(M)}=\\Ga$ whenever $\\att_M(\\Ga)=1$.\n\n\\begin{question}[{\\cite[Question~5.8]{RSparl2019}}]\\label{que:Sparl}\nIs it true that, if $\\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\\leq\\Aut(\\Ga)$, the graphs $\\Alt_M(\\Ga)$ and $\\Ga_{\\mathcal{B}(M)}$ are both arc-transitive?\n\\end{question}\n\nNote that any larger HAT group $N>M$ has the same orbits on $A(\\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\\Ga$ is $M$-HAT. Take $H\\leq\\Aut(\\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\\Ga$, and so $(M, H)$ is a maximal $(\\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\\{O_M^+(\\Ga),O_M^-(\\Ga)\\}$, and so we deduce from~\\cite[Proposition~5.7]{RSparl2019} that $\\Alt_M(\\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.\n\nOur main theorem characterizes tetravalent graphs admitting a maximal $(\\frac{1}{2}, t)$-pair with $t\\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \\emph{core} of $X$ in $Y$, denoted by $\\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.\nLet $\\Ga$ be a graph. Assume that $G\\leq\\Aut(\\Ga)$ is such that $G$ is transitive on the vertex set of $\\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\\Ga)$. The \\emph{normal quotient graph\\/} $\\Ga_N$ of $\\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\\Ga)$ and with two different orbits adjacent if there exists an edge in $\\Ga$ between the vertices lying in those two orbits. If $\\Ga_N$ and $\\Ga$ have the same valency, then $\\Ga$ is a \\emph{cover} of $\\Ga_N$.\n\nOur main theorem characterizes tetravalent graphs admitting a maximal $(\\frac{1}{2}, t)$-pair with $t\\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \\emph{core} of $X$ in $Y$, denoted by $\\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.\nLet $\\Ga$ be a graph. Assume that $G\\leq\\Aut(\\Ga)$ is such that $G$ is transitive on the vertex set of $\\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\\Ga)$. The \\emph{normal quotient graph\\/} $\\Ga_N$ of $\\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\\Ga)$ and with two different orbits adjacent if there exists an edge in $\\Ga$ between the vertices lying in those two orbits. If $\\Ga_N$ and $\\Ga$ have the same valency, then $\\Ga$ is a \\emph{cover} of $\\Ga_N$.\n\nSummarized in Section~\\ref{sec:sum}, the proof of Theorem~\\ref{Thm} is developed in Sections~\\ref{sec:proofofmain} and~\\ref{sec:examples}. More precisely, we first prove in Section~\\ref{sec:proofofmain} that one of~\\eqref{Thmcase:t=3}--\\eqref{Thmcase:t=1} must occur, and then in Section~\\ref{sec:examples} we construct examples for each of the cases~\\eqref{Thmcase:t=3},~\\eqref{Thmcase:t=2},~\\eqref{Thmcase:t=1_cover} and~\\eqref{Thmcase:t=1_cycle}. The example for~\\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\\ref{que:Sparl} of Rivera and \\v{S}parl~\\cite{RSparl2019}.\n\n\\begin{lemma}[{\\cite[Theorem~1.1]{AAMPS2016}}]\\label{lem:4HAT}\nLet $\\Ga$ be a tetravalent $G$-HAT graph, and let $N\\trianglelefteq G$ such that $N$ has at least three orbits on $V(\\Ga)$. Then one of the following holds:\n\\begin{enumerate}[{\\rm (a)}]\n \\item $\\Ga$ is a cover of $\\Ga_N$, $N$ is semiregular on $V(\\Ga)$ and is the kernel of $G$ acting on $V(\\Ga_N)$, and $\\Ga_N$ is $(G/N)$-HAT.\n \\item $\\Ga_N$ is an $r$-cycle with $r=|V(\\Ga_N)|$, and the induced permutation group of $G$ on $V(\\Ga_N)$ is $\\mz_r$ or $D_{2r}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:hat-2at}\nLet $\\Ga$ be a tetravalent graph, let $u\\in V(\\Ga)$, let $(M, H)$ be a maximal $(\\frac{1}{2}, t)$-pair of $\\Ga$ with $t\\geq2$, and let $K=\\mathrm{Core}_H(M)$. Then $\\Ga$ is a cover of $\\Ga_K$, and one of the following holds:\n\\begin{enumerate}[\\rm (a)]\n\\item $t=2$, $(H/K, M/K, H_{u}, M_{u})=(S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Gamma_K\\cong \\K_{5, 5}-5\\K_2$;\n\\item $t=3$, and $(H/K, M/K, H_{u}, M_{u})=(A_{72}, A_{71},S_4\\times S_3, \\mz_2)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:stab:=2}\nLet $\\Ga$ be a tetravalent graph, let $u\\in V(\\Ga)$, let $(M, H)$ be a maximal $(\\frac{1}{2},1)$-pair of $\\Ga$ such that $M_u^{\\Ga(u)}\\cong\\mz_2$ and $H_u^{\\Ga(u)}\\cong D_8$, and let $K=\\mathrm{Core}_H(M)$. Then $M_u^{\\Ga(u)}\\ntrianglelefteq H_u^{\\Ga(u)}$, $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$, and one of the following holds:\n\\begin{enumerate}[\\rm (a)]\n\\item\\label{lem:stab:=2-a} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$, and $\\Ga$ is a cover of $\\Ga_K$;\n\\item\\label{lem:stab:=2-b} $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, $M/K\\cong D_{2r}$, and $K$ is the kernel of $M$ acting on $V(\\Ga_K)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{example}\\label{exam:hat_in_3-trans_non-normal}\nLet $M=\\langle a,b\\mid a^2=b^3=(ab)^2=1\\rangle\\times\\langle c,d\\mid c^2=d^3=(cd)^4=1\\rangle\\cong S_3\\times S_4$, let $R$ be the faithful action of $M$ by right multiplication on $V:=[M\\,{:}\\,\\langle a(cd)^2\\rangle]$, let $X=\\mathrm{Alt}(V)\\cong A_{72}$, and let $Y=R(M'\\rtimes\\langle ac\\rangle)\\cong \\mz_3\\rtimes S_4$. Then $Y$ is a regular subgroup of $X$. Let $Z=R((\\langle b\\rangle\\times\\langle d\\rangle)\\rtimes\\langle ac^{dc}\\rangle)\\cong \\mz_3^2\\rtimes \\mz_2$. Computer searching shows that there exists an element $x$ of order $4$ in $\\N_X(Z)\\cap\\mathbf{C}_X(ac^{dc})$ satisfying the following conditions:\n\\begin{itemize}\n\\item $x^2=ac^{dc}\\in Z$;\n\\item $Y\\cap Y^x=Z$;\n\\item $X=\\langle Y,x\\rangle$;\n\\item $YxY=YS$ for some $S=\\{g,g^{-1},g^h,(g^h)^{-1}\\}$, where $g\\in X_v$ for some $v\\in V$ and $h$ is an involution in $X_v$. \\end{itemize}\n\nCorollaries~\\ref{cor-1} and~\\ref{cor-2} follow from Examples~\\ref{exam:hat_in_1-trans_altgra} and~\\ref{exam:hat_in_3-trans_non-normal}, respectively. Now we prove Theorem~\\ref{Thm}.\nLet $\\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\\frac{1}{2}, t)$-pair of $\\Ga$ with $t\\geq1$, let $K=\\mathrm{Core}_H(M)$, and let $u\\in V(\\Ga)$.\nIf $t\\geq2$, then Lemma~\\ref{lem:hat-2at} asserts that~\\eqref{Thmcase:t=3} or~\\eqref{Thmcase:t=2} holds.\nAssume that $t=1$. If $M_u^{\\Ga(u)}\\cong\\mz_2$ and $H_u^{\\Ga(u)}\\cong D_8$, then by Lemma~\\ref{lem:stab:=2}, $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and we have~\\eqref{Thmcase:t=1_cover} or~\\eqref{Thmcase:t=1_cycle}. If either $M_u^{\\Ga(u)}\\ncong\\mz_2$ or $H_u^{\\Ga(u)}\\ncong D_8$, then since $(M, H)$ is a $(\\frac{1}{2}, 1)$-pair of $\\Ga$, it follows that $M_u^{\\Ga(u)}$ is normal in\n$H_u^{\\Ga(u)}$ with $|H_u^{\\Ga(u)}:M_u^{\\Ga(u)}|=2$. In this case, Lemma~\\ref{lem:hat-1t}\\,\\eqref{lem:hat-1t-c} implies that $M$ is normal in $H$, as in case~\\eqref{Thmcase:t=1} of Theorem~\\ref{Thm}. Moreover, according to Examples~\\ref{exam:hat_in_1-trans_altgra},~\\ref{exam:hat_in_3-trans_non-normal},~\\ref{exam:hat_in_2-trans} and~\\ref{exam:hat_in_1-trans_cycle}, examples for each of the cases~\\eqref{Thmcase:t=3},~\\eqref{Thmcase:t=2},~\\eqref{Thmcase:t=1_cover} and~\\eqref{Thmcase:t=1_cycle} of Theorem~\\ref{Thm} exist. This completes the proof.",
    "post_theorem_intro_text_len": 2526,
    "post_theorem_intro_text": "Summarized in Section~\\ref{sec:sum}, the proof of Theorem~\\ref{Thm} is developed in Sections~\\ref{sec:proofofmain} and~\\ref{sec:examples}. More precisely, we first prove in Section~\\ref{sec:proofofmain} that one of~\\eqref{Thmcase:t=3}--\\eqref{Thmcase:t=1} must occur, and then in Section~\\ref{sec:examples} we construct examples for each of the cases~\\eqref{Thmcase:t=3},~\\eqref{Thmcase:t=2},~\\eqref{Thmcase:t=1_cover} and~\\eqref{Thmcase:t=1_cycle}. The example for~\\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\\ref{que:Sparl} of Rivera and \\v{S}parl~\\cite{RSparl2019}.\n\n\\begin{corollary}\\label{cor-1}\nThere exists a connected tetravalent graph $\\Ga$ admiting a maximal $(\\frac{1}{2}, 1)$-pair $(M,\\Aut(\\Gamma))$ such that $\\Alt_M(\\Ga)$ is half-arc-transitive.\n\\end{corollary}\n\nAnother byproduct of proving Theorem~\\ref{Thm} is the discovery of the first example of a tetravalent normal-edge-transitive but non-normal Cayley graph on a nonabelian simple group.\nGiven a group $G$ and an inverse-closed generating subset $S\\subseteq G\\setminus\\{1\\}$, the \\emph{Cayley graph} $\\Cay(G,S)$ on $G$ is the graph with vertex set $G$ and edge set $\\{\\{g, sg\\} \\mid g\\in G, s\\in S\\}$. Denote by $R_G$ the right multiplication action of $G$ on itself, and denote\n\\[\n\\Aut(G, S)=\\{\\a\\in\\Aut(G)\\mid S^\\a=S\\}.\n\\]\nIt is well known~\\cite{Godsil1981} that the normalizer of $R_G(G)$ in $\\Aut(\\Cay(G,S))$ is $R_G(G)\\rtimes\\Aut(G,S)$. We call $\\Cay(G,S)$ \\emph{normal-edge-transitive} if this normalizer is transitive on $E(\\Cay(G,S))$, and \\emph{normal} if this normalizer equals $\\Aut(\\Cay(G,S))$.\nIn 1999, Praeger~\\cite{Praeger1999} proved that every connected normal-edge-transitive Cayley graph of valency $3$ on a nonabelian simple group is normal. This result was recently extended to all prime valencies in~\\cite{Zhang-F-Z-Y}, where it was also shown that there exist connected normal-edge-transitive non-normal Cayley graphs of valency $8$ on nonabelian simple groups. However, it has remained open whether such graphs exist at smaller valencies, particularly valency $4$, the smallest possible valency, despite substantial work on tetravalent edge-transitive Cayley graphs on nonabelian simple groups~\\cite{FangLiXu,FangWZ}. Our example for Theorem~\\ref{Thm}\\,\\eqref{Thmcase:t=3} resolves this question by establishing the following:\n\n\\begin{corollary}\\label{cor-2}\nThere exists a connected tetravalent normal-edge-transitive non-normal Cayley graph on $A_{71}$.\n\\end{corollary}",
    "sketch": "The post-theorem introduction says that “the proof of Theorem~\\ref{Thm} is developed in Sections~\\ref{sec:proofofmain} and~\\ref{sec:examples}.” More precisely: (1) “we first prove in Section~\\ref{sec:proofofmain} that one of~\\eqref{Thmcase:t=3}--\\eqref{Thmcase:t=1} must occur,” and then (2) “in Section~\\ref{sec:examples} we construct examples for each of the cases~\\eqref{Thmcase:t=3},~\\eqref{Thmcase:t=2},~\\eqref{Thmcase:t=1_cover} and~\\eqref{Thmcase:t=1_cycle}.”",
    "expanded_sketch": "The post-theorem introduction says that “the proof of the main theorem is developed later.” More precisely: (1) “we first prove later that one of the following must occur: \\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}\\end{enumerate},” and then (2) “later we construct examples for each of the cases given by the items \\ref{Thmcase:t=3} and \\ref{Thmcase:t=2} above, and by the subcases \\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}.”",
    "expanded_theorem": "\\label{Thm}\nLet $\\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\\frac{1}{2}, t)$-pair of $\\Ga$ with $t\\geq1$, and let $K=\\mathrm{Core}_H(M)$. Take $u\\in V(\\Ga)$. Then one of the followings holds.\n\\begin{enumerate}[\\rm(a)]\n\\item\\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\\times S_3, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K$.\n\\item\\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \\AGL_1(5), A_4, \\mz_2)$, and $\\Ga$ is a cover of $\\Ga_K\\cong \\K_{5, 5}-5\\K_2$.\n\\item\\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\\Ga(u)}\\cong\\mz_2$ is non-normal in $H_u^{\\Ga(u)}\\cong D_8$ such that $K$ is semiregular on $V(\\Ga)$ with $|V(\\Ga_K)|\\neq2^\\ell$ for any integer $\\ell\\geq0$ and one of the followings holds:\n\\begin{enumerate}[\\rm (c1)]\n\\item\\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\\Ga_K)$, $\\Ga$ is a cover of $\\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\\frac{1}{2}, 1)$-pair of $\\Ga_K$;\n\\item\\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\\Ga_K)$, $\\Ga_K\\cong C_r$ with $r=|V(\\Ga_K)|$, and $M/K\\cong D_{2r}$.\n\\end{enumerate}\n\\end{enumerate}\nMoreover, examples for each of the above cases exist.,",
    "theorem_type": [
      "Classification or Bijection",
      "Universal"
    ],
    "mcq": {
      "question": "Let \\(\\Gamma\\) be a connected tetravalent graph. Suppose \\(M\\le H\\le \\operatorname{Aut}(\\Gamma)\\), \\(M\\) is a maximal subgroup of \\(H\\), \\(\\Gamma\\) is \\(M\\)-half-arc-transitive (that is, \\(M\\) is transitive on vertices and edges but not on arcs), and \\(\\Gamma\\) is \\((H,t)\\)-transitive for some integer \\(t\\ge 1\\); equivalently, \\((M,H)\\) is a maximal \\((\\tfrac12,t)\\)-pair. Let \\(K=\\operatorname{Core}_H(M)\\), the largest normal subgroup of \\(H\\) contained in \\(M\\), and fix \\(u\\in V(\\Gamma)\\). Write \\(\\Gamma(u)\\) for the neighbor set of \\(u\\), \\(H_u\\) and \\(M_u\\) for the stabilizers of \\(u\\), and \\(G_u^{\\Gamma(u)}\\) for the permutation group induced by \\(G_u\\) on \\(\\Gamma(u)\\). Since \\(K\\trianglelefteq H\\), let \\(\\Gamma_K\\) be the normal quotient graph whose vertices are the \\(K\\)-orbits on \\(V(\\Gamma)\\), with two \\(K\\)-orbits adjacent whenever some edge of \\(\\Gamma\\) joins them. A subgroup is semiregular on \\(V(\\Gamma)\\) if no nonidentity element fixes a vertex, and saying that \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\) means the natural quotient map is a covering projection. Which statement holds for every such \\(\\Gamma,M,H,K,u\\)?",
      "correct_choice": {
        "label": "A",
        "text": "One of the following holds. (a) \\(t=3\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(A_{72},\\,A_{71},\\,S_4\\times S_3,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\). (b) \\(t=2\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(S_5,\\,\\operatorname{AGL}_1(5),\\,A_4,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\cong K_{5,5}-5K_2\\) (the graph obtained from \\(K_{5,5}\\) by deleting a perfect matching). (c) \\(t=1\\), and either \\(M\\trianglelefteq H\\), or else \\(M_u^{\\Gamma(u)}\\cong\\mathbb Z_2\\) is non-normal in \\(H_u^{\\Gamma(u)}\\cong D_8\\), \\(K\\) is semiregular on \\(V(\\Gamma)\\), \\(|V(\\Gamma_K)|\\neq 2^\\ell\\) for every integer \\(\\ell\\ge 0\\), and in addition either (c1) \\(K\\) is the kernel of the action of \\(H\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\), and \\((M/K,H/K)\\) is a maximal \\((\\tfrac12,1)\\)-pair of \\(\\Gamma_K\\); or (c2) \\(K\\) is the kernel of the action of \\(M\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\cong C_r\\) with \\(r=|V(\\Gamma_K)|\\), and \\(M/K\\) is dihedral of order \\(2r\\). Moreover, examples realizing each listed case exist."
      },
      "choices": [
        {
          "label": "B",
          "text": "One of the following holds. (a) \\(t=3\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(A_{72},\\,A_{71},\\,S_4\\times S_3,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\). (b) \\(t=2\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(S_5,\\,\\operatorname{AGL}_1(5),\\,A_4,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\cong K_{5,5}-5K_2\\). (c) \\(t=1\\), and either \\(M\\trianglelefteq H\\), or else \\(M_u^{\\Gamma(u)}\\cong\\mathbb Z_2\\) is non-normal in \\(H_u^{\\Gamma(u)}\\cong D_8\\), \\(K\\) is semiregular on \\(V(\\Gamma)\\), \\(|V(\\Gamma_K)|\\neq 2^\\ell\\) for every integer \\(\\ell\\ge 1\\), and in addition either (c1) \\(K\\) is the kernel of the action of \\(H\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\), and \\((M/K,H/K)\\) is a maximal \\((\\tfrac12,1)\\)-pair of \\(\\Gamma_K\\); or (c2) \\(K\\) is the kernel of the action of \\(M\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\cong C_r\\) with \\(r=|V(\\Gamma_K)|\\), and \\(M/K\\) is dihedral of order \\(2r\\). Moreover, examples realizing each listed case exist."
        },
        {
          "label": "C",
          "text": "One of the following holds. (a) \\(t=3\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(A_{72},\\,A_{71},\\,S_4\\times S_3,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\). (b) \\(t=2\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(S_5,\\,\\operatorname{AGL}_1(5),\\,A_4,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\cong K_{5,5}-5K_2\\). (c) \\(t=1\\), and either \\(M\\trianglelefteq H\\), or else \\(M_u^{\\Gamma(u)}\\cong\\mathbb Z_2\\) is non-normal in \\(H_u^{\\Gamma(u)}\\cong D_8\\), \\(K\\) is semiregular on \\(V(\\Gamma)\\), and in addition either (c1) \\(K\\) is the kernel of the action of \\(H\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\), and \\((M/K,H/K)\\) is a maximal \\((\\tfrac12,1)\\)-pair of \\(\\Gamma_K\\); or (c2) \\(K\\) is the kernel of the action of \\(M\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\cong C_r\\) with \\(r=|V(\\Gamma_K)|\\), and \\(M/K\\) is dihedral of order \\(2r\\)."
        },
        {
          "label": "D",
          "text": "One of the following holds. (a) \\(t=3\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(A_{72},\\,A_{71},\\,S_4\\times S_3,\\,\\mathbb Z_2)\\), and \\(\\Gamma_K\\) is a cover of \\(\\Gamma\\). (b) \\(t=2\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(S_5,\\,\\operatorname{AGL}_1(5),\\,A_4,\\,\\mathbb Z_2)\\), and \\(\\Gamma_K\\cong K_{5,5}-5K_2\\) with \\(\\Gamma_K\\) a cover of \\(\\Gamma\\). (c) \\(t=1\\), and either \\(M\\trianglelefteq H\\), or else \\(M_u^{\\Gamma(u)}\\cong\\mathbb Z_2\\) is non-normal in \\(H_u^{\\Gamma(u)}\\cong D_8\\), \\(K\\) is semiregular on \\(V(\\Gamma)\\), \\(|V(\\Gamma_K)|\\neq 2^\\ell\\) for every integer \\(\\ell\\ge 0\\), and in addition either (c1) \\(K\\) is the kernel of the action of \\(H\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\) is a cover of \\(\\Gamma\\), and \\((M/K,H/K)\\) is a maximal \\((\\tfrac12,1)\\)-pair of \\(\\Gamma_K\\); or (c2) \\(K\\) is the kernel of the action of \\(M\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\cong C_r\\) with \\(r=|V(\\Gamma_K)|\\), and \\(M/K\\) is dihedral of order \\(2r\\). Moreover, examples realizing each listed case exist."
        },
        {
          "label": "E",
          "text": "One of the following holds. (a) \\(t=3\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(A_{72},\\,A_{71},\\,S_4\\times S_3,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\). (b) \\(t=2\\), \\((H/K,\\,M/K,\\,H_u,\\,M_u)=(S_5,\\,\\operatorname{AGL}_1(5),\\,A_4,\\,\\mathbb Z_2)\\), and \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\cong K_{5,5}-5K_2\\). (c) \\(t=1\\), and either \\(M\\trianglelefteq H\\), or else \\(M_u^{\\Gamma(u)}\\cong\\mathbb Z_2\\) is normal in \\(H_u^{\\Gamma(u)}\\cong D_8\\), \\(K\\) is semiregular on \\(V(\\Gamma)\\), \\(|V(\\Gamma_K)|\\neq 2^\\ell\\) for every integer \\(\\ell\\ge 0\\), and in addition either (c1) \\(K\\) is the kernel of the action of \\(H\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma\\) is a cover of \\(\\Gamma_K\\), and \\((M/K,H/K)\\) is a maximal \\((\\tfrac12,1)\\)-pair of \\(\\Gamma_K\\); or (c2) \\(K\\) is the kernel of the action of \\(M\\) on \\(V(\\Gamma_K)\\), \\(\\Gamma_K\\cong C_r\\) with \\(r=|V(\\Gamma_K)|\\), and \\(H/K\\) is dihedral of order \\(2r\\). Moreover, examples realizing each listed case exist."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "excluded_power_of_two_range",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "characteristic",
          "tampered_component": "dropped condition |V(\\Gamma_K)|\\neq 2^\\ell for all \\ell\\ge 0",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "geometric_construction",
          "tampered_component": "direction of covering projection to normal quotient",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "non-normal local action and quotient group in cycle case",
          "template_used": "property_confusion"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option; it only sets up the hypotheses and asks for the full classification. There is no explicit answer leakage, though the highly specialized notation makes it clear this is a theorem-recall item."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-statement recognition question: the task is to identify the exact classification theorem rather than derive a conclusion from competing mathematical principles."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to compare subtle logical differences among the options (off-by-one condition, omitted hypotheses, swapped kernels, disjunction vs conjunction). However, this is mostly precise recall and statement verification, not genuine generative mathematical problem solving."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect realistic failure modes such as weakened statements, boundary mistakes, and swapped subgroup-action roles."
      },
      "total_score": 5,
      "overall_assessment": "Well-constructed as a theorem-statement discrimination item with high-quality distractors and no answer leakage, but it is largely a tautological recall question rather than a test of generative reasoning."
    }
  },
  {
    "id": "2512.21950v1",
    "paper_link": "http://arxiv.org/abs/2512.21950v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:main}\n\tLet $G$ be an $n$-vertex digraph with $\\alpha^*(G) \\le s$.\n\tThen, $f(G) \\ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then\n\t$f(G) \\ge n^{5/9-o(1)}$.",
      "start_pos": 6914,
      "end_pos": 7131,
      "label": "t:main"
    },
    "ref_dict": {
      "l:l22": "\\begin{lemma}\\label{l:l22}\n\tLet $k = n^{\\Omega(1)}$ and $k \\ge s^7$. If an $n$-vertex orientation $G$ with $\\alpha(G) < s$ and $\\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \\ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$\n\twhere $q =  n's^6/k^{1-o(1)}$.\n\\end{lemma}",
      "t:main": "\\begin{theorem}\\label{t:main}\n\tLet $G$ be an $n$-vertex digraph with $\\alpha^*(G) \\le s$.\n\tThen, $f(G) \\ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then\n\t$f(G) \\ge n^{5/9-o(1)}$.\n\\end{theorem}",
      "l:l23": "\\begin{lemma}\\label{l:l23}\nSuppose $n^{1/3}s^{-2/3} \\le q \\le n^{1-\\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with\n$\\alpha(G) < s$ has a permutation $\\pi$ with $\\alpha(G_\\pi) \\le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\\chi(G_\\pi) \\ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 3365,
    "pre_theorem_intro_text": "Throughout this paper, the chromatic number $\\chi(G)$, the independence number $\\alpha(G)$, and the bipartite independence number $\\alpha^*(G)$\\,\\footnote{Recall that $\\alpha(G)$ is the largest $k$ for which there is an empty $k$-vertex set and that $\\alpha^*(G)$ is the largest $k$ for which there are two disjoint $k$-vertex sets with no edges between them.} of a digraph $G$ are inherited from the corresponding parameters of its underlying undirected graph.\nAn {\\em orientation} (a.k.a. oriented graph) is a digraph without directed cycles of length $2$. A {\\em tournament} is an orientation of a complete graph. A digraph is {\\em acyclic} if it has no directed cycles.\n\nGraph colorings and orientations are closely related topics whose origins date back to the\nclassical theorem of Gallai–Hasse–Roy–Vitaver \\cite{gallai-1968,hasse-1965,roy-1967,vitaver-1962}\nasserting that every orientation of a $k$-chromatic graph has a directed path with $k$ vertices.\nMany variations and related results connecting these concepts can be found in the survey \\cite{havet-2013}.\nA very general variant of the Gallai–Hasse–Roy–Vitaver Theorem was conjectured by Burr \\cite{burr-1980},\nstating that any $(2k-2)$-chromatic digraph contains every oriented tree on $k$ vertices.\nThe special case of Burr's conjecture for tournaments is known as\nSumner's conjecture which has been resolved, for $k$ sufficiently large, by K\\\"uhn, Mycroft, and Osthus\n\\cite{KMO-2011}. The best bound on the chromatic number of digraphs which guarantees the containment of every oriented tree on $k$ vertices\nis $O(k^{1.5})$ which was recently obtained by Bessy, Gon{\\c{c}}alves, and Reinald \\cite{BGR-2024}.\n\nBurr's conjecture is known to hold for acyclic digraphs. In fact, it was proved by\nAddario-Berry, Havet, Sales, Reed, and Thomass\\'e \\cite{addario+-2013} that every $k$-chromatic acyclic digraph \ncontains every oriented tree on $k$ vertices. It would therefore be of interest to prove that\ndigraphs have acyclic subgraphs that retain the original chromatic number to some nontrivial extent.\nIndeed, this problem was formulated in \\cite{addario+-2013}.\nFor a digraph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$\nand let $f(k)$ be the minimum of $f(G)$ taken over all $k$-chromatic digraphs.\nIt is a folklore argument (see below) that $f(G) \\ge \\sqrt{\\chi(G)}$ (whence $f(k) \\ge \\sqrt{k}$), and it was proved in \\cite{addario+-2013}\nthat $f(k) \\ge 1+\\lfloor \\sqrt{k-1} \\rfloor$. Trivially, $f(3)=2$ and it is known that $f(4)=3$ \\cite{NY-2019,shapira-2022}. The special case of this parameter for the case of tournaments was considered\nby Nassar and the author in \\cite{NY-2019}. Let $g(k)$ be the maximum integer such that every $k$-vertex tournament has an acyclic subgraph with chromatic number $g(k)$. It was conjectured in \\cite{NY-2019}\nthat $g(k) = \\omega(\\sqrt{k})$, namely that the folklore lower bound can be improved by more than a constant factor. This conjecture has recently been resolved by Fox, Kwan, and Sudakov \\cite{FKS-2021} who proved by an\ningenious argument that $g(k)=\\Omega(k^{5/9-o(1)})$. In that same paper, they have shown that the exponent $5/9$\ncannot be improved beyond $3/4$. Returning back to $f(k)$, this result suggests that $f(k)=\\omega(\\sqrt{k})$ as well.\nOur main result shows that this is the case in a dense regime.",
    "context": "Throughout this paper, the chromatic number $\\chi(G)$, the independence number $\\alpha(G)$, and the bipartite independence number $\\alpha^*(G)$\\,\\footnote{Recall that $\\alpha(G)$ is the largest $k$ for which there is an empty $k$-vertex set and that $\\alpha^*(G)$ is the largest $k$ for which there are two disjoint $k$-vertex sets with no edges between them.} of a digraph $G$ are inherited from the corresponding parameters of its underlying undirected graph.\nAn {\\em orientation} (a.k.a. oriented graph) is a digraph without directed cycles of length $2$. A {\\em tournament} is an orientation of a complete graph. A digraph is {\\em acyclic} if it has no directed cycles.\n\nGraph colorings and orientations are closely related topics whose origins date back to the\nclassical theorem of Gallai–Hasse–Roy–Vitaver \\cite{gallai-1968,hasse-1965,roy-1967,vitaver-1962}\nasserting that every orientation of a $k$-chromatic graph has a directed path with $k$ vertices.\nMany variations and related results connecting these concepts can be found in the survey \\cite{havet-2013}.\nA very general variant of the Gallai–Hasse–Roy–Vitaver Theorem was conjectured by Burr \\cite{burr-1980},\nstating that any $(2k-2)$-chromatic digraph contains every oriented tree on $k$ vertices.\nThe special case of Burr's conjecture for tournaments is known as\nSumner's conjecture which has been resolved, for $k$ sufficiently large, by K\\\"uhn, Mycroft, and Osthus\n\\cite{KMO-2011}. The best bound on the chromatic number of digraphs which guarantees the containment of every oriented tree on $k$ vertices\nis $O(k^{1.5})$ which was recently obtained by Bessy, Gon{\\c{c}}alves, and Reinald \\cite{BGR-2024}.\n\nBurr's conjecture is known to hold for acyclic digraphs. In fact, it was proved by\nAddario-Berry, Havet, Sales, Reed, and Thomass\\'e \\cite{addario+-2013} that every $k$-chromatic acyclic digraph \ncontains every oriented tree on $k$ vertices. It would therefore be of interest to prove that\ndigraphs have acyclic subgraphs that retain the original chromatic number to some nontrivial extent.\nIndeed, this problem was formulated in \\cite{addario+-2013}.\nFor a digraph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$\nand let $f(k)$ be the minimum of $f(G)$ taken over all $k$-chromatic digraphs.\nIt is a folklore argument (see below) that $f(G) \\ge \\sqrt{\\chi(G)}$ (whence $f(k) \\ge \\sqrt{k}$), and it was proved in \\cite{addario+-2013}\nthat $f(k) \\ge 1+\\lfloor \\sqrt{k-1} \\rfloor$. Trivially, $f(3)=2$ and it is known that $f(4)=3$ \\cite{NY-2019,shapira-2022}. The special case of this parameter for the case of tournaments was considered\nby Nassar and the author in \\cite{NY-2019}. Let $g(k)$ be the maximum integer such that every $k$-vertex tournament has an acyclic subgraph with chromatic number $g(k)$. It was conjectured in \\cite{NY-2019}\nthat $g(k) = \\omega(\\sqrt{k})$, namely that the folklore lower bound can be improved by more than a constant factor. This conjecture has recently been resolved by Fox, Kwan, and Sudakov \\cite{FKS-2021} who proved by an\ningenious argument that $g(k)=\\Omega(k^{5/9-o(1)})$. In that same paper, they have shown that the exponent $5/9$\ncannot be improved beyond $3/4$. Returning back to $f(k)$, this result suggests that $f(k)=\\omega(\\sqrt{k})$ as well.\nOur main result shows that this is the case in a dense regime.",
    "full_context": "Throughout this paper, the chromatic number $\\chi(G)$, the independence number $\\alpha(G)$, and the bipartite independence number $\\alpha^*(G)$\\,\\footnote{Recall that $\\alpha(G)$ is the largest $k$ for which there is an empty $k$-vertex set and that $\\alpha^*(G)$ is the largest $k$ for which there are two disjoint $k$-vertex sets with no edges between them.} of a digraph $G$ are inherited from the corresponding parameters of its underlying undirected graph.\nAn {\\em orientation} (a.k.a. oriented graph) is a digraph without directed cycles of length $2$. A {\\em tournament} is an orientation of a complete graph. A digraph is {\\em acyclic} if it has no directed cycles.\n\nGraph colorings and orientations are closely related topics whose origins date back to the\nclassical theorem of Gallai–Hasse–Roy–Vitaver \\cite{gallai-1968,hasse-1965,roy-1967,vitaver-1962}\nasserting that every orientation of a $k$-chromatic graph has a directed path with $k$ vertices.\nMany variations and related results connecting these concepts can be found in the survey \\cite{havet-2013}.\nA very general variant of the Gallai–Hasse–Roy–Vitaver Theorem was conjectured by Burr \\cite{burr-1980},\nstating that any $(2k-2)$-chromatic digraph contains every oriented tree on $k$ vertices.\nThe special case of Burr's conjecture for tournaments is known as\nSumner's conjecture which has been resolved, for $k$ sufficiently large, by K\\\"uhn, Mycroft, and Osthus\n\\cite{KMO-2011}. The best bound on the chromatic number of digraphs which guarantees the containment of every oriented tree on $k$ vertices\nis $O(k^{1.5})$ which was recently obtained by Bessy, Gon{\\c{c}}alves, and Reinald \\cite{BGR-2024}.\n\nBurr's conjecture is known to hold for acyclic digraphs. In fact, it was proved by\nAddario-Berry, Havet, Sales, Reed, and Thomass\\'e \\cite{addario+-2013} that every $k$-chromatic acyclic digraph \ncontains every oriented tree on $k$ vertices. It would therefore be of interest to prove that\ndigraphs have acyclic subgraphs that retain the original chromatic number to some nontrivial extent.\nIndeed, this problem was formulated in \\cite{addario+-2013}.\nFor a digraph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$\nand let $f(k)$ be the minimum of $f(G)$ taken over all $k$-chromatic digraphs.\nIt is a folklore argument (see below) that $f(G) \\ge \\sqrt{\\chi(G)}$ (whence $f(k) \\ge \\sqrt{k}$), and it was proved in \\cite{addario+-2013}\nthat $f(k) \\ge 1+\\lfloor \\sqrt{k-1} \\rfloor$. Trivially, $f(3)=2$ and it is known that $f(4)=3$ \\cite{NY-2019,shapira-2022}. The special case of this parameter for the case of tournaments was considered\nby Nassar and the author in \\cite{NY-2019}. Let $g(k)$ be the maximum integer such that every $k$-vertex tournament has an acyclic subgraph with chromatic number $g(k)$. It was conjectured in \\cite{NY-2019}\nthat $g(k) = \\omega(\\sqrt{k})$, namely that the folklore lower bound can be improved by more than a constant factor. This conjecture has recently been resolved by Fox, Kwan, and Sudakov \\cite{FKS-2021} who proved by an\ningenious argument that $g(k)=\\Omega(k^{5/9-o(1)})$. In that same paper, they have shown that the exponent $5/9$\ncannot be improved beyond $3/4$. Returning back to $f(k)$, this result suggests that $f(k)=\\omega(\\sqrt{k})$ as well.\nOur main result shows that this is the case in a dense regime.\n\nWe shall require the following result of Gallai and Milgram \\cite{GM-1960} (see also \\cite{diestel-2005}).\n\\begin{lemma}\\label{l:gm}{\\rm\\cite{GM-1960}}\n Let $G$ be a digraph with $\\alpha(G)=s$. Then there is a set of $s$ pairwise vertex-disjoint directed paths of $G$ that cover all its vertices.\n\\end{lemma}\n\nFor the rest of this paper, all digraphs are assumed to be orientations. Clearly this assumption does not change the chromatic number, independence number, or bipartite independence number.\nFor a subset $S$ of vertices of an oriented graph $G$, let $G[S]$ denote the induced subgraph of $G$ on $S$.\nWe say that $S$ is an {\\em acyclic $k$-set} if $|S|=k$ and $G[S]$ is acyclic.\n\\begin{lemma}\\label{l:prob}\n Let $G$ be a $k$-vertex orientation with $\\alpha(G) \\le s$. For a random permutation $\\pi$, the probability that $G_\\pi$ has no edges is at most $(se/k)^k$.\n\\end{lemma}\n\\begin{proof}\n By Lemma \\ref{l:gm}, $G$ has a set of $s$ pairwise vertex-disjoint paths $P_1,\\ldots,P_s$ that cover each vertex.\n Let $p_i$ denote the number of vertices of $P_i$ (possibly $p_i=0$) and observe that $\\sum_{i=1}^s p_i = k$.\n Notice that for $G_\\pi$ to have no edges, all vertices of $P_i$ must\n appear in the reverse order they appear in $P_i$, which, for a random $\\pi$, occurs with probability\n $1/p_i!$. As the $P_i$'s are pairwise vertex-disjoint, we have that the probability that\n $G_\\pi$ has no edges is therefore at most the reciprocal of $\\prod_{i=1}^s p_i!$.\n Finally note that by logarithmic convexity of $x^x$,\n $$\n \\prod_{i=1}^s p_i! \\ge \\prod_{i=1}^s \\left( \\frac{p_i}{e}\\right)^{p_i} \\ge \\left( \\frac{\\sum_{i=1}^s p_i}{se}\\right)^{\\sum_{i=1}^s p_i}=\\left( \\frac{k}{se}\\right)^{k}\\;.\n $$\n\\end{proof}\n\\begin{corollary}\\label{coro:prob}\n If $G$ is an $n$-vertex orientation with $\\alpha(G) \\le s$ and $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then there is a permutation $\\pi$ such that $\\alpha(G_\\pi) \\le k$. In particular, $\\chi(G_\\pi) \\ge n/k$.\n\\end{corollary}\n\\begin{proof}\n Consider a random $\\pi$. For a set $S$ of $k$ vertices, the\n probability that $S$ is an independent set in $G_\\pi$ is either $0$ if $S$ is not acyclic, or, by Lemma \\ref{l:prob}, at\n most $(se/k)^k$ if $S$ is acyclic. Hence, if $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then by the union bound, $\\alpha(G_\\pi) \\le k$ with positive probability.\n\\end{proof}\n\n\\begin{lemma}\\label{l:deg}\n Let $G$ be an $m$-vertex orientation with $\\alpha(G) \\le s$. Then $G$ has a vertex whose in-degree and out-degree are both at least $m/4s-1/2$.\n\\end{lemma}\n\\begin{proof}\n Let $t=m/4s-1/2$. Assume the claim is false and partition $V(G)$ into parts $V_1$ and $V_2$ where all vertices of $V_1$ have out-degree smaller than $t$ and all vertices of $V_2$ have in-degree smaller than $t$. Without loss of generality, $|V_1| \\ge m/2$. Consider the underlying undirected graph of $G[V_1]$. Since $\\alpha(G[V_1]) \\le s$, there is a subgraph of this undirected graph\n with minimum degree at least $|V_1|/s-1 \\ge m/2s-1$. Returning to the directed version, there must be some vertex in this subgraph whose out-degree is not smaller than its in-degree, hence its out-degree is at least $(m/2s-1)/2=t$, contradicting the definition of $V_1$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:l22}\n Let $k = n^{\\Omega(1)}$ and $k \\ge s^7$. If an $n$-vertex orientation $G$ with $\\alpha(G) < s$ and $\\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \\ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$\n where $q = n's^6/k^{1-o(1)}$.\n\\end{lemma}\n\n\\begin{lemma}\\label{l:l23}\nSuppose $n^{1/3}s^{-2/3} \\le q \\le n^{1-\\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with\n$\\alpha(G) < s$ has a permutation $\\pi$ with $\\alpha(G_\\pi) \\le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\\chi(G_\\pi) \\ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.\n\\end{lemma}\n\n\\begin{lemma}\\label{l:l24}\n Let $n$ be sufficiently large and let $G$ be an $n$-vertex orientation with $\\alpha(G) < s$ and $\\alpha^*(G) < s$ having at least\n $M$ acyclic $k$-sets. If $q = 6000s^6 n\\log^2n/k$, then there is $k'$ satisfying $k-3k/\\log n \\le k' \\le k$ such that $G$ has an induced $(q,6sk/\\log n)$-almost-acyclic subgraph $G'$ on at least $e^{-8}kM^{1/k}$ vertices having at least $e^{-6k}M$ acyclic $k'$-sets.\n\\end{lemma}\n\\begin{proof}\n Let $t$ be a positive integer parameter to be set later.\n For each $i=0,\\ldots,t$ we define vertex sets $W_i,V_i,V'_i$ such that\n \\begin{enumerate}\n \\item\n $W_i$ is an ordered acyclic $i$-set, $W_i \\subset W_{i+1}$, and the ordering of $W_{i+1}$ is consistent with the ordering of $W_i$.\n \\item\n $V_0'=V(G)$, $V_i' \\supseteq V_i \\supset V_{i+1}' \\supseteq V_{i+1}$ for $i=0,\\ldots,t-1$.\n \\item\n $V_i' \\cap W_i = \\emptyset$.\n \\end{enumerate}\n Describing the iterative procedure building these sets requires several notions.\n \\begin{itemize}\n \\item\n Let $H'_i$ be the $(k-i)$-uniform hypergraph with vertex set $V'_i$ whose edges are the\n $(k-i)$-subsets $S$ for which $G[S \\cup W_i]$\n is an acyclic $k$-set for which there is a topological sort $\\sigma_S$ consistent with the ordering of $W_i$.\n Let $N'_i$ denote the number of edges of $H'_i$ and observe that $N'_0 \\ge M$.\n \\item\n Let $H_i$ be an induced subhypergraph of $H'_i$ with minimum degree at least $N'_i/|V'_i|$ (it will always be the case that $N'_i > 0$). Such a subhypergraph can be obtained by repeatedly removing vertices with degree smaller than\n $N'_i/|V'_i|$ until none are left. Let $N_i$ be the number of edges of $H_i$\n and let $V_i$ be its vertex set.\n \\item\n Let the ordering of $W_i$ be $\\{w_1,\\ldots,w_i\\}$ so that if $(w_j,w_{j'}) \\in E(G)$, then $j < j'$.\n Consider some vertex $v \\in V$ and some edge $S$ of $H_i$ containing $v$.\n Recall that $G[S \\cup W_i]$ is an acyclic $k$-set for which $\\sigma_S$ is a topological sort consistent with the ordering of $W_i$. So if $(w_j,v) \\in E(G)$ and $(v, w_{j'}) \\in E(G)$, it must be that $j < j'$. Thus, all out-neighbors of $v$ in $W_i$ (if there are any) appear in $W_i$'s ordering after all in-neighbors of $v$ in $W_i$ (if there are any). We say that the set of {\\em allowed bins} of $v$ with respect to $W_i$ is $\\{j,\\ldots,j'\\}$\n if $(w_j,v) \\in E(G)$, $(v,w_{j'+1}) \\in E(G)$ and $v$ is not adjacent in $G$ to any of\n $w_{j+1},\\ldots,w_{j'}$. If $v$ has no in-neighbors in $W_i$, then let $j=0$ and if $v$ has no out-neighbors in $W_i$, then let $j'=i$. Notice that for any edge $S$ of $H_i$ containing $v$,\n $\\sigma_S$ places $v$ after some $w_\\ell$ and before some $w_{\\ell+1}$ where $j \\le \\ell \\le j'$ (if $\\ell = 0$, then $j=0$ so $v$ is placed before $w_1$ and if $\\ell =i$, then $j'=i$ so $v$ is placed after all $w_i$). We say that bin $\\ell$ is {\\em associated with $(\\sigma_S,v)$}. One may think of $\\{w_1,\\ldots,w_i\\}$ as labeling the consecutive internal walls of a set of $i+1$ bins labeled $0,\\ldots,i$.\n \\item\n Two vertices $x,y$ of $V_i$ are {\\em incompatible} if $(x,y) \\in E(G)$ and every allowed bin of $x$ with respect to $W_i$ is {\\em after} every allowed bin of $y$ with respect to $W_i$\n (in particular, no edge of $H_i$ contains both $x$ and $y$).\n \\end{itemize}\n For $0 \\le i \\le t-1$, we shall go from step $i$ to step $i+1$ by picking some (carefully selected) vertex $w \\in V_i$ and some (carefully selected) allowed bin $\\ell$ of $w$. We add $v$ to $W_i$ to obtain $W_{i+1}$ with the ordering of $W_{i+1}$ being $\\{w_1,\\ldots,w_\\ell,w,w_{\\ell+1},\\ldots,w_i\\}$.\n We remove from $V_i$ all vertices incompatible with $w$ to obtain $V'_{i+1}$. The edges of $H'_{i+1}$ are obtained by taking all edges $S$ of $H_i$ which contain $w$ and for which $(\\sigma_S,w)$ is associated with $\\ell$, and keeping $S \\setminus w$ as an edge of $H'_{i+1}$.",
    "post_theorem_intro_text_len": 3344,
    "post_theorem_intro_text": "\\noindent\nTheorem \\ref{t:main} extends the result of Fox, Kwan, and Sudakov which is the case $s=0$.\nNotice that if $\\alpha^*(G) \\le s$, then $\\alpha(G) \\le 2s+1$ and thus $\\chi(G) \\ge n/(2s+1)$.\nSo, Theorem \\ref{t:main} significantly improves upon the folklore $\\sqrt{\\chi(G)}$ lower bound\nfor graphs with relatively small bipartite independence number.\nTheorem \\ref{t:main} is particularly interesting in the setting of {\\em random graphs}.\nLet ${\\cal G}(n,p)$ denote the  Erd\\H{o}s–R\\'enyi random $n$-vertex graph with edge probability $p$.\nIt is well known that if $p = n^{-o(1)}$, then $G \\sim {\\cal G}(n,p)$ almost surely\\footnote{As usual, ``almost surely'' means with probability tending to $1$ as $n$ tends to infinity.} has\n$\\chi(G) =n^{1-o(1)}$ and $\\alpha^*(G)=n^{o(1)}$ (see \\cite{bollobas-1988} for sharp estimates when $p$ is constant). We therefore have:\n\\begin{corollary}\\label{coro:main}\n\tLet $p = n^{-o(1)}$ and $G \\sim {\\cal G}(n,p)$. Then almost surely,\n\tevery orientation $\\vec{G}$ of $G$ has $f(\\vec{G}) \\ge n^{5/9-o(1)}$. Using $p=1/2$ we have that almost all $n$-vertex graphs $G$ have the property that every orientation $\\vec{G}$ of $G$ has $f(\\vec{G}) \\ge n^{5/9-o(1)}$.\n\\end{corollary}\n\nTo prove Theorem \\ref{t:main}, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively\nreduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. In Section \\ref{sec:lem} we prove several lemmas needed for our proof.\nIn Section \\ref{sec:fks} we revisit the proof of Fox, Kwan and Sudakov, state variants of their\nLemmas 2.2 (which is Lemma \\ref{l:l22} here) and 2.3 (which is Lemma \\ref{l:l23} here) and prove\nLemma \\ref{l:l23}. Lemma \\ref{l:l22}, which is the main technical lemma, is proved in Section \\ref{sec:l22}. Throughout the paper, we ignore rounding issues as long as they do not affect the asymptotic claims.\n\nWe end the introduction recalling the aforementioned folklore lower bound. Clearly, in every red/blue edge coloring\nof a graph $G$, $\\chi(G_r) \\cdot \\chi(G_b) \\ge \\chi(G)$ where $G_r$ and $G_b$ respectively denote the red and blue subgraphs. For every digraph $G$ and every linear order of its vertices, we can color red the edges going from left to right and color blue the edges going from right to left, observing that the obtained $G_r$ and $G_b$ are acyclic. Hence $\\max(\\chi(G_r),\\chi(G_b)) \\ge \\sqrt{\\chi(G)}$.\nNevertheless, despite the similarity between acyclic orientations and red/blue edge colorings, note that an analogous result such as Theorem \\ref{t:main} is, in general, false for red/blue edge colorings. To see this, consider the complete $n/s$-partite\ngraph with each part of order $s$; clearly in this case $\\alpha^*(G) \\le \\alpha(G)=s$.\nFurther partition the parts into $\\sqrt{n/s}$ buckets, each bucket consisting of$\\sqrt{n/s}$ parts. Now color blue all the edges with endpoints in parts belonging to the same bucket\nand color red all remaining edges. It is immediate to verify that $G_r$ and $G_b$ are both $\\sqrt{n/s}$ chromatic.",
    "sketch": "To prove Theorem \\ref{t:main}, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. In Section \\ref{sec:lem} we prove several lemmas needed for our proof. In Section \\ref{sec:fks} we revisit the proof of Fox, Kwan and Sudakov, state variants of their Lemmas 2.2 (which is Lemma \\ref{l:l22} here) and 2.3 (which is Lemma \\ref{l:l23} here) and prove Lemma \\ref{l:l23}. Lemma \\ref{l:l22}, which is the main technical lemma, is proved in Section \\ref{sec:l22}.",
    "expanded_sketch": "To prove the main theorem, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. Next we prove several lemmas needed for our proof. We then revisit the proof of Fox, Kwan and Sudakov, state variants of their Lemmas 2.2 (which is\n\\begin{lemma}\\label{l:l22}\n\tLet $k = n^{\\Omega(1)}$ and $k \\ge s^7$. If an $n$-vertex orientation $G$ with $\\alpha(G) < s$ and $\\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \\ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$\n\twhere $q =  n's^6/k^{1-o(1)}$.\n\\end{lemma}\nhere) and 2.3 (which is\n\\begin{lemma}\\label{l:l23}\nSuppose $n^{1/3}s^{-2/3} \\le q \\le n^{1-\\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with\n$\\alpha(G) < s$ has a permutation $\\pi$ with $\\alpha(G_\\pi) \\le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\\chi(G_\\pi) \\ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.\n\\end{lemma}\nhere) and prove the lemma above. The lemma above, which is the main technical lemma, is proved later.",
    "expanded_theorem": "\\label{t:main}\n\tLet $G$ be an $n$-vertex digraph with $\\alpha^*(G) \\le s$.\n\tThen, $f(G) \\ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then\n\t$f(G) \\ge n^{5/9-o(1)}$.",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let $G$ be an $n$-vertex orientation (that is, a digraph with no directed $2$-cycle). The bipartite independence number $\\alpha^*(G)$ is the largest integer $k$ for which there exist two disjoint $k$-vertex sets with no edges between them. Let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$, where chromatic number is taken for the underlying undirected graph and acyclic means having no directed cycle. If $\\alpha^*(G) \\le s$, which quantitative lower bound holds for $f(G)$, and what special case follows when $\\alpha^*(G)=n^{o(1)}$?",
      "correct_choice": {
        "label": "A",
        "text": "$f(G) \\ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then $f(G) \\ge n^{5/9-o(1)}$."
      },
      "choices": [
        {
          "label": "B",
          "text": "$f(G) \\ge n^{5/9-o(1)}s^{-3/2}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then $f(G) \\ge n^{5/9-o(1)}$."
        },
        {
          "label": "C",
          "text": "$f(G) \\ge n^{1/2-o(1)}s^{-14/9}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then $f(G) \\ge n^{1/2-o(1)}$."
        },
        {
          "label": "D",
          "text": "$f(G) \\ge n^{5/9-o(1)}s^{-14/9}$ whenever $\\alpha(G) \\le s$. In particular, if $\\alpha(G)=n^{o(1)}$, then $f(G) \\ge n^{5/9-o(1)}$."
        },
        {
          "label": "E",
          "text": "$f(G) \\ge n^{3/4-o(1)}s^{-3/2}$. In particular, if $\\alpha^*(G)=n^{o(1)}$, then $f(G) \\ge n^{3/4-o(1)}$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "final s-exponent from combining technical lemmas",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "lowered n-exponent from 5/9 to 1/2",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "hypothesis uses bipartite independence number rather than independence number",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "imports intermediate 3/4 exponent from almost-acyclic lemma as final theorem",
          "template_used": "stronger_trap"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the correct option directly; it asks for a quantitative bound, so the answer is not leaked from the wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct restatement of a specific theorem/result, asking the test-taker to identify the exact bound rather than derive or compare substantive conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some discrimination is required among close exponents and parameter variants, but the task is mainly recall/recognition of the theorem statement rather than genuine mathematical generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible: they vary exponents, swap in a weaker true bound, confuse independence number with bipartite independence number, or promote an intermediate estimate to the final claim."
      },
      "total_score": 5,
      "overall_assessment": "A solid theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than generative understanding."
    }
  },
  {
    "id": "2512.21963v1",
    "paper_link": "http://arxiv.org/abs/2512.21963v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:main}\nLet $\\kappa \\in\\mathbb{Z}\\setminus \\{4\\}$.\nThen the followings hold:\n\\begin{enumerate}\n\\item[$(1)$]\n$(\\mathrm{Theorems}~\\ref{thm-n=1},\\ \\ref{thm:n=2 non-planarity} \\  \\text{and}\\ \\ref{thm-n=(p-1)/2})$ \nFor each such $\\kappa$ and for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/2$, \nthere exists an explicit $K_{3,3}$-subdivision in $\\Gkp$.\n\n\\item[$(2)$] \n$(\\mathrm{Theorem}~\\ref{thm:np density})$\nMoreover, for almost all $\\kappa$, \nthe natural density of primes $p$ for which $(1)$ holds is at least $13/16$.\n\n\\end{enumerate}",
      "start_pos": 336981,
      "end_pos": 337592,
      "label": "thm:main"
    },
    "ref_dict": {
      "thm-n=1": "\\begin{theorem}\n\\label{thm-n=1}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ \nif $\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\ne 0$ and $\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1$.\n\\end{theorem}",
      "def:Markoff-equation": "\\begin{equation}\n\\label{def:Markoff-equation}\nx^2+y^2+z^2=xyz   \n\\end{equation}",
      "eq-number": "\\begin{align}\n\\label{eq-number}\n    \\# \\Mkp\n=p^2+\\left(3+\\left(\\frac{\\kappa}{p}\\right)\\right)\\left(\\frac{\\kappa-4}{p}\\right)p+1,\n\\end{align}",
      "cor:cycles": "\\begin{cor}\n\\label{cor:cycles}\nThe graph $\\Gkp$ contains cycles \nof lengths $4n+2$, $6n+3$, and $8n+2$ if there exists $n\\in\\mathbb{Z}_{>0}$ such that $\\Knkpprop\\ne\\emptyset$.\n\\end{cor}",
      "thm:np density": "\\begin{theorem}\n\\label{thm:np density}\n\\begin{itemize}\n\\item[$(1)$]\nFor each $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ such that none of  \n$\\eta_{1,\\kappa}$,  \n$\\eta_{2,\\kappa}$, \n$5$, and $\\kappa-4$\nis a square in $\\mathbb{Z}$, \nand that the ratio of any two of them is not a square in $\\mathbb{Q}$, \nwe have\n\\begin{equation}\n\\label{for:lower bound of Dnp}\n\\delta_{\\kappa}\\ge \\frac{13}{16}.\n\\end{equation}\nThat is, $\\Gkp$ contains a $K_{3,3}$-subdivision\nfor at least $81.25\\%$ \nof primes $p$ asymptotically.\n\\item[$(2)$]\nThe natural density \nof $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ for which $(1)$ holds \nis equal to $1$.\n\\end{itemize}\n\\end{theorem}",
      "thm:disjoint-intro": "\\begin{theorem}[Theorem~\\ref{thm-disjoint-k}]\n\\label{thm:disjoint-intro}\nFor each $\\kappa \\in \\mathbb{Z} \\setminus \\{ 4\\}$, $\\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/4$ if $\\kappa=2$ and $1/2$ otherwise.\nMoreover, for almost all $\\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.\n\\end{theorem}",
      "cor:direct": "\\begin{cor} \n\\label{cor:direct}\nUnder the same assumptions on $\\kappa$ and $p$ of Theorem~\\ref{thm:main} $(1)$,\nthe followings hold:\n\\begin{enumerate}\n\\item[$(1)$] \n$(\\mathrm{Corollary}~\\ref{cor:sufficient condition non-planarity})$\n$\\Gkp$ is non-planar. \n\\item[$(2)$]\n$(\\mathrm{Corollary}~\\ref{cor:cycles})$\n$\\Gkp$ has cycles (i.e. closed paths with no repeated vertices or edges) of lengths $6,9,10,15$, and $18$, \nall of which can be constructed explicitly.  \n\\end{enumerate}\n\\end{cor}",
      "def:Markoff-type equation": "\\begin{equation}\n\\label{def:Markoff-type equation}\nx^2+y^2+z^2=xyz+\\kappa.   \n\\end{equation}",
      "cor:sufficient condition non-planarity": "\\begin{cor}\n\\label{cor:sufficient condition non-planarity}\nThe graph $\\Gkp$ is non-planar if there exists $n\\in\\mathbb{Z}_{>0}$ such that $\\Knkpprop\\ne\\emptyset$.\n\\qed\n\\end{cor}",
      "thm-disjoint-k": "\\begin{theorem}\n\\label{thm-disjoint-k}\n\\begin{enumerate}\n\\item[$(1)$] \nFor each $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$,\n$\\Gkp$ contains at least four mutually vertex-disjoint \n$K_{3,3}$-subdivisions for infinitely many primes $p$ of natural density at least $1/4$ if $\\kappa=2$ and $1/2$ otherwise. \nFurthermore, if $\\kappa$ is an integer such that \nnone of $\\eta_{1, \\kappa}$, $\\eta_{2,\\kappa}$ and $5$ is a square in $\\mathbb{Z}$ and that the ratio of any two of them is also not a square in $\\mathbb{Q}$,\nthen the natural density of such primes is at least $5/8$.\n\\item[$(2)$]\nThe natural density of $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$\nfor which the latter assertion of $(1)$ holds is equal to $1$.\n\\end{enumerate}\n\\end{theorem}",
      "thm:n=2 non-planarity": "\\begin{theorem}\n\\label{thm:n=2 non-planarity}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ if $\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\ne 0$, $5(\\kappa^2-73\\kappa+61)\\ne 0$, \nand either of the following conditions holds. \n\\begin{itemize}\n\\item[$\\mathrm{(A)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=1$ and \n$\\left(\\dfrac{5}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1$, or,\n\\\\[0pt]\n\\item[$\\mathrm{(B)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=-1$ and $\\left(\\dfrac{5}{p}\\right)=1$.\n\\end{itemize}\n\\end{theorem}",
      "thm:embedded-intro": "\\begin{theorem}[Theorem~\\ref{thm-proj}]\n\\label{thm:embedded-intro}\nLet $p>3$ be a prime. Then the graph $\\Gzp$ admits an embedding into the torus \n(i.e. the orientable closed surface of Euler characteristic zero) \nor the (real) projective plane \n(i.e. the non-orientable closed surface of Euler characteristic one) \nif and only if $p=7$. \n\\end{theorem}",
      "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $\\kappa \\in\\mathbb{Z}\\setminus \\{4\\}$.\nThen the followings hold:\n\\begin{enumerate}\n\\item[$(1)$]\n$(\\mathrm{Theorems}~\\ref{thm-n=1},\\ \\ref{thm:n=2 non-planarity} \\  \\text{and}\\ \\ref{thm-n=(p-1)/2})$ \nFor each such $\\kappa$ and for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/2$, \nthere exists an explicit $K_{3,3}$-subdivision in $\\Gkp$.\n\n\\item[$(2)$] \n$(\\mathrm{Theorem}~\\ref{thm:np density})$\nMoreover, for almost all $\\kappa$, \nthe natural density of primes $p$ for which $(1)$ holds is at least $13/16$.\n\n\\end{enumerate}\n\\end{theorem}",
      "thm-proj": "\\begin{theorem}\n\\label{thm-proj}\nLet $p>3$ be a prime. Then we have the followings.\n\\begin{enumerate}\n\\item[$(1)$] The graph $\\Gzp$ is toroidal if and only if $p=7$. \n\\item[$(2)$] The graph $\\Gzp$ is projective-planar if and only if $p=7$. \n\\end{enumerate}\n\\end{theorem}",
      "thm-n=(p-1)/2": "\\begin{theorem}\n\\label{thm-n=(p-1)/2}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ if $\\left(\\frac{\\kappa-4}{p}\\right)=1$.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 5542,
    "pre_theorem_intro_text": "The Diophantine equation\n\\begin{equation}\n\\label{def:Markoff-equation}\nx^2+y^2+z^2=xyz   \n\\end{equation}\nis called the {\\it Markoff equation},\nand a solution $(x,y,z)$ is called a {\\it Markoff triple}.\nFor a prime $p>3$, \nlet $\\Mzp$ denote the set of all Markoff triples over $\\Fp$, the finite field with $p$ elements.\nThe {\\it Markoff mod $p$ graph} $\\Gzp$ is a finite $3$-regular graph (possibly  with loops) whose vertex set is $\\Mzp \\setminus \\{(0,0,0)\\}$, and two triples are adjacent if and only if they are transformed into each other by one of the following {\\it Vieta involutions}:\n\\[\nR_1(x,y,z)=(yz-x,y,z), \\quad \nR_2(x,y,z)=(x,zx-y,z), \\quad \nR_3(x,y,z)=(x,y,xy-z). \n\\]\n\nThe graph $\\Gzp$ has been extensively  studied in the literature on ``strong approximation'' for the Markoff equation (\\ref{def:Markoff-equation}). \nIndeed, Bourgain, Gamburd, and Sarnak~\\cite{BGS2016} conjectured that $\\Gzp$ is connected for all primes $p>3$. \nRecently Chen~\\cite{C2024} made a breakthrough toward this conjecture by proving that $\\Gzp$ is connected for any sufficiently large prime $p$.\nA key result here is that the size of any component of $\\Gzp$ must be divisible by $p$. Furthermore, for any $\\varepsilon>0$, there are only $O(p^{\\varepsilon})$ vertices that are not contained in the giant component (\\cite{BGS2016}) while $\\Gzp$ has approximately $p^2$ vertices (see (\\ref{eq-number}) below).\nMartin~\\cite{Martin2025} provided an elementary proof of Chen's result concerning the  divisibility of the component sizes of $\\Gzp$.\nIn addition, Eddy, Fuchs, Litman, Martin, and Tripeny~\\cite{EFLMT2025} established an explicit lower bound for primes $p$ for which the connectivity of $\\Gzp$ is guaranteed, namely, $p>3.448\\times10^{392}$.\nThere are several algorithmic results towards the connectivity of $\\Gzp$, and it was verified in \\cite{B2025} that $\\Gzp$ is connected for $p<10^6$.\n\nOn the other hand, de Courcy-Ireland~\\cite{C2024} established the non-planarity of $\\Gzp$: $\\Gzp$ is planar if and only if $p=7$. This provides (indirect but positive) evidence for another well-known conjecture by Bourgain, Gamburd, and Sarnak~\\cite{BGS2016, BGS2026} asserting that $\\{\\Gzp\\}_{\\text{$p$:prime}}$ is in fact an expander family, which would imply higher connectivity for $\\Gzp$ (for further details on expander family, see e.g. \\cite{HLW2004, L2012}).\nThe proof of non-planarity consists of two parts: an asymptotic evaluation of the Euler characteristic\nof $\\Gzp$ and explicit constructions of subdivisions of the complete bipartite graph $K_{3,3}$. \nThe Kuratowski-Wagner theorem shows a complete characterization for planar graphs in terms of graph minors, that is, a graph is planar if and only if it contains neither $K_{3,3}$ nor the complete graph $K_5$ as a graph minor.\nSince $\\Gzp$ is a $3$-regular graph, $K_{3,3}$ is the unique obstruction to planarity of $\\Gzp$ in the sense that \na $3$-regular graph cannot contain a subdivision of $K_5$, and more generally, any $3$-regular graph containing $K_5$ as a minor must also contain a subdivision of $K_{3,3}$.   \n\nThere are several generalizations of \\eqref{def:Markoff-equation}, such as the following equation: for a (rational) integer $\\kappa$,\n\\begin{equation}\n\\label{def:Markoff-type equation}\nx^2+y^2+z^2=xyz+\\kappa.   \n\\end{equation}\nThroughout this paper, we naturally treat $\\kappa$ as an element of $\\Fp$ when considering the solutions of \\eqref{def:Markoff-type equation} modulo $p$.\nFor the set of all such solutions over $\\Fp$, denoted by $\\Mkp$, \nCarlitz \\cite{C1954} proved that \n\\begin{align}\n\\label{eq-number}\n    \\# \\Mkp\n=p^2+\\left(3+\\left(\\frac{\\kappa}{p}\\right)\\right)\\left(\\frac{\\kappa-4}{p}\\right)p+1,\n\\end{align}\nwhere $(\\frac{\\cdot}{p})$ is the Legendre symbol.\nThe {\\it generalized Markoff mod $p$ graph} $\\Gkp$ (with respect to \\eqref{def:Markoff-type equation}) is defined as a $3$-regular graph\nwhose vertex set is $\\Mkp$ \nwith the adjacency relation defined in the same manner as for $\\Gzp$ (hence $\\mathcal{G}_0(p)$ coincides with $\\Gzp \\sqcup \\{(0,0,0)\\}$, where $(0,0,0)$ is an isolated vertex).\nThe graph structure of $\\Gkp$ depends sensitively on $\\kappa$ and $p$ in general.\nFor example, in the case of $\\kappa=4$ (corresponding to a degenerate surface called the {\\it Cayley cubic}, cf. \\cite{CL2020}), the graph $\\mathcal{G}_4(p)$ possesses many small components.\nWhile $\\Gkp$ is no longer connected in general,\nit is still expected that for $\\kappa\\neq 4$,\nthere exists a unique {\\it giant component} $\\mathcal{C}_\\kappa(p)$ in  \n$\\Gkp$ such that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=o_\\kappa(p^2)$. In a recent preprint~\\cite{Mart2025}, Martin announced that this in fact holds for a majority of primes.\nMoreover, the main result of \\cite{Mart2025} would imply that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=O(1)$, representing a significant advance in the understanding of the global structure of $\\Gkp$.\nHowever, the precise structure of $\\Gkp$ (and $\\mathcal{C}_\\kappa(p)$) is still largely unexplored.\n\nIn this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.\nIn particular, by leveraging computational algebraic methods including resultants and Gr\\\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.\nThroughout this paper, the term {\\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.",
    "context": "The Diophantine equation\n\\begin{equation}\n\\label{def:Markoff-equation}\nx^2+y^2+z^2=xyz \n\\end{equation}\nis called the {\\it Markoff equation},\nand a solution $(x,y,z)$ is called a {\\it Markoff triple}.\nFor a prime $p>3$, \nlet $\\Mzp$ denote the set of all Markoff triples over $\\Fp$, the finite field with $p$ elements.\nThe {\\it Markoff mod $p$ graph} $\\Gzp$ is a finite $3$-regular graph (possibly with loops) whose vertex set is $\\Mzp \\setminus \\{(0,0,0)\\}$, and two triples are adjacent if and only if they are transformed into each other by one of the following {\\it Vieta involutions}:\n\\[\nR_1(x,y,z)=(yz-x,y,z), \\quad \nR_2(x,y,z)=(x,zx-y,z), \\quad \nR_3(x,y,z)=(x,y,xy-z). \n\\]\n\nThe graph $\\Gzp$ has been extensively studied in the literature on ``strong approximation'' for the Markoff equation (\\ref{def:Markoff-equation}). \nIndeed, Bourgain, Gamburd, and Sarnak~\\cite{BGS2016} conjectured that $\\Gzp$ is connected for all primes $p>3$. \nRecently Chen~\\cite{C2024} made a breakthrough toward this conjecture by proving that $\\Gzp$ is connected for any sufficiently large prime $p$.\nA key result here is that the size of any component of $\\Gzp$ must be divisible by $p$. Furthermore, for any $\\varepsilon>0$, there are only $O(p^{\\varepsilon})$ vertices that are not contained in the giant component (\\cite{BGS2016}) while $\\Gzp$ has approximately $p^2$ vertices (see (\\ref{eq-number}) below).\nMartin~\\cite{Martin2025} provided an elementary proof of Chen's result concerning the divisibility of the component sizes of $\\Gzp$.\nIn addition, Eddy, Fuchs, Litman, Martin, and Tripeny~\\cite{EFLMT2025} established an explicit lower bound for primes $p$ for which the connectivity of $\\Gzp$ is guaranteed, namely, $p>3.448\\times10^{392}$.\nThere are several algorithmic results towards the connectivity of $\\Gzp$, and it was verified in \\cite{B2025} that $\\Gzp$ is connected for $p<10^6$.\n\nOn the other hand, de Courcy-Ireland~\\cite{C2024} established the non-planarity of $\\Gzp$: $\\Gzp$ is planar if and only if $p=7$. This provides (indirect but positive) evidence for another well-known conjecture by Bourgain, Gamburd, and Sarnak~\\cite{BGS2016, BGS2026} asserting that $\\{\\Gzp\\}_{\\text{$p$:prime}}$ is in fact an expander family, which would imply higher connectivity for $\\Gzp$ (for further details on expander family, see e.g. \\cite{HLW2004, L2012}).\nThe proof of non-planarity consists of two parts: an asymptotic evaluation of the Euler characteristic\nof $\\Gzp$ and explicit constructions of subdivisions of the complete bipartite graph $K_{3,3}$. \nThe Kuratowski-Wagner theorem shows a complete characterization for planar graphs in terms of graph minors, that is, a graph is planar if and only if it contains neither $K_{3,3}$ nor the complete graph $K_5$ as a graph minor.\nSince $\\Gzp$ is a $3$-regular graph, $K_{3,3}$ is the unique obstruction to planarity of $\\Gzp$ in the sense that \na $3$-regular graph cannot contain a subdivision of $K_5$, and more generally, any $3$-regular graph containing $K_5$ as a minor must also contain a subdivision of $K_{3,3}$.\n\nThere are several generalizations of \\eqref{def:Markoff-equation}, such as the following equation: for a (rational) integer $\\kappa$,\n\\begin{equation}\n\\label{def:Markoff-type equation}\nx^2+y^2+z^2=xyz+\\kappa. \n\\end{equation}\nThroughout this paper, we naturally treat $\\kappa$ as an element of $\\Fp$ when considering the solutions of \\eqref{def:Markoff-type equation} modulo $p$.\nFor the set of all such solutions over $\\Fp$, denoted by $\\Mkp$, \nCarlitz \\cite{C1954} proved that \n\\begin{align}\n\\label{eq-number}\n \\# \\Mkp\n=p^2+\\left(3+\\left(\\frac{\\kappa}{p}\\right)\\right)\\left(\\frac{\\kappa-4}{p}\\right)p+1,\n\\end{align}\nwhere $(\\frac{\\cdot}{p})$ is the Legendre symbol.\nThe {\\it generalized Markoff mod $p$ graph} $\\Gkp$ (with respect to \\eqref{def:Markoff-type equation}) is defined as a $3$-regular graph\nwhose vertex set is $\\Mkp$ \nwith the adjacency relation defined in the same manner as for $\\Gzp$ (hence $\\mathcal{G}_0(p)$ coincides with $\\Gzp \\sqcup \\{(0,0,0)\\}$, where $(0,0,0)$ is an isolated vertex).\nThe graph structure of $\\Gkp$ depends sensitively on $\\kappa$ and $p$ in general.\nFor example, in the case of $\\kappa=4$ (corresponding to a degenerate surface called the {\\it Cayley cubic}, cf. \\cite{CL2020}), the graph $\\mathcal{G}_4(p)$ possesses many small components.\nWhile $\\Gkp$ is no longer connected in general,\nit is still expected that for $\\kappa\\neq 4$,\nthere exists a unique {\\it giant component} $\\mathcal{C}_\\kappa(p)$ in \n$\\Gkp$ such that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=o_\\kappa(p^2)$. In a recent preprint~\\cite{Mart2025}, Martin announced that this in fact holds for a majority of primes.\nMoreover, the main result of \\cite{Mart2025} would imply that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=O(1)$, representing a significant advance in the understanding of the global structure of $\\Gkp$.\nHowever, the precise structure of $\\Gkp$ (and $\\mathcal{C}_\\kappa(p)$) is still largely unexplored.\n\nIn this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.\nIn particular, by leveraging computational algebraic methods including resultants and Gr\\\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.\nThroughout this paper, the term {\\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.",
    "full_context": "The Diophantine equation\n\\begin{equation}\n\\label{def:Markoff-equation}\nx^2+y^2+z^2=xyz \n\\end{equation}\nis called the {\\it Markoff equation},\nand a solution $(x,y,z)$ is called a {\\it Markoff triple}.\nFor a prime $p>3$, \nlet $\\Mzp$ denote the set of all Markoff triples over $\\Fp$, the finite field with $p$ elements.\nThe {\\it Markoff mod $p$ graph} $\\Gzp$ is a finite $3$-regular graph (possibly with loops) whose vertex set is $\\Mzp \\setminus \\{(0,0,0)\\}$, and two triples are adjacent if and only if they are transformed into each other by one of the following {\\it Vieta involutions}:\n\\[\nR_1(x,y,z)=(yz-x,y,z), \\quad \nR_2(x,y,z)=(x,zx-y,z), \\quad \nR_3(x,y,z)=(x,y,xy-z). \n\\]\n\nThe graph $\\Gzp$ has been extensively studied in the literature on ``strong approximation'' for the Markoff equation (\\ref{def:Markoff-equation}). \nIndeed, Bourgain, Gamburd, and Sarnak~\\cite{BGS2016} conjectured that $\\Gzp$ is connected for all primes $p>3$. \nRecently Chen~\\cite{C2024} made a breakthrough toward this conjecture by proving that $\\Gzp$ is connected for any sufficiently large prime $p$.\nA key result here is that the size of any component of $\\Gzp$ must be divisible by $p$. Furthermore, for any $\\varepsilon>0$, there are only $O(p^{\\varepsilon})$ vertices that are not contained in the giant component (\\cite{BGS2016}) while $\\Gzp$ has approximately $p^2$ vertices (see (\\ref{eq-number}) below).\nMartin~\\cite{Martin2025} provided an elementary proof of Chen's result concerning the divisibility of the component sizes of $\\Gzp$.\nIn addition, Eddy, Fuchs, Litman, Martin, and Tripeny~\\cite{EFLMT2025} established an explicit lower bound for primes $p$ for which the connectivity of $\\Gzp$ is guaranteed, namely, $p>3.448\\times10^{392}$.\nThere are several algorithmic results towards the connectivity of $\\Gzp$, and it was verified in \\cite{B2025} that $\\Gzp$ is connected for $p<10^6$.\n\nOn the other hand, de Courcy-Ireland~\\cite{C2024} established the non-planarity of $\\Gzp$: $\\Gzp$ is planar if and only if $p=7$. This provides (indirect but positive) evidence for another well-known conjecture by Bourgain, Gamburd, and Sarnak~\\cite{BGS2016, BGS2026} asserting that $\\{\\Gzp\\}_{\\text{$p$:prime}}$ is in fact an expander family, which would imply higher connectivity for $\\Gzp$ (for further details on expander family, see e.g. \\cite{HLW2004, L2012}).\nThe proof of non-planarity consists of two parts: an asymptotic evaluation of the Euler characteristic\nof $\\Gzp$ and explicit constructions of subdivisions of the complete bipartite graph $K_{3,3}$. \nThe Kuratowski-Wagner theorem shows a complete characterization for planar graphs in terms of graph minors, that is, a graph is planar if and only if it contains neither $K_{3,3}$ nor the complete graph $K_5$ as a graph minor.\nSince $\\Gzp$ is a $3$-regular graph, $K_{3,3}$ is the unique obstruction to planarity of $\\Gzp$ in the sense that \na $3$-regular graph cannot contain a subdivision of $K_5$, and more generally, any $3$-regular graph containing $K_5$ as a minor must also contain a subdivision of $K_{3,3}$.\n\nThere are several generalizations of \\eqref{def:Markoff-equation}, such as the following equation: for a (rational) integer $\\kappa$,\n\\begin{equation}\n\\label{def:Markoff-type equation}\nx^2+y^2+z^2=xyz+\\kappa. \n\\end{equation}\nThroughout this paper, we naturally treat $\\kappa$ as an element of $\\Fp$ when considering the solutions of \\eqref{def:Markoff-type equation} modulo $p$.\nFor the set of all such solutions over $\\Fp$, denoted by $\\Mkp$, \nCarlitz \\cite{C1954} proved that \n\\begin{align}\n\\label{eq-number}\n \\# \\Mkp\n=p^2+\\left(3+\\left(\\frac{\\kappa}{p}\\right)\\right)\\left(\\frac{\\kappa-4}{p}\\right)p+1,\n\\end{align}\nwhere $(\\frac{\\cdot}{p})$ is the Legendre symbol.\nThe {\\it generalized Markoff mod $p$ graph} $\\Gkp$ (with respect to \\eqref{def:Markoff-type equation}) is defined as a $3$-regular graph\nwhose vertex set is $\\Mkp$ \nwith the adjacency relation defined in the same manner as for $\\Gzp$ (hence $\\mathcal{G}_0(p)$ coincides with $\\Gzp \\sqcup \\{(0,0,0)\\}$, where $(0,0,0)$ is an isolated vertex).\nThe graph structure of $\\Gkp$ depends sensitively on $\\kappa$ and $p$ in general.\nFor example, in the case of $\\kappa=4$ (corresponding to a degenerate surface called the {\\it Cayley cubic}, cf. \\cite{CL2020}), the graph $\\mathcal{G}_4(p)$ possesses many small components.\nWhile $\\Gkp$ is no longer connected in general,\nit is still expected that for $\\kappa\\neq 4$,\nthere exists a unique {\\it giant component} $\\mathcal{C}_\\kappa(p)$ in \n$\\Gkp$ such that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=o_\\kappa(p^2)$. In a recent preprint~\\cite{Mart2025}, Martin announced that this in fact holds for a majority of primes.\nMoreover, the main result of \\cite{Mart2025} would imply that $\\#(\\Gkp \\setminus \\mathcal{C}_\\kappa(p))=O(1)$, representing a significant advance in the understanding of the global structure of $\\Gkp$.\nHowever, the precise structure of $\\Gkp$ (and $\\mathcal{C}_\\kappa(p)$) is still largely unexplored.\n\nIn this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.\nIn particular, by leveraging computational algebraic methods including resultants and Gr\\\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.\nThroughout this paper, the term {\\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.\n\nIn this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.\nIn particular, by leveraging computational algebraic methods including resultants and Gr\\\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.\nThroughout this paper, the term {\\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.\n\nTheorem~\\ref{thm:main} yields several topological properties of $\\Gkp$.\nIn particular, the following corollary is an immediate consequence of Theorem~\\ref{thm:main}.\n\n\\begin{theorem}[Theorem~\\ref{thm-disjoint-k}]\n\\label{thm:disjoint-intro}\nFor each $\\kappa \\in \\mathbb{Z} \\setminus \\{ 4\\}$, $\\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/4$ if $\\kappa=2$ and $1/2$ otherwise.\nMoreover, for almost all $\\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:n=2 non-planarity}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ if $\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\ne 0$, $5(\\kappa^2-73\\kappa+61)\\ne 0$, \nand either of the following conditions holds. \n\\begin{itemize}\n\\item[$\\mathrm{(A)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=1$ and \n$\\left(\\dfrac{5}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1$, or,\n\\\\[0pt]\n\\item[$\\mathrm{(B)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=-1$ and $\\left(\\dfrac{5}{p}\\right)=1$.\n\\end{itemize}\n\\end{theorem}\n\n\\subsection{Natural densities}\n\\label{subsec:density}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nFrom Theorem~\\ref{thm-n=1}, Theorem~\\ref{thm:n=2 non-planarity} (B),\nor Theorem~\\ref{thm-n=(p-1)/2}, \nit follows from Dirichlet's theorem on arithmetic progressions that there are infinitely many primes $p$ for which\n$\\Gkp$ contains a $K_{3,3}$-subdivision, which yields topological properties of $\\Gkp$ such as the non-planarity and existence of short cycles by Corollaries~\\ref{cor:sufficient condition non-planarity} and \\ref{cor:cycles}, respectively.\nThis motivates us to consider the natural density of such primes $p$: \\[\n\\delta_{\\kappa}\n\\coloneq\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x\\,|\\,\\text{$\\Gkp$ contains a $K_{3,3}$-subdivision}\\}}{\\#\\{p\\le x\\}}.\n\\]\nIt holds that $\\delta_0=1$\nsince $\\mathcal{G}(p)$ is non-planar for any $p\\neq 7$\nby \\cite[p.115]{C2024}.\nFor any $\\kappa \\neq 4$, Theorem~\\ref{thm-n=1} and the Chebotarev density theorem (see, e.g.~\\cite{N1999}) \nimply that $ \\delta_{\\kappa}\\geq 1/2$.\nMoreover, combining the three theorems above,\nwe arrive at the following theorem.\n\n\\begin{theorem}\n\\label{thm:np density}\n\\begin{itemize}\n\\item[$(1)$]\nFor each $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ such that none of \n$\\eta_{1,\\kappa}$, \n$\\eta_{2,\\kappa}$, \n$5$, and $\\kappa-4$\nis a square in $\\mathbb{Z}$, \nand that the ratio of any two of them is not a square in $\\mathbb{Q}$, \nwe have\n\\begin{equation}\n\\label{for:lower bound of Dnp}\n\\delta_{\\kappa}\\ge \\frac{13}{16}.\n\\end{equation}\nThat is, $\\Gkp$ contains a $K_{3,3}$-subdivision\nfor at least $81.25\\%$ \nof primes $p$ asymptotically.\n\\item[$(2)$]\nThe natural density \nof $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ for which $(1)$ holds \nis equal to $1$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm-disjoint-k}\n\\begin{enumerate}\n\\item[$(1)$] \nFor each $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$,\n$\\Gkp$ contains at least four mutually vertex-disjoint \n$K_{3,3}$-subdivisions for infinitely many primes $p$ of natural density at least $1/4$ if $\\kappa=2$ and $1/2$ otherwise. \nFurthermore, if $\\kappa$ is an integer such that \nnone of $\\eta_{1, \\kappa}$, $\\eta_{2,\\kappa}$ and $5$ is a square in $\\mathbb{Z}$ and that the ratio of any two of them is also not a square in $\\mathbb{Q}$,\nthen the natural density of such primes is at least $5/8$.\n\\item[$(2)$]\nThe natural density of $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$\nfor which the latter assertion of $(1)$ holds is equal to $1$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nThe four mutually vertex-disjoint $K_{3,3}$-subdivisions above are not necessarily contained in the giant component $\\mathcal{C}_{\\kappa}(p)$ in general.\nIndeed, for $\\kappa=2$ and $3$, then \nthe $K_{3,3}$-subdivision $K\\in \\mathcal{K}^{\\mathrm{prop}}_{1,\\kappa}(p)$ \nobtained from Theorem~\\ref{thm-n=1} (along with its three images under double sign changes) is contained in $\\mathcal{S}_{2}(p)$ and $\\mathcal{S}_{3}(p)$, respectively, where $\\mathcal{S}_{2}(p)$ is a unique $16$-vertex component in Remark~\\ref{rem-k=2} and $\\mathcal{S}_{3}(p)$ denotes a unique $72$-vertex component in $\\mathcal{G}_3(p)$ corresponds to the orbit of $((1+\\sqrt{5})/2, 1, 1) \\in \\mathcal{M}_{3}(p)$ \nunder the group generated by Vieta involutions, \nprovided that $(\\frac{5}{p})=1$ \n(see Figure~\\ref{fig:k3smallcomponent}).\nOn the other hand, according to a recent preprint \\cite{Mart2025}, for $\\kappa\\in \\mathbb{Z} \\setminus \\{2, 3, 4\\}$, \nall the constructed $K_{3,3}$-subdivisions are expected to be contained within $\\mathcal{C}_\\kappa(p)$ of $\\Gkp$. \nFurthermore, even in the cases $\\kappa=2$ and $3$, \nthe $K_{3,3}$-subdivision with 24 vertices constructed in Theorem~\\ref{thm:n=2 non-planarity} $(n=2)$ must be contained in $\\mathcal{C}_\\kappa(p)$.\nThis follows from the proof of Theorem~\\ref{thm-disjoint-k}, together with a simple observation that for $\\kappa=2$ and $3$, \nany component other than $\\mathcal{C}_\\kappa(p)$ has at most $72$ vertices\nand hence lacks sufficient space to accommodate all four mutually vertex-disjoint copies of the subdivision.\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth, height=0.2\n\\textheight, keepaspectratio]{Small_component_k=3_colored.pdf}\n\\caption{The $72$-vertex component $\\mathcal{S}_3(p)$ of $\\mathcal{G}_3(p)$ containing four mutually vertex-disjoint $K_{3,3}$-subdivisions as established in Theorem~\\ref{thm-disjoint-k}}\n\\label{fig:k3smallcomponent}\n\\end{center}\n\\end{figure}\n\\end{remark}",
    "post_theorem_intro_text_len": 6058,
    "post_theorem_intro_text": "Theorem~\\ref{thm:main} yields several topological properties of $\\Gkp$.\nIn particular, the following corollary is an immediate consequence of Theorem~\\ref{thm:main}.\n\n\\begin{cor} \n\\label{cor:direct}\nUnder the same assumptions on $\\kappa$ and $p$ of Theorem~\\ref{thm:main} $(1)$,\nthe followings hold:\n\\begin{enumerate}\n\\item[$(1)$] \n$(\\mathrm{Corollary}~\\ref{cor:sufficient condition non-planarity})$\n$\\Gkp$ is non-planar. \n\\item[$(2)$]\n$(\\mathrm{Corollary}~\\ref{cor:cycles})$\n$\\Gkp$ has cycles (i.e. closed paths with no repeated vertices or edges) of lengths $6,9,10,15$, and $18$, \nall of which can be constructed explicitly.  \n\\end{enumerate}\n\\end{cor}\nNote that it would be interesting to obtain results analogous to Theorem~\\ref{thm:main} and Corollary~\\ref{cor:direct} for other generalizations of the Markoff mod $p$ graph $\\Gzp$ (e.g. \\cite{CLM2025}); these will be discussed in our subsequent works.\n\nRecall that $\\Gkp$ admits the following non-trivial graph automorphisms, known as {\\it double sign changes}:\n\\[\n(x,y,z) \\mapsto (x,-y,-z), \\quad \n(x,y,z) \\mapsto (-x,y,-z), \\quad \\text{and} \\quad \n(x,y,z) \\mapsto (-x,-y,z).\n\\]\nThese automorphisms, together with the identity, form a group isomorphic to $(\\mathbb{Z}/2\\mathbb{Z})^2$.\nCombining this symmetry of $\\Gkp$ \nwith Theorem~\\ref{thm:main},\nwe obtain an explicit construction of mutually vertex-disjoint $K_{3,3}$-subdivisions.\n\n\\begin{theorem}[Theorem~\\ref{thm-disjoint-k}]\n\\label{thm:disjoint-intro}\nFor each $\\kappa \\in \\mathbb{Z} \\setminus \\{ 4\\}$, $\\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/4$ if $\\kappa=2$ and $1/2$ otherwise.\nMoreover, for almost all $\\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.\n\\end{theorem}\n\nTheorem~\\ref{thm:disjoint-intro} strengthens Corollary~\\ref{cor:direct} (1). Consider an embedding of $\\Gkp$ into a closed surface (i.e. a compact topological surface without boundary); see e.g.~\\cite{GT1987, MT2001} for the details on surface embeddings of graphs.\nRecall that the Euler characteristic $\\chi$ of $\\Gkp$ is defined by $V-E+F$, where $V, E$, and $F$\ndenote the numbers of vertices, edges and faces, respectively.\nCombining \\cite[Theorem~1.1]{Mart2025} with Theorem~\\ref{thm:disjoint-intro}, \nit follows that $\\chi\\leq -6$\nfor $\\kappa$ and $p$ as in Theorem~\\ref{thm:disjoint-intro},\nwhereas Euler's formula requires  $\\chi=2$ if $\\Gkp$ is planar.\nMoreover, the existence of mutually vertex-disjoint $K_{3,3}$-subdivisions highlights the robustness of the non-planarity of $\\Gkp$ in the sense that, by Theorem~\\ref{thm:disjoint-intro}, one must remove at least four vertices to make the graph planar.\nIt is worth noting that, in light of \\cite[Theorem~1.1]{Mart2025}, our construction of $K_{3,3}$-subdivisions implies that the Euler characteristic of the giant component $\\mathcal{C}_\\kappa(p)$ is at most $-6$ for any $\\kappa\\neq 4$\nand infinitely many primes $p$ (see Section~\\ref{sect:disjoint} for details).\nFrom these structural perspectives, our results serve as heuristic evidence for the expander property of the giant component $\\mathcal{C}_\\kappa(p)$.\nAt least, $\\mathcal{C}_\\kappa(p)$ in fact possesses certain complex structure similar to that of $\\Gzp$.\n\nIt is also worth mentioning that in cryptography and number theory, studying the structure of $\\Gkp$ is crucial for understanding the security of the associated cryptographic hash functions proposed in \\cite{FLLT2021, Sil2025}, which are expected to be expander hash functions~\\cite{CLG2009}.\nIn particular, it is a challenging open problem to investigate the existence and distribution of short cycles in $\\Gkp$, as these properties directly affect the collision resistance of the hash functions (\\cite{D2023, FLLT2021}).\nCorollary~\\ref{cor:direct} (2) provides a modest step toward addressing this problem.\n\nIn addition, for $\\kappa=0$, \nwe give a complete characterization for primes $p$ for which the graph $\\Gzp$ admits an embedding into closed surfaces of minimal topological complexity beyond the plane. \n\\begin{theorem}[Theorem~\\ref{thm-proj}]\n\\label{thm:embedded-intro}\nLet $p>3$ be a prime. Then the graph $\\Gzp$ admits an embedding into the torus \n(i.e. the orientable closed surface of Euler characteristic zero) \nor the (real) projective plane \n(i.e. the non-orientable closed surface of Euler characteristic one) \nif and only if $p=7$. \n\\end{theorem}\n\nThis result is achieved through Theorem~\\ref{thm:main} by finding suitable forbidden minors for these embeddings, as well as the numerical verification of the connectivity of $\\Gzp$ for $p<10^6$ (\\cite{B2025}).\nNote that the graph $\\mathcal{G}(7)$ is planar, and hence one can draw it on any closed surface without edge crossings. \nConsequently, Theorem~\\ref{thm:embedded-intro} immediately implies the non-planarity of $\\Gzp$ for any $p\\neq 7$, and hence strengthens the main result of \\cite{C2024}.\n\nThe rest of this paper is organized as follows. \nSection~\\ref{sect:chebyshev} introduces the basic and relevant properties of  Chebyshev polynomials, which play a key role in constructing $K_{3, 3}$-subdivisions in $\\Gkp$.\nIn Section~\\ref{sec:Six triples}, we present a systematic construction of $K_{3,3}$-subdivisions based on careful computations of Chebyshev polynomials and their associated resultants.\nBriefly, we first reduce the construction to the problem of solving systems of algebraic equations via Gr\\\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.  \nThen we establish sufficient conditions for the ``properness'', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.\nBased on these arguments, Section~\\ref{sect:non-planarity} provides explicit $K_{3,3}$-subdivisions.\nFinally, Sections~\\ref{sect:disjoint} and \\ref{sect:embedded} are devoted to the proofs of Theorems~\\ref{thm:disjoint-intro} and \\ref{thm:embedded-intro}, respectively.",
    "sketch": "The introduction outlines the overall strategy leading to Theorem~\\ref{thm:main} as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\\\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \\`\\`properness\\'\\', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” yielding Theorem~\\ref{thm:main}.",
    "expanded_sketch": "The introduction outlines the overall strategy leading to the main theorem as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\\\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \\`\\`properness\\'\\'”, which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” thereby establishing the main theorem.",
    "expanded_theorem": "\\label{thm:main}\nLet $\\kappa \\in\\mathbb{Z}\\setminus \\{4\\}$.\nThen the followings hold:\n\\begin{enumerate}\n\\item[$(1)$]\nThe following three theorems provide sufficient conditions for the existence of an explicit $K_{3,3}$-subdivision in $\\Gkp$:\n\\begin{theorem}\n\\label{thm-n=1}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ \nif $\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\ne 0$ and $\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1$.\n\\end{theorem}\n\\begin{theorem}\n\\label{thm:n=2 non-planarity}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ if $\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\ne 0$, $5(\\kappa^2-73\\kappa+61)\\ne 0$, \nand either of the following conditions holds. \n\\begin{itemize}\n\\item[$\\mathrm{(A)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=1$ and \n$\\left(\\dfrac{5}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)\n=\\left(\\dfrac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1$, or,\n\\\\[0pt]\n\\item[$\\mathrm{(B)}$] \n$\\left(\\dfrac{\\eta_{2,\\kappa}}{p}\\right)=-1$ and $\\left(\\dfrac{5}{p}\\right)=1$.\n\\end{itemize}\n\\end{theorem}\n\\begin{theorem}\n\\label{thm-n=(p-1)/2}\nLet $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$.\nThen, there exists an explicit $K_{3,3}$-subdivision in $\\Gkp$ if $\\left(\\frac{\\kappa-4}{p}\\right)=1$.\n\\end{theorem}\nFor each such $\\kappa$ and for infinitely many primes $p$ (depending on $\\kappa$) of natural density at least $1/2$, \nthere exists an explicit $K_{3,3}$-subdivision in $\\Gkp$.\n\n\\item[$(2)$] \nMoreover, the following theorem implies that for almost all $\\kappa$, \nthe natural density of primes $p$ for which $(1)$ holds is at least $13/16$.\n\\begin{theorem}\n\\label{thm:np density}\n\\begin{itemize}\n\\item[$(1)$]\nFor each $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ such that none of  \n$\\eta_{1,\\kappa}$,  \n$\\eta_{2,\\kappa}$, \n$5$, and $\\kappa-4$\nis a square in $\\mathbb{Z}$, \nand that the ratio of any two of them is not a square in $\\mathbb{Q}$, \nwe have\n\\begin{equation}\n\\label{for:lower bound of Dnp}\n\\delta_{\\kappa}\\ge \\frac{13}{16}.\n\\end{equation}\nThat is, $\\Gkp$ contains a $K_{3,3}$-subdivision\nfor at least $81.25\\%$ \nof primes $p$ asymptotically.\n\\item[$(2)$]\nThe natural density \nof $\\kappa\\in\\mathbb{Z}\\setminus\\{4\\}$ for which $(1)$ holds \nis equal to $1$.\n\\end{itemize}\n\\end{theorem}\n\n\\end{enumerate},",
    "theorem_type": [
      "Existence",
      "Universal–Existential"
    ],
    "mcq": {
      "question": "Let \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\). For a prime \\(p>3\\), let \\(\\mathcal M_\\kappa(p)\\subset \\mathbb F_p^3\\) be the set of solutions to \\(x^2+y^2+z^2=xyz+\\kappa\\), and let \\(\\mathcal G_\\kappa(p)\\) be the graph with vertex set \\(\\mathcal M_\\kappa(p)\\), where two vertices are adjacent exactly when one is obtained from the other by a Vieta involution \\(R_1(x,y,z)=(yz-x,y,z)\\), \\(R_2(x,y,z)=(x,zx-y,z)\\), or \\(R_3(x,y,z)=(x,y,xy-z)\\). Write \\(\\left(\\frac{\\cdot}{p}\\right)\\) for the Legendre symbol, and let \\(\\xi_{1,\\kappa},\\eta_{1,\\kappa},\\xi_{2,\\kappa},\\eta_{2,\\kappa}\\) denote the auxiliary integers attached to \\(\\kappa\\). Which statement about subdivisions of \\(K_{3,3}\\) in \\(\\mathcal G_\\kappa(p)\\) holds?",
      "correct_choice": {
        "label": "A",
        "text": "An explicit \\(K_{3,3}\\)-subdivision exists in \\(\\mathcal G_\\kappa(p)\\) whenever at least one of the following holds: (i) \\(\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\neq 0\\) and \\(\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1\\); (ii) \\(\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\neq 0\\), \\(5(\\kappa^2-73\\kappa+61)\\neq 0\\), and either (A) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=1\\) and \\(\\left(\\frac{5}{p}\\right)=\\left(\\frac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=\\left(\\frac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1\\), or (B) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=-1\\) and \\(\\left(\\frac{5}{p}\\right)=1\\); (iii) \\(\\left(\\frac{\\kappa-4}{p}\\right)=1\\). Consequently, for every \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\), the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision is infinite and has natural density at least \\(1/2\\). Moreover, if none of \\(\\eta_{1,\\kappa},\\eta_{2,\\kappa},5,\\kappa-4\\) is a square in \\(\\mathbb Z\\) and the ratio of any two of them is not a square in \\(\\mathbb Q\\), then the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains a \\(K_{3,3}\\)-subdivision has natural density at least \\(13/16\\) (equivalently, at least \\(81.25\\%\\) of primes asymptotically), and the set of integers \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\) satisfying this arithmetic hypothesis has natural density \\(1\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "An explicit \\(K_{3,3}\\)-subdivision exists in \\(\\mathcal G_\\kappa(p)\\) whenever at least one of the following holds: (i) \\(\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\neq 0\\) and \\(\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1\\); (ii) \\(\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\neq 0\\), \\(5(\\kappa^2-73\\kappa+61)\\neq 0\\), and either (A) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=1\\) and \\(\\left(\\frac{5}{p}\\right)=\\left(\\frac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=\\left(\\frac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1\\), or (B) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=-1\\) and \\(\\left(\\frac{5}{p}\\right)=-1\\); (iii) \\(\\left(\\frac{\\kappa-4}{p}\\right)=1\\). Consequently, for every \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\), the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision is infinite and has natural density at least \\(1/2\\). Moreover, if none of \\(\\eta_{1,\\kappa},\\eta_{2,\\kappa},5,\\kappa-4\\) is a square in \\(\\mathbb Z\\) and the ratio of any two of them is not a square in \\(\\mathbb Q\\), then the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains a \\(K_{3,3}\\)-subdivision has natural density at least \\(13/16\\), and the set of integers \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\) satisfying this arithmetic hypothesis has natural density \\(1\\)."
        },
        {
          "label": "C",
          "text": "If at least one of the following holds for a prime \\(p>3\\): (i) \\(\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\neq 0\\) and \\(\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1\\); (ii) \\(\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\neq 0\\), \\(5(\\kappa^2-73\\kappa+61)\\neq 0\\), and either (A) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=1\\) together with \\(\\left(\\frac{5}{p}\\right)=\\left(\\frac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=\\left(\\frac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1\\), or (B) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=-1\\) and \\(\\left(\\frac{5}{p}\\right)=1\\); or (iii) \\(\\left(\\frac{\\kappa-4}{p}\\right)=1\\), then \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision. Consequently, for every \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\), there are infinitely many primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision."
        },
        {
          "label": "D",
          "text": "An explicit \\(K_{3,3}\\)-subdivision exists in \\(\\mathcal G_\\kappa(p)\\) whenever at least one of the following holds: (i) \\(\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\neq 0\\) and \\(\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1\\); (ii) \\(\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\neq 0\\), \\(5(\\kappa^2-73\\kappa+61)\\neq 0\\), and \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=1\\) with \\(\\left(\\frac{5}{p}\\right)=\\left(\\frac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=\\left(\\frac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1\\); (iii) \\(\\left(\\frac{\\kappa-4}{p}\\right)=1\\). Consequently, for every \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\), the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision is infinite and has natural density at least \\(1/2\\). Moreover, if none of \\(\\eta_{1,\\kappa},\\eta_{2,\\kappa},5,\\kappa-4\\) is a square in \\(\\mathbb Z\\) and the ratio of any two of them is not a square in \\(\\mathbb Q\\), then the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains a \\(K_{3,3}\\)-subdivision has natural density at least \\(13/16\\), and the set of integers \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\) satisfying this arithmetic hypothesis has natural density \\(1\\)."
        },
        {
          "label": "E",
          "text": "An explicit \\(K_{3,3}\\)-subdivision exists in \\(\\mathcal G_\\kappa(p)\\) whenever at least one of the following holds: (i) \\(\\xi_{1,\\kappa}\\eta_{1,\\kappa}\\neq 0\\) and \\(\\left(\\frac{\\eta_{1,\\kappa}}{p}\\right)=1\\); (ii) \\(\\xi_{2,\\kappa}\\eta_{2,\\kappa}\\neq 0\\), \\(5(\\kappa^2-73\\kappa+61)\\neq 0\\), and either (A) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=1\\) and \\(\\left(\\frac{5}{p}\\right)=\\left(\\frac{8\\kappa-17+2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=\\left(\\frac{8\\kappa-17-2\\sqrt{\\eta_{2,\\kappa}}}{p}\\right)=1\\), or (B) \\(\\left(\\frac{\\eta_{2,\\kappa}}{p}\\right)=-1\\) and \\(\\left(\\frac{5}{p}\\right)=1\\); (iii) \\(\\left(\\frac{\\kappa-4}{p}\\right)=1\\). Consequently, for every \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\), the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains an explicit \\(K_{3,3}\\)-subdivision is infinite and has natural density at least \\(1/2\\). Moreover, if none of \\(\\eta_{1,\\kappa},\\eta_{2,\\kappa},5,\\kappa-4\\) is a square in \\(\\mathbb Z\\) and the ratio of any two of them is not a square in \\(\\mathbb Q\\), then the set of primes \\(p\\) for which \\(\\mathcal G_\\kappa(p)\\) contains a \\(K_{3,3}\\)-subdivision has natural density at least \\(3/4\\), and the set of integers \\(\\kappa\\in\\mathbb Z\\setminus\\{4\\}\\) satisfying this arithmetic hypothesis has natural density \\(1\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "computational_check",
          "tampered_component": "Legendre-sign condition in case (ii)(B)",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "density conclusions beyond infinitude",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "omission of alternative case (ii)(B)",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "lower-density bound 13/16 replaced by 3/4",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly or implicitly identify the correct option. It asks for the valid existence statement, and the alternatives are close enough that the answer is not leaked by wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the student is asked to choose which full existence statement from the paper is correct. It is very close to a direct restatement rather than an application or inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "Some careful comparison is required, since the distractors differ by subtle sign changes, omitted subcases, or weakened conclusions. However, the task mainly tests precise recall/textual matching, not genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and well-targeted: they reflect realistic failure modes such as dropping a case, flipping a Legendre-symbol sign, or weakening the theorem. They are distinct without being obviously absurd."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall/comparison MCQ with strong distractors, but it scores poorly on tautology avoidance and only modestly on generative reasoning because it mainly asks for recognition of the paper’s theorem statement."
    }
  },
  {
    "id": "2512.22459v1",
    "paper_link": "http://arxiv.org/abs/2512.22459v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{main} Let $G$ be a primitive permutation group with socle $\\hbox{\\rm PSU}(3,q)$, where $q=p^m\\ge 7$ for an odd prime $p$. Suppose a point-stabilizer\n  $M$ of $G$ contains $\\hbox{\\rm PSO}(3,q)$. Then $b(G)=2$ and either (i) $G=\\hbox{\\rm PSU}(3,q)$ and $M=\\hbox{\\rm SO}(3,q)$; or (ii) $G=\\hbox{\\rm PSU}(3,q)\\ZZ_{m_1}$ and  $M=\\hbox{\\rm SO}(3,q)\\ZZ_{m_1}$ $(m_1\\mid 2m$ and  $m_1\\ne  1)$. Further, the BG-Conjecture is valid for  case (i);\n  and for   $\\sqrt{q}\\geq 17$ or $45$,  if $3\\nmid (q+1)$  and $3\\bigm| (q+1)$, respectively in case (ii).",
      "start_pos": 13243,
      "end_pos": 13762,
      "label": "main"
    },
    "ref_dict": {
      "main": "\\begin{theorem} \\label{main} Let $G$ be a primitive permutation group with socle $\\PSU(3,q)$, where $q=p^m\\ge 7$ for an odd prime $p$. Suppose a point-stabilizer\n  $M$ of $G$ contains $\\PSO(3,q)$. Then $b(G)=2$ and either (i) $G=\\PSU(3,q)$ and $M=\\SO(3,q)$; or (ii) $G=\\PSU(3,q)\\ZZ_{m_1}$ and  $M=\\SO(3,q)\\ZZ_{m_1}$ $(m_1\\mid 2m$ and  $m_1\\ne  1)$. Further, the BG-Conjecture is valid for  case (i);\n  and for   $\\sqrt{q}\\geq 17$ or $45$,  if $3\\nmid (q+1)$  and $3\\di (q+1)$, respectively in case (ii).\n  \\end{theorem}",
      "main2": "\\begin{theorem} \\label{main2}  Let $G=\\PSU(3,q)\\ZZ_{m_1}$ and\n    $\\a=M=\\PSO(3,q)\\ZZ_{m_1}\\le G$, with  $q=p^m\\ge 7$ for an odd prime $p$,  $m_1\\mid 2m$ and $m_1\\ne 1$.\nUnder the primitive right multiplication action of $G$ on $\\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\\sqrt{q}\\geq 17$ or $45$ if $3\\nmid (q+1)$ and $3\\di (q+1)$, respectively.\n  \\end{theorem}",
      "main1": "\\begin{theorem} \\label{main1}  Let $G=\\PSU(3,q)$ and $M=\\PSO(3,q),$ a maximal subgroup of $G$, where $q=p^m\\ge 7$ for an odd prime $p$.\nConsider the primitive right multiplication action  of $G$ on the set of right cosets $\\Omega =[G:M]$.  Then $b(G)=2$ and the BG-Conjecture holds.\n  \\end{theorem}",
      "BG": "\\begin{conj}{\\rm \\cite{BG}} \\label{BG}\nLet $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\\Sigma(G)$ have a common neighbour.\n\\end{conj}"
    },
    "pre_theorem_intro_text_len": 6106,
    "pre_theorem_intro_text": "A {\\em base} for a finite permutation group $G$ on a set $\\Omega$ is a subset $\\Delta\\subset \\Omega$ whose pointwise  stabilizer $G_{(\\Delta)}$ is trivial.  This  concept  is a natural generalization of basis for  vector spaces. The {\\em base size} $b(G)$ of $G$  is the minimal cardinality $|\\Delta|$, where $\\Delta$ goes  over all bases for $G$, while  a  base $\\Delta '$ with $|\\Delta '|=b(G)$  is called a {\\em base size set}. With a history dating to the 19th century, these concepts remain vital to contemporary algebra and combinatorics; see \\cite{Bailey-Cameron,LS} and \\cite[$\\S5$]{B} for overviews and \\cite{G,S} for applications.\n\nLet $G\\leq \\hbox{\\rm Sym}(\\Omega)$ and $\\Delta$  a base for $G$. If $x,y\\in G$, then $\\alpha^x=\\alpha^y$ for all $\\alpha\\in\\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\\Delta$. In particular, if $\\Omega$ is finite, then $|G|\\leq |\\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \\cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.\n Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \\cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.\n\nPrimitive permutation groups with a base of size $1$ are cyclic groups $\\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.\nSaxl's ``base-two programme\", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \\cite{BG} introduced the {\\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\\Sigma(G)$ (denoted by $\\Sigma$, simply) of $G$ is just\n$\\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\\Sigma(G)$ is connected, but the converse is not true.\n  In \\cite{BG}, the authors discussed  the valency, connectivity, hamiltonicity and the independence number of $\\Sigma(G)$, and   proposed the following conjecture.\n\\vskip -10pt\n\\begin{conj}{\\rm \\cite{BG}} \\label{BG}\nLet $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\\Sigma(G)$ have a common neighbour.\n\\end{conj}\n\\vskip -5pt\n\n From now on, we assume that $G\\leq \\hbox{\\rm Sym}(\\Omega)$ acts transitively on $\\Omega$. Fixing a point $\\alpha\\in \\Omega$, we call the orbits of $G_\\alpha$ the {\\em suborbits} of $G$ relative to $\\alpha$; the trivial suborbit is $\\{\\alpha\\}$ itself.\n   Now $\\{\\alpha,\\beta\\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\\alpha}$ acts regularly on the suborbit containing $\\beta$.\n   It follows that the neighborhood $\\si_1(\\alpha)$ of $\\alpha$ in\n  $\\Sigma(G)$ is the union $\\Gamma$ of all regular suborbits of $G$ relative to $\\alpha$. Consequently,\n  Conjecture~\\ref{BG} can be restated in terms of $\\Gamma$ as follows, which we refer to as the BG-Conjecture.\n\\begin{conj}$($BG-Conjecture$)$ \\label{BG1}\n Every finite primitive permutation group $G$ with $b(G)=2$ has the property  that\n   $\\Gamma\\cap \\Gamma^g\\ne \\emptyset$ for any $g\\in G$, where $\\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\\alpha$.\n\\end{conj}\n\\vskip -5pt\n\nUsing a probabilistic approach, Burness and Giudici \\cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \\cite{HPHD}).\nFurthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \\cite{BG} also confirms the conjecture for many sporadic simple groups.\nBurness and Huang \\cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \\cite{BH2}.\nLee and Popiel \\cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs\nand Saxl hypergraphs in\n\\cite{FH, LP1}.\n\nSo far this conjecture remains open for all  primitive  groups $G$ whose socle is a simple group of  Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.\n  This class of groups contains four families, with  respective socles $\\hbox{\\rm PSL}(2,q)$,   $\\hbox{\\rm PSU}(3,q)$, $\\hbox{\\rm Ree}(q)$ and $\\hbox{\\rm Sz}(q)$.\nHaving established the conjecture for groups with socle $\\hbox{\\rm PSL}(2,q)$ in \\cite{BH,Chen-Du},\n  we now address the case of $\\hbox{\\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\\hbox{\\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\\ref{main}). Part II (i.e.~\\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \\cite{Chen-Du2}) will present the cases  $\\hbox{\\rm soc}(G)\\in \\{\\hbox{\\rm Ree}(q)$, $\\hbox{\\rm Sz}(q)$\\}.\nWe now state the main theorem of this paper.",
    "context": "Let $G\\leq \\hbox{\\rm Sym}(\\Omega)$ and $\\Delta$ a base for $G$. If $x,y\\in G$, then $\\alpha^x=\\alpha^y$ for all $\\alpha\\in\\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\\Delta$. In particular, if $\\Omega$ is finite, then $|G|\\leq |\\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \\cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.\n Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \\cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.\n\nPrimitive permutation groups with a base of size $1$ are cyclic groups $\\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.\nSaxl's ``base-two programme\", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \\cite{BG} introduced the {\\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\\Sigma(G)$ (denoted by $\\Sigma$, simply) of $G$ is just\n$\\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\\Sigma(G)$ is connected, but the converse is not true.\n In \\cite{BG}, the authors discussed the valency, connectivity, hamiltonicity and the independence number of $\\Sigma(G)$, and proposed the following conjecture.\n\\vskip -10pt\n\\begin{conj}{\\rm \\cite{BG}} \\label{BG}\nLet $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\\Sigma(G)$ have a common neighbour.\n\\end{conj}\n\\vskip -5pt\n\nFrom now on, we assume that $G\\leq \\hbox{\\rm Sym}(\\Omega)$ acts transitively on $\\Omega$. Fixing a point $\\alpha\\in \\Omega$, we call the orbits of $G_\\alpha$ the {\\em suborbits} of $G$ relative to $\\alpha$; the trivial suborbit is $\\{\\alpha\\}$ itself.\n Now $\\{\\alpha,\\beta\\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\\alpha}$ acts regularly on the suborbit containing $\\beta$.\n It follows that the neighborhood $\\si_1(\\alpha)$ of $\\alpha$ in\n $\\Sigma(G)$ is the union $\\Gamma$ of all regular suborbits of $G$ relative to $\\alpha$. Consequently,\n Conjecture~\\ref{BG} can be restated in terms of $\\Gamma$ as follows, which we refer to as the BG-Conjecture.\n\\begin{conj}$($BG-Conjecture$)$ \\label{BG1}\n Every finite primitive permutation group $G$ with $b(G)=2$ has the property that\n $\\Gamma\\cap \\Gamma^g\\ne \\emptyset$ for any $g\\in G$, where $\\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\\alpha$.\n\\end{conj}\n\\vskip -5pt\n\nUsing a probabilistic approach, Burness and Giudici \\cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \\cite{HPHD}).\nFurthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \\cite{BG} also confirms the conjecture for many sporadic simple groups.\nBurness and Huang \\cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \\cite{BH2}.\nLee and Popiel \\cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs\nand Saxl hypergraphs in\n\\cite{FH, LP1}.\n\nSo far this conjecture remains open for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.\n This class of groups contains four families, with respective socles $\\hbox{\\rm PSL}(2,q)$, $\\hbox{\\rm PSU}(3,q)$, $\\hbox{\\rm Ree}(q)$ and $\\hbox{\\rm Sz}(q)$.\nHaving established the conjecture for groups with socle $\\hbox{\\rm PSL}(2,q)$ in \\cite{BH,Chen-Du},\n we now address the case of $\\hbox{\\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\\hbox{\\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\\ref{main}). Part II (i.e.~\\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \\cite{Chen-Du2}) will present the cases $\\hbox{\\rm soc}(G)\\in \\{\\hbox{\\rm Ree}(q)$, $\\hbox{\\rm Sz}(q)$\\}.\nWe now state the main theorem of this paper.",
    "full_context": "Let $G\\leq \\hbox{\\rm Sym}(\\Omega)$ and $\\Delta$ a base for $G$. If $x,y\\in G$, then $\\alpha^x=\\alpha^y$ for all $\\alpha\\in\\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\\Delta$. In particular, if $\\Omega$ is finite, then $|G|\\leq |\\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \\cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.\n Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \\cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.\n\nPrimitive permutation groups with a base of size $1$ are cyclic groups $\\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.\nSaxl's ``base-two programme\", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \\cite{BG} introduced the {\\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\\Sigma(G)$ (denoted by $\\Sigma$, simply) of $G$ is just\n$\\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\\Sigma(G)$ is connected, but the converse is not true.\n In \\cite{BG}, the authors discussed the valency, connectivity, hamiltonicity and the independence number of $\\Sigma(G)$, and proposed the following conjecture.\n\\vskip -10pt\n\\begin{conj}{\\rm \\cite{BG}} \\label{BG}\nLet $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\\Sigma(G)$ have a common neighbour.\n\\end{conj}\n\\vskip -5pt\n\nFrom now on, we assume that $G\\leq \\hbox{\\rm Sym}(\\Omega)$ acts transitively on $\\Omega$. Fixing a point $\\alpha\\in \\Omega$, we call the orbits of $G_\\alpha$ the {\\em suborbits} of $G$ relative to $\\alpha$; the trivial suborbit is $\\{\\alpha\\}$ itself.\n Now $\\{\\alpha,\\beta\\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\\alpha}$ acts regularly on the suborbit containing $\\beta$.\n It follows that the neighborhood $\\si_1(\\alpha)$ of $\\alpha$ in\n $\\Sigma(G)$ is the union $\\Gamma$ of all regular suborbits of $G$ relative to $\\alpha$. Consequently,\n Conjecture~\\ref{BG} can be restated in terms of $\\Gamma$ as follows, which we refer to as the BG-Conjecture.\n\\begin{conj}$($BG-Conjecture$)$ \\label{BG1}\n Every finite primitive permutation group $G$ with $b(G)=2$ has the property that\n $\\Gamma\\cap \\Gamma^g\\ne \\emptyset$ for any $g\\in G$, where $\\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\\alpha$.\n\\end{conj}\n\\vskip -5pt\n\nUsing a probabilistic approach, Burness and Giudici \\cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \\cite{HPHD}).\nFurthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \\cite{BG} also confirms the conjecture for many sporadic simple groups.\nBurness and Huang \\cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \\cite{BH2}.\nLee and Popiel \\cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs\nand Saxl hypergraphs in\n\\cite{FH, LP1}.\n\nSo far this conjecture remains open for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.\n This class of groups contains four families, with respective socles $\\hbox{\\rm PSL}(2,q)$, $\\hbox{\\rm PSU}(3,q)$, $\\hbox{\\rm Ree}(q)$ and $\\hbox{\\rm Sz}(q)$.\nHaving established the conjecture for groups with socle $\\hbox{\\rm PSL}(2,q)$ in \\cite{BH,Chen-Du},\n we now address the case of $\\hbox{\\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\\hbox{\\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\\ref{main}). Part II (i.e.~\\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \\cite{Chen-Du2}) will present the cases $\\hbox{\\rm soc}(G)\\in \\{\\hbox{\\rm Ree}(q)$, $\\hbox{\\rm Sz}(q)$\\}.\nWe now state the main theorem of this paper.\n\nUsing a probabilistic approach, Burness and Giudici \\cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \\cite{HPHD}).\nFurthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \\cite{BG} also confirms the conjecture for many sporadic simple groups.\nBurness and Huang \\cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \\cite{BH2}.\nLee and Popiel \\cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs\nand Saxl hypergraphs in\n\\cite{FH, LP1}.\n\nRemark that for the reason of the length of the paper, we are not able to improve the lower bound $\\sqrt{q}\\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\\ref{main}.\n Hopefully, these small cases may be verified directly with a sufficiently powerful computer.\n\nCombining Theorem~\\ref{main} and the main results of \\cite{BH,Chen-Du,Chen-Du2}, we get\n\\begin{theorem} \\label{main'} Let $G$ be a primitive group whose socle is a simple group of Lie-type of rank $1$ such that $b(G)=2$.\n Then the BG-Conjecture is valid, except for $G=\\PSU(3,q)\\ZZ_{m_1}$ and $M=\\SO(3,q)\\ZZ_{m_1}$ $(m_1\\mid 2m$ and $m_1\\ne 1)$, where\n $\\sqrt{q}\\le 17$ or $45$, if $3\\nmid (q+1)$ and $3\\di (q+1)$, respectively.\n \\end{theorem}\n\n\\section{$\\PSL(3,q)\\lvertneqq G$ and $\\SO(3,q)\\lvertneqq M$}\\label{ell}\n In this section, we mainly prove the following theorem.\n\\vskip -7pt \\begin{theorem} \\label{main2} Let $G=\\PSU(3,q)\\ZZ_{m_1}$ and\n $\\a=M=\\PSO(3,q)\\ZZ_{m_1}\\le G$, with $q=p^m\\ge 7$ for an odd prime $p$, $m_1\\mid 2m$ and $m_1\\ne 1$.\nUnder the primitive right multiplication action of $G$ on $\\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\\sqrt{q}\\geq 17$ or $45$ if $3\\nmid (q+1)$ and $3\\di (q+1)$, respectively.\n \\end{theorem}\n\n\\section{A lower bound of $|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|$}\n Theorem~\\ref{main21} is the main result of this section.\n\\begin{theorem}\\label{main21}\nUsing the notation from Section~\\ref{ell},\nconsider the right multiplication action of $T$ on the cosets $[T:M_0]$, where $d=(3, q+1)$ and $q\\geq 7$ is odd. Suppose that $\\sqrt{q}\\geq 15d$. Then\n$|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|\\ge \\frac{4}{9}q^5-\\frac{1}{3}q^4\\sqrt{q}-22.1555q^4$ for $d=1$; and\n$|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|\\ge \\frac{4}{27}q^5-\\frac{31}{27}q^4\\sqrt{q}-16.8372q^4$ for $d=3$.\n\\end{theorem}\n\\demo From now on, set\n$G=T=\\PSU(3,q)$ and $M=M_0=\\PSO(3,q)$. Since $M$ is maximal in $G$, we identify each coset $Mg$ with its point stabilizer $M^g$ so that $\\O=\\{ M^g\\mid g\\in G\\}$, with the conjugacy action. Set $\\a=M$.\n Let $\\a':=M'=M^{g'}$ for any $g'\\in G$. Then we shall get a lower bound of $|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|$.\n \\vskip 3mm\nThe possible isomorphic types for $M\\cap M'$ are:\n \\vspace{-5pt}\\begin{equation}\\label{A_i} A_q=\\ZZ_p^m.\\ZZ_2,\\, A_{q\\pm 1}=\\D_{2(q\\pm 1)}, \\, A_3=\\D_4, A_4=\\Alt(4)\\, ({\\rm if}\\, 3\\mid q+1), A_2=\\ZZ_2 \\, {\\rm or}\\, A_1=1.\\end{equation}\nAll of these groups $A_\\imath$ contain at least one involution, except for $A_1$. Therefore, set\n\\vspace{-5pt}\\begin{equation}\\label{I} I=\\{ g\\in M\\mid |g|=2\\}\\quad {\\rm and}\\quad I'=\\{ g\\in M'\\mid |g|=2\\}.\\end{equation}\nClearly, $0 \\le |I \\cap I'| \\le q+2$, giving\n \\vspace{-5pt}\\begin{equation}\\label{lw}\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')=\\{M_\\imath\\in \\O \\setminus \\{M, M'\\}\\mid t, t'\\in M_\\imath,\\,\\, {\\rm for \\, some}\\,\\, t\\in I, t'\\in I' \\}.\\end{equation}\n For the reasen of a simplicity, we turn to consider a subset ${\\cal W}$ of $\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')$, where\n \\vspace{-5pt}\\begin{equation}\\label{W_1} \\cal W=\\{ M_\\imath\\in \\O \\setminus \\{M, M'\\}\\mid t, t'\\in M_\\imath\\,\\, {\\rm for \\, some}\\, \\, t\\in I, t'\\in I' \\,\\, {\\rm but}\\,\\, t, t'\\notin I\\cap I'\\}.\\end{equation}\nTo determine $|{\\cal W}|$, we introduce a set of incident triples (under the inclusion relation)\n \\vspace{-5pt}\\begin{equation}\\label{E} \\cal E=\\{(t, M_\\imath, t')\\mid M_\\imath\\in {\\cal W}, t\\in M\\cap M_\\imath, t'\\in M'\\cap M_\\imath, t, t'\\notin I\\cap I'\\},\\end{equation}\n\\f while for any given $ t\\in I\\setminus I'$ and $M_\\imath\\in {\\cal W}$, we introduce two respective sets of pairs\n\\vspace{-5pt}\\begin{equation}\\label{E(t)} \\cal E(t)=\\{(M_\\imath,t')\\mid (t, M_\\imath, t')\\in \\cal E\\}\\,\\, {\\rm and}\\,\\, \\cal E(M_\\imath)=\\{(t,t')\\mid (t, M_\\imath, t')\\in \\cal E\\},\\end{equation}\nso that we may count $|\\cal E|$ by two different ways, that is,\n\\vspace{-5pt}\\begin{equation} \\label{2way} \\sum_{t\\in I\\setminus I'}|\\cal E(t)|=|\\cal E|=\\sum_{M_\\imath\\in {\\cal W}}|\\cal E(M_\\imath)|.\\end{equation}\nFurther, according to $M_\\imath\\cap M\\cong A_j$, $M_\\imath\\cap M'\\cong A_k$, $M_\\imath\\cap M\\cap M'\\cong A_l$, where $j, k,l\\in \\{1,2,3,4,q,q\\pm1\\}$, as mentioned in Eq(\\ref{A_i}),\n the set ${\\cal W}$ is the union of the following subsets ${\\cal W}_{j, k, l}$, where\n\\vspace{-5pt}\\begin{equation}\\label{jkl}\\begin{array}{lll}\n {\\cal W}_{j,k, l}&=&\\{M_\\imath\\in {\\cal W}\\mid M_\\imath\\cap M\\cong A_j, M_\\imath\\cap M'\\cong A_k, M_\\imath\\cap M\\cap M'=A_l\\}\\end{array}\\end{equation}\nand $j,k,l\\in\\{1,2,3,4,q,q\\pm1\\}$,\nreminding that $(M_\\imath\\cap M) \\cap (M_\\imath\\cap M')=M\\cap M_\\imath\\cap M'$. Moreover, set\n\\vspace{-5pt}\\begin{equation}\\label{n}\n n_{j, k, l}=|{\\cal W}_{j,k,l}|.\\end{equation}\n Then by Eq(\\ref{2way}) and Eq(\\ref{jkl}), we have\n \\vspace{-5pt}$$\\begin{array}{ll}\n &\\sum_{t\\in I\\setminus I'}|\\cal E(t)|=\\sum_{j,k, l} n_{j,k, l} |\\{\\cal E(M_\\imath): M_\\imath\\in {\\cal W}_{j,k, l}\\}|,\n \\end{array}$$\n\\begin{equation}\\label{w1}\n\\begin{array}{lll}|{\\cal W}|&=&\\sum\\limits_{j,k, l}n_{j,k, l}=\\sum\\limits_{t\\in I\\setminus I'}|\\cal E(t)|\n-(\\sum\\limits_{t\\in I\\setminus I'}|\\cal E(t)|-\\sum\\limits_{j,k, l}n_{j,k,l})\\\\\n&=&\\sum\\limits_{t\\in I\\setminus I'}|\\cal E(t)|\n-\\sum\\limits_{j,k, l} n_{j,k,l}(|\\{\\cal E(M_\\imath): M_\\imath\\in {\\cal W}_{j,k, l}\\}|-1)\\\\\n&=&{\\hm A}-{\\hm B}-{\\hm C}-{\\hm D},\\quad {\\rm where} \\end{array}\n\\end{equation}\n$$\\begin{array}{ll}\n&{\\hm A}=\\sum\\limits_{t\\in I\\setminus I'}|\\cal E(t)|,\\hskip 2cm\n{\\hm B}=\\sum\\limits_{4, q\\not\\in \\{j, k\\}}n_{j, k, 1}(|\\{\\cal E(M_\\imath): M_\\imath\\in{\\cal W}_{j,k,1}\\}|-1),\\\\\n&{\\hm C}=\\sum\\limits_{4, q\\not\\in \\{j, k\\},l\\ne 1}n_{j, k, l}(|\\{\\cal E(M_\\imath): M_\\imath\\in{\\cal W}_{j,k,l}\\}|-1),\\\\\n&{\\hm D}=\\sum\\limits_{\\{4, q\\}\\cap \\{j, k\\}\\ne \\emptyset} n_{j, k, l}(|\\{\\cal E(M_\\imath): M_\\imath\\in{\\cal W}_{j,k, l}\\}|-1).\\\\\n\\end{array}$$\n\\f In the following Lemmas~\\ref{AB},~\\ref{C} and~\\ref{D} and~\\ref{DD}, it will be respectively shown that\n\\vspace{-5pt}$$\\begin{array}{lll}\n{\\hm A}&\\geq &q^5-1.0355 q^4,\\, \\hbox{$d=1;$}\\,\\, \\frac 13q^5-0.3342 q^4, \\, \\hbox{$d=3;$}\\,\\\\\n{\\hm B} &\\leq & \\frac{5}{9}q^5+\\frac{1}{3}q^4\\sqrt{q}+15.25 q^4, \\, \\hbox{$d=1$;} \\,\n \\frac{5}{27}q^5+\\frac{31}{27}q^4\\sqrt{q}+10.74 q^4, \\, \\hbox{$d=3$;}\\\\\n {\\hm C}&\\leq& 3.85q^4, \\hbox{$d=1$;} \\, 3.26 q^4, \\hbox{$d=3$};\\\\\n {\\hm D}&\\leq& 2.02 q^4, \\, \\hbox{$d=1$;} \\, 2.503 q^4,\\, \\hbox{$d=3$}. \\\\\n\\end{array}$$\n\\f By using Eq(\\ref{lw}), Eq(\\ref{W_1}) and Eq(\\ref{w1}), we get $|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|\\ge|{\\cal W}|= {\\hm A}-{\\hm B}-{\\hm C}-{\\hm D}$. Inserting these bounds of ${\\hm A}$, ${\\hm B}$, ${\\hm C}$, ${\\hm D}$, we get a lower bound for $|\\Gamma_{nr}(\\a)\\cap \\Gamma_{nr}(\\a')|$ as shown in Theorem~\\ref{main21}, omitting the details.\n\\qed",
    "post_theorem_intro_text_len": 2069,
    "post_theorem_intro_text": "Remark that for the reason of the length of the paper, we are not able to  improve the lower bound  $\\sqrt{q}\\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\\ref{main}.\n Hopefully,  these  small cases may be verified directly with a sufficiently powerful computer.\n\nCombining Theorem~\\ref{main} and the main results of \\cite{BH,Chen-Du,Chen-Du2}, we get\n\\begin{theorem} \\label{main'} Let $G$ be  a primitive  group whose socle is a simple group of  Lie-type of rank $1$ such that $b(G)=2$.\n Then  the BG-Conjecture is valid, except for $G=\\hbox{\\rm PSU}(3,q)\\ZZ_{m_1}$ and  $M=\\hbox{\\rm SO}(3,q)\\ZZ_{m_1}$ $(m_1\\mid 2m$ and  $m_1\\ne  1)$, where\n    $\\sqrt{q}\\le 17$ or $45$,  if $3\\nmid (q+1)$  and $3\\bigm| (q+1)$, respectively.\n  \\end{theorem}\n\n\\vskip 3mm\n\\noindent {\\bf  Outline  of the proof of Theorem~\\ref{main}}: From now on, let $G$ be an almost simple group with the socle $T=\\hbox{\\rm PSU}(3,q)$, so that $T\\leq G\\leq \\hbox{\\rm Aut\\,}(T)$, where $q=p^m$.\n Let $\\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\\hbox{\\rm PSO}(3,q)$, up to conjugacy. Consider the primitive permutation representations of $G$ on the right cosets $[G:M]$.\n Then $q\\ge7$ and either: (i) $G=T$ and $M=M_0$; or (ii) $G=T\\ZZ_{m_1}$ and  $M=M_0\\ZZ_{m_1}$, where $m_1\\mid 2m$ and $m_1\\ne 1$.\nFor the first case, we exhibit a common  neighbor of $\\alpha $ and $\\alpha^g$ in the Saxl graph for any $g\\in G$ via a direct geometric construction in Section 2.\nHowever, we are not able to extend  this geometric construction   to the second case, which necessitates a new approach developed in Section 3.\nConsequently, the proof of Theorem~\\ref{main} is obtained by combining Theorems~\\ref{main1} and \\ref{main2}.\nMoreover, to prove Theorem~\\ref{main2} crucially depends on estimates of $|\\Gamma_r(\\alpha)|$ and $|\\Gamma_{nr}(\\alpha) \\cap \\Gamma_{nr}(\\alpha')|$, developed in Sections 4 and 5 respectively. Recall that $\\Gamma_r(\\alpha)$ (resp. $\\Gamma_{nr}(\\alpha)$) is the union of all regular (resp. non-regular) suborbits of $G$ relative to $\\alpha$.\n\\vskip -7pt",
    "sketch": "Outline of the proof of Theorem~\\ref{main}: Let $G$ be almost simple with socle $T=\\hbox{\\rm PSU}(3,q)$, so $T\\le G\\le \\hbox{\\rm Aut}(T)$, $q=p^m$. Let $\\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\\hbox{\\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\\!:\\!M]$. Then $q\\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\\ZZ_{m_1}$ and $M=M_0\\ZZ_{m_1}$ with $m_1\\mid 2m$ and $m_1\\ne 1$. In case (i), a common neighbor of $\\alpha$ and $\\alpha^g$ in the Saxl graph (for any $g\\in G$) is exhibited by a direct geometric construction in Section 2. This geometric construction does not extend to case (ii), so a new approach is developed in Section 3. Theorem~\\ref{main} then follows by combining Theorems~\\ref{main1} and~\\ref{main2}. The proof of Theorem~\\ref{main2} depends crucially on estimates for $|\\Gamma_r(\\alpha)|$ and $|\\Gamma_{nr}(\\alpha)\\cap \\Gamma_{nr}(\\alpha')|$, developed in Sections 4 and 5, where $\\Gamma_r(\\alpha)$ (resp. $\\Gamma_{nr}(\\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\\alpha$.",
    "expanded_sketch": "Outline of the proof of the main theorem: Let $G$ be almost simple with socle $T=\\hbox{\\rm PSU}(3,q)$, so $T\\le G\\le \\hbox{\\rm Aut}(T)$, $q=p^m$. Let $\\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\\hbox{\\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\\!:\\!M]$. Then $q\\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\\ZZ_{m_1}$ and $M=M_0\\ZZ_{m_1}$ with $m_1\\mid 2m$ and $m_1\\ne 1$. In case (i), a common neighbor of $\\alpha$ and $\\alpha^g$ in the Saxl graph (for any $g\\in G$) is exhibited by a direct geometric construction developed earlier. This geometric construction does not extend to case (ii), so a new approach is developed later.\n\nWe first prove the following theorem.\n\\begin{theorem} \\label{main1}  Let $G=\\PSU(3,q)$ and $M=\\PSO(3,q),$ a maximal subgroup of $G$, where $q=p^m\\ge 7$ for an odd prime $p$.\nConsider the primitive right multiplication action  of $G$ on the set of right cosets $\\Omega =[G:M]$.  Then $b(G)=2$ and the BG-Conjecture holds.\n  \\end{theorem}\n\nWe also prove the following theorem.\n\\begin{theorem} \\label{main2}  Let $G=\\PSU(3,q)\\ZZ_{m_1}$ and\n    $\\a=M=\\PSO(3,q)\\ZZ_{m_1}\\le G$, with  $q=p^m\\ge 7$ for an odd prime $p$,  $m_1\\mid 2m$ and $m_1\\ne 1$.\nUnder the primitive right multiplication action of $G$ on $\\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\\sqrt{q}\\geq 17$ or $45$ if $3\\nmid (q+1)$ and $3\\di (q+1)$, respectively.\n  \\end{theorem}\n\nIn establishing the main theorem, we combine Theorem~\\ref{main1} and Theorem~\\ref{main2}. The proof of Theorem~\\ref{main2} depends crucially on estimates for $|\\Gamma_r(\\alpha)|$ and $|\\Gamma_{nr}(\\alpha)\\cap \\Gamma_{nr}(\\alpha')|$, developed later, where $\\Gamma_r(\\alpha)$ (resp. $\\Gamma_{nr}(\\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\\alpha$.",
    "expanded_theorem": "\\label{main} Let $G$ be a primitive permutation group with socle $\\hbox{\\rm PSU}(3,q)$, where $q=p^m\\ge 7$ for an odd prime $p$. Suppose a point-stabilizer\n  $M$ of $G$ contains $\\hbox{\\rm PSO}(3,q)$. Then $b(G)=2$ and either (i) $G=\\hbox{\\rm PSU}(3,q)$ and $M=\\hbox{\\rm SO}(3,q)$; or (ii) $G=\\hbox{\\rm PSU}(3,q)\\ZZ_{m_1}$ and  $M=\\hbox{\\rm SO}(3,q)\\ZZ_{m_1}$ $(m_1\\mid 2m$ and  $m_1\\ne  1)$. Further, the BG-Conjecture is valid for  case (i);\n  and for   $\\sqrt{q}\\geq 17$ or $45$,  if $3\\nmid (q+1)$  and $3\\bigm| (q+1)$, respectively in case (ii).",
    "theorem_type": [
      "Implication",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $G\\leq \\mathrm{Sym}(\\Omega)$ be a finite primitive permutation group with socle $\\mathrm{PSU}(3,q)$, where $q=p^m\\ge 7$ for an odd prime $p$, and let $M=G_\\alpha$ be a point-stabilizer containing $\\mathrm{PSO}(3,q)$. Here $b(G)$ is the minimal size of a base, i.e. a subset $\\Delta\\subseteq \\Omega$ whose pointwise stabilizer in $G$ is trivial. When $b(G)=2$, the Saxl graph $\\Sigma(G)$ is the graph with vertex set $\\Omega$ in which two vertices are adjacent exactly when they form a base; the BG-Conjecture asserts that every two vertices of $\\Sigma(G)$ have a common neighbour. Under these assumptions, which statement about $G$, $M$, $b(G)$, and the BG-Conjecture holds?",
      "correct_choice": {
        "label": "A",
        "text": "One has $b(G)=2$, and either (i) $G=\\mathrm{PSU}(3,q)$ and $M=\\mathrm{SO}(3,q)$, or (ii) $G=\\mathrm{PSU}(3,q)\\mathbb Z_{m_1}$ and $M=\\mathrm{SO}(3,q)\\mathbb Z_{m_1}$ for some $m_1\\mid 2m$ with $m_1\\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\\sqrt q\\ge 17$ if $3\\nmid(q+1)$, and whenever $\\sqrt q\\ge 45$ if $3\\mid(q+1)$."
      },
      "choices": [
        {
          "label": "B",
          "text": "One has $b(G)=2$, and either (i) $G=\\mathrm{PSU}(3,q)$ and $M=\\mathrm{SO}(3,q)$, or (ii) $G=\\mathrm{PSU}(3,q)\\mathbb Z_{m_1}$ and $M=\\mathrm{SO}(3,q)\\mathbb Z_{m_1}$ for some $m_1\\mid 2m$ with $m_1\\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $q\\ge 17$ if $3\\nmid(q+1)$, and whenever $q\\ge 45$ if $3\\mid(q+1)$."
        },
        {
          "label": "C",
          "text": "One has $b(G)=2$, and either (i) $G=\\mathrm{PSU}(3,q)$ and $M=\\mathrm{SO}(3,q)$, or (ii) $G=\\mathrm{PSU}(3,q)\\mathbb Z_{m_1}$ and $M=\\mathrm{SO}(3,q)\\mathbb Z_{m_1}$ for some $m_1\\mid 2m$ with $m_1\\ne 1$."
        },
        {
          "label": "D",
          "text": "One has $b(G)=2$, and either (i) $G=\\mathrm{PSU}(3,q)$ and $M=\\mathrm{SO}(3,q)$, or (ii) $G=\\mathrm{PSU}(3,q)\\mathbb Z_{m_1}$ and $M=\\mathrm{SO}(3,q)\\mathbb Z_{m_1}$ for some $m_1\\mid 2m$. Moreover, the BG-Conjecture holds in both cases (i) and (ii)."
        },
        {
          "label": "E",
          "text": "One has $b(G)=2$, and either (i) $G=\\mathrm{PSU}(3,q)$ and $M=\\mathrm{SO}(3,q)$, or (ii) $G=\\mathrm{PSU}(3,q)\\mathbb Z_{m_1}$ and $M=\\mathrm{SO}(3,q)\\mathbb Z_{m_1}$ for some $m_1\\mid m$ with $m_1\\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\\sqrt q\\ge 17$ if $3\\nmid(q+1)$, and whenever $\\sqrt q\\ge 45$ if $3\\mid(q+1)$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "sqrtq_thresholds_in_case_ii",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "BG-Conjecture_conclusion_removed",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "conditional_validity_of_BG_in_case_ii",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "outer_extension_parameter_m1_divides_2m",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It states the setup and asks for the full classification, but the exact divisibility condition and BG-Conjecture thresholds must still be selected from the choices."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific classification theorem and its corollary. The question asks for the theorem statement itself rather than requiring application to a new situation."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish subtle variants in the hypotheses and conclusions, such as whether m1 divides 2m or m, whether m1 ≠ 1 is required, and the conditional BG thresholds. However, this is mainly theorem recognition/recall rather than genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and carefully constructed: they alter boundary conditions, omit a necessary exclusion, weaken the BG conclusion, or overstate it. These align with realistic mathematical errors and are not trivially dismissible."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall question with strong distractors, but it is largely theorem restatement rather than a true reasoning-based MCQ."
    }
  },
  {
    "id": "2512.22613v1",
    "paper_link": "http://arxiv.org/abs/2512.22613v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in Lemma \\ref{lem:disper-log}. Then for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\mathbb{T}^d$ and any $t\\in\\mathbb{R}$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\mathbb{Z}\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\mathbb{Z}\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\mathbb{Z}\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\mathbb{Z}) \\ .\n\t\t\\end{equation}",
      "start_pos": 9179,
      "end_pos": 9715,
      "label": "thm:disper"
    },
    "ref_dict": {
      "eq: nonlinear": "\\begin{equation}\\label{eq: nonlinear}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)= \\pm |u|^{p-1}u,\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\mathbb{R}_{>0},\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}",
      "thm:Strichartz": "\\begin{theorem}\\label{thm:Strichartz}\n\n        Fix\n\t\t$0<\\tau<\\tfrac13$ and $F\\in L^{\\tilde q'}(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))$. Let $(q,r)$ and $(\\tilde q,\\tilde r)$ be two $\\tau$--admissible pairs with $q,\\tilde q<\\infty.$  Then the Cauchy\n\t\tproblem \\eqref{eq: inhomogegeneous} has a unique global solution\n\t\t$\n\t\tu\\in C(\\mathbb R;\\ell^2(\\mathbb Z))\n\t\t$\n\t\tand \\begin{equation}\\label{eq:inhom-Strichartz}\n\t\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\n\t\t\t\\;\\le\\; C\\Bigl(\n\t\t\t\\|\\varphi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|\\psi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|F\\|_{L^{\\tilde q'}_t(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))}\n\t\t\t\\Bigr),\n\t\t\\end{equation}\n\t\twhere $\\tilde q'$ and $\\tilde r'$ denote the Hölder conjugates of\n\t\t$\\tilde q$ and $\\tilde r$, respectively, and the constant $C$ is\n\t\tindependent of $\\theta$, $\\varphi$, $\\psi$ and $F$. \n\n\t\\end{theorem}",
      "thm:nonlinear": "\\begin{theorem}\\label{thm:nonlinear}\n\t\tLet $p>7$. There exist $\\varepsilon > 0$ and a constant $C$, so that whenever\n\t\t$\\| \\varphi \\|_{\\ell^2\\left(\\Z\\right)}$ and $\\| \\psi\\|_{\\ell^2\\left(\\Z\\right)} \\leq \\varepsilon\n\t\t$, there exists a unique global solution to \\eqref{eq: nonlinear}, satisfying\n\t\t\\begin{equation*}\n\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\\leq C\\varepsilon,\n\t\t\\end{equation*}\n\t\tfor all admissible pairs $(q, r)$. In particular, $\\| u(n,t) \\|_{\\ell^r(\\Z)} \\to 0$ as $t \\to \\infty$ for every $r > 2$.  \n\t\\end{theorem}",
      "lem:disper-log": "\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}",
      "eq:DKG": "\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}",
      "eq: inhomogegeneous": "\\begin{equation}\\label{eq: inhomogegeneous}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=F(n,t),\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}",
      "thm:disper": "\\begin{theorem}\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in Lemma \\ref{lem:disper-log}. Then for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\T^d$ and any $t\\in\\R$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\Z\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\Z\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\Z\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\Z) \\ .\n\t\t\\end{equation}\n\t\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3331,
    "pre_theorem_intro_text": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n\t\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere $G_{\\theta}$ is defined as\n\t\\begin{equation*}\n\t\t(G_{\\theta}u)(n)\n\t\t:=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n\t\t\\qquad n\\in\\mathbb{Z},\n\t\\end{equation*}\n\tand the discrete Laplacian is given by\n\t$\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n\tHere $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n    \\begin{equation}\\label{dio}\n\t\t\\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n\t\t>\\frac{\\gamma}{|k|^{\\eta}},\n\t\t\\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n    \\end{equation}\n    where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n    \\begin{equation*}\n        \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n    \\end{equation*}\n    In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}.  We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n\t\\begin{equation}\\label{eq:dispersive}\n\t\t\\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n\t\t\\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n\t\t\\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n\t\\end{equation}\n\twhere $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\t\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive. \n\n\tOur main result states as follows.",
    "context": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n \\begin{equation}\\label{eq:DKG}\n \\begin{cases}\n \\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n \\end{cases}\n \\end{equation}\n where $G_{\\theta}$ is defined as\n \\begin{equation*}\n (G_{\\theta}u)(n)\n :=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n \\qquad n\\in\\mathbb{Z},\n \\end{equation*}\n and the discrete Laplacian is given by\n $\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n Here $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n \\begin{equation}\\label{dio}\n \\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n >\\frac{\\gamma}{|k|^{\\eta}},\n \\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n \\end{equation}\n where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n \\begin{equation*}\n \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n \\end{equation*}\n In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}. We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n \\begin{equation}\\label{eq:dispersive}\n \\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n \\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n \\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n \\end{equation}\n where $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive.\n\nOur main result states as follows.\n\n\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}",
    "full_context": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n \\begin{equation}\\label{eq:DKG}\n \\begin{cases}\n \\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n \\end{cases}\n \\end{equation}\n where $G_{\\theta}$ is defined as\n \\begin{equation*}\n (G_{\\theta}u)(n)\n :=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n \\qquad n\\in\\mathbb{Z},\n \\end{equation*}\n and the discrete Laplacian is given by\n $\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n Here $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n \\begin{equation}\\label{dio}\n \\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n >\\frac{\\gamma}{|k|^{\\eta}},\n \\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n \\end{equation}\n where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n \\begin{equation*}\n \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n \\end{equation*}\n In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}. We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n \\begin{equation}\\label{eq:dispersive}\n \\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n \\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n \\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n \\end{equation}\n where $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive.\n\nOur main result states as follows.\n\n\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}\n\nBy \\eqref{infinito}, we have\n \\begin{equation}\\label{eq:u(t)_int}\n |u(n,t)|\\leq\\frac{1}{\\pi}\\left|\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right) \\rho^{\\prime}\\d E\\right|+\\varepsilon_0^{\\frac{\\sigma^2}{10}}\\|u(\\cdot,t)\\|_{\\ell^\\infty\\left(\\Z\\right)} \\ .\n \\end{equation}\n To estimate $\\|u(\\cdot,t)\\|_{\\ell^\\infty\\left(\\Z\\right)}$, it suffices to control the integral term by taking $\\varepsilon_{0}$ to be small. Substituting the expression of $\\mathcal{S}(u(t))$ into the integral, and recalling the definitions of $g^{\\varphi}$ and $g^{\\psi}$, we obtain\n \\begin{equation}\\label{eq:appro_u}\n \\begin{aligned}\n &\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E\\\\\n =&\\frac{1}{2} \\int_\\Sigma e^{{\\rm i}\\Omega(E)t} \\sum_{k\\in\\Z} \\left(\\varphi(k) + \\frac{\\psi(k)}{{\\rm i}\\Omega(E)}\\right) \\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E \\\\\n &+\\frac{1}{2} \\int_\\Sigma e^{-{\\rm i}\\Omega(E)t} \\sum_{k\\in\\Z} \\left(\\varphi(k) - \\frac{\\psi(k)}{{\\rm i}\\Omega(E)}\\right) \\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E \\ .\n \\end{aligned}\n \\end{equation}\n Note that the term $\\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)$ can be expanded as\n \\begin{equation}\\label{eq:K,J expansions}\n \\sum_{k_{\\Delta}, n_{\\Delta}} \\beta_{k,k_{\\Delta}} \\beta_{n,n_{\\Delta}} \\cos(k_{\\Delta}-n_{\\Delta})\\rho(E) \\ .\n \\end{equation}\n Since $m>0$, the factor $\\frac{1}{\\Omega(E)}= \\frac{1}{\\sqrt{2+m^2+E}}$ is smooth and bounded on the spectrum of $G_{\\theta}+m^2$. By the symmetry of the complex conjugate terms in \\eqref{eq:appro_u}, the dispersive decay estimates for both integrals depend on the oscillatory behavior of the phase function. Thus, the problem reduces to establishing the estimates for the oscillatory integral associated with the kernel\n $e^{-{\\rm i} t \\sqrt{2+m^2+E}}$. For simplicity, we denote\n $$\\mathbb{I}(S;k_{\\Delta},n_{\\Delta},\\rho):=\\int_{S} \\beta_{k,k_{\\Delta}}\\beta_{n,n_{\\Delta}}\\cos\n \\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-\\mathrm{i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime}\\d E, \\quad \\text{for}\\ S\\subset\\R,$$ \n and\n $$\\mathbb{I}^{J}(S;k_{\\Delta},n_{\\Delta},\\rho):=\\int_{S} \\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\cos\n \\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-\\mathrm{i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime} \\d E, \\quad \\text{for}\\ S\\subset\\R.$$ \n Whenever no confusion can arise, we simply write $\\mathbb{I}(E)$ and $\\mathbb{I}^{J}(E)$ respectively.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:disper} assuming Lemma \\ref{lem:disper-log}]\n By the expansions \\eqref{eq:K,J expansions}, we apply Lemma \\ref{lem:disper-log} to equation \\eqref{eq:appro_u} and derive\n \\begin{equation}\\label{eq:L infty estimate}\n \\begin{aligned}\n &\\left|\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E\\right|\\\\\n \\leq&C\\left(m\\right)\\cdot\\frac{\\left|\\ln\\varepsilon_{0}\\right|^{82000\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}d}}{\\JP{t}^{\\frac{1}{3}}}\\left(\\sum_{k\\in\\Z}\\left|\\varphi\\left(k\\right)\\right|+\\frac{1}{\\inf_{E\\in\\Sigma}\\left|\\Omega\\left(E\\right)\\right|}\\sum_{k\\in\\Z}\\left|\\psi\\left(k\\right)\\right|\\right),\n \\end{aligned}\n \\end{equation}\n where the absolute constant $C_{1}$ relates to the estimate of $e^{{\\rm i}\\Omega\\left(\\lambda\\right)t}$ and so does $C_{2}$ for the estimate related to $e^{-{\\rm i}\\Omega\\left(\\lambda\\right)t}$. Since $\\epsilon_{0}<1$ is small, and by \\eqref{eq:u(t)_int}, \\eqref{eq:L infty estimate} we deduce\n \\begin{equation*}\n \\norm{u\\left(t\\right)}_{\\ell^{\\infty}\\left(\\Z\\right)}\\leq\\frac{C\\left(m\\right)}{1-\\varepsilon_0^{\\frac{\\sigma^2}{10}}}\\frac{\\left|\\ln\\varepsilon_{0}\\right|^{82000\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}d}}{\\JP{t}^{\\frac{1}{3}}}\\left(\\norm{\\varphi}_{\\ell^{1}\\left(\\Z\\right)}+\\frac{1}{\\inf_{E\\in\\Sigma}\\left|\\Omega\\left(E\\right)\\right|}\\norm{\\psi}_{\\ell^{1}\\left(\\Z\\right)}\\right).\n \\end{equation*}\n This completes the proof.\n \\end{proof}\n\nIn the case of $|k_{\\Delta}-n_{\\Delta}|$ is large, we derive the following bound. \n\\begin{lemma}\\label{lem:large}\nFor any $t \\in \\R$, the following estimate holds, \n\\begin{equation*}\n \\left|\\mathbb{I}^{J}(\\Sigma)\\right|\\leq\\frac{18}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}+\\frac{3\\Theta_{1}\\JP{t}}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left(\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}\\right)^{2}+\\varepsilon_{j}^{6\\sigma}\\right)\\left(1+\\frac{1}{\\Theta_{2}}\\right).\\\\\n\\end{equation*} \n\\end{lemma}\n\\begin{proof}\nBy Proposition~\\ref{propsana1}, for each $0\\leq j\\leq J$ we can write\n\\begin{equation*}\n \\Gamma_{j}^{(J)}\n =\\mathcal{R}_{j}\\cup\\Bigl(\\,\\bigsqcup_{s^{J}_{j}}I_{s^{J}_{j}}\\Bigr),\n\\end{equation*}\nwhere $\\mathcal{R}_{j}$ is a finite set of points (possibly empty) and each $I_{s^{J}_{j}}$ is an open interval in $\\Gamma_{j}^{(J)}$. Recall that $\\rho$ is absolutely continuous. Then\n \\begin{equation}\\label{eq:Ij-M large}\n \\begin{aligned}\n \\mathbb{I}^{J}(\\Sigma)=&\\int_{\\inf\\Sigma}^{\\sup\\Sigma}\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\cos\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime} \\d E\\\\\n =&\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)\\Big|_{\\inf I_{s_{j}^{J}}}^{\\sup I_{s_{j}^{J}}}\\\\\n &\\quad-\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\int_{I_{s_{j}^{J}}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)^{\\prime}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)\\d E.\\\\\n \\end{aligned}\n \\end{equation}\n Using Proposition \\ref{propsana1} again, we derive the upper bound for connected components of $\\bigcup \\Gamma_{j}^{(J)}$:\n \\begin{equation*}\n \\sum_{j=0}^{J}\\#\\left(s_{j}^{J}\\right)\\leq \\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}.\n \\end{equation*}\n For the first term in equation \\eqref{eq:Ij-M large}, by the $L^{\\infty}$-estimate and inequality \\eqref{esti_beta_0} that\n \\begin{equation}\\label{eq:first term}\n \\begin{aligned}\n &\\left|\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)\\Big|_{\\inf I_{s_{j}^{J}}}^{\\sup I_{s_{j}^{J}}}\\right|\\\\\n \\leq&\\frac{2}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}\\cdot\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}+\\varepsilon_{0}^{3\\sigma}\\right)^{2}\\\\\n \\leq&\\frac{18}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}.\n \\end{aligned}\n \\end{equation}\nFor the second term of the result in equation \\eqref{eq:Ij-M large}, we need the following direct estimate\n\\begin{equation}\\label{eq:betac1}\n \\left|\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\right|_{\\mathcal{C}^{1}\\left(\\Gamma^{\\left(J\\right)}_{j}\\right)}\\leq\n \\begin{cases}\n 3\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}\\right)^{2}\\;{\\rm when}\\;j=0;\\\\\n 3\\varepsilon_{j}^{6\\sigma}\\qquad{\\rm when}\\;1\\leq j\\leq J.\\\\\n \\end{cases}\n\\end{equation}\n Using inequality\n \\eqref{eq:betac1}, we deduce",
    "post_theorem_intro_text_len": 6353,
    "post_theorem_intro_text": "For our Theorem \\ref{thm:disper}, we have the following remarks.\n\n\t\\begin{itemize}\n\t\t\\item[($\\mathbf{R}_1$)] Stefanov and Kevrekidis~\\cite{SK05} first established a sharp dispersive decay rate  \n\t\t$\n\t\t\\frac{1}{3}\n\t\t$ \n\t\tfor the free discrete Klein-Gordon equation (i.e., equation \\eqref{eq:DKG} with $V=0$) on $\\mathbb Z$ via Van der Corput's lemma.  \n\t\tRecently, Borovyk and Goldberg \\cite{BG17} derived the corresponding rate on $\\mathbb Z^{2}$, namely  $\\langle t\\rangle^{-\\frac{3}{4}}$. Later, Cuenin and Ikromov~\\cite{CI21} derived the rates on~$\\mathbb Z^{d}$ for $d=3,4$, namely $\n\t\t\\langle t\\rangle^{-\\frac{7}{6}}\\ (d=3)$, and\n\t\t$\\langle t\\rangle^{-\\frac{3}{2}}\\log \\bigl(2+|t|\\bigr)\\ (d=4)$. The obstacle to extending Cuenin-Ikromov to higher dimensions is that the finite classification of stable phase singularities (Thom's catastrophes), which is essential for deriving sharp decay rates, breaks down in dimensions $d \\ge 5$. Our result can be seen as a perturbation result for Stefanov and Kevrekidis, which shows that the decay rate $(\\frac{1}{3})^{-}$ persists under small quasi-periodic perturbations of the potentials. \n\n\t\t\\item[($\\mathbf{R}_2$)] While several studies (e.g., \\cite{CT09, GPP13, PS08, KPS09, KKK06}) have addressed dispersive equations with potentials whose continuous spectrum is a union of disjoint intervals, generic behavior of the operator $H_{\\theta}$ is markedly different: its spectrum is purely absolutely continuous and forms a Cantor set; see, e.g., \\cite{AD08, Avi08, E92}. Our approach builds heavily on recent advances in the spectral theory of quasi-periodic Schr\\\"odinger operators; see, for example, \\cite{Z16}. Specifically, we employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals. Using a modified  spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay. We emphasize that a similar strategy appears in the study of ballistic motion and dispersive estimates for discrete Schr\\\"odinger equations with quasi-periodic potentials; see, e.g., \\cite{BZ20, Z16}. \n\n        \\item[($\\mathbf{R}_3$)] For discrete wave equation with quasi-periodic potentials (i.e., equation \\eqref{eq:DKG} with $m=0$), our method does not work. In fact, there does not exist $\\ell^1 \\to \\ell^{\\infty}$ estimates for fundamental solution of the free wave equation. We point out the following development regarding the dispersive estimate for the free discrete wave equation. This phenomenon also holds on $\\mathbb{R}^1$. Schultz \\cite{Sch98} proved dispersive estimates for the wave equation on lattice graphs $\\mathbb{Z}^d$ for $d=2,3$, namely  $\\langle t\\rangle^{-\\frac{3}{4}}\\ (d=2)$ and $\\langle t\\rangle^{-\\frac{7}{6}}\\ (d=3)$. Combining the singularity theory\n\t\twith results in uniform estimates of oscillatory integrals, Bi-Chen-Hua \\cite{BCH1} extended this result  to $d=4$: they proved that the optimal\n\t\ttime decay rate is $|t|^{-\\frac{3}{2}} \\log |t|$. By tools from Newton polyhedra, Bi-Chen-Hua \\cite{BCH2} further extended the result to $d=5$: the sharp decay rate of the fundamental solution of the wave equation on $\\mathbb{Z}^5$ is $|t|^{-\\frac{6}{11}}$.\n    \\end{itemize}\n\n\tA typical application for dispersive estimates is the derivation of Strichartz inequalities. For the inhomogeneous discrete Klein-Gordon equation, \n\t\\begin{equation}\\label{eq: inhomogegeneous}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=F(n,t),\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}\n\twe have the following mixed space-time estimates via the standard Keel-Tao argument. For convenience, we say   \n\ta pair of exponents $(q,r)$ with\n\t$\n\t2\\le q,r\\le\\infty\n\t$\n\tis called \\emph{$\\tau$--admissible} if\n\t$\n\t\\frac{2}{q}\n\t=\\tau\\Bigl(1-\\frac{2}{r}\\Bigr).\n\t$ Here $0<\\tau<\\tfrac13$ is fixed as in Theorem~\\ref{thm:disper}.\n\n\t\\begin{theorem}\\label{thm:Strichartz}\n\n        Fix\n\t\t$0<\\tau<\\tfrac13$ and $F\\in L^{\\tilde q'}(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))$. Let $(q,r)$ and $(\\tilde q,\\tilde r)$ be two $\\tau$--admissible pairs with $q,\\tilde q<\\infty.$  Then the Cauchy\n\t\tproblem \\eqref{eq: inhomogegeneous} has a unique global solution\n\t\t$\n\t\tu\\in C(\\mathbb R;\\ell^2(\\mathbb Z))\n\t\t$\n\t\tand \\begin{equation}\\label{eq:inhom-Strichartz}\n\t\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\n\t\t\t\\;\\le\\; C\\Bigl(\n\t\t\t\\|\\varphi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|\\psi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|F\\|_{L^{\\tilde q'}_t(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))}\n\t\t\t\\Bigr),\n\t\t\\end{equation}\n\t\twhere $\\tilde q'$ and $\\tilde r'$ denote the Hölder conjugates of\n\t\t$\\tilde q$ and $\\tilde r$, respectively, and the constant $C$ is\n\t\tindependent of $\\theta$, $\\varphi$, $\\psi$ and $F$. \n\n\t\\end{theorem}\n\n\tUsing Strichartz estimates, we establish the global well-posedness of the nonlinear Klein-Gordon equation  \\begin{equation}\\label{eq: nonlinear}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)= \\pm |u|^{p-1}u,\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\mathbb{R}_{>0},\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}\nwith small initial data.\t\n\t\\begin{theorem}\\label{thm:nonlinear}\n\t\tLet $p>7$. There exist $\\varepsilon > 0$ and a constant $C$, so that whenever\n\t\t$\\| \\varphi \\|_{\\ell^2\\left(\\mathbb{Z}\\right)}$ and $\\| \\psi\\|_{\\ell^2\\left(\\mathbb{Z}\\right)} \\leq \\varepsilon\n\t\t$, there exists a unique global solution to \\eqref{eq: nonlinear}, satisfying\n\t\t\\begin{equation*}\n\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\\leq C\\varepsilon,\n\t\t\\end{equation*}\n\t\tfor all admissible pairs $(q, r)$. In particular, $\\| u(n,t) \\|_{\\ell^r(\\mathbb{Z})} \\to 0$ as $t \\to \\infty$ for every $r > 2$.  \n\t\\end{theorem}\n\nThe remainder of the paper is organized as follows. Section \\ref{sec:2} reviews the structure of the spectrum of the quasi-periodic Schr\\\"odinger operator and presents a spectral transform that will be used throughout. Section \\ref{sec:3} establishes Theorem \\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum. In Section \\ref{sec:4}, we prove Theorems \\ref{thm:Strichartz} and \\ref{thm:nonlinear} as applications of the dispersive estimate.",
    "sketch": "A proof strategy for Theorem~\\ref{thm:disper} is outlined in Remark $\\mathbf{R}_2$ and the paper organization: the authors \"employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals.\" Then, \"[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay.\" Concretely, Section~\\ref{sec:3} \"establishes Theorem~\\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum.\"",
    "expanded_sketch": "A proof strategy for the main theorem is outlined in Remark $\\mathbf{R}_2$ and the paper organization: the authors \"employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals.\" Then, \"[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay.\" Concretely, next the authors establish the main theorem by analysing a specific oscillatory integral on the spectrum.",
    "expanded_theorem": "\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in the following lemma.\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}\n\n\t\tThen for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\mathbb{T}^d$ and any $t\\in\\mathbb{R}$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\mathbb{Z}\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\mathbb{Z}\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\mathbb{Z}\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\mathbb{Z}) \\ .\n\t\t\\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Consider the discrete Klein--Gordon equation on \\(\\mathbb Z\\)\n\\[\n\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad (n,t)\\in \\mathbb Z\\times (0,\\infty),\n\\]\nwith initial data\n\\[\nu(\\cdot,0)=\\varphi\\in \\ell^{1}(\\mathbb Z),\\qquad \\partial_t u(\\cdot,0)=\\psi\\in \\ell^{1}(\\mathbb Z),\n\\]\nwhere\n\\[\n(G_{\\theta}u)(n)=-\\Delta_{\\mathrm{disc}}u(n)+V(\\theta+n\\omega)u(n),\n\\qquad\n\\Delta_{\\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).\n\\]\nAssume \\(\\theta\\in \\mathbb T^d\\), \\(d\\ge 1\\), \\(m>0\\), \\(\\omega\\in\\mathbb R^d\\) is Diophantine, and \\(V:\\mathbb T^d\\to\\mathbb R\\) extends to a bounded analytic function on \\(\\{z\\in\\mathbb C^d:|\\operatorname{Im} z|<r\\}\\). Let\n\\[\n\\varepsilon_0:=|V|_r=\n\\sup_{\\{z\\in\\mathbb C^d:|\\operatorname{Im} z|<r\\}} |V(z)|,\n\\]\nand assume \\(\\varepsilon_0<\\varepsilon_*\\). Also write \\(\\langle t\\rangle:=\\sqrt{1+t^2}\\). Under these assumptions, which quantitative \\(\\ell^1\\to\\ell^\\infty\\) dispersive estimate is valid for the solution \\(u\\)?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac13\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant \\(K_1=K_1(\\varepsilon_0)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-1/3}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant \\(K_1=K_1(\\varepsilon_0)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac13\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau,\\theta)\\) such that, for every \\(t\\in\\mathbb R\\) and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac12\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "endpoint decay exponent from oscillatory-integral bound",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped time-decay factor \\(\\langle t\\rangle^{-\\tau}\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity in phase parameter \\(\\theta\\)",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "allowed range of admissible decay exponents",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state the conclusion or explicitly reveal the correct option. It gives hypotheses and asks for the resulting dispersive estimate, so there is no direct answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "This is largely a theorem-recall item: the hypotheses are presented and the learner is asked to identify the theorem's conclusion. The nearby variants prevent it from being a completely trivial restatement, but it is still close to direct recall rather than a genuinely independent inference task."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options differ in subtle but important ways: open vs. closed endpoint, uniformity in θ, exact endpoint decay, and a weaker existential claim. However, the problem mainly tests memory of the exact theorem statement, not substantial generative reasoning."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and target common failure modes, especially endpoint inclusion and loss of uniformity in θ. However, option C is a weaker true statement if A is true, so the item is not cleanly single-correct unless the stem asks for the strongest statement. That weakens distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "A reasonably well-constructed theorem-discrimination MCQ with no answer leakage and mostly strong distractors, but it is close to theorem restatement and is weakened by the inclusion of a weaker true option, making the single-best-answer format somewhat ambiguous."
    }
  },
  {
    "id": "2512.22613v1",
    "paper_link": "http://arxiv.org/abs/2512.22613v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in Lemma \\ref{lem:disper-log}. Then for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\mathbb{T}^d$ and any $t\\in\\mathbb{R}$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\mathbb{Z}\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\mathbb{Z}\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\mathbb{Z}\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\mathbb{Z}) \\ .\n\t\t\\end{equation}",
      "start_pos": 9179,
      "end_pos": 9715,
      "label": "thm:disper"
    },
    "ref_dict": {
      "eq: nonlinear": "\\begin{equation}\\label{eq: nonlinear}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)= \\pm |u|^{p-1}u,\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\mathbb{R}_{>0},\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}",
      "thm:Strichartz": "\\begin{theorem}\\label{thm:Strichartz}\n\n        Fix\n\t\t$0<\\tau<\\tfrac13$ and $F\\in L^{\\tilde q'}(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))$. Let $(q,r)$ and $(\\tilde q,\\tilde r)$ be two $\\tau$--admissible pairs with $q,\\tilde q<\\infty.$  Then the Cauchy\n\t\tproblem \\eqref{eq: inhomogegeneous} has a unique global solution\n\t\t$\n\t\tu\\in C(\\mathbb R;\\ell^2(\\mathbb Z))\n\t\t$\n\t\tand \\begin{equation}\\label{eq:inhom-Strichartz}\n\t\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\n\t\t\t\\;\\le\\; C\\Bigl(\n\t\t\t\\|\\varphi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|\\psi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|F\\|_{L^{\\tilde q'}_t(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))}\n\t\t\t\\Bigr),\n\t\t\\end{equation}\n\t\twhere $\\tilde q'$ and $\\tilde r'$ denote the Hölder conjugates of\n\t\t$\\tilde q$ and $\\tilde r$, respectively, and the constant $C$ is\n\t\tindependent of $\\theta$, $\\varphi$, $\\psi$ and $F$. \n\n\t\\end{theorem}",
      "thm:nonlinear": "\\begin{theorem}\\label{thm:nonlinear}\n\t\tLet $p>7$. There exist $\\varepsilon > 0$ and a constant $C$, so that whenever\n\t\t$\\| \\varphi \\|_{\\ell^2\\left(\\Z\\right)}$ and $\\| \\psi\\|_{\\ell^2\\left(\\Z\\right)} \\leq \\varepsilon\n\t\t$, there exists a unique global solution to \\eqref{eq: nonlinear}, satisfying\n\t\t\\begin{equation*}\n\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\\leq C\\varepsilon,\n\t\t\\end{equation*}\n\t\tfor all admissible pairs $(q, r)$. In particular, $\\| u(n,t) \\|_{\\ell^r(\\Z)} \\to 0$ as $t \\to \\infty$ for every $r > 2$.  \n\t\\end{theorem}",
      "lem:disper-log": "\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}",
      "eq:DKG": "\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}",
      "eq: inhomogegeneous": "\\begin{equation}\\label{eq: inhomogegeneous}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=F(n,t),\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}",
      "thm:disper": "\\begin{theorem}\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in Lemma \\ref{lem:disper-log}. Then for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\T^d$ and any $t\\in\\R$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\Z\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\Z\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\Z\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\Z) \\ .\n\t\t\\end{equation}\n\t\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 3331,
    "pre_theorem_intro_text": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n\t\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere $G_{\\theta}$ is defined as\n\t\\begin{equation*}\n\t\t(G_{\\theta}u)(n)\n\t\t:=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n\t\t\\qquad n\\in\\mathbb{Z},\n\t\\end{equation*}\n\tand the discrete Laplacian is given by\n\t$\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n\tHere $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n    \\begin{equation}\\label{dio}\n\t\t\\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n\t\t>\\frac{\\gamma}{|k|^{\\eta}},\n\t\t\\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n    \\end{equation}\n    where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n    \\begin{equation*}\n        \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n    \\end{equation*}\n    In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}.  We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n\t\\begin{equation}\\label{eq:dispersive}\n\t\t\\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n\t\t\\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n\t\t\\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n\t\\end{equation}\n\twhere $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\t\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive. \n\n\tOur main result states as follows.",
    "context": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n \\begin{equation}\\label{eq:DKG}\n \\begin{cases}\n \\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n \\end{cases}\n \\end{equation}\n where $G_{\\theta}$ is defined as\n \\begin{equation*}\n (G_{\\theta}u)(n)\n :=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n \\qquad n\\in\\mathbb{Z},\n \\end{equation*}\n and the discrete Laplacian is given by\n $\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n Here $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n \\begin{equation}\\label{dio}\n \\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n >\\frac{\\gamma}{|k|^{\\eta}},\n \\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n \\end{equation}\n where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n \\begin{equation*}\n \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n \\end{equation*}\n In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}. We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n \\begin{equation}\\label{eq:dispersive}\n \\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n \\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n \\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n \\end{equation}\n where $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive.\n\nOur main result states as follows.\n\n\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}",
    "full_context": "We consider the Klein--Gordon equation with quasi-periodic potentials on the one-dimensional lattice $\\mathbb{Z}$:\n \\begin{equation}\\label{eq:DKG}\n \\begin{cases}\n \\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\mathbb{Z}\\right),\\quad n\\in\\mathbb{Z},\\\\\n \\end{cases}\n \\end{equation}\n where $G_{\\theta}$ is defined as\n \\begin{equation*}\n (G_{\\theta}u)(n)\n :=-\\Delta_{\\text{disc}}u(n)+V(\\theta+n\\omega)u(n),\n \\qquad n\\in\\mathbb{Z},\n \\end{equation*}\n and the discrete Laplacian is given by\n $\\Delta_{\\text{disc}}u(n):=u(n+1)+u(n-1)-2u(n)$.\n Here $\\theta \\in \\mathbb{T}^d$, the potential $V\\colon\\mathbb{T}^{d}\\to\\mathbb{R}$ is a real-analytic with $d\\geq 1$, and the frequency vector $\\omega\\in\\mathbb{R}^{d}$ is \\textbf{Diophantine}; i.e., there exist constants $\\gamma>0$ and $\\eta>d-1$ such that\n \\begin{equation}\\label{dio}\n \\inf_{j\\in\\mathbb{Z}}\\bigl|\\langle k,\\omega\\rangle_{\\mathbb{R}^{d}}-j\\pi\\bigr|\n >\\frac{\\gamma}{|k|^{\\eta}},\n \\qquad \\forall\\,k\\in\\mathbb{Z}^{d}\\setminus\\{0\\},\n \\end{equation}\n where $\\langle\\cdot{,}\\cdot\\rangle_{\\mathbb{R}^{d}}$ denotes the standard scalar product on $\\mathbb{R}^{d}$. We assume that $V$ admits a bounded analytic complex extension to $\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}$. We also denote\n \\begin{equation*}\n \\varepsilon_{0}:=\\left|V\\right|_{r}=\\sup_{\\left\\{z\\in\\mathbb{C}^{d}:\\left|{\\rm Im}z\\right|<r\\right\\}}\\left|V\\left(z\\right)\\right|.\n \\end{equation*}\n In the literature, one often sets $H_{\\theta}=G_{\\theta}-2$ to be the \\textbf{quasi-periodic Schr\\\"odinger operator}. We are interested in the dispersive estimates for the equation \\eqref{eq:DKG}, namely the $\\ell^{1}\\!\\to\\!\\ell^{\\infty}$ estimates of the form\n \\begin{equation}\\label{eq:dispersive}\n \\norm{u(\\cdot,t)}_{\\ell^{\\infty}(\\mathbb{Z})}\n \\leq C\\JP{t}^{-\\tau}\\bigl(\\norm{\\varphi}_{\\ell^{1}(\\mathbb{Z})}+\\norm{\\psi}_{\\ell^{1}(\\mathbb{Z})}\\bigr),\n \\qquad \\forall\\,\\varphi,\\psi\\in\\ell^{1}(\\mathbb{Z}),\n \\end{equation}\n where $\\JP{t}:=\\sqrt{1+t^{2}}$ is the Japanese bracket, $C>0$ is a constant independent of $t$ and $\\tau$ is so called dispersive decay rate.\n\n\\textbf{Motivation} The goal of this work is to understand to what extent dispersive properties of lattice Klein–Gordon dynamics are stable under introducing a quasi-periodic medium. On the one hand, the free discrete Klein–Gordon equation on $\\mathbb Z$ is known to exhibit a precise dispersive decay rate (see \\cite{SK05}), which plays a central role in both the linear and nonlinear theory. On the other hand, quasi-periodic Schr\\\"odinger operators $H_\\theta$ are canonical models for wave propagation in quasicrystals and are known, in the small analytic and Diophantine regime, to have purely absolutely continuous spectrum of Cantor type (see for example \\cite{AD08, Avi08}). From a dispersive PDE viewpoint, this spectral picture is highly singular and lies completely outside the standard setting of the fine continuous spectrum given by a finite union of intervals, so it is a priori unclear whether any robust $\\ell^1\\to\\ell^\\infty$ decay should survive.\n\nOur main result states as follows.\n\n\\begin{equation}\\label{eq:DKG}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad\\qquad\\quad(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\\n            u(\\cdot,0)=\\varphi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad \\partial_{t}u(\\cdot,0)=\\psi(\\cdot)\\in\\ell^{1}\\left(\\Z\\right),\\quad n\\in\\mathbb{Z},\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}\n\nBy \\eqref{infinito}, we have\n \\begin{equation}\\label{eq:u(t)_int}\n |u(n,t)|\\leq\\frac{1}{\\pi}\\left|\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right) \\rho^{\\prime}\\d E\\right|+\\varepsilon_0^{\\frac{\\sigma^2}{10}}\\|u(\\cdot,t)\\|_{\\ell^\\infty\\left(\\Z\\right)} \\ .\n \\end{equation}\n To estimate $\\|u(\\cdot,t)\\|_{\\ell^\\infty\\left(\\Z\\right)}$, it suffices to control the integral term by taking $\\varepsilon_{0}$ to be small. Substituting the expression of $\\mathcal{S}(u(t))$ into the integral, and recalling the definitions of $g^{\\varphi}$ and $g^{\\psi}$, we obtain\n \\begin{equation}\\label{eq:appro_u}\n \\begin{aligned}\n &\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E\\\\\n =&\\frac{1}{2} \\int_\\Sigma e^{{\\rm i}\\Omega(E)t} \\sum_{k\\in\\Z} \\left(\\varphi(k) + \\frac{\\psi(k)}{{\\rm i}\\Omega(E)}\\right) \\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E \\\\\n &+\\frac{1}{2} \\int_\\Sigma e^{-{\\rm i}\\Omega(E)t} \\sum_{k\\in\\Z} \\left(\\varphi(k) - \\frac{\\psi(k)}{{\\rm i}\\Omega(E)}\\right) \\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E \\ .\n \\end{aligned}\n \\end{equation}\n Note that the term $\\left({\\cK}_k(E){\\cK}_n(E)+{\\cJ}_k(E){\\cJ}_n(E)\\right)$ can be expanded as\n \\begin{equation}\\label{eq:K,J expansions}\n \\sum_{k_{\\Delta}, n_{\\Delta}} \\beta_{k,k_{\\Delta}} \\beta_{n,n_{\\Delta}} \\cos(k_{\\Delta}-n_{\\Delta})\\rho(E) \\ .\n \\end{equation}\n Since $m>0$, the factor $\\frac{1}{\\Omega(E)}= \\frac{1}{\\sqrt{2+m^2+E}}$ is smooth and bounded on the spectrum of $G_{\\theta}+m^2$. By the symmetry of the complex conjugate terms in \\eqref{eq:appro_u}, the dispersive decay estimates for both integrals depend on the oscillatory behavior of the phase function. Thus, the problem reduces to establishing the estimates for the oscillatory integral associated with the kernel\n $e^{-{\\rm i} t \\sqrt{2+m^2+E}}$. For simplicity, we denote\n $$\\mathbb{I}(S;k_{\\Delta},n_{\\Delta},\\rho):=\\int_{S} \\beta_{k,k_{\\Delta}}\\beta_{n,n_{\\Delta}}\\cos\n \\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-\\mathrm{i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime}\\d E, \\quad \\text{for}\\ S\\subset\\R,$$ \n and\n $$\\mathbb{I}^{J}(S;k_{\\Delta},n_{\\Delta},\\rho):=\\int_{S} \\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\cos\n \\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-\\mathrm{i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime} \\d E, \\quad \\text{for}\\ S\\subset\\R.$$ \n Whenever no confusion can arise, we simply write $\\mathbb{I}(E)$ and $\\mathbb{I}^{J}(E)$ respectively.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:disper} assuming Lemma \\ref{lem:disper-log}]\n By the expansions \\eqref{eq:K,J expansions}, we apply Lemma \\ref{lem:disper-log} to equation \\eqref{eq:appro_u} and derive\n \\begin{equation}\\label{eq:L infty estimate}\n \\begin{aligned}\n &\\left|\\int_\\Sigma \\left(g_1(E,t){\\cK}_n(E)+g_2(E,t){\\cJ}_n(E)\\right)\\, \\rho^{\\prime}\\d E\\right|\\\\\n \\leq&C\\left(m\\right)\\cdot\\frac{\\left|\\ln\\varepsilon_{0}\\right|^{82000\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}d}}{\\JP{t}^{\\frac{1}{3}}}\\left(\\sum_{k\\in\\Z}\\left|\\varphi\\left(k\\right)\\right|+\\frac{1}{\\inf_{E\\in\\Sigma}\\left|\\Omega\\left(E\\right)\\right|}\\sum_{k\\in\\Z}\\left|\\psi\\left(k\\right)\\right|\\right),\n \\end{aligned}\n \\end{equation}\n where the absolute constant $C_{1}$ relates to the estimate of $e^{{\\rm i}\\Omega\\left(\\lambda\\right)t}$ and so does $C_{2}$ for the estimate related to $e^{-{\\rm i}\\Omega\\left(\\lambda\\right)t}$. Since $\\epsilon_{0}<1$ is small, and by \\eqref{eq:u(t)_int}, \\eqref{eq:L infty estimate} we deduce\n \\begin{equation*}\n \\norm{u\\left(t\\right)}_{\\ell^{\\infty}\\left(\\Z\\right)}\\leq\\frac{C\\left(m\\right)}{1-\\varepsilon_0^{\\frac{\\sigma^2}{10}}}\\frac{\\left|\\ln\\varepsilon_{0}\\right|^{82000\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}d}}{\\JP{t}^{\\frac{1}{3}}}\\left(\\norm{\\varphi}_{\\ell^{1}\\left(\\Z\\right)}+\\frac{1}{\\inf_{E\\in\\Sigma}\\left|\\Omega\\left(E\\right)\\right|}\\norm{\\psi}_{\\ell^{1}\\left(\\Z\\right)}\\right).\n \\end{equation*}\n This completes the proof.\n \\end{proof}\n\nIn the case of $|k_{\\Delta}-n_{\\Delta}|$ is large, we derive the following bound. \n\\begin{lemma}\\label{lem:large}\nFor any $t \\in \\R$, the following estimate holds, \n\\begin{equation*}\n \\left|\\mathbb{I}^{J}(\\Sigma)\\right|\\leq\\frac{18}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}+\\frac{3\\Theta_{1}\\JP{t}}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left(\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}\\right)^{2}+\\varepsilon_{j}^{6\\sigma}\\right)\\left(1+\\frac{1}{\\Theta_{2}}\\right).\\\\\n\\end{equation*} \n\\end{lemma}\n\\begin{proof}\nBy Proposition~\\ref{propsana1}, for each $0\\leq j\\leq J$ we can write\n\\begin{equation*}\n \\Gamma_{j}^{(J)}\n =\\mathcal{R}_{j}\\cup\\Bigl(\\,\\bigsqcup_{s^{J}_{j}}I_{s^{J}_{j}}\\Bigr),\n\\end{equation*}\nwhere $\\mathcal{R}_{j}$ is a finite set of points (possibly empty) and each $I_{s^{J}_{j}}$ is an open interval in $\\Gamma_{j}^{(J)}$. Recall that $\\rho$ is absolutely continuous. Then\n \\begin{equation}\\label{eq:Ij-M large}\n \\begin{aligned}\n \\mathbb{I}^{J}(\\Sigma)=&\\int_{\\inf\\Sigma}^{\\sup\\Sigma}\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\cos\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\cdot\\rho^{\\prime} \\d E\\\\\n =&\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)\\Big|_{\\inf I_{s_{j}^{J}}}^{\\sup I_{s_{j}^{J}}}\\\\\n &\\quad-\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\int_{I_{s_{j}^{J}}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)^{\\prime}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)\\d E.\\\\\n \\end{aligned}\n \\end{equation}\n Using Proposition \\ref{propsana1} again, we derive the upper bound for connected components of $\\bigcup \\Gamma_{j}^{(J)}$:\n \\begin{equation*}\n \\sum_{j=0}^{J}\\#\\left(s_{j}^{J}\\right)\\leq \\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}.\n \\end{equation*}\n For the first term in equation \\eqref{eq:Ij-M large}, by the $L^{\\infty}$-estimate and inequality \\eqref{esti_beta_0} that\n \\begin{equation}\\label{eq:first term}\n \\begin{aligned}\n &\\left|\\frac{1}{k_{\\Delta}-n_{\\Delta}}\\sum_{j=0}^{J}\\sum_{s_{j}^{J}}\\big(\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\sin\\big((k_{\\Delta}-n_{\\Delta})\\rho\\left(E\\right)\\big)e^{-{\\rm i}t\\Omega\\left(E\\right)}\\big)\\Big|_{\\inf I_{s_{j}^{J}}}^{\\sup I_{s_{j}^{J}}}\\right|\\\\\n \\leq&\\frac{2}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}\\cdot\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}+\\varepsilon_{0}^{3\\sigma}\\right)^{2}\\\\\n \\leq&\\frac{18}{\\left|k_{\\Delta}-n_{\\Delta}\\right|}\\left|\\ln \\varepsilon_{0}\\right|^{2J^{2}d}.\n \\end{aligned}\n \\end{equation}\nFor the second term of the result in equation \\eqref{eq:Ij-M large}, we need the following direct estimate\n\\begin{equation}\\label{eq:betac1}\n \\left|\\beta^J_{k,k_{\\Delta}} \\beta^J_{n,n_{\\Delta}}\\right|_{\\mathcal{C}^{1}\\left(\\Gamma^{\\left(J\\right)}_{j}\\right)}\\leq\n \\begin{cases}\n 3\\left(1+\\varepsilon_{0}^{\\frac{1}{4}}\\right)^{2}\\;{\\rm when}\\;j=0;\\\\\n 3\\varepsilon_{j}^{6\\sigma}\\qquad{\\rm when}\\;1\\leq j\\leq J.\\\\\n \\end{cases}\n\\end{equation}\n Using inequality\n \\eqref{eq:betac1}, we deduce",
    "post_theorem_intro_text_len": 6353,
    "post_theorem_intro_text": "For our Theorem \\ref{thm:disper}, we have the following remarks.\n\n\t\\begin{itemize}\n\t\t\\item[($\\mathbf{R}_1$)] Stefanov and Kevrekidis~\\cite{SK05} first established a sharp dispersive decay rate  \n\t\t$\n\t\t\\frac{1}{3}\n\t\t$ \n\t\tfor the free discrete Klein-Gordon equation (i.e., equation \\eqref{eq:DKG} with $V=0$) on $\\mathbb Z$ via Van der Corput's lemma.  \n\t\tRecently, Borovyk and Goldberg \\cite{BG17} derived the corresponding rate on $\\mathbb Z^{2}$, namely  $\\langle t\\rangle^{-\\frac{3}{4}}$. Later, Cuenin and Ikromov~\\cite{CI21} derived the rates on~$\\mathbb Z^{d}$ for $d=3,4$, namely $\n\t\t\\langle t\\rangle^{-\\frac{7}{6}}\\ (d=3)$, and\n\t\t$\\langle t\\rangle^{-\\frac{3}{2}}\\log \\bigl(2+|t|\\bigr)\\ (d=4)$. The obstacle to extending Cuenin-Ikromov to higher dimensions is that the finite classification of stable phase singularities (Thom's catastrophes), which is essential for deriving sharp decay rates, breaks down in dimensions $d \\ge 5$. Our result can be seen as a perturbation result for Stefanov and Kevrekidis, which shows that the decay rate $(\\frac{1}{3})^{-}$ persists under small quasi-periodic perturbations of the potentials. \n\n\t\t\\item[($\\mathbf{R}_2$)] While several studies (e.g., \\cite{CT09, GPP13, PS08, KPS09, KKK06}) have addressed dispersive equations with potentials whose continuous spectrum is a union of disjoint intervals, generic behavior of the operator $H_{\\theta}$ is markedly different: its spectrum is purely absolutely continuous and forms a Cantor set; see, e.g., \\cite{AD08, Avi08, E92}. Our approach builds heavily on recent advances in the spectral theory of quasi-periodic Schr\\\"odinger operators; see, for example, \\cite{Z16}. Specifically, we employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals. Using a modified  spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay. We emphasize that a similar strategy appears in the study of ballistic motion and dispersive estimates for discrete Schr\\\"odinger equations with quasi-periodic potentials; see, e.g., \\cite{BZ20, Z16}. \n\n        \\item[($\\mathbf{R}_3$)] For discrete wave equation with quasi-periodic potentials (i.e., equation \\eqref{eq:DKG} with $m=0$), our method does not work. In fact, there does not exist $\\ell^1 \\to \\ell^{\\infty}$ estimates for fundamental solution of the free wave equation. We point out the following development regarding the dispersive estimate for the free discrete wave equation. This phenomenon also holds on $\\mathbb{R}^1$. Schultz \\cite{Sch98} proved dispersive estimates for the wave equation on lattice graphs $\\mathbb{Z}^d$ for $d=2,3$, namely  $\\langle t\\rangle^{-\\frac{3}{4}}\\ (d=2)$ and $\\langle t\\rangle^{-\\frac{7}{6}}\\ (d=3)$. Combining the singularity theory\n\t\twith results in uniform estimates of oscillatory integrals, Bi-Chen-Hua \\cite{BCH1} extended this result  to $d=4$: they proved that the optimal\n\t\ttime decay rate is $|t|^{-\\frac{3}{2}} \\log |t|$. By tools from Newton polyhedra, Bi-Chen-Hua \\cite{BCH2} further extended the result to $d=5$: the sharp decay rate of the fundamental solution of the wave equation on $\\mathbb{Z}^5$ is $|t|^{-\\frac{6}{11}}$.\n    \\end{itemize}\n\n\tA typical application for dispersive estimates is the derivation of Strichartz inequalities. For the inhomogeneous discrete Klein-Gordon equation, \n\t\\begin{equation}\\label{eq: inhomogegeneous}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=F(n,t),\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\left(0,\\infty\\right),\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}\n\twe have the following mixed space-time estimates via the standard Keel-Tao argument. For convenience, we say   \n\ta pair of exponents $(q,r)$ with\n\t$\n\t2\\le q,r\\le\\infty\n\t$\n\tis called \\emph{$\\tau$--admissible} if\n\t$\n\t\\frac{2}{q}\n\t=\\tau\\Bigl(1-\\frac{2}{r}\\Bigr).\n\t$ Here $0<\\tau<\\tfrac13$ is fixed as in Theorem~\\ref{thm:disper}.\n\n\t\\begin{theorem}\\label{thm:Strichartz}\n\n        Fix\n\t\t$0<\\tau<\\tfrac13$ and $F\\in L^{\\tilde q'}(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))$. Let $(q,r)$ and $(\\tilde q,\\tilde r)$ be two $\\tau$--admissible pairs with $q,\\tilde q<\\infty.$  Then the Cauchy\n\t\tproblem \\eqref{eq: inhomogegeneous} has a unique global solution\n\t\t$\n\t\tu\\in C(\\mathbb R;\\ell^2(\\mathbb Z))\n\t\t$\n\t\tand \\begin{equation}\\label{eq:inhom-Strichartz}\n\t\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\n\t\t\t\\;\\le\\; C\\Bigl(\n\t\t\t\\|\\varphi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|\\psi\\|_{\\ell^2(\\mathbb Z)}\n\t\t\t+\\|F\\|_{L^{\\tilde q'}_t(\\mathbb R;\\ell^{\\tilde r'}(\\mathbb Z))}\n\t\t\t\\Bigr),\n\t\t\\end{equation}\n\t\twhere $\\tilde q'$ and $\\tilde r'$ denote the Hölder conjugates of\n\t\t$\\tilde q$ and $\\tilde r$, respectively, and the constant $C$ is\n\t\tindependent of $\\theta$, $\\varphi$, $\\psi$ and $F$. \n\n\t\\end{theorem}\n\n\tUsing Strichartz estimates, we establish the global well-posedness of the nonlinear Klein-Gordon equation  \\begin{equation}\\label{eq: nonlinear}\n\t\t\\begin{cases}\n\t\t\t\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)= \\pm |u|^{p-1}u,\n\t\t\t&(n,t)\\in\\mathbb{Z}\\times\\mathbb{R}_{>0},\\\\[2mm]\n\t\t\tu(n,0)=\\varphi(n),\\quad \\partial_{t}u(n,0)=\\psi(n),\n\t\t\t&n\\in\\mathbb{Z},\n\t\t\\end{cases}\n\t\\end{equation}\nwith small initial data.\t\n\t\\begin{theorem}\\label{thm:nonlinear}\n\t\tLet $p>7$. There exist $\\varepsilon > 0$ and a constant $C$, so that whenever\n\t\t$\\| \\varphi \\|_{\\ell^2\\left(\\mathbb{Z}\\right)}$ and $\\| \\psi\\|_{\\ell^2\\left(\\mathbb{Z}\\right)} \\leq \\varepsilon\n\t\t$, there exists a unique global solution to \\eqref{eq: nonlinear}, satisfying\n\t\t\\begin{equation*}\n\t\t\\|u\\|_{L^q_t(\\mathbb R;\\ell^r(\\mathbb Z))}\\leq C\\varepsilon,\n\t\t\\end{equation*}\n\t\tfor all admissible pairs $(q, r)$. In particular, $\\| u(n,t) \\|_{\\ell^r(\\mathbb{Z})} \\to 0$ as $t \\to \\infty$ for every $r > 2$.  \n\t\\end{theorem}\n\nThe remainder of the paper is organized as follows. Section \\ref{sec:2} reviews the structure of the spectrum of the quasi-periodic Schr\\\"odinger operator and presents a spectral transform that will be used throughout. Section \\ref{sec:3} establishes Theorem \\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum. In Section \\ref{sec:4}, we prove Theorems \\ref{thm:Strichartz} and \\ref{thm:nonlinear} as applications of the dispersive estimate.",
    "sketch": "A proof strategy for Theorem~\\ref{thm:disper} is outlined in Remark $\\mathbf{R}_2$ and the paper organization: the authors \"employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals.\" Then, \"[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay.\" Concretely, Section~\\ref{sec:3} \"establishes Theorem~\\ref{thm:disper} by analysing a specific oscillatory integral on the spectrum.\"",
    "expanded_sketch": "A proof strategy for the main theorem is outlined in Remark $\\mathbf{R}_2$ and the paper organization: the authors \"employ a KAM-based perturbative construction to approximate the rotation number and decompose the spectrum into many small intervals.\" Then, \"[u]sing a modified spectral transform, we represent the solution as an oscillatory integral and then apply Van der Corput's lemma on each interval to establish dispersive decay.\" Concretely, next the authors establish the main theorem by analysing a specific oscillatory integral on the spectrum.",
    "expanded_theorem": "\\label{thm:disper}\n\t\tAssume $\\varepsilon_0<\\varepsilon_*$, with $\\varepsilon_*$ as in the following lemma.\n\n\\begin{lemma}\\label{lem:disper-log}\n\t\tAssume that $\\left|V\\right|_{r}=\\varepsilon_{0}\\leq \\varepsilon_{\\ast}$ with $\\varepsilon_{\\ast}$ defined in Proposition \\ref{propsana}. For any $k,k_{\\Delta},n,n_{\\Delta}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\left|\\mathbb{I}\\left(\\Sigma\\right)\\right|\\leq\\frac{C\\left(m\\right)\\left|\\ln\\varepsilon_{0}\\right|^{82000d\\left(\\ln\\ln\\left(2+\\JP{t}\\right)\\right)^{2}}}{\\JP{t}^{\\frac{1}{3}}},\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tfor any $t\\in\\R$ where $m>0$ and the constant $C\\left(m\\right)$ only depends on $m$.\n\t\\end{lemma}\n\n\t\tThen for any given $0<\\tau<\\frac13$, there exists $K_1=K_1(\\varepsilon_0,\\tau)$ such that for any $\\theta\\in\\mathbb{T}^d$ and any $t\\in\\mathbb{R}$,\n\\begin{equation}\\label{linest}\n\t\t\t\\norm{u\\left( \\cdot ,t\\right)}_{\\ell^\\infty\\left(\\mathbb{Z}\\right)}\\leq K_1\\JP{t}^{-\\tau}(\\norm{\\varphi}_{\\ell^1\\left(\\mathbb{Z}\\right)}+\\norm{\\psi}_{\\ell^1\\left(\\mathbb{Z}\\right)}) \\ , \\quad \\forall \\ \\varphi, \\psi \\in\\ell^1(\\mathbb{Z}) \\ .\n\t\t\\end{equation}",
    "theorem_type": [
      "Existential–Universal",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Consider the discrete Klein--Gordon equation on \\(\\mathbb Z\\)\n\\[\n\\bigl(\\partial_{tt}+G_{\\theta}+m^{2}\\bigr)u(n,t)=0,\\qquad (n,t)\\in \\mathbb Z\\times (0,\\infty),\n\\]\nwith initial data\n\\[\nu(\\cdot,0)=\\varphi\\in \\ell^{1}(\\mathbb Z),\\qquad \\partial_t u(\\cdot,0)=\\psi\\in \\ell^{1}(\\mathbb Z),\n\\]\nwhere\n\\[\n(G_{\\theta}u)(n)=-\\Delta_{\\mathrm{disc}}u(n)+V(\\theta+n\\omega)u(n),\n\\qquad\n\\Delta_{\\mathrm{disc}}u(n)=u(n+1)+u(n-1)-2u(n).\n\\]\nAssume \\(\\theta\\in \\mathbb T^d\\), \\(d\\ge 1\\), \\(m>0\\), \\(\\omega\\in\\mathbb R^d\\) is Diophantine, and \\(V:\\mathbb T^d\\to\\mathbb R\\) extends to a bounded analytic function on \\(\\{z\\in\\mathbb C^d:|\\operatorname{Im} z|<r\\}\\). Let\n\\[\n\\varepsilon_0:=|V|_r=\n\\sup_{\\{z\\in\\mathbb C^d:|\\operatorname{Im} z|<r\\}} |V(z)|,\n\\]\nand assume \\(\\varepsilon_0<\\varepsilon_*\\). Also write \\(\\langle t\\rangle:=\\sqrt{1+t^2}\\). Under these assumptions, which quantitative \\(\\ell^1\\to\\ell^\\infty\\) dispersive estimate is valid for the solution \\(u\\)?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac13\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists a constant \\(K_1=K_1(\\varepsilon_0)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-1/3}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "C",
          "text": "There exists a constant \\(K_1=K_1(\\varepsilon_0)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "D",
          "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac13\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau,\\theta)\\) such that, for every \\(t\\in\\mathbb R\\) and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        },
        {
          "label": "E",
          "text": "For every \\(\\tau\\) with \\(0<\\tau<\\tfrac12\\), there exists a constant \\(K_1=K_1(\\varepsilon_0,\\tau)\\) such that, for every \\(\\theta\\in\\mathbb T^d\\), every \\(t\\in\\mathbb R\\), and all \\(\\varphi,\\psi\\in \\ell^1(\\mathbb Z)\\), the corresponding solution satisfies\n\\[\n\\|u(\\cdot,t)\\|_{\\ell^\\infty(\\mathbb Z)}\n\\le K_1\\langle t\\rangle^{-\\tau}\n\\bigl(\\|\\varphi\\|_{\\ell^1(\\mathbb Z)}+\\|\\psi\\|_{\\ell^1(\\mathbb Z)}\\bigr).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "endpoint decay exponent from oscillatory-integral bound",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped time-decay factor \\(\\langle t\\rangle^{-\\tau}\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "uniformity in phase parameter \\(\\theta\\)",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "allowed range of admissible decay exponents",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives hypotheses and asks for the valid dispersive estimate, but it does not state or strongly hint at the precise decay rate, admissible range of exponents, or uniformity in \u0003b8. The correct option is not leaked."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the question presents the assumptions and asks for the exact conclusion of the dispersive estimate. The options are small perturbations of the theorem statement rather than genuinely different mathematical outcomes."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish subtle variants (endpoint \\(1/3\\) vs. \\(\\tau<1/3\\), uniform vs. \\(\\theta\\)-dependent constants, stronger vs. weaker decay), but the task mainly rewards recalling the precise theorem statement rather than generating a conclusion from reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: they reflect common failure modes such as claiming the endpoint decay, dropping decay entirely, weakening uniformity in \\(\\theta\\), or overstating the admissible exponent range. They are plausible and mathematically distinct."
      },
      "total_score": 5,
      "overall_assessment": "A solid multiple-choice item with no answer leakage and high-quality distractors, but it is mostly a direct theorem-restatement question and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.22662v1",
    "paper_link": "http://arxiv.org/abs/2512.22662v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.",
      "start_pos": 15706,
      "end_pos": 15914,
      "label": "thm"
    },
    "ref_dict": {
      "co": "\\begin{proof} For every $p\\in M^k$ the set $\\cal{S}(p)$ has by Lemma~\\ref{l5} an $(r+1)$-step fibering over $C$. As in the proof of Lemma~\\ref{l5}, this yields a finite set $E$ of tuples \n$$\\big( f_1,\\dots, f_{r+1}, f_{r+2}; m_1,\\dots,m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nwhere for $j=1,\\dots, r+2$: $f_j$ is a definable map $ \\Sigma_j\\to M^{n_i}$ with (definable) domain \n$\\Sigma_j\\subseteq M^l\\times M^{m_1}\\times\\cdots\\times M^{m_{j-1}}\\times M^{m+n_1+\\cdots + n_{j-1}}$, and for each $p\\in M^k$ there is \na tuple in $E$ as displayed and $q\\in M^{l}$\nsuch that $\\cal{S}(p)=\\Sigma_1(q)$ and the tuple\n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1}, n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.  \n\nConsider one such tuple $\\tau\\in E$ as displayed above. Let $Q$ be the definable\nset of $q\\in M^{l}$ such that\nfor some $p\\in M^k$ we have $\\cal{S}(p)=\\Sigma_1(q)$ and \n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.\nEach $q\\in Q$ yields a map\n$f_1(q,-): \\Sigma_1(q)\\to M^{n_1}$ such that its preimages of points $y\\in M^{n_1}$ are in $\\cC_r$. The maps $f_1(q,-)$ with $q\\in Q$ make up a definable family $\\cal{F}_\\tau$ of maps from subsets of $M^m$ into $M^{n_1}$ as described just before Lemma~\\ref{l2+}. That lemma then tells us that $\\{\\mu\\big(\\Sigma_1(q)\\big): q\\in Q\\}$ is finite, and that\n$\\{q\\in Q:\\ \\mu\\big(\\Sigma_1(q)\\big)=a\\}$ is definable for every $a$. This holds for all $\\tau\\in E$, which yields the\ndesired result.\n\\end{proof} \n\n\\noindent\nThis finishes the inductive proof of Theorem~\\ref{thm} in the case of $\\omega$-saturated \n$\\cM$. \n\n\\subsection*{Reduction to the $\\omega$-saturated case}  Here $C$ is stably embedded in $\\cM$, but we do not assume $\\cM$ is $\\omega$-saturated. \n\nLet $\\cM^*$ be an elementary extension of $\\cM$. We define an $A$-valued Fubini measure $\\mu^*$ on  the full subcategory \n$\\Def(C^*)$ of $\\Def(\\cM^*)$: Let $Y\\subseteq (C^*)^n$ be definable in $\\cM^*$; take an $\\cal{L}$-formula $\\phi(x,y)$ with $x=(x_1,\\dots, x_m)$, $y=(y_1,\\dots, y_n)$, and a point $p_Y\\in (C^*)^m$ such that $Y=\\{q\\in (C^*)^n: \\cM^*\\models \\phi(p_Y,q)\\}$. To simplify notation, set $\\phi(p, C^n):=\\{q\\in C^n:\\ \\cM\\models \\phi(p,q)\\}$ for $p\\in C^m$, and likewise define\n$\\phi\\big(p, (C^*)^n\\big)\\subseteq (C^*)^n$ for $p\\in (C^*)^m$; in particular, $Y=\\phi\\big(p_Y, (C^*)^n\\big)$. \nNow $C^m$ is the disjoint union of the definable sets $X_a:=\\{p\\in C^m:\\ \\mu\\big(\\phi(p,C^m)\\big)=a\\}$ (which is empty for all but finitely many $a$). \nSet $\\mu^*(Y):=a$, where $a$ is unique such that $p_Y\\in (X_a)^*$. It is routine to verify that this particular $a$ does not depend on the choice of $\\phi$ and that it yields indeed a Fubini measure on $\\Def(C^*)$. \n\nNow take such an elementary extension $\\cM^*$ to be $\\omega$-saturated. Then $\\mu^*$ extends uniquely to an $A$-valued Fubini measure $\\mu^*$ on the full subcategory $\\Def(C^*)^{\\f}$ of $\\Def(\\cM^*)$. \nFor $X\\in \\Def(C)^{\\f}$ we have $X^*\\in \\Def(C^*)^{\\f}$; set $\\mu(X):= \\mu^*(X^*)$.\nThis extends the original $A$-valued Fubini measure $\\mu$ on $\\Def(C)$ to an $A$-valued Fubini measure $\\mu$ on $\\Def(C)^{\\f}$.  As to uniqueness, suppose $X$ is fiberable over $C$, and take an $r$-step fibering $(f_1,\\dots, f_r, f_{r+1}; m_1,\\dots, m_{r}, n_1,\\dots, n_{r+1})$\nof $X$ over $C$. \nFor $r=0$ the definable map $f_1: X \\to C^{n_1}$ is injective, and so by Fubini\n$\\mu(X)$ is uniquely determined by $\\mu: \\Def(C)\\to A$. For $r\\ge 1$ the definable map $f_1: X\\to M^{n_1}$ is such that\n$f_1(X)$ and every\n$f_1^{-1}(y)$ with $y\\in f_1(X)$ is $(r-1)$-step fiberable over $C$, and we can assume inductively that the values of \n$\\mu\\big(f(X)\\big)$ and $\\mu\\big(f_1^{-1}(y)\\big)$ for\n$y\\in f_1(X)$ are uniquely determined by $\\mu: \\Def(C)\\to A$, which again by Fubini uniquely deternines $\\mu(X)$. \nThis concludes the proof of Theorem~\\ref{thm}. \n\n\\section{Fiberability and Co-analyzability}\\label{co}\n\n\\noindent\nFiberability is a somewhat ad hoc notion, but agrees under natural model-theoretic conditions with the more intrinsic notion of co-analyzability from \\cite{HHM}, as we shall see. {\\it Throughout this section $X\\subseteq M^m$ is definable}.  \n\n\\subsection*{Fiberability implies co-analyzability}\nWe recall here the recursion on $r$ that defines ``$X$ is co-analyzable in $r$ steps\" (tacitly: relative to $C$ in $\\cM$), where $\\cM$ is assumed to be $\\omega$-saturated: \\begin{enumerate}\n\\item  $X$ is co-analyzable in $0$ steps iff $X$ is finite;\n\\item  $X$ is co-analyzable in $r+1$ steps iff for some definable set $R \\subseteq M^m\\times C$ we have: $\\breve{R}(y)$ is co-analyzable in $r$ steps for every $y\\in C$, and \n$X:= \\bigcup_{y\\in C}\\breve{R}(y)$. \n\\end{enumerate} \n See \\cite[p. 35]{ADH} for how this extends to not necessarily $\\omega$-saturated $\\cM$ in such a way that\n for any elementary extension $\\cM^*$ of $\\cM$: $X$ is co-analyzable in $r$ steps relative to $C$ in $\\cM$ iff $X^*$ is co-analyzable in $r$ steps relative to $C^*$ in $\\cM^*$.  Call $X$ {\\em co-analyzable\\/} if $X$ is co-analyzable in $r$ steps for some $r$. \n\n By \\cite{HHM} this notion can be characterized as follows:\n\n \\begin{prop}\\label{hhm} \nLet $T$ be a complete $L$-theory, where $L$ is a countable one-sorted language. Let $y$ be a single variable and $D(y)$ an $L$-formula that defines in each model $\\cal{N}=(N;\\cdots)$ of $T$ a set $D(N)$ with $|D(N)|\\ge 2$.\nThen the following conditions on an $L$-formula $\\varphi(x)$ with $x=(x_1,\\dots,x_m)$ are equivalent:\n\\begin{enumerate}\n\\item[\\textup{(i)}] for some  $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(ii)}] for all $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(iii)}] for all  $\\cal{N}\\models T$, if \n$D(N)$ is countable, then so is $\\varphi(N^m)$;\n\\item[\\textup{(iv)}] for all $\\cal{N}\\preceq\\cal{N}^*\\models T$, if\n$D(N)=D(N^*)$, then $\\varphi(N^m)=\\varphi((N^*)^m)$.\n\\end{enumerate}\n\\end{prop}\n\n\\noindent\nThe equivalences (i)$\\Leftrightarrow$(ii)$\\Leftrightarrow$(iv) hold without assuming that $L$ is countable:\nFor (i)$\\Leftrightarrow$(ii)$\\Rightarrow$(iv), reduce to countable $L$ by taking suitable reducts. For (iv)$\\Rightarrow$(ii), use also that (iv) amounts to $T_2\\models \\sigma$ where $T_2$ is the theory of elementary pairs of models of $T$ and $\\sigma$ is a certain sentence, depending on $\\varphi$, in the language of $T_2$. Note that in addition we can restrict $\\cal{N}$ and $\\cal{N}^*$ in (iv) to be  $\\omega$-saturated.\n\n\\begin{lemma}\\label{fico} If $X$ is fiberable over $C$, then  $X$ is co-analyzable relative to $C$. \n\\end{lemma} \n\\begin{proof} \nWe arrange that $\\cM$ is $\\omega$-saturated. \nBelow we prove by induction on $r$:  if $X$ is $r$-step fiberable over $C$, then $X$ is co-analyzable relative to $C$. \nFor $r=0$, if $f: X\\to C^n$ is definable and injective, then $X$ is co-analyzable in $n$ steps relative to $C$, by an easy induction on $n$. Thus the claimed implication holds for $r=0$. Assume this implication holds for a certain $r$, and let\n$X$ be $(r+1)$-step fiberable over $C$.\nTo show $X$ is co-analyzable relative to $C$ in $\\cM$ we \naugment $\\cal{L}$ by names for finitely many elements of $M$ to arrange that $X$ is $0$-definable, so $X=\\varphi(M^m)$ with $\\varphi(x)$ an $\\cal{L}$-formula, $x=(x_1,\\dots, x_m)$. We now apply (iv)$\\Rightarrow$(ii) of Proposition~\\ref{hhm} without assuming $L=\\cal{L}$ is countable, with $T:=\\Th(\\cM)$ and $D(y)$ an $\\cal{L}$-formula such that $D(M)=C$. \nSo let $\\cal{N}\\preceq \\cal{N}^*\\models T$ with $D(N)=D(N^*)$; it suffices to show that then $\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$.\nWe also assume $\\cal{N}$ is $\\omega$-saturated, as we may by a remark following Proposition~\\ref{hhm}. Since $\\cal{M}\\equiv \\cal{N}$, the set $\\varphi(N^m)$ is $(r+1)$-step fiberable over $C^{\\cal{N}}:= D(N)$ in $\\cal{N}$. Take $f: \\varphi(N^m)\\to Y$ where $Y\\subseteq N^n$ and $f$ are definable in $\\cal{N}$, such that $Y$ is $r$-step fiberable over $C^{\\cal{N}}$ and for all\n$y\\in Y$ the fiber $f^{-1}(y)$ is $r$-step fiberable over $C^{\\cal{N}}$. Thus by our inductive assumption and Proposition~\\ref{hhm}, $Y$ and the fibers\n$f^{-1}(y)$ for $y\\in Y$ do not change in passing from $\\cal{N}$ to $\\cal{N}^*$. Hence the same is true for $\\varphi(N^m)$, that is,\n$\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$. \n\\end{proof}",
      "proof": "\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}",
      "ms": "\\label{ms}  \n\n\\noindent\nSo far we assumed $\\cM$ to be one-sorted. But many structures are more naturally viewed as many-sorted; for example, valued fields as $3$-sorted structures with residue field a",
      "prelim": "\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below",
      "thm": "\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}",
      "special": "\\begin{proof} Take definable $f: Y\\to M^m$ with $Y\\subseteq C^\\nu$, $\\nu\\in \\N$, such that $X=f(Y)$. Since $(C;\\cM)$ has EI, we have a definable map $g:Y\\to C^n $ such that for all $y,z\\in Y$ we have $f(y)=f(z)\\Leftrightarrow g(y)=g(z)$. This yields a definable bijection $$\\qquad\\qquad \\qquad \\qquad f(y)\\mapsto g(y)\\ :\\  X\\to g(Y),\\qquad (y\\in Y). \\qquad \\qquad \\qquad \\qquad \\qedhere$$\n\\end{proof} \n\n\\subsection*{Analyzability} \nThe concept of {\\em analyzability\\/} is more popular in model theory than co-analyzability, but is usually defined for types\nin the environment of a monster model, instead of definable sets. I have been told that for definable sets and $\\omega$-saturated $\\cM$ its recursive definition should be as follows: \n\n{\\em $X$ is $0$-step analyzable in $C$ iff $X$ is internal to $C$, and $X$ is $(r+1)$-step analyzable in $C$ iff there is a $\\Def(\\cM)$-morphism $f: X \\to Y$ such that $Y$ is $r$-step analyzable over $C$ and all fibers $f^{-1}(y)$, $y\\in Y$, are internal to $C$. Finally, $X$ is analyzable in $C$ iff $X$ is $r$-step analyzable in $C$ for some $r$}. \n\nThis extends to not necessarily $\\omega$-saturated $\\cM$ by requiring in the $(r+1)$-clause the existence of a definable \nfamily of maps from subsets of $C^n$ (for a fixed $n$) into $M^m$ such that for each $y\\in Y$ some member of the family has image $f^{-1}(y)$. \nWith this extended notion and any elementary extension $\\cM^*$ of $\\cM$, \n$$X \\text{ is $r$-step analyzable in }C\\ \\Longleftrightarrow\\  X^* \\text{ is $r$-step analyzable in }C^*.$$ \nUsing this it follows from Proposition~\\ref{hhm} that if $X$ is analyzable in $C$, then $X$ is co-analyzable relative to $C$. Ben Castle and Rahim Moosa have assured me that the converse should also hold under some natural model-theoretic conditions to be satisfied in particular by differentially closed fields (and probably by $\\T$) with $C$ its constant field. I have not pursued this, but it would be good to have this confirmed. One motivation for this: it might help in showing that for $\\T$ and for differentially closed fields the notions of  fiberability over $C$ and co-analyzability relative to $C$ do not collapse for any $r$ to $r$-step fiberability over $C$ and co-analyzability in $r$-steps relative to $C$, respectively. This is plausible in view of \\cite{J} where Jin constructs in a ``big\" differentially closed field for any $r\\ge 1$ a $1$-type that is $r$-step analyzable but not $(r-1)$-step analyzable (in the constant field). \n\n\\section{Application to $\\T$ and to differentially closed fields} \\label{special} \n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$. \n\nAll this holds also for any $\\upo$-free, newtonian, Liouville closed $H$-field $K$ and its constant field instead of\n$\\T$ and its constant field $\\R$; this includes all ordered differential fields $K \\equiv\\T$.  The extended Euler characteristic now yields an answer to a test question raised in \\cite[p. 34, end of Section 5]{ADH}:\n\n\\begin{cor} With $K$ as above, let $X\\subseteq K$ be definable in $K$. Then $X$ with the ordering induced by $K$\nis not elementarily equivalent to the ordered set $\\omega$. \n\\end{cor}\n\\begin{proof} Suppose $X\\equiv \\omega$. Then $X$ is discrete as a subspace of $K$ with its order topology. Hence $X$ is fiberable\nover $C$ by \\cite{ADH}. Now $X\\equiv \\omega$ also yields  a definable bijection between $X$ and $X\\setminus \\{\\text{the least element of $X$}\\}$, but the extended Euler characteristic then gives $E(X)=E(X)-1$, a contradiction. \n\\end{proof}"
    },
    "pre_theorem_intro_text_len": 5546,
    "pre_theorem_intro_text": "\\label{int}\n\n\\noindent\nSemialgebraic sets $Y\\subseteq  {\\bbold R} ^n$ have a dimension $\\dim(Y)\\in  {\\bbold N} \\cup\\{-\\infty\\}$ and an Euler characteristic $E(Y)\\in  {\\bbold Z} $ with good properties; see \\cite[Ch. 4]{D}. In \\cite{ADH} we asked for a natural extension of this dimension and Euler characteristic to the definable sets $X\\subseteq  {\\bbold T} ^m$\nthat are ``fiberable by $ {\\bbold R} $\" as defined there; here $ {\\bbold T} $ is the real closed differential field of transseries, which has its subfield $ {\\bbold R} $ as constant field. \n\nThere are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).   \n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise.  For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in  {\\bbold N} $ is part of  specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$.  For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$. \n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)]  $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq  M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet  $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in  \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in  \\cal{C}$:  $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property);  for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using  notation of (F2). \n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are  semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers.  Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity. \n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$.  Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq  \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c)  imposed earlier on $\\cal{C}$.",
    "context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.",
    "full_context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\Def(C)$ be the full subcategory of $\\Def(\\cM)$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\Def(C)^{\\f}\\supseteq \\Def(C)$ of $\\Def(\\cM)$. The categories $\\Def(C)$ and $\\Def(C)^{\\f}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cC$.\n\n\\noindent\nSuppose $\\cM$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in \\N$ we define ``$X$ is $r$-step fiberable over $C$\": this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\n\\begin{lemma}\\label{smfub} If $\\cM$ is strongly minimal $($which includes $M$ being infinite$)$, then $X\\mapsto \\operatorname{MR}(X)$ is an $\\N_{\\trop}$-valued Fubini measure on $\\Def(\\cM)$. \n\\end{lemma}\n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$.\n\n\\begin{prop}\\label{uniq} If $\\operatorname{char}(\\k)>0$, then there is no $\\Z$-valued Fubini measure on $\\Def(\\k)$. If \n$\\operatorname{char}(\\k)=0$, then there is a unique $\\Z$-valued Fubini measure on $\\Def(\\k)$. \n\\end{prop} \n\\begin{proof} Suppose $\\mu$ is a $\\Z$-valued Fubini measure on $\\Def(\\k)$. Then $\\mu(\\k)=1$: \n if $\\operatorname{char}(\\k)\\ne 2$, the fibers of \n $y\\mapsto y^2: \\k\\to \\k$\nhave size $2$ except at $0$, so \n$$\\mu(\\k)\\ =\\ 1+2\\big(\\mu(\\k)-1\\big),$$\n and thus $\\mu(\\k)=1$; if $\\operatorname{char}(\\k)=2$, the map $y\\mapsto y^3: \\k\\to \\k$ gives likewise $\\mu(\\k)=1+3\\big(\\mu(\\k)-1\\big)$, again resulting in $\\mu(\\k)=1$.\n\n\\noindent\nLet now $\\k$ be a proper subfield of the algebraically closed field $K$, and $\\cM:=(K, \\k)$. Then\n$\\cM$ is $\\omega$-stable with $\\text{MR}(\\k)=1$, $\\text{MR}(K)=\\omega$, and a set $Y\\subseteq \\k^n$ is definable in $\\cM$ iff $Y$ is constructible in the sense of the algebraically closed field $\\k$. Thus $\\k$ is stably embedded in $\\cM$. \nLet $X\\subseteq K^m$ be definable. Then\n$$X\\text{ is almost internal to }\\k\\\n\\Leftrightarrow\\ X \\text{ is co-analyzable relative to }\\k\\\n \\Leftrightarrow\\ \\operatorname{MR}(X)<\\omega.$$\nThese equivalences are in \\cite{AD}, which also has an example of a definable $X\\subseteq K$ that is almost internal to $\\k$ but not internal to $\\k$. Another proof of the first equivalence using \\cite{P} was pointed out to me by Pillay. Lemmas~\\ref{fico} and ~\\ref{cofi} also give: \n$$X \\text{ is fiberable over }\\k\\ \\Longleftrightarrow\\ X \\text{ is co-analyzable relative to }\\k.$$ \n By Lemma~\\ref{smfub} the function $Y\\mapsto \\text{MR}(Y)$\non $\\Def(\\k)$ is a Fubini measure with values in $\\N_{\\trop}$. By Theorem~\\ref{thm} it extends uniquely to\nan $\\N_{\\trop}$-valued Fubini measures on $\\Def(\\k)^{\\f}$; this extension is necessarily\n$X\\mapsto \\text{MR}(X)$: the above relation to almost internality implies that it is a Fubini measure on $\\Def(\\k)^{\\f}$.\n\n\\medskip\\noindent\nNow assume $\\operatorname{char}(\\k)=0$. Then we have the unique $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\k)$ from the proof of Proposition~\\ref{uniq}. By Theorem~\\ref{thm} it extends uniquely to a $\\Z$-valued Fubini measure on \n$\\Def(\\k)^{\\f}$. \nAlthough \"fiberable over $\\k$\" collapses here to ``almost internal to $\\k$\", we still need all of Section~\\ref{prelim} and much of Section~\\ref{proof} to obtain this extension result.\n\n\\medskip\n{\\em If $\\cC$ is stably embedded in $\\cM$ and $\\mu$ is an $A$-valued Fubini measure on $\\cC$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\cC^{\\f}$}.\n\n\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}",
    "post_theorem_intro_text_len": 5494,
    "post_theorem_intro_text": "\\noindent\nSuppose $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in  {\\bbold N} $ we define ``$X$ is $r$-step fiberable over $C$\":  this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\nNow drop the assumption that $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable.\nThen the definition of ``$X$ is $r$-step fiberable\nover $C$'' (with $\\mathbf{M}$ as ambient structure) is more complicated, see end of Section~\\ref{proof}, but will be such that it is equivalent to ``$X^*$ is $r$-step fiberable  over $C^*$\" with $\\mathbf{M}^*$ as ambient structure, where $\\mathbf{M}^*=(M^*,\\dots)$ \nis any $\\omega$-saturated elementary extension of $\\mathbf{M}$. Define $X$ to be {\\em fiberable over $C$\\/} if  it is $r$-step fiberable over $C$ for some $r$. \n\nUsing this equivalence the proof of Theorem~\\ref{thm} reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated, which is very convenient. The idea is to show by induction on $r$ that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$. In the step from $r$ to $r+1$ we consider an $(r+1)$-step fiberable $X\\subseteq M^m$ over $C$. This is witnessed by a definable $f: X\\to M^n$ such that $f(X)$ and all $f^{-1}(y)$ are $r$-step fiberable over $C$. A carefully chosen inductive assumption will tell us that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and takes only finitely many values for $y\\in f(X)$. Then we define \n$\\mu_f(X)$ as a suitably weighted sum of these values and show that $\\mu_f(X)$ does not depend on $f$, but only on $X$, and that the resulting $\\mu(X)=\\mu_f(X)$ defined for such $X$ preserves the Fubini property. We also have to show that the inductive assumption is preserved in this step from $r$ to $r+1$. All this is carried out in detail in Section~\\ref{proof}.  \n\nSection~\\ref{prelim} contains preliminary results. In section~\\ref{co} we relate fiberability over $C$ to the more intrinsic notion of {\\em co-analyzability relative to $C$\\/}  from \\cite{HHM}, and show their equivalence under various model theoretic conditions. These conditions are satisfied by $ {\\bbold T} $ and by differentially closed fields, with $C$ their constant field. This is discussed in Section~\\ref{special}, where it is used to answer questions left open in \\cite{ADH}. In the short last section~\\ref{ms} we state for the record a many-sorted version of Theorem~\\ref{thm}. \n\n\\medskip\\noindent\nThe  measures from  \\cite{MS, AMSW} also take values in semirings and satisfy {\\em Fubini}, but seem to be of a rather different nature. Nevertheless, some variant of Theorem~\\ref{thm} might hold in that setting. One could also take a look at motivic measures as in ~\\cite{HK}, and consider their ``Fubini\" nature, or lack of it. \n\n\\medskip\\noindent\nI thank Ben Castle, Artem Chernikov, Rahim Moosa, Anand Pillay, and Charles Steinhorn for useful feedback after a talk I gave on an earlier version of this paper. \n\n\\subsection*{Notations and Terminology} We let $d,e, k, l, m,n,r$, sometimes subscripted, range over $ {\\bbold N} =\\{0,1,2,\\dots\\}$.  For sets $P, Q$, and $\\mathcal{S}\\subseteq P\\times Q$ and $p\\in P$ we set $$\\mathcal{S}(p)\\ :=\\ \\{q\\in Q:\\ (p,q)\\in \\mathcal{S}\\}. $$  (Think of $\\mathcal{S}$ as describing the family $\\big(\\mathcal{S}(p)\\big)_{p\\in P}$ of subsets of $Q$.) Throughout, $\\mathbf{M}=(M;\\dots)$ is a one-sorted $\\mathcal{L}$-structure. When $C$ gets mentioned, we assume $C\\subseteq M$ is $0$-definable and $|C|\\ge 2$. Recall, for example from \\cite{HHM}, that $C$ is said to be {\\em stably embedded in $\\mathbf{M}$\\/} if for every $0$-definable $\\mathcal{S}\\subseteq M^{m}\\times M^n$ there is a $0$-definable $\\mathcal{T}\\subseteq M^{\\mu}\\times M^n$, $\\mu\\in  {\\bbold N} $, such that for all $p\\in M^m$ there exists $c\\in C^{\\mu}$ for which $\\mathcal{S}(p)\\cap C^n = \\mathcal{T}(c)\\cap C^n$. \n  For $\\omega$-saturated $\\mathbf{M}$, this is equivalent to all definable $X\\subseteq C^n$ being $C$-definable. \n\nSuppose $X\\subseteq M^m$ and $Y\\subseteq M^n$ are definable. Then a {\\em definable family of maps from subsets of $X$ into $Y$\\/} is a definable set\n$\\mathcal{F}\\subseteq P\\times X \\times Y$, with definable $P\\subseteq M^k$ such that for each ``parameter\" $p\\in P$ the set\n$\\mathcal{F}(p)\\subseteq X\\times Y$ is the graph of a map $f_p$ with domain a subset of $X$ and codomain $Y$.  We think of such $\\mathcal{F}$ as the family $(f_p)_{p\\in P}$. Abusing language we write $f\\in \\mathcal{F}$ to indicate that $f=f_p$ for some $p\\in P$, and to state that the set of $f\\in \\mathcal{F}$ with a certain property is definable means that the set of $p\\in P$ such that $f_p$ has that property is definable. Given $\\mathcal{S}\\subseteq P\\times Q$, $p\\in P$, and a map $f: \\mathcal{S} \\to Z$, we let $f(p,-)$ denote the map $q\\mapsto f(p,q): \\mathcal{S}(p)\\to Z$. \n\nOur definition of ``$r$-step fiberable over $C$'' differs from that of ``fiberable by $C$ in $r$ steps'' \nin \\cite{ADH} for a Liouville closed $H$-field $K$ and its constant field $C$. But, in that case,  ``fiberable over $C$'' as defined here is  equivalent to ``fiberable by $C$ in $r$ steps for some $r$\" as defined in \\cite{ADH}.",
    "sketch": "Using the equivalence (for an $\u0005omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of Theorem~\\ref{thm} “reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated”. One then proves the extension by “induction on $r$”, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to M^n$ with $f(X)$ and all fibers $f^{-1}(y)$ ($y\\in M^n$) $r$-step fiberable over $C$. A “carefully chosen inductive assumption” gives that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and “takes only finitely many values for $y\\in f(X)$”. Define $\\mu_f(X)$ as a “suitably weighted sum of these values”, and show $\\mu_f(X)$ “does not depend on $f$, but only on $X$”, so one may set $\\mu(X)=\\mu_f(X)$. Finally, verify that this $\\mu$ “preserves the Fubini property” and that the inductive assumption is preserved from $r$ to $r+1$. (Details are said to be in Section~\\ref{proof}.)",
    "expanded_sketch": "Using the equivalence (for an $\\omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of the main theorem reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated. One then proves the extension by induction on $r$, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nMore precisely, one introduces by recursion on $r$ a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n$\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to Y$ with $Y\\in \\cC_r$ and all fibers $f^{-1}(y)$ ($y\\in Y$) lying in $\\cC_r$. The inductive setup yields that $\\mu$ is already defined on $Y$ and on each fiber $f^{-1}(y)$. To ensure the finiteness/definability needed to form a Fubini-type sum, one uses the following lemma.\n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\nUsing this, one obtains that $\\mu\\big(f^{-1}(y)\\big)$ takes only finitely many values as $y$ ranges over $Y=f(X)$. One then defines $\\mu_f(X)$ as the corresponding suitably weighted sum of these values and shows that $\\mu_f(X)$ does not depend on the choice of witness $f$ but only on $X$, allowing one to set $\\mu(X)=\\mu_f(X)$. Finally, one checks that this definition preserves the Fubini property and that the inductive hypotheses propagate from $r$ to $r+1$.\n\nThe details are carried out later.\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}",
    "expanded_theorem": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.",
    "theorem_type": [
      "Implication",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(\\mathbf M\\) be a one-sorted \\(\\mathcal L\\)-structure, let \\(C\\subseteq M\\) be a \\(0\\)-definable set with more than one element, and let \\(A\\) be a commutative semiring. Write \\(\\operatorname{Def}(C)\\) for the full subcategory of definable sets whose objects are the definable subsets of powers \\(C^n\\). An \\(A\\)-valued Fubini measure on a full subcategory assigns to each object an element of \\(A\\), sends the empty set to \\(0\\) and singletons to \\(1\\), is additive on disjoint definable unions, and satisfies the stated Fubini fiber conditions for definable maps. Let \\(\\operatorname{Def}(C)^{\\operatorname f}\\) be the full subcategory of definable sets fiberable over \\(C\\), equivalently the union of the classes \\(\\mathcal C_r\\) defined by \\(\\mathcal C_0=\\operatorname{Def}(C)^{\\operatorname{iso}}\\) and \\(X\\in \\mathcal C_{r+1}\\) iff there is a definable map \\(f:X\\to Y\\) with \\(Y\\in \\mathcal C_r\\) and every fiber \\(f^{-1}(y)\\in \\mathcal C_r\\). Assume that \\(C\\) is stably embedded in \\(\\mathbf M\\), and let \\(\\mu\\) be an \\(A\\)-valued Fubini measure on \\(\\operatorname{Def}(C)\\). Which statement holds about extending \\(\\mu\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); equivalently, \\(\\mu\\) extends uniquely to an \\(A\\)-valued Fubini measure on \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists exactly one \\(A\\)-valued semi-Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); in general one cannot require \\(\\widetilde\\mu\\) to satisfy the full Fubini axiom on all of \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
        },
        {
          "label": "C",
          "text": "There exists an \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\)."
        },
        {
          "label": "D",
          "text": "For every \\(r\\in \\mathbb N\\), there exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu_r\\) on \\(\\mathcal C_r\\) extending \\(\\mu\\), but in general these extensions need not glue to a single \\(A\\)-valued Fubini measure on the whole union \\(\\operatorname{Def}(C)^{\\operatorname f}=\\bigcup_r \\mathcal C_r\\)."
        },
        {
          "label": "E",
          "text": "There exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on the full subcategory of all definable sets \\(X\\subseteq M^m\\) for which there is a definable map \\(X\\to C^n\\) with all fibers finite, such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); equivalently, \\(\\mu\\) extends uniquely to \\(\\mathcal C_1\\), and this already exhausts \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "promotion from semi-Fubini to full Fubini in the inductive step",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "uniqueness of the extension",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "witness-independence and compatibility across the chain \\(\\mathcal C_r\\)",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "all fiberable sets are already in \\(\\mathcal C_1\\)",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option or uniquely encode choice A. It gives the hypotheses in full, but the conclusion still has to be selected from several closely related possibilities."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the stem presents the assumptions of an extension theorem and asks for the valid conclusion, with one option matching the theorem statement almost verbatim."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish exact existence-and-uniqueness from weaker, stronger, or hypothesis-dropping variants, but the item mainly tests recognition/recall of the theorem rather than generating a conclusion from first principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening existence to uniqueness only, dropping stable embeddedness, confusing semi-Fubini with full Fubini, and asserting non-uniqueness from witness-based construction."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically careful MCQ with strong distractors, but it is largely a theorem restatement rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.22662v1",
    "paper_link": "http://arxiv.org/abs/2512.22662v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.",
      "start_pos": 15706,
      "end_pos": 15914,
      "label": "thm"
    },
    "ref_dict": {
      "co": "\\begin{proof} For every $p\\in M^k$ the set $\\cal{S}(p)$ has by Lemma~\\ref{l5} an $(r+1)$-step fibering over $C$. As in the proof of Lemma~\\ref{l5}, this yields a finite set $E$ of tuples \n$$\\big( f_1,\\dots, f_{r+1}, f_{r+2}; m_1,\\dots,m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nwhere for $j=1,\\dots, r+2$: $f_j$ is a definable map $ \\Sigma_j\\to M^{n_i}$ with (definable) domain \n$\\Sigma_j\\subseteq M^l\\times M^{m_1}\\times\\cdots\\times M^{m_{j-1}}\\times M^{m+n_1+\\cdots + n_{j-1}}$, and for each $p\\in M^k$ there is \na tuple in $E$ as displayed and $q\\in M^{l}$\nsuch that $\\cal{S}(p)=\\Sigma_1(q)$ and the tuple\n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1}, n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.  \n\nConsider one such tuple $\\tau\\in E$ as displayed above. Let $Q$ be the definable\nset of $q\\in M^{l}$ such that\nfor some $p\\in M^k$ we have $\\cal{S}(p)=\\Sigma_1(q)$ and \n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.\nEach $q\\in Q$ yields a map\n$f_1(q,-): \\Sigma_1(q)\\to M^{n_1}$ such that its preimages of points $y\\in M^{n_1}$ are in $\\cC_r$. The maps $f_1(q,-)$ with $q\\in Q$ make up a definable family $\\cal{F}_\\tau$ of maps from subsets of $M^m$ into $M^{n_1}$ as described just before Lemma~\\ref{l2+}. That lemma then tells us that $\\{\\mu\\big(\\Sigma_1(q)\\big): q\\in Q\\}$ is finite, and that\n$\\{q\\in Q:\\ \\mu\\big(\\Sigma_1(q)\\big)=a\\}$ is definable for every $a$. This holds for all $\\tau\\in E$, which yields the\ndesired result.\n\\end{proof} \n\n\\noindent\nThis finishes the inductive proof of Theorem~\\ref{thm} in the case of $\\omega$-saturated \n$\\cM$. \n\n\\subsection*{Reduction to the $\\omega$-saturated case}  Here $C$ is stably embedded in $\\cM$, but we do not assume $\\cM$ is $\\omega$-saturated. \n\nLet $\\cM^*$ be an elementary extension of $\\cM$. We define an $A$-valued Fubini measure $\\mu^*$ on  the full subcategory \n$\\Def(C^*)$ of $\\Def(\\cM^*)$: Let $Y\\subseteq (C^*)^n$ be definable in $\\cM^*$; take an $\\cal{L}$-formula $\\phi(x,y)$ with $x=(x_1,\\dots, x_m)$, $y=(y_1,\\dots, y_n)$, and a point $p_Y\\in (C^*)^m$ such that $Y=\\{q\\in (C^*)^n: \\cM^*\\models \\phi(p_Y,q)\\}$. To simplify notation, set $\\phi(p, C^n):=\\{q\\in C^n:\\ \\cM\\models \\phi(p,q)\\}$ for $p\\in C^m$, and likewise define\n$\\phi\\big(p, (C^*)^n\\big)\\subseteq (C^*)^n$ for $p\\in (C^*)^m$; in particular, $Y=\\phi\\big(p_Y, (C^*)^n\\big)$. \nNow $C^m$ is the disjoint union of the definable sets $X_a:=\\{p\\in C^m:\\ \\mu\\big(\\phi(p,C^m)\\big)=a\\}$ (which is empty for all but finitely many $a$). \nSet $\\mu^*(Y):=a$, where $a$ is unique such that $p_Y\\in (X_a)^*$. It is routine to verify that this particular $a$ does not depend on the choice of $\\phi$ and that it yields indeed a Fubini measure on $\\Def(C^*)$. \n\nNow take such an elementary extension $\\cM^*$ to be $\\omega$-saturated. Then $\\mu^*$ extends uniquely to an $A$-valued Fubini measure $\\mu^*$ on the full subcategory $\\Def(C^*)^{\\f}$ of $\\Def(\\cM^*)$. \nFor $X\\in \\Def(C)^{\\f}$ we have $X^*\\in \\Def(C^*)^{\\f}$; set $\\mu(X):= \\mu^*(X^*)$.\nThis extends the original $A$-valued Fubini measure $\\mu$ on $\\Def(C)$ to an $A$-valued Fubini measure $\\mu$ on $\\Def(C)^{\\f}$.  As to uniqueness, suppose $X$ is fiberable over $C$, and take an $r$-step fibering $(f_1,\\dots, f_r, f_{r+1}; m_1,\\dots, m_{r}, n_1,\\dots, n_{r+1})$\nof $X$ over $C$. \nFor $r=0$ the definable map $f_1: X \\to C^{n_1}$ is injective, and so by Fubini\n$\\mu(X)$ is uniquely determined by $\\mu: \\Def(C)\\to A$. For $r\\ge 1$ the definable map $f_1: X\\to M^{n_1}$ is such that\n$f_1(X)$ and every\n$f_1^{-1}(y)$ with $y\\in f_1(X)$ is $(r-1)$-step fiberable over $C$, and we can assume inductively that the values of \n$\\mu\\big(f(X)\\big)$ and $\\mu\\big(f_1^{-1}(y)\\big)$ for\n$y\\in f_1(X)$ are uniquely determined by $\\mu: \\Def(C)\\to A$, which again by Fubini uniquely deternines $\\mu(X)$. \nThis concludes the proof of Theorem~\\ref{thm}. \n\n\\section{Fiberability and Co-analyzability}\\label{co}\n\n\\noindent\nFiberability is a somewhat ad hoc notion, but agrees under natural model-theoretic conditions with the more intrinsic notion of co-analyzability from \\cite{HHM}, as we shall see. {\\it Throughout this section $X\\subseteq M^m$ is definable}.  \n\n\\subsection*{Fiberability implies co-analyzability}\nWe recall here the recursion on $r$ that defines ``$X$ is co-analyzable in $r$ steps\" (tacitly: relative to $C$ in $\\cM$), where $\\cM$ is assumed to be $\\omega$-saturated: \\begin{enumerate}\n\\item  $X$ is co-analyzable in $0$ steps iff $X$ is finite;\n\\item  $X$ is co-analyzable in $r+1$ steps iff for some definable set $R \\subseteq M^m\\times C$ we have: $\\breve{R}(y)$ is co-analyzable in $r$ steps for every $y\\in C$, and \n$X:= \\bigcup_{y\\in C}\\breve{R}(y)$. \n\\end{enumerate} \n See \\cite[p. 35]{ADH} for how this extends to not necessarily $\\omega$-saturated $\\cM$ in such a way that\n for any elementary extension $\\cM^*$ of $\\cM$: $X$ is co-analyzable in $r$ steps relative to $C$ in $\\cM$ iff $X^*$ is co-analyzable in $r$ steps relative to $C^*$ in $\\cM^*$.  Call $X$ {\\em co-analyzable\\/} if $X$ is co-analyzable in $r$ steps for some $r$. \n\n By \\cite{HHM} this notion can be characterized as follows:\n\n \\begin{prop}\\label{hhm} \nLet $T$ be a complete $L$-theory, where $L$ is a countable one-sorted language. Let $y$ be a single variable and $D(y)$ an $L$-formula that defines in each model $\\cal{N}=(N;\\cdots)$ of $T$ a set $D(N)$ with $|D(N)|\\ge 2$.\nThen the following conditions on an $L$-formula $\\varphi(x)$ with $x=(x_1,\\dots,x_m)$ are equivalent:\n\\begin{enumerate}\n\\item[\\textup{(i)}] for some  $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(ii)}] for all $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(iii)}] for all  $\\cal{N}\\models T$, if \n$D(N)$ is countable, then so is $\\varphi(N^m)$;\n\\item[\\textup{(iv)}] for all $\\cal{N}\\preceq\\cal{N}^*\\models T$, if\n$D(N)=D(N^*)$, then $\\varphi(N^m)=\\varphi((N^*)^m)$.\n\\end{enumerate}\n\\end{prop}\n\n\\noindent\nThe equivalences (i)$\\Leftrightarrow$(ii)$\\Leftrightarrow$(iv) hold without assuming that $L$ is countable:\nFor (i)$\\Leftrightarrow$(ii)$\\Rightarrow$(iv), reduce to countable $L$ by taking suitable reducts. For (iv)$\\Rightarrow$(ii), use also that (iv) amounts to $T_2\\models \\sigma$ where $T_2$ is the theory of elementary pairs of models of $T$ and $\\sigma$ is a certain sentence, depending on $\\varphi$, in the language of $T_2$. Note that in addition we can restrict $\\cal{N}$ and $\\cal{N}^*$ in (iv) to be  $\\omega$-saturated.\n\n\\begin{lemma}\\label{fico} If $X$ is fiberable over $C$, then  $X$ is co-analyzable relative to $C$. \n\\end{lemma} \n\\begin{proof} \nWe arrange that $\\cM$ is $\\omega$-saturated. \nBelow we prove by induction on $r$:  if $X$ is $r$-step fiberable over $C$, then $X$ is co-analyzable relative to $C$. \nFor $r=0$, if $f: X\\to C^n$ is definable and injective, then $X$ is co-analyzable in $n$ steps relative to $C$, by an easy induction on $n$. Thus the claimed implication holds for $r=0$. Assume this implication holds for a certain $r$, and let\n$X$ be $(r+1)$-step fiberable over $C$.\nTo show $X$ is co-analyzable relative to $C$ in $\\cM$ we \naugment $\\cal{L}$ by names for finitely many elements of $M$ to arrange that $X$ is $0$-definable, so $X=\\varphi(M^m)$ with $\\varphi(x)$ an $\\cal{L}$-formula, $x=(x_1,\\dots, x_m)$. We now apply (iv)$\\Rightarrow$(ii) of Proposition~\\ref{hhm} without assuming $L=\\cal{L}$ is countable, with $T:=\\Th(\\cM)$ and $D(y)$ an $\\cal{L}$-formula such that $D(M)=C$. \nSo let $\\cal{N}\\preceq \\cal{N}^*\\models T$ with $D(N)=D(N^*)$; it suffices to show that then $\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$.\nWe also assume $\\cal{N}$ is $\\omega$-saturated, as we may by a remark following Proposition~\\ref{hhm}. Since $\\cal{M}\\equiv \\cal{N}$, the set $\\varphi(N^m)$ is $(r+1)$-step fiberable over $C^{\\cal{N}}:= D(N)$ in $\\cal{N}$. Take $f: \\varphi(N^m)\\to Y$ where $Y\\subseteq N^n$ and $f$ are definable in $\\cal{N}$, such that $Y$ is $r$-step fiberable over $C^{\\cal{N}}$ and for all\n$y\\in Y$ the fiber $f^{-1}(y)$ is $r$-step fiberable over $C^{\\cal{N}}$. Thus by our inductive assumption and Proposition~\\ref{hhm}, $Y$ and the fibers\n$f^{-1}(y)$ for $y\\in Y$ do not change in passing from $\\cal{N}$ to $\\cal{N}^*$. Hence the same is true for $\\varphi(N^m)$, that is,\n$\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$. \n\\end{proof}",
      "proof": "\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}",
      "ms": "\\label{ms}  \n\n\\noindent\nSo far we assumed $\\cM$ to be one-sorted. But many structures are more naturally viewed as many-sorted; for example, valued fields as $3$-sorted structures with residue field a",
      "prelim": "\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below",
      "thm": "\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}",
      "special": "\\begin{proof} Take definable $f: Y\\to M^m$ with $Y\\subseteq C^\\nu$, $\\nu\\in \\N$, such that $X=f(Y)$. Since $(C;\\cM)$ has EI, we have a definable map $g:Y\\to C^n $ such that for all $y,z\\in Y$ we have $f(y)=f(z)\\Leftrightarrow g(y)=g(z)$. This yields a definable bijection $$\\qquad\\qquad \\qquad \\qquad f(y)\\mapsto g(y)\\ :\\  X\\to g(Y),\\qquad (y\\in Y). \\qquad \\qquad \\qquad \\qquad \\qedhere$$\n\\end{proof} \n\n\\subsection*{Analyzability} \nThe concept of {\\em analyzability\\/} is more popular in model theory than co-analyzability, but is usually defined for types\nin the environment of a monster model, instead of definable sets. I have been told that for definable sets and $\\omega$-saturated $\\cM$ its recursive definition should be as follows: \n\n{\\em $X$ is $0$-step analyzable in $C$ iff $X$ is internal to $C$, and $X$ is $(r+1)$-step analyzable in $C$ iff there is a $\\Def(\\cM)$-morphism $f: X \\to Y$ such that $Y$ is $r$-step analyzable over $C$ and all fibers $f^{-1}(y)$, $y\\in Y$, are internal to $C$. Finally, $X$ is analyzable in $C$ iff $X$ is $r$-step analyzable in $C$ for some $r$}. \n\nThis extends to not necessarily $\\omega$-saturated $\\cM$ by requiring in the $(r+1)$-clause the existence of a definable \nfamily of maps from subsets of $C^n$ (for a fixed $n$) into $M^m$ such that for each $y\\in Y$ some member of the family has image $f^{-1}(y)$. \nWith this extended notion and any elementary extension $\\cM^*$ of $\\cM$, \n$$X \\text{ is $r$-step analyzable in }C\\ \\Longleftrightarrow\\  X^* \\text{ is $r$-step analyzable in }C^*.$$ \nUsing this it follows from Proposition~\\ref{hhm} that if $X$ is analyzable in $C$, then $X$ is co-analyzable relative to $C$. Ben Castle and Rahim Moosa have assured me that the converse should also hold under some natural model-theoretic conditions to be satisfied in particular by differentially closed fields (and probably by $\\T$) with $C$ its constant field. I have not pursued this, but it would be good to have this confirmed. One motivation for this: it might help in showing that for $\\T$ and for differentially closed fields the notions of  fiberability over $C$ and co-analyzability relative to $C$ do not collapse for any $r$ to $r$-step fiberability over $C$ and co-analyzability in $r$-steps relative to $C$, respectively. This is plausible in view of \\cite{J} where Jin constructs in a ``big\" differentially closed field for any $r\\ge 1$ a $1$-type that is $r$-step analyzable but not $(r-1)$-step analyzable (in the constant field). \n\n\\section{Application to $\\T$ and to differentially closed fields} \\label{special} \n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$. \n\nAll this holds also for any $\\upo$-free, newtonian, Liouville closed $H$-field $K$ and its constant field instead of\n$\\T$ and its constant field $\\R$; this includes all ordered differential fields $K \\equiv\\T$.  The extended Euler characteristic now yields an answer to a test question raised in \\cite[p. 34, end of Section 5]{ADH}:\n\n\\begin{cor} With $K$ as above, let $X\\subseteq K$ be definable in $K$. Then $X$ with the ordering induced by $K$\nis not elementarily equivalent to the ordered set $\\omega$. \n\\end{cor}\n\\begin{proof} Suppose $X\\equiv \\omega$. Then $X$ is discrete as a subspace of $K$ with its order topology. Hence $X$ is fiberable\nover $C$ by \\cite{ADH}. Now $X\\equiv \\omega$ also yields  a definable bijection between $X$ and $X\\setminus \\{\\text{the least element of $X$}\\}$, but the extended Euler characteristic then gives $E(X)=E(X)-1$, a contradiction. \n\\end{proof}"
    },
    "pre_theorem_intro_text_len": 5546,
    "pre_theorem_intro_text": "\\label{int}\n\n\\noindent\nSemialgebraic sets $Y\\subseteq  {\\bbold R} ^n$ have a dimension $\\dim(Y)\\in  {\\bbold N} \\cup\\{-\\infty\\}$ and an Euler characteristic $E(Y)\\in  {\\bbold Z} $ with good properties; see \\cite[Ch. 4]{D}. In \\cite{ADH} we asked for a natural extension of this dimension and Euler characteristic to the definable sets $X\\subseteq  {\\bbold T} ^m$\nthat are ``fiberable by $ {\\bbold R} $\" as defined there; here $ {\\bbold T} $ is the real closed differential field of transseries, which has its subfield $ {\\bbold R} $ as constant field. \n\nThere are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).   \n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise.  For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in  {\\bbold N} $ is part of  specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$.  For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$. \n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)]  $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq  M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet  $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in  \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in  \\cal{C}$:  $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property);  for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using  notation of (F2). \n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are  semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers.  Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity. \n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$.  Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq  \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c)  imposed earlier on $\\cal{C}$.",
    "context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.",
    "full_context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\Def(C)$ be the full subcategory of $\\Def(\\cM)$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\Def(C)^{\\f}\\supseteq \\Def(C)$ of $\\Def(\\cM)$. The categories $\\Def(C)$ and $\\Def(C)^{\\f}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cC$.\n\n\\noindent\nSuppose $\\cM$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in \\N$ we define ``$X$ is $r$-step fiberable over $C$\": this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\n\\begin{lemma}\\label{smfub} If $\\cM$ is strongly minimal $($which includes $M$ being infinite$)$, then $X\\mapsto \\operatorname{MR}(X)$ is an $\\N_{\\trop}$-valued Fubini measure on $\\Def(\\cM)$. \n\\end{lemma}\n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$.\n\n\\begin{prop}\\label{uniq} If $\\operatorname{char}(\\k)>0$, then there is no $\\Z$-valued Fubini measure on $\\Def(\\k)$. If \n$\\operatorname{char}(\\k)=0$, then there is a unique $\\Z$-valued Fubini measure on $\\Def(\\k)$. \n\\end{prop} \n\\begin{proof} Suppose $\\mu$ is a $\\Z$-valued Fubini measure on $\\Def(\\k)$. Then $\\mu(\\k)=1$: \n if $\\operatorname{char}(\\k)\\ne 2$, the fibers of \n $y\\mapsto y^2: \\k\\to \\k$\nhave size $2$ except at $0$, so \n$$\\mu(\\k)\\ =\\ 1+2\\big(\\mu(\\k)-1\\big),$$\n and thus $\\mu(\\k)=1$; if $\\operatorname{char}(\\k)=2$, the map $y\\mapsto y^3: \\k\\to \\k$ gives likewise $\\mu(\\k)=1+3\\big(\\mu(\\k)-1\\big)$, again resulting in $\\mu(\\k)=1$.\n\n\\noindent\nLet now $\\k$ be a proper subfield of the algebraically closed field $K$, and $\\cM:=(K, \\k)$. Then\n$\\cM$ is $\\omega$-stable with $\\text{MR}(\\k)=1$, $\\text{MR}(K)=\\omega$, and a set $Y\\subseteq \\k^n$ is definable in $\\cM$ iff $Y$ is constructible in the sense of the algebraically closed field $\\k$. Thus $\\k$ is stably embedded in $\\cM$. \nLet $X\\subseteq K^m$ be definable. Then\n$$X\\text{ is almost internal to }\\k\\\n\\Leftrightarrow\\ X \\text{ is co-analyzable relative to }\\k\\\n \\Leftrightarrow\\ \\operatorname{MR}(X)<\\omega.$$\nThese equivalences are in \\cite{AD}, which also has an example of a definable $X\\subseteq K$ that is almost internal to $\\k$ but not internal to $\\k$. Another proof of the first equivalence using \\cite{P} was pointed out to me by Pillay. Lemmas~\\ref{fico} and ~\\ref{cofi} also give: \n$$X \\text{ is fiberable over }\\k\\ \\Longleftrightarrow\\ X \\text{ is co-analyzable relative to }\\k.$$ \n By Lemma~\\ref{smfub} the function $Y\\mapsto \\text{MR}(Y)$\non $\\Def(\\k)$ is a Fubini measure with values in $\\N_{\\trop}$. By Theorem~\\ref{thm} it extends uniquely to\nan $\\N_{\\trop}$-valued Fubini measures on $\\Def(\\k)^{\\f}$; this extension is necessarily\n$X\\mapsto \\text{MR}(X)$: the above relation to almost internality implies that it is a Fubini measure on $\\Def(\\k)^{\\f}$.\n\n\\medskip\\noindent\nNow assume $\\operatorname{char}(\\k)=0$. Then we have the unique $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\k)$ from the proof of Proposition~\\ref{uniq}. By Theorem~\\ref{thm} it extends uniquely to a $\\Z$-valued Fubini measure on \n$\\Def(\\k)^{\\f}$. \nAlthough \"fiberable over $\\k$\" collapses here to ``almost internal to $\\k$\", we still need all of Section~\\ref{prelim} and much of Section~\\ref{proof} to obtain this extension result.\n\n\\medskip\n{\\em If $\\cC$ is stably embedded in $\\cM$ and $\\mu$ is an $A$-valued Fubini measure on $\\cC$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\cC^{\\f}$}.\n\n\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}",
    "post_theorem_intro_text_len": 5494,
    "post_theorem_intro_text": "\\noindent\nSuppose $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in  {\\bbold N} $ we define ``$X$ is $r$-step fiberable over $C$\":  this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\nNow drop the assumption that $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable.\nThen the definition of ``$X$ is $r$-step fiberable\nover $C$'' (with $\\mathbf{M}$ as ambient structure) is more complicated, see end of Section~\\ref{proof}, but will be such that it is equivalent to ``$X^*$ is $r$-step fiberable  over $C^*$\" with $\\mathbf{M}^*$ as ambient structure, where $\\mathbf{M}^*=(M^*,\\dots)$ \nis any $\\omega$-saturated elementary extension of $\\mathbf{M}$. Define $X$ to be {\\em fiberable over $C$\\/} if  it is $r$-step fiberable over $C$ for some $r$. \n\nUsing this equivalence the proof of Theorem~\\ref{thm} reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated, which is very convenient. The idea is to show by induction on $r$ that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$. In the step from $r$ to $r+1$ we consider an $(r+1)$-step fiberable $X\\subseteq M^m$ over $C$. This is witnessed by a definable $f: X\\to M^n$ such that $f(X)$ and all $f^{-1}(y)$ are $r$-step fiberable over $C$. A carefully chosen inductive assumption will tell us that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and takes only finitely many values for $y\\in f(X)$. Then we define \n$\\mu_f(X)$ as a suitably weighted sum of these values and show that $\\mu_f(X)$ does not depend on $f$, but only on $X$, and that the resulting $\\mu(X)=\\mu_f(X)$ defined for such $X$ preserves the Fubini property. We also have to show that the inductive assumption is preserved in this step from $r$ to $r+1$. All this is carried out in detail in Section~\\ref{proof}.  \n\nSection~\\ref{prelim} contains preliminary results. In section~\\ref{co} we relate fiberability over $C$ to the more intrinsic notion of {\\em co-analyzability relative to $C$\\/}  from \\cite{HHM}, and show their equivalence under various model theoretic conditions. These conditions are satisfied by $ {\\bbold T} $ and by differentially closed fields, with $C$ their constant field. This is discussed in Section~\\ref{special}, where it is used to answer questions left open in \\cite{ADH}. In the short last section~\\ref{ms} we state for the record a many-sorted version of Theorem~\\ref{thm}. \n\n\\medskip\\noindent\nThe  measures from  \\cite{MS, AMSW} also take values in semirings and satisfy {\\em Fubini}, but seem to be of a rather different nature. Nevertheless, some variant of Theorem~\\ref{thm} might hold in that setting. One could also take a look at motivic measures as in ~\\cite{HK}, and consider their ``Fubini\" nature, or lack of it. \n\n\\medskip\\noindent\nI thank Ben Castle, Artem Chernikov, Rahim Moosa, Anand Pillay, and Charles Steinhorn for useful feedback after a talk I gave on an earlier version of this paper. \n\n\\subsection*{Notations and Terminology} We let $d,e, k, l, m,n,r$, sometimes subscripted, range over $ {\\bbold N} =\\{0,1,2,\\dots\\}$.  For sets $P, Q$, and $\\mathcal{S}\\subseteq P\\times Q$ and $p\\in P$ we set $$\\mathcal{S}(p)\\ :=\\ \\{q\\in Q:\\ (p,q)\\in \\mathcal{S}\\}. $$  (Think of $\\mathcal{S}$ as describing the family $\\big(\\mathcal{S}(p)\\big)_{p\\in P}$ of subsets of $Q$.) Throughout, $\\mathbf{M}=(M;\\dots)$ is a one-sorted $\\mathcal{L}$-structure. When $C$ gets mentioned, we assume $C\\subseteq M$ is $0$-definable and $|C|\\ge 2$. Recall, for example from \\cite{HHM}, that $C$ is said to be {\\em stably embedded in $\\mathbf{M}$\\/} if for every $0$-definable $\\mathcal{S}\\subseteq M^{m}\\times M^n$ there is a $0$-definable $\\mathcal{T}\\subseteq M^{\\mu}\\times M^n$, $\\mu\\in  {\\bbold N} $, such that for all $p\\in M^m$ there exists $c\\in C^{\\mu}$ for which $\\mathcal{S}(p)\\cap C^n = \\mathcal{T}(c)\\cap C^n$. \n  For $\\omega$-saturated $\\mathbf{M}$, this is equivalent to all definable $X\\subseteq C^n$ being $C$-definable. \n\nSuppose $X\\subseteq M^m$ and $Y\\subseteq M^n$ are definable. Then a {\\em definable family of maps from subsets of $X$ into $Y$\\/} is a definable set\n$\\mathcal{F}\\subseteq P\\times X \\times Y$, with definable $P\\subseteq M^k$ such that for each ``parameter\" $p\\in P$ the set\n$\\mathcal{F}(p)\\subseteq X\\times Y$ is the graph of a map $f_p$ with domain a subset of $X$ and codomain $Y$.  We think of such $\\mathcal{F}$ as the family $(f_p)_{p\\in P}$. Abusing language we write $f\\in \\mathcal{F}$ to indicate that $f=f_p$ for some $p\\in P$, and to state that the set of $f\\in \\mathcal{F}$ with a certain property is definable means that the set of $p\\in P$ such that $f_p$ has that property is definable. Given $\\mathcal{S}\\subseteq P\\times Q$, $p\\in P$, and a map $f: \\mathcal{S} \\to Z$, we let $f(p,-)$ denote the map $q\\mapsto f(p,q): \\mathcal{S}(p)\\to Z$. \n\nOur definition of ``$r$-step fiberable over $C$'' differs from that of ``fiberable by $C$ in $r$ steps'' \nin \\cite{ADH} for a Liouville closed $H$-field $K$ and its constant field $C$. But, in that case,  ``fiberable over $C$'' as defined here is  equivalent to ``fiberable by $C$ in $r$ steps for some $r$\" as defined in \\cite{ADH}.",
    "sketch": "Using the equivalence (for an $\u0005omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of Theorem~\\ref{thm} “reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated”. One then proves the extension by “induction on $r$”, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to M^n$ with $f(X)$ and all fibers $f^{-1}(y)$ ($y\\in M^n$) $r$-step fiberable over $C$. A “carefully chosen inductive assumption” gives that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and “takes only finitely many values for $y\\in f(X)$”. Define $\\mu_f(X)$ as a “suitably weighted sum of these values”, and show $\\mu_f(X)$ “does not depend on $f$, but only on $X$”, so one may set $\\mu(X)=\\mu_f(X)$. Finally, verify that this $\\mu$ “preserves the Fubini property” and that the inductive assumption is preserved from $r$ to $r+1$. (Details are said to be in Section~\\ref{proof}.)",
    "expanded_sketch": "Using the equivalence (for an $\\omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of the main theorem reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated. One then proves the extension by induction on $r$, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nMore precisely, one introduces by recursion on $r$ a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n$\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to Y$ with $Y\\in \\cC_r$ and all fibers $f^{-1}(y)$ ($y\\in Y$) lying in $\\cC_r$. The inductive setup yields that $\\mu$ is already defined on $Y$ and on each fiber $f^{-1}(y)$. To ensure the finiteness/definability needed to form a Fubini-type sum, one uses the following lemma.\n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\nUsing this, one obtains that $\\mu\\big(f^{-1}(y)\\big)$ takes only finitely many values as $y$ ranges over $Y=f(X)$. One then defines $\\mu_f(X)$ as the corresponding suitably weighted sum of these values and shows that $\\mu_f(X)$ does not depend on the choice of witness $f$ but only on $X$, allowing one to set $\\mu(X)=\\mu_f(X)$. Finally, one checks that this definition preserves the Fubini property and that the inductive hypotheses propagate from $r$ to $r+1$.\n\nThe details are carried out later.\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}",
    "expanded_theorem": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.",
    "theorem_type": [
      "Implication",
      "Uniqueness"
    ],
    "mcq": {
      "question": "Let \\(\\mathbf M\\) be a one-sorted \\(\\mathcal L\\)-structure, let \\(C\\subseteq M\\) be a \\(0\\)-definable set with more than one element, and let \\(A\\) be a commutative semiring. Write \\(\\operatorname{Def}(C)\\) for the full subcategory of definable sets whose objects are the definable subsets of powers \\(C^n\\). An \\(A\\)-valued Fubini measure on a full subcategory assigns to each object an element of \\(A\\), sends the empty set to \\(0\\) and singletons to \\(1\\), is additive on disjoint definable unions, and satisfies the stated Fubini fiber conditions for definable maps. Let \\(\\operatorname{Def}(C)^{\\operatorname f}\\) be the full subcategory of definable sets fiberable over \\(C\\), equivalently the union of the classes \\(\\mathcal C_r\\) defined by \\(\\mathcal C_0=\\operatorname{Def}(C)^{\\operatorname{iso}}\\) and \\(X\\in \\mathcal C_{r+1}\\) iff there is a definable map \\(f:X\\to Y\\) with \\(Y\\in \\mathcal C_r\\) and every fiber \\(f^{-1}(y)\\in \\mathcal C_r\\). Assume that \\(C\\) is stably embedded in \\(\\mathbf M\\), and let \\(\\mu\\) be an \\(A\\)-valued Fubini measure on \\(\\operatorname{Def}(C)\\). Which statement holds about extending \\(\\mu\\)?",
      "correct_choice": {
        "label": "A",
        "text": "There exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); equivalently, \\(\\mu\\) extends uniquely to an \\(A\\)-valued Fubini measure on \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "There exists exactly one \\(A\\)-valued semi-Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); in general one cannot require \\(\\widetilde\\mu\\) to satisfy the full Fubini axiom on all of \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
        },
        {
          "label": "C",
          "text": "There exists an \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on \\(\\operatorname{Def}(C)^{\\operatorname f}\\) such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\)."
        },
        {
          "label": "D",
          "text": "For every \\(r\\in \\mathbb N\\), there exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu_r\\) on \\(\\mathcal C_r\\) extending \\(\\mu\\), but in general these extensions need not glue to a single \\(A\\)-valued Fubini measure on the whole union \\(\\operatorname{Def}(C)^{\\operatorname f}=\\bigcup_r \\mathcal C_r\\)."
        },
        {
          "label": "E",
          "text": "There exists exactly one \\(A\\)-valued Fubini measure \\(\\widetilde\\mu\\) on the full subcategory of all definable sets \\(X\\subseteq M^m\\) for which there is a definable map \\(X\\to C^n\\) with all fibers finite, such that \\(\\widetilde\\mu(X)=\\mu(X)\\) for every object \\(X\\in \\operatorname{Def}(C)\\); equivalently, \\(\\mu\\) extends uniquely to \\(\\mathcal C_1\\), and this already exhausts \\(\\operatorname{Def}(C)^{\\operatorname f}\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "regularity",
          "tampered_component": "promotion from semi-Fubini to full Fubini in the inductive step",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "uniqueness of the extension",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "witness-independence and compatibility across the chain \\(\\mathcal C_r\\)",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "all fiberable sets are already in \\(\\mathcal C_1\\)",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option and does not contain a giveaway phrase matching choice A. It sets up definitions and hypotheses, but the conclusion still has to be identified."
      },
      "TAS": {
        "score": 1,
        "justification": "This is very close to a theorem-recall item: the correct option is essentially the theorem’s conclusion stated verbatim. However, it is not pure tautology because the student must distinguish it from nearby variants involving existence vs. uniqueness and weaker extension claims."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to select the strongest valid extension statement and reject weaker or subtly false alternatives. Still, the task is mainly recognition of the exact theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are strong: B confuses full Fubini with semi-Fubini, C is a weaker true statement, D challenges compatibility/gluing across the inductive classes, and E incorrectly truncates the hierarchy at C1. These are plausible and mathematically meaningful failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and little answer leakage, but it remains fairly close to direct theorem recall rather than testing deep generative reasoning."
    }
  },
  {
    "id": "2512.22900v1",
    "paper_link": "http://arxiv.org/abs/2512.22900v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}",
      "start_pos": 7567,
      "end_pos": 7851,
      "label": null
    },
    "ref_dict": {
      "lemma:main": "\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}",
      "lemma:groups of order 8": "\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}",
      "lemma:small subset": "\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 4601,
    "pre_theorem_intro_text": "\\label{section:Introduction}\nLet $G$ be a finite group.  A subset $A \\subseteq G$ is called a \\textit{Lagrange subset} if $\\lvert A\\rvert$ divides $\\lvert G\\rvert$.  If there exists a subset $B \\subseteq G$ such that every element $x\\in G$ can be written \\textit{uniquely} in the form\n\\[\nx = ab,\n\\quad a\\in A,\\; b\\in B,\n\\]\nthen $A$ is said to be a (left) \\textit{factor} of $G$.  In this situation one immediately has\n\\[\n\\lvert G\\rvert = \\lvert A\\rvert \\,\\lvert B\\rvert,\n\\]\nand both $A$ and $B$ are Lagrange subsets of $G$.\n\nThe problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}.  \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}.  \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena.  A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups.  A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$.  \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}.  Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$.  \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.  \n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:",
    "context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:",
    "full_context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.\n\\end{abstract}\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\nThe authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n \\item Let $G$ be an elementary abelian $2$-group.\n Every subset of size $4$ is a factor of $G$.\n \\item\\,Let $G$ be an elementary abelian $3$-group.\n Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of \\textup{(i)}]\nFirst suppose that the order of $a$ is even, say $|a| = 2m$ for some positive integer $m$.\nLet $H = \\langle a\\rangle$.\nSet\n\\[\nB = \\{e,a^2,\\ldots,a^{2m-2}\\}.\n\\]\nClearly $H = \\{e,a\\} B$, and it is straightforward to check that\n$B \\cap aB = \\varnothing$.\nThus $\\{e,a\\}$ is a factor of $H$.\nBy Lemma~\\ref{lemma:subset of subgroup}, $\\{e,a\\}$ is then a factor of $G$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\\begin{proof}\n Suppose to the contrary that there is an element $x\\in G$ such that\n \\[\n G=A\\cup Ax \\text{ and } A\\cap Ax=\\varnothing.\n \\]\n Then $e\\in Ax$ and hence $x=a^{-i}$ for some $i\\in\\{1,2,3\\}$, or $x=b^{-1}$.\n If $x=a^{-i}$, then $a\\in A\\cap A x$, contradicting disjointness.\n If $x=b^{-1}=b^3$, then\n \\[\n Ax=\\{ab^3,a^2b^3,a^3b^3,e\\}.\n \\]\n Since $b^3\\notin A$ and also $b^3\\notin A x$,\n the element $b^3$ is not contained in $A\\cup A x=G$, a clear contradiction.\n\\end{proof}\n\nFirst, suppose that $|G|$ is even. By Lemma~\\ref{lemma:small subset}\\,(i), \nevery non-identity element of $G$ must have even order. \nIt follows that $G$ is a $2$-group.\nSince $G$ cannot have a proper subgroup of order greater than $4$ (by Lemma~\\ref{lemma:main}), \nthe order of $G$ must be $4$ or $8$.\nIf $|G|=4$, $G$ is either $C_4$ or $C_2^2$, both of which appear in the theorem's list.\nIf $|G|=8$, Lemma~\\ref{lemma:groups of order 8} implies that $G$ cannot contain \ntwo distinct cyclic subgroups of order $4$. \nThis eliminates the quaternion group $Q_8$ and the product $C_4 \\times C_2$. \nSince $C_2^3$ is a solution, the only remaining candidates to check are $D_4$ and $C_8$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}",
    "post_theorem_intro_text_len": 1209,
    "post_theorem_intro_text": "The authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n  \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n  \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n  \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\vspace{1ex}\n\\noindent\\textbf{Notation.} Throughout, $G$ is a finite group with identity $e$. For any $A \\subseteq G$ and $x \\in G$, we denote $Ax = \\{ax \\mid a \\in A\\}$. We write $\\langle g \\rangle$ for the cyclic subgroup generated by $g$. A subset $A$ is a \\textit{left factor} if $G = \\bigsqcup_{b \\in B} Ab$ for some $B \\subseteq G$. \n\nWith these conventions in place, we turn in Section \\ref{section:lemmas} to our lemmas, \nand then in Section \\ref{section:proof theorem} to the completion of the proof.",
    "sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups} (Lemma \\ref{lemma:main});\n  \\item \\emph{Groups of even order containing elements of odd order} (Lemma \\ref{lemma:small subset});\n  \\item \\emph{Investigation of groups of order $8$} (Lemma \\ref{lemma:groups of order 8});\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas in Section \\ref{section:lemmas} and then completing the proof in Section \\ref{section:proof theorem}.",
    "expanded_sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups}. We first prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{Groups of even order containing elements of odd order}. We next prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\n  \\item \\emph{Investigation of groups of order $8$}. We then prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas next and then completing the proof later.",
    "expanded_theorem": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $G$ be a nontrivial finite group. A subset $A\\subseteq G$ is called a Lagrange subset if $|A|$ divides $|G|$. Call $A$ a factor of $G$ if there exists a subset $B\\subseteq G$ such that the multiplication map $A\\times B\\to G$, $(a,b)\\mapsto ab$, is bijective. The group $G$ is said to have the strong CFS property if every Lagrange subset of $G$ is a factor of $G$. Which finite groups $G$ have the strong CFS property?",
      "correct_choice": {
        "label": "A",
        "text": "Exactly the cyclic groups of prime order, together with the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly the cyclic groups of prime order, together with the five groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$."
        },
        {
          "label": "C",
          "text": "At least the cyclic groups of prime order and the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$ have the strong CFS property."
        },
        {
          "label": "D",
          "text": "Exactly the finite abelian groups in which every nontrivial proper divisor of $|G|$ is $2$, $3$, or a prime, namely the cyclic groups of prime order together with $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$."
        },
        {
          "label": "E",
          "text": "Exactly the cyclic groups of prime order, together with the groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and all groups of order $8$ having a unique cyclic subgroup of order $4$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "computational_check",
          "tampered_component": "final direct-check exclusion of C_9",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the 'only if' classification/exhaustiveness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "confuses the specific classified groups with a broader abelian/order-divisibility condition",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "misuses the order-8 lemma by turning elimination of groups with two distinct order-4 cyclic subgroups into a positive classification criterion",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the strong CFS property but does not reveal or strongly hint at the classification in the correct option. There is no explicit answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct classification-theorem recall question: it asks for the statement equivalent to the defined property, and the correct choice is just the exact classification. That makes it close to a restatement rather than a genuinely new inferential task."
      },
      "GPS": {
        "score": 1,
        "justification": "The student must distinguish the exact classification from several near-miss alternatives, so some reasoning or careful theorem recall is needed. However, the task mainly tests recognition of the theorem rather than substantial generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they include a weaker true overgeneralization, boundary-case insertions like C9, an overextension to all elementary abelian 2-groups, and a nonabelian candidate D4. These align well with common classification mistakes."
      },
      "total_score": 5,
      "overall_assessment": "A solid MCQ in terms of answer concealment and distractor design, but it is largely a theorem-restatement/recognition item rather than one that strongly tests generative reasoning."
    }
  },
  {
    "id": "2512.22900v1",
    "paper_link": "http://arxiv.org/abs/2512.22900v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "theorem",
      "content": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}",
      "start_pos": 7567,
      "end_pos": 7851,
      "label": null
    },
    "ref_dict": {
      "lemma:main": "\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}",
      "lemma:groups of order 8": "\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}",
      "lemma:small subset": "\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 4601,
    "pre_theorem_intro_text": "\\label{section:Introduction}\nLet $G$ be a finite group.  A subset $A \\subseteq G$ is called a \\textit{Lagrange subset} if $\\lvert A\\rvert$ divides $\\lvert G\\rvert$.  If there exists a subset $B \\subseteq G$ such that every element $x\\in G$ can be written \\textit{uniquely} in the form\n\\[\nx = ab,\n\\quad a\\in A,\\; b\\in B,\n\\]\nthen $A$ is said to be a (left) \\textit{factor} of $G$.  In this situation one immediately has\n\\[\n\\lvert G\\rvert = \\lvert A\\rvert \\,\\lvert B\\rvert,\n\\]\nand both $A$ and $B$ are Lagrange subsets of $G$.\n\nThe problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}.  \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}.  \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena.  A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups.  A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$.  \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}.  Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$.  \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.  \n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:",
    "context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:",
    "full_context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.\n\\end{abstract}\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\nThe authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n \\item Let $G$ be an elementary abelian $2$-group.\n Every subset of size $4$ is a factor of $G$.\n \\item\\,Let $G$ be an elementary abelian $3$-group.\n Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of \\textup{(i)}]\nFirst suppose that the order of $a$ is even, say $|a| = 2m$ for some positive integer $m$.\nLet $H = \\langle a\\rangle$.\nSet\n\\[\nB = \\{e,a^2,\\ldots,a^{2m-2}\\}.\n\\]\nClearly $H = \\{e,a\\} B$, and it is straightforward to check that\n$B \\cap aB = \\varnothing$.\nThus $\\{e,a\\}$ is a factor of $H$.\nBy Lemma~\\ref{lemma:subset of subgroup}, $\\{e,a\\}$ is then a factor of $G$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\\begin{proof}\n Suppose to the contrary that there is an element $x\\in G$ such that\n \\[\n G=A\\cup Ax \\text{ and } A\\cap Ax=\\varnothing.\n \\]\n Then $e\\in Ax$ and hence $x=a^{-i}$ for some $i\\in\\{1,2,3\\}$, or $x=b^{-1}$.\n If $x=a^{-i}$, then $a\\in A\\cap A x$, contradicting disjointness.\n If $x=b^{-1}=b^3$, then\n \\[\n Ax=\\{ab^3,a^2b^3,a^3b^3,e\\}.\n \\]\n Since $b^3\\notin A$ and also $b^3\\notin A x$,\n the element $b^3$ is not contained in $A\\cup A x=G$, a clear contradiction.\n\\end{proof}\n\nFirst, suppose that $|G|$ is even. By Lemma~\\ref{lemma:small subset}\\,(i), \nevery non-identity element of $G$ must have even order. \nIt follows that $G$ is a $2$-group.\nSince $G$ cannot have a proper subgroup of order greater than $4$ (by Lemma~\\ref{lemma:main}), \nthe order of $G$ must be $4$ or $8$.\nIf $|G|=4$, $G$ is either $C_4$ or $C_2^2$, both of which appear in the theorem's list.\nIf $|G|=8$, Lemma~\\ref{lemma:groups of order 8} implies that $G$ cannot contain \ntwo distinct cyclic subgroups of order $4$. \nThis eliminates the quaternion group $Q_8$ and the product $C_4 \\times C_2$. \nSince $C_2^3$ is a solution, the only remaining candidates to check are $D_4$ and $C_8$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}",
    "post_theorem_intro_text_len": 1209,
    "post_theorem_intro_text": "The authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n  \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n  \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n  \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\vspace{1ex}\n\\noindent\\textbf{Notation.} Throughout, $G$ is a finite group with identity $e$. For any $A \\subseteq G$ and $x \\in G$, we denote $Ax = \\{ax \\mid a \\in A\\}$. We write $\\langle g \\rangle$ for the cyclic subgroup generated by $g$. A subset $A$ is a \\textit{left factor} if $G = \\bigsqcup_{b \\in B} Ab$ for some $B \\subseteq G$. \n\nWith these conventions in place, we turn in Section \\ref{section:lemmas} to our lemmas, \nand then in Section \\ref{section:proof theorem} to the completion of the proof.",
    "sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups} (Lemma \\ref{lemma:main});\n  \\item \\emph{Groups of even order containing elements of odd order} (Lemma \\ref{lemma:small subset});\n  \\item \\emph{Investigation of groups of order $8$} (Lemma \\ref{lemma:groups of order 8});\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas in Section \\ref{section:lemmas} and then completing the proof in Section \\ref{section:proof theorem}.",
    "expanded_sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups}. We first prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{Groups of even order containing elements of odd order}. We next prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\n  \\item \\emph{Investigation of groups of order $8$}. We then prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas next and then completing the proof later.",
    "expanded_theorem": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let $G$ be a nontrivial finite group. A subset $A\\subseteq G$ is called a Lagrange subset if $|A|$ divides $|G|$. Call $A$ a factor of $G$ if there exists a subset $B\\subseteq G$ such that the multiplication map $A\\times B\\to G$, $(a,b)\\mapsto ab$, is bijective. The group $G$ is said to have the strong CFS property if every Lagrange subset of $G$ is a factor of $G$. Which finite groups $G$ have the strong CFS property?",
      "correct_choice": {
        "label": "A",
        "text": "Exactly the cyclic groups of prime order, together with the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$."
      },
      "choices": [
        {
          "label": "B",
          "text": "Exactly the cyclic groups of prime order, together with the five groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$."
        },
        {
          "label": "C",
          "text": "At least the cyclic groups of prime order and the four groups $C_4$, $C_2^2$, $C_2^3$, and $C_3^2$ have the strong CFS property."
        },
        {
          "label": "D",
          "text": "Exactly the finite abelian groups in which every nontrivial proper divisor of $|G|$ is $2$, $3$, or a prime, namely the cyclic groups of prime order together with $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and $C_9$."
        },
        {
          "label": "E",
          "text": "Exactly the cyclic groups of prime order, together with the groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, and all groups of order $8$ having a unique cyclic subgroup of order $4$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "computational_check",
          "tampered_component": "final direct-check exclusion of C_9",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the 'only if' classification/exhaustiveness",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "confuses the specific classified groups with a broader abelian/order-divisibility condition",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "misuses the order-8 lemma by turning elimination of groups with two distinct order-4 cyclic subgroups into a positive classification criterion",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only defines the strong CFS property and asks for a classification; it does not state or strongly hint at the listed exceptional groups."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially asking for the statement of a classification theorem, so it is somewhat theorem-recall-based, though the answer choices are competing formulations rather than a verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Selecting the correct option requires some reasoning about exactness versus overreach (e.g., excluding C_9 and rejecting weaker true statements), but it mainly tests recognition of the classification rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are nuanced and plausible: one is weaker-but-true, others reflect natural overgeneralizations or boundary-case mistakes, and they are clearly distinct."
      },
      "total_score": 6,
      "overall_assessment": "A solid classification-style MCQ with no answer leakage and strong distractors, but it leans more toward theorem recognition than genuinely generative mathematical reasoning."
    }
  },
  {
    "id": "2512.22960v1",
    "paper_link": "http://arxiv.org/abs/2512.22960v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "theo",
      "content": "\\label{Th-SynthesExistenceSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Denote by $\\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of \\cref{Eq-StationaryAGP} in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$,\n    \\begin{itemize}\n        \\item $\\lambda>0$ and $\\omega>-\\mu_0$,\n        \\item $\\lambda = 0$ and $\\omega=\\mu_0$,\n        \\item $\\lambda<0$ and $\\omega<-\\mu_0$.\n    \\end{itemize}\n    In every other cases, there is no non-trivial non-negative solution in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$.",
      "start_pos": 92486,
      "end_pos": 93076,
      "label": "Th-SynthesExistenceSW"
    },
    "ref_dict": {
      "Eq-GP": "\\begin{cases}\\complexI\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-GP}\n\\end{equation} \nby a white noise potential. In the study of Bose-Einstein condensation, \\cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \\cite{BEC_Pitaevskii}. Adding a random potential in \\cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \\cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \\cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \\cref{Eq:AGP} has two conserved quantities physically relevant, the mass \n$$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\nand the energy\n$$\\mathcal{E}(u)=\\frac{1}{2}\\langle-(H+\\xi)u,u\\rangle - \\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\nStanding waves of \\cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\\omega\\in\\R$ such that\n$$-(H+\\xi)\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0$$\nand $u(t,x)=\\e^{i\\omega t}\\phi(x)$ is solution of \\cref{Eq:AGP}.\\\\\n\nIn dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\e^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form. \n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\L^2(\\R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\L^2(\\R),xu\\in\\L^2(\\R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n    \\begin{cases}\\complexI\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases}",
      "Eq:AGP": "\\begin{cases}\\complexI\\p_t u+ H u + \\xi u+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq:AGP}\n\\end{equation} \nof the form $u(t,x)=\\e^{i\\omega t}\\phi(x)$, where $H=\\Delta-|x|^2$ is the Hermite operator, $\\xi$ is a real spatial white noise, $\\lambda\\in\\R$ and $\\gamma>0$. Such a solution is called a standing wave of \\cref{Eq:AGP} of frequency $\\omega\\in\\R$ and profile $\\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation. \nIn fact, \\cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation\n\\begin{equation}{~}\n    \\begin{cases}\\complexI\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases}",
      "Th-EnergyGS": "\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. For $u\\in\\D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\D^{-1,2},\\D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\e^{\\complexI\\theta}\\phi_m,\\; \\theta\\in\\R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}",
      "Eq-StationaryAGP": "\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}",
      "Eq-AGP": "\\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\e^{\\complexI\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\p"
    },
    "pre_theorem_intro_text_len": 6990,
    "pre_theorem_intro_text": "In this paper, we are interested in particular solutions of the Anderson-Gross-Pitaevskii\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ H u + \\xi u+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq:AGP}\n\\end{equation} \nof the form $u(t,x)=\\mathrm{e}^{i\\omega t}\\phi(x)$, where $H=\\Delta-|x|^2$ is the Hermite operator, $\\xi$ is a real spatial white noise, $\\lambda\\in\\mathbb R$ and $\\gamma>0$. Such a solution is called a standing wave of \\cref{Eq:AGP} of frequency $\\omega\\in\\mathbb R$ and profile $\\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation. \nIn fact, \\cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-GP}\n\\end{equation} \nby a white noise potential. In the study of Bose-Einstein condensation, \\cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \\cite{BEC_Pitaevskii}. Adding a random potential in \\cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \\cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \\cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \\cref{Eq:AGP} has two conserved quantities physically relevant, the mass \n$$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\mathbb R^d}|u|^2\\mathrm{d} x$$\nand the energy\n$$\\mathcal{E}(u)=\\frac{1}{2}\\langle-(H+\\xi)u,u\\rangle - \\frac{\\lambda}{2\\gamma+2}\\int_{\\mathbb R^d}|u|^{2\\gamma+2}\\mathrm{d} x.$$\nStanding waves of \\cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\\omega\\in\\mathbb R$ such that\n$$-(H+\\xi)\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0$$\nand $u(t,x)=\\mathrm{e}^{i\\omega t}\\phi(x)$ is solution of \\cref{Eq:AGP}.\\\\\n\nIn dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form. \n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties. \n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.",
    "context": "In dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form.\n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties.\n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.",
    "full_context": "In dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form.\n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties.\n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.\n\nIn order to do so, we use constrained minimization methods. In \\cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.\n\nLet $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise. Let $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. Denote by $\\mu_0$ the smallest eigenvalue of $-A$. Let $\\omega>-\\mu_0$ and define $$\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$$ and $$\\mathcal{N}_\\omega =\\left\\{u\\in\\D^{1,2}\\backslash\\{0\\},\\; \\langle \\nabla\\mathcal{S}_\\omega u,u \\rangle_{\\D^{-1,2},\\D^{1,2}}=0\\right\\}.$$\n Then, there exists $\\psi_\\omega\\in\\displaystyle\\argmin_{\\mathcal{N}_\\omega}\\mathcal{S}_\\omega$ such that $\\psi_\\omega>0$.\n\n\\begin{itemize}\n \\item Assume $\\gamma>\\frac{2}{d}$, let $v\\in\\W^{1,2}\\backslash\\{0\\}$ and $u=\\rho v$, then for any $\\alpha>0$, \\cref{Eq-EnergyVside} implies\n \\begin{align*}\n \\mathcal{E}(u_\\alpha) &= - \\frac{\\lambda\\alpha^{d\\gamma}}{2\\gamma+2}\\int_{\\R^d}|v(x)|^{2\\gamma+2}\\rho(x/\\alpha)^{2\\gamma+2}\\d x &(I)\\\\\n &+ \\frac{\\alpha^2}{2}\\int_{\\R^d}\\rho(x/\\alpha)^2|\\nabla v(x)|^2\\d x &(II)\\\\\n &+ \\frac{1}{2\\alpha^2}\\int_{\\R^d} (1-Y(x/\\alpha))\\rho(x/\\alpha)^2|xv(x)|^2\\d x &(III)\\\\\n &-\\frac{1}{2}\\langle Z,|u_\\alpha|^2\\rangle. &(IV)\n \\end{align*}\n \\cref{Lem-EstimScalTransf,Lem-ProductRule} imply $(IV)=O(\\alpha^s)$ as $\\alpha$ goes to infinity, for any $s\\in(0,2)$, as $Z\\in\\W^{0-,\\infty}$. By dominated convergence, $(III)=O(\\alpha^{-2})$ as $\\alpha$ goes to infinity. Finally, $(II)=O(\\alpha^2)$ and\n $$(I)\\sim - \\frac{\\lambda\\alpha^{d\\gamma}\\rho(0)^{2\\gamma+2}}{2\\gamma+2}\\int_{\\R^d}|v(x)|^{2\\gamma+2}\\d x.$$\n Chose $v = \\sqrt{2m}\\rho(0)^{-1}h_0$ ($m>0$), then $\\mathcal{M}(u_\\alpha)$ converges to $m$ whereas $\\mathcal{E}(u_\\alpha)$ goes to $-\\infty$, because $d\\gamma>2$. Thus, \\cref{Lem-AsympEnergy} implies\n $$\\forall m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n \\item Assume $\\gamma=\\frac{2}{d}$ and let $Q$ be the unique positive and radial solution of \n $$\\Delta Q + \\lambda Q^{2\\gamma+1} = Q.$$\n Recall $\\mathcal{M}(Q)=m_\\lambda^*=\\frac{1}{2}\\left(\\frac{(\\gamma+1)J}{\\lambda}\\right)^{\\frac{1}{\\gamma}}$. For $c>0$, let $u=c\\rho Q$. For any $\\alpha>0$, it holds\n \\begin{align*}\n \\mathcal{E}(u_\\alpha) &= \\frac{\\alpha^2}{2}\\int_{\\R^d}\\rho(x/\\alpha)^2|c\\nabla Q(x)|^2\\d x - \\frac{\\lambda\\alpha^{2}}{2\\gamma+2}\\int_{\\R^d}|cQ(x)|^{2\\gamma+2}\\rho(x/\\alpha)^{2\\gamma+2}\\d x &(I)\\\\\n &+ \\frac{c^2}{2\\alpha^2}\\int_{\\R^d} (1-Y(x/\\alpha))\\rho(x/\\alpha)^2|xQ(x)|^2\\d x -\\frac{1}{2}\\langle Z,|u_\\alpha|^2\\rangle. &(II)\n \\end{align*}\n As previously $(II)=O(\\alpha^s)$ as $\\alpha$ goes to infinity, for any $s\\in(0,2)$, and for $c\\rho(0)\\neq 1$,\n $$(I)\\sim\\frac{\\alpha^2\\lambda(c\\rho(0))^2}{2\\gamma+2}\\left(1-(c\\rho(0))^{2\\gamma}\\right)\\int_{\\R^d}|Q(x)|^{2\\gamma+2}\\d x$$\n as $\\alpha$ goes to infinity, using \\cref{Eq-PohozaevQ}. Thus, for $c\\rho(0)>1$, \\cref{Lem-AsympEnergy} implies\n $$\\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=c\\rho(0)m^*_\\lambda}}\\mathcal{E}(u)=-\\infty,$$\n that is\n $$\\forall m>m^*_\\lambda,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n Let $m'>m>0$ and $\\phi\\in\\D^{1,2}$ such that $\\mathcal{M}(\\phi)=m$. Let $\\psi=\\sqrt{\\frac{m'}{m}}\\phi$. Then,\n \\begin{align*}\n \\mathcal{E}(\\phi) &= \\frac{m}{m'}\\left[\\frac{1}{2}\\langle -A\\psi,\\psi\\rangle - \\frac{\\lambda}{2\\gamma+2}\\left(\\frac{m}{m'}\\right)^{\\gamma}\\int_{\\R^d}|\\psi|^{2\\gamma+2}\\d x\\right]\\\\\n &=\\frac{m}{m'}\\mathcal{E}(\\psi) + \\frac{m\\lambda}{m'(2\\gamma+2)}\\left[1 - \\left(\\frac{m}{m'}\\right)^{\\gamma}\\right]\\int_{\\R^d}|\\phi|^{2\\gamma+2}\\d x\\\\\n &\\geqslant \\frac{m}{m'}\\mathcal{E}(\\psi).\n \\end{align*}\n Thus,\n $$m'\\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u) \\geqslant m \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}'}\\mathcal{E}(u).$$\n In particular, if $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m}\\mathcal{E}(u)=-\\infty$, then $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m'}\\mathcal{E}(u)=-\\infty$. Similarly, if $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m'}\\mathcal{E}(u)>-\\infty$, then $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m}\\mathcal{E}(u)>-\\infty$. Hence, we can define\n \\begin{align*}\n m^*_\\lambda(\\Xi) &= \\inf\\left\\{m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty\\right\\}\\\\\n &= \\sup\\left\\{m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)>-\\infty\\right\\} \\in [0,m^*_\\lambda]\n \\end{align*}\n To conclude, we only have to prove a lower bound for $m^*_\\lambda(\\Xi)$.\n\nOur aim is now to prove \\cref{Th-ACtionGS}. Showing directly that the minimization problem\n\\begin{equation}\n \\inf_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}\\label{Eq-MinAction}\n\\end{equation}\nhas non-negative solutions may be a bit technical as it is not straightforward that\n$$u\\in\\argmin_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}\\Rightarrow |u|\\in\\argmin_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}.$$\nThus, we will introduce two other, but equivalent, minimization problems which are easier to study. Remark if $u$ solves \\cref{Eq-StationaryAGP} for some $\\lambda>0$, then $\\alpha u$ solves \\cref{Eq-StationaryAGP} for $\\lambda'=\\frac{\\lambda}{|\\alpha|^{2\\gamma}}$. Thus, the parameter $\\lambda$ can be seen as a Lagrange multiplier, and solving \\cref{Eq-StationaryAGP} for some $\\lambda>0$ is equivalent to solving \\cref{Eq-StationaryAGP} for all $\\lambda>0$. Following an idea of \\cite{ROSE1988207}, we define for $u\\in\\D^{1,2}$\n\\begin{equation}\n \\mathcal{P}_\\omega(u)=\\langle (-A+\\omega) u, u\\rangle,\\label{Eq-POmega}\n\\end{equation}\n\\begin{equation}\n \\mathcal{Q}(u) = \\int_\\R |u|^{2(\\gamma+1)}\\d x\\label{Eq-Q(u)}\n\\end{equation}\nand if $u\\neq0,$\n\\begin{equation}\n \\mathcal{J}_\\omega(u) = \\frac{\\mathcal{P}_\\omega(u)}{\\mathcal{Q}(u)^{1/(\\gamma+1)}}.\\label{Eq-JOmega}\n\\end{equation}\nLet us stress out $\\mathcal{J}_\\omega(\\alpha u)=\\mathcal{J}_\\omega(u)$ for any $\\alpha\\in\\C^*$, thus if $u$ minimizes $\\mathcal{J}_\\omega$, $\\alpha u$ minimizes $\\mathcal{J}_\\omega$ too. Using standard argument of variational calculus, one can show minimizers of both\n\\begin{equation}\n \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{Q}(u)=q}}\\mathcal{P}_\\omega(u)\\label{Eq-MinP/Q=q}\n\\end{equation}\nand\n\\begin{equation}\n \\inf_{\\D^{1,2}\\backslash\\{0\\}}\\mathcal{J}_\\omega\\label{Eq-MinJ}\n\\end{equation}\nare solutions of \\cref{Eq-StationaryAGP} for some $\\lambda>0$. For $\\lambda>0$, one can rewrite\n$$\\mathcal{S}_{\\omega,\\lambda}(u)=\\frac{1}{2}\\mathcal{P}_\\omega(u)-\\frac{\\lambda}{2(\\gamma+1)}\\mathcal{Q}(u)$$\nand\n\\begin{equation}\n \\mathcal{N}_{\\omega,\\lambda} = \\left\\{u\\in\\D^{1,2}\\backslash\\{0\\},\\; \\mathcal{J}_\\omega(u) = \\lambda \\mathcal{Q}(u)^{\\gamma/(\\gamma+1)} \\right\\}.\\label{Eq-NehariJQ}\n\\end{equation}\nAs remarked in \\cite{ROSE1988207}, minimizers of \\cref{Eq-MinP/Q=q,Eq-MinJ} are essentially the same, that is up to a multiplicative factor, due to the the homogeneity of $\\mathcal{J}_\\omega$. The following proposition shows minimization problems \\cref{Eq-MinAction,Eq-MinP/Q=q,Eq-MinJ} are equivalent, up to a multiplicative factor (see also \\cite{LiuRotatingActionGS}).",
    "post_theorem_intro_text_len": 4412,
    "post_theorem_intro_text": "In order to do so, we use constrained minimization methods. In \\cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.\n\n\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\mathcal D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\mathcal D^{-1,2}$ its dual. For $u\\in\\mathcal D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\mathbb R^d}|u|^2\\mathrm{d} x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\mathcal D^{-1,2},\\mathcal D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\mathbb R^d}|u|^{2\\gamma+2}\\mathrm{d} x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\mathcal D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\mathrm{e}^{\\mathrm{i}\\theta}\\phi_m,\\; \\theta\\in\\mathbb R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\mathcal D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\mathcal D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}\n\nIt is classical that the set of minimal energy states of fixed mass is stable in $\\mathcal D^{1,2}$ for \\cref{Eq-AGP}. In particular, \\cref{Th-EnergyGS} implies the orbital stability of the ground-state $\\phi_m$ in the defocusing case ($\\lambda<0$). \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \\cref{Eq-StationaryAGP} for any possible value of $\\omega$. We also prove the small mass asymptotic of $\\phi_m$ and its frequency $\\omega_m$.\\\\\n\nIn \\cref{Sec-ActionGS}, we investigate the minimization of action $\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$. In the focusing case ($\\lambda> 0$), one easily verifies the action is not lowerbounded. We thus introduce the Nehari manifold and show the following result.\n\n\\begin{theo}\\label{Th-ACtionGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$, $\\gamma>0$ and $A$ the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise. Let $\\mathcal D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\mathcal D^{-1,2}$ its dual. Denote by $\\mu_0$ the smallest eigenvalue of $-A$. Let $\\omega>-\\mu_0$ and define $$\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$$ and $$\\mathcal{N}_\\omega =\\left\\{u\\in\\mathcal D^{1,2}\\backslash\\{0\\},\\; \\langle \\nabla\\mathcal{S}_\\omega u,u \\rangle_{\\mathcal D^{-1,2},\\mathcal D^{1,2}}=0\\right\\}.$$\n    Then, there exists $\\psi_\\omega\\in\\displaystyle\\argmin_{\\mathcal{N}_\\omega}\\mathcal{S}_\\omega$ such that $\\psi_\\omega>0$.\n\n\\end{theo}\n\nLet us stress out an important property of solution of \\cref{Eq-StationaryAGP} compared to the noiseless case. In presence of white noise, standing waves have a limited Sobolev regularity.\n\\begin{theo}\\label{Th-MaxRegSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Almost surely any solution $u$ of \\cref{Eq-StationaryAGP} belongs to $\\mathcal W^{2-\\frac{d}{2}-,q}$ for all $q\\in[2,+\\infty)$, where $\\mathcal W^{s,q}$ spaces are defined in \\cref{Sec-ConfiningSobolev}. Moreover, almost surely any solution $u$ of \\cref{Eq-StationaryAGP} which belongs to $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ for some open set $U\\subset\\mathbb R^d$ and $q\\in[2,+\\infty]$ vanishes everywhere in $U$.\n\\end{theo}",
    "sketch": "In order to prove Theorem~\\ref{Th-SynthesExistenceSW}, the text says they \"use constrained minimization methods\".\n\n- In \\cref{Sec-EnergyGS}, they \"study the minimization of energy at fixed mass\" (Theorem~\\ref{Th-EnergyGS}). It is then stated that \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} \"show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \\cref{Eq-StationaryAGP} for any possible value of $\\omega$\".\n\n- In \\cref{Sec-ActionGS}, they instead \"investigate the minimization of action $\\mathcal{S}_\\omega=\\mathcal{E}+\\omega\\mathcal{M}$\". In the focusing case ($\\lambda>0$), they note that \"one easily verifies the action is not lowerbounded\", so they \"introduce the Nehari manifold\" and obtain an action minimizer on it (Theorem~\\ref{Th-ACtionGS}) for $\\omega>-\\mu_0$.\n\n- They also \"stress out\" that, in the presence of white noise, standing waves have \"limited Sobolev regularity\" (Theorem~\\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ on some open set $U$, then it \"vanishes everywhere in $U$\").",
    "expanded_sketch": "In order to prove the main theorem, the text says they \"use constrained minimization methods\".\n\nNext, they \"study the minimization of energy at fixed mass\" and use the following result.\n\n\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. For $u\\in\\D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\D^{-1,2},\\D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\e^{\\complexI\\theta}\\phi_m,\\; \\theta\\in\\R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schr\u0006dinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}\n\nIt is then stated that \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provides positive solutions of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nfor any possible value of $\\omega$.\n\nLater, they instead \"investigate the minimization of action $\\mathcal{S}_\\omega=\\mathcal{E}+\\omega\\mathcal{M}$\". In the focusing case ($\\lambda>0$), they note that \"one easily verifies the action is not lowerbounded\", so they \"introduce the Nehari manifold\" and obtain an action minimizer on it (Theorem~\\ref{Th-ACtionGS}) for $\\omega>-\\mu_0$.\n\nThey also \"stress out\" that, in the presence of white noise, standing waves have \"limited Sobolev regularity\" (Theorem~\\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ on some open set $U$, then it \"vanishes everywhere in $U$\").",
    "expanded_theorem": "\\label{Th-SynthesExistenceSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Denote by $\\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\n in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$,\n    \\begin{itemize}\n        \\item $\\lambda>0$ and $\\omega>-\\mu_0$,\n        \\item $\\lambda = 0$ and $\\omega=\\mu_0$,\n        \\item $\\lambda<0$ and $\\omega<-\\mu_0$.\n    \\end{itemize}\n    In every other cases, there is no non-trivial non-negative solution in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$.,",
    "theorem_type": [
      "Existence",
      "Nonexistence"
    ],
    "mcq": {
      "question": "Let \\(d\\in\\{1,2\\}\\), let \\(A=H+\\xi\\) be the Anderson Hamiltonian in dimension \\(d\\), and let \\(\\mu_0\\) denote the lowest eigenvalue of \\(-A\\). For \\(\\lambda,\\omega\\in\\mathbb R\\) and \\(\\gamma>0\\), consider the stationary equation\n\\[\n-A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega\\phi=0.\n\\]\nWhich statement about positive or nonnegative solutions \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) is valid?",
      "correct_choice": {
        "label": "A",
        "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\); \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega\\ge -\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\); \\(\\lambda<0\\) and \\(\\omega\\le -\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."
        },
        {
          "label": "C",
          "text": "If \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\), or if \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\), or if \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\), then there exists a non-trivial non-negative solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\)."
        },
        {
          "label": "D",
          "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists whenever \\(\\lambda>0\\) and \\(\\omega\\in\\mathbb R\\), whenever \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\), and whenever \\(\\lambda<0\\) and \\(\\omega< -\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."
        },
        {
          "label": "E",
          "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega\\ge \\mu_0\\); \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "strict_thresholds_at_\\pm\\mu_0",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "case_split",
          "tampered_component": "dropped_positivity_and_uniqueness_of_exclusion_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "case_split",
          "tampered_component": "focusing_case_requires_\\omega>-\\mu_0",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "linear_case_only_at_\\omega=\\mu_0",
          "template_used": "quantifier_dependence"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives the operator, equation, and parameter names, but it does not reveal the correct parameter regime or the full existence/nonexistence classification. There is no explicit answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is close to asking for the theorem’s exact conclusion, so it is partly a theorem-recall question. However, the options differ in logical strength, boundary cases, and converse claims, so it is not a pure verbatim restatement."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish strict vs. non-strict thresholds, sign-dependent regimes, and the stronger full classification from a merely weaker true statement. Still, the task is mainly recognition/recall of the correct theorem statement rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: one mishandles boundary cases, one is a weaker true statement, one flips the sign regime, and one overgeneralizes the classification. These align well with common failure modes."
      },
      "total_score": 6,
      "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it mainly tests precise theorem recall/comparison rather than deep generative reasoning."
    }
  },
  {
    "id": "2512.23079v1",
    "paper_link": "http://arxiv.org/abs/2512.23079v1",
    "theorems_cnt": 1,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:main}\n    Let $\\Lambda_\\alpha$ be a Delone set associated with an $\\alpha$-Kakutani tiling of $\\R$. Then $\\Lambda_\\alpha$ is uniformly spread if and only if \n    \\begin{equation}\\label{eq:special values}\n            r_\\alpha:=\\frac{\\log(\\min\\{\\alpha,1-\\alpha\\})}{\\log(1-\\min\\{\\alpha,1-\\alpha\\})}\\in\\left\\{1,\\frac{3}{2},2,3,4  \\right\\}.\n    \\end{equation}",
      "start_pos": 132739,
      "end_pos": 133125,
      "label": "thm:main"
    },
    "ref_dict": {
      "eq:special values": "\\begin{equation}\\label{eq:special values}\n            r_\\alpha:=\\frac{\\log(\\min\\{\\alpha,1-\\alpha\\})}{\\log(1-\\min\\{\\alpha,1-\\alpha\\})}\\in\\left\\{1,\\frac{3}{2},2,3,4  \\right\\}.\n    \\end{equation}"
    },
    "pre_theorem_intro_text_len": 2420,
    "pre_theorem_intro_text": "A set $\\Lambda\\subset\\R^d$ is \\emph{Delone} if it is \\emph{uniformly discrete} and \\emph{relatively dense}, that is, if there exist constants $0<r,R<\\infty$ so that $B(x,r)\\cap\\Lambda=x$ and $B(y,R)\\cap\\Lambda\\neq\\emptyset$ for all $x\\in\\Lambda$ and $y\\in \\R^d$, respectively. clearly lattices and their translations are Delone sets in $\\R^d$, and there are numerous non-lattice examples that are not periodic in any way. Given such a non-lattice Delone set, a natural question is to ask how ``close'' it is to a lattice in $\\R^d$. One natural interpretation of such a question involves the concept of \\emph{bounded displacement (BD) equivalence}, where two Delone sets $\\Lambda,\\Gamma\\subset\\R^d$ are BD equivalent if there exists a bijection $\\varphi:\\Lambda\\rightarrow \\Gamma$ satisfying $\\sup_{x\\in\\Lambda}\\|x-\\varphi(x)\\|<\\infty$. \nA Delone set $\\Lambda\\subset\\R^d$ is \\emph{uniformly spread} if it is BD equivalent to some lattice in $\\R^d$, or equivalently, by a simple application of Hall's marriage theorem, if there is some $c>0$ for which $\\Lambda$ and $c\\cdot\\Z^d$ are BD equivalent. See \\cite{SmilanskySolomon-2025} for proofs and a comprehensive discussion. \n\nThe Delone sets considered in this manuscript are those associated with $\\alpha$-Kakutani tilings for $\\alpha\\in(0,1)$, which form a simple class of multiscale substitution tilings \\cite{SmilanskySolomon-2021}. These tilings are defined by a substitution rule on a closed interval $I\\subset\\R$ of unit length that splits $I$ into two subintervals, a left one of length $\\alpha$ and a right one of length $1-\\alpha$. Starting with a copy of $I$, which we think of as a \\emph{tile}, we apply a uniform inflation and substitute any tile of length greater than $1$. This results in a growing family of patches that exhausts the real line, and limits taken in an appropriate way define a space of tilings of $\\R$, all made of interval-tiles of lengths between $\\min\\{\\alpha,1-\\alpha\\}$ and $1$. Given such a tiling, an \\emph{associated} Delone set is obtained by selecting one point from each tile, with the requirements that distinct tiles yield distinct points and that the resulting set is uniformly discrete. A natural choice is to take the left endpoint of each tile. However, since all Delone sets associated with a fixed tiling are BD equivalent, the particular choice of points is immaterial. \n\nOur main result is the following classification.",
    "context": "A set $\\Lambda\\subset\\R^d$ is \\emph{Delone} if it is \\emph{uniformly discrete} and \\emph{relatively dense}, that is, if there exist constants $0<r,R<\\infty$ so that $B(x,r)\\cap\\Lambda=x$ and $B(y,R)\\cap\\Lambda\\neq\\emptyset$ for all $x\\in\\Lambda$ and $y\\in \\R^d$, respectively. clearly lattices and their translations are Delone sets in $\\R^d$, and there are numerous non-lattice examples that are not periodic in any way. Given such a non-lattice Delone set, a natural question is to ask how ``close'' it is to a lattice in $\\R^d$. One natural interpretation of such a question involves the concept of \\emph{bounded displacement (BD) equivalence}, where two Delone sets $\\Lambda,\\Gamma\\subset\\R^d$ are BD equivalent if there exists a bijection $\\varphi:\\Lambda\\rightarrow \\Gamma$ satisfying $\\sup_{x\\in\\Lambda}\\|x-\\varphi(x)\\|<\\infty$. \nA Delone set $\\Lambda\\subset\\R^d$ is \\emph{uniformly spread} if it is BD equivalent to some lattice in $\\R^d$, or equivalently, by a simple application of Hall's marriage theorem, if there is some $c>0$ for which $\\Lambda$ and $c\\cdot\\Z^d$ are BD equivalent. See \\cite{SmilanskySolomon-2025} for proofs and a comprehensive discussion.\n\nThe Delone sets considered in this manuscript are those associated with $\\alpha$-Kakutani tilings for $\\alpha\\in(0,1)$, which form a simple class of multiscale substitution tilings \\cite{SmilanskySolomon-2021}. These tilings are defined by a substitution rule on a closed interval $I\\subset\\R$ of unit length that splits $I$ into two subintervals, a left one of length $\\alpha$ and a right one of length $1-\\alpha$. Starting with a copy of $I$, which we think of as a \\emph{tile}, we apply a uniform inflation and substitute any tile of length greater than $1$. This results in a growing family of patches that exhausts the real line, and limits taken in an appropriate way define a space of tilings of $\\R$, all made of interval-tiles of lengths between $\\min\\{\\alpha,1-\\alpha\\}$ and $1$. Given such a tiling, an \\emph{associated} Delone set is obtained by selecting one point from each tile, with the requirements that distinct tiles yield distinct points and that the resulting set is uniformly discrete. A natural choice is to take the left endpoint of each tile. However, since all Delone sets associated with a fixed tiling are BD equivalent, the particular choice of points is immaterial.\n\nOur main result is the following classification.",
    "full_context": "A set $\\Lambda\\subset\\R^d$ is \\emph{Delone} if it is \\emph{uniformly discrete} and \\emph{relatively dense}, that is, if there exist constants $0<r,R<\\infty$ so that $B(x,r)\\cap\\Lambda=x$ and $B(y,R)\\cap\\Lambda\\neq\\emptyset$ for all $x\\in\\Lambda$ and $y\\in \\R^d$, respectively. clearly lattices and their translations are Delone sets in $\\R^d$, and there are numerous non-lattice examples that are not periodic in any way. Given such a non-lattice Delone set, a natural question is to ask how ``close'' it is to a lattice in $\\R^d$. One natural interpretation of such a question involves the concept of \\emph{bounded displacement (BD) equivalence}, where two Delone sets $\\Lambda,\\Gamma\\subset\\R^d$ are BD equivalent if there exists a bijection $\\varphi:\\Lambda\\rightarrow \\Gamma$ satisfying $\\sup_{x\\in\\Lambda}\\|x-\\varphi(x)\\|<\\infty$. \nA Delone set $\\Lambda\\subset\\R^d$ is \\emph{uniformly spread} if it is BD equivalent to some lattice in $\\R^d$, or equivalently, by a simple application of Hall's marriage theorem, if there is some $c>0$ for which $\\Lambda$ and $c\\cdot\\Z^d$ are BD equivalent. See \\cite{SmilanskySolomon-2025} for proofs and a comprehensive discussion.\n\nThe Delone sets considered in this manuscript are those associated with $\\alpha$-Kakutani tilings for $\\alpha\\in(0,1)$, which form a simple class of multiscale substitution tilings \\cite{SmilanskySolomon-2021}. These tilings are defined by a substitution rule on a closed interval $I\\subset\\R$ of unit length that splits $I$ into two subintervals, a left one of length $\\alpha$ and a right one of length $1-\\alpha$. Starting with a copy of $I$, which we think of as a \\emph{tile}, we apply a uniform inflation and substitute any tile of length greater than $1$. This results in a growing family of patches that exhausts the real line, and limits taken in an appropriate way define a space of tilings of $\\R$, all made of interval-tiles of lengths between $\\min\\{\\alpha,1-\\alpha\\}$ and $1$. Given such a tiling, an \\emph{associated} Delone set is obtained by selecting one point from each tile, with the requirements that distinct tiles yield distinct points and that the resulting set is uniformly discrete. A natural choice is to take the left endpoint of each tile. However, since all Delone sets associated with a fixed tiling are BD equivalent, the particular choice of points is immaterial.\n\nOur main result is the following classification.\n\nOur main result is the following classification.\n\n\\subsection{Commensurable tilings as primitive substitution tilings}\\label{subsec:construction of rho_alpha} Recall our assumption that $\\alpha=\\min\\{\\alpha,1-\\alpha\\}\\in(0,1/2]$, and consider a commensurable $\\alpha$-Kakutani rule. Let $n\\geq m>0$ be coprime integers so that \n\\begin{equation}\\label{eq:rational ratio}\n \\frac{n}{m}=r_\\alpha=\\frac{\\log\\alpha}{\\log(1-\\alpha)}. \n\\end{equation}\nIf $n=m$ then $\\alpha=1/2$, and any $\\alpha$-Kakutani tiling is a translation of a lattice tiling in $\\R$, with all associated Delone sets uniformly spread. Assume from now on that $n> m$, and consider the associated graph $G_\\alpha$. By \\eqref{eq:rational ratio} we have \n$$\ng_\\alpha :=\\frac{1}{n}\\log\\frac{1}{\\alpha}=\\frac{1}{m}\\log\\frac{1}{1-\\alpha}.\n$$\nThen by adding vertices to $G_\\alpha$, the loop of length $\\log(1/\\alpha)$ and the loop of length $\\log(1/(1-\\alpha))$ can be partitioned into $n$ and $m$ edges of equal length $g_\\alpha$, respectively. This defines a directed graph $G'_\\alpha$ with $n+m-1$ vertices and $n+m$ edges of equal length. Label the original vertex with $1$, and assign labels $2,\\ldots,n$ and $n+1,\\ldots,n+m-1$ to the $(n-1)+(m-1)$ new vertices that are added along the two loops.\n\n\\begin{prop}\\label{prop:generating patches}\n Consider a commensurable $\\alpha$-Kakutani substitution rule with $r_\\alpha=n/m$ for $n,m\\in\\N$ coprime. Let $\\ell\\in\\Z_{\\geq 0}$ and put \n \\begin{equation}\\label{eq:t_l}\n t_\\ell:= \\ell\\cdot g_\\alpha=\\ell\\cdot \\frac{1}{n}\\log\\frac{1}{\\alpha}.\n \\end{equation}\n Then the (Kakutani) generating patch $F_{t_\\ell}(I)$ and the (primitive substitution) generating patch $(\\xi\\rho_\\alpha)^\\ell(I)$ are geometrically the same, where $\\xi=\\alpha^{-1/n}$.\n\\end{prop}\n\\begin{proof}\n We prove by induction on $\\ell\\geq0$. For $\\ell=0$, both patches contain only the single tile $I$ and so are geometrically the same. Assume the claim holds for some $\\ell\\geq0$. Tiles in $F_{t_\\ell}(I)$ and $(\\xi\\rho_\\alpha)^\\ell(I)$ are either intervals of unit length, that is, copies of $I$, or intervals of length at most $|T_n|=|T_{n+m-1}|=\\alpha^{1/n}$. By construction, the patch $F_{t_{\\ell+1}}(I)$ is defined by inflating the patch $F_{t_\\ell}(I)$ by $\\alpha^{-1/n}$ and then substituting every tile of length strictly greater than $1$ according to the $\\alpha$-Kakutani rule. The tiles that are substituted are exactly the unit intervals from $F_{t_\\ell}(I)$. On the other hand, the tiles in $(\\xi\\rho_\\alpha)^{\\ell+1}(I)$ are the result of one application of $\\xi\\rho_\\alpha$ on the tiles in $(\\xi\\rho_\\alpha)^\\ell(I)$. For copies of $I$ this amounts to an application of the $\\alpha$-Kakutani rule once and inflation by $\\xi=\\alpha^{-1/n}$, while all other tiles in the patch are simply inflated by the same constant, and so $F_{t_{\\ell+1}}(I)$ and $(\\xi\\rho_\\alpha)^{\\ell+1}(I)$ are the same.\n\\end{proof}\n\nWe focus here on Delone sets in $\\R$. \nFor a detailed discussion about equivalence relations on Delone sets in $\\R^d$ we refer to the book chapter \\cite{SmilanskySolomon-2025}.\nLet $|U|$ denote the Lebesgue measure of a bounded measurable set $U\\subset \\R$. A Delone set $\\Lambda\\subset\\R$ has \\emph{asymptotic density} $d_\\Lambda$ if\n$$\n\\lim_{n\\rightarrow\\infty}\\frac{\\#(\\Lambda\\cap U_n)}{|U_n|}\n$$\nexists and is equal to $d_\\Lambda$ for any sequence \\emph{van Hove sequence} $(U_n)$, that is, a sequence of bounded measurable sets so that $|(\\partial U_n)^{+\\varepsilon}|/|U_n|\\rightarrow0$ for all $\\varepsilon>0$, where $A^{+\\varepsilon}$ is the $\\varepsilon$ neighbourhood of $A\\subset\\R$. The \\emph{discrepancy} of $\\Lambda$ with respect to the parameter $\\beta>0$ and a measurable set $B\\subset\\R$ is defined as \n$$\n\\disc{\\Lambda}{\\beta}{U}:=\\left|\\#(\\Lambda \\cap U)-\\beta\\cdot|U|\\right|.\n$$\nA criterion established by Laczkovich \\cite{Laczkovich-1992} provides a necessary and sufficient condition for a Delone set in $\\R^d$ to be uniformly spread. In the case of a Delone set $\\Lambda\\subset \\R$ with asymptotic density $d_\\Lambda$, Laczkovich's criterion can be stated in the following way.\n\\begin{thm}\\label{thm:Laczkovich}\n A Delone set $\\Lambda\\subset \\R$ with asymptotic density $d_\\Lambda$ is uniformly spread if and only if\n $$\n \\disc{\\Lambda}{d_\\Lambda}{U}\\leq C\\cdot|\\partial U|^{+1}\n $$\n for all bounded measurable sets $U\\subset\\R$.\n\\end{thm}\n\n\\begin{thm}\\label{thm:BD_substitution}\n Let $\\rho$ be a primitive substitution rule in $\\R^d$ on a set of prototiles that are biLipschitz homeomorphic to closed balls, and let $\\Lambda$ be a Delone set associated with a tiling generated by $\\rho$. Let $\\ell\\ge 2$ be the minimal index such that the corresponding eigenspace satisfies $W_{\\lambda_\\ell} \\nsubseteq \\mathbf{1}^\\perp$, where $\\mathbf{1}=(1,\\ldots,1)\\in \\R^k$. Then \n\\begin{enumerate}\n\\item\nIf $\\absolute{\\lambda_\\ell}<\\lambda_1^{\\frac{d-1}{d}}$ then $\\Lambda$ is uniformly spread.\n\\item\nIf $\\absolute{\\lambda_\\ell}>\\lambda_1^{\\frac{d-1}{d}}$ then $\\Lambda$ is not uniformly spread.\n\\end{enumerate}\n\\end{thm}\nWe note that if $\\absolute{\\lambda_\\ell}=\\lambda_1^{\\frac{d-1}{d}}$ then $\\Lambda$ may exhibit either property, for an example see \\cite{FrettlohSmilanskySolomon-2021}.\nIn the case $d=1$, checking the conditions in Theorem \\ref{thm:BD_substitution} is reduced to determining if $\\lambda_\\ell$ is in the interior of the closed unit disk or in its complement.\n\nThe discrepancy can be large however, and by \\cite[\\S 6]{SmilanskySolomon-2025} there exist arbitrarily large intervals $U$ for which \n\\begin{equation}\\label{eq:incomm_discrepancy}\n \\disc{\\Lambda_\\alpha}{d_{\\Lambda_\\alpha}}{U}=\\Omega\\left(\\frac{|U|}{\\log(|U|)}\\right).\n\\end{equation}\nBy Laczkovich's criterion as stated in Theorem \\ref{thm:Laczkovich}, if $\\Lambda_\\alpha$ is uniformly spread then there exists a constant $C>0$ so that for any interval $U$ the discrepancy is bounded by $\\disc{\\Lambda_\\alpha}{d_{\\Lambda_\\alpha}}{U}\\leq 2C$. By \\eqref{eq:incomm_discrepancy} this is clearly not the case, and so $\\Lambda_\\alpha$ is not uniformly spread. In fact, for any fixed incommensurable $\\alpha$-Kakutani rule there are continuously many associated Delone sets that are pairwise BD non-equivalent, see \\cite{SmilanskySolomon1-2022}.\n\nAssume now that $r_\\alpha=n/m$ with $n>m>0$ coprime integers. By Proposition \\ref{prop:primitive cover}, any $\\alpha$-Kakutani tiling is also generated by a primitive substitution rule $\\rho_\\alpha$ with substitution matrix $M_\\alpha$ as in \\eqref{eq:substitution matrix}. All tiles are closed interval and are therefore closed balls in $\\R$, and so we can apply Solomon's criterion for uniform spreadness, stated above as Theorem \\ref{thm:BD_substitution}, with $d=1$. By\nProposition \\ref{prop:eigenspacesnotperp} and the preceding discussion, the eigenvalue $\\lambda_2$ of $M_\\alpha$ is non-zero and satisfies $W_{\\lambda_2}\\nsubseteq{\\bf 1}^\\perp$, and so $\\ell=2$. Since $\\gcd(n-m,n)=\\gcd(m,n)=1$, Proposition \\ref{prop:nounitroots} applies to $f_\\alpha(x)$ as defined in \\eqref{eq:charpoly}, and so $|\\lambda_2|\\neq1$. Combining the above, we deduce that an associated Delone set $\\Lambda_\\alpha$ is uniformly spread if and only if $|\\lambda_2|<1$. By Proposition \\ref{prop:PVpolynomials}, for this to hold $f_\\alpha(x)$ must be one of the four polynomials listed in \\eqref{eq:PVpoly}. We thus conclude that the values of $\\alpha\\in(0,1/2)$ for which $\\Lambda_\\alpha$ is uniformly spread are given by\n\\begin{equation}\\label{eq:PVvalues}\n r_\\alpha=\\frac{n}{m}\\in\\left\\{\\frac32,\\frac21,\\frac31,\\frac41\\right\\}, \n\\end{equation}\nand the proof is complete. \n\\end{proof}",
    "post_theorem_intro_text_len": 3333,
    "post_theorem_intro_text": "For simplicity of presentation, we will assume throughout that $\\alpha\\in(0,1/2]$, and so $\\alpha=\\min\\{\\alpha,1-\\alpha\\}$. We note that if $r_\\alpha=1$ then $\\alpha=1/2$, and the left endpoints of tiles in any corresponding tiling are simply translated lattices in $\\R$. When $r_\\alpha=2$ then $\\alpha=1/\\phi^2$, where $\\phi$ is the \\emph{golden ratio}. The other values of $\\alpha$ are harder to present explicitly, though the cases $r_\\alpha=3/2$ and $r_\\alpha=3$ are related to the \\emph{supergolden ratio} and the \\emph{plastic ratio}, respectively. The latter is the smallest \\emph{Pisot–Vijayaraghavan (PV) number}, and the case $r_\\alpha=4$ is related to the second smallest PV number, see \\S \\ref{sec:remarks}. For all other values of $\\alpha$ the corresponding Delone sets are never uniformly spread. \n\n\\subsection{Key steps of the proof} \n\nA complete proof requires some further preparation, but  we can aready outline the main steps of the argument. First, in the \\emph{incommensurable} case $r_\\alpha\\notin\\Q$, the discrepancy in $\\alpha$-Kakutani tilings, that is, the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density), can be shown to be large. This, when combined with Laczkovich's criterion for uniform spreadness \\cite{Laczkovich-1992}, is enough to establish that any associated Delone set is not BD equivalent to a lattice. This is a special case of \\cite[Theorem 8.2]{SmilanskySolomon-2021}, which states that any Delone set associated with an incommensurable multiscale substitution tiling in $\\R^d$ is not uniformly spread.\n\nThe case $r_\\alpha\\in\\Q$, namely the \\emph{commensurable} case, is rather different, and is the new result contained in this contribution. \nThe first step is to observe that, by introducing labelled prototiles and modifying the substitution rule in a suitable way, one can construct a standard primitive substitution that ``covers'' the commensurable multiscale substitution construction, in the sense that it generates substitution tilings that are geometrically identical to the corresponding $\\alpha$-Kakutani tilings described above.\n The problem of uniform spreadness of Delone set associated with primitive substitution tilings can be approached using Solomon's criterion \\cite{Solomon-2014}. This criterion, which is a careful application of the aforementioned Laczkovich criterion, is given in terms of a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues. In the special case considered here, this becomes a question of classification of certain \\emph{Pisot–Vijayaraghavan (PV) polynomials} \\cite{DubickasJankauskas-2014}, from which the special values that appear in \\eqref{eq:special values} can be extracted.\n\nWe complete our discussion by showing that such scarcity of uniformly spread constructions does not necessarily carry over when moving away from $\\alpha$-Kakutani tilings to more complicated multiscale substitution tilings. For example, when considering $(\\alpha,\\beta)$-Kakutani tilings, generated by a substitution rule on $I$, now split into three tiles of lengths $\\alpha,\\beta,1-\\alpha-\\beta$, we present a countable set of parameters $(\\alpha,\\beta)$ for which the associated Delone sets are uniformly spread.",
    "sketch": "A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.\n\n1. **Assume** $\\alpha\\in(0,1/2]$, so $\\alpha=\\min\\{\\alpha,1-\\alpha\\}$.\n\n2. **Incommensurable case** ($r_\\alpha\\notin\\Q$): show that the *discrepancy* in $\\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of \\cite[Theorem 8.2]{SmilanskySolomon-2021}.\n\n3. **Commensurable case** ($r_\\alpha\\in\\Q$): this is described as “rather different” and the “new result.”\n   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\\alpha$-Kakutani tilings.\n   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”\n   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in \\eqref{eq:special values} can be extracted,” yielding exactly the listed set $\\left\\{1,\\frac{3}{2},2,3,4\\right\\}$.\n\n4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\\alpha,\\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\\alpha,\\beta)$ for which the associated Delone sets are uniformly spread.”",
    "expanded_sketch": "A complete proof requires some further preparation, but the post-theorem introduction outlines these main steps.\n\n1. **Assume** $\\alpha\\in(0,1/2]$, so $\\alpha=\\min\\{\\alpha,1-\\alpha\\}$.\n\n2. **Incommensurable case** ($r_\\alpha\\notin\\Q$): show that the *discrepancy* in $\\alpha$-Kakutani tilings—“the fluctuation of the number of tiles that appear in large patches relative to their expected number (namely, the density)”—“can be shown to be large.” Combine this with “Laczkovich's criterion for uniform spreadness” to conclude that any associated Delone set “is not BD equivalent to a lattice,” hence is not uniformly spread. This is noted as a special case of Smilansky–Solomon, *[title and year not provided]*, Theorem 8.2.\n\n3. **Commensurable case** ($r_\\alpha\\in\\Q$): this is described as “rather different” and the “new result.”\n   - **Cover by a standard primitive substitution**: “by introducing labelled prototiles and modifying the substitution rule in a suitable way,” construct “a standard primitive substitution that ‘covers’ the commensurable multiscale substitution construction,” meaning it generates substitution tilings “geometrically identical” to the $\\alpha$-Kakutani tilings.\n   - **Apply Solomon's criterion**: study uniform spreadness for the resulting primitive substitution tilings via “Solomon's criterion,” formulated as “a comparison between the leading Perron-Frobenius eigenvalue of the associated substitution matrix, and the subsequent eigenvalues.”\n   - **Reduce to PV polynomial classification**: “In the special case considered here, this becomes a question of classification of certain Pisot–Vijayaraghavan (PV) polynomials,” and “from [this] the special values that appear in\n\\begin{equation}\\label{eq:special values}\n            r_\\alpha:=\\frac{\\log(\\min\\{\\alpha,1-\\alpha\\})}{\\log(1-\\min\\{\\alpha,1-\\alpha\\})}\\in\\left\\{1,\\frac{3}{2},2,3,4  \\right\\}.\n    \\end{equation}\ncan be extracted,” yielding exactly the listed set $\\left\\{1,\\frac{3}{2},2,3,4\\right\\}$.\n\n4. **Further remark beyond the theorem**: they add that this “scarcity” need not persist for more general multiscale substitution tilings (e.g. $(\\alpha,\\beta)$-Kakutani tilings), where they “present a countable set of parameters $(\\alpha,\\beta)$ for which the associated Delone sets are uniformly spread.”",
    "expanded_theorem": "\\label{thm:main}\n    Let $\\Lambda_\\alpha$ be a Delone set associated with an $\\alpha$-Kakutani tiling of $\\R$. Then $\\Lambda_\\alpha$ is uniformly spread if and only if \n    \\begin{equation}\\label{eq:special values}\n            r_\\alpha:=\\frac{\\log(\\min\\{\\alpha,1-\\alpha\\})}{\\log(1-\\min\\{\\alpha,1-\\alpha\\})}\\in\\left\\{1,\\frac{3}{2},2,3,4  \\right\\}.\n    \\end{equation}",
    "theorem_type": [
      "Biconditional or Equivalence",
      "Classification or Bijection"
    ],
    "mcq": {
      "question": "Let \\(\\alpha\\in(0,1)\\), and let \\(\\Lambda_\\alpha\\) be a Delone set associated with an \\(\\alpha\\)-Kakutani tiling of \\(\\mathbb{R}\\) (for example, by choosing one point from each tile so that the resulting set is uniformly discrete). Say that \\(\\Lambda_\\alpha\\) is uniformly spread if it is bounded-displacement equivalent to a lattice in \\(\\mathbb{R}\\). Define\n\\[\nr_\\alpha:=\\frac{\\log(\\min\\{\\alpha,1-\\alpha\\})}{\\log(1-\\min\\{\\alpha,1-\\alpha\\})}.\n\\]\nWhich condition on \\(r_\\alpha\\) is exactly equivalent to \\(\\Lambda_\\alpha\\) being uniformly spread?",
      "correct_choice": {
        "label": "A",
        "text": "\\(\\Lambda_\\alpha\\) is uniformly spread if and only if \\(r_\\alpha\\in\\left\\{1,\\tfrac{3}{2},2,3,4\\right\\}.\\)"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\(\\Lambda_\\alpha\\) is uniformly spread if and only if \\(r_\\alpha\\in\\mathbb{Q}.\\)"
        },
        {
          "label": "C",
          "text": "If \\(\\Lambda_\\alpha\\) is uniformly spread, then \\(r_\\alpha\\in\\mathbb{Q}.\\)"
        },
        {
          "label": "D",
          "text": "\\(\\Lambda_\\alpha\\) is uniformly spread if and only if \\(r_\\alpha\\in\\left\\{\\tfrac{3}{2},2,3,4\\right\\}.\\)"
        },
        {
          "label": "E",
          "text": "\\(\\Lambda_\\alpha\\) is uniformly spread if and only if \\(r_\\alpha\\in\\left\\{1,\\tfrac{3}{2},2,3\\right\\}.\\)"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "PV-classification refinement from rational to finite exceptional set",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "characteristic",
          "tampered_component": "replace exact finite classification by necessary commensurability only",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "exceptional value r_alpha=1",
          "template_used": "boundary_range"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "top endpoint value 4 in the extracted special set",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not reveal the finite exceptional set or otherwise point directly to the correct option; it only defines the quantity and asks for the exact criterion."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a classification theorem: the correct choice states the exact equivalence being asked for, with little reformulation beyond theorem-to-MCQ conversion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some discrimination is needed among nearby alternatives (all rationals, one-way implication, omitted exceptional values), but success depends mainly on recall of the theorem rather than substantial reasoning or synthesis."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they reflect common errors such as overgeneralizing to all rationals, weakening an iff to a necessary condition, or missing boundary exceptional values."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall-based MCQ with no answer leakage and strong distractors, but it is largely a direct theorem restatement and does not strongly test generative reasoning."
    }
  },
  {
    "id": "2512.23164v1",
    "paper_link": "http://arxiv.org/abs/2512.23164v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Theorem ID of alpha Cauchy}\nThe $\\alpha$-Cauchy variable $\\mathcal{C}_\\alpha$ is infinitely divisible if  $1 < \\alpha \\leq 6/5$.",
      "start_pos": 8835,
      "end_pos": 8992,
      "label": "Theorem ID of alpha Cauchy"
    },
    "ref_dict": {
      "Theorem ID of half alpha Cauchy": "\\begin{theorem}\\label{Theorem ID of half alpha Cauchy} \nLet $p \\in (0, \\infty)$. \nThe random variable $|\\mathcal{C}_\\a |^{ \\varepsilon p}, \\varepsilon = \\pm 1,$ is infinitely divisible if \n\\begin{equation}\\label{eq range p}\np \\geq \n    \\begin{cases}\n   (\\a + 1)/3 ,& \\quad \\varepsilon = 1, \\, 1 < \\a \\leq 2. \\\\\n     \\a/2 ,& \\quad \\varepsilon = 1, \\, \\a > 2. \\\\\n     \\a/2 ,& \\quad \\varepsilon = - 1, \\, 1 < \\a \\leq 2. \\\\\n     (2\\a - 1)/3 ,& \\quad \\varepsilon = -1, \\, \\a > 2. \\\\\n    \\end{cases}\n\\end{equation}\n\\end{theorem}",
      "Proposition Steutel coro VI.4.8": "\\begin{proposition}\n    \\label{Proposition Steutel coro VI.4.8}{\\cite[Corollary VI.4.8]{SH04}}\n    A random variable $X$ with $X \\elaw Z \\times Y$ is infinitely divisible\nif $Z$ and $Y$ are independent, $Z$ has a symmetric distribution and $Y$ has the \ndensity $f$ given by \n$$ f(x) = \\frac{1}{2}(1+x)e^{-x}, \\qquad x >0.  $$\n\\end{proposition}",
      "Kristiansen's Theorem": "\\begin{theorem}\\label{Kristiansen's Theorem}{\\cite{Kri94}}\n    The independent product $\\G_2 \\times \\W $ is infinitely divisible for any non-negative random variable $\\W$. See the beginning of section \\ref{section Gamma type} for the definition of $\\G_2 $.\n\\end{theorem}",
      "Introduction X": "\\begin{equation}\\label{Introduction X}\n        \\mathbb{E}(X^s) =  \\frac{\\sin(\\pi/\\a)}{\\pi}\\frac{ \\Gamma\\left(\\frac{1}{\\a} +\\frac{ \\varepsilon p s}{\\a}\\right)\\Gamma\\left(1 - \\frac{1}{\\a} - \\frac{ \\varepsilon p s}{\\a}\\right)}{ \\Gamma(2 + s) }, \n    \\end{equation}",
      "Theorem ID of alpha Cauchy": "\\begin{theorem}\\label{Theorem ID of alpha Cauchy}\nThe $\\alpha$-Cauchy variable $\\mathcal{C}_\\a$ is infinitely divisible if  $1 < \\a \\leq 6/5$. \n\\end{theorem}",
      "thm Gamma type": "\\begin{theorem}\\label{thm Gamma type}\n    For positive $a,b,c,d$, we consider the random variable $\\X_{a,b,c,d}$ satisfying \n       \\begin{equation}\\label{def Xabcd}\n        \\mathbb{E}(\\X_{a,b,c,d}^s) = \\frac{ \\Gamma(c) }{ \\Gamma(a)\\Gamma(b)} \\frac{ \\Gamma(a + s)\\Gamma(b-s)}{ \\Gamma(c+ d s) }, \\quad -a < s < b.\n    \\end{equation} \n\\begin{itemize}\n    \\item[(I)] $\\X_{a,b,c,d}$ does not exist if one of the following conditions holds: \\vspace{1mm}\n    \\begin{itemize}\n        \\item[(1)]  $d > 2$; \\vspace{1mm}\n        \\item[(2)]   $c < ad$; \\vspace{1mm}\n        \\item[(3)]  $3a + b > c, \\; d=2$;  \\vspace{1mm}\n        \\item[(4)] $1 < d \\leq 2,\\; a +b \\geq 1, \\;c < ad -d + f(d)$.\n    \\end{itemize}\n    \\vspace{1mm}\n    \\item[(II)] $\\X_{a,b,c,d}$ exists if one of the following conditions holds: \\vspace{1mm}\n    \\begin{itemize}\n        \\item[(1)]  $d \\leq 1, c \\geq ad$;\\vspace{1mm}\n        \\item[(2)]  $1 < d \\leq 2,\\; a +b \\leq 1, \\;c \\geq ad -d + f(d)$;\\vspace{1mm}\n        \\item[(3)] $1 < d \\leq 2,\\; \\frac{2c}{d} \\geq 3a+b, \\;2(\\frac{c}{d}-a)(\\frac{c}{d}+\\frac{1}{2}-a)\\geq a+b$.\\vspace{1mm}\n    \\end{itemize}\n\\end{itemize}\n where $f$ is the function defined in Theorem \\ref{Theorem admissible parameter domain}.  \n\\end{theorem}",
      "Introduction Xabcd": "\\begin{equation}\\label{Introduction Xabcd}\n        \\mathbb{E}(\\X_{a,b,c,d}^s) =  \\frac{ \\Gamma(c) }{ \\Gamma(a)\\Gamma(b)} \\frac{ \\Gamma(a + s)\\Gamma(b-s)}{ \\Gamma(c+ d s) }, \\quad -a < s < b,\n    \\end{equation}"
    },
    "pre_theorem_intro_text_len": 3509,
    "pre_theorem_intro_text": "For $\\alpha > 1$, the $\\alpha$-Cauchy random variable, denoted by $\\mathcal{C}_\\alpha$, was introduced by Yano, Yano and Yor \\cite{YYY09} in 2009. The density of $\\mathcal{C}_\\alpha$ is \n\\begin{equation}\\label{eq density alpha Cauchy}\n    f_{\\mathcal{C}_\\alpha}(t) = \\frac{\\sin(\\pi/\\alpha)}{2\\pi/\\alpha}\\frac{1}{1+|t|^\\alpha}, \\quad \\alpha > 1, \\,\\, t \\in \\mathbb{R}. \n\\end{equation}\nIn particular, when $\\alpha = 2$, it is \nthe standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.\nIt is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\\mathcal{C}_\\alpha$. \n\nA random variable is infinitely divisible (ID)  if for any positive integer $n$, it is the sum of $n$ independent and identically distributed random variables. Typical examples of infinitely divisible distributions are normal, Cauchy, half Cauchy, Stable, Gamma, Poisson, and Student-t. \nThe theory of infinitely divisible distribution is a classic topic in probability theory. In 1929, de Finetti first introduced the concept of infinite divisibility, and then Kolmogorov, Lévy, and Khintchine further developed the theory. Infinitely divisible distributions are closely related to limit theorems and the theory of Lévy processes and have important applications in finance, insurance, biology, physics, and signal processing. Proving or disproving infinite divisibility of a certain distribution can sometimes be quite sophisticated. In some cases, it is more convenient to consider subclasses of infinitely divisible distributions, like self-decomposable (SD) distributions, generalized gamma convolutions (GGC) and hyperbolically completely monotone (HCM) distributions. We refer to Steutel and van Harn \\cite{SH04}, Sato \\cite{Sat99} and Bondesson \\cite{Bon92} for abundant properties of these subclasses. \n\n In 1987, \n Bondesson \\cite{Bon87} proved that the half Cauchy variable $ |\\mathcal{C}_2|$ is ID. In 1998, Di\\'edhiou \\cite{Die98} proved that $ |\\mathcal{C}_2|$ is SD. \nYano, Yano, and Yor proved $ |\\mathcal{C}_\\alpha|$ is ID for any $1 < \\alpha \\leq 2$ by Bondesson's argument, see, Theorem 2.7 in \\cite{YYY09}. Furthermore, they \nproposed three open questions (\\cite[Remark 2.9]{YYY09}):\n\\begin{enumerate}\n    \\item Is $\\mathcal{C}_\\alpha$ SD (or ID at least)?\n    \\item  Is $|\\mathcal{C}_\\alpha|$  SD (or ID at least)?\n    \\item Is $|\\mathcal{C}_\\alpha|^{-p}$ SD (or ID at least) for $p > 0$?\n\\end{enumerate}\n\n There are very few results on the above questions. \nIn the present paper, we study the infinite divisibility of\n$ \\mathcal{C}_\\alpha$ and $ |\\mathcal{C}_\\alpha|^{p}, \\, p \\in \\mathbb{R}$, and partially give positive answers to the questions of Yano, Yano, and Yor. \n\n\\subsection{Infinite divisibility of \\texorpdfstring{$\\alpha$}{}-Cauchy variables}\nIt is well-known that the Cauchy variable $\\mathcal{C}_2$ is infinitely divisible and even stable. But when $\\alpha \\neq 2$,  there is no known result on the infinite divisibility of $\\mathcal{C}_\\alpha$. The following theorem gives a positive answer to this question when $1 < \\alpha \\leq 6/5$.",
    "context": "For $\\alpha > 1$, the $\\alpha$-Cauchy random variable, denoted by $\\mathcal{C}_\\alpha$, was introduced by Yano, Yano and Yor \\cite{YYY09} in 2009. The density of $\\mathcal{C}_\\alpha$ is \n\\begin{equation}\\label{eq density alpha Cauchy}\n f_{\\mathcal{C}_\\alpha}(t) = \\frac{\\sin(\\pi/\\alpha)}{2\\pi/\\alpha}\\frac{1}{1+|t|^\\alpha}, \\quad \\alpha > 1, \\,\\, t \\in \\mathbb{R}. \n\\end{equation}\nIn particular, when $\\alpha = 2$, it is \nthe standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.\nIt is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\\mathcal{C}_\\alpha$.\n\nA random variable is infinitely divisible (ID) if for any positive integer $n$, it is the sum of $n$ independent and identically distributed random variables. Typical examples of infinitely divisible distributions are normal, Cauchy, half Cauchy, Stable, Gamma, Poisson, and Student-t. \nThe theory of infinitely divisible distribution is a classic topic in probability theory. In 1929, de Finetti first introduced the concept of infinite divisibility, and then Kolmogorov, Lévy, and Khintchine further developed the theory. Infinitely divisible distributions are closely related to limit theorems and the theory of Lévy processes and have important applications in finance, insurance, biology, physics, and signal processing. Proving or disproving infinite divisibility of a certain distribution can sometimes be quite sophisticated. In some cases, it is more convenient to consider subclasses of infinitely divisible distributions, like self-decomposable (SD) distributions, generalized gamma convolutions (GGC) and hyperbolically completely monotone (HCM) distributions. We refer to Steutel and van Harn \\cite{SH04}, Sato \\cite{Sat99} and Bondesson \\cite{Bon92} for abundant properties of these subclasses.\n\nIn 1987, \n Bondesson \\cite{Bon87} proved that the half Cauchy variable $ |\\mathcal{C}_2|$ is ID. In 1998, Di\\'edhiou \\cite{Die98} proved that $ |\\mathcal{C}_2|$ is SD. \nYano, Yano, and Yor proved $ |\\mathcal{C}_\\alpha|$ is ID for any $1 < \\alpha \\leq 2$ by Bondesson's argument, see, Theorem 2.7 in \\cite{YYY09}. Furthermore, they \nproposed three open questions (\\cite[Remark 2.9]{YYY09}):\n\\begin{enumerate}\n \\item Is $\\mathcal{C}_\\alpha$ SD (or ID at least)?\n \\item Is $|\\mathcal{C}_\\alpha|$ SD (or ID at least)?\n \\item Is $|\\mathcal{C}_\\alpha|^{-p}$ SD (or ID at least) for $p > 0$?\n\\end{enumerate}\n\nThere are very few results on the above questions. \nIn the present paper, we study the infinite divisibility of\n$ \\mathcal{C}_\\alpha$ and $ |\\mathcal{C}_\\alpha|^{p}, \\, p \\in \\mathbb{R}$, and partially give positive answers to the questions of Yano, Yano, and Yor.\n\n\\subsection{Infinite divisibility of \\texorpdfstring{$\\alpha$}{}-Cauchy variables}\nIt is well-known that the Cauchy variable $\\mathcal{C}_2$ is infinitely divisible and even stable. But when $\\alpha \\neq 2$, there is no known result on the infinite divisibility of $\\mathcal{C}_\\alpha$. The following theorem gives a positive answer to this question when $1 < \\alpha \\leq 6/5$.",
    "full_context": "For $\\alpha > 1$, the $\\alpha$-Cauchy random variable, denoted by $\\mathcal{C}_\\alpha$, was introduced by Yano, Yano and Yor \\cite{YYY09} in 2009. The density of $\\mathcal{C}_\\alpha$ is \n\\begin{equation}\\label{eq density alpha Cauchy}\n f_{\\mathcal{C}_\\alpha}(t) = \\frac{\\sin(\\pi/\\alpha)}{2\\pi/\\alpha}\\frac{1}{1+|t|^\\alpha}, \\quad \\alpha > 1, \\,\\, t \\in \\mathbb{R}. \n\\end{equation}\nIn particular, when $\\alpha = 2$, it is \nthe standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.\nIt is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\\mathcal{C}_\\alpha$.\n\nA random variable is infinitely divisible (ID) if for any positive integer $n$, it is the sum of $n$ independent and identically distributed random variables. Typical examples of infinitely divisible distributions are normal, Cauchy, half Cauchy, Stable, Gamma, Poisson, and Student-t. \nThe theory of infinitely divisible distribution is a classic topic in probability theory. In 1929, de Finetti first introduced the concept of infinite divisibility, and then Kolmogorov, Lévy, and Khintchine further developed the theory. Infinitely divisible distributions are closely related to limit theorems and the theory of Lévy processes and have important applications in finance, insurance, biology, physics, and signal processing. Proving or disproving infinite divisibility of a certain distribution can sometimes be quite sophisticated. In some cases, it is more convenient to consider subclasses of infinitely divisible distributions, like self-decomposable (SD) distributions, generalized gamma convolutions (GGC) and hyperbolically completely monotone (HCM) distributions. We refer to Steutel and van Harn \\cite{SH04}, Sato \\cite{Sat99} and Bondesson \\cite{Bon92} for abundant properties of these subclasses.\n\nIn 1987, \n Bondesson \\cite{Bon87} proved that the half Cauchy variable $ |\\mathcal{C}_2|$ is ID. In 1998, Di\\'edhiou \\cite{Die98} proved that $ |\\mathcal{C}_2|$ is SD. \nYano, Yano, and Yor proved $ |\\mathcal{C}_\\alpha|$ is ID for any $1 < \\alpha \\leq 2$ by Bondesson's argument, see, Theorem 2.7 in \\cite{YYY09}. Furthermore, they \nproposed three open questions (\\cite[Remark 2.9]{YYY09}):\n\\begin{enumerate}\n \\item Is $\\mathcal{C}_\\alpha$ SD (or ID at least)?\n \\item Is $|\\mathcal{C}_\\alpha|$ SD (or ID at least)?\n \\item Is $|\\mathcal{C}_\\alpha|^{-p}$ SD (or ID at least) for $p > 0$?\n\\end{enumerate}\n\nThere are very few results on the above questions. \nIn the present paper, we study the infinite divisibility of\n$ \\mathcal{C}_\\alpha$ and $ |\\mathcal{C}_\\alpha|^{p}, \\, p \\in \\mathbb{R}$, and partially give positive answers to the questions of Yano, Yano, and Yor.\n\n\\subsection{Infinite divisibility of \\texorpdfstring{$\\alpha$}{}-Cauchy variables}\nIt is well-known that the Cauchy variable $\\mathcal{C}_2$ is infinitely divisible and even stable. But when $\\alpha \\neq 2$, there is no known result on the infinite divisibility of $\\mathcal{C}_\\alpha$. The following theorem gives a positive answer to this question when $1 < \\alpha \\leq 6/5$.\n\n\\begin{abstract} \n We study the infinite divisibility of the $\\alpha$-Cauchy variable $\\mathcal{C}_\\alpha$, $\\alpha > 1$. The distribution of $\\mathcal{C}_2$ is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when $\\alpha \\neq 2$, there is no known result on the infinite divisibility of $\\mathcal{C}_\\alpha$. \n In this paper, we prove that $\\mathcal{C}_\\alpha$ is infinitely divisible if $1 < \\alpha \\leq 6/5$, and we give some sufficient conditions for $|\\mathcal{C}_\\alpha|^p, \\, p\\in \\mathbb{R},$ to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs, \n a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence. \n\\end{abstract}\n\n\\section{Introduction}\nFor $\\alpha > 1$, the $\\alpha$-Cauchy random variable, denoted by $\\mathcal{C}_\\a$, was introduced by Yano, Yano and Yor \\cite{YYY09} in 2009. The density of $\\mathcal{C}_\\a$ is \n\\begin{equation}\\label{eq density alpha Cauchy}\n f_{\\mathcal{C}_\\a}(t) = \\frac{\\sin(\\pi/\\a)}{2\\pi/\\a}\\frac{1}{1+|t|^\\a}, \\quad \\alpha > 1, \\,\\, t \\in \\mathbb{R}. \n\\end{equation}\nIn particular, when $\\alpha = 2$, it is \nthe standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.\nIt is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\\mathcal{C}_\\a$.\n\n\\subsection{Infinite divisibility of \\texorpdfstring{$\\a$}{}-Cauchy variables}\nIt is well-known that the Cauchy variable $\\mathcal{C}_2$ is infinitely divisible and even stable. But when $\\alpha \\neq 2$, there is no known result on the infinite divisibility of $\\mathcal{C}_\\a$. The following theorem gives a positive answer to this question when $1 < \\a \\leq 6/5$.\n\nYano, Yano and Yor introduced the $\\alpha$-Cauchy variable to study the first hitting times of point $T_{\\{a\\}}(X_\\alpha)$ for one-dimensional \\textbf{symmetric} stable L\\'evy process of index $\\alpha$ ($1< \\alpha \\leq 2$) starting from zero, denoted by $X_\\alpha = (X_\\alpha(t): t\\geq 0)$.\nIn 2019, Letemplier and Simon \\cite{LS19} studied the first hitting time of zero $\\tau(\\alpha, \\rho)$ for a real strictly $\\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\\tau(\\alpha, \\frac{1}{2})$ coincides with $T_{\\{-1\\}}(X_\\alpha)$ mentioned above. They conjectured that $\\tau(\\alpha, \\rho)$ is infinitely divisible for $\\alpha \\in (1,2]$.\n\nWe note that both $\\mathcal{C}_\\a$ and $|\\mathcal{C}_\\a|$ are not infinitely divisible when $\\a$ is large enough. Indeed, \nthe fractional moments of the half $\\alpha$-Cauchy variable $ |\\mathcal{C}_\\a|$ are of the form\n\\begin{equation*}\n \\mathbb{E}\\left[|\\mathcal{C}_\\a|^s\\right] = \\frac{\\sin(\\pi/\\a)}{\\pi} \\Gamma\\left(\\frac{1}{\\a} +\\frac{s}{\\a}\\right)\\Gamma\\left(1 - \\frac{1}{\\a} - \\frac{s}{\\a}\\right), \\quad -1 < s < \\a-1.\n\\end{equation*}\nThe right hand side tends to $1/(1+s)$ as $\\a$ tends to infinity, thus \n\\begin{equation}\n |\\mathcal{C}_\\a| \\claw \\mathcal{U}, \\qquad \\text{as} \\, \\,\\,\\a \\rightarrow \\infty, \n\\end{equation}\nwhere $\\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \\mathcal{U} $ is bounded, $\\mathcal{U}$ is not ID; see Sato \\cite[Corollary 24.4]{Sat99}. Similarly, as $\\alpha$ tends to infinity, $\\mathcal{C}_\\a$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID. \nIt is probable that $\\mathcal{C}_\\a$ is ID if and only if $1 < \\a \\leq 2$.\n\n\\begin{theorem}\\label{Theorem ID of half alpha Cauchy} \nLet $p \\in (0, \\infty)$. \nThe random variable $|\\mathcal{C}_\\a |^{ \\varepsilon p}, \\varepsilon = \\pm 1,$ is infinitely divisible if \n\\begin{equation}\\label{eq range p}\np \\geq \n \\begin{cases}\n (\\a + 1)/3 ,& \\quad \\varepsilon = 1, \\, 1 < \\a \\leq 2. \\\\\n \\a/2 ,& \\quad \\varepsilon = 1, \\, \\a > 2. \\\\\n \\a/2 ,& \\quad \\varepsilon = - 1, \\, 1 < \\a \\leq 2. \\\\\n (2\\a - 1)/3 ,& \\quad \\varepsilon = -1, \\, \\a > 2. \\\\\n \\end{cases}\n\\end{equation}\n\\end{theorem}\n\n\\section{Proofs of Theorems \\ref{Theorem ID of alpha Cauchy} and \\ref{Theorem ID of half alpha Cauchy}}\n\\label{section proofs}\n\\subsection{Proof of Theorem \\ref{Theorem ID of alpha Cauchy}}\nWe apply Proposition \\ref{Proposition Steutel coro VI.4.8} to prove Theorem \\ref{Theorem ID of alpha Cauchy}. Let $Y$ be a positive random variable with density $f$ given by \n$$ f(x) = \\frac{1}{2}(1+x)e^{-x}, \\qquad x >0. $$\nThe fractional moments of $Y$ are \n\\begin{equation*}\n \\mathbb{E}[Y^s] = \\frac{1}{2}(s+2)\\Gamma(s+1), \\quad s > -1. \n\\end{equation*}\n Let $1 < \\alpha \\leq 2$. \nRecall that \n\\begin{equation}\\label{eq Mellin alpha Cauchy}\n \\mathbb{E}\\left[|\\mathcal{C}_\\a|^s\\right] = \\frac{\\sin(\\pi/\\a)}{\\pi} \\Gamma\\left(\\frac{1}{\\a} +\\frac{s}{\\a}\\right)\\Gamma\\left(1 - \\frac{1}{\\a} - \\frac{s}{\\a}\\right), \\quad -1 < s < \\a-1.\n\\end{equation}\nTherefore, by Proposition \\ref{Proposition Steutel coro VI.4.8}, if there exists a symmetric distribution \n$V_\\alpha$ having fractional moments,\n\\begin{equation}\n \\mathbb{E} [|V_\\alpha|^s] = \\mathbb{E}\\left[|\\mathcal{C}_\\a|^s\\right] / \\mathbb{E}[Y^s] = \\frac{2\\alpha \\sin(\\pi/\\a)}{\\pi} \\frac{\\Gamma\\left(1+\\frac{1}{\\a} +\\frac{s}{\\a}\\right)\\Gamma\\left(1 - \\frac{1}{\\a} - \\frac{s}{\\a}\\right)}{\\Gamma(3+s)}, \n\\end{equation}\nthen $\\mathcal{C}_\\alpha$ is infinitely divisible. Compared with \\eqref{def Xabcd}, we see that \n\\begin{equation}\n |V_\\alpha| \\elaw \\X_{1+1/\\alpha, 1-1/\\alpha, 3, \\alpha}^{1/\\alpha}. \n\\end{equation}\nBy \\eqref{fact equiv}, the existence of $V_\\alpha$ is equivalent to the non-negativity of $E_{\\alpha, 2+\\alpha}^{2}(-t)$ for any $t>0$. \nBy formula (5.1.31) in \\cite{GKMR20}, we have\n\\begin{equation}\n\\label{LT of 3ML}\n \\int_0^{\\infty} e^{-st} e^{-t} t^{\\beta-1} E^\\gamma_{\\alpha, \\beta}(-t^\\alpha)dt = \\frac{(s+1)^{-\\beta}}{(1+(s+1)^{-\\alpha})^\\gamma}\n\\end{equation}\nwhere $\\alpha > 0, \\beta > 0, \\gamma > 0, s > 0$. \nTwo integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero, see, e.g. Proposition 1.2 in \\cite{SSV12}. Moreover, a function is completely monotone if and only if it is the Laplace transform of positive measure on $[0, \\infty)$ (see Theorem 1.4 in \\cite{SSV12}), and we have the following equivalence\n$$ \\text{ $E_{\\alpha, 2+\\alpha}^{2}(-t) \\geq 0$ for any $t>0$ $\\Longleftrightarrow$ \n$ \\frac{(s+1)^{-2-\\alpha}}{(1+(s+1)^{-\\alpha})^2} $ is completely monotone. } $$\nThe multiplication of completely monotone functions is still completely monotone, see \\cite[Corollary 1.6]{SSV12}. Therefore, the \ncomplete monotonicity of $\\frac{(s+1)^{-1 - \\alpha/2}}{1+(s+1)^{-\\alpha} }$ implies that of $\\frac{(s+1)^{-2-\\alpha}}{(1+(s+1)^{-\\alpha})^2}$. \n Using again \\eqref{LT of 3ML}, we obtain that \n$$ \\text{$E_{\\alpha, 1+\\alpha/2}(-t) \\geq 0$ for any $t>0$ } \\text{$\\Longleftrightarrow$ \n$\\frac{(s+1)^{-1 - \\alpha/2}}{1+(s+1)^{-\\alpha} }$ is completely monotone.} $$\nIf $1< \\alpha \\leq 6/5$, then $1+\\alpha/2 \\geq 4\\alpha/3$, by Theorem \\ref{Theorem admissible parameter domain}, we have \n$E_{\\alpha, 1+\\alpha/2}(-t) \\geq 0$ for any $t>0$. \nThis completes the proof.\n\nLet $\\T_{\\nu}$ be the Student-t random variable with density\n$$ \\frac{\\Gamma(\\frac{\\nu +1}{2})}{\\sqrt{\\pi \\nu} \\Gamma(\\frac{\\nu }{2})} \\left( 1 + \\frac{t^2}{\\nu}\\right)^{-\\frac{\\nu +1}{2}}, \\quad \\nu > 0, \\; t \\in \\mathbb{R}.$$\nNote that $|\\T_{1}| \\elaw |\\mathcal{C}_2|$. \n\\begin{theorem}\\label{thm half student}\n For $0 < \\nu \\leq 1$, the half Student-t random variable $|\\T_\\nu|$ is infinitely divisible.\n\\end{theorem}",
    "post_theorem_intro_text_len": 6056,
    "post_theorem_intro_text": "Yano, Yano and Yor introduced the $\\alpha$-Cauchy variable to study the first hitting times of point $T_{\\{a\\}}(X_\\alpha)$ for one-dimensional \\textbf{symmetric} stable L\\'evy process of index $\\alpha$ ($1< \\alpha \\leq 2$) starting from zero, denoted by $X_\\alpha = (X_\\alpha(t): t\\geq 0)$.\nIn 2019, Letemplier and Simon \\cite{LS19} studied the first hitting time of zero $\\tau(\\alpha, \\rho)$ for a real strictly $\\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\\tau(\\alpha, \\frac{1}{2})$ coincides with $T_{\\{-1\\}}(X_\\alpha)$ mentioned above. They conjectured that $\\tau(\\alpha, \\rho)$ is infinitely divisible for $\\alpha \\in (1,2]$. \n\nMore precisely, by formula (5.12) in \\cite{YYY09}, the relation between the first hitting time of point for stable processes and the alpha Cauchy distributions is as follows\n$$\\hat{X}_\\alpha ( T_{\\{a\\}}(X_\\alpha)) \\stackrel{(d)}{=} |a| \\mathcal{C}_\\alpha, $$\nwhere $\\hat{X}_\\alpha$ is an independent copy of $X_\\alpha$. Using Bochner’s subordination -see e.g. Theorem 30.1 in \\cite{Sat99}- the infinite divisibility of $T_{\\{a\\}}(X_\\alpha)$ would entail the infinite divisibility of $ \\mathcal{C}_\\alpha$. Vise versa, the infinite divisibility of $\\mathcal{C}_\\alpha$ would support the conjecture on the infinite divisibility of $T_{\\{a\\}}(X_\\alpha)$. \n\nWe note that both $\\mathcal{C}_\\alpha$ and $|\\mathcal{C}_\\alpha|$ are not infinitely divisible when $\\alpha$ is large enough. Indeed, \nthe fractional moments of the half $\\alpha$-Cauchy variable $ |\\mathcal{C}_\\alpha|$ are of the form\n\\begin{equation*}\n    \\mathbb{E}\\left[|\\mathcal{C}_\\alpha|^s\\right] =  \\frac{\\sin(\\pi/\\alpha)}{\\pi} \\Gamma\\left(\\frac{1}{\\alpha} +\\frac{s}{\\alpha}\\right)\\Gamma\\left(1 - \\frac{1}{\\alpha} - \\frac{s}{\\alpha}\\right), \\quad -1 < s < \\alpha-1.\n\\end{equation*}\nThe right hand side tends to $1/(1+s)$ as $\\alpha$ tends to infinity, thus \n\\begin{equation}\n    |\\mathcal{C}_\\alpha| \\stackrel{(d)}{\\longrightarrow} \\mathcal{U}, \\qquad \\text{as} \\, \\,\\,\\alpha \\rightarrow \\infty, \n\\end{equation}\nwhere $\\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \\mathcal{U} $ is bounded, $\\mathcal{U}$ is not ID; see Sato \\cite[Corollary 24.4]{Sat99}. Similarly, as $\\alpha$ tends to infinity, $\\mathcal{C}_\\alpha$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID. \nIt is probable that $\\mathcal{C}_\\alpha$ is ID if and only if  $1 < \\alpha \\leq 2$. \n\n\\subsection{Infinite divisibility of powers of half \\texorpdfstring{$\\alpha$}{}-Cauchy variable} \nWe know from Bondesson \\cite{Bon87} that $ |\\mathcal{C}_\\alpha|$ is ID for $1 < \\alpha \\leq 2$. We have seen that $|\\mathcal{C}_\\alpha|$ is not infinitely divisible when $\\alpha$ is large enough. It is also probable that $|\\mathcal{C}_\\alpha|$ is ID if and only if  $1 < \\alpha \\leq 2$. \n\nThe following theorem describes the infinite divisibility of powers of half $\\alpha$-Cauchy variable. \n\n\\begin{theorem}\\label{Theorem ID of half alpha Cauchy} \nLet $p \\in (0, \\infty)$. \nThe random variable $|\\mathcal{C}_\\alpha |^{ \\varepsilon p}, \\varepsilon = \\pm 1,$ is infinitely divisible if \n\\begin{equation}\\label{eq range p}\np \\geq \n    \\begin{cases}\n   (\\alpha + 1)/3 ,& \\quad \\varepsilon = 1, \\, 1 < \\alpha \\leq 2. \\\\\n     \\alpha/2 ,& \\quad \\varepsilon = 1, \\, \\alpha > 2. \\\\\n     \\alpha/2 ,& \\quad \\varepsilon = - 1, \\, 1 < \\alpha \\leq 2. \\\\\n     (2\\alpha - 1)/3 ,& \\quad \\varepsilon = -1, \\, \\alpha > 2. \\\\\n    \\end{cases}\n\\end{equation}\n\\end{theorem}\n\nIn particular, this theorem recovers and extends the result of Bondesson mentioned above.\n\n\\subsection{Sketch of the proofs} We first sketch the proof of Theorem \\ref{Theorem ID of alpha Cauchy}. \nWe will prove the infinite divisibility of $\\alpha$-Cauchy ($1<\\alpha \\leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition \\ref{Proposition Steutel coro VI.4.8}. \nThis motivates us to study the existence condition of a class of positive random variables having moments of Gamma type satisfying\n       \\begin{equation}\\label{Introduction Xabcd}\n        \\mathbb{E}(\\X_{a,b,c,d}^s) =  \\frac{ \\Gamma(c) }{ \\Gamma(a)\\Gamma(b)} \\frac{ \\Gamma(a + s)\\Gamma(b-s)}{ \\Gamma(c+ d s) }, \\quad -a < s < b,\n    \\end{equation} \nfor positive $a,b,c,d$. We finish the proof by non-negativity conditions of the Mittag-Leffler function. \n\nWe then sketch the proof of Theorem \\ref{Theorem ID of half alpha Cauchy}.\nBy Kristiansen's Theorem (see Theorem \\ref{Kristiansen's Theorem}), the existence of a random variable $X$ satisfying \n         \\begin{equation}\\label{Introduction X}\n        \\mathbb{E}(X^s) =  \\frac{\\sin(\\pi/\\alpha)}{\\pi}\\frac{ \\Gamma\\left(\\frac{1}{\\alpha} +\\frac{ \\varepsilon p s}{\\alpha}\\right)\\Gamma\\left(1 - \\frac{1}{\\alpha} - \\frac{ \\varepsilon p s}{\\alpha}\\right)}{ \\Gamma(2 + s) }, \n    \\end{equation} \nfor $s$ in some interval, will imply the infinite divisibility of $|\\mathcal{C}_\\alpha|^{\\varepsilon p}, \\varepsilon = \\pm 1$. Therefore, we thoroughly investigate the conditions for the existence of random variables of the form \\eqref{Introduction Xabcd}, and formulate our results as Theorem \\ref{thm Gamma type}. Applying this theorem to \\eqref{Introduction X}, we can prove Theorem \\ref{Theorem ID of half alpha Cauchy}. \n\nThe rest of this paper is organized as follows. \n In Section \\ref{section Preliminaries}, we introduce the three-parametric Mittag-Leffler function. We then state some\nuseful known propositions and theorems that will be used for the proofs of our theorems. \nIn Section \\ref{section Gamma type},\nwe explicitly state and prove some necessary conditions and some sufficient conditions for the existence of some random variables having moments of Gamma type satisfying \\eqref{Introduction Xabcd}, which serve as the foundation for proving the above two theorems. In Section \\ref{section proofs}, we prove Theorems \\ref{Theorem ID of alpha Cauchy} and \\ref{Theorem ID of half alpha Cauchy}. In section \\ref{section apllication of new Gamma type}, we give three other applications of Theorem \\ref{thm Gamma type}.",
    "sketch": "To prove Theorem~\\ref{Theorem ID of alpha Cauchy}, they \"prove the infinite divisibility of $\\alpha$-Cauchy ($1<\\alpha \\leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition~\\ref{Proposition Steutel coro VI.4.8}.\" This leads them to \"study the existence condition of a class of positive random variables having moments of Gamma type\" with\n\\[\n\\mathbb{E}(\\X_{a,b,c,d}^s) =  \\frac{ \\Gamma(c) }{ \\Gamma(a)\\Gamma(b)} \\frac{ \\Gamma(a + s)\\Gamma(b-s)}{ \\Gamma(c+ d s) }, \\quad -a < s < b,\n\\]\nfor positive $a,b,c,d$. They \"finish the proof by non-negativity conditions of the Mittag-Leffler function.\"",
    "expanded_sketch": "To prove the main theorem, they \"prove the infinite divisibility of $\\alpha$-Cauchy ($1<\\alpha \\leq 6/5$) by proving that it is a sym-gamma(2) mixture\". We use the following proposition.\n\n\\begin{proposition}\n    \\label{Proposition Steutel coro VI.4.8}{\\cite[Corollary VI.4.8]{SH04}}\n    A random variable $X$ with $X \\elaw Z \\times Y$ is infinitely divisible\nif $Z$ and $Y$ are independent, $Z$ has a symmetric distribution and $Y$ has the \ndensity $f$ given by \n$$ f(x) = \\frac{1}{2}(1+x)e^{-x}, \\qquad x >0.  $$\n\\end{proposition}\n\nThis leads them to \"study the existence condition of a class of positive random variables having moments of Gamma type\" with\n\\[\n\\mathbb{E}(\\X_{a,b,c,d}^s) =  \\frac{ \\Gamma(c) }{ \\Gamma(a)\\Gamma(b)} \\frac{ \\Gamma(a + s)\\Gamma(b-s)}{ \\Gamma(c+ d s) }, \\quad -a < s < b,\n\\]\nfor positive $a,b,c,d$. They \"finish the proof by non-negativity conditions of the Mittag-Leffler function.\"",
    "expanded_theorem": "\\label{Theorem ID of alpha Cauchy}\nThe $\\alpha$-Cauchy variable $\\mathcal{C}_\\alpha$ is infinitely divisible if $1 < \\alpha \\leq 6/5$.",
    "theorem_type": [
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(\\alpha\\) satisfy \\(1<\\alpha\\le 6/5\\), and let \\(\\mathcal C_\\alpha\\) denote the \\(\\alpha\\)-Cauchy random variable with density\n\\[\nf_{\\mathcal C_\\alpha}(t)=\\frac{\\sin(\\pi/\\alpha)}{2\\pi/\\alpha}\\frac{1}{1+|t|^\\alpha},\\qquad t\\in\\mathbb R.\n\\]\nWhich of the following conclusions about \\(\\mathcal C_\\alpha\\) is valid under these hypotheses?",
      "correct_choice": {
        "label": "A",
        "text": "The random variable \\(\\mathcal C_\\alpha\\) is infinitely divisible; that is, for every positive integer \\(n\\), its distribution can be written as the distribution of a sum of \\(n\\) independent and identically distributed random variables."
      },
      "choices": [
        {
          "label": "B",
          "text": "The random variable \\(\\mathcal C_\\alpha\\) is infinitely divisible for every \\(\\alpha>1\\); that is, for every positive integer \\(n\\), its distribution can be written as the distribution of a sum of \\(n\\) independent and identically distributed random variables."
        },
        {
          "label": "C",
          "text": "There exists at least one positive integer \\(n\\ge 2\\) such that the distribution of \\(\\mathcal C_\\alpha\\) can be written as the distribution of a sum of \\(n\\) independent and identically distributed random variables."
        },
        {
          "label": "D",
          "text": "The random variable \\(\\mathcal C_\\alpha\\) is self-decomposable; equivalently, for every \\(c\\in(0,1)\\), there exists an independent random variable \\(Y_c\\) such that \\(\\mathcal C_\\alpha\\stackrel{d}=c\\mathcal C_\\alpha+Y_c\\)."
        },
        {
          "label": "E",
          "text": "For each \\(1<\\alpha\\le 6/5\\), there exists a decomposition \\(\\mathcal C_\\alpha\\stackrel{d}=Z\\times Y\\) with \\(Z\\) and \\(Y\\) independent, \\(Y\\) having density \\(\\tfrac12(1+x)e^{-x}\\) on \\((0,\\infty)\\), and \\(Z\\) symmetric and itself infinitely divisible."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "characteristic",
          "tampered_component": "alpha-range-6/5-threshold",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "for-every-n quantifier in infinite divisibility weakened to existence of one n",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "regularity",
          "tampered_component": "ID conclusion replaced by stronger self-decomposability",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "Steutel product criterion only requires symmetric Z, not Z itself infinitely divisible",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem states only the definition and parameter range; it does not explicitly reveal infinite divisibility or otherwise give away choice A. The threshold 6/5 may hint at a known theorem, but there is no direct answer leakage."
      },
      "TAS": {
        "score": 0,
        "justification": "Choice A is essentially a direct restatement of the underlying theorem for the given hypothesis range, so the item mainly tests recognition/recall rather than selecting among genuinely competing derived conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because one must distinguish the exact theorem statement from a stronger overgeneralization (B), a weaker true statement (C), and stronger/adjacent properties (D, E). However, the main task is still theorem identification rather than substantial generation of a conclusion."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and target common errors: extending the parameter range too far, settling for a weaker consequence, confusing infinite divisibility with self-decomposability, or accepting an overly strong technical decomposition statement."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically polished item with strong distractors and little answer leakage, but it is largely tautological because the correct option restates the theorem almost verbatim."
    }
  },
  {
    "id": "2512.23224v1",
    "paper_link": "http://arxiv.org/abs/2512.23224v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "ithm",
      "content": "[$=$\\,Theorem~\\ref{thm:QK-Whitney_A}\\,$+$\\,Theorem~\\ref{thm:QK-Whitney_B}] \\label{ithm:QK-Whitney}\n\\begin{enu}\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_1_intro}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2_intro}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3_intro}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\underline{\\BC}^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\n\\end{enu}",
      "start_pos": 10898,
      "end_pos": 12139,
      "label": "ithm:QK-Whitney"
    },
    "ref_dict": {
      "thm:QK-Whitney_B": "\\begin{thm} \\label{thm:QK-Whitney_B}\nIn $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\\end{thm}",
      "eq:QK-Whitney_1_intro": "\\begin{equation} \\label{eq:QK-Whitney_1_intro}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}",
      "thm:Whitney_presentation": "\\begin{thm} \\label{thm:Whitney_presentation}\nThere exists an $R(T)$-algebra isomorphism $R_{Q}^{\\SW}/I_{Q}^{\\SW} \\xrightarrow{\\sim} QK_{T}(\\FlC{n})$ such that \n\\begin{align}\nF_{d}^{k} & \\mapsto \\left[ \\extprod^{d} \\CS_{k} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nG_{d}^{k} & \\mapsto \\left[ \\extprod^{d} \\CS_{k}^{\\vee} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nx_{j} & \\mapsto [\\CS_{j}/\\CS_{j-1}] & & \\text{for $1 \\le j \\le n$}, \\\\ \ny_{j} & \\mapsto [(\\CS_{j}/\\CS_{j-1})^{\\vee}] & & \\text{for $1 \\le j \\le n$}. \n\\end{align}\n\\end{thm}",
      "thm:QK-Whitney_A": "\\begin{thm} \\label{thm:QK-Whitney_A}\n\\begin{enu}\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_1}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\\end{enu}\n\\end{thm}",
      "prop:QK-Borel_rel": "\\begin{prop}[{\\cite[Corollary~6.4]{KN2}}] \\label{prop:QK-Borel_rel}\nIn $QK_{T}(\\FlC{n})$, for $0 \\le d \\le 2n$, we have \n\\begin{equation} \\label{eq:QK-Borel_rel}\n\\sum_{\\substack{J \\subset [1, \\overline{1}] \\\\ |J|=d}} \\left( \\prod_{1 \\le j \\le \\overline{1}} \\varphi_{J}(j) \\right) \\left( \\qprod_{j \\in J} [\\CO_{G/B}(-\\ve_{j})] \\right) = \\left[ \\extprod^{d} \\udBC^{2n} \\right]. \n\\end{equation}\n\\end{prop}",
      "cor:multiple_line_bundles": "\\begin{cor} \\label{cor:multiple_line_bundles}\nAssume that $G = \\Sp_{2n}(\\BC)$. \nLet $2 \\le k \\le n$ and take $J \\subset [1, k-1]$. Then we have \n\\begin{align}\n[\\CO_{G/B}(-\\ve_{J \\sqcup \\{k\\}})] &= \\begin{cases} \\dfrac{1}{1-Q_{k-1}} [\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(-\\ve_{k})] & \\text{if $k-1 \\in J$}, \\\\[5mm]\n[\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(-\\ve_{k})] & \\text{if $k-1 \\notin J$}, \\end{cases} \\label{eq:multiple_1} \\\\ \n[\\CO_{G/B}(\\ve_{J \\sqcup \\{k\\}})] &= \\begin{cases} \\dfrac{1}{1-Q_{k-1}} [\\CO_{G/B}(\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{k})] & \\text{if $k-1 \\in J$}, \\\\[5mm]\n[\\CO_{G/B}(\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{k})] & \\text{if $k-1 \\notin J$}. \\end{cases} \\label{eq:multiple_2}\n\\end{align}\n\\end{cor}",
      "eq:Whitney": "\\begin{equation} \\label{eq:Whitney}\n\\lambda_{y}(\\CE_{1}) \\cdot \\lambda_{y}(\\CE_{3}) = \\lambda_{y}(\\CE_{2}) \n\\end{equation}",
      "eq:QK-Whitney_2_intro": "\\begin{equation} \\label{eq:QK-Whitney_2_intro}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}",
      "eq:QK-Whitney_typeA": "\\begin{equation} \\label{eq:QK-Whitney_typeA}\n\\lambda_{y}(\\CS_{j}) \\star \\lambda_{y}(\\CS_{j+1}/\\CS_{j}) = \\lambda_{y}(\\CS_{j+1}) - y^{r_{j+1}-r_{j}} \\dfrac{Q_{j}}{1-Q_{j}} \\det(\\CS_{j+1}/\\CS_{j}) \\star (\\lambda_{y}(\\CS_{j}) - \\lambda_{y}(\\CS_{j-1})) \n\\end{equation}",
      "thm:Nakayama": "\\begin{thm}[{\\cite[Theorem~4.1]{GMSXZZ2}}] \\label{thm:Nakayama}\n\nAssume that we are given an isomorphism of $R(T)$-algebras\n\\begin{equation}\n\\Phi: R(T)[m_{1}, \\ldots, m_{s}]/\\langle P_{1}, \\ldots, P_{r} \\rangle \\xrightarrow{\\sim} K_{T}(G/P)\n\\end{equation}\nwhere $m_{1}, \\ldots, m_{s}$ are indeterminates, and the $P_{i}$ are polynomials in the variables $m_{j}$ with coefficients in $R(T)$. \nConsider any power series\n\\begin{equation}\n\\ti{P}_{i} = \\ti{P}_{i}(m_{1}, \\ldots, m_{s}; Q_{1}, \\ldots, Q_{k}) \\in (R(T)[m_{1}, \\ldots, m_{s}])\\bra{Q_{1}, \\ldots, Q_{k}}\n\\end{equation}\nsuch that \n\\begin{itemize}\n\\item $\\ti{P}_{i}(m_{1}, \\ldots, m_{s}; 0, \\ldots, 0) = P_{i}(m_{1}, \\ldots, m_{s})$ for all $m_{1}, \\ldots, m_{s}$ and all $i$, and \n\\item $\\ti{P}_{i}(m_{1}, \\ldots, m_{s}; Q_{1}, \\ldots, Q_{k}) = 0$ in $QK_{T}(G/P)$, where the $m_{i}$ are regarded as elements in $QK_{T}(G/P)$ via the isomorphism $\\Phi$. \n\\end{itemize}\nThen $\\Phi$ lifts to an isomorphism of $R(T)\\bra{Q_{1}, \\ldots, Q_{k}}$-algebras\n\\begin{equation}\n\\ti{\\Phi}: (R(T)\\bra{Q_{1}, \\ldots, Q_{k}})[m_{1}, \\ldots, m_{s}]/\\langle \\ti{P}_{1}, \\ldots, \\ti{P}_{r} \\rangle \\xrightarrow{\\sim} QK_{T}(G/P). \n\\end{equation}\n\\end{thm}",
      "eq:QK-Whitney_3_intro": "\\begin{equation} \\label{eq:QK-Whitney_3_intro}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}",
      "ithm:Whitney_presentation": "\\begin{ithm}[$=$\\,Theorem~\\ref{thm:Whitney_presentation}] \\label{ithm:Whitney_presentation}\nDefine a ring $R_{Q}^{\\SW}$ by \n\\begin{equation}\nR_{Q}^{\\SW} := (R(T)\\bra{Q_{1}, \\ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \\mid 1 \\le k \\le n, \\ 1 \\le d \\le k, \\ 1 \\le j \\le n]. \n\\end{equation}\nTake an ideal $I_{Q}^{\\SW}$ of $R_{Q}^{\\SW}$ generated by \n\\begin{align}\n& F_{d}^{k} - \\left( F_{d}^{k-1} + \\frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \\right) \\quad \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \n& G_{d}^{k} - \\left( G_{d}^{k-1} + \\frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \\right) \\quad \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \n\\begin{split}\n&\\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \\Biggl( e_{d}(T_{1}, T_{2}, \\ldots, T_{n}, T_{n}^{-1}, \\ldots, T_{2}^{-1}, T_{1}^{-1}) \\\\ \n& \\quad - \\sum_{p = 1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} \\sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \\Biggr) \\quad \\text{for $1 \\le d \\le n$}, \n\\end{split} \\\\ \n& F_{1}^{1} - x_{1}, \\\\\n& G_{1}^{1} - y_{1}, \\\\ \n& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \\quad \\text{for $1 \\le j \\le n$}; \n\\end{align}\nhere, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \\in R(T)$ for $1 \\le k \\le 2n$ are obtained from the decomposition of $\\BC^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \\ldots$, $T_{2n} = T_{1}^{-1}$. \nIn addition, we understand that \n$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \\le k \\le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \\le k \\le n$. \n\nThen there exists an $R(T)$-algebra isomorphism $R_{Q}^{\\SW}/I_{Q}^{\\SW} \\xrightarrow{\\sim} QK_{T}(\\FlC{n})$ such that \n\\begin{align}\nF_{d}^{k} & \\mapsto \\left[ \\extprod^{d} \\CS_{k} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nG_{d}^{k} & \\mapsto \\left[ \\extprod^{d} \\CS_{k}^{\\vee} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nx_{j} & \\mapsto [\\CS_{j}/\\CS_{j-1}] & & \\text{for $1 \\le j \\le n$}, \\\\ \ny_{j} & \\mapsto [(\\CS_{j}/\\CS_{j-1})^{\\vee}] & & \\text{for $1 \\le j \\le n$}. \n\\end{align}\n\\end{ithm}",
      "ithm:QK-Whitney": "\\begin{ithm}[$=$\\,Theorem~\\ref{thm:QK-Whitney_A}\\,$+$\\,Theorem~\\ref{thm:QK-Whitney_B}] \\label{ithm:QK-Whitney}\n\\begin{enu}\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_1_intro}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2_intro}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3_intro}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\n\\end{enu}\n\\end{ithm}"
    },
    "pre_theorem_intro_text_len": 5747,
    "pre_theorem_intro_text": "Let $X$ be a partial flag manifold of arbitrary type. \nGivental \\cite{Givental} and Lee \\cite{Lee} introduced the \\emph{torus-equivariant quantum $K$-ring} $QK_{T}(X)$ of the flag manifold $X$, \nwhich is a deformation of the ordinary torus-equivariant $K$-ring $K_{T}(X)$; here $T$ denotes the torus. \nThe product $\\star$ of $QK_{T}(X)$, called the \\emph{quantum product}, is defined by using \\emph{$K$-theoretic Gromov--Witten invariants}. \nTo understand the quantum product, presentations of $QK_{T}(X)$ with generators and relations have been studied, including Borel-type, Coulomb branch, and Whitney presentations. In this paper, we focus on the Whitney presentation. \n\nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $X$, we denote by $[\\mathcal{E}]$ the class of $\\mathcal{E}$ in $K_{T}(X)$. \nThe \\emph{Hurzebruch $\\lambda_{y}$-class} $\\lambda_{y}(\\mathcal{E}) \\in K_{T}(X)[y]$ of a $T$-equivariant vector bundle $\\mathcal{E}$, introduced in \\cite{Hirzebruch}, is defined as \n\\begin{equation}\n\\lambda_{y}(\\mathcal{E}) := 1 + y [\\mathcal{E}] + y^{2} \\left[ \\bigwedge\\nolimits^{2} \\mathcal{E} \\right] + \\cdots + y^{\\rk(\\mathcal{E})} \\left[ \\bigwedge\\nolimits^{\\rk(\\mathcal{E})} \\mathcal{E} \\right],  \n\\end{equation}\nwhere $\\rk(\\mathcal{E})$ denotes the rank of $\\mathcal{E}$. \nFor a short exact sequence $0 \\rightarrow \\CE_{1} \\rightarrow \\CE_{2} \\rightarrow \\CE_{3} \\rightarrow 0$ of torus-equivariant vector bundles over $X$, we have \n\\begin{equation} \\label{eq:Whitney}\n\\lambda_{y}(\\CE_{1}) \\cdot \\lambda_{y}(\\CE_{3}) = \\lambda_{y}(\\CE_{2}) \n\\end{equation}\nin $K_{T}(X)[y]$, where $\\cdot$ denotes the product of $K_{T}(X)$ (or $K_{T}(X)[y]$).\nThe identity \\eqref{eq:Whitney} is called the \\emph{$K$-theoretic Whitney relation}. \n\nOur interest is to quantize the above $K$-theoretic Whitney relations.\nMore precisely, we study the quantum product $\\lambda_{y}(\\CE_{1}) \\star \\lambda_{y}(\\CE_{3})$ in $QK_{T}(X)[y]$. \nLet us consider the case $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ is a $k$-step flag manifold of type $A$ parametrizing all sequences $(0 \\subset V_{1} \\subset \\cdots \\subset V_{k} \\subset \\mathbb{C}^{n})$ of vector subspaces of $\\mathbb{C}^{n}$ with $\\dim V_{i} = r_{i}$ for $1 \\le i \\le k$.\nThen we can take a tautological filtration $0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{k} \\subset \\underline{\\BC}^{n}$ of the trivial bundle $\\underline{\\BC}^{n} = X \\times \\mathbb{C}^{n}$. \nGu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}) conjectured the result of the quantum product of $\\lambda_{y}$-classes associated to tautological vector bundles as follows: \n\\begin{equation} \\label{eq:QK-Whitney_typeA}\n\\lambda_{y}(\\CS_{j}) \\star \\lambda_{y}(\\CS_{j+1}/\\CS_{j}) = \\lambda_{y}(\\CS_{j+1}) - y^{r_{j+1}-r_{j}} \\dfrac{Q_{j}}{1-Q_{j}} \\det(\\CS_{j+1}/\\CS_{j}) \\star (\\lambda_{y}(\\CS_{j}) - \\lambda_{y}(\\CS_{j-1})) \n\\end{equation}\nfor $1 \\le j \\le k$, here $Q_{1}, \\ldots, Q_{k}$ are Novikov variables of $QK_{T}(X)$, and $\\lambda_{y}(\\CS_{n+1}) = \\lambda_{y}(\\underline{\\BC}^{n}) = \\prod_{1 \\le j \\le n} (1 + yT_{j})$, where $T_{j} \\in R(T)$ for $1 \\le j \\le n$ are given by the decomposition of $\\mathbb{C}^{n}$ into one-dimensional $T$-modules (see \\cite[Conjecture~1.1]{GMSXZZ}). \nThis identity is called the \\emph{quantum $K$-theoretic Whitney relation} (the \\emph{$QK$-Whitney relation} for short). \nThis $QK$-Whitney relation was proved for Grassmannians $X = \\mathrm{Gr}(k, n) := \\Fl(k; n)$ by Gu, Mihalcea, Sharpe, and Zou (\\cite{GMSZ2}), for $1 \\le k \\le n$, for the incidence variety $X = \\Fl(1, n-1; n)$ by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}), and for arbitrary partial flag manifolds in type $A$ by Huq-Kuruvilla (\\cite{HK}). \nIn addition, Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou showed in \\cite[Theorem~1.4]{GMSXZZ} that this conjecture holds under the quantum $K$-theoretic divisor axiom stated by Buch and Mihalcea, the identity for three-pointed $K$-theoretic Gromov--Witten invariants. \nRecently, Lenart, Naito, Sagaki, and Xu proved the quantum $K$-theoretic divisor axiom in \\cite{LNSX} (not only for $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ but also for arbitrary $X$). \nAs a benefit of describing $QK$-Whitney relations, we can obtain a presentation of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}; n))$ as a quotient ring of a polynomial ring. In fact, $QK$-Whitney relations give a complete set of defining relations of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}, n))$ (\\cite[Corollary~5.4]{GMSXZZ2}). \n\nSince the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \\cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$. \nLet $X = \\FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences \n\\begin{equation}\n(0 \\subset V_{1} \\subset \\cdots \\subset V_{n} \\subset \\mathbb{C}^{2n})\n\\end{equation}\nof isotropic subspaces $V_{1}, \\ldots, V_{n}$ with respect to a fixed symplectic form with $\\dim V_{k} = k$ for $1 \\le k \\le n$. \nThen we can take a tautological filtration\n\\begin{equation}\n0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{n} \\subset \\underline{\\BC}^{2n}\n\\end{equation}\nof subbundles of the trivial bundle $\\underline{\\BC}^{2n} = \\FlC{n} \\times \\mathbb{C}^{2n}$ over $\\FlC{n}$ such that $\\rk(\\CS_{k}) = k$ for $1 \\le k \\le n$. \nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $\\FlC{n}$, we denote by $\\mathcal{E}^{\\vee}$ its dual bundle. \nWe denote by $Q_{1}, \\ldots, Q_{n}$ the Novikov variables of $QK_{T}(\\FlC{n})$. We understand that $Q_{0} = 0$ and $\\CS_{-1} = 0$. \nOur main results are as follows.",
    "context": "Let $X$ be a partial flag manifold of arbitrary type. \nGivental \\cite{Givental} and Lee \\cite{Lee} introduced the \\emph{torus-equivariant quantum $K$-ring} $QK_{T}(X)$ of the flag manifold $X$, \nwhich is a deformation of the ordinary torus-equivariant $K$-ring $K_{T}(X)$; here $T$ denotes the torus. \nThe product $\\star$ of $QK_{T}(X)$, called the \\emph{quantum product}, is defined by using \\emph{$K$-theoretic Gromov--Witten invariants}. \nTo understand the quantum product, presentations of $QK_{T}(X)$ with generators and relations have been studied, including Borel-type, Coulomb branch, and Whitney presentations. In this paper, we focus on the Whitney presentation.\n\nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $X$, we denote by $[\\mathcal{E}]$ the class of $\\mathcal{E}$ in $K_{T}(X)$. \nThe \\emph{Hurzebruch $\\lambda_{y}$-class} $\\lambda_{y}(\\mathcal{E}) \\in K_{T}(X)[y]$ of a $T$-equivariant vector bundle $\\mathcal{E}$, introduced in \\cite{Hirzebruch}, is defined as \n\\begin{equation}\n\\lambda_{y}(\\mathcal{E}) := 1 + y [\\mathcal{E}] + y^{2} \\left[ \\bigwedge\\nolimits^{2} \\mathcal{E} \\right] + \\cdots + y^{\\rk(\\mathcal{E})} \\left[ \\bigwedge\\nolimits^{\\rk(\\mathcal{E})} \\mathcal{E} \\right], \n\\end{equation}\nwhere $\\rk(\\mathcal{E})$ denotes the rank of $\\mathcal{E}$. \nFor a short exact sequence $0 \\rightarrow \\CE_{1} \\rightarrow \\CE_{2} \\rightarrow \\CE_{3} \\rightarrow 0$ of torus-equivariant vector bundles over $X$, we have \n\\begin{equation} \\label{eq:Whitney}\n\\lambda_{y}(\\CE_{1}) \\cdot \\lambda_{y}(\\CE_{3}) = \\lambda_{y}(\\CE_{2}) \n\\end{equation}\nin $K_{T}(X)[y]$, where $\\cdot$ denotes the product of $K_{T}(X)$ (or $K_{T}(X)[y]$).\nThe identity \\eqref{eq:Whitney} is called the \\emph{$K$-theoretic Whitney relation}.\n\nOur interest is to quantize the above $K$-theoretic Whitney relations.\nMore precisely, we study the quantum product $\\lambda_{y}(\\CE_{1}) \\star \\lambda_{y}(\\CE_{3})$ in $QK_{T}(X)[y]$. \nLet us consider the case $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ is a $k$-step flag manifold of type $A$ parametrizing all sequences $(0 \\subset V_{1} \\subset \\cdots \\subset V_{k} \\subset \\mathbb{C}^{n})$ of vector subspaces of $\\mathbb{C}^{n}$ with $\\dim V_{i} = r_{i}$ for $1 \\le i \\le k$.\nThen we can take a tautological filtration $0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{k} \\subset \\underline{\\BC}^{n}$ of the trivial bundle $\\underline{\\BC}^{n} = X \\times \\mathbb{C}^{n}$. \nGu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}) conjectured the result of the quantum product of $\\lambda_{y}$-classes associated to tautological vector bundles as follows: \n\\begin{equation} \\label{eq:QK-Whitney_typeA}\n\\lambda_{y}(\\CS_{j}) \\star \\lambda_{y}(\\CS_{j+1}/\\CS_{j}) = \\lambda_{y}(\\CS_{j+1}) - y^{r_{j+1}-r_{j}} \\dfrac{Q_{j}}{1-Q_{j}} \\det(\\CS_{j+1}/\\CS_{j}) \\star (\\lambda_{y}(\\CS_{j}) - \\lambda_{y}(\\CS_{j-1})) \n\\end{equation}\nfor $1 \\le j \\le k$, here $Q_{1}, \\ldots, Q_{k}$ are Novikov variables of $QK_{T}(X)$, and $\\lambda_{y}(\\CS_{n+1}) = \\lambda_{y}(\\underline{\\BC}^{n}) = \\prod_{1 \\le j \\le n} (1 + yT_{j})$, where $T_{j} \\in R(T)$ for $1 \\le j \\le n$ are given by the decomposition of $\\mathbb{C}^{n}$ into one-dimensional $T$-modules (see \\cite[Conjecture~1.1]{GMSXZZ}). \nThis identity is called the \\emph{quantum $K$-theoretic Whitney relation} (the \\emph{$QK$-Whitney relation} for short). \nThis $QK$-Whitney relation was proved for Grassmannians $X = \\mathrm{Gr}(k, n) := \\Fl(k; n)$ by Gu, Mihalcea, Sharpe, and Zou (\\cite{GMSZ2}), for $1 \\le k \\le n$, for the incidence variety $X = \\Fl(1, n-1; n)$ by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}), and for arbitrary partial flag manifolds in type $A$ by Huq-Kuruvilla (\\cite{HK}). \nIn addition, Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou showed in \\cite[Theorem~1.4]{GMSXZZ} that this conjecture holds under the quantum $K$-theoretic divisor axiom stated by Buch and Mihalcea, the identity for three-pointed $K$-theoretic Gromov--Witten invariants. \nRecently, Lenart, Naito, Sagaki, and Xu proved the quantum $K$-theoretic divisor axiom in \\cite{LNSX} (not only for $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ but also for arbitrary $X$). \nAs a benefit of describing $QK$-Whitney relations, we can obtain a presentation of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}; n))$ as a quotient ring of a polynomial ring. In fact, $QK$-Whitney relations give a complete set of defining relations of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}, n))$ (\\cite[Corollary~5.4]{GMSXZZ2}).\n\nSince the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \\cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$. \nLet $X = \\FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences \n\\begin{equation}\n(0 \\subset V_{1} \\subset \\cdots \\subset V_{n} \\subset \\mathbb{C}^{2n})\n\\end{equation}\nof isotropic subspaces $V_{1}, \\ldots, V_{n}$ with respect to a fixed symplectic form with $\\dim V_{k} = k$ for $1 \\le k \\le n$. \nThen we can take a tautological filtration\n\\begin{equation}\n0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{n} \\subset \\underline{\\BC}^{2n}\n\\end{equation}\nof subbundles of the trivial bundle $\\underline{\\BC}^{2n} = \\FlC{n} \\times \\mathbb{C}^{2n}$ over $\\FlC{n}$ such that $\\rk(\\CS_{k}) = k$ for $1 \\le k \\le n$. \nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $\\FlC{n}$, we denote by $\\mathcal{E}^{\\vee}$ its dual bundle. \nWe denote by $Q_{1}, \\ldots, Q_{n}$ the Novikov variables of $QK_{T}(\\FlC{n})$. We understand that $Q_{0} = 0$ and $\\CS_{-1} = 0$. \nOur main results are as follows.",
    "full_context": "Let $X$ be a partial flag manifold of arbitrary type. \nGivental \\cite{Givental} and Lee \\cite{Lee} introduced the \\emph{torus-equivariant quantum $K$-ring} $QK_{T}(X)$ of the flag manifold $X$, \nwhich is a deformation of the ordinary torus-equivariant $K$-ring $K_{T}(X)$; here $T$ denotes the torus. \nThe product $\\star$ of $QK_{T}(X)$, called the \\emph{quantum product}, is defined by using \\emph{$K$-theoretic Gromov--Witten invariants}. \nTo understand the quantum product, presentations of $QK_{T}(X)$ with generators and relations have been studied, including Borel-type, Coulomb branch, and Whitney presentations. In this paper, we focus on the Whitney presentation.\n\nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $X$, we denote by $[\\mathcal{E}]$ the class of $\\mathcal{E}$ in $K_{T}(X)$. \nThe \\emph{Hurzebruch $\\lambda_{y}$-class} $\\lambda_{y}(\\mathcal{E}) \\in K_{T}(X)[y]$ of a $T$-equivariant vector bundle $\\mathcal{E}$, introduced in \\cite{Hirzebruch}, is defined as \n\\begin{equation}\n\\lambda_{y}(\\mathcal{E}) := 1 + y [\\mathcal{E}] + y^{2} \\left[ \\bigwedge\\nolimits^{2} \\mathcal{E} \\right] + \\cdots + y^{\\rk(\\mathcal{E})} \\left[ \\bigwedge\\nolimits^{\\rk(\\mathcal{E})} \\mathcal{E} \\right], \n\\end{equation}\nwhere $\\rk(\\mathcal{E})$ denotes the rank of $\\mathcal{E}$. \nFor a short exact sequence $0 \\rightarrow \\CE_{1} \\rightarrow \\CE_{2} \\rightarrow \\CE_{3} \\rightarrow 0$ of torus-equivariant vector bundles over $X$, we have \n\\begin{equation} \\label{eq:Whitney}\n\\lambda_{y}(\\CE_{1}) \\cdot \\lambda_{y}(\\CE_{3}) = \\lambda_{y}(\\CE_{2}) \n\\end{equation}\nin $K_{T}(X)[y]$, where $\\cdot$ denotes the product of $K_{T}(X)$ (or $K_{T}(X)[y]$).\nThe identity \\eqref{eq:Whitney} is called the \\emph{$K$-theoretic Whitney relation}.\n\nOur interest is to quantize the above $K$-theoretic Whitney relations.\nMore precisely, we study the quantum product $\\lambda_{y}(\\CE_{1}) \\star \\lambda_{y}(\\CE_{3})$ in $QK_{T}(X)[y]$. \nLet us consider the case $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ is a $k$-step flag manifold of type $A$ parametrizing all sequences $(0 \\subset V_{1} \\subset \\cdots \\subset V_{k} \\subset \\mathbb{C}^{n})$ of vector subspaces of $\\mathbb{C}^{n}$ with $\\dim V_{i} = r_{i}$ for $1 \\le i \\le k$.\nThen we can take a tautological filtration $0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{k} \\subset \\underline{\\BC}^{n}$ of the trivial bundle $\\underline{\\BC}^{n} = X \\times \\mathbb{C}^{n}$. \nGu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}) conjectured the result of the quantum product of $\\lambda_{y}$-classes associated to tautological vector bundles as follows: \n\\begin{equation} \\label{eq:QK-Whitney_typeA}\n\\lambda_{y}(\\CS_{j}) \\star \\lambda_{y}(\\CS_{j+1}/\\CS_{j}) = \\lambda_{y}(\\CS_{j+1}) - y^{r_{j+1}-r_{j}} \\dfrac{Q_{j}}{1-Q_{j}} \\det(\\CS_{j+1}/\\CS_{j}) \\star (\\lambda_{y}(\\CS_{j}) - \\lambda_{y}(\\CS_{j-1})) \n\\end{equation}\nfor $1 \\le j \\le k$, here $Q_{1}, \\ldots, Q_{k}$ are Novikov variables of $QK_{T}(X)$, and $\\lambda_{y}(\\CS_{n+1}) = \\lambda_{y}(\\underline{\\BC}^{n}) = \\prod_{1 \\le j \\le n} (1 + yT_{j})$, where $T_{j} \\in R(T)$ for $1 \\le j \\le n$ are given by the decomposition of $\\mathbb{C}^{n}$ into one-dimensional $T$-modules (see \\cite[Conjecture~1.1]{GMSXZZ}). \nThis identity is called the \\emph{quantum $K$-theoretic Whitney relation} (the \\emph{$QK$-Whitney relation} for short). \nThis $QK$-Whitney relation was proved for Grassmannians $X = \\mathrm{Gr}(k, n) := \\Fl(k; n)$ by Gu, Mihalcea, Sharpe, and Zou (\\cite{GMSZ2}), for $1 \\le k \\le n$, for the incidence variety $X = \\Fl(1, n-1; n)$ by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ}), and for arbitrary partial flag manifolds in type $A$ by Huq-Kuruvilla (\\cite{HK}). \nIn addition, Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou showed in \\cite[Theorem~1.4]{GMSXZZ} that this conjecture holds under the quantum $K$-theoretic divisor axiom stated by Buch and Mihalcea, the identity for three-pointed $K$-theoretic Gromov--Witten invariants. \nRecently, Lenart, Naito, Sagaki, and Xu proved the quantum $K$-theoretic divisor axiom in \\cite{LNSX} (not only for $X = \\Fl(r_{1}, \\ldots, r_{k}; n)$ but also for arbitrary $X$). \nAs a benefit of describing $QK$-Whitney relations, we can obtain a presentation of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}; n))$ as a quotient ring of a polynomial ring. In fact, $QK$-Whitney relations give a complete set of defining relations of $QK_{T}(\\Fl(r_{1}, \\ldots, r_{k}, n))$ (\\cite[Corollary~5.4]{GMSXZZ2}).\n\nSince the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \\cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$. \nLet $X = \\FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences \n\\begin{equation}\n(0 \\subset V_{1} \\subset \\cdots \\subset V_{n} \\subset \\mathbb{C}^{2n})\n\\end{equation}\nof isotropic subspaces $V_{1}, \\ldots, V_{n}$ with respect to a fixed symplectic form with $\\dim V_{k} = k$ for $1 \\le k \\le n$. \nThen we can take a tautological filtration\n\\begin{equation}\n0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{n} \\subset \\underline{\\BC}^{2n}\n\\end{equation}\nof subbundles of the trivial bundle $\\underline{\\BC}^{2n} = \\FlC{n} \\times \\mathbb{C}^{2n}$ over $\\FlC{n}$ such that $\\rk(\\CS_{k}) = k$ for $1 \\le k \\le n$. \nFor a $T$-equivariant vector bundle $\\mathcal{E}$ over $\\FlC{n}$, we denote by $\\mathcal{E}^{\\vee}$ its dual bundle. \nWe denote by $Q_{1}, \\ldots, Q_{n}$ the Novikov variables of $QK_{T}(\\FlC{n})$. We understand that $Q_{0} = 0$ and $\\CS_{-1} = 0$. \nOur main results are as follows.\n\nSince the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \\cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$. \nLet $X = \\FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences \n\\begin{equation}\n(0 \\subset V_{1} \\subset \\cdots \\subset V_{n} \\subset \\BC^{2n})\n\\end{equation}\nof isotropic subspaces $V_{1}, \\ldots, V_{n}$ with respect to a fixed symplectic form with $\\dim V_{k} = k$ for $1 \\le k \\le n$. \nThen we can take a tautological filtration\n\\begin{equation}\n0 = \\CS_{0} \\subset \\CS_{1} \\subset \\cdots \\subset \\CS_{n} \\subset \\udBC^{2n}\n\\end{equation}\nof subbundles of the trivial bundle $\\udBC^{2n} = \\FlC{n} \\times \\BC^{2n}$ over $\\FlC{n}$ such that $\\rk(\\CS_{k}) = k$ for $1 \\le k \\le n$. \nFor a $T$-equivariant vector bundle $\\CE$ over $\\FlC{n}$, we denote by $\\CE^{\\vee}$ its dual bundle. \nWe denote by $Q_{1}, \\ldots, Q_{n}$ the Novikov variables of $QK_{T}(\\FlC{n})$. We understand that $Q_{0} = 0$ and $\\CS_{-1} = 0$. \nOur main results are as follows.\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2_intro}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3_intro}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\nThen we observe that relations \\eqref{eq:QK-Whitney_1_intro}, \\eqref{eq:QK-Whitney_2_intro}, and \\eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\\FlC{n})$, which we call the \\emph{Whitney-type presentation}.\n\nNote that the above identities are similar to those for the type $A$ flag manifold; see \\eqref{eq:QK-Whitney_typeA}. \nOn the other hand, the following identity, analogous to \\eqref{eq:Whitney_classical_3}, is specific to the type $C$ case. \n\\begin{thm} \\label{thm:QK-Whitney_B}\nIn $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\\end{thm}\nWe call identities \\eqref{eq:QK-Whitney_1}, \\eqref{eq:QK-Whitney_2}, and \\eqref{eq:QK-Whitney_3} \\emph{quantum $K$-theoretic Whitney relations} (\\emph{$QK$-Whitney relations} for short) for $\\FlC{n}$.\n\n\\subsection{Proof of Theorem~\\ref{thm:QK-Whitney_A}}\nFirst, we give a proof \\eqref{eq:QK-Whitney_1}. \nFor this purpose, we show the following identity. \n\\begin{lem}\nIn $QK_{T}(\\FlC{n})$, for $1 \\le k \\le n$ and $1 \\le d \\le k$, we have \n\\begin{equation} \\label{eq:QK-Whitney_lem_1}\n\\left[ \\extprod^{d} \\CS_{k} \\right] = \\left[ \\extprod^{d} \\CS_{k-1} \\right] + \\frac{1}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star \\left( \\left[ \\extprod^{d-1} \\CS_{k-1} \\right] - Q_{k-1} \\left[ \\extprod^{d-1} \\CS_{k-2} \\right] \\right). \n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe compute as follows: \n\\begin{align}\n& \\left[ \\extprod^{d} \\CS_{k} \\right] \\\\ \n& = \\sum_{\\substack{J \\subset [1, k] \\\\ |J| = d}} [\\CO_{G/B}(-\\ve_{J})] \\\\ \n& = \\sum_{\\substack{J \\subset [1, k-1] \\\\ |J| = d}} [\\CO_{G/B}(-\\ve_{J})] + \\sum_{\\substack{J \\subset [1, k-1] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{J \\sqcup \\{k\\}})] \\\\ \n\\begin{split}\n& = \\left[ \\extprod^{d} \\CS_{k-1} \\right] + \\sum_{\\substack{J \\subset [1, k-2] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{k})] \\star [\\CO_{G/B}(-\\ve_{J})] \\\\ \n& \\qquad + \\frac{1}{1-Q_{k-1}} \\sum_{\\substack{J \\subset [1, k-1] \\\\ k-1 \\in J \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{k})] \\star [\\CO_{G/B}(-\\ve_{J})]\n\\end{split} \\qquad \\text{by \\eqref{eq:multiple_1}} \\\\ \n\\begin{split}\n& = \\left[ \\extprod^{d} \\CS_{k-1} \\right] + \\frac{1}{1-Q_{k-1}} [\\CO_{G/B}(-\\ve_{k})] \\star \\sum_{\\substack{J \\subset [1, k-1] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{J})] \\\\ \n& \\qquad - \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CO_{G/B}(-\\ve_{k})] \\star \\sum_{\\substack{J \\subset [1, k-2] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{J})]\n\\end{split} \\\\ \n\\begin{split}\n& = \\left[ \\extprod^{d} \\CS_{k-1} \\right] \\\\ \n& \\qquad + \\frac{1}{1-Q_{k-1}} [\\CO_{G/B}(-\\ve_{k})] \\star \\left( \\sum_{\\substack{J \\subset [1, k-1] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{J})] - Q_{k-1} \\sum_{\\substack{J \\subset [1, k-2] \\\\ |J| = d-1}} [\\CO_{G/B}(-\\ve_{J})] \\right)\n\\end{split} \\\\ \n&= \\left[ \\extprod^{d} \\CS_{k-1} \\right] + \\frac{1}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star \\left( \\left[ \\extprod^{d-1} \\CS_{k-1} \\right] - Q_{k-1} \\left[ \\extprod^{d-1} \\CS_{k-2} \\right] \\right), \n\\end{align}\nas desired. \n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:QK-Whitney_B}}\nWe prove \\eqref{eq:QK-Whitney_3} by comparing coefficients of $y^{d}$ of both sides for each $1 \\le d \\le 2n$. \n\\begin{lem}\nIn $QK_{T}(\\FlC{n})$, for $1 \\le d \\le 2n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_computation_lem1}\n\\begin{split}\n& \\sum_{\\substack{k, l \\ge 0 \\\\ k+l = d}} \\left[ \\extprod^{k} \\CS_{n} \\right] \\star \\left[ \\extprod^{l} \\CS_{n}^{\\vee} \\right] = \\left[ \\extprod^{d} \\udBC^{2n} \\right] \\\\ \n& \\hspace{20mm} - \\sum_{p = 1}^{n} \\frac{Q_{p}Q_{p+1} \\cdots Q_{n}}{1 - Q_{p}} \\sum_{\\substack{J, K \\subset [1, n] \\\\ |J| + |K| = d \\\\ \\max J = \\max K = p}} [\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{K})]. \n\\end{split}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nIn what follows, we set $\\overline{K} := \\{\\overline{j} \\mid j \\in K \\}$ for $K \\subset [1, n]$. \nLet $J, K \\subset [1, n]$. \nBy \\eqref{eq:multiple_1}, we have \n\\begin{align}\n[\\CO_{G/B}(-\\ve_{J})] &= \\left( \\prod_{\\substack{2 \\le j \\le n \\\\ j-1 \\in J, j \\in J}} \\frac{1}{1-Q_{j-1}} \\right) \\left( \\qprod_{j \\in J} [\\CO_{G/B}(-\\ve_{j})] \\right) \\\\ \n&= \\left( \\prod_{1 \\le j \\le n} \\varphi_{J}(j) \\right) \\left( \\qprod_{j \\in J} [\\CO_{G/B}(-\\ve_{j})] \\right). \n\\end{align}\nAlso, by \\eqref{eq:multiple_2}, we have \n\\begin{align}\n[\\CO_{G/B}(\\ve_{K})] &= \\left( \\prod_{\\substack{2 \\le j \\le n \\\\ j-1\\in K, j \\in K}} \\frac{1}{1-Q_{j-1}} \\right) \\left( \\qprod_{j \\in K} [\\CO_{G/B}(\\ve_{j})] \\right) \\\\ \n&= \\left( \\prod_{\\overline{n} \\le j \\le \\overline{1}} \\varphi_{\\overline{K}}(j) \\right) \\left( \\qprod_{j \\in \\overline{K}} [\\CO_{G/B}(-\\ve_{j})] \\right). \n\\end{align}\nHence we have \n\\begin{equation}\n[\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{K})] = \\left( \\prod_{1 \\le j \\le n} \\varphi_{J}(j) \\right) \\left( \\prod_{\\overline{n} \\le j \\le \\overline{1}} \\varphi_{\\overline{K}}(j) \\right) \\left( \\ \\qprod_{j \\in J \\sqcup \\overline{K}} [\\CO_{G/B}(-\\ve_{j})] \\right). \n\\end{equation}\n\n\\begin{prop} \nIn $QK_{T}(\\FlC{n})$, for $1 \\le d \\le 2n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_lem_3}\n\\begin{split}\n& \\sum_{\\substack{k, l \\ge 0 \\\\ k+l = d}} \\left[ \\extprod^{k} \\CS_{n} \\right] \\star \\left[ \\extprod^{l} \\CS_{n}^{\\vee} \\right] = \\left[ \\extprod^{d} \\udBC^{2n} \\right] \\\\ \n& \\qquad - \\sum_{p = 1}^{n} \\frac{Q_{p} Q_{p+1} \\cdots Q_{n}}{1 - Q_{p-1}} \\sum_{\\substack{k, l \\ge 0 \\\\ k+l = d-2}} \\left( \\left[ \\extprod^{k} \\CS_{p-1} \\right] - Q_{p-1} \\left[ \\extprod^{k} \\CS_{p-2} \\right] \\right) \\\\ \n& \\hspace{80mm} \\star \\left( \\left[ \\extprod^{l} \\CS_{p-1}^{\\vee} \\right] - Q_{p-1} \\left[ \\extprod^{l} \\CS_{p-2}^{\\vee} \\right] \\right). \n\\end{split}\n\\end{equation}\n\\end{prop}",
    "post_theorem_intro_text_len": 5688,
    "post_theorem_intro_text": "While \\eqref{eq:QK-Whitney_1_intro} and \\eqref{eq:QK-Whitney_2_intro} are analogous to \\eqref{eq:QK-Whitney_typeA}, \\eqref{eq:QK-Whitney_3_intro} is specific to the type $C$ case. \n\nThen we observe that relations \\eqref{eq:QK-Whitney_1_intro}, \\eqref{eq:QK-Whitney_2_intro}, and \\eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\\FlC{n})$, which we call the \\emph{Whitney-type presentation}. \n\n\\begin{ithm}[$=$\\,Theorem~\\ref{thm:Whitney_presentation}] \\label{ithm:Whitney_presentation}\nDefine a ring $R_{Q}^{\\mathsf{W}}$ by \n\\begin{equation}\nR_{Q}^{\\mathsf{W}} := (R(T)\\bra{Q_{1}, \\ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \\mid 1 \\le k \\le n, \\ 1 \\le d \\le k, \\ 1 \\le j \\le n]. \n\\end{equation}\nTake an ideal $I_{Q}^{\\mathsf{W}}$ of $R_{Q}^{\\mathsf{W}}$ generated by \n\\begin{align}\n& F_{d}^{k} - \\left( F_{d}^{k-1} + \\frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \\right) \\quad \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \n& G_{d}^{k} - \\left( G_{d}^{k-1} + \\frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \\right) \\quad \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \n\\begin{split}\n&\\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \\Biggl( e_{d}(T_{1}, T_{2}, \\ldots, T_{n}, T_{n}^{-1}, \\ldots, T_{2}^{-1}, T_{1}^{-1}) \\\\ \n& \\quad - \\sum_{p = 1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} \\sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \\Biggr) \\quad \\text{for $1 \\le d \\le n$}, \n\\end{split} \\\\ \n& F_{1}^{1} - x_{1}, \\\\\n& G_{1}^{1} - y_{1}, \\\\ \n& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \\quad \\text{for $1 \\le j \\le n$}; \n\\end{align}\nhere, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \\in R(T)$ for $1 \\le k \\le 2n$ are obtained from the decomposition of $\\mathbb{C}^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \\ldots$, $T_{2n} = T_{1}^{-1}$. \nIn addition, we understand that \n$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \\le k \\le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \\le k \\le n$. \n\nThen there exists an $R(T)$-algebra isomorphism $R_{Q}^{\\mathsf{W}}/I_{Q}^{\\mathsf{W}} \\xrightarrow{\\sim} QK_{T}(\\FlC{n})$ such that \n\\begin{align}\nF_{d}^{k} & \\mapsto \\left[ \\bigwedge\\nolimits^{d} \\CS_{k} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nG_{d}^{k} & \\mapsto \\left[ \\bigwedge\\nolimits^{d} \\CS_{k}^{\\vee} \\right] & & \\text{for $1 \\le k \\le n$ and $1 \\le d \\le k$}, \\\\ \nx_{j} & \\mapsto [\\CS_{j}/\\CS_{j-1}] & & \\text{for $1 \\le j \\le n$}, \\\\ \ny_{j} & \\mapsto [(\\CS_{j}/\\CS_{j-1})^{\\vee}] & & \\text{for $1 \\le j \\le n$}. \n\\end{align}\n\\end{ithm}\n\nTo prove Theorem~\\ref{ithm:QK-Whitney}, we use the theory of semi-infinite flag manifolds. \nKato (\\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules. \nThrough this isomorphism, we can translate certain quantum products in $QK_{T}(\\FlC{n})$ that appear in the computation of the left-hand side of $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold. \nSince tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\\FlC{n})$ (Corollary~\\ref{cor:multiple_line_bundles}). \nThese identities enable us to compute $\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1})$ and $\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee})$. \nTo consider the product $\\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee})$, we use an additional method, the Borel-type presentation given in \\cite{KN2}. This gives a relation in $QK_{T}(\\FlC{n})$ (Proposition~\\ref{prop:QK-Borel_rel}) which enables us to simplify such a product. \n\nTo prove Theorem~\\ref{ithm:Whitney_presentation}, we use Nakayama-type lemma established by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\\cite{GMSXZZ2}). This lemma shows that the isomorphism for the classical $K$-ring $K_{T}(X)$ lifts to that for the quantum $K$-ring $QK_{T}(X)$ under conditions (Theorem~\\ref{thm:Nakayama}). By applying this lemma, our Whitney-type presentation reduces to the well-known presentation of $K_{T}(\\FlC{n})$, and hence we obtain the theorem. \n\nThis paper is organized as follows. In Section~\\ref{sec:preliminaries}, we fix notation for root systems; in particular, for the type $C$ root system. \nAlso, we briefly review the flag manifold and its $K$-ring. \nIn Section~\\ref{sec:QK-Whitney}, we give explicit descriptions of $QK$-Whitney relations in $QK_{T}(\\FlC{n})$ (Theorem~\\ref{ithm:QK-Whitney}) and the Whitney-type presentation of $QK_{T}(\\FlC{n})$ (Theorem~\\ref{ithm:Whitney_presentation}), which are the main results of this paper. \nIn Section~\\ref{sec:method}, we review the semi-infinite flag manifold and its $K$-group. Then we compute some quantum products necessary to study $QK$-Whitney relations, where some proofs are deferred to Appendix~\\ref{sec:appendix}. \nIn Section~\\ref{sec:QK-Whitney_computation}, we give a proof of Theorem~\\ref{ithm:QK-Whitney}. In the computation of $\\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee})$, we review Borel-type relations of $QK_{T}(\\FlC{n})$. \nIn Section~\\ref{sec:Nakayama}, we review the Nakayama-type lemma and give a proof of Theorem~\\ref{ithm:Whitney_presentation}. \n\n\\subsection*{Acknowledgments}\nThe author is grateful to Takeshi Ikeda and Satoshi Naito for helpful discussions. \nThis work was supported by JSPS Grant-in-Aid for Research Activity Start-up 24K22842.",
    "sketch": "To prove Theorem~\\ref{ithm:QK-Whitney}, we use the theory of semi-infinite flag manifolds. Kato (\\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules. Through this isomorphism, we can translate certain quantum products in $QK_{T}(\\FlC{n})$ that appear in the computation of the left-hand side of $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold. Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\\FlC{n})$ (Corollary~\\ref{cor:multiple_line_bundles}). These identities enable us to compute $\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1})$ and $\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee})$. To consider the product $\\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee})$, we use an additional method, the Borel-type presentation given in \\cite{KN2}. This gives a relation in $QK_{T}(\\FlC{n})$ (Proposition~\\ref{prop:QK-Borel_rel}) which enables us to simplify such a product.",
    "expanded_sketch": "To prove the main theorem, we use the theory of semi-infinite flag manifolds. Kato (\\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules. Through this isomorphism, we can translate certain quantum products in $QK_{T}(\\FlC{n})$ that appear in the computation of the left-hand side of the $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold. Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\\FlC{n})$. We first record the following corollary.\n\n\\begin{cor} \\label{cor:multiple_line_bundles}\nAssume that $G = \\Sp_{2n}(\\BC)$. \nLet $2 \\le k \\le n$ and take $J \\subset [1, k-1]$. Then we have \n\\begin{align}\n[\\CO_{G/B}(-\\ve_{J \\sqcup \\{k\\}})] &= \\begin{cases} \\dfrac{1}{1-Q_{k-1}} [\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(-\\ve_{k})] & \\text{if $k-1 \\in J$}, \\\\[5mm]\n[\\CO_{G/B}(-\\ve_{J})] \\star [\\CO_{G/B}(-\\ve_{k})] & \\text{if $k-1 \\notin J$}, \\end{cases} \\label{eq:multiple_1} \\\\ \n[\\CO_{G/B}(\\ve_{J \\sqcup \\{k\\}})] &= \\begin{cases} \\dfrac{1}{1-Q_{k-1}} [\\CO_{G/B}(\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{k})] & \\text{if $k-1 \\in J$}, \\\\[5mm]\n[\\CO_{G/B}(\\ve_{J})] \\star [\\CO_{G/B}(\\ve_{k})] & \\text{if $k-1 \\notin J$}. \\end{cases} \\label{eq:multiple_2}\n\\end{align}\n\\end{cor}\n\nThese identities enable us to compute $\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1})$ and $\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee})$. To consider the product $\\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee})$, we use an additional method, the Borel-type presentation. This gives a relation in $QK_{T}(\\FlC{n}) enabling us to simplify such a product; more precisely, we use the following proposition.\n\n\\begin{prop}[{\\cite[Corollary~6.4]{KN2}}] \\label{prop:QK-Borel_rel}\nIn $QK_{T}(\\FlC{n})$, for $0 \\le d \\le 2n$, we have \n\\begin{equation} \\label{eq:QK-Borel_rel}\n\\sum_{\\substack{J \\subset [1, \\overline{1}] \\\\ |J|=d}} \\left( \\prod_{1 \\le j \\le \\overline{1}} \\varphi_{J}(j) \\right) \\left( \\qprod_{j \\in J} [\\CO_{G/B}(-\\ve_{j})] \\right) = \\left[ \\extprod^{d} \\udBC^{2n} \\right]. \n\\end{equation}\n\\end{prop}",
    "expanded_theorem": "[$=$\\,\\begin{thm} \\label{thm:QK-Whitney_A}\n\\begin{enu}\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_1}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\\end{enu}\n\\end{thm}\\,+$\\,\\begin{thm} \\label{thm:QK-Whitney_B}\nIn $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\udBC^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\\end{thm}] \\label{ithm:QK-Whitney}\n\\begin{enu}\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have \n\\begin{equation} \\label{eq:QK-Whitney_1_intro}\n\\lambda_{y}(\\CS_{k-1}) \\star \\lambda_{y}(\\CS_{k}/\\CS_{k-1}) = \\lambda_{y}(\\CS_{k}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [\\CS_{k}/\\CS_{k-1}] \\star (\\lambda_{y}(\\CS_{k-1}) - \\lambda_{y}(\\CS_{k-2})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, for $1 \\le k \\le n$, we have\n\\begin{equation} \\label{eq:QK-Whitney_2_intro}\n\\lambda_{y}(\\CS_{k-1}^{\\vee}) \\star \\lambda_{y}((\\CS_{k}/\\CS_{k-1})^{\\vee}) = \\lambda_{y}(\\CS_{k}^{\\vee}) - y \\frac{Q_{k-1}}{1-Q_{k-1}} [(\\CS_{k}/\\CS_{k-1})^{\\vee}] \\star (\\lambda_{y}(\\CS_{k-1}^{\\vee}) - \\lambda_{y}(\\CS_{k-2}^{\\vee})). \n\\end{equation}\n\n\\item In $QK_{T}(\\FlC{n})[y]$, we have \n\\begin{equation} \\label{eq:QK-Whitney_3_intro}\n\\begin{split}\n& \\lambda_{y}(\\CS_{n}) \\star \\lambda_{y}(\\CS_{n}^{\\vee}) = \\lambda_{y}(\\underline{\\BC}^{2n}) \\\\ \n& \\qquad - y^{2} \\sum_{p=1}^{n} \\frac{Q_{p} \\cdots Q_{n}}{1-Q_{p-1}} (\\lambda_{y}(\\CS_{p-1}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2})) \\star (\\lambda_{y}(\\CS_{p-1}^{\\vee}) - Q_{p-1} \\lambda_{y}(\\CS_{p-2}^{\\vee})). \n\\end{split}\n\\end{equation}\n\n\\end{enu}",
    "theorem_type": "unknown",
    "mcq": {
      "question": "Let \\(X=\\mathrm{Fl}^C_n\\) be the type \\(C_n\\) flag manifold parametrizing complete isotropic flags \\(0\\subset V_1\\subset\\cdots\\subset V_n\\subset \\mathbb C^{2n}\\), and let \\(0=\\mathcal S_0\\subset \\mathcal S_1\\subset\\cdots\\subset \\mathcal S_n\\) be its tautological filtration. In the torus-equivariant quantum \\(K\\)-ring \\(QK_T(X)\\) with quantum product \\(\\star\\) and Novikov variables \\(Q_i\\), define the Hirzebruch \\(\\lambda_y\\)-class of a \\(T\\)-equivariant vector bundle \\(\\mathcal E\\) by \\(\\lambda_y(\\mathcal E)=\\sum_{i=0}^{\\operatorname{rk}(\\mathcal E)} y^i[\\wedge^i\\mathcal E]\\). Which explicit identities hold in \\(QK_T(X)[y]\\) for the tautological bundles, their quotients, and their duals?",
      "correct_choice": {
        "label": "A",
        "text": "The following three identities hold in \\(QK_T(\\mathrm{Fl}^C_n)[y]\\):\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1})\\star \\lambda_y(\\mathcal S_k/\\mathcal S_{k-1})\n= \\lambda_y(\\mathcal S_k)\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[\\mathcal S_k/\\mathcal S_{k-1}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1})-\\lambda_y(\\mathcal S_{k-2})\\bigr),\n\\]\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1}^{\\vee})\\star \\lambda_y\\bigl((\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}\\bigr)\n= \\lambda_y(\\mathcal S_k^{\\vee})\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[(\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1}^{\\vee})-\\lambda_y(\\mathcal S_{k-2}^{\\vee})\\bigr),\n\\]\nand\n\\[\n\\lambda_y(\\mathcal S_n)\\star \\lambda_y(\\mathcal S_n^{\\vee})\n= \\lambda_y(\\underline{\\mathbb C}^{2n})\n- y^2\\sum_{p=1}^n \\frac{Q_p\\cdots Q_n}{1-Q_{p-1}}\n\\bigl(\\lambda_y(\\mathcal S_{p-1})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2})\\bigr)\n\\star\n\\bigl(\\lambda_y(\\mathcal S_{p-1}^{\\vee})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2}^{\\vee})\\bigr).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "The following three identities hold in \\(QK_T(\\mathrm{Fl}^C_n)[y]\\):\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1})\\star \\lambda_y(\\mathcal S_k/\\mathcal S_{k-1})\n= \\lambda_y(\\mathcal S_k)\n- y\\,\\frac{Q_k}{1-Q_k}\\,[\\mathcal S_k/\\mathcal S_{k-1}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1})-\\lambda_y(\\mathcal S_{k-2})\\bigr),\n\\]\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1}^{\\vee})\\star \\lambda_y\\bigl((\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}\\bigr)\n= \\lambda_y(\\mathcal S_k^{\\vee})\n- y\\,\\frac{Q_k}{1-Q_k}\\,[(\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1}^{\\vee})-\\lambda_y(\\mathcal S_{k-2}^{\\vee})\\bigr),\n\\]\nand\n\\[\n\\lambda_y(\\mathcal S_n)\\star \\lambda_y(\\mathcal S_n^{\\vee})\n= \\lambda_y(\\underline{\\mathbb C}^{2n})\n- y^2\\sum_{p=1}^n \\frac{Q_p\\cdots Q_n}{1-Q_p}\n\\bigl(\\lambda_y(\\mathcal S_{p-1})-Q_p\\lambda_y(\\mathcal S_{p-2})\\bigr)\n\\star\n\\bigl(\\lambda_y(\\mathcal S_{p-1}^{\\vee})-Q_p\\lambda_y(\\mathcal S_{p-2}^{\\vee})\\bigr).\n\\]"
        },
        {
          "label": "C",
          "text": "The following two identities hold in \\(QK_T(\\mathrm{Fl}^C_n)[y]\\):\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1})\\star \\lambda_y(\\mathcal S_k/\\mathcal S_{k-1})\n= \\lambda_y(\\mathcal S_k)\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[\\mathcal S_k/\\mathcal S_{k-1}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1})-\\lambda_y(\\mathcal S_{k-2})\\bigr),\n\\]\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1}^{\\vee})\\star \\lambda_y\\bigl((\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}\\bigr)\n= \\lambda_y(\\mathcal S_k^{\\vee})\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[(\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1}^{\\vee})-\\lambda_y(\\mathcal S_{k-2}^{\\vee})\\bigr).\n\\]"
        },
        {
          "label": "D",
          "text": "The following three identities hold in \\(QK_T(\\mathrm{Fl}^C_n)[y]\\):\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1})\\star \\lambda_y(\\mathcal S_k/\\mathcal S_{k-1})\n= \\lambda_y(\\mathcal S_k)\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[\\mathcal S_k/\\mathcal S_{k-1}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1})-\\lambda_y(\\mathcal S_{k-2})\\bigr),\n\\]\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1}^{\\vee})\\star \\lambda_y\\bigl((\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}\\bigr)\n= \\lambda_y(\\mathcal S_k^{\\vee})\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[(\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1}^{\\vee})-\\lambda_y(\\mathcal S_{k-2}^{\\vee})\\bigr),\n\\]\nand\n\\[\n\\lambda_y(\\mathcal S_n)\\star \\lambda_y(\\mathcal S_n^{\\vee})\n= \\lambda_y(\\underline{\\mathbb C}^{2n})\n- y\\sum_{p=1}^n \\frac{Q_p\\cdots Q_n}{1-Q_{p-1}}\n\\bigl(\\lambda_y(\\mathcal S_{p-1})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2})\\bigr)\n\\star\n\\bigl(\\lambda_y(\\mathcal S_{p-1}^{\\vee})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2}^{\\vee})\\bigr).\n\\]"
        },
        {
          "label": "E",
          "text": "The following three identities hold in \\(QK_T(\\mathrm{Fl}^C_n)[y]\\):\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1})\\star \\lambda_y(\\mathcal S_k/\\mathcal S_{k-1})\n= \\lambda_y(\\mathcal S_k)\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[\\mathcal S_k/\\mathcal S_{k-1}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1})-\\lambda_y(\\mathcal S_{k-2})\\bigr),\n\\]\n\\[\n\\text{for }1\\le k\\le n,\\qquad\n\\lambda_y(\\mathcal S_{k-1}^{\\vee})\\star \\lambda_y\\bigl((\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}\\bigr)\n= \\lambda_y(\\mathcal S_k^{\\vee})\n- y\\,\\frac{Q_{k-1}}{1-Q_{k-1}}\\,[(\\mathcal S_k/\\mathcal S_{k-1})^{\\vee}]\\star\\bigl(\\lambda_y(\\mathcal S_{k-1}^{\\vee})-\\lambda_y(\\mathcal S_{k-2}^{\\vee})\\bigr),\n\\]\nand\n\\[\n\\lambda_y(\\mathcal S_n)\\star \\lambda_y(\\mathcal S_n^{\\vee})\n= \\lambda_y(\\underline{\\mathbb C}^{2n})\n- y^2\\sum_{p=1}^n \\frac{Q_p\\cdots Q_n}{1-Q_{p-1}}\n\\bigl(\\lambda_y(\\mathcal S_{p-1})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2})\\bigr)\n\\star\n\\bigl(\\lambda_y(\\mathcal S_{p-1})-Q_{p-1}\\lambda_y(\\mathcal S_{p-2})\\bigr).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "shifted-index denominator and Novikov dependence",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped the third identity for \\(\\lambda_y(\\mathcal S_n)\\star\\lambda_y(\\mathcal S_n^{\\vee})\\)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "power of \\(y\\) in the global correction term",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "property_confusion",
          "tampered_component": "dual factor in the second parenthesis of the third identity",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It defines the setting and asks which identities hold, but does not contain the identities themselves or a direct hint toward choice A."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct theorem-recall item: the correct choice reproduces the explicit identities verbatim, and the task is to recognize the exact statement rather than infer a conclusion from premises."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning/pattern-checking needed because the distractors alter indices, powers of y, omitted identities, or dual factors. However, the item mainly tests precise recall or formula matching, not genuine generation of a mathematical conclusion."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and distinct: shifted Novikov indices, a weaker-true partial statement, an incorrect power of y, and confusion between a bundle and its dual. These align with realistic error modes."
      },
      "total_score": 5,
      "overall_assessment": "Technically well-constructed in terms of distractors and lack of answer leakage, but it is mostly a verbatim theorem-recognition question rather than a non-tautological or genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.23266v1",
    "paper_link": "http://arxiv.org/abs/2512.23266v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$",
      "start_pos": 6513,
      "end_pos": 6711,
      "label": "t:main"
    },
    "ref_dict": {
      "Ding-2": "\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}",
      "alpha:polynomial": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}",
      "t:main": "\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}",
      "s:upper": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for $\\L(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for $\\L(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.\n\n\\section{Proofs of the upper bounds in Theorems \\ref{t:main} and \\ref{t:main2}}\n\\label{s:upper}\n\nLet $(S_1,S_2,\\ldots)$ denote a random walk with step distribution $X$ of \\eqref{def:X}, starting at $S_0=0$. For any $\\ell\\ge 1$ and Borel set $A\\subset \\r^\\ell$, \n\\begin{equation}\\label{comp:moment1}\n\\e\\Big[\\#\\{\\gamma \\in {\\mathcal P}_\\ell\\colon S(\\gamma) \\in A\\}\\Big] \\le n^{\\ell+1}\\p\\Big((S_1,\\ldots,S_\\ell) \\in A\\Big), \n\\end{equation}",
      "t:main2": "\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}",
      "s:lower": "\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main2}] Let $n$ be large and  $a \\in (0, 1)$. Consider $\\ell= \\lceil ne^{- \\frac{\\pi}{\\sqrt{ 2 \\beta_n}} (1-a)}\\rceil$,  $r=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-\\frac{a}2)}$ and  let $m'_\\Delta(L)$ be the number of $i\\in \\lb 0,L\\rb$ such that $\\inf_{j \\in \\lb i,L\\rb} S_j\\le S_i- \\Delta$ with $\\Delta:=\\ln r$.  Let $b_n=\\ln n + \\ln\\ln n$.  By the same argument leading to \\eqref{eq:L>ell-bn}, but using Corollary \\ref{cor:jump} (ii) in addition, we obtain that  for any $M>2$, $$\\p\\Big(\\L(n, \\beta_n) \\ge \\ell\\Big) \\le  \\sum_{L=\\ell}^{2\\ell} \\widetilde a_L + \\frac{2}{M}+ o(1),$$\n\n\\noindent where for $L\\in \\lb \\ell, 2\\ell \\rb$, $$ \\widetilde a_L:= \\p\\Big(\\exists \\gamma\\in {\\mathcal P}_L: S_L(\\gamma) \\le \\beta_n L, m'_\\Delta(L) \\le M \\frac{n}{r},\\, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) >-  b_n \\Big).$$\n\nBy \\eqref{comp:moment1} and \\eqref{eq:many-to-one}, \\begin{align*}    \\widetilde a_L & \\le n \\, e^{\\beta_n L}  \\p\\Big( Y_L\\le \\beta_n L , \\, m_\\Delta(L) \\le M \\frac{n}{r},\\,  \\min_{0\\le i \\le j \\le L} (Y_j - Y_i ) >- b_n\\Big)\n\\\\\n& \\le n \\, e^{\\beta_n L}  \\p\\Big(   \\max_{i \\in \\lb 0, L\\rb} Y_i \\le 2 \\beta_n L, \\,  m_\\Delta(L) \\le M \\frac{n}{r}\\Big)\n ,\n\\end{align*} where $m_\\Delta(L)$ is as defined in Lemma \\ref{main2:upper} and in the last inequality, we have used the fact that $\\max_{i \\in \\lb 0, L\\rb} Y_i\\le Y_L+b_n \\le \\beta_n L+ b_n \\le 2 \\beta_n L$. Choose $d\\in (0, \\frac{\\pi^2}2)$ \n  such that $\\eta:= \\frac{2d}{\\pi^2 (1-\\frac{a}2)^2} - 1>0$. Fix an arbitrary $c\\in (0, 1)$. If $\\varepsilon>0$ is chosen small enough in Theorem \\ref{t:main2}, we have $L\\ge \\Delta^{2+c}$, $M \\frac{n}{r} \\le L \\Delta^{-2-c}$ and $2 \\beta_n L \\le L \\Delta^{-1-c}$.  Note that $\\Delta$ is arbitrarily large when $\\varepsilon$ goes to $0$.   Lemma \\ref{main2:upper} yields that $$ \\widetilde a_L \\le n e^{\\beta_n L} e^{- d \\frac{L}{\\Delta^2}} = n e^{- \\eta \\,\\beta_n L}. $$\n\n Notice that for all large $n$, $\\beta_n L \\ge \\beta_n \\ell \\ge n^{a'}$ for some positive constant $a'$. Hence   $\\sum_{L=\\ell}^{2\\ell} \\widetilde a_L \\le n^2  e^{- \\eta\\, n^{a'}}\\to 0$ and we conclude the proof by letting $n \\to\\infty$ then $M\\to\\infty$.  \\end{proof}\n\n\\section{Proofs of the lower bounds}\n\\label{s:lower}\n\nWe consider a branching random walk $\\V$ on $\\r$. It starts with a certain number of particles at $0$. At each discrete time $k\\ge 0$, each particle of the current population splits into an infinite number of children, whose displacements from their parent  are given by a copy of \n\\begin{equation} \\label{eq:BRW-V}   \n\\sum_{i=1}^\\infty {\\delta}_{\\{e T_i -1\\}} \n\\end{equation}\n\n\\noindent where $0<T_1<T_2<\\ldots$ are the atoms of a Poisson point process of intensity $1$ on $\\r_+$. The position of a particle $u$ is denoted by $\\V(u)$.\nThe offspring distribution corresponds to the limiting law  of $\\sum_{u=2}^{n} \\delta_{X(1,u)}$, so that, when $n$ is large, exploring the graph around a given vertex locally approximates the branching random walk $\\V$. To make it precise, we will  couple labels $X(u,v)$ and copies of the Poisson point process $(T_i)_{i\\ge 1}$ as follows. \n\n\\medskip\n\nConsider $(X_i)_{1\\le i \\le n-1}$ a family of $(n-1)$ i.i.d. copies of $X= n e \\mbox{Exp}(1)-1$.  Its increasing order statistics $ X_{(0)}:=-1 < X_{(1)}< ...< X_{(n-1)}$ verify that the variables $X_{(i)}-X_{(i-1)}$ are independent, distributed as $\\frac{n e }{n-i}\\text{Exp}(1)$.   One can couple it with a Poisson point process $T_1<T_2<\\ldots$ of intensity $1$, by \n\\[     X_{(i)}-X_{(i-1)} = \\frac{n e}{n-i} (T_{i}-T_{i-1}), \\qquad i\\in \\lb 1, n-1\\rb, \n\\]where we set $T_0=0$.  In particular,  for all $i\\in \\lb 1, n-1\\rb$, \\begin{equation}\\label{coupling}\n   e T_i-1 \\le  X_{(i)} \\le   \\frac{n}{n-i} (e T_{i}-1) + \\frac{i}{n-i}.\n\\end{equation}\n\nFor each $u\\in K_n$, this gives a coupling between $(X(u,v))_{v\\neq u}$ and a copy of $(T_i)_{i\\ge 1}$.  We further need the following simple observation.  Let $m \\in \\lb 1, n-1 \\rb$. If we choose   $m$  points uniformly from $(X_{(i)})_{1\\le i \\le n-1}$ and arrange them in increasing order, then they have the same distribution as  the increasing order statistics of $(X_i)_{1\\le i \\le m}$. In other words, there exists a coupling in the sense of \\eqref{coupling}, between $X(u, v)$ when $v$ varies over a subset of $K_n$ with $m$ vertices, and the family of $m$  points  chosen uniformly from $(T_i)_{1\\le i \\le n-1}$.  \n\nLet $N$ be an integer between $1$ and $n-1$.  We denote by $\\V^N$ the branching random walk $\\V$  in which, at each step, we only keep the $N$ leftmost particles of the next generation. For a particle $u$ in $\\V^N$, let   $|u|$ denote its generation in the genealogical tree of $\\V^N$, and let $\\V^N(u)$ denote its position in $\\r$.\n\n\\begin{proposition}\\label{p:first_moment} Fix $a\\in [0, 1]$ and $t>0$.    If $\\V^N$ starts with $\\lfloor N^a\\rfloor$ particles located at $0$, then for any sequence $\\ell_N$ satisfying $\\frac{\\ell_N}{\\ln^3 N} \\to t$,  \\begin{equation}  \\frac{1}{\\ln N}  \\min_{|u|=\\ell_N} \\V^N(u)   \\,\\cvL2\\, -a+\\frac{\\pi^2}{2}t  , \\qquad N \\to\\infty. \\label{Nbrw-cvL2}\\end{equation}\n\\end{proposition}\n\nWe will apply the above proposition to two cases: $a=1$ and $a=0$.\nThe proof of Proposition \\ref{p:first_moment} is postponed to Section \\ref{sub:brw}. \n\nIn what follows, we construct a coupling between $\\V^N$ and an exploration process on $K_n$. \nWe will color particles of $\\V^{N}$ in blue, red or purple. Blue particles will correspond to  vertices of $K_{n}$ and will be the particles of interest. \n\n\\subsection{Construction of an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$ in $K_n$ and its coupling with $\\V^N$.}\n \\label{s:exploration}\n\nFor each $j\\ge 0$, we define a  set $Z_j$ of so-called {\\it active} vertices of $K_n$ and a set of paths ${\\mathcal Q}_j\\subset {\\mathcal P}_j$  such that each vertex  in $Z_j$ is associated with a unique  path in ${\\mathcal Q}_j$ ending at this point. The set $Z_j$ will correspond to the blue particles of $\\V^{N}$ at time $j$. We will therefore write $u$ either for the blue particle or its corresponding  vertex in $K_n$.  For each active vertex $u$ at step $j$, we will write $V(u)$  for $S_j(\\gamma)$ where $\\gamma$ is the path in ${\\mathcal Q}_j$ ending at $u$.  \n\nLet $G_0$ be a complete subgraph of $K_n$. Let $Z_0\\subset G_0$ be a set of $N_0\\le N$ active vertices at time $0$. We start  the branching random walk $\\V$ with  $N_0$ blue particles,  each particle corresponding to an active vertex in $Z_0$.  We let $\\mathcal Q_0$ be the collection $\\{\\{u\\},\\, u\\in Z_0\\}$.  We denote by $\\#G_0$ the number of vertices in $G_0$. Let $m:=\\#G_0- \\#Z_0$.   Each $u\\in Z_0$ corresponds to a blue particle in $\\V$, still denoted by $u$. We have $\\V(u)=V(u)=0$. We  choose $m$ children $v$ of $u$ in $\\V$ uniformly among the $n-1$ leftmost ones and color them {\\it blue.}  By the coupling described above, for each blue particle $u$, $\\V(v)-\\V(u)$ is coupled with $X(u, v)$ for $v \\in G_0 \\backslash Z_0$ (where, as for $u$, the variable $v$ denotes both a particle in $\\V$ and a vertex in $G_0$).  Let $i_v$ be the index of the point $T_i$ corresponding to $v$ in the coupling \\eqref{coupling}. We have  $$X(u,v) = X_{(i_v)} \\le \\frac{n}{n-i_v} e T_{i_v} -1 =   \\frac{n}{n-i_v} (\\V(v)-\\V(u))+\\frac{i_v}{n-i_v}.$$ We also have $X(u,v) \\ge \\V(v)-\\V(u)$. \nWe will see that $i_v \\le N$ for all $v$ of interest.  We color the remaining children of $u$ in $\\V$ {\\it red.}  We then retain the $N$ leftmost particles of $\\V$ in generation $1$ which is in fact the population of $\\V^N$ at generation $1$.  This population may contain red and blue particles.  If a blue particle $v$  is selected in $\\V^N$, we select the corresponding edge $(u, v)$ in $G_0$. Note that in this case we indeed have $i_v \\le N$. We also remark that several selected edges may point to the same vertex $v$. If this happens, we keep blue the corresponding particle in $\\V^N$ which has the minimal position in $\\V$,  and color the other particles corresponding to $v$ {\\it purple.} We then define $Z_1$ to be the set of blue particles in ${\\mathcal V}^N$ at generation $1$, after removing all purple particles. We identify $Z_1$ with a subset of $G_0$: a blue particle of $Z_1$ in $\\V^N$ corresponds exactly to one (active) vertex of $G_0$, and the paths $(u, v)_{u\\in Z_0, v \\in Z_1}$ form  ${\\mathcal Q}_1$. Since $i_v\\le N$ if $(u,v)\\in {\\mathcal Q}_1$,  we have $\\V^N(v)-\\V^N(u)= \\V(v)-\\V(u)$ and \n \\begin{equation}\\label{eq:index}\n  V(v)-V(u):=X(u,v) \\le \\frac{n}{n-N} (\\V^N(v)-\\V^N(u)) + \\frac{N}{n-N}.\n\\end{equation} \n Finally,  $G_1$ is the sub-complete graph formed by  removing $Z_0$ and all incident edges in $G_0$.  \n\nFor the next steps, we proceed by induction on $j$ as long as $Z_j\\neq \\emptyset$. See Figure \\ref{f:coupling}.\n\n \\begin{figure}[ht!]\n\\centering\n \\begin{tikzpicture}[scale=0.9, \n    >=Stealth,\n    vertex/.style={circle,draw,inner sep=1pt,minimum size=5mm},\n    bluev/.style={vertex,fill=blue!30},\n    redv/.style={vertex,fill=red!50},\n    purplev/.style={vertex,fill=violet!55},\n    thin edge/.style={line width=0.05pt},\n    dottedbox/.style={dotted,thick},\n    small/.style={font=\\scriptsize}\n]\n\n\\node at (0,4) {$G_j$};\n\n\\node[bluev]   (u1) at (-2,2) {$u_1$};\n\\node[bluev]   (u2) at (-0.3,3) {$u_2$};\n\\node[bluev]   (u3) at (2,2) {$u_3$};\n\n\\node[vertex] (v1)  at (-2,0.5) {$v_1$};\n\\node[vertex] (v2)  at (-1,-0.8) {$v_2$};\n\\node[vertex] (v3)  at (0,-1.2) {$v_3$};\n\\node[vertex] (v4)  at (1,-0.8) {$v_4$};\n\\node[vertex] (v5)  at (2,0.5) {$v_5$};\n\n\\foreach \\x/\\y in {u1/u2,u2/u3,u1/u3,u1/v1,u1/v2,u1/v3,u2/v1,u2/v2,u2/v3,u3/v1,u3/v2,u3/v3, v1/v2, v1/v3, v2/v3, u1/v4, u2/v4, u3/v4, v1/v4, v2/v4, v4/v4, u1/v5, u2/v5, u3/v5, v1/v5, v2/v5, v3/v5, v4/v5}\n  \\draw[thin edge] (\\x) -- (\\y);\n\n\\node[small] at (0,-2) {$Z_j=\\{u_1,u_2,u_3\\}$};\n\\node[small] at (0,-2.5) {$G_j\\setminus Z_j=\\{v_1,v_2,v_3, v_4, v_5\\}$};\n\n\\node at (6.8,4) {\\small $\\V^N$ at time $j$};\n\n\\node[bluev] (r1) at (6.3,3) {$u_1$};\n\\node[bluev] (r2) at (7,3) {$u_2$};\n\\node[bluev] (r3) at (8.4,3) {$u_3$};\n\\node[purplev] (r4) at (7.7,3) {$\\cdot$};\n\\node[redv] (r5) at (5.6,3) {$\\cdot$};\n\n\\draw[dottedbox] ($(r5)+(-0.5,-0.5)$) rectangle ($(r3)+(0.5,0.5)$);\n\n\\node[bluev] (a1) at (5.6,0) {$v_1$};\n\\node[redv]  (a2) at (6.3,0) {$\\cdot$};\n\n\\node[purplev] (b1) at (7,0) {$v_1$};\n\\node[bluev]   (b2) at (7.7,0) {$v_2$};\n\n\\node[redv] (c1) at (8.4,0) {$\\cdot$};\n\n\\node[bluev]  (c2) at (9.35,0) {$v_5$};\n\\node[bluev]  (c3) at (10.05,0) {$v_4$};\n\\node[redv]  (c4) at (10.75,0) {$\\cdot$};\n\n\\draw (r1)--(a1);\n\\draw (r1)--(b2);\n\\draw [dashed] (r1)--(a2);\n\n\\draw (r2)--(b1);\n\n\\draw  (r3)--(c2);\n\\draw  (r3)--(c3);\n\\draw [dashed] (r5)--(c4);\n\n\\draw [dashed](r4)--(c1);\n\n\\draw[dottedbox] ($(a1)+(-0.5,-0.5)$) rectangle ($(c1)+(0.5,0.5)$);\n\n\\node[small] at (6.8,-0.8) {$\\mathcal V^N$ at time $j+1$};\n\\node[small] at (6.8,-1.2) {by keeping the $N$ leftmost particles};\n\n\\node[small] (Zj1) at (6.8,-2) {$Z_{j+1}=\\{v_1,v_2\\}$};\n\\node[small] (Zj1) at (6.8,-2.5) {$G_{j+1}=\\{v_1, v_2, v_3, v_4, v_5\\}$};\n\n\\end{tikzpicture}\n\\caption{\\small  Schematic illustration of the coupling between $(Z_j, G_j)$ and $\\V^N$.}\n\\label{f:coupling}\n\\end{figure}\n\n Suppose we are at step $j$ and have constructed the graph $G_j$, the set $Z_j\\subset G_j$ and the paths in $\\mathcal Q_j$.  Each active vertex  $u\\in Z_j$ corresponds to a blue particle $u$ of $\\V^N$ at time $j$. We randomly color $\\#G_j-\\#Z_j$ children of $u$ in $\\V$ blue, among the $n-1$ leftmost ones, so that each blue child $v$ corresponds to an edge in $\\{u\\}\\times (G_j\\setminus Z_j)$.  The remaining children of $u$ in $\\V$ are colored {\\it red}.  We couple the family $(X(u,v))_{v\\in G_j\\setminus Z_j}$ with the displacements of the blue particles in $\\V$ via the same coupling described above. \n\n Purple and red particles at time $j$ give birth only to red particles. We then retain the $N$ leftmost particles of the population at generation $j+1$, which form the population of $\\V^N$ at generation $j+1$.  It is composed of blue/red particles. If a blue particle $|v|=j+1$ with parent $u$ is selected in $\\V^N$, we select the corresponding edge $(u,v)$ in $G_j$.  In the case several selected edges  point to the same vertex $v$ in $G_j \\setminus Z_j$, so that multiple blue particles at time $j+1$ correspond to the same vertex $v$, we keep the one with minimal position in $\\V$ {\\it blue} and color the other particles {\\it purple.}  By definition, $V(v):=V(u)+X(u,v)$ is the weight $S_{j+1}(\\gamma,v)$ of the path $(\\gamma,v)$ in $K_n$ obtained by  adding the edge $e=(u,v)$ to the unique path $\\gamma \\in {\\mathcal Q}_j$ ending at $u$. The blue particles  $v$ in $\\V^N$ at time $j+1$ form the set $Z_{j+1}$, and the paths $(\\gamma,v)$ the set ${\\mathcal Q}_{j+1}$.  Finally, we remove $Z_j$ and all incident edges to obtain the graph $G_{j+1}$.  We note that \\eqref{eq:index} also holds for any $v \\in Z_{j+1}$ (if it exists). Summing these inequalities, we obtain that \\begin{equation}\\label{eq:index-2}\nV(v) \\le \\frac{n}{n-N}  \\V^N(v)+\\frac{N}{n-N}(j+1) , \\qquad \\forall\\, v \\in Z_{j+1}.\n\\end{equation} \n\n\\noindent Therefore, we have constructed an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$, together with its coupling with $\\V^N$,  up to the first index $j$ for which $Z_j=\\emptyset$.  \\hfill {\\footnotesize $\\Box$}\n\n\\medskip\n   Let $|u|=j$ be a blue particle in $\\V^N$ at time $j$ and $v$ be a child of $u$ in $\\V$ among the $n-1$ leftmost children. Conditionally on the whole branching random walk $\\V$ and     on the exploration process up to time $j$, the probability for $v$ to be colored  red is \n\\[ 1-\\frac{ \\#G_{j+1}}{n-1}= \\frac{n-1-\\#G_0+\\#Z_0 + \\#Z_1+\\ldots+\\#Z_j }{n-1}\\le \\frac{n-\\#G_0+N(j+1)}{n-1}.\n\\]  \n\nNow, we estimate the probability that $v$ is colored purple, given it is not red. In particular, $v$ belongs to $\\V^N$ at time $j+1$. To avoid confusion, write temporarily $w$ for the vertex in $K_n$ associated to the particle $v$ in $\\V^N$. Whenever $v$ is colored purple,  there must exist some particle $u'\\neq u$  in $\\V^N$ at time $j$ and some child $v'$ of $u'$ in $\\V^N$ such that  $\\V^N(v')-\\V^N(u')$  is less than $x:=\\V^N(v)- \\V^N(u')$, and $v'$ is coupled with the vertex $w$. Let  ${\\mathcal N}_{u'}$ be the set of children of $u'$ that are not red and  whose displacement (in $\\V$) with respect to $u'$ is less than $x$.  By our coupling, we identify ${\\mathcal N}_{u'}$ with a set of vertices in $G_{j+1}$. In particular, $w\\in {\\mathcal N}_{u'}$.\n  By symmetry, conditionally on $\\V$, on the exploration process up to time $j$, and on the red/non-red particles at time $j+1$,   the probability that $w\\in {\\mathcal N}_{u'}$ is $\\frac{\\#{\\mathcal N}_{u'}}{\\#G_{j+1}}$.   Since $\\sum_{u'} \\#{\\mathcal N}_{u'} \\le N$, we obtain that the conditional probability that $v$ is colored purple, given it is not colored red, is less than  \n \\[\n  \\frac{N}{\\#G_{j+1}}\\le \\frac{N}{\\#G_0-N(j+1)}\n  \\]\nas long as $N(j+1)<\\#G_0$.  Since $\\p(v \\textrm{ is not blue} \\mid \\V) = \\p(u \\textrm{ is not blue} \\mid \\V)+\\p(u \\textrm{ is blue}, v \\textrm{ is not blue} \\mid \\V)$, we find by induction that\n \\begin{equation}\\label{proba_red}\n       \\p( v \\textrm{ is not blue} \\mid \\V) \\le \\sum_{k=0}^j \\Big(\\frac{n-\\#G_0+N(k+1)}{n-1}  +\\frac{N}{\\#G_0-N(k+1)}\\Big).\n  \\end{equation}\n\n\\subsection{Lower bound in Theorem \\ref{t:main}}\nThe lower bound of Theorem \\ref{t:main} is a consequence  of the following lemma.\n\\begin{lemma}\n Let $\\alpha < \\frac{\\pi^2}{2}$ and $t<\\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$. Choose $a\\in (0,1)$ close enough to $1$ such that \n\\[\n- a + \\frac{\\pi^2}{2 a^2} t < \\alpha t .\n\\]\nApply the above exploration process with $G_0=K_n$ and $N_0=N=\\lfloor n^a\\rfloor$. With probability tending to $1$ as $n\\to\\infty$, there is an active vertex  at time $\\lfloor t\\ln^3 n\\rfloor $ such that $V(u)\\le \\alpha t\\ln n$.  \n\\end{lemma}\n \\begin{proof}\n   For simplicity, suppose that $N=n^a$ and that $L:=t\\ln^3 n= \\frac{t}{a^3} \\ln^3 N$ is an integer.  Note that $(-1+ \\frac{\\pi^2}2 \\frac{t}{a^3}) \\ln N= (-a + \\frac{\\pi^2}{2 a^2} t) \\ln n $. Fix   $b\\in (0,\\alpha)$ such that $- a + \\frac{\\pi^2}{2 a^2} t<  b t$. By Proposition \\ref{p:first_moment} (with $a=1$ there), there is  a particle $|u|=L$ in $\\V^N$ with $\\V^N(u)\\le b t \\ln n$ with probability going to $1$ as $n\\to\\infty$. By \\eqref{proba_red}, the probability for $u$ to be blue also tends to $1$ as $n\\to\\infty$.  It remains to prove that $V(u)\\le \\alpha t \\ln n$. By \\eqref{eq:index-2},  $V(u)\\le (bt \\ln n)  \\frac{n}{n-N} + \\frac{N}{n-N} L \\le \\alpha t \\ln n$ for $n$ large enough indeed. \n\\end{proof}",
      "s:rw-brw": "\\begin{equation}    V(u^*_{j}) \\le \\frac{n}{n-N} \\sum_{i=1}^{j} \\eta_i + \\frac{Nj}{n-N}, \\qquad \\forall\\, j< \\tau_1.\\label{control-Vuj} \\end{equation}  \n\n\\noindent Hence $${\\tt s}_1 \\le \\frac{n}{n-N} \\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0) + \\frac{N (\\Delta-1) }{n-N}.$$\n\n The term $\\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0)$ is stochastically smaller than the sum of $\\Delta-1$ independent exponential r.v. with mean $2e$, as long as $G_{\\Delta}$ has more than $\\frac{n}{2}$ vertices. This condition is indeed satisfied thanks to \\eqref{card_G}. Then for some positive constant $c$,  $$  \\e_{G_{\\tau_{i-1}}}[({\\tt s}_1+\\Delta)^2] \\le c \\Delta^2.$$ \n\nBy Cauchy--Schwarz inequality and  \\eqref{bound_eventE}, we obtain that $$  \\e_{G_{\\tau_{i-1}}}[ ({\\tt s}_1+\\Delta) {\\bf 1}_{{\\mathcal E^c_1}}]\n\\le\nc^{1/2} \\Delta \\Big( (\\frac{2L}{n-1} +\\frac{1}{n-2NL})N\\Delta\\Big)^{1/2}\n\\le \\Delta^{-4}, \n$$\n\n\\noindent where the last inequality holds for all small $\\varepsilon$ and large $n$ (see \\eqref{eq:NLDelta}). It follows from \\eqref{cond-Gi} that $$ \\e \\Big[\\sum_{i=1}^{a_1L/\\Delta}   ({\\tt s}_i+\\Delta)   {\\bf 1}_{{\\mathcal E}_i^c}\\Big]\n\\le a_1 L \\Delta^{-3}.\n$$\n\n\\noindent Since $L \\Delta^{-3}= o(\\beta_n L)$, the Markov inequality yields \\eqref{sum-S-tau<Delta-2} \nand completes the proof of the lemma.\n\\end{proof}\n\n\\section{Proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper},  and of Proposition  \\ref{p:first_moment}}\\label{s:rw-brw}\n\nRecall that $(Y_n)_{n\\ge 0}$ is a centered random walk starting at $0$ with step distribution $\\mbox{Exp}(1)-1$. In the first subsection, we present several estimates on $(Y_n)_{n\\ge 0}$ and provide  the proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper}. The second subsection is devoted to the study of the branching random walks $\\V$ and $\\V^N$, and to the proof of Proposition  \\ref{p:first_moment}. \n\n\\subsection{Estimates of random walks: Proofs of Lemmas \\ref{main:upper} and \\ref{main2:upper}}\\label{sub:rw}\n\n\\begin{lemma}\\label{l:mogulski}\n Let $\\upsilon>0$ and $0<d<\\frac{\\pi^2}{2}$. There exists $\\Delta_0>0$ such that for all $\\Delta\\ge \\Delta_0$,   $r_0\\le 0\\le r_0+\\Delta$ and  $\\ell\\ge 1$,\n\\begin{equation}\n    \\p(r_0  \\le Y_j\\le r_0+\\Delta,\\, \\forall \\, j\\in \\lb 1,  \\ell\\rb) \\le e^{\\Delta^\\upsilon-\\frac{d}{\\Delta^2}\\ell}. \\label{mogulskii_upper0} \n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3616,
    "pre_theorem_intro_text": "We consider the complete graph $K_n$ of the set of vertices $\\llbracket 1, n\\rrbracket=\\{1,2,\\ldots,n\\}$. Following Aldous \\cite{aldous}, we  assign each edge  an independent exponential random variable with mean $n$. In this setting, Aldous proved the existence of a phase transition for the size of  the largest tree with average weight below a given threshold \\cite[Theorem 1]{aldous}. We are interested in the analogous result for paths, rather than trees, established in  \\cite[Proposition 3.7]{aldous}. In order to state this result, we introduce the following notation. \nFor ease of notation, in this paper, we will suppose that the labels $X(u,v)_{u\\neq v \\in \\llbracket 1, n\\rrbracket}$  on the (non-oriented) edges of $K_n$ are i.i.d. copies of \n\\begin{equation}\\label{def:X}\nX=ne \\mathrm{Exp}(1)-1\n\\end{equation} \nwhere $\\mathrm{Exp(1)}$ denotes an exponential random variable with mean $1$. \nA path $\\gamma$ is a sequence of vertices $(\\gamma_0,\\gamma_1,\\ldots,\\gamma_\\ell) \\in  \\llbracket 1, n\\rrbracket^{\\ell+1}$ which is made of distinct adjacent edges $(\\gamma_i,\\gamma_{i+1})$. We call $|\\gamma|=\\ell$ the length of the path, and let ${\\mathcal P}_\\ell$ be the collection of paths of length $\\ell$. A path of length $0$ is reduced to a single vertex.\n\nFor any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation}    \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad  i\\in \\llbracket 1,  \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation}    \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in  {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\n  Translated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and  ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$:  there exist some constants $c'> c >0$ and $C'>C$ such that \n\n (i) if $ -1\\le \\alpha \\le c $, then  \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty;  \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which  the longest path undergoes a transition from  order $\\ln^3 n$ to   polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le   n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1.  \\label{Ding-Goswami}\\end{equation} \n\n  The following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.",
    "context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.",
    "full_context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\p\\Big( C \\ln^3 n \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of $\\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\p\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le \\L(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nNote that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.\n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty} \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$. As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\, -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large.\n\n\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main}] Recall by Claim 2.4 of \\cite{ding} that if there is a path of length greater than $\\ell$ with average weight below $\\lambda$, then there is a path of length in $\\lb \\ell,2\\ell\\rb$ with average weight below $\\lambda$. Then for $\\lambda= \\frac{\\alpha}{\\ln^2 n}$, $t> \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$, and $\\ell= \\lfloor t \\ln^3 n\\rfloor$, \\begin{align} & \\p\\Big(\\L(n, \\lambda) \\ge \\ell\\Big) \n \\nonumber \\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L \\}\\Big) \\nonumber \n\\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) > -b_n\\}\\Big) + o(1), \\label{eq:L>ell-bn}\n\\end{align}\n\n\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\n\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}",
    "post_theorem_intro_text_len": 4434,
    "post_theorem_intro_text": "Note that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.  \n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln {\\mathscr L}(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\mathbb P\\Big(\\frac{1}{n}{\\mathscr L}(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation}    {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for ${\\mathscr L}(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.",
    "sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields Theorem~\\ref{t:main} (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem~\\ref{t:main2}). A “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”",
    "expanded_sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields the main theorem (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields the following theorem.\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\nA “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”",
    "expanded_theorem": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$,",
    "theorem_type": [
      "Implication",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Fix a real number \\(\\alpha<\\pi^2/2\\). For \\(\\lambda\\in(-1,\\infty)\\), define\n\\[\n{\\mathscr L}(n,\\lambda):=\\sup\\{\\ell\\ge 1:\\ \\exists\\text{ a path }\\gamma=(\\gamma_0,\\ldots,\\gamma_\\ell)\\text{ with }S_i(\\gamma)=\\sum_{k=1}^i X(\\gamma_{k-1},\\gamma_k)\\text{ and }S_\\ell(\\gamma)\\le \\lambda \\ell\\}.\n\\]\nThus \\({\\mathscr L}(n,\\lambda)\\) is the maximum path length for which there exists a path whose average weight \\(S_\\ell(\\gamma)/\\ell\\) is at most \\(\\lambda\\). Under these assumptions, which statement about \\({\\mathscr L}(n,\\alpha/\\ln^2 n)\\) holds as \\(n\\to\\infty\\)?",
      "correct_choice": {
        "label": "A",
        "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}+\\alpha}.\n\\]"
        },
        {
          "label": "C",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)}{\\ln^3 n}=O_{\\mathbb P}(1).\n\\]"
        },
        {
          "label": "D",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^2 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every fixed \\(\\alpha<\\pi^2/2\\), there exists a deterministic constant \\(C=C(\\alpha)>0\\) such that, with probability tending to \\(1\\) as \\(n\\to\\infty\\),\n\\[\n{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)= C\\ln^3 n+o(1).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "critical-drift denominator sign",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "explicit limiting constant",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "correct logarithmic scaling exponent",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "mode of approximation upgraded from relative/probabilistic scale to additive deterministic asymptotics",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the quantity and asks for the asymptotic regime, but it does not state the limit, scaling exponent, or mode of convergence. The condition \u001calpha < \\pi^2/2\u001d gives contextual information rather than leaking the answer."
      },
      "TAS": {
        "score": 1,
        "justification": "This is close to a theorem-recall item: it essentially asks for the exact limiting statement under the stated hypothesis. However, it is not a pure verbatim restatement because the student must distinguish among several nearby conclusions (sign, scaling, strength, and convergence mode)."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to select the strongest correct conclusion among close alternatives, especially to reject the wrong logarithmic scaling and the almost-sure version. Still, the item mainly tests recognition/recall of a known asymptotic result rather than deeper derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically meaningful and target common failure modes: sign error in the constant, weaker-but-true asymptotic boundedness, incorrect logarithmic exponent, and overly strong convergence mode. They are distinct and plausible, though choice E has a minor typo."
      },
      "total_score": 6,
      "overall_assessment": "A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall than genuine generative mathematical reasoning."
    }
  },
  {
    "id": "2512.23266v1",
    "paper_link": "http://arxiv.org/abs/2512.23266v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$",
      "start_pos": 6513,
      "end_pos": 6711,
      "label": "t:main"
    },
    "ref_dict": {
      "Ding-2": "\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}",
      "alpha:polynomial": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}",
      "t:main": "\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}",
      "s:upper": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for $\\L(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for $\\L(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.\n\n\\section{Proofs of the upper bounds in Theorems \\ref{t:main} and \\ref{t:main2}}\n\\label{s:upper}\n\nLet $(S_1,S_2,\\ldots)$ denote a random walk with step distribution $X$ of \\eqref{def:X}, starting at $S_0=0$. For any $\\ell\\ge 1$ and Borel set $A\\subset \\r^\\ell$, \n\\begin{equation}\\label{comp:moment1}\n\\e\\Big[\\#\\{\\gamma \\in {\\mathcal P}_\\ell\\colon S(\\gamma) \\in A\\}\\Big] \\le n^{\\ell+1}\\p\\Big((S_1,\\ldots,S_\\ell) \\in A\\Big), \n\\end{equation}",
      "t:main2": "\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}",
      "s:lower": "\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main2}] Let $n$ be large and  $a \\in (0, 1)$. Consider $\\ell= \\lceil ne^{- \\frac{\\pi}{\\sqrt{ 2 \\beta_n}} (1-a)}\\rceil$,  $r=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-\\frac{a}2)}$ and  let $m'_\\Delta(L)$ be the number of $i\\in \\lb 0,L\\rb$ such that $\\inf_{j \\in \\lb i,L\\rb} S_j\\le S_i- \\Delta$ with $\\Delta:=\\ln r$.  Let $b_n=\\ln n + \\ln\\ln n$.  By the same argument leading to \\eqref{eq:L>ell-bn}, but using Corollary \\ref{cor:jump} (ii) in addition, we obtain that  for any $M>2$, $$\\p\\Big(\\L(n, \\beta_n) \\ge \\ell\\Big) \\le  \\sum_{L=\\ell}^{2\\ell} \\widetilde a_L + \\frac{2}{M}+ o(1),$$\n\n\\noindent where for $L\\in \\lb \\ell, 2\\ell \\rb$, $$ \\widetilde a_L:= \\p\\Big(\\exists \\gamma\\in {\\mathcal P}_L: S_L(\\gamma) \\le \\beta_n L, m'_\\Delta(L) \\le M \\frac{n}{r},\\, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) >-  b_n \\Big).$$\n\nBy \\eqref{comp:moment1} and \\eqref{eq:many-to-one}, \\begin{align*}    \\widetilde a_L & \\le n \\, e^{\\beta_n L}  \\p\\Big( Y_L\\le \\beta_n L , \\, m_\\Delta(L) \\le M \\frac{n}{r},\\,  \\min_{0\\le i \\le j \\le L} (Y_j - Y_i ) >- b_n\\Big)\n\\\\\n& \\le n \\, e^{\\beta_n L}  \\p\\Big(   \\max_{i \\in \\lb 0, L\\rb} Y_i \\le 2 \\beta_n L, \\,  m_\\Delta(L) \\le M \\frac{n}{r}\\Big)\n ,\n\\end{align*} where $m_\\Delta(L)$ is as defined in Lemma \\ref{main2:upper} and in the last inequality, we have used the fact that $\\max_{i \\in \\lb 0, L\\rb} Y_i\\le Y_L+b_n \\le \\beta_n L+ b_n \\le 2 \\beta_n L$. Choose $d\\in (0, \\frac{\\pi^2}2)$ \n  such that $\\eta:= \\frac{2d}{\\pi^2 (1-\\frac{a}2)^2} - 1>0$. Fix an arbitrary $c\\in (0, 1)$. If $\\varepsilon>0$ is chosen small enough in Theorem \\ref{t:main2}, we have $L\\ge \\Delta^{2+c}$, $M \\frac{n}{r} \\le L \\Delta^{-2-c}$ and $2 \\beta_n L \\le L \\Delta^{-1-c}$.  Note that $\\Delta$ is arbitrarily large when $\\varepsilon$ goes to $0$.   Lemma \\ref{main2:upper} yields that $$ \\widetilde a_L \\le n e^{\\beta_n L} e^{- d \\frac{L}{\\Delta^2}} = n e^{- \\eta \\,\\beta_n L}. $$\n\n Notice that for all large $n$, $\\beta_n L \\ge \\beta_n \\ell \\ge n^{a'}$ for some positive constant $a'$. Hence   $\\sum_{L=\\ell}^{2\\ell} \\widetilde a_L \\le n^2  e^{- \\eta\\, n^{a'}}\\to 0$ and we conclude the proof by letting $n \\to\\infty$ then $M\\to\\infty$.  \\end{proof}\n\n\\section{Proofs of the lower bounds}\n\\label{s:lower}\n\nWe consider a branching random walk $\\V$ on $\\r$. It starts with a certain number of particles at $0$. At each discrete time $k\\ge 0$, each particle of the current population splits into an infinite number of children, whose displacements from their parent  are given by a copy of \n\\begin{equation} \\label{eq:BRW-V}   \n\\sum_{i=1}^\\infty {\\delta}_{\\{e T_i -1\\}} \n\\end{equation}\n\n\\noindent where $0<T_1<T_2<\\ldots$ are the atoms of a Poisson point process of intensity $1$ on $\\r_+$. The position of a particle $u$ is denoted by $\\V(u)$.\nThe offspring distribution corresponds to the limiting law  of $\\sum_{u=2}^{n} \\delta_{X(1,u)}$, so that, when $n$ is large, exploring the graph around a given vertex locally approximates the branching random walk $\\V$. To make it precise, we will  couple labels $X(u,v)$ and copies of the Poisson point process $(T_i)_{i\\ge 1}$ as follows. \n\n\\medskip\n\nConsider $(X_i)_{1\\le i \\le n-1}$ a family of $(n-1)$ i.i.d. copies of $X= n e \\mbox{Exp}(1)-1$.  Its increasing order statistics $ X_{(0)}:=-1 < X_{(1)}< ...< X_{(n-1)}$ verify that the variables $X_{(i)}-X_{(i-1)}$ are independent, distributed as $\\frac{n e }{n-i}\\text{Exp}(1)$.   One can couple it with a Poisson point process $T_1<T_2<\\ldots$ of intensity $1$, by \n\\[     X_{(i)}-X_{(i-1)} = \\frac{n e}{n-i} (T_{i}-T_{i-1}), \\qquad i\\in \\lb 1, n-1\\rb, \n\\]where we set $T_0=0$.  In particular,  for all $i\\in \\lb 1, n-1\\rb$, \\begin{equation}\\label{coupling}\n   e T_i-1 \\le  X_{(i)} \\le   \\frac{n}{n-i} (e T_{i}-1) + \\frac{i}{n-i}.\n\\end{equation}\n\nFor each $u\\in K_n$, this gives a coupling between $(X(u,v))_{v\\neq u}$ and a copy of $(T_i)_{i\\ge 1}$.  We further need the following simple observation.  Let $m \\in \\lb 1, n-1 \\rb$. If we choose   $m$  points uniformly from $(X_{(i)})_{1\\le i \\le n-1}$ and arrange them in increasing order, then they have the same distribution as  the increasing order statistics of $(X_i)_{1\\le i \\le m}$. In other words, there exists a coupling in the sense of \\eqref{coupling}, between $X(u, v)$ when $v$ varies over a subset of $K_n$ with $m$ vertices, and the family of $m$  points  chosen uniformly from $(T_i)_{1\\le i \\le n-1}$.  \n\nLet $N$ be an integer between $1$ and $n-1$.  We denote by $\\V^N$ the branching random walk $\\V$  in which, at each step, we only keep the $N$ leftmost particles of the next generation. For a particle $u$ in $\\V^N$, let   $|u|$ denote its generation in the genealogical tree of $\\V^N$, and let $\\V^N(u)$ denote its position in $\\r$.\n\n\\begin{proposition}\\label{p:first_moment} Fix $a\\in [0, 1]$ and $t>0$.    If $\\V^N$ starts with $\\lfloor N^a\\rfloor$ particles located at $0$, then for any sequence $\\ell_N$ satisfying $\\frac{\\ell_N}{\\ln^3 N} \\to t$,  \\begin{equation}  \\frac{1}{\\ln N}  \\min_{|u|=\\ell_N} \\V^N(u)   \\,\\cvL2\\, -a+\\frac{\\pi^2}{2}t  , \\qquad N \\to\\infty. \\label{Nbrw-cvL2}\\end{equation}\n\\end{proposition}\n\nWe will apply the above proposition to two cases: $a=1$ and $a=0$.\nThe proof of Proposition \\ref{p:first_moment} is postponed to Section \\ref{sub:brw}. \n\nIn what follows, we construct a coupling between $\\V^N$ and an exploration process on $K_n$. \nWe will color particles of $\\V^{N}$ in blue, red or purple. Blue particles will correspond to  vertices of $K_{n}$ and will be the particles of interest. \n\n\\subsection{Construction of an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$ in $K_n$ and its coupling with $\\V^N$.}\n \\label{s:exploration}\n\nFor each $j\\ge 0$, we define a  set $Z_j$ of so-called {\\it active} vertices of $K_n$ and a set of paths ${\\mathcal Q}_j\\subset {\\mathcal P}_j$  such that each vertex  in $Z_j$ is associated with a unique  path in ${\\mathcal Q}_j$ ending at this point. The set $Z_j$ will correspond to the blue particles of $\\V^{N}$ at time $j$. We will therefore write $u$ either for the blue particle or its corresponding  vertex in $K_n$.  For each active vertex $u$ at step $j$, we will write $V(u)$  for $S_j(\\gamma)$ where $\\gamma$ is the path in ${\\mathcal Q}_j$ ending at $u$.  \n\nLet $G_0$ be a complete subgraph of $K_n$. Let $Z_0\\subset G_0$ be a set of $N_0\\le N$ active vertices at time $0$. We start  the branching random walk $\\V$ with  $N_0$ blue particles,  each particle corresponding to an active vertex in $Z_0$.  We let $\\mathcal Q_0$ be the collection $\\{\\{u\\},\\, u\\in Z_0\\}$.  We denote by $\\#G_0$ the number of vertices in $G_0$. Let $m:=\\#G_0- \\#Z_0$.   Each $u\\in Z_0$ corresponds to a blue particle in $\\V$, still denoted by $u$. We have $\\V(u)=V(u)=0$. We  choose $m$ children $v$ of $u$ in $\\V$ uniformly among the $n-1$ leftmost ones and color them {\\it blue.}  By the coupling described above, for each blue particle $u$, $\\V(v)-\\V(u)$ is coupled with $X(u, v)$ for $v \\in G_0 \\backslash Z_0$ (where, as for $u$, the variable $v$ denotes both a particle in $\\V$ and a vertex in $G_0$).  Let $i_v$ be the index of the point $T_i$ corresponding to $v$ in the coupling \\eqref{coupling}. We have  $$X(u,v) = X_{(i_v)} \\le \\frac{n}{n-i_v} e T_{i_v} -1 =   \\frac{n}{n-i_v} (\\V(v)-\\V(u))+\\frac{i_v}{n-i_v}.$$ We also have $X(u,v) \\ge \\V(v)-\\V(u)$. \nWe will see that $i_v \\le N$ for all $v$ of interest.  We color the remaining children of $u$ in $\\V$ {\\it red.}  We then retain the $N$ leftmost particles of $\\V$ in generation $1$ which is in fact the population of $\\V^N$ at generation $1$.  This population may contain red and blue particles.  If a blue particle $v$  is selected in $\\V^N$, we select the corresponding edge $(u, v)$ in $G_0$. Note that in this case we indeed have $i_v \\le N$. We also remark that several selected edges may point to the same vertex $v$. If this happens, we keep blue the corresponding particle in $\\V^N$ which has the minimal position in $\\V$,  and color the other particles corresponding to $v$ {\\it purple.} We then define $Z_1$ to be the set of blue particles in ${\\mathcal V}^N$ at generation $1$, after removing all purple particles. We identify $Z_1$ with a subset of $G_0$: a blue particle of $Z_1$ in $\\V^N$ corresponds exactly to one (active) vertex of $G_0$, and the paths $(u, v)_{u\\in Z_0, v \\in Z_1}$ form  ${\\mathcal Q}_1$. Since $i_v\\le N$ if $(u,v)\\in {\\mathcal Q}_1$,  we have $\\V^N(v)-\\V^N(u)= \\V(v)-\\V(u)$ and \n \\begin{equation}\\label{eq:index}\n  V(v)-V(u):=X(u,v) \\le \\frac{n}{n-N} (\\V^N(v)-\\V^N(u)) + \\frac{N}{n-N}.\n\\end{equation} \n Finally,  $G_1$ is the sub-complete graph formed by  removing $Z_0$ and all incident edges in $G_0$.  \n\nFor the next steps, we proceed by induction on $j$ as long as $Z_j\\neq \\emptyset$. See Figure \\ref{f:coupling}.\n\n \\begin{figure}[ht!]\n\\centering\n \\begin{tikzpicture}[scale=0.9, \n    >=Stealth,\n    vertex/.style={circle,draw,inner sep=1pt,minimum size=5mm},\n    bluev/.style={vertex,fill=blue!30},\n    redv/.style={vertex,fill=red!50},\n    purplev/.style={vertex,fill=violet!55},\n    thin edge/.style={line width=0.05pt},\n    dottedbox/.style={dotted,thick},\n    small/.style={font=\\scriptsize}\n]\n\n\\node at (0,4) {$G_j$};\n\n\\node[bluev]   (u1) at (-2,2) {$u_1$};\n\\node[bluev]   (u2) at (-0.3,3) {$u_2$};\n\\node[bluev]   (u3) at (2,2) {$u_3$};\n\n\\node[vertex] (v1)  at (-2,0.5) {$v_1$};\n\\node[vertex] (v2)  at (-1,-0.8) {$v_2$};\n\\node[vertex] (v3)  at (0,-1.2) {$v_3$};\n\\node[vertex] (v4)  at (1,-0.8) {$v_4$};\n\\node[vertex] (v5)  at (2,0.5) {$v_5$};\n\n\\foreach \\x/\\y in {u1/u2,u2/u3,u1/u3,u1/v1,u1/v2,u1/v3,u2/v1,u2/v2,u2/v3,u3/v1,u3/v2,u3/v3, v1/v2, v1/v3, v2/v3, u1/v4, u2/v4, u3/v4, v1/v4, v2/v4, v4/v4, u1/v5, u2/v5, u3/v5, v1/v5, v2/v5, v3/v5, v4/v5}\n  \\draw[thin edge] (\\x) -- (\\y);\n\n\\node[small] at (0,-2) {$Z_j=\\{u_1,u_2,u_3\\}$};\n\\node[small] at (0,-2.5) {$G_j\\setminus Z_j=\\{v_1,v_2,v_3, v_4, v_5\\}$};\n\n\\node at (6.8,4) {\\small $\\V^N$ at time $j$};\n\n\\node[bluev] (r1) at (6.3,3) {$u_1$};\n\\node[bluev] (r2) at (7,3) {$u_2$};\n\\node[bluev] (r3) at (8.4,3) {$u_3$};\n\\node[purplev] (r4) at (7.7,3) {$\\cdot$};\n\\node[redv] (r5) at (5.6,3) {$\\cdot$};\n\n\\draw[dottedbox] ($(r5)+(-0.5,-0.5)$) rectangle ($(r3)+(0.5,0.5)$);\n\n\\node[bluev] (a1) at (5.6,0) {$v_1$};\n\\node[redv]  (a2) at (6.3,0) {$\\cdot$};\n\n\\node[purplev] (b1) at (7,0) {$v_1$};\n\\node[bluev]   (b2) at (7.7,0) {$v_2$};\n\n\\node[redv] (c1) at (8.4,0) {$\\cdot$};\n\n\\node[bluev]  (c2) at (9.35,0) {$v_5$};\n\\node[bluev]  (c3) at (10.05,0) {$v_4$};\n\\node[redv]  (c4) at (10.75,0) {$\\cdot$};\n\n\\draw (r1)--(a1);\n\\draw (r1)--(b2);\n\\draw [dashed] (r1)--(a2);\n\n\\draw (r2)--(b1);\n\n\\draw  (r3)--(c2);\n\\draw  (r3)--(c3);\n\\draw [dashed] (r5)--(c4);\n\n\\draw [dashed](r4)--(c1);\n\n\\draw[dottedbox] ($(a1)+(-0.5,-0.5)$) rectangle ($(c1)+(0.5,0.5)$);\n\n\\node[small] at (6.8,-0.8) {$\\mathcal V^N$ at time $j+1$};\n\\node[small] at (6.8,-1.2) {by keeping the $N$ leftmost particles};\n\n\\node[small] (Zj1) at (6.8,-2) {$Z_{j+1}=\\{v_1,v_2\\}$};\n\\node[small] (Zj1) at (6.8,-2.5) {$G_{j+1}=\\{v_1, v_2, v_3, v_4, v_5\\}$};\n\n\\end{tikzpicture}\n\\caption{\\small  Schematic illustration of the coupling between $(Z_j, G_j)$ and $\\V^N$.}\n\\label{f:coupling}\n\\end{figure}\n\n Suppose we are at step $j$ and have constructed the graph $G_j$, the set $Z_j\\subset G_j$ and the paths in $\\mathcal Q_j$.  Each active vertex  $u\\in Z_j$ corresponds to a blue particle $u$ of $\\V^N$ at time $j$. We randomly color $\\#G_j-\\#Z_j$ children of $u$ in $\\V$ blue, among the $n-1$ leftmost ones, so that each blue child $v$ corresponds to an edge in $\\{u\\}\\times (G_j\\setminus Z_j)$.  The remaining children of $u$ in $\\V$ are colored {\\it red}.  We couple the family $(X(u,v))_{v\\in G_j\\setminus Z_j}$ with the displacements of the blue particles in $\\V$ via the same coupling described above. \n\n Purple and red particles at time $j$ give birth only to red particles. We then retain the $N$ leftmost particles of the population at generation $j+1$, which form the population of $\\V^N$ at generation $j+1$.  It is composed of blue/red particles. If a blue particle $|v|=j+1$ with parent $u$ is selected in $\\V^N$, we select the corresponding edge $(u,v)$ in $G_j$.  In the case several selected edges  point to the same vertex $v$ in $G_j \\setminus Z_j$, so that multiple blue particles at time $j+1$ correspond to the same vertex $v$, we keep the one with minimal position in $\\V$ {\\it blue} and color the other particles {\\it purple.}  By definition, $V(v):=V(u)+X(u,v)$ is the weight $S_{j+1}(\\gamma,v)$ of the path $(\\gamma,v)$ in $K_n$ obtained by  adding the edge $e=(u,v)$ to the unique path $\\gamma \\in {\\mathcal Q}_j$ ending at $u$. The blue particles  $v$ in $\\V^N$ at time $j+1$ form the set $Z_{j+1}$, and the paths $(\\gamma,v)$ the set ${\\mathcal Q}_{j+1}$.  Finally, we remove $Z_j$ and all incident edges to obtain the graph $G_{j+1}$.  We note that \\eqref{eq:index} also holds for any $v \\in Z_{j+1}$ (if it exists). Summing these inequalities, we obtain that \\begin{equation}\\label{eq:index-2}\nV(v) \\le \\frac{n}{n-N}  \\V^N(v)+\\frac{N}{n-N}(j+1) , \\qquad \\forall\\, v \\in Z_{j+1}.\n\\end{equation} \n\n\\noindent Therefore, we have constructed an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$, together with its coupling with $\\V^N$,  up to the first index $j$ for which $Z_j=\\emptyset$.  \\hfill {\\footnotesize $\\Box$}\n\n\\medskip\n   Let $|u|=j$ be a blue particle in $\\V^N$ at time $j$ and $v$ be a child of $u$ in $\\V$ among the $n-1$ leftmost children. Conditionally on the whole branching random walk $\\V$ and     on the exploration process up to time $j$, the probability for $v$ to be colored  red is \n\\[ 1-\\frac{ \\#G_{j+1}}{n-1}= \\frac{n-1-\\#G_0+\\#Z_0 + \\#Z_1+\\ldots+\\#Z_j }{n-1}\\le \\frac{n-\\#G_0+N(j+1)}{n-1}.\n\\]  \n\nNow, we estimate the probability that $v$ is colored purple, given it is not red. In particular, $v$ belongs to $\\V^N$ at time $j+1$. To avoid confusion, write temporarily $w$ for the vertex in $K_n$ associated to the particle $v$ in $\\V^N$. Whenever $v$ is colored purple,  there must exist some particle $u'\\neq u$  in $\\V^N$ at time $j$ and some child $v'$ of $u'$ in $\\V^N$ such that  $\\V^N(v')-\\V^N(u')$  is less than $x:=\\V^N(v)- \\V^N(u')$, and $v'$ is coupled with the vertex $w$. Let  ${\\mathcal N}_{u'}$ be the set of children of $u'$ that are not red and  whose displacement (in $\\V$) with respect to $u'$ is less than $x$.  By our coupling, we identify ${\\mathcal N}_{u'}$ with a set of vertices in $G_{j+1}$. In particular, $w\\in {\\mathcal N}_{u'}$.\n  By symmetry, conditionally on $\\V$, on the exploration process up to time $j$, and on the red/non-red particles at time $j+1$,   the probability that $w\\in {\\mathcal N}_{u'}$ is $\\frac{\\#{\\mathcal N}_{u'}}{\\#G_{j+1}}$.   Since $\\sum_{u'} \\#{\\mathcal N}_{u'} \\le N$, we obtain that the conditional probability that $v$ is colored purple, given it is not colored red, is less than  \n \\[\n  \\frac{N}{\\#G_{j+1}}\\le \\frac{N}{\\#G_0-N(j+1)}\n  \\]\nas long as $N(j+1)<\\#G_0$.  Since $\\p(v \\textrm{ is not blue} \\mid \\V) = \\p(u \\textrm{ is not blue} \\mid \\V)+\\p(u \\textrm{ is blue}, v \\textrm{ is not blue} \\mid \\V)$, we find by induction that\n \\begin{equation}\\label{proba_red}\n       \\p( v \\textrm{ is not blue} \\mid \\V) \\le \\sum_{k=0}^j \\Big(\\frac{n-\\#G_0+N(k+1)}{n-1}  +\\frac{N}{\\#G_0-N(k+1)}\\Big).\n  \\end{equation}\n\n\\subsection{Lower bound in Theorem \\ref{t:main}}\nThe lower bound of Theorem \\ref{t:main} is a consequence  of the following lemma.\n\\begin{lemma}\n Let $\\alpha < \\frac{\\pi^2}{2}$ and $t<\\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$. Choose $a\\in (0,1)$ close enough to $1$ such that \n\\[\n- a + \\frac{\\pi^2}{2 a^2} t < \\alpha t .\n\\]\nApply the above exploration process with $G_0=K_n$ and $N_0=N=\\lfloor n^a\\rfloor$. With probability tending to $1$ as $n\\to\\infty$, there is an active vertex  at time $\\lfloor t\\ln^3 n\\rfloor $ such that $V(u)\\le \\alpha t\\ln n$.  \n\\end{lemma}\n \\begin{proof}\n   For simplicity, suppose that $N=n^a$ and that $L:=t\\ln^3 n= \\frac{t}{a^3} \\ln^3 N$ is an integer.  Note that $(-1+ \\frac{\\pi^2}2 \\frac{t}{a^3}) \\ln N= (-a + \\frac{\\pi^2}{2 a^2} t) \\ln n $. Fix   $b\\in (0,\\alpha)$ such that $- a + \\frac{\\pi^2}{2 a^2} t<  b t$. By Proposition \\ref{p:first_moment} (with $a=1$ there), there is  a particle $|u|=L$ in $\\V^N$ with $\\V^N(u)\\le b t \\ln n$ with probability going to $1$ as $n\\to\\infty$. By \\eqref{proba_red}, the probability for $u$ to be blue also tends to $1$ as $n\\to\\infty$.  It remains to prove that $V(u)\\le \\alpha t \\ln n$. By \\eqref{eq:index-2},  $V(u)\\le (bt \\ln n)  \\frac{n}{n-N} + \\frac{N}{n-N} L \\le \\alpha t \\ln n$ for $n$ large enough indeed. \n\\end{proof}",
      "s:rw-brw": "\\begin{equation}    V(u^*_{j}) \\le \\frac{n}{n-N} \\sum_{i=1}^{j} \\eta_i + \\frac{Nj}{n-N}, \\qquad \\forall\\, j< \\tau_1.\\label{control-Vuj} \\end{equation}  \n\n\\noindent Hence $${\\tt s}_1 \\le \\frac{n}{n-N} \\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0) + \\frac{N (\\Delta-1) }{n-N}.$$\n\n The term $\\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0)$ is stochastically smaller than the sum of $\\Delta-1$ independent exponential r.v. with mean $2e$, as long as $G_{\\Delta}$ has more than $\\frac{n}{2}$ vertices. This condition is indeed satisfied thanks to \\eqref{card_G}. Then for some positive constant $c$,  $$  \\e_{G_{\\tau_{i-1}}}[({\\tt s}_1+\\Delta)^2] \\le c \\Delta^2.$$ \n\nBy Cauchy--Schwarz inequality and  \\eqref{bound_eventE}, we obtain that $$  \\e_{G_{\\tau_{i-1}}}[ ({\\tt s}_1+\\Delta) {\\bf 1}_{{\\mathcal E^c_1}}]\n\\le\nc^{1/2} \\Delta \\Big( (\\frac{2L}{n-1} +\\frac{1}{n-2NL})N\\Delta\\Big)^{1/2}\n\\le \\Delta^{-4}, \n$$\n\n\\noindent where the last inequality holds for all small $\\varepsilon$ and large $n$ (see \\eqref{eq:NLDelta}). It follows from \\eqref{cond-Gi} that $$ \\e \\Big[\\sum_{i=1}^{a_1L/\\Delta}   ({\\tt s}_i+\\Delta)   {\\bf 1}_{{\\mathcal E}_i^c}\\Big]\n\\le a_1 L \\Delta^{-3}.\n$$\n\n\\noindent Since $L \\Delta^{-3}= o(\\beta_n L)$, the Markov inequality yields \\eqref{sum-S-tau<Delta-2} \nand completes the proof of the lemma.\n\\end{proof}\n\n\\section{Proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper},  and of Proposition  \\ref{p:first_moment}}\\label{s:rw-brw}\n\nRecall that $(Y_n)_{n\\ge 0}$ is a centered random walk starting at $0$ with step distribution $\\mbox{Exp}(1)-1$. In the first subsection, we present several estimates on $(Y_n)_{n\\ge 0}$ and provide  the proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper}. The second subsection is devoted to the study of the branching random walks $\\V$ and $\\V^N$, and to the proof of Proposition  \\ref{p:first_moment}. \n\n\\subsection{Estimates of random walks: Proofs of Lemmas \\ref{main:upper} and \\ref{main2:upper}}\\label{sub:rw}\n\n\\begin{lemma}\\label{l:mogulski}\n Let $\\upsilon>0$ and $0<d<\\frac{\\pi^2}{2}$. There exists $\\Delta_0>0$ such that for all $\\Delta\\ge \\Delta_0$,   $r_0\\le 0\\le r_0+\\Delta$ and  $\\ell\\ge 1$,\n\\begin{equation}\n    \\p(r_0  \\le Y_j\\le r_0+\\Delta,\\, \\forall \\, j\\in \\lb 1,  \\ell\\rb) \\le e^{\\Delta^\\upsilon-\\frac{d}{\\Delta^2}\\ell}. \\label{mogulskii_upper0} \n\\end{equation}"
    },
    "pre_theorem_intro_text_len": 3616,
    "pre_theorem_intro_text": "We consider the complete graph $K_n$ of the set of vertices $\\llbracket 1, n\\rrbracket=\\{1,2,\\ldots,n\\}$. Following Aldous \\cite{aldous}, we  assign each edge  an independent exponential random variable with mean $n$. In this setting, Aldous proved the existence of a phase transition for the size of  the largest tree with average weight below a given threshold \\cite[Theorem 1]{aldous}. We are interested in the analogous result for paths, rather than trees, established in  \\cite[Proposition 3.7]{aldous}. In order to state this result, we introduce the following notation. \nFor ease of notation, in this paper, we will suppose that the labels $X(u,v)_{u\\neq v \\in \\llbracket 1, n\\rrbracket}$  on the (non-oriented) edges of $K_n$ are i.i.d. copies of \n\\begin{equation}\\label{def:X}\nX=ne \\mathrm{Exp}(1)-1\n\\end{equation} \nwhere $\\mathrm{Exp(1)}$ denotes an exponential random variable with mean $1$. \nA path $\\gamma$ is a sequence of vertices $(\\gamma_0,\\gamma_1,\\ldots,\\gamma_\\ell) \\in  \\llbracket 1, n\\rrbracket^{\\ell+1}$ which is made of distinct adjacent edges $(\\gamma_i,\\gamma_{i+1})$. We call $|\\gamma|=\\ell$ the length of the path, and let ${\\mathcal P}_\\ell$ be the collection of paths of length $\\ell$. A path of length $0$ is reduced to a single vertex.\n\nFor any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation}    \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad  i\\in \\llbracket 1,  \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation}    \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in  {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\n  Translated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and  ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$:  there exist some constants $c'> c >0$ and $C'>C$ such that \n\n (i) if $ -1\\le \\alpha \\le c $, then  \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty;  \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which  the longest path undergoes a transition from  order $\\ln^3 n$ to   polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le   n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1.  \\label{Ding-Goswami}\\end{equation} \n\n  The following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.",
    "context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.",
    "full_context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\p\\Big( C \\ln^3 n \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of $\\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\p\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le \\L(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nNote that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.\n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty} \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$. As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\, -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large.\n\n\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main}] Recall by Claim 2.4 of \\cite{ding} that if there is a path of length greater than $\\ell$ with average weight below $\\lambda$, then there is a path of length in $\\lb \\ell,2\\ell\\rb$ with average weight below $\\lambda$. Then for $\\lambda= \\frac{\\alpha}{\\ln^2 n}$, $t> \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$, and $\\ell= \\lfloor t \\ln^3 n\\rfloor$, \\begin{align} & \\p\\Big(\\L(n, \\lambda) \\ge \\ell\\Big) \n \\nonumber \\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L \\}\\Big) \\nonumber \n\\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) > -b_n\\}\\Big) + o(1), \\label{eq:L>ell-bn}\n\\end{align}\n\n\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\n\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}",
    "post_theorem_intro_text_len": 4434,
    "post_theorem_intro_text": "Note that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.  \n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln {\\mathscr L}(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\mathbb P\\Big(\\frac{1}{n}{\\mathscr L}(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation}    {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for ${\\mathscr L}(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.",
    "sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields Theorem~\\ref{t:main} (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem~\\ref{t:main2}). A “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”",
    "expanded_sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields the main theorem (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields the following theorem.\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\nA “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”",
    "expanded_theorem": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$,",
    "theorem_type": [
      "Implication",
      "Asymptotic or Limit"
    ],
    "mcq": {
      "question": "Fix a real number \\(\\alpha<\\pi^2/2\\). For \\(\\lambda\\in(-1,\\infty)\\), define\n\\[\n{\\mathscr L}(n,\\lambda):=\\sup\\{\\ell\\ge 1:\\ \\exists\\text{ a path }\\gamma=(\\gamma_0,\\ldots,\\gamma_\\ell)\\text{ with }S_i(\\gamma)=\\sum_{k=1}^i X(\\gamma_{k-1},\\gamma_k)\\text{ and }S_\\ell(\\gamma)\\le \\lambda \\ell\\}.\n\\]\nThus \\({\\mathscr L}(n,\\lambda)\\) is the maximum path length for which there exists a path whose average weight \\(S_\\ell(\\gamma)/\\ell\\) is at most \\(\\lambda\\). Under these assumptions, which statement about \\({\\mathscr L}(n,\\alpha/\\ln^2 n)\\) holds as \\(n\\to\\infty\\)?",
      "correct_choice": {
        "label": "A",
        "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}+\\alpha}.\n\\]"
        },
        {
          "label": "C",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)}{\\ln^3 n}=O_{\\mathbb P}(1).\n\\]"
        },
        {
          "label": "D",
          "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^2 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"
        },
        {
          "label": "E",
          "text": "For every fixed \\(\\alpha<\\pi^2/2\\), there exists a deterministic constant \\(C=C(\\alpha)>0\\) such that, with probability tending to \\(1\\) as \\(n\\to\\infty\\),\n\\[\n{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)= C\\ln^3 n+o(1).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "critical-drift denominator sign",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "explicit limiting constant",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "correct logarithmic scaling exponent",
          "template_used": "stronger_trap"
        },
        {
          "label": "E",
          "sketch_hook_type": "regularity",
          "tampered_component": "mode of approximation upgraded from relative/probabilistic scale to additive deterministic asymptotics",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the quantity and asks for its asymptotic behavior, but it does not reveal the limiting constant, scaling, or the correct option. There is no explicit answer leakage."
      },
      "TAS": {
        "score": 1,
        "justification": "The item is essentially asking the test-taker to recognize the exact theorem conclusion under the stated hypotheses. The options do vary in meaningful ways, so it is not a pure verbatim restatement, but it is still close to theorem recall rather than a genuinely rederived conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning involved in comparing the logarithmic scale, the constant, and the mode of convergence. However, the problem mainly tests recall/recognition of a known asymptotic statement rather than substantial generative reasoning from the stem alone."
      },
      "DQS": {
        "score": 1,
        "justification": "Most distractors are mathematically plausible and target common mistakes: wrong sign in the denominator, wrong log exponent, and an overly strong deterministic asymptotic. However, option C is a weaker statement that is also true if A is true, which creates ambiguity in a single-answer MCQ and lowers distractor quality."
      },
      "total_score": 5,
      "overall_assessment": "No answer leakage and several plausible distractors, but the item is theorem-recognition-heavy and weakened by the presence of a weaker true alternative, making it somewhat ambiguous as a single-correct-answer MCQ."
    }
  },
  {
    "id": "2512.23276v1",
    "paper_link": "http://arxiv.org/abs/2512.23276v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "thm",
      "content": "\\label{thm:1.1} For $G=\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$, the type 1 chamber zeta function $Z_{\\Gamma}(u)$ converges for sufficiently $u$, and it is given by \n$$Z_{\\Gamma}(u)=\\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}\n.$$",
      "start_pos": 15782,
      "end_pos": 16098,
      "label": "thm:1.1"
    },
    "ref_dict": {
      "thm:1.1": "\\begin{thm}\\label{thm:1.1} For $G=\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$, the type 1 chamber zeta function $Z_{\\Gamma}(u)$ converges for sufficiently $u$, and it is given by \n$$Z_{\\Gamma}(u)=\\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}\n.$$\n\\end{thm}",
      "coro:cycle_intro": "\\begin{coro}[Counting closed galleries]\\label{coro:cycle_intro} Let $\\Gamma$ be the discrete subgroup $\\operatorname{PGL}(3,\\mathbb{F}_q[t])$ of $\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $N_n(\\Gamma\\backslash\\mathcal{B})$ be defined as above. Then, we have\n$$N_{m}=\\left\\{\\begin{array}{ll}3q^{3r}-3q^{2r} & \\textrm{ if }n=3r\\textrm{ and }n\\not\\equiv 0 \\,(\\operatorname{mod}\\,{6}) \\\\ \n3q^{6r}-9q^{4r}+3q^{3r} & \\textrm{ if }n=6r \\\\ 0 \n& \\textrm{ otherwise}\\end{array}\\right..$$\\end{coro}"
    },
    "pre_theorem_intro_text_len": 5664,
    "pre_theorem_intro_text": "\\label{sec:1}\nIn this paper we define and compute a chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat--Tits building of $\\mathrm{PGL}_3$. Our zeta function is defined via weighted tailless galleries of chambers and admits an Ihara--Bass type determinant formula in terms of a natural chamber transfer operator.\n\nIhara introduced a zeta function associated with prime elements in discrete subgroups of rank one $p$-adic groups \\cite{I}. This zeta function has been generalized by Sunada \\cite{Su}, Hashimoto \\cite{Has}, and Bass \\cite{B} to uncover the connections between number theory and representation theory. \n\n In recent years, the extension of zeta functions to infinite graphs has been studied \\cite{CS,Cla,GZ,CJK,DK18}. In particular, Deitmar and Kang considered cycles obtained from geodesics in the universal covering tree of infinite weighted graphs. We defined a Selberg zeta function of graphs of groups and compared it with the zeta function introduced by Deitmar and Kang \\cite{HK2}. \n\n Building on Ihara's insights, Kang and Li defined a zeta function of a finite quotient of a Bruhat-Tits building associated with $\\PGL_3(F)$ over a non-archimedean field by a discrete cocompact torsion-free subgroup $\\Gamma\\subset \\PGL_3(F)$, and found a necessary and sufficient condition for Ramanujan complexes \\cite{KL}.  \n This article aims to generalize zeta functions of infinite graphs and complexes of groups simultaneously. \n\nIn \\cite{KLW10} and \\cite{KL}, the authors considered not only edge zeta function but also the chamber zeta function. Given a torsion-free cocompact lattice $\\Gamma$ of $\\operatorname{PGL}(3,F)$, the type 1 chamber zeta function for $\\Gamma$ is defined as follows. Let $X=\\Gamma\\backslash \\mathcal{B}$ be the quotient complex. Every vertex $v$ has a color $\\tau(v)$ (see Section \\ref{sec:2}). The action of $\\Gamma$ does not preserve the color of vertices whereas it preserves the color difference between the endpoints of each directed edge. This enables us to define the type of oriented edges and pointed chambers. A pointed chamber $c=\\{v_1,v_2,v_3\\}$ is called \\textit{type} $k$ if  for any $i\\in \\mathbb{Z}/3\\mathbb{Z}$, $$\\tau(v_{i+1})-\\tau(v_i)=k.$$ \nA sequence of type $k$ pointed chambers $(c_1,\\cdots,c_n)$ is called a \\textit{type $k$ gallery} if the sequence consists of the edge-adjacent pointed chambers, i.e. for any $ i$, $$v_{2,i}=v_{1,i+1},v_{3,i}=v_{2,i+1},$$\nwhere $c_i=\\{v_{1,i},v_{2,i},v_{3,i}\\}.$ A gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\textit{tailless} if $v_{1,i}\\neq v_{3,i+1}$ for any $i$.\nA gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\emph{closed} if $v_{1,1}=v_{2,n}$ and $v_{2,1}=v_{3,n}.$ The \\emph{shift map} of a gallery is defined by $\\sigma(\\mathcal{G})=(c_2,\\ldots,c_n,c_1)$.  Two closed galleries $\\mathcal{G}_1$ and $\\mathcal{G}_2$ are equivalent if $\\mathcal{G}_1=\\sigma^i(\\mathcal{G}_2)$ for some $i$. An equivalence class $\\mathcal{C}=[\\mathcal{G}]$ of closed galleries is called a\n\\emph{closed gallery class}, or simply a \\emph{class} when no confusion arises. The power $\\mathcal{C}^n$ of a closed gallery class $\\mathcal{C}$ is the class obtained by winding around the same gallery for $n$ times. A class $\\mathcal{C}_0$ is called \\emph{primitive} if it is not a power of a shorter one. For each class $\\mathcal{C}$, there exists a unique primitive class $\\mathcal{C}_0$ satisfying $\\mathcal{C}=\\mathcal{C}_0^m$. The number $m=m(\\mathcal{C})$ is called the \\emph{multiplicity} of $\\mathcal{C}$. As an analogy of edge zeta function, the \\emph{type 1 chamber zeta function} for $\\Gamma$ is defined by\n$$Z_{\\Gamma}:=\\prod_{\\mathcal{C}}\\left(1-u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product runs over all tailless type 1 primitive classes and and $\\ell(\\mathcal{C})$ is the length (the number of chambers) of the class $\\mathcal{C}$. Let $T$ be the edge adjacency operator on $L^2$-space on the set of type 1 pointed chambers in $X$ defined by\n$$Tf(c):=\\sum_{c'}f(c'),$$\nwhere the sum is taken over the edge adjacency type 1 pointed chamber of $c$ in $X$.\nThe authors of \\cite{KLW10} and \\cite{KL} proved that the type 1 chamber zeta function and the operator $T$ also satisfy the determinant formula\n$$Z_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}.$$\n\nBy extending the idea in \\cite{DK18} to higher dimension, we have defined the edge zeta function for non-compact weighted complexes.\nCombining the ideas in \\cite{DK18} and \\cite{KL}, we regarded classes of galleries as image of the tailless galleries in the Bruhat-Tits building $\\mathcal{B}_3$ associated to $\\operatorname{PGL}_3(F).$ Similar to \\cite{HK1} and \\cite{HK3}, we define the weight $w$ of two type 1 pointed chambers.  \nFor a pointed chamber $\\widetilde{c}$ of $\\mathcal{B}_3$ and a pointed chamber $c'$ of $X$, let\n$$w(\\widetilde{c},c'):=\\#\\{\\widetilde{c}'\\colon\\widetilde{c}\\textrm{ is a lift }\\textrm {of }c',(\\widetilde{c},\\widetilde{c}') \\textrm{ is a talless gallery in }\\mathcal{B}\\}.$$ \nFor a pointed chamber $c$ of $X$, let $\\widetilde{c}$ be a lift of $c$ in $\\mathcal{B}$ and $w(c,c')=w(\\widetilde{c},c')$. The weight $w(c,c')$ does not depend on the specific choice of the preimage $\\widetilde{c}$. Thus it is well-defined.\n\nThe weight of a closed gallery $\\mathcal{G}=(c_1,c_2,\\ldots,c_n)$ is defined by\n$$w(\\mathcal{G}):=\\prod_{j\\textrm{ mod }n}w(c_j,c_{j+1}).$$ \nThen, the type 1 chamber zeta function for $\\Gamma$ is defined by\n$$Z_{\\Gamma}(u):=\\prod_C\\left(1-w(\\mathcal{C})u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product is taken over all primitive type 1 closed gallery classes, and $\\ell(\\mathcal{C})$ is the length of the class $\\mathcal{C}$.",
    "context": "\\label{sec:1}\nIn this paper we define and compute a chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat--Tits building of $\\mathrm{PGL}_3$. Our zeta function is defined via weighted tailless galleries of chambers and admits an Ihara--Bass type determinant formula in terms of a natural chamber transfer operator.\n\nIhara introduced a zeta function associated with prime elements in discrete subgroups of rank one $p$-adic groups \\cite{I}. This zeta function has been generalized by Sunada \\cite{Su}, Hashimoto \\cite{Has}, and Bass \\cite{B} to uncover the connections between number theory and representation theory.\n\nBuilding on Ihara's insights, Kang and Li defined a zeta function of a finite quotient of a Bruhat-Tits building associated with $\\PGL_3(F)$ over a non-archimedean field by a discrete cocompact torsion-free subgroup $\\Gamma\\subset \\PGL_3(F)$, and found a necessary and sufficient condition for Ramanujan complexes \\cite{KL}. \n This article aims to generalize zeta functions of infinite graphs and complexes of groups simultaneously.\n\nIn \\cite{KLW10} and \\cite{KL}, the authors considered not only edge zeta function but also the chamber zeta function. Given a torsion-free cocompact lattice $\\Gamma$ of $\\operatorname{PGL}(3,F)$, the type 1 chamber zeta function for $\\Gamma$ is defined as follows. Let $X=\\Gamma\\backslash \\mathcal{B}$ be the quotient complex. Every vertex $v$ has a color $\\tau(v)$ (see Section \\ref{sec:2}). The action of $\\Gamma$ does not preserve the color of vertices whereas it preserves the color difference between the endpoints of each directed edge. This enables us to define the type of oriented edges and pointed chambers. A pointed chamber $c=\\{v_1,v_2,v_3\\}$ is called \\textit{type} $k$ if for any $i\\in \\mathbb{Z}/3\\mathbb{Z}$, $$\\tau(v_{i+1})-\\tau(v_i)=k.$$ \nA sequence of type $k$ pointed chambers $(c_1,\\cdots,c_n)$ is called a \\textit{type $k$ gallery} if the sequence consists of the edge-adjacent pointed chambers, i.e. for any $ i$, $$v_{2,i}=v_{1,i+1},v_{3,i}=v_{2,i+1},$$\nwhere $c_i=\\{v_{1,i},v_{2,i},v_{3,i}\\}.$ A gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\textit{tailless} if $v_{1,i}\\neq v_{3,i+1}$ for any $i$.\nA gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\emph{closed} if $v_{1,1}=v_{2,n}$ and $v_{2,1}=v_{3,n}.$ The \\emph{shift map} of a gallery is defined by $\\sigma(\\mathcal{G})=(c_2,\\ldots,c_n,c_1)$. Two closed galleries $\\mathcal{G}_1$ and $\\mathcal{G}_2$ are equivalent if $\\mathcal{G}_1=\\sigma^i(\\mathcal{G}_2)$ for some $i$. An equivalence class $\\mathcal{C}=[\\mathcal{G}]$ of closed galleries is called a\n\\emph{closed gallery class}, or simply a \\emph{class} when no confusion arises. The power $\\mathcal{C}^n$ of a closed gallery class $\\mathcal{C}$ is the class obtained by winding around the same gallery for $n$ times. A class $\\mathcal{C}_0$ is called \\emph{primitive} if it is not a power of a shorter one. For each class $\\mathcal{C}$, there exists a unique primitive class $\\mathcal{C}_0$ satisfying $\\mathcal{C}=\\mathcal{C}_0^m$. The number $m=m(\\mathcal{C})$ is called the \\emph{multiplicity} of $\\mathcal{C}$. As an analogy of edge zeta function, the \\emph{type 1 chamber zeta function} for $\\Gamma$ is defined by\n$$Z_{\\Gamma}:=\\prod_{\\mathcal{C}}\\left(1-u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product runs over all tailless type 1 primitive classes and and $\\ell(\\mathcal{C})$ is the length (the number of chambers) of the class $\\mathcal{C}$. Let $T$ be the edge adjacency operator on $L^2$-space on the set of type 1 pointed chambers in $X$ defined by\n$$Tf(c):=\\sum_{c'}f(c'),$$\nwhere the sum is taken over the edge adjacency type 1 pointed chamber of $c$ in $X$.\nThe authors of \\cite{KLW10} and \\cite{KL} proved that the type 1 chamber zeta function and the operator $T$ also satisfy the determinant formula\n$$Z_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}.$$\n\nBy extending the idea in \\cite{DK18} to higher dimension, we have defined the edge zeta function for non-compact weighted complexes.\nCombining the ideas in \\cite{DK18} and \\cite{KL}, we regarded classes of galleries as image of the tailless galleries in the Bruhat-Tits building $\\mathcal{B}_3$ associated to $\\operatorname{PGL}_3(F).$ Similar to \\cite{HK1} and \\cite{HK3}, we define the weight $w$ of two type 1 pointed chambers. \nFor a pointed chamber $\\widetilde{c}$ of $\\mathcal{B}_3$ and a pointed chamber $c'$ of $X$, let\n$$w(\\widetilde{c},c'):=\\#\\{\\widetilde{c}'\\colon\\widetilde{c}\\textrm{ is a lift }\\textrm {of }c',(\\widetilde{c},\\widetilde{c}') \\textrm{ is a talless gallery in }\\mathcal{B}\\}.$$ \nFor a pointed chamber $c$ of $X$, let $\\widetilde{c}$ be a lift of $c$ in $\\mathcal{B}$ and $w(c,c')=w(\\widetilde{c},c')$. The weight $w(c,c')$ does not depend on the specific choice of the preimage $\\widetilde{c}$. Thus it is well-defined.\n\nThe weight of a closed gallery $\\mathcal{G}=(c_1,c_2,\\ldots,c_n)$ is defined by\n$$w(\\mathcal{G}):=\\prod_{j\\textrm{ mod }n}w(c_j,c_{j+1}).$$ \nThen, the type 1 chamber zeta function for $\\Gamma$ is defined by\n$$Z_{\\Gamma}(u):=\\prod_C\\left(1-w(\\mathcal{C})u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product is taken over all primitive type 1 closed gallery classes, and $\\ell(\\mathcal{C})$ is the length of the class $\\mathcal{C}$.",
    "full_context": "\\label{sec:1}\nIn this paper we define and compute a chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat--Tits building of $\\mathrm{PGL}_3$. Our zeta function is defined via weighted tailless galleries of chambers and admits an Ihara--Bass type determinant formula in terms of a natural chamber transfer operator.\n\nIhara introduced a zeta function associated with prime elements in discrete subgroups of rank one $p$-adic groups \\cite{I}. This zeta function has been generalized by Sunada \\cite{Su}, Hashimoto \\cite{Has}, and Bass \\cite{B} to uncover the connections between number theory and representation theory.\n\nBuilding on Ihara's insights, Kang and Li defined a zeta function of a finite quotient of a Bruhat-Tits building associated with $\\PGL_3(F)$ over a non-archimedean field by a discrete cocompact torsion-free subgroup $\\Gamma\\subset \\PGL_3(F)$, and found a necessary and sufficient condition for Ramanujan complexes \\cite{KL}. \n This article aims to generalize zeta functions of infinite graphs and complexes of groups simultaneously.\n\nIn \\cite{KLW10} and \\cite{KL}, the authors considered not only edge zeta function but also the chamber zeta function. Given a torsion-free cocompact lattice $\\Gamma$ of $\\operatorname{PGL}(3,F)$, the type 1 chamber zeta function for $\\Gamma$ is defined as follows. Let $X=\\Gamma\\backslash \\mathcal{B}$ be the quotient complex. Every vertex $v$ has a color $\\tau(v)$ (see Section \\ref{sec:2}). The action of $\\Gamma$ does not preserve the color of vertices whereas it preserves the color difference between the endpoints of each directed edge. This enables us to define the type of oriented edges and pointed chambers. A pointed chamber $c=\\{v_1,v_2,v_3\\}$ is called \\textit{type} $k$ if for any $i\\in \\mathbb{Z}/3\\mathbb{Z}$, $$\\tau(v_{i+1})-\\tau(v_i)=k.$$ \nA sequence of type $k$ pointed chambers $(c_1,\\cdots,c_n)$ is called a \\textit{type $k$ gallery} if the sequence consists of the edge-adjacent pointed chambers, i.e. for any $ i$, $$v_{2,i}=v_{1,i+1},v_{3,i}=v_{2,i+1},$$\nwhere $c_i=\\{v_{1,i},v_{2,i},v_{3,i}\\}.$ A gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\textit{tailless} if $v_{1,i}\\neq v_{3,i+1}$ for any $i$.\nA gallery $\\mathcal{G}=(c_1,\\cdots,c_n)$ is called \\emph{closed} if $v_{1,1}=v_{2,n}$ and $v_{2,1}=v_{3,n}.$ The \\emph{shift map} of a gallery is defined by $\\sigma(\\mathcal{G})=(c_2,\\ldots,c_n,c_1)$. Two closed galleries $\\mathcal{G}_1$ and $\\mathcal{G}_2$ are equivalent if $\\mathcal{G}_1=\\sigma^i(\\mathcal{G}_2)$ for some $i$. An equivalence class $\\mathcal{C}=[\\mathcal{G}]$ of closed galleries is called a\n\\emph{closed gallery class}, or simply a \\emph{class} when no confusion arises. The power $\\mathcal{C}^n$ of a closed gallery class $\\mathcal{C}$ is the class obtained by winding around the same gallery for $n$ times. A class $\\mathcal{C}_0$ is called \\emph{primitive} if it is not a power of a shorter one. For each class $\\mathcal{C}$, there exists a unique primitive class $\\mathcal{C}_0$ satisfying $\\mathcal{C}=\\mathcal{C}_0^m$. The number $m=m(\\mathcal{C})$ is called the \\emph{multiplicity} of $\\mathcal{C}$. As an analogy of edge zeta function, the \\emph{type 1 chamber zeta function} for $\\Gamma$ is defined by\n$$Z_{\\Gamma}:=\\prod_{\\mathcal{C}}\\left(1-u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product runs over all tailless type 1 primitive classes and and $\\ell(\\mathcal{C})$ is the length (the number of chambers) of the class $\\mathcal{C}$. Let $T$ be the edge adjacency operator on $L^2$-space on the set of type 1 pointed chambers in $X$ defined by\n$$Tf(c):=\\sum_{c'}f(c'),$$\nwhere the sum is taken over the edge adjacency type 1 pointed chamber of $c$ in $X$.\nThe authors of \\cite{KLW10} and \\cite{KL} proved that the type 1 chamber zeta function and the operator $T$ also satisfy the determinant formula\n$$Z_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}.$$\n\nBy extending the idea in \\cite{DK18} to higher dimension, we have defined the edge zeta function for non-compact weighted complexes.\nCombining the ideas in \\cite{DK18} and \\cite{KL}, we regarded classes of galleries as image of the tailless galleries in the Bruhat-Tits building $\\mathcal{B}_3$ associated to $\\operatorname{PGL}_3(F).$ Similar to \\cite{HK1} and \\cite{HK3}, we define the weight $w$ of two type 1 pointed chambers. \nFor a pointed chamber $\\widetilde{c}$ of $\\mathcal{B}_3$ and a pointed chamber $c'$ of $X$, let\n$$w(\\widetilde{c},c'):=\\#\\{\\widetilde{c}'\\colon\\widetilde{c}\\textrm{ is a lift }\\textrm {of }c',(\\widetilde{c},\\widetilde{c}') \\textrm{ is a talless gallery in }\\mathcal{B}\\}.$$ \nFor a pointed chamber $c$ of $X$, let $\\widetilde{c}$ be a lift of $c$ in $\\mathcal{B}$ and $w(c,c')=w(\\widetilde{c},c')$. The weight $w(c,c')$ does not depend on the specific choice of the preimage $\\widetilde{c}$. Thus it is well-defined.\n\nThe weight of a closed gallery $\\mathcal{G}=(c_1,c_2,\\ldots,c_n)$ is defined by\n$$w(\\mathcal{G}):=\\prod_{j\\textrm{ mod }n}w(c_j,c_{j+1}).$$ \nThen, the type 1 chamber zeta function for $\\Gamma$ is defined by\n$$Z_{\\Gamma}(u):=\\prod_C\\left(1-w(\\mathcal{C})u^{\\ell(\\mathcal{C})}\\right)^{-1},$$\nwhere the product is taken over all primitive type 1 closed gallery classes, and $\\ell(\\mathcal{C})$ is the length of the class $\\mathcal{C}$.\n\nHence, we have\n$$Z_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}=\\exp\\left(\\sum_{n=1}^{\\infty}\\frac{N_n(\\Gamma\\backslash\\mathcal{B})}{n}u^{n}\\right)$$ which implies the following corollary. (See Section~\\ref{sec:4} for the detail.)\n\\begin{coro}[Counting closed galleries]\\label{coro:cycle_intro} Let $\\Gamma$ be the discrete subgroup $\\operatorname{PGL}(3,\\mathbb{F}_q[t])$ of $\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $N_n(\\Gamma\\backslash\\mathcal{B})$ be defined as above. Then, we have\n$$N_{m}=\\left\\{\\begin{array}{ll}3q^{3r}-3q^{2r} & \\textrm{ if }n=3r\\textrm{ and }n\\not\\equiv 0 \\,(\\operatorname{mod}\\,{6}) \\\\ \n3q^{6r}-9q^{4r}+3q^{3r} & \\textrm{ if }n=6r \\\\ 0 \n& \\textrm{ otherwise}\\end{array}\\right..$$\\end{coro}\n\nThe trace of $T^n$ is described by\n$$\\operatorname{Tr}(T^n)=\\sum_{c }\\sum_{\\mathcal{C}\\colon c\\in \\mathcal{C}}w(\\mathcal{C}),$$\nwhere $c$ runs over the type 1 pointed chambers and $\\mathcal{C}$ runs over the closed gallery classes containing $c$. \nFor given a class $\\mathcal{C}$ of length $n$, the weight of $\\mathcal{C}$ appears $\\ell(\\mathcal{C}_0)$ times in the above sum. Thus we have the proof of Lemma \\ref{lem:4.1}\n\\end{proof}\n\\begin{prop}\\label{prop:4.2} The type 1 chamber zeta function is formally described by\n$$Z_\\Gamma(u)=\\exp\\biggr(\\sum_{n=1}^\\infty \\frac{\\operatorname{Tr}(T^n)}{n}u^n\\biggr).$$\n\\end{prop}\n\\begin{proof}Given a closed gallery class $\\mathcal{C}$, let $\\mathcal{C}_0$ be the primitive class of $\\mathcal{C}$. As in \\cite{DK18} and \\cite{HK3}, we have \n\\begin{equation*}\n\\begin{split}\nZ_{\\Gamma}(u)^{-1}=&\\,\\prod_{\\mathcal{C}_0}\\bigr(1-w(\\mathcal{C}_0)u^{\\ell(\\mathcal{C}_0)}\\bigr)=\\exp\\left(\\sum_{\\mathcal{C}_0}\\log(1-w(\\mathcal{C}_0)u^{\\ell(\\mathcal{C}_0)})\\right)\\\\\n=&\\,\\exp\\left(-\\sum_{\\mathcal{C}_0}\\sum_{n=1}^{\\infty}\\frac{w(\\mathcal{C}_0)^nu^{n\\ell(\\mathcal{C}_0)}}{n}\\right)=\\exp\\left(-\\sum_{\\mathcal{C}}\\frac{w(\\mathcal{C})u^{\\ell(\\mathcal{C})}}{\\ell(\\mathcal{C})}\\ell(\\mathcal{C}_0)\\right)\\\\=&\\,\\exp\\left(-\\sum_{n=1}^{\\infty}\\frac{u^{n}}{n}\\sum_{\\mathcal{C}\\colon \\ell(\\mathcal{C})=n}w(\\mathcal{C})\\ell(\\mathcal{C}_0)\\right)=\\exp\\biggr(-\\sum_{n=1}^\\infty \\frac{\\operatorname{Tr}(T^n)}{n}u^n\\biggr).\n\\end{split}\n\\end{equation*}\nThis proves Proposition \\ref{prop:4.2}.\n\\end{proof}\nFollowing the proof of Lemma 4.2 in \\cite{DK18} and Proposition 4.3 in \\cite{HK3}, we have the following proposition.\n\\begin{prop}For sufficiently small $u$, the type 1 chamber zeta function satisfies\n$$Z_\\Gamma(u)=\\frac{1}{\\det(I-uT)}.$$\n\\end{prop}\n\n\\begin{coro}\\label{coro:rationality}\nFor $\\Gamma=\\mathrm{PGL}_3(\\mathbb{F}_q[t])$ acting on the Bruhat--Tits building $\\mathcal{B}$ of $G=\\mathrm{PGL}_3(\\mathbb{F}_q(\\!(t^{-1})\\!))$, the type~1 chamber zeta function\n\\[\nZ_{\\Gamma}(u)\n=\n\\prod_{\\mathcal{C}}\n\\bigl(1-w(\\mathcal{C})u^{\\ell(\\mathcal{C})}\\bigr)^{-1}\n\\]\nconverges for $|u|$ sufficiently small and extends to a rational function in $u$. In particular, $Z_{\\Gamma}(u)$ admits meromorphic continuation to the projective line $\\mathbb{P}^1(\\mathbb{C})$, with possible poles contained in the spectrum of the chamber transfer operator $T$.\n\\end{coro}\n\n\\subsection{The determinant of $I-uT$} For any $t\\geq 2$, we obtain\n$$a_{k,(1,t)}(u)=q^3u^4a_{k,(1,t-1)}(u)=q^6u^8a_{k,(1,t-2)}(u)=\\cdots=q^{3t-3}u^{4t-4}a_{k,(1,1)}(u).$$\nUsing this, we have\n\\begin{equation*}\n\\begin{split}\na_{k,(1,1)}(u)&=-(q-1)u-q^2(q-1)u^4+a_{k,(1,1)}\\sum_{i=1}^k q^{3i}(q-1)u^{6i}\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\na_{k,(1,1)}(u)=\\frac{-(q-1)u-q^2(q-1)u^4}{1-\\displaystyle\\sum_{i=1}^kq^{3i}(q-1)u^{6i}}.\n\\end{equation*}\nThe sequence $a_{k,(1,1)}$ converges to\n$$a_{(1,1)}=\\frac{-(1-q^3u^6)((q-1)u+q^2(q-1)u^4)}{1-q^4u^6}.$$\nFor any $s,t,$ with $s\\geq 2$ and $t< s,$\n$$a_{k,(s,t)}(u)=q^3u^4a_{k,(s,t-1)}(u)=q^6u^8a_{k,(1,t-2)}(u)=\\cdots=q^{3t-3}u^{4t-4}a_{k,(s,1)}(u).$$\nFor any $s,t$ with $s\\geq 2$ and $t\\geq s$,\n\\begin{equation*}\n\\begin{split}\na_{k,(s,t)}(u)&=q^3u^4a_{k,(s,t-1)}(u)=\\cdots=q^{3(t-s)}u^{4(t-s)}a_{k,(s,s)}(u)\\\\\n&=-q^{3(t-s)}(q-1)u^{4(t-s)+1}+q^{3(t-s+1)}u^{4(t-s+1)}a_{k,(s,s-1)}(u)\\\\\n&=-q^{3(t-s)}(q-1)u^{4(t-s)+1}+q^{3(t-1)}u^{4(t-1)}a_{k,(s,1)}(u).\n\\end{split}\n\\end{equation*}\nSimilarly, \n\\begin{equation*}\n\\begin{split}\na_{k,(s,1)}=&-q^{2}(q-1)u^{2s+2}+\\displaystyle\\sum_{i=1}^k a_{k,(s,i)}q^{3}(q-1)u^{2i+4}\\\\\n=&-q^{2}(q-1)u^{2s+2}+\\displaystyle\\sum_{i=1}^k a_{k,(s,1)}q^{3i}(q-1)u^{6i}\\\\\n&-\\sum_{s}^kq^{3(i-s+1)}(q-1)^2u^{6i-4s+5}\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\na_{k,(s,1)}=\\frac{-q^{2}(q-1)u^{2s+2}-\\displaystyle\\sum_{s}^kq^{3(i-s+1)}(q-1)u^{6i-4s+5}}{1-\\displaystyle\\sum_{i=1}^k q^{3i}(q-1)^2u^{6i}}.\n\\end{equation*}\nThe sequence $a_{k,(s,1)}$ converges to \n\\begin{equation*}\na_{(s,1)}=\\frac{-q^2(q-1)u^{2s+2}(1-q^3u^6)-q^3(q-1)^2u^{2s+5}}{1-q^4u^6}.\n\\end{equation*}\nUsing the limit of $\\det(A_{k,0})$, the determinant of $I-uT$ is \n\\begin{equation*}\n\\lim_{k\\rightarrow \\infty}\\det(A_{k,0})=\\frac{(1-q^3u^6)(1-q^3u^3)}{(1-q^4u^6)(1-q^2u^3)}.\n\\end{equation*}\nBy determinant formula, we obtain the following theorem:\n\\begin{thm}\\label{thm:5.1} Let $\\Gamma=\\PGL_3(\\mathbb{F}_q[t])$. The type 1 chamber zeta function $Z_\\Gamma(u)$ converges for sufficiently small $u$ and it is given by\n$$Z_{\\Gamma}(u)=\\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}.$$\n\\end{thm}\nRecall that we defined $$N_n(\\Gamma\\backslash\\mathcal{B})=\\sum_{c\\colon\\ell(\\mathcal{C})=n}w(\\mathcal{C})\\ell(\\mathcal{C}_0)$$ by the weighted number of type $1$ closed galleries in $\\Gamma\\backslash \\mathcal{B}$ of length $n$.\nThe trace of the operator $T^m$ coincides with $N_m(\\Gamma\\backslash\\mathcal{B})$. By Proposition \\ref{prop:4.2}, we have\n\\begin{equation}\\label{eq:5.1}\nu\\frac{d}{du}\\log Z_{\\Gamma}(u)=u\\frac{Z'_{\\Gamma}(u)}{Z_{\\Gamma}(u)}=\\sum_{m=1}^{\\infty} N_m(\\Gamma\\backslash \\mathcal{B})u^m.\n\\end{equation}\n\\begin{coro} Let $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$ and $N_m(\\Gamma\\backslash\\mathcal{B})$ as above. Then, we have\n$$N_m(\\Gamma\\backslash\\mathcal{B})=\\left\\{\\begin{array}{ll}3q^{3r}-3q^{2r} & \\textrm{ if }m=3r\\textrm{ and }m\\not\\equiv 0 \\,(\\operatorname{mod}\\,{6}) \\\\ \n3q^{6r}-9q^{4r}+6q^{3r} & \\textrm{ if }m=6r \\\\ 0 \n& \\textrm{ otherwise}\\end{array}\\right..$$\n\\end{coro}\n\\begin{proof}\nBy Theorem~5.1, we have\n\\[\nZ_\\Gamma(u)\n=\n\\frac{(1-q^{4}u^{6})(1-q^{2}u^{3})}{(1-q^{3}u^{6})(1-q^{3}u^{3})}.\n\\]\nRecall from \\eqref{eq:5.1} that\n\\[\nu\\frac{d}{du}\\log Z_\\Gamma(u)\n=\n\\sum_{m\\ge1} N_m(\\Gamma\\backslash B)\\,u^m.\n\\]\nTaking the logarithmic derivative of $Z_\\Gamma(u)$, we obtain\n\\begin{align*}\nu\\frac{d}{du}\\log Z_\\Gamma(u)\n&=\nu\\frac{d}{du}\\Bigl(\n\\log(1-q^{4}u^{6})\n+\n\\log(1-q^{2}u^{3})\n-\n\\log(1-q^{3}u^{6})\n-\n\\log(1-q^{3}u^{3})\n\\Bigr) \\\\\n&=\n-\\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}\n-\\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}\n+\\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}\n+\\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}.\n\\end{align*}\nEquivalently,\n\\begin{equation}\\label{eq:Nm-generating}\n\\sum_{m\\ge1} N_m(\\Gamma\\backslash B)\\,u^m\n=\n\\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}\n-\n\\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}\n+\n\\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}\n-\n\\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}.\n\\end{equation}\nEach term on the right-hand side admits a geometric series expansion:\n\\[\n\\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}\n=\n3\\sum_{r\\ge1} q^{3r}u^{3r},\n\\qquad\n\\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}\n=\n3\\sum_{r\\ge1} q^{2r}u^{3r},\n\\]\n\\[\n\\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}\n=\n6\\sum_{r\\ge1} q^{3r}u^{6r},\n\\qquad\n\\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}\n=\n6\\sum_{r\\ge1} q^{4r}u^{6r}.\n\\]\nIt follows immediately that $N_m(\\Gamma\\backslash B)=0$ unless $3\\,|\\, m$.\nIf $m=3r$ and $m\\not\\equiv 0 \\pmod{6}$, then only the first two series contribute,\nand we obtain\n\\[\nN_{3r}(\\Gamma\\backslash B)\n=\n3q^{3r}-3q^{2r}.\n\\]\nIf $m=6r$, then all four series in \\eqref{eq:Nm-generating} contribute, yielding\n\\[\nN_{6r}(\\Gamma\\backslash B)\n=\n\\bigl(3q^{6r}-3q^{4r}\\bigr)\n+\n\\bigl(6q^{3r}-6q^{4r}\\bigr)\n=\n3q^{6r}+6q^{3r}-9q^{4r}.\n\\]\nThis completes the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 7136,
    "post_theorem_intro_text": "\\begin{proof}[Idea of Proof] Given a class $\\mathcal{C}$, we obtain that $\\mathcal{C}=\\mathcal{C}_0^n$ for some primitive class $\\mathcal{C}_0$. The type 1 chamber zeta function formally satisfies\n\\begin{align*}\nZ_{\\Gamma}(u)^{-1}=\\exp\\left(-\\sum_{n=1}^{\\infty}\\frac{u^{n}}{n}\\sum_{\\mathcal{C}\\colon \\ell(\\mathcal{C})=n}w(\\mathcal{C})\\ell(\\mathcal{C}_0)\\right).\n\\end{align*}\nLet $\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})$ be the set of all type 1 pointed chambers in $\\Gamma\\backslash \\mathcal{B}$ and let $S(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B}))$ be the formal vector space defined by\n$$S(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})):=\\displaystyle\\bigoplus_{c\\in\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})}\\mathbb{C}c.$$\nThe operator $T\\colon S(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B}))\\to S(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B}))$ is given by\n$$Tc=\\sum_{c'}w(c,c')c'$$ where the sum runs over all type 1 pointed chambers $c'$ adjacent to $c$. We will prove the operator $T^n$ is traceable, and the trace is represented by\n$$\\operatorname{Tr}(T_k^n)=\\sum_{c\\colon\\ell(\\mathcal{C})=n}w(\\mathcal{C})\\ell(\\mathcal{C}_0).$$\nThus, we obtain that $Z_{\\Gamma,k}(q^{-s})^{-1}=\\exp\\bigr(\\!\\operatorname{Tr}(\\log(1-uT))\\bigr)$. Fubini trick and truncation by certain horizontal direction allows us to compute this determinant, which enables us to complete the proof.\n\\end{proof}\n\n\\noindent\n\\begin{remark}[Idea of the traceable argument]\nThe traceable property relies on a monotonicity phenomenon along the cuspidal directions of the quotient complex. Certain directed steps force the height to strictly increase (or decrease), and once such a step occurs, returning to the starting point requires traversing a path whose length is bounded below by a uniform constant. Consequently, for each fixed length $n$, only finitely many closed admissible gallery classes can occur.\n\\end{remark}\n\nThe trace of the operator $T^n$ coincides with the weighted number\n$$N_n(\\Gamma\\backslash\\mathcal{B})=\\sum_{c\\colon\\ell(\\mathcal{C})=n}w(\\mathcal{C})\\ell(\\mathcal{C}_0)$$ of type~$1$ closed gallery classes in $\\Gamma\\backslash \\mathcal{B}$ of length $n$. \n\nHence, we have\n$$Z_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}=\\exp\\left(\\sum_{n=1}^{\\infty}\\frac{N_n(\\Gamma\\backslash\\mathcal{B})}{n}u^{n}\\right)$$ which implies the following corollary. (See Section~\\ref{sec:4} for the detail.)\n\\begin{coro}[Counting closed galleries]\\label{coro:cycle_intro} Let $\\Gamma$ be the discrete subgroup $\\operatorname{PGL}(3,\\mathbb{F}_q[t])$ of $\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $N_n(\\Gamma\\backslash\\mathcal{B})$ be defined as above. Then, we have\n$$N_{m}=\\left\\{\\begin{array}{ll}3q^{3r}-3q^{2r} & \\textrm{ if }n=3r\\textrm{ and }n\\not\\equiv 0 \\,(\\operatorname{mod}\\,{6}) \\\\ \n3q^{6r}-9q^{4r}+3q^{3r} & \\textrm{ if }n=6r \\\\ 0 \n& \\textrm{ otherwise}\\end{array}\\right..$$\\end{coro}\n\n\\begin{remark}[Position within the higher-rank zeta program]\nThe present work fits into a broader ``higher-rank zeta'' program that originates from the Ihara--Bass theory for regular graphs and its extensions to graphs of groups and Bruhat--Tits buildings. Very roughly, one can distinguish two axes:\n\\begin{itemize}\n\\item[(i)] the nature of the quotient $\\Gamma\\backslash X$ (finite/cocompact versus non-uniform of finite volume);\n\\item[(ii)] the combinatorial level on which the zeta function is defined (vertex/edge versus higher-dimensional cells such as chambers).\n\\end{itemize}\nOn the cocompact side, Ihara's original zeta function and its refinements may be viewed as edge zeta functions for finite regular graphs, while the works of Kang--Li and Kang--Li--Wang treat chamber zeta functions for finite quotients of the $\\widetilde{A}_2$-building by cocompact lattices in $\\mathrm{PGL}_3(F)$ \\cite{KL,KLW10,KLW18}. On the non-uniform side, the edge zeta function for the standard non-uniform complex attached to\n\\[\nG=\\mathrm{PGL}_3\\bigl(\\mathbb{F}_q(\\!(t^{-1})\\!)\\bigr),\\qquad\n\\Gamma=\\mathrm{PGL}_3(\\mathbb{F}_q[t]),\n\\]\nwas recently analyzed in detail, providing explicit determinant formulas and rational expressions \\cite{HK3}.\n\nThe present paper occupies the remaining natural corner of this picture: we define and study the type~1 chamber zeta function for the same standard non-uniform complex $\\Gamma\\backslash\\mathcal{B}$ and show that it admits an Ihara--Bass type determinant formula together with an explicit rational expression. From this point of view, one may regard our result as completing the first explicit higher-rank case in which both edge and chamber zeta functions are understood for a non-uniform arithmetic quotient of a Bruhat--Tits building. This suggests several possible generalizations, for instance to higher rank groups such as $\\mathrm{PGL}_d$ with $d\\ge4$ or to other non-uniform complexes of groups, where systematic comparisons between edge- and chamber-level zeta functions remain largely unexplored.\n\\end{remark}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[>=latex, node distance=3.4cm, font=\\small]\n\n\\node at (-4.8,3.6) {quotient type};\n\\node[rotate=90] at (-4.95,1.7) {non-uniform};\n\\node[rotate=90] at (-4.9,-1.8) {finite / cocompact};\n\n\\node at (-2.0,-3.6) {edge-level};\n\\node at (3.7,-3.6) {chamber-level};\n\n\\draw[dashed] (0.77,4) -- (0.77,-4);\n\n\\draw[dashed] (-5.2,0) -- (6.2,0);\n\n\\node[anchor=south] at (-2,3.3) {edge zeta};\n\\node[anchor=south] at (3.4,3.3) {chamber zeta};\n\n\\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q1) at (-2,-1.8)\n{finite graphs,\\\\\n(regular or irregular)\\\\\n(Ihara, Bass, Hashimoto\\\\\nSunada, etc.)};\n\n\\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q2) at (3.5,-1.8)\n{$\\widetilde{A}_2$-buildings,\\\\\ncocompact lattices in\\\\\n$\\mathrm{PGL}_3(F)$, chamber zeta\\\\\n(\\cite{KL},\\cite{KLW10},\\cite{KLW18})};\n\n\\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q3) at (-2,1.8)\n{weighted graphs (\\cite{DK18}),\\\\standard non-uniform\\\\\ncomplex for $\\mathrm{PGL}_3$\\\\\n (\\cite{HK3})\n};\n\n\\node[draw, rounded corners, thick, align=center, minimum width=3.2cm, minimum height=1.6cm, fill=gray!10] (Q4) at (3.5,1.8)\n{standard non-uniform\\\\\ncomplex for $\\mathrm{PGL}_3$,\\\\\ntype~1 chamber zeta\\\\\n\\textbf{(this paper)}};\n\n\\draw[->, thick] (Q3.east) -- (Q4.west);\n\\draw[->, thick] (Q2.north) -- (Q4.south);\n\n\\end{tikzpicture}\n\\caption{A schematic position of this work among edge and chamber zeta functions for cocompact and non-uniform quotients.}\n\\end{figure}\n\nThis paper is organized as follows. We provide the preliminary notions on Bruhat-Tits building for $\\operatorname{PGL}_3$ in Section~\\ref{sec:2}. \nIn Section~\\ref{sec:3}, we deal with the fundamental domain of $\\Gamma$ and the image of tailless galleries in the quotient space by the standard non-uniform lattice $\\Gamma$ of $G$. In Section~\\ref{sec:4}, we define the type 1 chamber zeta function for $\\Gamma\\backslash \\mathcal{B}$ and prove that the zeta function satisfies the determinant formula. In Section \\ref{sec:5}, we present the proof of Theorem \\ref{thm:1.1} and Corollary \\ref{coro:cycle_intro}.",
    "sketch": "Given a class $\\mathcal{C}$, one has $\\mathcal{C}=\\mathcal{C}_0^n$ for some primitive class $\\mathcal{C}_0$, and the type 1 chamber zeta function \"formally satisfies\"\n\\[\nZ_{\\Gamma}(u)^{-1}=\\exp\\left(-\\sum_{n=1}^{\\infty}\\frac{u^{n}}{n}\\sum_{\\mathcal{C}:\\, \\ell(\\mathcal{C})=n}w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0)\\right).\n\\]\nIntroduce the formal vector space\n\\[\nS(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})):=\\bigoplus_{c\\in\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})}\\mathbb{C}c\n\\]\nof type 1 pointed chambers, and define an operator $T$ by\n\\[\nTc=\\sum_{c'} w(c,c')\\,c'\n\\]\nsumming over adjacent type 1 pointed chambers $c'$. One \"will prove the operator $T^n$ is traceable\" and that its trace is\n\\[\n\\operatorname{Tr}(T_k^n)=\\sum_{\\mathcal{C}:\\,\\ell(\\mathcal{C})=n} w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0).\n\\]\nHence $Z_{\\Gamma,k}(q^{-s})^{-1}=\\exp\\bigl(\\operatorname{Tr}(\\log(1-uT))\\bigr)$, yielding the determinant expression\n\\[\nZ_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}=\\exp\\left(\\sum_{n=1}^{\\infty}\\frac{N_n(\\Gamma\\backslash\\mathcal{B})}{n}u^{n}\\right),\\qquad\nN_n(\\Gamma\\backslash\\mathcal{B})=\\sum_{\\mathcal{C}:\\,\\ell(\\mathcal{C})=n}w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0).\n\\]\nTo justify traceability, the \"traceable property relies on a monotonicity phenomenon along the cuspidal directions\": certain directed steps force a height to strictly increase/decrease, and returning to the start then requires a path of uniformly bounded-below length, so for each fixed $n$ \"only finitely many closed admissible gallery classes can occur.\" Finally, a \"Fubini trick and truncation by certain horizontal direction allows us to compute this determinant,\" which completes the proof of Theorem~\\ref{thm:1.1} (producing the explicit rational expression).",
    "expanded_sketch": "Given a class $\\mathcal{C}$, one has $\\mathcal{C}=\\mathcal{C}_0^n$ for some primitive class $\\mathcal{C}_0$, and the type 1 chamber zeta function \"formally satisfies\"\n\\[\nZ_{\\Gamma}(u)^{-1}=\\exp\\left(-\\sum_{n=1}^{\\infty}\\frac{u^{n}}{n}\\sum_{\\mathcal{C}:\\, \\ell(\\mathcal{C})=n}w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0)\\right).\n\\]\nIntroduce the formal vector space\n\\[\nS(\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})):=\\bigoplus_{c\\in\\operatorname{C}(\\Gamma\\backslash \\mathcal{B})}\\mathbb{C}c\n\\]\nof type 1 pointed chambers, and define an operator $T$ by\n\\[\nTc=\\sum_{c'} w(c,c')\\,c'\n\\]\nsumming over adjacent type 1 pointed chambers $c'$. One \"will prove the operator $T^n$ is traceable\" and that its trace is\n\\[\n\\operatorname{Tr}(T_k^n)=\\sum_{\\mathcal{C}:\\,\\ell(\\mathcal{C})=n} w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0).\n\\]\nHence $Z_{\\Gamma,k}(q^{-s})^{-1}=\\exp\\bigl(\\operatorname{Tr}(\\log(1-uT))\\bigr)$, yielding the determinant expression\n\\[\nZ_{\\Gamma}(u)=\\frac{1}{\\det(I-uT)}=\\exp\\left(\\sum_{n=1}^{\\infty}\\frac{N_n(\\Gamma\\backslash\\mathcal{B})}{n}u^{n}\\right),\\qquad\nN_n(\\Gamma\\backslash\\mathcal{B})=\\sum_{\\mathcal{C}:\\,\\ell(\\mathcal{C})=n}w(\\mathcal{C})\\,\\ell(\\mathcal{C}_0).\n\\]\nTo justify traceability, the \"traceable property relies on a monotonicity phenomenon along the cuspidal directions\": certain directed steps force a height to strictly increase/decrease, and returning to the start then requires a path of uniformly bounded-below length, so for each fixed $n$ \"only finitely many closed admissible gallery classes can occur.\" Finally, a \"Fubini trick and truncation by certain horizontal direction allows us to compute this determinant,\" which completes the proof of the main theorem (producing the explicit rational expression).",
    "expanded_theorem": "\\label{thm:1.1} For $G=\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$ and $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$, the type 1 chamber zeta function $Z_{\\Gamma}(u)$ converges for sufficiently $u$, and it is given by \n$$Z_{\\Gamma}(u)=\\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}\n.$$",
    "theorem_type": [
      "Existence",
      "Algorithmic or Constructive"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal B\\) be the Bruhat--Tits building of \\(G=\\operatorname{PGL}(3,\\mathbb F_q((t^{-1})))\\), and let \\(\\Gamma=\\operatorname{PGL}(3,\\mathbb F_q[t])\\). For the quotient \\(\\Gamma\\backslash \\mathcal B\\), define the type 1 chamber zeta function by\n\\[\nZ_\\Gamma(u)=\\prod_{\\mathcal C}\\left(1-w(\\mathcal C)u^{\\ell(\\mathcal C)}\\right)^{-1},\n\\]\nwhere the product runs over tailless type 1 primitive closed gallery classes \\(\\mathcal C\\), \\(\\ell(\\mathcal C)\\) is the gallery length, and for a representative closed gallery \\((c_1,\\dots,c_n)\\), its weight is \\(w(\\mathcal C)=\\prod_j w(c_j,c_{j+1})\\), with \\(w(c,c')\\) the number of lifts of \\(c'\\) that form a tailless type 1 gallery with a lift of \\(c\\). Which statement is valid for \\(Z_\\Gamma(u)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "\\(Z_\\Gamma(u)\\) converges for sufficiently small \\(u\\), and\n\\[\nZ_\\Gamma(u)=\\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}.\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\(Z_\\Gamma(u)\\) converges for sufficiently small \\(u\\), and\n\\[\nZ_\\Gamma(u)=\\frac{(1-q^3u^6)(1-q^2u^3)}{(1-q^4u^6)(1-q^3u^3)}.\n\\]"
        },
        {
          "label": "C",
          "text": "\\(Z_\\Gamma(u)\\) converges for sufficiently small \\(u\\), and \\(Z_\\Gamma(u)\\) is a rational function of \\(u\\)."
        },
        {
          "label": "D",
          "text": "\\(Z_\\Gamma(u)\\) converges for sufficiently small \\(u\\), and\n\\[\nZ_\\Gamma(u)=\\frac{1}{\\det(I-uT)},\n\\]\nwhere \\(T\\) is the weighted type 1 chamber transfer operator on type 1 pointed chambers of \\(\\Gamma\\backslash\\mathcal B\\)."
        },
        {
          "label": "E",
          "text": "\\(Z_\\Gamma(u)\\) converges for sufficiently small \\(u\\), and\n\\[\nZ_\\Gamma(u)=\\frac{(1-q^4u^3)(1-q^2u^6)}{(1-q^3u^3)(1-q^3u^6)}.\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "explicit determinant evaluation factors placed on wrong sides",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "dropped the explicit rational formula while keeping convergence and rationality",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "formal determinant identity for transfer operator treated as the final theorem in place of the computed closed formula",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "length divisibility structure in exponents 3 and 6 interchanged",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the formula or uniquely signal choice A. It names the objects and asks for the explicit statement, but does not leak the answer beyond identifying the theorem's topic."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct recall of a specific explicit theorem/formula for the zeta function in the stated setting. The task is mainly to recognize the exact closed form rather than derive or compare genuinely different mathematical conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because several choices are close variants of the correct formula, and one option is a weaker true statement. However, solving it mainly depends on remembering the exact identity rather than generating a conclusion from underlying principles."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically targeted: they alter exponents, swap numerator/denominator factors, omit numerator terms, or offer a weaker true statement. These align with realistic failure modes in recalling or manipulating the formula."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed recall-style MCQ with strong distractors, but it is largely theorem-restatement rather than a genuine test of generative mathematical reasoning."
    }
  },
  {
    "id": "2512.23302v1",
    "paper_link": "http://arxiv.org/abs/2512.23302v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "cor",
      "content": "Assume GRH for the non-principal character modulo $4$. Then, the natural density of the set $ \\mathcal{P}_{1/2}(4;3,1)=\\{ x\\ge 2\\, :\\, \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}$\n    exists and equals $1$.",
      "start_pos": 8575,
      "end_pos": 8799,
      "label": null
    },
    "ref_dict": {
      "main": "\\begin{thm}\\label{main}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\M(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$, and the logarithmic density of the set \n    \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\M(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$.\n\\end{thm}",
      "pi_1/2": "\\begin{equation}\\label{pi_1/2}\n    \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}",
      "DRH": "\\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}",
      "DRH-estimate": "\\begin{equation}\\label{DRH-estimate}\n    \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\M(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}",
      "ing": "\\begin{lem}\\label{ing}\n    If \\text{GRH} holds for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$, then:\\begin{enumerate}\n        \\item Uniformly for $Y\\ge 1$: \\[\\bigl| G(Y;t) \\bigr|\\ll_t \\log Y\\,  .\\]\n        \\item There exists $L\\in \\C$ such that uniformly for $Y \\ge1$: \\[ \\int_{\\log 2}^Y\\frac{\\Delta(y;t)}{y}\\mathrm{d}y=L+O_t\\left(\\frac{\\log Y}{Y}\\right)\\, .\\]\n        \\item For all $\\varepsilon>0 $, we have uniformly for $Y\\ge 1$: \\[\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+\\varepsilon}}\\mathrm{d}y \\ll_{\\varepsilon,t} 1\\, .\\]\n        \\item For all $\\varepsilon>0$, we have \\[\\lim_{Y\\to \\infty} \\frac{1}{Y} \\int_{\\log 2}^Y\\frac{|\\Delta(u;t)|^2}{u^\\varepsilon}\\mathrm{d}u=0\\, .\\]\n    \\end{enumerate}\n\\end{lem}"
    },
    "pre_theorem_intro_text_len": 6126,
    "pre_theorem_intro_text": "In a letter~\\cite{Chebyshev} sent to Fuss in 1853, Chebyshev observed that primes congruent to $3\\bmod\\,  4$ seem to appear more often than those congruent to $1\\bmod\\,  4$. Let $q\\ge3$ and let $a\\in \\mathbb Z$ be coprime to $q$, and define $\\pi(x;q,a)=\\#\\{ p\\le x\\, :\\,  p\\equiv a\\bmod\\,  q\\}$. Although the prime number theorem in arithmetic progressions asserts that primes are equidistributed among invertible classes modulo $q$, it gives no control over the difference $\\pi(x;q,a)-\\pi(x;q,b)$ when $b\\in\\mathbb Z$ is coprime to $q$. In 1914, Littlewood~\\cite{Littlewood} proved that the function $x\\mapsto\\pi(x;4,1)-\\pi(x;4,3)$ changes sign infinitely often. This naturally raised the question of the existence of the natural density of the set $\\mathcal{P}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi(x;q,a)>\\pi(x;q,b)\\}$, namely the limit \\[ \\lim_{X\\to \\infty}\\frac{\\boldsymbol{m}\\{2\\le x\\le X\\, :\\, x\\in \\mathcal{P}(q;a,b) \\}}{X},\\] where $\\boldsymbol{m}$ is the Lebesgue measure on $\\mathbb R$. This question was first addressed by Kaczorowski~\\cite{Kac} who proved under $\\text{GRH}$ that the natural density of $\\mathcal{P}(4;3,1)$ does not exist. \n\nIn 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n    \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n    \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n    where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if  \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller. \n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$, and the logarithmic density of the set \n    \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\,  q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:",
    "context": "In 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n exists and equals $1$, and the logarithmic density of the set \n \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\, q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:",
    "full_context": "In 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n exists and equals $1$, and the logarithmic density of the set \n \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\, q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:\n\nIn 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nOur last main result concerns the mean $\\frac{1}{x}\\int_2^x(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b))\\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\\eqref{DRH-estimate}:\n\n\\end{proof}\n\\section{Proofs of the main theorems}\nAs explained at the beginning of section~\\ref{sec:Prel}, we will prove our theorems for a general function $t:(\\Z/q\\Z)^\\times\\to \\C$ satisfying $\\langle t,\\chi_0\\rangle=0$. Our main theorems are obtained by replacing $t$ with $\\mathds{1}_{\\{a\\}}-\\mathds{1}_{\\{b\\}}$. In what follows, we assume GRH for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$.\n\\begin{proof}[Proof of Theorem~\\ref{main}]\n We begin with a summation by parts, yielding, for all $x\\ge 2$,\n\\[\\pi_{1/2}(x;t)=\\frac{\\pi(x;t)}{\\sqrt{x}}+\\frac{1}{2}\\int_2^x\\frac{\\pi(u;t)}{u^{3/2}}\\mathrm{d}u\\, .\\]\nThus, for all $y\\ge 1$\n\\begin{align*}\\pi_{1/2}(e^y;t)&=\\frac{\\pi(e^y;t)}{\\exp(y/2)}+\\frac{1}{2}\\int_{\\log 2}^y\\frac{\\pi(e^u;t)}{\\exp(u/2)}\\mathrm{d}u\\\\\n&=\\frac{1}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)} \\right)+\\frac{1}{2}\\int_{\\log 2}^y\\frac{1}{u}\\left(\\Delta(u;t)-\\frac{\\M(t)}{\\varphi(q)}\\right)\\mathrm{d}u\\, .\n\\end{align*}\nHence, by the second estimate of Lemma~\\ref{ing}, denoting $C=(\\varphi(q)L+\\M(t)\\log \\log 2)/(2\\varphi(q))$, we have \n\\begin{equation}\\label{main-form}\n \\pi_{1/2}(e^y;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log y-C=\\frac{1}{y}\\left( \\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\n\\end{equation}\nwhere $|R(y)|\\ll \\log y$. Let $Y_0=Y_0(t)$ be a large positive constant such that for all $y\\ge Y_0$ we have $|R(y)-\\M(t)/\\varphi(q)|<\\sqrt{y}/2$. Denote by $\\m$ the Lebesgue measure on $\\R$, fix $\\varepsilon>0$, and define \\[E:=\\left\\{ x\\ge 2\\, :\\, \\left|\\pi_{1/2}(x;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log \\log x-C\\right|>\\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\, .\\]\nLet $Y>Y_0$, by the triangle inequality and Markov's inequality\n\\begin{align*}\n \\m \\left\\{ \\log 2\\le y\\le Y\\, :\\, e^y\\in E\\right\\} &\\le Y_0+ \\m\\left\\{ Y_0\\le y\\le Y\\, :\\, \\left| \\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right|>y^{\\frac{1}{2}+\\varepsilon} \\right\\}\\\\\n &\\le Y_0+ \\m\\left\\{ Y_0\\le y\\le Y\\, :\\, \\bigl| \\Delta(y;t) \\bigr|>\\frac{y^{\\frac{1}{2}+\\varepsilon}}{2}\\right\\}\\\\\n &\\le Y_0+ 4 \\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+2\\varepsilon}}\\mathrm{d}y\\ll1\\, .\n\\end{align*}\nWe just proved that \\[ \\int_{\\log 2}^\\infty \\mathds{1}_E(e^y)\\mathrm{d}y <\\infty \\, . \\]\nThus, by~\\eqref{nat-density}, the natural density of the set $E$ exists and equals $0$. Hence, the natural density of its complement exists and equals $1$. We proceed similarly for the statement regarding the logarithmic density. Define \n\\[E':=\\left\\{ x\\ge 2\\, :\\, \\left|\\pi_{1/2}(x;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log \\log x-C\\right|>\\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\, .\\]\nLet $Y_1>\\log 2$ be large enough so that for all $y\\ge Y_1$ we have $|R(y)-\\M(t)/\\varphi(q)|<y^\\varepsilon/2$. We deduce that \n\\[\\m \\left\\{ \\log 2\\le y\\le Y\\, :\\, e^y\\in E'\\right\\}\\le Y_1+4\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{2\\varepsilon}}\\mathrm{d}y\\, .\\]\nThis proves that the logarithmic density of the set $E'$ exists and equals $0$, hence the logarithmic density of its complement exists and equals $1$.\n\\end{proof}\nWe now move to Theorem~\\ref{GRHalmimpliesDRH}\n\\begin{proof}[Proof of Theorem~\\ref{GRHalmimpliesDRH}]\n Define, for $x\\ge 2$, $F(x)=(\\log x)^{m_\\chi} \\prod_{p\\le x} \\bigl(\\,1-\\chi(p)/\\sqrt{p}\\,\\bigr)$. Following the proof of~\\cite{Ao-Ko}*{Proposition 2.1}, and using the following form of Mertens' theorem \\[ \\sum_{p\\le x} \\frac{\\chi(p^2)}{p}=\\langle \\chi,r\\rangle \\log \\log x+c+O\\left(\\frac{1}{\\log x}\\right) \\, ,\\]\n where $c\\in \\C$ is a constant, we deduce that there exists a constant $C'\\in \\C$ such that \n \\[ \\log F(x)= \\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x+C'+O\\left(\\frac{1}{\\log x}\\right)\\, .\\]\n Assume that $t=\\chi$, and let $\\varepsilon>0$ and $C$ be the constant given by~\\eqref{main-form}. Define $\\ell_\\chi =\\exp(C+C')\\ne 0$, and fix $x\\ge 2$ sufficiently large satisfying the inequality\n \\[ \\left|\\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x-C\\right|<\\frac{1}{(\\log x)^{(1-\\varepsilon)/2}}\\, . \\]\n Using the inequality $|\\exp(z)-1|\\le 2|z|$ for $|z|<1/2$, we deduce that \\begin{align*} |F(x)-\\ell_\\chi|&=|\\ell_\\chi|\\,\\bigl|\\exp(\\log F(x)-C-C')-1\\bigr|\\\\\n &\\le 2|\\ell_\\chi|\\, \\left|\\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x-C+O\\left(\\frac{1}{\\log x}\\right)\\right|\\\\\n &< \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\, ,\n \\end{align*}\n since $x$ is sufficiently large. Since $\\mathcal{E}_\\chi$ contains all sufficiently large values of a set of natural density $1$. It follows that the natural density of $\\mathcal{E}_\\chi$ exists and equals $1$. \n\\end{proof}\nWe now prove Theorem~\\ref{mean}.\n\\begin{proof}[Proof of Theorem~\\ref{mean}]\n Let $x\\ge 2$ and set $Y=\\log x$.\n From~\\eqref{main-form} we deduce that \n \\[\\int_{\\log 2}^Y e^y\\pi_{1/2}(e^y;t)\\mathrm{d}y=\\int_{\\log 2}^Y\\left(-\\frac{\\M(t)}{2\\varphi(q)}e^y \\log y+Ce^y\\right)\\mathrm{d}y+\\int_{\\log 2}^Y\\frac{e^y}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\\mathrm{d}y\\, .\\]\n We now use Lemma~\\ref{ing} together with the bound $ \\int_{\\log 2}^Y\\tfrac{e^y}{y}\\mathrm{d}y\\ll e^Y/Y$ to deduce that \\[ \\left| \\int_{\\log 2}^Y \\frac{e^y \\Delta(y;t)}{y}\\mathrm{d}y\\right|=\\left| \\frac{e^Y}{Y}G(Y;t)-\\int_{\\log 2}^Y G(y;t)\\frac{ye^y-e^y}{y^2}\\mathrm{d}y\\right|\\ll \\log Y \\frac{e^Y}{Y}\\, .\\]\n Since $|R(y)|\\ll \\log y$, we have\n \\[ \\left|\\int_{\\log 2}^Y \\frac{e^y}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\\mathrm{d}y\\right|\\ll \\log Y\\frac{e^Y}{Y}\\, .\\]\n Using the classical estimate \n \\[ \\int_{\\log 2}^Y e^y \\log y\\, \\mathrm{d}y=e^Y \\log Y+O\\left(\\frac{e^Y}{Y}\\right)\\, , \\]\n we conclude that\n \\[\\int_2^x\\pi_{1/2}(u;t)\\mathrm{d}u =\\int_{\\log 2}^Ye^y\\pi_{1/2}(e^y;t) \\mathrm{d}y=-\\frac{\\M(t)}{2\\varphi(q)}e^Y\\log Y+Ce^Y+O\\left(\\frac{e^Y \\log Y}{Y}\\right)\\, .\\]\n The result follows by substituting $Y=\\log x$.\n\\end{proof}",
    "post_theorem_intro_text_len": 3578,
    "post_theorem_intro_text": "Our second main result shows that if GRH holds, then the equality~\\eqref{DRH} holds for a set of natural density $1$.\n\\begin{thm}\\label{GRHalmimpliesDRH}\n    Assume GRH for a non-principal Dirichlet character $\\chi$ modulo $q$. Then, there exists $\\ell_\\chi\\ne 0$ such that the natural density of the set \n    \\begin{equation}\\label{Echi}\n        \\mathcal{E}_\\chi:=\\left\\{x\\ge 2\\, :\\, \\left|  (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}-\\ell_\\chi  \\right|\\le \\frac{1}{(\\log x)^{1/2 - \\varepsilon}} \\right\\}\n    \\end{equation}\n    exists and equals $1$.\n\\end{thm}\n\nOur last main result concerns the mean $\\frac{1}{x}\\int_2^x(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b))\\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\\eqref{DRH-estimate}:\n\n\\begin{thm}\\label{mean}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $a,b$ be distinct invertible residue classes modulo $q$, we have \n    \\[\\frac{1}{x}\\int_{ 2}^x \\bigl(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b)\\bigr)\\mathrm{d}u=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x+C+O\\left(\\frac{\\log \\log x}{\\log x}\\right)\\, .\\]\n\\end{thm}\nAlthough our arguments extend straightforwardly to the context of global fields, we restrict our focus to the classical case of primes in arithmetic progressions for the sake of clarity and to highlight the underlying ideas.\nTo establish our results, we rely on the explicit formula for prime counting functions. However, a direct application of this formula to the natural auxiliary Chebyshev function $\\psi_{1/2}$ associated with the weighted function $\\pi_{1/2}$ would be inconclusive: the resulting error term is of order $O(\\log x)$, while its main term is not expected to be that large; in fact that main term is expected to be of order $O((\\log \\log \\log x)^2)$ by Montgomery's conjecture~\\cite{Montg}. Instead, our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts. We show (Lemma~\\ref{ing}) that, thanks to the explicit formula for $\\pi$, the integral term arising from this summation yields the main bias term\n\\[-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x +C + O\\left(\\frac{\\log \\log x}{\\log x}\\right).\\]\nThe remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$. Relying on the fact that the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance, we apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$. Since our estimates are naturally formulated on a logarithmic scale, we deduce the existence of the natural density, by showing that the set of large deviations has finite logarithmic measure. Indeed, any Borel set $A\\subset (1,\\infty)$ satisfying $\\int_0^\\infty \\mathds{1}_A(e^y)\\mathrm{d}y<\\infty$ has zero natural density. To see this, note that for any $L>1$,\n \\begin{equation}\\label{nat-density}\n    0\\le \\limsup_{x\\to \\infty}\\frac{1}{x} \\int_1^x \\mathds{1}_A(u)\\mathrm{d}u\\le \\limsup_{x\\to \\infty}\\int_L^x \\mathds{1}_A(u) \\frac{\\mathrm{d}u}{u}=\\int_{\\log L}^\\infty \\mathds{1}_{A}\\bigl(e^y\\bigr)\\mathrm{d}y\\, .\n\\end{equation}\nThe result follows by letting $L\\to \\infty$.\n\\subsection*{Acknowledgments} I would like to thank Florent Jouve and Daniel Fiorilli for their constant encouragement and for their suggestions, which were essential to the development of this paper. I am also grateful to Alexandre Bailleul for his detailed reading and for several important remarks.",
    "sketch": "To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\\log x)$ while the main term “is expected to be of order $O((\\log\\log\\log x)^2)$.”\n\nInstead, “our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts.” They show (Lemma~\\ref{ing}) that, using the explicit formula for $\\pi$, “the integral term arising from this summation yields the main bias term”\n\\[\n-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log\\log x + C + O\\left(\\frac{\\log\\log x}{\\log x}\\right).\n\\]\nThe “remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$.” Using that “the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$.”\n\nFinally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\\subset(1,\\infty)$ with $\\int_0^\\infty \\mathds{1}_A(e^y)\\,dy<\\infty$ has zero natural density, via the bound\n\\[\n0\\le \\limsup_{x\\to\\infty}\\frac1x\\int_1^x \\mathds{1}_A(u)\\,du \\le \\int_{\\log L}^\\infty \\mathds{1}_A(e^y)\\,dy\n\\]\n(and “letting $L\\to\\infty$”).",
    "expanded_sketch": "To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\\log x)$ while the main term “is expected to be of order $O((\\log\\log\\log x)^2)$.”\n\nInstead, “our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts.” They show the following lemma.\n\n\\begin{lem}\\label{ing}\n    If \\text{GRH} holds for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$, then:\\begin{enumerate}\n        \\item Uniformly for $Y\\ge 1$: \\[\\bigl| G(Y;t) \\bigr|\\ll_t \\log Y\\,  .\\]\n        \\item There exists $L\\in \\C$ such that uniformly for $Y \\ge1$: \\[ \\int_{\\log 2}^Y\\frac{\\Delta(y;t)}{y}\\mathrm{d}y=L+O_t\\left(\\frac{\\log Y}{Y}\\right)\\, .\\]\n        \\item For all $\\varepsilon>0 $, we have uniformly for $Y\\ge 1$: \\[\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+\\varepsilon}}\\mathrm{d}y \\ll_{\\varepsilon,t} 1\\, .\\]\n        \\item For all $\\varepsilon>0$, we have \\[\\lim_{Y\\to \\infty} \\frac{1}{Y} \\int_{\\log 2}^Y\\frac{|\\Delta(u;t)|^2}{u^\\varepsilon}\\mathrm{d}u=0\\, .\\]\n    \\end{enumerate}\n\\end{lem}\n\nUsing the explicit formula for $\\pi$, they then argue that “the integral term arising from this summation yields the main bias term”\n\\[\n-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log\\log x + C + O\\left(\\frac{\\log\\log x}{\\log x}\\right).\n\\]\nThe “remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$.” Using that “the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$.”\n\nFinally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\\subset(1,\\infty)$ with $\\int_0^\\infty \\mathds{1}_A(e^y)\\,dy<\\infty$ has zero natural density, via the bound\n\\[\n0\\le \\limsup_{x\\to\\infty}\\frac1x\\int_1^x \\mathds{1}_A(u)\\,du \\le \\int_{\\log L}^\\infty \\mathds{1}_A(e^y)\\,dy\n\\]\n(and “letting $L\\to\\infty$”).",
    "expanded_theorem": "Assume GRH for the non-principal character modulo $4$. Then, the natural density of the set $ \\mathcal{P}_{1/2}(4;3,1)=\\{ x\\ge 2\\, :\\, \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}$\n    exists and equals $1$.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let\n\\[\n\\pi_{1/2}(x;4,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\!\\!\\pmod 4}}\\frac{1}{\\sqrt p}\n\\]\nfor invertible residue classes $a\\bmod 4$, where the sum is over primes $p\\le x$. Assume the Generalized Riemann Hypothesis for the non-principal Dirichlet character modulo $4$. Consider\n\\[\n\\mathcal P_{1/2}(4;3,1)=\\{x\\ge 2:\\ \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}.\n\\]\nWhich statement holds about this set, where its natural density means\n\\[\n\\lim_{X\\to\\infty}\\frac{1}{X}\\int_2^X \\mathbf 1_{\\mathcal P_{1/2}(4;3,1)}(u)\\,du\n\\]\nwhen the limit exists?",
      "correct_choice": {
        "label": "A",
        "text": "The natural density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The natural density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $\\tfrac12$."
        },
        {
          "label": "C",
          "text": "The upper natural density of $\\mathcal P_{1/2}(4;3,1)$ is equal to $1$."
        },
        {
          "label": "D",
          "text": "The logarithmic density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$, but the natural density need not exist."
        },
        {
          "label": "E",
          "text": "There exists a subset $S\\subseteq \\mathcal P_{1/2}(4;3,1)$ with natural density $1$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "main bias forces eventual dominance toward residue class $3$ rather than symmetry",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "existence of the full natural density limit",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "upgrade from logarithmic-scale control to natural density via finite logarithmic measure of exceptions",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replacing density of the whole set by density of a dense subset",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the objects carefully and asks which existence statement holds, but it does not explicitly reveal or strongly hint at the correct density conclusion."
      },
      "TAS": {
        "score": 0,
        "justification": "The item is essentially asking for the exact theorem-level conclusion about the specific set \\(\\mathcal{P}_{1/2}(4;3,1)\\). This makes it close to a direct restatement rather than a problem requiring selection among independently derived consequences."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure because the options distinguish natural density, upper natural density, logarithmic density, and strict versus non-strict inequalities. However, the question mainly tests recall/recognition of the precise result rather than substantial derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic confusions: symmetry suggesting density \\(1/2\\), weakening to upper density, mixing logarithmic and natural density, and incorrectly upgrading a density-one non-strict inequality statement to strict inequality on the original set."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically careful MCQ with strong distractors and no answer leakage, but it is largely a theorem-recall item and only moderately tests generative reasoning."
    }
  },
  {
    "id": "2512.23302v1",
    "paper_link": "http://arxiv.org/abs/2512.23302v1",
    "theorems_cnt": 4,
    "theorem": {
      "env_name": "cor",
      "content": "Assume GRH for the non-principal character modulo $4$. Then, the natural density of the set $ \\mathcal{P}_{1/2}(4;3,1)=\\{ x\\ge 2\\, :\\, \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}$\n    exists and equals $1$.",
      "start_pos": 8575,
      "end_pos": 8799,
      "label": null
    },
    "ref_dict": {
      "main": "\\begin{thm}\\label{main}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\M(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$, and the logarithmic density of the set \n    \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\M(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$.\n\\end{thm}",
      "pi_1/2": "\\begin{equation}\\label{pi_1/2}\n    \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}",
      "DRH": "\\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}",
      "DRH-estimate": "\\begin{equation}\\label{DRH-estimate}\n    \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\M(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}",
      "ing": "\\begin{lem}\\label{ing}\n    If \\text{GRH} holds for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$, then:\\begin{enumerate}\n        \\item Uniformly for $Y\\ge 1$: \\[\\bigl| G(Y;t) \\bigr|\\ll_t \\log Y\\,  .\\]\n        \\item There exists $L\\in \\C$ such that uniformly for $Y \\ge1$: \\[ \\int_{\\log 2}^Y\\frac{\\Delta(y;t)}{y}\\mathrm{d}y=L+O_t\\left(\\frac{\\log Y}{Y}\\right)\\, .\\]\n        \\item For all $\\varepsilon>0 $, we have uniformly for $Y\\ge 1$: \\[\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+\\varepsilon}}\\mathrm{d}y \\ll_{\\varepsilon,t} 1\\, .\\]\n        \\item For all $\\varepsilon>0$, we have \\[\\lim_{Y\\to \\infty} \\frac{1}{Y} \\int_{\\log 2}^Y\\frac{|\\Delta(u;t)|^2}{u^\\varepsilon}\\mathrm{d}u=0\\, .\\]\n    \\end{enumerate}\n\\end{lem}"
    },
    "pre_theorem_intro_text_len": 6126,
    "pre_theorem_intro_text": "In a letter~\\cite{Chebyshev} sent to Fuss in 1853, Chebyshev observed that primes congruent to $3\\bmod\\,  4$ seem to appear more often than those congruent to $1\\bmod\\,  4$. Let $q\\ge3$ and let $a\\in \\mathbb Z$ be coprime to $q$, and define $\\pi(x;q,a)=\\#\\{ p\\le x\\, :\\,  p\\equiv a\\bmod\\,  q\\}$. Although the prime number theorem in arithmetic progressions asserts that primes are equidistributed among invertible classes modulo $q$, it gives no control over the difference $\\pi(x;q,a)-\\pi(x;q,b)$ when $b\\in\\mathbb Z$ is coprime to $q$. In 1914, Littlewood~\\cite{Littlewood} proved that the function $x\\mapsto\\pi(x;4,1)-\\pi(x;4,3)$ changes sign infinitely often. This naturally raised the question of the existence of the natural density of the set $\\mathcal{P}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi(x;q,a)>\\pi(x;q,b)\\}$, namely the limit \\[ \\lim_{X\\to \\infty}\\frac{\\boldsymbol{m}\\{2\\le x\\le X\\, :\\, x\\in \\mathcal{P}(q;a,b) \\}}{X},\\] where $\\boldsymbol{m}$ is the Lebesgue measure on $\\mathbb R$. This question was first addressed by Kaczorowski~\\cite{Kac} who proved under $\\text{GRH}$ that the natural density of $\\mathcal{P}(4;3,1)$ does not exist. \n\nIn 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n    \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n    \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n    where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if  \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller. \n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$, and the logarithmic density of the set \n    \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n    exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\,  q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:",
    "context": "In 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n exists and equals $1$, and the logarithmic density of the set \n \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\, q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:",
    "full_context": "In 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nMore recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \\begin{equation}\\label{pi_1/2}\n \\pi_{1/2}(x;q,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\bmod\\, {q}}}\\frac{1}{\\sqrt{p}}\n\\end{equation}\nand ask whether the natural density of the set $\\mathcal{P}_{1/2}(q;a,b):=\\{x\\ge 2\\, :\\, \\pi_{1/2}(x;q,a)>\\pi_{1/2}(x;q,b)\\}$ exists. In their framework, and that of subsequent authors~\\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \\begin{equation}\\label{DRH} (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}=\\ell_\\chi+o(1) \\quad(x\\to \\infty) ,\\end{equation}\nwhere $\\ell_\\chi\\ne 0$ and $m_\\chi:=\\text{ord}_{s=1/2}L(s,\\chi)$ is the order of vanishing of $L(s,\\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\\sum_{p\\le x} \\chi(p^2)/p$, leads to an estimate of $\\sum_{p\\le x} \\chi(p)/\\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \\begin{equation}\\label{DRH-estimate}\n \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b)=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x+C+o(1) \\, ,\n\\end{equation}\n where $\\text{M}(q;a,b)=r(a)-r(b)+2\\sum_{\\chi \\ne \\chi _0}(\\chi(a)-\\chi(b))m_\\chi$, $r:(\\mathbb Z/q\\mathbb Z)^\\times\\to \\Z_{\\ge 0}$ defined by $r(a)=\\#\\{ x\\in (\\mathbb Z/q\\mathbb Z)^\\times\\, :\\, x^2=a\\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\\cite{Su} studied the weighted Chebyshev-type function $\\psi_{1/2}(x):=\\sum_{n\\le x}\\Lambda(n)/\\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if \n\\[\\sqrt{x}\\left(\\sum_{\\substack{p\\le x\\\\ p\\equiv 1\\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log\\left(\\frac{x}{p}\\right)-\\sum_{ \\substack{p\\le x\\\\ p\\equiv 3 \\bmod\\, 4}}\\frac{\\log p}{\\sqrt{p}}\\log \\left(\\frac{x}{p}\\right) \\right)\\sim-(\\log x)^2 \\frac{\\sqrt{x}}{4}\\quad (x\\to \\infty)\\, .\\]\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nIn this paper, we show that GRH is sufficient to prove that the equality~\\eqref{DRH-estimate} holds for a set of $x\\ge 2$ of natural density $1$. More precisely: \n\\begin{thm}\\label{main}\n Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\\, a,$ and $b$, such that the natural density of the set \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\]\n exists and equals $1$, and the logarithmic density of the set \n \\[\\left\\{x\\ge 2\\, :\\, \\left| \\pi_{1/2}(x;q,a)-\\pi_{1/2}(x;q,b) +\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x-C\\right|\\le \\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\]\n exists and equals $1$.\n\\end{thm}\nAssuming that $m_\\chi=0$ for all $\\chi \\bmod\\, q$, we have $\\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:\n\nIn 1994, Rubinstein and Sarnak~\\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \\[ \\lim_{X\\to \\infty} \\frac{1}{\\log X}\\int_{2}^X \\mathds{1}_{\\mathcal{P}(q;a,b)}(u)\\frac{\\mathrm{d}u}{u}\\]\nof the set $\\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\\mathcal{P}(4;3,1)$ is approximately $0.9959\\dots$\n\nThis suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\\log p \\log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.\n\nOur last main result concerns the mean $\\frac{1}{x}\\int_2^x(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b))\\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\\eqref{DRH-estimate}:\n\n\\end{proof}\n\\section{Proofs of the main theorems}\nAs explained at the beginning of section~\\ref{sec:Prel}, we will prove our theorems for a general function $t:(\\Z/q\\Z)^\\times\\to \\C$ satisfying $\\langle t,\\chi_0\\rangle=0$. Our main theorems are obtained by replacing $t$ with $\\mathds{1}_{\\{a\\}}-\\mathds{1}_{\\{b\\}}$. In what follows, we assume GRH for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$.\n\\begin{proof}[Proof of Theorem~\\ref{main}]\n We begin with a summation by parts, yielding, for all $x\\ge 2$,\n\\[\\pi_{1/2}(x;t)=\\frac{\\pi(x;t)}{\\sqrt{x}}+\\frac{1}{2}\\int_2^x\\frac{\\pi(u;t)}{u^{3/2}}\\mathrm{d}u\\, .\\]\nThus, for all $y\\ge 1$\n\\begin{align*}\\pi_{1/2}(e^y;t)&=\\frac{\\pi(e^y;t)}{\\exp(y/2)}+\\frac{1}{2}\\int_{\\log 2}^y\\frac{\\pi(e^u;t)}{\\exp(u/2)}\\mathrm{d}u\\\\\n&=\\frac{1}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)} \\right)+\\frac{1}{2}\\int_{\\log 2}^y\\frac{1}{u}\\left(\\Delta(u;t)-\\frac{\\M(t)}{\\varphi(q)}\\right)\\mathrm{d}u\\, .\n\\end{align*}\nHence, by the second estimate of Lemma~\\ref{ing}, denoting $C=(\\varphi(q)L+\\M(t)\\log \\log 2)/(2\\varphi(q))$, we have \n\\begin{equation}\\label{main-form}\n \\pi_{1/2}(e^y;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log y-C=\\frac{1}{y}\\left( \\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\n\\end{equation}\nwhere $|R(y)|\\ll \\log y$. Let $Y_0=Y_0(t)$ be a large positive constant such that for all $y\\ge Y_0$ we have $|R(y)-\\M(t)/\\varphi(q)|<\\sqrt{y}/2$. Denote by $\\m$ the Lebesgue measure on $\\R$, fix $\\varepsilon>0$, and define \\[E:=\\left\\{ x\\ge 2\\, :\\, \\left|\\pi_{1/2}(x;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log \\log x-C\\right|>\\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\right\\}\\, .\\]\nLet $Y>Y_0$, by the triangle inequality and Markov's inequality\n\\begin{align*}\n \\m \\left\\{ \\log 2\\le y\\le Y\\, :\\, e^y\\in E\\right\\} &\\le Y_0+ \\m\\left\\{ Y_0\\le y\\le Y\\, :\\, \\left| \\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right|>y^{\\frac{1}{2}+\\varepsilon} \\right\\}\\\\\n &\\le Y_0+ \\m\\left\\{ Y_0\\le y\\le Y\\, :\\, \\bigl| \\Delta(y;t) \\bigr|>\\frac{y^{\\frac{1}{2}+\\varepsilon}}{2}\\right\\}\\\\\n &\\le Y_0+ 4 \\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+2\\varepsilon}}\\mathrm{d}y\\ll1\\, .\n\\end{align*}\nWe just proved that \\[ \\int_{\\log 2}^\\infty \\mathds{1}_E(e^y)\\mathrm{d}y <\\infty \\, . \\]\nThus, by~\\eqref{nat-density}, the natural density of the set $E$ exists and equals $0$. Hence, the natural density of its complement exists and equals $1$. We proceed similarly for the statement regarding the logarithmic density. Define \n\\[E':=\\left\\{ x\\ge 2\\, :\\, \\left|\\pi_{1/2}(x;t)+\\frac{\\M(t)}{2\\varphi(q)}\\log \\log x-C\\right|>\\frac{1}{(\\log x)^{1-\\varepsilon}}\\right\\}\\, .\\]\nLet $Y_1>\\log 2$ be large enough so that for all $y\\ge Y_1$ we have $|R(y)-\\M(t)/\\varphi(q)|<y^\\varepsilon/2$. We deduce that \n\\[\\m \\left\\{ \\log 2\\le y\\le Y\\, :\\, e^y\\in E'\\right\\}\\le Y_1+4\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{2\\varepsilon}}\\mathrm{d}y\\, .\\]\nThis proves that the logarithmic density of the set $E'$ exists and equals $0$, hence the logarithmic density of its complement exists and equals $1$.\n\\end{proof}\nWe now move to Theorem~\\ref{GRHalmimpliesDRH}\n\\begin{proof}[Proof of Theorem~\\ref{GRHalmimpliesDRH}]\n Define, for $x\\ge 2$, $F(x)=(\\log x)^{m_\\chi} \\prod_{p\\le x} \\bigl(\\,1-\\chi(p)/\\sqrt{p}\\,\\bigr)$. Following the proof of~\\cite{Ao-Ko}*{Proposition 2.1}, and using the following form of Mertens' theorem \\[ \\sum_{p\\le x} \\frac{\\chi(p^2)}{p}=\\langle \\chi,r\\rangle \\log \\log x+c+O\\left(\\frac{1}{\\log x}\\right) \\, ,\\]\n where $c\\in \\C$ is a constant, we deduce that there exists a constant $C'\\in \\C$ such that \n \\[ \\log F(x)= \\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x+C'+O\\left(\\frac{1}{\\log x}\\right)\\, .\\]\n Assume that $t=\\chi$, and let $\\varepsilon>0$ and $C$ be the constant given by~\\eqref{main-form}. Define $\\ell_\\chi =\\exp(C+C')\\ne 0$, and fix $x\\ge 2$ sufficiently large satisfying the inequality\n \\[ \\left|\\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x-C\\right|<\\frac{1}{(\\log x)^{(1-\\varepsilon)/2}}\\, . \\]\n Using the inequality $|\\exp(z)-1|\\le 2|z|$ for $|z|<1/2$, we deduce that \\begin{align*} |F(x)-\\ell_\\chi|&=|\\ell_\\chi|\\,\\bigl|\\exp(\\log F(x)-C-C')-1\\bigr|\\\\\n &\\le 2|\\ell_\\chi|\\, \\left|\\pi_{1/2}(x,\\chi)+\\frac{\\M(\\chi)}{2\\varphi(q)}\\log \\log x-C+O\\left(\\frac{1}{\\log x}\\right)\\right|\\\\\n &< \\frac{1}{(\\log x)^{\\frac{1}{2}-\\varepsilon}}\\, ,\n \\end{align*}\n since $x$ is sufficiently large. Since $\\mathcal{E}_\\chi$ contains all sufficiently large values of a set of natural density $1$. It follows that the natural density of $\\mathcal{E}_\\chi$ exists and equals $1$. \n\\end{proof}\nWe now prove Theorem~\\ref{mean}.\n\\begin{proof}[Proof of Theorem~\\ref{mean}]\n Let $x\\ge 2$ and set $Y=\\log x$.\n From~\\eqref{main-form} we deduce that \n \\[\\int_{\\log 2}^Y e^y\\pi_{1/2}(e^y;t)\\mathrm{d}y=\\int_{\\log 2}^Y\\left(-\\frac{\\M(t)}{2\\varphi(q)}e^y \\log y+Ce^y\\right)\\mathrm{d}y+\\int_{\\log 2}^Y\\frac{e^y}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\\mathrm{d}y\\, .\\]\n We now use Lemma~\\ref{ing} together with the bound $ \\int_{\\log 2}^Y\\tfrac{e^y}{y}\\mathrm{d}y\\ll e^Y/Y$ to deduce that \\[ \\left| \\int_{\\log 2}^Y \\frac{e^y \\Delta(y;t)}{y}\\mathrm{d}y\\right|=\\left| \\frac{e^Y}{Y}G(Y;t)-\\int_{\\log 2}^Y G(y;t)\\frac{ye^y-e^y}{y^2}\\mathrm{d}y\\right|\\ll \\log Y \\frac{e^Y}{Y}\\, .\\]\n Since $|R(y)|\\ll \\log y$, we have\n \\[ \\left|\\int_{\\log 2}^Y \\frac{e^y}{y}\\left(\\Delta(y;t)-\\frac{\\M(t)}{\\varphi(q)}+R(y)\\right)\\mathrm{d}y\\right|\\ll \\log Y\\frac{e^Y}{Y}\\, .\\]\n Using the classical estimate \n \\[ \\int_{\\log 2}^Y e^y \\log y\\, \\mathrm{d}y=e^Y \\log Y+O\\left(\\frac{e^Y}{Y}\\right)\\, , \\]\n we conclude that\n \\[\\int_2^x\\pi_{1/2}(u;t)\\mathrm{d}u =\\int_{\\log 2}^Ye^y\\pi_{1/2}(e^y;t) \\mathrm{d}y=-\\frac{\\M(t)}{2\\varphi(q)}e^Y\\log Y+Ce^Y+O\\left(\\frac{e^Y \\log Y}{Y}\\right)\\, .\\]\n The result follows by substituting $Y=\\log x$.\n\\end{proof}",
    "post_theorem_intro_text_len": 3578,
    "post_theorem_intro_text": "Our second main result shows that if GRH holds, then the equality~\\eqref{DRH} holds for a set of natural density $1$.\n\\begin{thm}\\label{GRHalmimpliesDRH}\n    Assume GRH for a non-principal Dirichlet character $\\chi$ modulo $q$. Then, there exists $\\ell_\\chi\\ne 0$ such that the natural density of the set \n    \\begin{equation}\\label{Echi}\n        \\mathcal{E}_\\chi:=\\left\\{x\\ge 2\\, :\\, \\left|  (\\log x)^{m_\\chi} \\prod_{p\\le x}\\left(1-\\frac{\\chi(p)}{p^{1/2}}\\right)^{-1}-\\ell_\\chi  \\right|\\le \\frac{1}{(\\log x)^{1/2 - \\varepsilon}} \\right\\}\n    \\end{equation}\n    exists and equals $1$.\n\\end{thm}\n\nOur last main result concerns the mean $\\frac{1}{x}\\int_2^x(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b))\\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\\eqref{DRH-estimate}:\n\n\\begin{thm}\\label{mean}\n    Let $q\\ge 3$ and assume \\text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $a,b$ be distinct invertible residue classes modulo $q$, we have \n    \\[\\frac{1}{x}\\int_{ 2}^x \\bigl(\\pi_{1/2}(u;q,a)-\\pi_{1/2}(u;q,b)\\bigr)\\mathrm{d}u=-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log \\log x+C+O\\left(\\frac{\\log \\log x}{\\log x}\\right)\\, .\\]\n\\end{thm}\nAlthough our arguments extend straightforwardly to the context of global fields, we restrict our focus to the classical case of primes in arithmetic progressions for the sake of clarity and to highlight the underlying ideas.\nTo establish our results, we rely on the explicit formula for prime counting functions. However, a direct application of this formula to the natural auxiliary Chebyshev function $\\psi_{1/2}$ associated with the weighted function $\\pi_{1/2}$ would be inconclusive: the resulting error term is of order $O(\\log x)$, while its main term is not expected to be that large; in fact that main term is expected to be of order $O((\\log \\log \\log x)^2)$ by Montgomery's conjecture~\\cite{Montg}. Instead, our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts. We show (Lemma~\\ref{ing}) that, thanks to the explicit formula for $\\pi$, the integral term arising from this summation yields the main bias term\n\\[-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)} \\log \\log x +C + O\\left(\\frac{\\log \\log x}{\\log x}\\right).\\]\nThe remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$. Relying on the fact that the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance, we apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$. Since our estimates are naturally formulated on a logarithmic scale, we deduce the existence of the natural density, by showing that the set of large deviations has finite logarithmic measure. Indeed, any Borel set $A\\subset (1,\\infty)$ satisfying $\\int_0^\\infty \\mathds{1}_A(e^y)\\mathrm{d}y<\\infty$ has zero natural density. To see this, note that for any $L>1$,\n \\begin{equation}\\label{nat-density}\n    0\\le \\limsup_{x\\to \\infty}\\frac{1}{x} \\int_1^x \\mathds{1}_A(u)\\mathrm{d}u\\le \\limsup_{x\\to \\infty}\\int_L^x \\mathds{1}_A(u) \\frac{\\mathrm{d}u}{u}=\\int_{\\log L}^\\infty \\mathds{1}_{A}\\bigl(e^y\\bigr)\\mathrm{d}y\\, .\n\\end{equation}\nThe result follows by letting $L\\to \\infty$.\n\\subsection*{Acknowledgments} I would like to thank Florent Jouve and Daniel Fiorilli for their constant encouragement and for their suggestions, which were essential to the development of this paper. I am also grateful to Alexandre Bailleul for his detailed reading and for several important remarks.",
    "sketch": "To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\\log x)$ while the main term “is expected to be of order $O((\\log\\log\\log x)^2)$.”\n\nInstead, “our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts.” They show (Lemma~\\ref{ing}) that, using the explicit formula for $\\pi$, “the integral term arising from this summation yields the main bias term”\n\\[\n-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log\\log x + C + O\\left(\\frac{\\log\\log x}{\\log x}\\right).\n\\]\nThe “remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$.” Using that “the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$.”\n\nFinally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\\subset(1,\\infty)$ with $\\int_0^\\infty \\mathds{1}_A(e^y)\\,dy<\\infty$ has zero natural density, via the bound\n\\[\n0\\le \\limsup_{x\\to\\infty}\\frac1x\\int_1^x \\mathds{1}_A(u)\\,du \\le \\int_{\\log L}^\\infty \\mathds{1}_A(e^y)\\,dy\n\\]\n(and “letting $L\\to\\infty$”).",
    "expanded_sketch": "To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\\log x)$ while the main term “is expected to be of order $O((\\log\\log\\log x)^2)$.”\n\nInstead, “our strategy is to relate $\\pi_{1/2}$ to $\\pi$ via summation by parts.” They show the following lemma.\n\n\\begin{lem}\\label{ing}\n    If \\text{GRH} holds for all Dirichlet characters $\\chi$ modulo $q$ satisfying $\\langle t,\\chi\\rangle \\ne 0$, then:\\begin{enumerate}\n        \\item Uniformly for $Y\\ge 1$: \\[\\bigl| G(Y;t) \\bigr|\\ll_t \\log Y\\,  .\\]\n        \\item There exists $L\\in \\C$ such that uniformly for $Y \\ge1$: \\[ \\int_{\\log 2}^Y\\frac{\\Delta(y;t)}{y}\\mathrm{d}y=L+O_t\\left(\\frac{\\log Y}{Y}\\right)\\, .\\]\n        \\item For all $\\varepsilon>0 $, we have uniformly for $Y\\ge 1$: \\[\\int_{\\log 2}^Y \\frac{|\\Delta(y;t)|^2}{y^{1+\\varepsilon}}\\mathrm{d}y \\ll_{\\varepsilon,t} 1\\, .\\]\n        \\item For all $\\varepsilon>0$, we have \\[\\lim_{Y\\to \\infty} \\frac{1}{Y} \\int_{\\log 2}^Y\\frac{|\\Delta(u;t)|^2}{u^\\varepsilon}\\mathrm{d}u=0\\, .\\]\n    \\end{enumerate}\n\\end{lem}\n\nUsing the explicit formula for $\\pi$, they then argue that “the integral term arising from this summation yields the main bias term”\n\\[\n-\\frac{\\text{M}(q;a,b)}{2\\varphi(q)}\\log\\log x + C + O\\left(\\frac{\\log\\log x}{\\log x}\\right).\n\\]\nThe “remaining non-integral term is essentially $(\\pi(x;q,a)-\\pi(x;q,b))/\\sqrt{x}$.” Using that “the limiting distribution of $y\\mapsto y(\\pi(e^y;q,a)-\\pi(e^y;q,b))/\\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\\ge 2$ for which the deviation exceeds $1/(\\log x)^{1/2-\\varepsilon}$.”\n\nFinally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\\subset(1,\\infty)$ with $\\int_0^\\infty \\mathds{1}_A(e^y)\\,dy<\\infty$ has zero natural density, via the bound\n\\[\n0\\le \\limsup_{x\\to\\infty}\\frac1x\\int_1^x \\mathds{1}_A(u)\\,du \\le \\int_{\\log L}^\\infty \\mathds{1}_A(e^y)\\,dy\n\\]\n(and “letting $L\\to\\infty$”).",
    "expanded_theorem": "Assume GRH for the non-principal character modulo $4$. Then, the natural density of the set $ \\mathcal{P}_{1/2}(4;3,1)=\\{ x\\ge 2\\, :\\, \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}$\n    exists and equals $1$.",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let\n\\[\n\\pi_{1/2}(x;4,a)=\\sum_{\\substack{p\\le x\\\\ p\\equiv a\\!\\!\\pmod 4}}\\frac{1}{\\sqrt p}\n\\]\nfor invertible residue classes $a\\bmod 4$, where the sum is over primes $p\\le x$. Assume the Generalized Riemann Hypothesis for the non-principal Dirichlet character modulo $4$. Consider\n\\[\n\\mathcal P_{1/2}(4;3,1)=\\{x\\ge 2:\\ \\pi_{1/2}(x;4,3)>\\pi_{1/2}(x;4,1)\\}.\n\\]\nWhich statement holds about this set, where its natural density means\n\\[\n\\lim_{X\\to\\infty}\\frac{1}{X}\\int_2^X \\mathbf 1_{\\mathcal P_{1/2}(4;3,1)}(u)\\,du\n\\]\nwhen the limit exists?",
      "correct_choice": {
        "label": "A",
        "text": "The natural density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$."
      },
      "choices": [
        {
          "label": "B",
          "text": "The natural density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $\\tfrac12$."
        },
        {
          "label": "C",
          "text": "The upper natural density of $\\mathcal P_{1/2}(4;3,1)$ is equal to $1$."
        },
        {
          "label": "D",
          "text": "The logarithmic density of $\\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$, but the natural density need not exist."
        },
        {
          "label": "E",
          "text": "There exists a subset $S\\subseteq \\mathcal P_{1/2}(4;3,1)$ with natural density $1$."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "trace_identity",
          "tampered_component": "main bias forces eventual dominance toward residue class $3$ rather than symmetry",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "existence of the full natural density limit",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "finiteness",
          "tampered_component": "upgrade from logarithmic-scale control to natural density via finite logarithmic measure of exceptions",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "finiteness",
          "tampered_component": "replacing density of the whole set by density of a dense subset",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not state or strongly hint at the correct conclusion; it only defines the weighted prime race and asks which density statement is true under GRH."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a direct specialization of a theorem-sized statement: once the relevant result is known, the correct option is just the theorem instantiated at modulus 4 and weight 1/2."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some pressure to distinguish between natural density, upper natural density, and logarithmic density, and to choose the strongest valid conclusion. However, the item mainly tests recognition/application of a known result rather than genuine multi-step reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and mathematically meaningful: symmetry-based confusion (1/2), a weaker true statement, density-type confusion, and a dense-subset wildcard. They reflect common failure modes and are clearly distinct."
      },
      "total_score": 5,
      "overall_assessment": "A mathematically coherent MCQ with strong distractors and no obvious answer leakage, but it is largely a theorem-recall/specialization item rather than a genuinely generative reasoning question."
    }
  },
  {
    "id": "2512.23359v1",
    "paper_link": "http://arxiv.org/abs/2512.23359v1",
    "theorems_cnt": 3,
    "theorem": {
      "env_name": "theorem",
      "content": "[Local boundedness]\\label{thm.bdd}\nLet  $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak subsolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{a.bound}, and let $B_{r}\\Subset \\Omega$ be a ball with $r\\le1$.\n\\begin{enumerate}\n\\item We have \n\\begin{equation}\\label{est:bdd1}\n\\esssup_{B_{r/2}} u \\le  c\\,r^{-(t-s)/\\beta} \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ r^{sp} T(u_+;r/2,r)\\right]^{1/(p-1)}\n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L)>0$, where \n\\begin{equation}\\label{def.beta}\n\\beta \\coloneqq \\min\\{sp/n,p-1\\}.\n\\end{equation} \n\\item Assume further that $b(\\cdot,\\cdot)$ satisfies \\eqref{a.holder} and \\eqref{assumption.hol}. Then we have\n\\begin{equation}\\label{est:bdd2}\n\\begin{aligned}\n\\esssup_{B_{r/2}} u & \\le c \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ \\frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\\right]^{1/(p-1)} \\\\\n& = c\\left(\\mean{B_{r}}u_{+}^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))\n\\end{aligned}\n\\end{equation}\nwhenever $r \\le R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice} and $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.  \n\\end{enumerate}\nConsequently, if $u \\in \\mathcal{A}(\\Omega)\\cap \\mathcal{T}(\\mathbb{R}^{n})$ is a weak solution to \\eqref{main.eq}, then $u \\in L^{\\infty}_{\\loc}(\\Omega)$ and estimate \\eqref{est:bdd1} or \\eqref{est:bdd2} holds with $|u|$ in each case.",
      "start_pos": 20242,
      "end_pos": 21657,
      "label": "thm.bdd"
    },
    "ref_dict": {
      "tail.pm": "\\begin{lemma}\\label{tail.pm}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot) \\equiv 1$, which is nonnegative in a ball $B_{R}\\equiv B_{R}(x_{0}) \\Subset \\Omega$ with $R\\le R_{0}$. Then \n\\[ T(u_{+};r,2r) \\le cg_{B_{r}}\\left(\\sup_{B_{r}}u\\right) + cT(u_{-};R,r) \\]\nholds for any $r\\in(0,R/2]$, where $c\\equiv c(\\data)>0$.\n\\end{lemma}",
      "Harnack": "\\begin{equation}\\label{Harnack}\n\\sup_{B_{r}}u \\le c\\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r)) \n\\end{equation}",
      "kernel.growth": "\\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}",
      "Caccio.bounded": "\\begin{lemma}[{\\cite[Lemma 6.1]{MR1962933}}]\\label{techlem}\nLet $Z:[r_{0},r_{1}] \\rightarrow \\mathbb{R}$ be a nonnegative and bounded function; let $\\vartheta \\in (0,1)$ and $C_{1},C_{2},\\chi > 0$ be numbers. Assume that\n\\begin{equation*}\nZ(\\varrho_{0}) \\le \\vartheta Z(\\varrho_{1}) + \\frac{C_{1}}{(\\varrho_{1}-\\varrho_{0})^{\\chi}} + C_{2}\n\\end{equation*}\nholds for every choice of $\\varrho_{0}$ and $\\varrho_{1}$ such that $r_{0} \\le \\varrho_{0} < \\varrho_{1} \\le r_{1}$. Then the following inequality holds with $c\\equiv c(\\vartheta,\\chi)>0$:\n\\begin{equation*}\nZ(r_{0}) \\le c\\left[\\frac{C_{1}}{(r_{1}-r_{0})^{\\chi}} + C_{2}\\right].\n\\end{equation*}\n\\end{lemma}\n\n\\section{Proof of Theorem~\\ref{thm.bdd}}\\label{Caccio.bounded}\nThe following Caccioppoli type estimate can be proved in a standard way, see for instance \\cite[Lemma~4.2]{BOS}.\n\n\\begin{lemma}\\label{caccioppoli2}\nLet $u\\in\\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{a.bound}. \nThere exists a constant $c\\equiv c(n,s,t,p,\\nu,L)>0$ such that\n\\begin{equation}\\label{caccio.2}\n\\begin{aligned}\n& \\int_{B_{\\rho}}\\int_{B_{\\rho}}\\left(a(x,y)\\frac{|w_{\\pm}(x)-w_{\\pm}(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{|w_{\\pm}(x)-w_{\\pm}(y)|^{p}}{|x-y|^{n+tp}}\\right)\\,dx\\,dy  \\\\\n& \\le \\frac{c}{(r-\\rho)^{p}}\\int_{B_{r}}\\int_{B_{r}}\\left(a(x,y)\\frac{w_{\\pm}^{p}(x)}{|x-y|^{n+(s-1)p}}+ b(x,y)\\frac{w_{\\pm}^{p}(x)}{|x-y|^{n+(t-1)p}}\\right)\\,dx\\,dy \\\\\n& \\quad\\; + c\\left(\\frac{r}{r-\\rho}\\right)^{n+tp}T(w_{\\pm};r,r)\\int_{B_{r}}w_{\\pm}\\,dx\n\\end{aligned}\n\\end{equation}\nholds whenever $B_{\\rho} \\subset  B_{r} \\Subset \\Omega$ are concentric balls, where $w_{\\pm} \\coloneqq (u-k)_{\\pm}$ for any $k\\in\\mathbb{R}$.\nMoreover, estimate \\eqref{caccio.2} continues to hold for $w_{+}$ (resp. $w_{-}$) when $u$ is merely a weak subsolution (resp. supersolution) to \\eqref{main.eq}.\n\\end{lemma}",
      "alt.p": "\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}",
      "thm.harnack": "\\begin{theorem}[Harnack inequality]\\label{thm.harnack}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot)\\equiv 1$. If $u$ is nonnegative in a ball $B_{10r} \\Subset \\Omega$ with $10r \\leq R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, then we have\n\\begin{equation}\\label{Harnack}\n\\sup_{B_{r}}u \\le c\\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r)) \n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}",
      "est:bdd1": "\\begin{equation}\\label{est:bdd1}\n\\esssup_{B_{r/2}} u \\le  c\\,r^{-(t-s)/\\beta} \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ r^{sp} T(u_+;r/2,r)\\right]^{1/(p-1)}\n\\end{equation}",
      "model.one": "\\begin{equation}\\label{model.one}\n\\begin{aligned}\n\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} \\left(\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\\right)\\,dy = 0\n\\end{aligned}\n\\end{equation}",
      "setting": "\\begin{aligned}\n\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} \\left(\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\\right)\\,dy = 0\n\\end{aligned}\n\\end{equation}\nwith $s,t \\in (0,1)$ and $1<p\\le q < \\infty$, were first considered by De Filippis and Palatucci \\cite{DP19}. More precisely, they proved H\\\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \\cite{BKK,fang2021weak,SM} for various results for \\eqref{model.one}. The papers \\cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\\mathbb{R}^{n}$ and are under the assumption that $t\\le s$, which implies that the second term in \\eqref{model.one} plays a role as a lower order term. Under these settings, $b(\\cdot,\\cdot)$ is imposed to be bounded and discontinuous.\n\nOn the contrary, in the paper \\cite{BOS}, joint with Byun, the second and third authors of this paper proved the local boundedness and H\\\"older continuity of weak solutions to \\eqref{model.one} in the case $s\\le t$, under natural assumptions on the powers and the modulating coefficients. Specifically, it was proved that if\n\\[  b \\in C^{0,\\alpha}(\\mathbb{R}^{n}\\times\\mathbb{R}^{n}) \\;\\; \\text{for some}\\;\\; \\alpha\\in (0,1] \\quad \\text{and} \\quad tq \\le sp + \\alpha, \\]\nthen every locally bounded weak solution to \\eqref{model.one} is locally H\\\"older continuous. \nWe emphasize that when $s \\le t$, the second term in \\eqref{model} or \\eqref{model.one} has a higher order and, as in the local case, the interplay between the H\\\"older regularity of $b(\\cdot,\\cdot)$ and the growth/differentiability condition of the problem plays a key role in the analysis. \nMoreover, in light of the Lavrentiev type phenomena considered in \\cite{Balci}, \nthe above assumption is essentially sharp.\nWe also refer to the paper \\cite{BLS} concerned with regularity results and Harnack inequality for mixed local and nonlocal double phase problems.\n\nIn this paper, we prove the local boundedness and H\\\"older continuity of weak solutions to the general equation \\eqref{main.eq}. Moreover, in the case of one modulating coefficient (i.e. when $a(\\cdot,\\cdot)\\equiv 1$), we further prove a Harnack inequality for weak solutions to \\eqref{main.eq}. \nTo our knowledge, each of our regularity results is the first one for \\eqref{main.eq}; \nmoreover, our Harnack inequality given in Theorem~\\ref{thm.harnack} below is the first one for nonlocal nonautonomous problems.\n\n\\subsection{Assumptions and main results}\\label{setting}\n\nWe say that a function $F:\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ is symmetric if $F(x,y)=F(y,x)$ for every $x,y \\in \\mathbb{R}^{n}$.\n\nThe kernels $K_{sp},K_{tp}:\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ are measurable, symmetric functions that satisfy \n\\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}\nfor a.e. $x,y \\in\\mathbb{R}^{n}$ with $x\\neq y$, where $0 < \\nu \\le 1 \\le L < \\infty$ and\n\\begin{equation}\\label{power}\n 0 < s < t < 1 < p < \\infty.\n\\end{equation} \nThe modulating coefficients $a,b:\\mathbb{R}^{n}\\times\\mathbb{R}^{n} \\rightarrow \\mathbb{R}$ are nonnegative, measurable and symmetric functions that satisfy the bound\n\\begin{equation}\\label{a.bound}\n\\nu \\le a(x,y) + b(x,y) \\le L\n\\end{equation} \nfor a.e. $x,y \\in \\ern$.  \n\nIn addition, for H\\\"older regularity and Harnack inequality, \nwe further assume the H\\\"older continuity of $b(\\cdot,\\cdot)$ as follows: There exist two numbers $[b]_{\\alpha}\\geq 0$ and $\\alpha \\in (0,1]$ such that\n\\begin{equation}\\label{a.holder}\n|b(x_{1},y_{1}) - b(x_{2},y_{2})| \\leq [b]_{\\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\\alpha}\n\\end{equation}\nfor every $(x_{1},y_{1}),(x_{2},y_{2})\\in\\ern\\times\\ern$, and \n\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}\n\n\\begin{remark}\nFor any ball $B_{r}\\subset \\Omega$, we set\n\\begin{equation*}\nb^{+}_{B_{r}} \\coloneqq \\sup_{x,y \\in B_{r}}b(x,y), \\qquad b^{-}_{B_{r}} \\coloneqq \\inf_{x,y \\in B_{r}}b(x,y).\n\\end{equation*}\nMoreover, when \\eqref{a.holder} is in force, we choose a small radius $R_{0} \\equiv R_{0}(\\nu,\\alpha,[b]_{\\alpha}) \\in (0,1)$ satisfying\n\\begin{equation}\\label{R0.choice}\n    [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}\nWe observe that, when $r \\le R_{0}$, \n\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}\nAccordingly, if we set\n\\begin{equation}\\label{G.def}\nG_{B_{r}}(\\tau) \\coloneqq \\sup_{x,y\\in B_{r}}\\left(a(x,y)\\frac{\\tau^{p}}{r^{sp}} + b(x,y)\\frac{\\tau^{p}}{r^{tp}}\\right), \\qquad g_{B_{r}}(\\tau) \\coloneqq \\frac{G_{B_{r}}(\\tau)}{\\tau}, \\qquad \\tau \\ge 0.\n\\end{equation}\nthen we have\n\\begin{align}\\label{alt.dG}\nb^{+}_{B_r} \\le  \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{sp}} + b^+_{B_r}\\frac{\\tau^p}{r^{tp}}\\approx \\frac{\\tau^p}{r^{sp}} + b^-_{B_r}\\frac{\\tau^p}{r^{tp}},\n\\end{align}\n\\begin{align}\\label{alt.G}\nb^{+}_{B_r} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{tp}}.\n\\end{align}\n\\end{remark}\n\nWe next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations.  \nFor $f:\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$, $x_{0} \\in \\mathbb{R}^{n}$ and $r,\\rho>0$, we denote\n\\begin{equation}\\label{def.T}\nT(f;x_{0},r,\\rho) \\coloneqq \\sup_{x \\in B_{\\rho}(x_{0})}\\int_{\\mathbb{R}^{n}\\setminus B_{r}(x_{0})}\\left(a(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy.\n\\end{equation}\nWe will omit the point $x_{0}$ if it is clear from context. \n\nWith the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results. \n\n\\begin{theorem}[Local boundedness]\\label{thm.bdd}\nLet  $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak subsolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{a.bound}, and let $B_{r}\\Subset \\Omega$ be a ball with $r\\le1$.\n\\begin{enumerate}\n\\item We have \n\\begin{equation}\\label{est:bdd1}\n\\esssup_{B_{r/2}} u \\le  c\\,r^{-(t-s)/\\beta} \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ r^{sp} T(u_+;r/2,r)\\right]^{1/(p-1)}\n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L)>0$, where \n\\begin{equation}\\label{def.beta}\n\\beta \\coloneqq \\min\\{sp/n,p-1\\}.\n\\end{equation} \n\\item Assume further that $b(\\cdot,\\cdot)$ satisfies \\eqref{a.holder} and \\eqref{assumption.hol}. Then we have\n\\begin{equation}\\label{est:bdd2}\n\\begin{aligned}\n\\esssup_{B_{r/2}} u & \\le c \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ \\frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\\right]^{1/(p-1)} \\\\\n& = c\\left(\\mean{B_{r}}u_{+}^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))\n\\end{aligned}",
      "R0.choice": "\\begin{equation}\\label{R0.choice}\n    [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}",
      "thm.hol": "\\begin{theorem}[H\\\"older regularity]\\label{thm.hol}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol}. Then there exists an exponent $\\gamma \\equiv \\gamma(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha}) \\in (0,1)$ such that $u \\in C^{0,\\gamma}_{\\loc}(\\Omega)$. Moreover, for any ball $B_{r} \\Subset \\Omega$ with $r\\le R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, we have\n\\begin{equation}\\label{est:hol}\n[u]_{C^{0,\\gamma}(B_{r/2})} \\le cr^{-\\gamma}\\left[\\left(\\mean{B_{r}}|u|^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u;r/2,r))\\right]\n\\end{equation}\nfor a constant $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}",
      "a.bound": "\\begin{equation}\\label{a.bound}\n\\nu \\le a(x,y) + b(x,y) \\le L\n\\end{equation}",
      "Holder": "\\begin{aligned}\n\\mean{B_{\\rho}}G((u-k)_{+})\\,dx\n& \\leq c \\left(\\mean{B_{\\sigma}} \\left[\\frac{(u-h)_{+}}{k-h}\\right]^p\\,dx\\right)^{sp/n}  \\mean{B_{\\sigma}}G((u-h)_{+})\\,dx \\\\\n& \\quad \\;\\cdot \\left[\\frac{r^{p}}{(\\sigma-\\rho)^{p}}+ \\frac{r^{n+tp}}{(\\sigma-\\rho)^{n+tp}}\\frac{1}{(k-h)^{p-1}G(1)}T((u-k)_{+};r/2,r)\\right] \\\\\n& \\quad + c \\left(\\mean{B_{\\sigma}} \\left[\\frac{(u-h)_{+}}{k-h}\\right]^p\\,dx\\right)^{p-1}  \\mean{B_{\\sigma}}G((u-h)_{+})\\,dx.\n\\end{aligned}\n\\end{equation*}\nDividing both sides of the above inequality by $r^{-sp}+b^{+}_{B_r}r^{-tp}$, we obtain the estimate \\eqref{cacciokh} with $r^{p+(s-t)p}$ and $r^{sp}$ replaced by $r^{p}$ and $1/G(1)$, respectively. Hence we can conclude with the desired estimate \\eqref{est:bdd2}, and the proof is complete. \n\\end{proof}\n\n\\section{Expansion of positivity}\\label{Holder}\n\nThroughout this section, we assume that the modulating coefficient $b(\\cdot,\\cdot)$ satisfies \\eqref{a.holder} with \\eqref{assumption.hol}. \nWe start this section with the following logarithmic type estimate, whose proof is analogous to those of \\cite[Corollary~3.2]{DKP16} and \\cite[Corollary~5.2]{BOS}.\n\n\\begin{lemma}\\label{log.lemma}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak supersolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol}, which is nonnegative in a ball $B_{R} \\equiv B_{R}(x_{0})$ with $R \\le R_{0}$. For any $d,\\zeta > 0$ and $\\xi>1$, define\n\\begin{equation*}\nv \\coloneqq \\min\\{(\\log(\\zeta+d)-\\log(u+d))_{+}, \\log\\xi\\}.\n\\end{equation*} \nThen for any $r \\in (0, R/2]$ and $d>0$, we have\n\\begin{equation}\\label{log.est}\n\\mean{B_{r}}|v-(v)_{B_{r}}|\\,dx\n\\leq c + cd^{1-p}\\frac{1}{G_{B_{2r}}(1)}T(u_{-};R,2r)\n\\end{equation}\nfor a constant $c \\equiv c(\\data)>0$.\n\\end{lemma}\n\\begin{proof}\nIn the case $b^{+}_{B_{R}} \\le \\nu/4$, since $\\nu/4 \\le a(x,y) \\le L$, the estimate is proved in \\cite[Lemma~5.1]{BOS} with $q=p$. Note that, in the setting of the present paper, \\cite[(5.10)]{BOS} can be replaced by\n\\begin{equation*}\n\\begin{aligned}\n& \\int_{B_{3r/2}}\\int_{\\mathbb{R}^{n}\\setminus B_{R}}\\frac{1}{g_{B_{2r}}(u(x)+d)}\\left(a(x,y)\\frac{(u_{-}(y))^{p-1}}{|x-y|^{n+sp}}+b(x,y)\\frac{(u_{-}(y))^{p-1}}{|x-y|^{n+tp}}\\right)\\,dy\\,dx \\\\\n& \\le \\frac{cr^{n}}{g_{B_{2r}}(d)}\\sup_{x\\in B_{2r}}\\int_{\\mathbb{R}^{n}\\setminus B_{R}}\\left(a(x,y)\\frac{(u_{-}(y))^{p-1}}{|y-x_{0}|^{n+sp}}+b(x,y)\\frac{(u_{-}(y))^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy \\\\\n& = \\frac{cr^{n}d^{1-p}}{G_{B_{2r}}(1)}T(u_{-};R,2r).\n\\end{aligned}",
      "assumption.hol": "\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}",
      "model": "\\begin{equation}\\label{model}\n\\begin{aligned}\n&\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\\left(\\frac{a(x,y)}{|x-y|^{n+sp}}+\\frac{b(x,y)}{|x-y|^{n+tp}}\\right)\\,dy = 0 \\quad \\text{in }\\Omega.\n\\end{aligned}\n\\end{equation}",
      "est:bdd2": "\\begin{equation}\\label{est:bdd2}\n\\begin{aligned}\n\\esssup_{B_{r/2}} u & \\le c \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ \\frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\\right]^{1/(p-1)} \\\\\n& = c\\left(\\mean{B_{r}}u_{+}^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))\n\\end{aligned}\n\\end{equation}",
      "thm.bdd": "\\begin{theorem}[Local boundedness]\\label{thm.bdd}\nLet  $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak subsolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{a.bound}, and let $B_{r}\\Subset \\Omega$ be a ball with $r\\le1$.\n\\begin{enumerate}\n\\item We have \n\\begin{equation}\\label{est:bdd1}\n\\esssup_{B_{r/2}} u \\le  c\\,r^{-(t-s)/\\beta} \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ r^{sp} T(u_+;r/2,r)\\right]^{1/(p-1)}\n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L)>0$, where \n\\begin{equation}\\label{def.beta}\n\\beta \\coloneqq \\min\\{sp/n,p-1\\}.\n\\end{equation} \n\\item Assume further that $b(\\cdot,\\cdot)$ satisfies \\eqref{a.holder} and \\eqref{assumption.hol}. Then we have\n\\begin{equation}\\label{est:bdd2}\n\\begin{aligned}\n\\esssup_{B_{r/2}} u & \\le c \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ \\frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\\right]^{1/(p-1)} \\\\\n& = c\\left(\\mean{B_{r}}u_{+}^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))\n\\end{aligned}\n\\end{equation}\nwhenever $r \\le R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice} and $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.  \n\\end{enumerate}\nConsequently, if $u \\in \\mathcal{A}(\\Omega)\\cap \\mathcal{T}(\\mathbb{R}^{n})$ is a weak solution to \\eqref{main.eq}, then $u \\in L^{\\infty}_{\\loc}(\\Omega)$ and estimate \\eqref{est:bdd1} or \\eqref{est:bdd2} holds with $|u|$ in each case.\n\\end{theorem}",
      "alt.dp": "\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}",
      "a.holder": "\\begin{equation}\\label{a.holder}\n|b(x_{1},y_{1}) - b(x_{2},y_{2})| \\leq [b]_{\\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\\alpha}\n\\end{equation}",
      "main.eq": "\\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}",
      "weak.harnack": "\\begin{lemma}\\label{weak.harnack}\nLet $u\\in\\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak supersolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot)\\equiv 1$, which is nonnegative in a ball $B_{10r} \\equiv B_{10r}(x_{0}) \\Subset \\Omega$ with $10r \\le R_{0}$. Then there exist constants $p_{0} \\in (0,1)$ and $c \\ge 1$, both depending only on $\\data$, such that \n\\begin{equation*}\n\\left(\\mean{B_{2r}}u^{p_{0}}\\,dx\\right)^{1/p_{0}} \\le c \\essinf_{B_{r}}u + cg_{B_{6r}}^{-1}(T(u_{-};10r,10r)). \n\\end{equation*}\n\\end{lemma}"
    },
    "pre_theorem_intro_text_len": 11413,
    "pre_theorem_intro_text": "This paper is concerned with a class of nonlinear nonlocal equations whose differentiability order shows a drastic change depending on the point. More precisely, with $\\Omega\\subset \\mathbb{R}^{n}$ being a bounded domain, we consider the following equation\n\\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}\nwhose leading operator $\\mathcal{L}(\\cdot)$ is given by\n\\begin{equation*}\n\\begin{aligned}\n\\mathcal{L}u(x) &\\coloneqq \\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\\left[a(x,y)K_{sp}(x,y)+b(x,y)K_{tp}(x,y)\\right]\\,dy\n\\end{aligned}\n\\end{equation*}\nfor $x\\in\\mathbb{R}^n$. Here, $K_{sp},K_{tp}: \\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ are measurable kernels of orders $(s,p)$ and $(t,p)$, respectively, for some $0<s < t <1< p <\\infty$, and $a,b:\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ are nonnegative functions. We refer to Section~\\ref{setting} below for the detailed assumptions. \n\nThere have been several researches on various kinds of anisotropic nonlocal equations, see for instance  \n \\cite{CK20,FKV15,KW24} for the linear case and \\cite{BOS,CK22} for the nonlinear case.   \n The equation \\eqref{main.eq} is modeled on the following example, which is the case where $K_{sp}(x,y) \\equiv |x-y|^{-n-sp}$ and $K_{tp}(x,y) \\equiv |x-y|^{-n-tp}$:\n\\begin{equation}\\label{model}\n\\begin{aligned}\n&\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\\left(\\frac{a(x,y)}{|x-y|^{n+sp}}+\\frac{b(x,y)}{|x-y|^{n+tp}}\\right)\\,dy = 0 \\quad \\text{in }\\Omega.\n\\end{aligned}\n\\end{equation}\nIt is straightforward to see that \\eqref{model} is the Euler-Lagrange equation of the functional\n\\begin{equation}\\label{functional}\nv \\mapsto \\iint_{\\mathcal{C}_{\\Omega}}\\left(a(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+tp}}\\right)\\,dx\\,dy,\n\\end{equation}\nwhere\n\\begin{equation}\\label{COmega}\n\\mathcal{C}_{\\Omega} \\coloneqq (\\mathbb{R}^{n}\\times\\mathbb{R}^{n})\\setminus((\\mathbb{R}^{n}\\setminus\\Omega)\\times(\\mathbb{R}^{n}\\setminus\\Omega)).\n\\end{equation}\nA main point in \\eqref{main.eq} is that the (possibly degenerate) coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$ make the associated integro-differential operator $\\mathcal{L}(\\cdot)$ switch between two different fractional elliptic phases. \nIn this respect, our problem is closely related to the following local equation \n\\[ \\mathrm{div}\\left(a(x)|Du|^{p-2}Du + b(x)|Du|^{q-2}Du \\right) = 0, \\]\nwhere $1 < p < q < \\infty$ and $a(\\cdot), b(\\cdot) \\ge 0$ satisfy $\\nu \\le a(x) + b(x) \\le L$ for some constants $0 < \\nu \\le L < \\infty$. This is called a double phase problem with two modulating coefficients. Since the pioneering works of Colombo and Mingione \\cite{CM15a, CM15b}, the regularity theory for local double phase problems with a single modulating coefficient (i.e., the case $a(\\cdot)\\equiv 1$ in the equation above) has been extensively developed; see, for instance, \\cite{BCM1, BCM2, CM16, DM20} and the references therein. More general classes of problems, including the double phase type, have subsequently been investigated in \\cite{HO22, KO24}. In particular, \\cite{KO24} studies various regularity results for equations of the form above, including the case in which $a(\\cdot)$ may be degenerate, meaning that $a(\\cdot)$ can vanish. A key idea in \\cite{KO24} is the following observation: when $a(x)$ is close to zero, the equation can be regarded as a $q$‑Laplace equation with a lower‑order perturbation of $p$‑Laplace type, whereas when $a(x)$ stays away from zero, the equation behaves genuinely as a double phase problem with a single modulating coefficient.\n\nWe now turn to regularity results for nonlinear nonlocal equations. The De Giorgi-Nash-Moser theory for the $(s,p)$-Laplacian was first investigated by Di Castro, Kuusi and Palatucci \\cite{DKP14,DKP16}; they proved local regularity and Harnack inequality for weak solutions to fractional $p$-Laplacian type equations involving measurable kernels. Cozzi \\cite{coz} extended such results to minimizers of non-differentiable functionals with lower-order dependencies, via a slightly different approach using fractional De Giorgi classes. We  also refer to \\cite{BDNS,Lind} and references therein for similar results concerned with more general equations.\n\nVery recently, the results and techniques in \\cite{coz,DKP14,DKP16} were further developed and extended to nonlocal problems with nonstandard growth and/or differentiability conditions. Similarly to the case of local problems, one can consider several typical examples, such as Orlicz growth \\cite{BKO,CKW2}, variable growth \\cite{CK23,Ok23}, and double phase \\cite{BOS,DP19,fang2021weak}. These papers are concerned with local boundedness and H\\\"older continuity of weak solutions. \nMoreover, in the case of Orlicz growth, Harnack inequalities were also proved in \\cite{BKS,CKW1}. \nWe also mention the paper \\cite{OS} concerned with regularity results and Harnack inequalities for nonlinear nonlocal equations with kernels of general differentiability order.\n\nNonlocal double phase problems with one modulating coefficient, whose prototype is\n\\begin{equation}\\label{model.one}\n\\begin{aligned}\n\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} \\left(\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\\right)\\,dy = 0\n\\end{aligned}\n\\end{equation}\nwith $s,t \\in (0,1)$ and $1<p\\le q < \\infty$, were first considered by De Filippis and Palatucci \\cite{DP19}. More precisely, they proved H\\\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \\cite{BKK,fang2021weak,SM} for various results for \\eqref{model.one}. The papers \\cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\\mathbb{R}^{n}$ and are under the assumption that $t\\le s$, which implies that the second term in \\eqref{model.one} plays a role as a lower order term. Under these settings, $b(\\cdot,\\cdot)$ is imposed to be bounded and discontinuous.\n\nOn the contrary, in the paper \\cite{BOS}, joint with Byun, the second and third authors of this paper proved the local boundedness and H\\\"older continuity of weak solutions to \\eqref{model.one} in the case $s\\le t$, under natural assumptions on the powers and the modulating coefficients. Specifically, it was proved that if\n\\[  b \\in C^{0,\\alpha}(\\mathbb{R}^{n}\\times\\mathbb{R}^{n}) \\;\\; \\text{for some}\\;\\; \\alpha\\in (0,1] \\quad \\text{and} \\quad tq \\le sp + \\alpha, \\]\nthen every locally bounded weak solution to \\eqref{model.one} is locally H\\\"older continuous. \nWe emphasize that when $s \\le t$, the second term in \\eqref{model} or \\eqref{model.one} has a higher order and, as in the local case, the interplay between the H\\\"older regularity of $b(\\cdot,\\cdot)$ and the growth/differentiability condition of the problem plays a key role in the analysis. \nMoreover, in light of the Lavrentiev type phenomena considered in \\cite{Balci}, \nthe above assumption is essentially sharp.\nWe also refer to the paper \\cite{BLS} concerned with regularity results and Harnack inequality for mixed local and nonlocal double phase problems.\n\nIn this paper, we prove the local boundedness and H\\\"older continuity of weak solutions to the general equation \\eqref{main.eq}. Moreover, in the case of one modulating coefficient (i.e. when $a(\\cdot,\\cdot)\\equiv 1$), we further prove a Harnack inequality for weak solutions to \\eqref{main.eq}. \nTo our knowledge, each of our regularity results is the first one for \\eqref{main.eq}; \nmoreover, our Harnack inequality given in Theorem~\\ref{thm.harnack} below is the first one for nonlocal nonautonomous problems.\n\n\\subsection{Assumptions and main results}\\label{setting}\n\nWe say that a function $F:\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ is symmetric if $F(x,y)=F(y,x)$ for every $x,y \\in \\mathbb{R}^{n}$.\n\nThe kernels $K_{sp},K_{tp}:\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$ are measurable, symmetric functions that satisfy \n\\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}\nfor a.e. $x,y \\in\\mathbb{R}^{n}$ with $x\\neq y$, where $0 < \\nu \\le 1 \\le L < \\infty$ and\n\\begin{equation}\\label{power}\n 0 < s < t < 1 < p < \\infty.\n\\end{equation} \nThe modulating coefficients $a,b:\\mathbb{R}^{n}\\times\\mathbb{R}^{n} \\rightarrow \\mathbb{R}$ are nonnegative, measurable and symmetric functions that satisfy the bound\n\\begin{equation}\\label{a.bound}\n\\nu \\le a(x,y) + b(x,y) \\le L\n\\end{equation} \nfor a.e. $x,y \\in \\mathbb{R}^n$.  \n\nIn addition, for H\\\"older regularity and Harnack inequality, \nwe further assume the H\\\"older continuity of $b(\\cdot,\\cdot)$ as follows: There exist two numbers $[b]_{\\alpha}\\geq 0$ and $\\alpha \\in (0,1]$ such that\n\\begin{equation}\\label{a.holder}\n|b(x_{1},y_{1}) - b(x_{2},y_{2})| \\leq [b]_{\\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\\alpha}\n\\end{equation}\nfor every $(x_{1},y_{1}),(x_{2},y_{2})\\in\\mathbb{R}^n\\times\\mathbb{R}^n$, and \n\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}\n\n\\begin{remark}\nFor any ball $B_{r}\\subset \\Omega$, we set\n\\begin{equation*}\nb^{+}_{B_{r}} \\coloneqq \\sup_{x,y \\in B_{r}}b(x,y), \\qquad b^{-}_{B_{r}} \\coloneqq \\inf_{x,y \\in B_{r}}b(x,y).\n\\end{equation*}\nMoreover, when \\eqref{a.holder} is in force, we choose a small radius $R_{0} \\equiv R_{0}(\\nu,\\alpha,[b]_{\\alpha}) \\in (0,1)$ satisfying\n\\begin{equation}\\label{R0.choice}\n    [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}\nWe observe that, when $r \\le R_{0}$, \n\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}\nAccordingly, if we set\n\\begin{equation}\\label{G.def}\nG_{B_{r}}(\\tau) \\coloneqq \\sup_{x,y\\in B_{r}}\\left(a(x,y)\\frac{\\tau^{p}}{r^{sp}} + b(x,y)\\frac{\\tau^{p}}{r^{tp}}\\right), \\qquad g_{B_{r}}(\\tau) \\coloneqq \\frac{G_{B_{r}}(\\tau)}{\\tau}, \\qquad \\tau \\ge 0.\n\\end{equation}\nthen we have\n\\begin{align}\\label{alt.dG}\nb^{+}_{B_r} \\le  \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{sp}} + b^+_{B_r}\\frac{\\tau^p}{r^{tp}}\\approx \\frac{\\tau^p}{r^{sp}} + b^-_{B_r}\\frac{\\tau^p}{r^{tp}},\n\\end{align}\n\\begin{align}\\label{alt.G}\nb^{+}_{B_r} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{tp}}.\n\\end{align}\n\\end{remark}\n\nWe next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations.  \nFor $f:\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$, $x_{0} \\in \\mathbb{R}^{n}$ and $r,\\rho>0$, we denote\n\\begin{equation}\\label{def.T}\nT(f;x_{0},r,\\rho) \\coloneqq \\sup_{x \\in B_{\\rho}(x_{0})}\\int_{\\mathbb{R}^{n}\\setminus B_{r}(x_{0})}\\left(a(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy.\n\\end{equation}\nWe will omit the point $x_{0}$ if it is clear from context. \n\nWith the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.",
    "context": "There have been several researches on various kinds of anisotropic nonlocal equations, see for instance \n \\cite{CK20,FKV15,KW24} for the linear case and \\cite{BOS,CK22} for the nonlinear case. \n The equation \\eqref{main.eq} is modeled on the following example, which is the case where $K_{sp}(x,y) \\equiv |x-y|^{-n-sp}$ and $K_{tp}(x,y) \\equiv |x-y|^{-n-tp}$:\n\\begin{equation}\\label{model}\n\\begin{aligned}\n&\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\\left(\\frac{a(x,y)}{|x-y|^{n+sp}}+\\frac{b(x,y)}{|x-y|^{n+tp}}\\right)\\,dy = 0 \\quad \\text{in }\\Omega.\n\\end{aligned}\n\\end{equation}\nIt is straightforward to see that \\eqref{model} is the Euler-Lagrange equation of the functional\n\\begin{equation}\\label{functional}\nv \\mapsto \\iint_{\\mathcal{C}_{\\Omega}}\\left(a(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+tp}}\\right)\\,dx\\,dy,\n\\end{equation}\nwhere\n\\begin{equation}\\label{COmega}\n\\mathcal{C}_{\\Omega} \\coloneqq (\\mathbb{R}^{n}\\times\\mathbb{R}^{n})\\setminus((\\mathbb{R}^{n}\\setminus\\Omega)\\times(\\mathbb{R}^{n}\\setminus\\Omega)).\n\\end{equation}\nA main point in \\eqref{main.eq} is that the (possibly degenerate) coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$ make the associated integro-differential operator $\\mathcal{L}(\\cdot)$ switch between two different fractional elliptic phases. \nIn this respect, our problem is closely related to the following local equation \n\\[ \\mathrm{div}\\left(a(x)|Du|^{p-2}Du + b(x)|Du|^{q-2}Du \\right) = 0, \\]\nwhere $1 < p < q < \\infty$ and $a(\\cdot), b(\\cdot) \\ge 0$ satisfy $\\nu \\le a(x) + b(x) \\le L$ for some constants $0 < \\nu \\le L < \\infty$. This is called a double phase problem with two modulating coefficients. Since the pioneering works of Colombo and Mingione \\cite{CM15a, CM15b}, the regularity theory for local double phase problems with a single modulating coefficient (i.e., the case $a(\\cdot)\\equiv 1$ in the equation above) has been extensively developed; see, for instance, \\cite{BCM1, BCM2, CM16, DM20} and the references therein. More general classes of problems, including the double phase type, have subsequently been investigated in \\cite{HO22, KO24}. In particular, \\cite{KO24} studies various regularity results for equations of the form above, including the case in which $a(\\cdot)$ may be degenerate, meaning that $a(\\cdot)$ can vanish. A key idea in \\cite{KO24} is the following observation: when $a(x)$ is close to zero, the equation can be regarded as a $q$‑Laplace equation with a lower‑order perturbation of $p$‑Laplace type, whereas when $a(x)$ stays away from zero, the equation behaves genuinely as a double phase problem with a single modulating coefficient.\n\nNonlocal double phase problems with one modulating coefficient, whose prototype is\n\\begin{equation}\\label{model.one}\n\\begin{aligned}\n\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} \\left(\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\\right)\\,dy = 0\n\\end{aligned}\n\\end{equation}\nwith $s,t \\in (0,1)$ and $1<p\\le q < \\infty$, were first considered by De Filippis and Palatucci \\cite{DP19}. More precisely, they proved H\\\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \\cite{BKK,fang2021weak,SM} for various results for \\eqref{model.one}. The papers \\cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\\mathbb{R}^{n}$ and are under the assumption that $t\\le s$, which implies that the second term in \\eqref{model.one} plays a role as a lower order term. Under these settings, $b(\\cdot,\\cdot)$ is imposed to be bounded and discontinuous.\n\n\\begin{remark}\nFor any ball $B_{r}\\subset \\Omega$, we set\n\\begin{equation*}\nb^{+}_{B_{r}} \\coloneqq \\sup_{x,y \\in B_{r}}b(x,y), \\qquad b^{-}_{B_{r}} \\coloneqq \\inf_{x,y \\in B_{r}}b(x,y).\n\\end{equation*}\nMoreover, when \\eqref{a.holder} is in force, we choose a small radius $R_{0} \\equiv R_{0}(\\nu,\\alpha,[b]_{\\alpha}) \\in (0,1)$ satisfying\n\\begin{equation}\\label{R0.choice}\n [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}\nWe observe that, when $r \\le R_{0}$, \n\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}\nAccordingly, if we set\n\\begin{equation}\\label{G.def}\nG_{B_{r}}(\\tau) \\coloneqq \\sup_{x,y\\in B_{r}}\\left(a(x,y)\\frac{\\tau^{p}}{r^{sp}} + b(x,y)\\frac{\\tau^{p}}{r^{tp}}\\right), \\qquad g_{B_{r}}(\\tau) \\coloneqq \\frac{G_{B_{r}}(\\tau)}{\\tau}, \\qquad \\tau \\ge 0.\n\\end{equation}\nthen we have\n\\begin{align}\\label{alt.dG}\nb^{+}_{B_r} \\le \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{sp}} + b^+_{B_r}\\frac{\\tau^p}{r^{tp}}\\approx \\frac{\\tau^p}{r^{sp}} + b^-_{B_r}\\frac{\\tau^p}{r^{tp}},\n\\end{align}\n\\begin{align}\\label{alt.G}\nb^{+}_{B_r} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{tp}}.\n\\end{align}\n\\end{remark}\n\nWe next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations. \nFor $f:\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$, $x_{0} \\in \\mathbb{R}^{n}$ and $r,\\rho>0$, we denote\n\\begin{equation}\\label{def.T}\nT(f;x_{0},r,\\rho) \\coloneqq \\sup_{x \\in B_{\\rho}(x_{0})}\\int_{\\mathbb{R}^{n}\\setminus B_{r}(x_{0})}\\left(a(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy.\n\\end{equation}\nWe will omit the point $x_{0}$ if it is clear from context.\n\nWith the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.",
    "full_context": "There have been several researches on various kinds of anisotropic nonlocal equations, see for instance \n \\cite{CK20,FKV15,KW24} for the linear case and \\cite{BOS,CK22} for the nonlinear case. \n The equation \\eqref{main.eq} is modeled on the following example, which is the case where $K_{sp}(x,y) \\equiv |x-y|^{-n-sp}$ and $K_{tp}(x,y) \\equiv |x-y|^{-n-tp}$:\n\\begin{equation}\\label{model}\n\\begin{aligned}\n&\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\\left(\\frac{a(x,y)}{|x-y|^{n+sp}}+\\frac{b(x,y)}{|x-y|^{n+tp}}\\right)\\,dy = 0 \\quad \\text{in }\\Omega.\n\\end{aligned}\n\\end{equation}\nIt is straightforward to see that \\eqref{model} is the Euler-Lagrange equation of the functional\n\\begin{equation}\\label{functional}\nv \\mapsto \\iint_{\\mathcal{C}_{\\Omega}}\\left(a(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+tp}}\\right)\\,dx\\,dy,\n\\end{equation}\nwhere\n\\begin{equation}\\label{COmega}\n\\mathcal{C}_{\\Omega} \\coloneqq (\\mathbb{R}^{n}\\times\\mathbb{R}^{n})\\setminus((\\mathbb{R}^{n}\\setminus\\Omega)\\times(\\mathbb{R}^{n}\\setminus\\Omega)).\n\\end{equation}\nA main point in \\eqref{main.eq} is that the (possibly degenerate) coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$ make the associated integro-differential operator $\\mathcal{L}(\\cdot)$ switch between two different fractional elliptic phases. \nIn this respect, our problem is closely related to the following local equation \n\\[ \\mathrm{div}\\left(a(x)|Du|^{p-2}Du + b(x)|Du|^{q-2}Du \\right) = 0, \\]\nwhere $1 < p < q < \\infty$ and $a(\\cdot), b(\\cdot) \\ge 0$ satisfy $\\nu \\le a(x) + b(x) \\le L$ for some constants $0 < \\nu \\le L < \\infty$. This is called a double phase problem with two modulating coefficients. Since the pioneering works of Colombo and Mingione \\cite{CM15a, CM15b}, the regularity theory for local double phase problems with a single modulating coefficient (i.e., the case $a(\\cdot)\\equiv 1$ in the equation above) has been extensively developed; see, for instance, \\cite{BCM1, BCM2, CM16, DM20} and the references therein. More general classes of problems, including the double phase type, have subsequently been investigated in \\cite{HO22, KO24}. In particular, \\cite{KO24} studies various regularity results for equations of the form above, including the case in which $a(\\cdot)$ may be degenerate, meaning that $a(\\cdot)$ can vanish. A key idea in \\cite{KO24} is the following observation: when $a(x)$ is close to zero, the equation can be regarded as a $q$‑Laplace equation with a lower‑order perturbation of $p$‑Laplace type, whereas when $a(x)$ stays away from zero, the equation behaves genuinely as a double phase problem with a single modulating coefficient.\n\nNonlocal double phase problems with one modulating coefficient, whose prototype is\n\\begin{equation}\\label{model.one}\n\\begin{aligned}\n\\mathrm{P.V.}\\int_{\\mathbb{R}^{n}} \\left(\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\\right)\\,dy = 0\n\\end{aligned}\n\\end{equation}\nwith $s,t \\in (0,1)$ and $1<p\\le q < \\infty$, were first considered by De Filippis and Palatucci \\cite{DP19}. More precisely, they proved H\\\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \\cite{BKK,fang2021weak,SM} for various results for \\eqref{model.one}. The papers \\cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\\mathbb{R}^{n}$ and are under the assumption that $t\\le s$, which implies that the second term in \\eqref{model.one} plays a role as a lower order term. Under these settings, $b(\\cdot,\\cdot)$ is imposed to be bounded and discontinuous.\n\n\\begin{remark}\nFor any ball $B_{r}\\subset \\Omega$, we set\n\\begin{equation*}\nb^{+}_{B_{r}} \\coloneqq \\sup_{x,y \\in B_{r}}b(x,y), \\qquad b^{-}_{B_{r}} \\coloneqq \\inf_{x,y \\in B_{r}}b(x,y).\n\\end{equation*}\nMoreover, when \\eqref{a.holder} is in force, we choose a small radius $R_{0} \\equiv R_{0}(\\nu,\\alpha,[b]_{\\alpha}) \\in (0,1)$ satisfying\n\\begin{equation}\\label{R0.choice}\n [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}\nWe observe that, when $r \\le R_{0}$, \n\\begin{align}\nb^{+}_{B_{r}} \\leq \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \\frac{3\\nu}{4} \\leq a(\\cdot,\\cdot)\\leq L \\ \\text{ in } \\ B_r\\times B_r, \\label{alt.dp} \\\\\nb^{+}_{B_{r}} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; \nb^{+}_{B_{r}} \\overset{\\eqref{a.holder}}{\\le} b^{-}_{B_{r}} + [b]_{\\alpha}(2r)^{\\alpha} \\overset{\\eqref{R0.choice}}{\\le} 2b^{-}_{B_{r}}. \\label{alt.p}\n\\end{align}\nAccordingly, if we set\n\\begin{equation}\\label{G.def}\nG_{B_{r}}(\\tau) \\coloneqq \\sup_{x,y\\in B_{r}}\\left(a(x,y)\\frac{\\tau^{p}}{r^{sp}} + b(x,y)\\frac{\\tau^{p}}{r^{tp}}\\right), \\qquad g_{B_{r}}(\\tau) \\coloneqq \\frac{G_{B_{r}}(\\tau)}{\\tau}, \\qquad \\tau \\ge 0.\n\\end{equation}\nthen we have\n\\begin{align}\\label{alt.dG}\nb^{+}_{B_r} \\le \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{sp}} + b^+_{B_r}\\frac{\\tau^p}{r^{tp}}\\approx \\frac{\\tau^p}{r^{sp}} + b^-_{B_r}\\frac{\\tau^p}{r^{tp}},\n\\end{align}\n\\begin{align}\\label{alt.G}\nb^{+}_{B_r} > \\frac{\\nu}{4} & \\;\\; \\Longrightarrow \\;\\; G_{B_r}(\\tau) \\approx \\frac{\\tau^p}{r^{tp}}.\n\\end{align}\n\\end{remark}\n\nWe next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations. \nFor $f:\\mathbb{R}^{n}\\rightarrow \\mathbb{R}$, $x_{0} \\in \\mathbb{R}^{n}$ and $r,\\rho>0$, we denote\n\\begin{equation}\\label{def.T}\nT(f;x_{0},r,\\rho) \\coloneqq \\sup_{x \\in B_{\\rho}(x_{0})}\\int_{\\mathbb{R}^{n}\\setminus B_{r}(x_{0})}\\left(a(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy.\n\\end{equation}\nWe will omit the point $x_{0}$ if it is clear from context.\n\nWith the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.\n\n\\begin{theorem}[Harnack inequality]\\label{thm.harnack}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot)\\equiv 1$. If $u$ is nonnegative in a ball $B_{10r} \\Subset \\Omega$ with $10r \\leq R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, then we have\n\\begin{equation}\\label{Harnack}\n\\sup_{B_{r}}u \\le c\\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r)) \n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}\n\\begin{remark}\nIn Theorem~\\ref{thm.harnack}, the condition $a(\\cdot,\\cdot)\\equiv 1$ can be weakened as $\\inf_{\\mathbb{R}^n \\times \\mathbb{R}^n}a(\\cdot,\\cdot)>0$. Such a restriction is concerned with the tail estimate given in Lemma~\\ref{tail.pm} below. \nIndeed, even if $a(\\cdot,\\cdot)\\not\\equiv1$, we can prove an estimate similar to \\eqref{Harnack} which involves nonlocal tails of $u$ instead of $u_{-}$. \nHowever, in view of the natural Harnack inequalities obtained in \\cite{coz,DKP14,Kas11}, we confine ourselves to the case $a(\\cdot,\\cdot)\\equiv1$ in Theorem~\\ref{thm.harnack}. \n\\end{remark}\n\n\\begin{lemma}\\label{caccioppoli2}\nLet $u\\in\\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{a.bound}. \nThere exists a constant $c\\equiv c(n,s,t,p,\\nu,L)>0$ such that\n\\begin{equation}\\label{caccio.2}\n\\begin{aligned}\n& \\int_{B_{\\rho}}\\int_{B_{\\rho}}\\left(a(x,y)\\frac{|w_{\\pm}(x)-w_{\\pm}(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{|w_{\\pm}(x)-w_{\\pm}(y)|^{p}}{|x-y|^{n+tp}}\\right)\\,dx\\,dy \\\\\n& \\le \\frac{c}{(r-\\rho)^{p}}\\int_{B_{r}}\\int_{B_{r}}\\left(a(x,y)\\frac{w_{\\pm}^{p}(x)}{|x-y|^{n+(s-1)p}}+ b(x,y)\\frac{w_{\\pm}^{p}(x)}{|x-y|^{n+(t-1)p}}\\right)\\,dx\\,dy \\\\\n& \\quad\\; + c\\left(\\frac{r}{r-\\rho}\\right)^{n+tp}T(w_{\\pm};r,r)\\int_{B_{r}}w_{\\pm}\\,dx\n\\end{aligned}\n\\end{equation}\nholds whenever $B_{\\rho} \\subset B_{r} \\Subset \\Omega$ are concentric balls, where $w_{\\pm} \\coloneqq (u-k)_{\\pm}$ for any $k\\in\\mathbb{R}$.\nMoreover, estimate \\eqref{caccio.2} continues to hold for $w_{+}$ (resp. $w_{-}$) when $u$ is merely a weak subsolution (resp. supersolution) to \\eqref{main.eq}.\n\\end{lemma}\n\n\\begin{lemma}\\label{density.improve}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak supersolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol}, which is nonnegative in a ball $B_{R} \\equiv B_{R}(x_{0}) \\Subset \\Omega$ with $R \\le R_{0}$. Let $k > 0$ and assume that there exists $\\sigma \\in (0,1]$ satisfying\n\\begin{equation}\\label{density.assumption}\n|B_{2r} \\cap \\{ u \\ge k \\}| \\ge \\sigma|B_{2r}| \\qquad \\text{for some} \\quad r \\in (0,R/4].\n\\end{equation}\nThen there exists a constant $\\delta \\equiv \\delta(\\data,\\sigma) \\in (0,1/4)$ such that if\n\\begin{equation}\\label{T.assumption}\nd \\coloneqq \\left[\\frac{1}{G_{B_{2r}}(1)}T(u_{-};R,2r) \\right]^{1/(p-1)} = g_{B_{2r}}^{-1}(T(u_{-};R,2r)) \\le \\delta k,\n\\end{equation}\nthen\n\\begin{equation}\\label{lower.bound}\n\\essinf_{B_{r}}u \\ge \\delta k.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe divide the proof into two steps.\n\nUsing Lemmas~\\ref{density.improve} and \\ref{KrySaf}, we obtain a weak Harnack type estimate.\n\\begin{lemma}\\label{weak.harnack}\nLet $u\\in\\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak supersolution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot)\\equiv 1$, which is nonnegative in a ball $B_{10r} \\equiv B_{10r}(x_{0}) \\Subset \\Omega$ with $10r \\le R_{0}$. Then there exist constants $p_{0} \\in (0,1)$ and $c \\ge 1$, both depending only on $\\data$, such that \n\\begin{equation*}\n\\left(\\mean{B_{2r}}u^{p_{0}}\\,dx\\right)^{1/p_{0}} \\le c \\essinf_{B_{r}}u + cg_{B_{6r}}^{-1}(T(u_{-};10r,10r)). \n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe set $\\bar{\\delta} \\coloneqq 2^{-n-1}$ and $T_{0}\\coloneqq c_{2}g_{B_{6r}}^{-1}(T(u_{-};10r,10r))$, with $c_{2}\\equiv c_{2}(\\data)$ being a constant whose precise value will be determined in \\eqref{dT} below. \nFor each $i \\in \\mathbb{N}\\cup\\{0\\}$ and $\\tau>0$, we define\n\\begin{equation*}\nA^{i}_{\\tau} \\coloneqq \\left\\{ x\\in B_{2r}:u(x)>\\tau\\delta^{i}-\\frac{T_{0}}{1-\\delta}\\right\\},\n\\end{equation*}\nwhere $\\delta \\equiv \\delta(\\data) \\in (0,1/4)$ is the constant determined in Lemma~\\ref{density.improve} in the case $\\sigma=\\bar{\\delta}/6^{n}$. \nIt is obvious that $A^{i-1}_{\\tau} \\subset A^{i}_{\\tau}$.\n\nThe following lemma allows us to capture the precise tail contribution in the Harnack inequality.\n\\begin{lemma}\\label{tail.pm}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot) \\equiv 1$, which is nonnegative in a ball $B_{R}\\equiv B_{R}(x_{0}) \\Subset \\Omega$ with $R\\le R_{0}$. Then \n\\[ T(u_{+};r,2r) \\le cg_{B_{r}}\\left(\\sup_{B_{r}}u\\right) + cT(u_{-};R,r) \\]\nholds for any $r\\in(0,R/2]$, where $c\\equiv c(\\data)>0$.\n\\end{lemma}\n\\begin{proof}\nWe set $w \\coloneqq u-2k$ for $k \\coloneqq \\sup_{B_{r}}u$, and take a cut-off function $\\phi \\in C^{\\infty}_{0}(B_{r})$ such that $\\supp\\,\\phi \\subseteq B_{3r/4}$, $0 \\le \\phi \\le 1$, $\\phi \\equiv 1$ in $B_{r/2}$ and $|D\\phi| \\le 8/r$. Recalling the notation \\eqref{phip}, we test \\eqref{main.eq} with $w\\phi^{p}$ in order to have\n\\begin{equation}\\label{test}\n\\begin{aligned}\n0 & = \\int_{B_{r}}\\int_{B_{r}}\\Phi_{p}(u(x)-u(y))(w(x)\\phi^{p}(x)-w(y)\\phi^{p}(y))\\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\\right] \\,dx\\,dy \\\\\n& \\quad\\; + 2\\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\int_{B_{r}}\\Phi_{p}(u(x)-u(y))w(x)\\phi^{p}(x)\\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\\right]\\,dx\\,dy \\\\\n& \\eqqcolon I_{1} + I_{2}.\n\\end{aligned}\n\\end{equation}\nWe first estimate $I_{1}$. By the calculations done in the proof of \\cite[Lemma 4.2]{BOS}, we have\n\\begin{equation*}\n\\begin{aligned}\n& \\Phi_{p}(w(x)-w(y))(w(x)\\phi^{p}(x)-w(y)\\phi^{p}(y)) \\\\\n& \\ge \\frac{1}{4}|w(x)-w(y)|^{p}(\\phi^{p}(x)+\\phi^{p}(y)) - c|\\phi(x)-\\phi(y)|^{p}(w(x)+w(y))^{p} \\\\\n& \\ge -ck^{p}\\frac{|x-y|^{p}}{r^{p}}\n\\end{aligned}\n\\end{equation*}\nand therefore\n\\begin{equation}\\label{I1}\n\\begin{aligned}\nI_{1} & \\ge -c\\int_{B_{r}}\\int_{B_{r}}\\left(\\frac{k^{p}}{r^{p}}\\frac{1}{|x-y|^{n+(s-1)p}} + b(x,y)\\frac{k^{p}}{r^{p}}\\frac{1}{|x-y|^{n+(t-1)p}}\\right)\\,dx\\,dy \\\\\n& \\ge -c\\left(k^{p}r^{n-sp} + b^{+}_{B_{r}}k^{p}r^{n-tp}\\right) \\\\\n& \\ge -cr^{n}G_{B_{r}}(k).\n\\end{aligned}\n\\end{equation}\nWe next split $I_{2}$ as\n\\begin{equation}\\label{I2.split}\n\\begin{aligned}\nI_{2} & \\ge 2 \\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\int_{B_{r}}k(u(y)-k)_{+}^{p-1}\\phi^{p}(x)\\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\\right]\\,dx\\,dy \\\\\n& \\quad\\; -2\\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\int_{B_{r}}2k\\chi_{\\{u(y)<k\\}}(u(x)-u(y))_{+}^{p-1}\\phi^{p}(x)\\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\\right]\\,dx\\,dy \\\\\n& \\eqqcolon I_{2,1} - I_{2,2}.\n\\end{aligned}\n\\end{equation}\nWe note that \n\\[ \\frac{1}{4}|y-x_{0}| \\le |y-x_{0}| - |x-x_{0}| \\le |x-y| \\le |x-x_{0}| + |y-x_{0}| \\le \\frac{7}{4}|y-x_{0}|\\] \nfor any $x\\in B_{3r/4}$ and $y\\in\\mathbb{R}^{n}\\setminus B_{r}$,\nin order to estimate $I_{2,1}$ as\n\\begin{equation}\\label{I21}\n\\begin{aligned}\nI_{2,1} & \\ge \\frac{k}{c}\\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\int_{B_{r}}\\left(\\frac{(u_{+}(y))^{p-1}}{|x-y|^{n+sp}}+b(x,y)\\frac{(u_{+}(y))^{p-1}}{|x-y|^{n+tp}}\\right)\\phi^{p}(x)\\,dx\\,dy \\\\\n& \\quad\\; - c\\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\int_{B_{r}}\\left(\\frac{k^{p}}{|x-y|^{n+sp}} + b(x,y)\\frac{k^{p}}{|x-y|^{n+tp}}\\right)\\phi^{p}(x)\\,dx\\,dy \\\\\n& \\ge \\frac{kr^{n}}{c}\\int_{\\mathbb{R}^{n}\\setminus B_{r}}\\left(\\frac{(u_{+}(y))^{p-1}}{|y-x_{0}|^{n+sp}} + \\inf_{B_{r/2}}b(\\cdot,y)\\frac{(u_{+}(y))^{p-1}}{|y-x_{0}|^{n+tp}}\\right)\\,dy - cr^{n}G_{B_{r}}(k).\n\\end{aligned}\n\\end{equation}\nOn the other hand, we observe that\n\\begin{itemize}\n\\item[(i)] If $x\\in B_{r}$ and $y \\in B_{R}$, then $(u(x)-u(y))_{+}^{p-1} \\le (u(x))^{p-1}$;",
    "post_theorem_intro_text_len": 5579,
    "post_theorem_intro_text": "\\begin{theorem}[H\\\"older regularity]\\label{thm.hol}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol}. Then there exists an exponent $\\gamma \\equiv \\gamma(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha}) \\in (0,1)$ such that $u \\in C^{0,\\gamma}_{\\loc}(\\Omega)$. Moreover, for any ball $B_{r} \\Subset \\Omega$ with $r\\le R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, we have\n\\begin{equation}\\label{est:hol}\n[u]_{C^{0,\\gamma}(B_{r/2})} \\le cr^{-\\gamma}\\left[\\left(\\mean{B_{r}}|u|^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u;r/2,r))\\right]\n\\end{equation}\nfor a constant $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}\n\n\\begin{theorem}[Harnack inequality]\\label{thm.harnack}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\eqref{main.eq} under assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} with $a(\\cdot,\\cdot)\\equiv 1$. If $u$ is nonnegative in a ball $B_{10r} \\Subset \\Omega$ with $10r \\leq R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, then we have\n\\begin{equation}\\label{Harnack}\n\\sup_{B_{r}}u \\le c\\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r)) \n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}\n\\begin{remark}\nIn Theorem~\\ref{thm.harnack}, the condition $a(\\cdot,\\cdot)\\equiv 1$ can be weakened as $\\inf_{\\mathbb{R}^n \\times \\mathbb{R}^n}a(\\cdot,\\cdot)>0$. Such a restriction is concerned with the tail estimate given in Lemma~\\ref{tail.pm} below. \nIndeed, even if $a(\\cdot,\\cdot)\\not\\equiv1$, we can prove an estimate similar to \\eqref{Harnack} which involves nonlocal tails of $u$ instead of $u_{-}$. \nHowever, in view of the natural Harnack inequalities obtained in \\cite{coz,DKP14,Kas11}, we confine ourselves to the case $a(\\cdot,\\cdot)\\equiv1$ in Theorem~\\ref{thm.harnack}. \n\\end{remark}\n\nOur proof of regularity results in Theorems~\\ref{thm.bdd} and \\ref{thm.hol} is based on the Moser type approach in \\cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \\cite{BOS}. \nMoreover, for the Harnack inequality in Theorem~\\ref{thm.harnack}, we additionally modify and develop the techniques used in \\cite{DKP14} adapted to our problem. \nWe stress that, in obtaining both regularity results and Harnack inequalities for \\eqref{main.eq}, these processes exhibit several substantial difficulties which did not appear in \\cite{BOS}. \nFirst, while we only had to consider the case $sp \\le n$ in \\cite{BOS}, here we also treat the case $sp > n$ as well; note that in the latter case we automatically obtain the local boundedness of weak solutions via the fractional Sobolev-Morrey embedding. However, in both cases, we have to obtain explicit and precise local sup-estimates which extend the one in \\cite[Theorem~1.1]{DKP16}. \nSecond, since we are dealing with weak solutions which are not necessarily bounded in $\\mathbb{R}^{n}$, we need a delicate analysis to handle the nonlocal tails in proceeding towards the proof of our Harnack inequality. \nMore precisely, with the same spirit as in the case of fractional $p$-Laplacian \\cite{coz,DKP14}, we aim to prove a Harnack inequality involving the tail of $u_{-} \\coloneqq \\max\\{-u,0\\}$ only, which reduces to the classical Harnack inequality when dealing with solutions nonnegative in the whole $\\mathbb{R}^{n}$. This is indeed the most problematic issue in the case of nonlocal nonautonomous problems. In order to address this issue, we establish a weak Harnack type estimate (Lemma~\\ref{weak.harnack}) and a tail estimate (Lemma~\\ref{tail.pm}), and then combine them with the local sup-estimate \\eqref{est:bdd2}. All of these estimates encode the long-range interactions of solutions in an optimal way.\n\nIn our proofs, we utilize assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} to deal with the two modulating coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$ in \\eqref{main.eq}. Specifically, with $B_r \\subset \\Omega$ being a fixed ball, we divide cases as follows: \n\\begin{itemize}\n\\item[(i)] If $b^{+}_{B_r}\\leq \\nu/4$, then \\eqref{alt.dp} implies that $a(x,y)$ satisfies uniform ellipticity in the sense that the first term $a(x,y)K_{sp}(x,y) \\approx K_{sp}(x,y)$  in $B_r \\times B_r$, while the second term $b(x,y)K_{tp}(x,y)$ has a higher order with $b(x,y)$ being possibly degenerate in $B_r \\times B_r$. As a consequence, the equation \\eqref{main.eq} features a nonlocal double phase structure analogous to \\eqref{model.one} with $p=q$.\n\\item[(ii)] If $b^{+}_{B_r}>\\nu/4$, then \\eqref{alt.p} implies that $b(x,y)$ satisfies uniform ellipticity in $B_r \\times B_r$ with ellipticity ratio $2b^-_{B_{r}}/b^-_{B_{r}}=2$. In other words, the term $b(x,y)K_{tp}(x,y) \\approx K_{tp}(x,y)$ becomes a dominating term and $a(x,y)K_{sp}(x,y)$ is regarded as a lower order term in $B_{r} \\times B_r$. In this case, the equation \\eqref{main.eq} becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.\n\\end{itemize}\n\nThe remaining part of this paper is organized as follows. In the next section, we introduce basic notation and function spaces which will be used throughout this paper. In Section~\\ref{Caccio.bounded}, we employ a Caccioppoli type estimate to prove Theorem~\\ref{thm.bdd}. After obtaining a logarithmic lemma and an expansion of positivity lemma in Section~\\ref{Holder}, we finally prove Theorems~\\ref{thm.hol} and \\ref{thm.harnack} in Section~\\ref{sec.mainthm}.",
    "sketch": "Our proof of regularity results in Theorems~\\ref{thm.bdd} and \\ref{thm.hol} is based on the Moser type approach in \\cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \\cite{BOS}. The authors note additional difficulties compared to \\cite{BOS}: they treat both cases $sp\\le n$ and $sp>n$ (in the latter, local boundedness follows from the fractional Sobolev--Morrey embedding), but “in both cases” they “have to obtain explicit and precise local sup-estimates which extend the one in \\cite[Theorem~1.1]{DKP16}.”\n\nThey also explain how the assumptions \\eqref{kernel.growth}--\\eqref{assumption.hol} are used “to deal with the two modulating coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$”: for a fixed ball $B_r\\subset\\Omega$ they split into two regimes:\n\\begin{itemize}\n\\item[(i)] If $b^{+}_{B_r}\\le \\nu/4$, then $a(x,y)$ is uniformly elliptic in $B_r\\times B_r$ so that $a(x,y)K_{sp}(x,y)\\approx K_{sp}(x,y)$, while $b(x,y)K_{tp}(x,y)$ is “higher order” and may be degenerate; thus \\eqref{main.eq} has a nonlocal double phase structure analogous to the model with $p=q$.\n\\item[(ii)] If $b^{+}_{B_r}>\\nu/4$, then $b(x,y)$ is uniformly elliptic in $B_r\\times B_r$ (ellipticity ratio $2$), so $b(x,y)K_{tp}(x,y)\\approx K_{tp}(x,y)$ is dominating and $a(x,y)K_{sp}(x,y)$ is “lower order”; hence \\eqref{main.eq} becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.\n\\end{itemize}\n\nFinally, the paper roadmap gives the proof structure for Theorem~\\ref{thm.bdd}: “In Section~\\ref{Caccio.bounded}, we employ a Caccioppoli type estimate to prove Theorem~\\ref{thm.bdd}.”",
    "expanded_sketch": "Our proof of regularity results in To prove the main theorem, and in the following theorem.\n\nWe first state the following theorem.\n\\begin{theorem}[H\\\"older regularity]\\label{thm.hol}\nLet $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak solution to \\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}\n under assumptions \\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}\n--\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}\n Then there exists an exponent $\\gamma \\equiv \\gamma(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha}) \\in (0,1)$ such that $u \\in C^{0,\\gamma}_{\\loc}(\\Omega)$. Moreover, for any ball $B_{r} \\Subset \\Omega$ with $r\\le R_{0}$, where $R_{0}$ is given in \\eqref{R0.choice}, we have\n\\begin{equation}\\label{est:hol}\n[u]_{C^{0,\\gamma}(B_{r/2})} \\le cr^{-\\gamma}\\left[\\left(\\mean{B_{r}}|u|^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u;r/2,r))\\right]\n\\end{equation}\nfor a constant $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{theorem}\n\nOur proof is based on the Moser type approach in \\cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \\cite{BOS}. The authors note additional difficulties compared to \\cite{BOS}: they treat both cases $sp\\le n$ and $sp>n$ (in the latter, local boundedness follows from the fractional Sobolev--Morrey embedding), but “in both cases” they “have to obtain explicit and precise local sup-estimates which extend the one in \\cite[Theorem~1.1]{DKP16}.”\n\nThey also explain how the assumptions \\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}\n--\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}\n are used “to deal with the two modulating coefficients $a(\\cdot,\\cdot)$ and $b(\\cdot,\\cdot)$”: for a fixed ball $B_r\\subset\\Omega$ they split into two regimes:\n\\begin{itemize}\n\\item[(i)] If $b^{+}_{B_r}\\le \\nu/4$, then $a(x,y)$ is uniformly elliptic in $B_r\\times B_r$ so that $a(x,y)K_{sp}(x,y)\\approx K_{sp}(x,y)$, while $b(x,y)K_{tp}(x,y)$ is “higher order” and may be degenerate; thus \\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}\n has a nonlocal double phase structure analogous to the model with $p=q$.\n\\item[(ii)] If $b^{+}_{B_r}>\\nu/4$, then $b(x,y)$ is uniformly elliptic in $B_r\\times B_r$ (ellipticity ratio $2$), so $b(x,y)K_{tp}(x,y)\\approx K_{tp}(x,y)$ is dominating and $a(x,y)K_{sp}(x,y)$ is “lower order”; hence \\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}\n becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.\n\\end{itemize}\n\nFinally, the paper roadmap gives the proof structure for Theorem~\\ref{thm.bdd}: Next, we employ a Caccioppoli type estimate to prove the main theorem.",
    "expanded_theorem": "[Local boundedness]\\label{thm.bdd}\nLet  $u \\in \\mathcal{A}(\\Omega) \\cap \\mathcal{T}(\\mathbb{R}^{n})$ be a weak subsolution to\n\\begin{equation}\\label{main.eq}\n\\mathcal{L}u = 0 \\quad \\text{in } \\Omega,\n\\end{equation}\nunder assumptions\n\\begin{equation}\\label{kernel.growth}\n\\frac{\\nu}{|x-y|^{n+sp}} \\leq K_{sp}(x,y) \\leq \\frac{L}{|x-y|^{n+sp}}, \\qquad\n\\frac{\\nu}{|x-y|^{n+tp}} \\leq K_{tp}(x,y) \\leq \\frac{L}{|x-y|^{n+tp}}\n\\end{equation}\nand\n\\begin{equation}\\label{a.bound}\n\\nu \\le a(x,y) + b(x,y) \\le L\n\\end{equation}\nand let $B_{r}\\Subset \\Omega$ be a ball with $r\\le1$.\n\\begin{enumerate}\n\\item We have \n\\begin{equation}\\label{est:bdd1}\n\\esssup_{B_{r/2}} u \\le  c\\,r^{-(t-s)/\\beta} \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ r^{sp} T(u_+;r/2,r)\\right]^{1/(p-1)}\n\\end{equation}\nfor a constant $c \\equiv c(n,s,t,p,\\nu,L)>0$, where \n\\begin{equation}\\label{def.beta}\n\\beta \\coloneqq \\min\\{sp/n,p-1\\}.\n\\end{equation} \n\\item Assume further that $b(\\cdot,\\cdot)$ satisfies\n\\begin{equation}\\label{a.holder}\n|b(x_{1},y_{1}) - b(x_{2},y_{2})| \\leq [b]_{\\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\\alpha}\n\\end{equation}\nand\n\\begin{equation}\\label{assumption.hol}\n\\quad (t-s)p \\le \\alpha.\n\\end{equation}\nThen we have\n\\begin{equation}\\label{est:bdd2}\n\\begin{aligned}\n\\esssup_{B_{r/2}} u & \\le c \\left(\\mean{B_{r}}u_{+}^p\\,dx\\right)^{1/p} + \\left[ \\frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\\right]^{1/(p-1)} \\\\\n& = c\\left(\\mean{B_{r}}u_{+}^{p}\\,dx\\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))\n\\end{aligned}\n\\end{equation}\nwhenever $r \\le R_{0}$, where $R_{0}$ is given in\n\\begin{equation}\\label{R0.choice}\n    [b]_{\\alpha}(2R_{0})^{\\alpha} \\le \\frac{\\nu}{8}.\n\\end{equation}\nand $c\\equiv c(n,s,t,p,\\nu,L,\\alpha,[b]_{\\alpha})>0$.\n\\end{enumerate}\nConsequently, if $u \\in \\mathcal{A}(\\Omega)\\cap \\mathcal{T}(\\mathbb{R}^{n})$ is a weak solution to the equation above, then $u \\in L^{\\infty}_{\\loc}(\\Omega)$ and the estimate \\eqref{est:bdd1} or \\eqref{est:bdd2} holds with $|u|$ in each case.",
    "theorem_type": [
      "Inequality or Bound",
      "Universal"
    ],
    "mcq": {
      "question": "Let\n\\[\n\\mathcal L u(x):=\\operatorname{P.V.}\\int_{\\mathbb R^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))\\big(a(x,y)K_{sp}(x,y)+b(x,y)K_{tp}(x,y)\\big)\\,dy,\n\\]\nand let \\(u\\) belong to the admissible class \\(\\mathcal A(\\Omega)\\cap \\mathcal T(\\mathbb R^n)\\) used for weak solutions of this nonlocal equation. Assume that \\(u\\) is a weak subsolution of \\(\\mathcal Lu=0\\) in \\(\\Omega\\), and that for some \\(\\nu,L>0\\),\n\\[\n\\frac{\\nu}{|x-y|^{n+sp}}\\le K_{sp}(x,y)\\le \\frac{L}{|x-y|^{n+sp}},\\qquad\n\\frac{\\nu}{|x-y|^{n+tp}}\\le K_{tp}(x,y)\\le \\frac{L}{|x-y|^{n+tp}},\n\\]\nwith\n\\[\n\\nu\\le a(x,y)+b(x,y)\\le L.\n\\]\nLet \\(B_r\\Subset \\Omega\\) be a ball with \\(r\\le 1\\), write \\(u_+=\\max\\{u,0\\}\\),\n\\(\\fint_{B_r}f\\,dx=|B_r|^{-1}\\int_{B_r}f\\,dx\\),\n\\[\n\\beta:=\\min\\{sp/n,\\,p-1\\},\n\\]\nand\n\\[\nG_{B_r}(\\tau):=\\sup_{x,y\\in B_r}\\left(a(x,y)\\frac{\\tau^p}{r^{sp}}+b(x,y)\\frac{\\tau^p}{r^{tp}}\\right),\n\\qquad g_{B_r}(\\tau):=\\frac{G_{B_r}(\\tau)}{\\tau}.\n\\]\nIf, in addition, \\(b\\) is Hölder continuous in the sense that\n\\[\n|b(x_1,y_1)-b(x_2,y_2)|\\le [b]_\\alpha\\,(|x_1-x_2|+|y_1-y_2|)^\\alpha,\n\\]\nand \\((t-s)p\\le \\alpha\\), with \\(R_0\\in(0,1)\\) chosen so that\n\\[\n[b]_\\alpha(2R_0)^\\alpha\\le \\nu/8,\n\\]\nwhich statement holds for every such \\(u\\)?",
      "correct_choice": {
        "label": "A",
        "text": "One always has\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\,r^{-(t-s)/\\beta}\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\big[r^{sp}T(u_+;r/2,r)\\big]^{1/(p-1)},\n\\]\nwhere \\(c=c(n,s,t,p,\\nu,L)>0\\). If, moreover, \\(b\\) satisfies the Hölder condition above and \\((t-s)p\\le \\alpha\\), then for every \\(r\\le R_0\\),\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\left[\\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\\right]^{1/(p-1)}\n= c\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}+g_{B_r}^{-1}(T(u_+;r/2,r)),\n\\]\nwith \\(c=c(n,s,t,p,\\nu,L,\\alpha,[b]_\\alpha)>0\\). Consequently, every weak solution \\(u\\in \\mathcal A(\\Omega)\\cap\\mathcal T(\\mathbb R^n)\\) of \\(\\mathcal Lu=0\\) belongs to \\(L^\\infty_{\\mathrm{loc}}(\\Omega)\\), and the corresponding estimate above holds with \\(|u|\\) in place of \\(u\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "One always has\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\,r^{-(t-s)/(p-1)}\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\big[r^{sp}T(u_+;r/2,r)\\big]^{1/(p-1)},\n\\]\nwhere \\(c=c(n,s,t,p,\\nu,L)>0\\). If, moreover, \\(b\\) satisfies the Hölder condition above and \\((t-s)p\\le \\alpha\\), then for every \\(r\\le R_0\\),\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\left[\\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\\right]^{1/(p-1)},\n\\]\nwith \\(c=c(n,s,t,p,\\nu,L,\\alpha,[b]_\\alpha)>0\\). Consequently, every weak solution \\(u\\in \\mathcal A(\\Omega)\\cap\\mathcal T(\\mathbb R^n)\\) of \\(\\mathcal Lu=0\\) belongs to \\(L^\\infty_{\\mathrm{loc}}(\\Omega)\\), and the corresponding estimate above holds with \\(|u|\\) in place of \\(u\\)."
        },
        {
          "label": "C",
          "text": "One always has\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\,r^{-(t-s)/\\beta}\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\big[r^{sp}T(u_+;r/2,r)\\big]^{1/(p-1)},\n\\]\nwhere \\(c=c(n,s,t,p,\\nu,L)>0\\). Consequently, every weak solution \\(u\\in \\mathcal A(\\Omega)\\cap\\mathcal T(\\mathbb R^n)\\) of \\(\\mathcal Lu=0\\) belongs to \\(L^\\infty_{\\mathrm{loc}}(\\Omega)\\), and this estimate holds with \\(|u|\\) in place of \\(u\\)."
        },
        {
          "label": "D",
          "text": "One always has\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\,r^{-(t-s)/\\beta}\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\big[r^{sp}T(u_+;r/2,r)\\big]^{1/(p-1)},\n\\]\nwhere \\(c=c(n,s,t,p,\\nu,L)>0\\). If, moreover, \\(b\\) satisfies the Hölder condition above and \\((t-s)p\\le \\alpha\\), then there exists a radius \\(R_0\\in(0,1)\\), depending only on \\(\\nu,\\alpha,[b]_\\alpha\\), such that for every ball \\(B_r\\Subset\\Omega\\) with \\(r\\le 1\\),\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\left[\\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\\right]^{1/(p-1)},\n\\]\nwith \\(c=c(n,s,t,p,\\nu,L,\\alpha,[b]_\\alpha)>0\\). Consequently, every weak solution \\(u\\in \\mathcal A(\\Omega)\\cap\\mathcal T(\\mathbb R^n)\\) of \\(\\mathcal Lu=0\\) belongs to \\(L^\\infty_{\\mathrm{loc}}(\\Omega)\\), and the corresponding estimate above holds with \\(|u|\\) in place of \\(u\\)."
        },
        {
          "label": "E",
          "text": "One always has\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\,r^{-(t-s)/\\beta}\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+\\big[r^{tp}T(u_+;r/2,r)\\big]^{1/(p-1)},\n\\]\nwhere \\(c=c(n,s,t,p,\\nu,L)>0\\). If, moreover, \\(b\\) satisfies the Hölder condition above and \\((t-s)p\\le \\alpha\\), then for every \\(r\\le R_0\\),\n\\[\n\\operatorname*{ess\\,sup}_{B_{r/2}} u\n\\le c\\left(\\fint_{B_r}u_+^p\\,dx\\right)^{1/p}\n+g_{B_r}^{-1}(T(u_+;r/2,r)),\n\\]\nwith \\(c=c(n,s,t,p,\\nu,L,\\alpha,[b]_\\alpha)>0\\). Consequently, every weak solution \\(u\\in \\mathcal A(\\Omega)\\cap\\mathcal T(\\mathbb R^n)\\) of \\(\\mathcal Lu=0\\) belongs to \\(L^\\infty_{\\mathrm{loc}}(\\Omega)\\), and the corresponding estimate above holds with \\(|u|\\) in place of \\(u\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "case_split",
          "tampered_component": "definition_of_beta_in_scaling_exponent",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "regularity",
          "tampered_component": "dropped_Holder_refined_estimate_part_(2)",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "small_scale_restriction_r_le_R0_for_refined_bound",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "case_split",
          "tampered_component": "tail_scaling_sp_replaced_by_tp_in_general_estimate",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly state the conclusion or reproduce the exact final estimate; it only gives the hypotheses and asks which bound follows. There is no direct answer leakage from the stem itself."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the question asks for the exact quantitative local boundedness estimate under a list of hypotheses. The correct option is basically the theorem statement rather than a derived or competing conclusion."
      },
      "GPS": {
        "score": 1,
        "justification": "Some reasoning is needed to distinguish subtle variants involving the exponent, tail scaling, the small-scale restriction, and the inverse relation with g_{B_r}. However, the task is still mostly recognition of the exact statement, not genuine generative mathematical reasoning."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic errors: wrong scaling exponent, omission of part of the theorem, incorrect dependence on the Hölder data/smallness scale, and confusion between G_{B_r}, g_{B_r}, and their inverses."
      },
      "total_score": 5,
      "overall_assessment": "A solid recall/recognition MCQ with strong distractors, but it is largely a direct theorem-statement identification problem rather than a genuinely generative reasoning task."
    }
  },
  {
    "id": "2512.23615v1",
    "paper_link": "http://arxiv.org/abs/2512.23615v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{Thm:IGP}\n    For any prime $p$ for which there exists an $\\ell \\in \\cL$ with the Kronecker symbol $\\legendre{\\ell}{p}$ equal to $-1$, the group $\\PSL_2(\\F_{p^2})$ is a \\emph{regular} Galois group over $\\Q$.\\footnote{Concretely, this implies that $\\PSL_2(\\F_{p^2})$ is realisable as a regular Galois group over $\\Q$ for all $p<3,267,289$ (compared to the previous implied bound $p<1,009$).}",
      "start_pos": 66896,
      "end_pos": 67321,
      "label": "Thm:IGP"
    },
    "ref_dict": {
      "Thm:IGP": "\\begin{theorem}\\label{Thm:IGP}\n    For any prime $p$ for which there exists an $\\ell \\in \\cL$ with the Kronecker symbol $\\legendre{\\ell}{p}$ equal to $-1$, the group $\\PSL_2(\\F_{p^2})$ is a \\emph{regular} Galois group over $\\Q$.\\footnote{Concretely, this implies that $\\PSL_2(\\F_{p^2})$ is realisable as a regular Galois group over $\\Q$ for all $p<3,267,289$ (compared to the previous implied bound $p<1,009$).}\n\\end{theorem}",
      "Thm:cover": "\\begin{theorem}\\label{Thm:cover}\n    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\\mathfrak o$, and let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$.\n\\end{theorem}",
      "Rem:shih": "\\begin{remark}\\label{Rem:shih}\nTheorem~\\ref{Thm:cover} should be compared to the analogous cover $X_0(p)\\to X(1)$ of modular curves corresponding to the level structure $\\Gamma_0(p)$.\n\nIn the one-dimensional setting, $X_0(p)$ has field of definition $\\Q(\\zeta_p)$ and one only obtains a $\\PGL_2(\\F_p)$-cover. Under certain congruence conditions, Atkin-Lehner involutions can be used to turn this into a $\\PSL_2(\\F_p)$-cover by what has become known as Shih's trick \\cite{Shih}. In our situation such a trick is unnecessary.\n\\end{remark}",
      "Thm:K3regular": "\\begin{theorem}\\label{Thm:K3regular}\n     Let $X/\\Q$ be the Hilbert modular surface of discriminant $D\\in \\cS_+$ (resp.\\ $\\cS_-$) and principal genus (resp.\\ non-principal genus) at base level. Then the natural image of $X(\\Q(t))$ in $X_{\\bar \\Q(t)}$ is non-thin.\n\\end{theorem}",
      "Thm:K3": "\\begin{theorem}\\label{Thm:K3}\n Let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\\Q)$ is not thin.\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2083,
    "pre_theorem_intro_text": "The following popular question, going back to at least Noether, has remained open for over a century:\n\\begin{question}[Inverse Galois Problem]\n    Is every finite group $G$ the Galois group of a finite extension of $\\Q$?\n\\end{question}\n\nWe also have the variant:\n\\begin{question}[Regular Inverse Galois Problem]\n    Is every finite group $G$ the Galois group of a finite geometrically integral cover $C \\to \\P^1_{\\Q}$?\n\\end{question}\n\nBy the Hilbert Irreducibility Theorem (a positive answer to) the regular Inverse Galois Problem implies (a positive answer to) the classical one. On the other hand, the regular version has various interesting implications: for instance, it gives not just one $G$-extension of $\\Q$ but infinitely many linearly disjoint $G$-extensions over any number field by specialisation.\n\nA fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \\cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\\PSL_2(\\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\\ell\\in\\{2,3,5,7\\}$ with $\\legendre{\\ell}{p}=-1$ \\cite{Shih, Malle} (although non-regularly for all $p$ \\cite{Zywina}) and for $k=2$ when $\\legendre{5}{p}=-1$ \\cite{Feit, Mestre, Mestre2}, when $\\legendre{2}{p}=-1$ or  $\\legendre{3}{p}=-1$ \\cite{Shiina}, and finally when $\\legendre{-3}{p}=-1$ or $\\legendre{-7}{p}=-1$ \\cite{DW} (although non-regularly for more $p$ \\cite{DV}).\n\nIn this paper we introduce new methods for the Inverse Galois Problem to prove the following. Let\n\\[\n\\cL:=\\{\\ell\\ \\text { prime}: \\ell \\leq 41, \\ell \\neq 31\\}.\n\\]",
    "context": "The following popular question, going back to at least Noether, has remained open for over a century:\n\\begin{question}[Inverse Galois Problem]\n Is every finite group $G$ the Galois group of a finite extension of $\\Q$?\n\\end{question}\n\nWe also have the variant:\n\\begin{question}[Regular Inverse Galois Problem]\n Is every finite group $G$ the Galois group of a finite geometrically integral cover $C \\to \\P^1_{\\Q}$?\n\\end{question}\n\nBy the Hilbert Irreducibility Theorem (a positive answer to) the regular Inverse Galois Problem implies (a positive answer to) the classical one. On the other hand, the regular version has various interesting implications: for instance, it gives not just one $G$-extension of $\\Q$ but infinitely many linearly disjoint $G$-extensions over any number field by specialisation.\n\nA fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \\cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\\PSL_2(\\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\\ell\\in\\{2,3,5,7\\}$ with $\\legendre{\\ell}{p}=-1$ \\cite{Shih, Malle} (although non-regularly for all $p$ \\cite{Zywina}) and for $k=2$ when $\\legendre{5}{p}=-1$ \\cite{Feit, Mestre, Mestre2}, when $\\legendre{2}{p}=-1$ or $\\legendre{3}{p}=-1$ \\cite{Shiina}, and finally when $\\legendre{-3}{p}=-1$ or $\\legendre{-7}{p}=-1$ \\cite{DW} (although non-regularly for more $p$ \\cite{DV}).\n\nIn this paper we introduce new methods for the Inverse Galois Problem to prove the following. Let\n\\[\n\\cL:=\\{\\ell\\ \\text { prime}: \\ell \\leq 41, \\ell \\neq 31\\}.\n\\]",
    "full_context": "The following popular question, going back to at least Noether, has remained open for over a century:\n\\begin{question}[Inverse Galois Problem]\n Is every finite group $G$ the Galois group of a finite extension of $\\Q$?\n\\end{question}\n\nWe also have the variant:\n\\begin{question}[Regular Inverse Galois Problem]\n Is every finite group $G$ the Galois group of a finite geometrically integral cover $C \\to \\P^1_{\\Q}$?\n\\end{question}\n\nBy the Hilbert Irreducibility Theorem (a positive answer to) the regular Inverse Galois Problem implies (a positive answer to) the classical one. On the other hand, the regular version has various interesting implications: for instance, it gives not just one $G$-extension of $\\Q$ but infinitely many linearly disjoint $G$-extensions over any number field by specialisation.\n\nA fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \\cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\\PSL_2(\\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\\ell\\in\\{2,3,5,7\\}$ with $\\legendre{\\ell}{p}=-1$ \\cite{Shih, Malle} (although non-regularly for all $p$ \\cite{Zywina}) and for $k=2$ when $\\legendre{5}{p}=-1$ \\cite{Feit, Mestre, Mestre2}, when $\\legendre{2}{p}=-1$ or $\\legendre{3}{p}=-1$ \\cite{Shiina}, and finally when $\\legendre{-3}{p}=-1$ or $\\legendre{-7}{p}=-1$ \\cite{DW} (although non-regularly for more $p$ \\cite{DV}).\n\nIn this paper we introduce new methods for the Inverse Galois Problem to prove the following. Let\n\\[\n\\cL:=\\{\\ell\\ \\text { prime}: \\ell \\leq 41, \\ell \\neq 31\\}.\n\\]\n\nA fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \\cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\\PSL_2(\\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\\ell\\in\\{2,3,5,7\\}$ with $\\legendre{\\ell}{p}=-1$ \\cite{Shih, Malle} (although non-regularly for all $p$ \\cite{Zywina}) and for $k=2$ when $\\legendre{5}{p}=-1$ \\cite{Feit, Mestre, Mestre2}, when $\\legendre{2}{p}=-1$ or $\\legendre{3}{p}=-1$ \\cite{Shiina}, and finally when $\\legendre{-3}{p}=-1$ or $\\legendre{-7}{p}=-1$ \\cite{DW} (although non-regularly for more $p$ \\cite{DV}).\n\nIn particular, this realises the group $\\PSL_2(\\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\\ell \\leq 17, \\ell \\neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \\cite{EK} have proved $\\Q$-rationality. For the primes $\\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \\cite{CZ}, and further studied by the first-named author \\cite{Demeio} and by the second-named author in collaboration with Mezzedimi \\cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.\n\nIn fact, we prove an even stronger statement, namely that $X(\\Q(t))$ is non-thin in $X_{\\bar \\Q(t)}(\\bar \\Q(t))$ (Theorem \\ref{Thm:K3regular}), thus producing an abundance of {\\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.\nThe application to the Inverse Galois Problem is found through the following\n\\begin{theorem}\\label{Thm:cover}\n Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\\mathfrak o$, and let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$.\n\\end{theorem}\n\nThe particular Shimura varieties under our consideration {\\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \\to X$ to {\\em these} rational curves does not produce surjective representations $\\Gal(\\overline{\\Q(t)}/\\Q(t)) \\to \\PSL_2(\\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \\emph{not} sufficient to construct Galois representations of $\\PSL_2(\\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \\ref{Thm:K3}.\n\n\\begin{proposition}\\label{Prop331regular}\n Let $k$ be a field of characteristic $0$, and $k(V)$ be a regular finitely generated transcendental extension of $k$. Let $\\phi:Y\\to X$ be a Galois cover with group $G$ of geometrically integral $k(V)$-varieties. Let $S \\subset X(k(V))$ be a subset whose natural image in $X(\\bar k(V))=X_{\\bar k(V)}(\\bar k(V))$ is a non-thin set of points of the base-changed variety $X_{\\bar k(V)}$. Then there are Zariski-dense $x \\in S$ for which the fibre $\\phi^{-1}(x)$ is irreducible with regular function field over $k$, and with $\\Gal(\\phi^{-1}(x)/k(V))\\simeq G$.\n\\end{proposition}\n\nWe write $\\H^+$ and $\\H^-$ for the upper and lower half-plane. For each genus $g \\in \\Cl^+K/2$ of $\\mathfrak o$ we may pick a representative fractional ideal $\\mathfrak a$ (of norm $A$) and define, for each finite index subgroup $\\Gamma \\subseteq \\PSL(\\mathfrak o \\oplus \\mathfrak a)$, the \\emph{Hilbert modular surface}\n\\[\nX_g(\\Gamma):=\\Gamma \\backslash (\\H^+)^2,\n\\]\nwhere the quotient is by the properly discontinuous action:\n\\[\n\\begin{pmatrix}\n a & b \\\\ c&d\n\\end{pmatrix}\\cdot (z_1,z_2) = \\left( \\frac{az_1+b}{cz_1+d} , \\frac{a'z_2+b'}{c'z_2+d'} \\right).\n\\]\nLet $\\Lambda=\\mathfrak o \\oplus \\mathfrak a$. We call the surface $X_g:= X_g(\\PSL(\\Lambda))$ of \\emph{base level}.\\footnote{One could take further quotients of $(\\H^+)^2$, the maximum being by the symmetrised Hurwitz-Maass extension of $\\PSL(\\Lambda)$, but we shall not be interested in these in this paper.} The isomorphism class of $X_g$ is independent of the choice of representative $\\mathfrak a$.\n\nThe classes in $\\Cl(K)$ are in bijection with $\\PSL(\\Lambda)$-orbits of points in $\\P^1(K)$ via the map sending $(x_0:x_1)\\in\\P^1(K)$ to the class $[x_0\\cdot \\mathfrak o + x_1\\cdot \\mathfrak a]\\in\\Cl(K)$. Fix a class $[\\mathfrak c]\\in\\Cl(K)$ and write shorthand $s=s_{[\\mathfrak c]}$. To $s$ we associate $M_s=\\mathfrak c^{-2}\\mathfrak a^{-1}$ viewed as an \\emph{ordered} $\\mathfrak o$-module, i.e.\\ equipped with an order on $M_s\\otimes_\\sigma \\R$ for each embedding $\\sigma:K\\hookrightarrow\\R$. The isomorphism class of $M_s$ does not depend on the chosen representative $\\mathfrak c$. \n\\begin{proposition}\\label{Prop:CoverQ}\\hfill\n \\begin{enumerate}\n \\item The action of $\\Gamma_\\Q$ on $X_g^\\infty(\\ov\\Q)$ is trivial, in other words all cusps are defined over $\\Q$.\n \\item The exceptional divisor $E_s$ of the minimal resolution $\\tilde X_g\\to X_g$ over any cusp $s$ is a cycle of smooth rational curves intersecting transversally.\n \\end{enumerate}\n\\end{proposition}\n\nCombining the above, we proved the theorem for all $p$ for which there exists $D_p$ in\n\\[\n\\cD \\coloneqq \\{5,8,12,13,17,21,24,28,29,33,37,40,41,44,56,57,69,105\\}\n\\]\nwith $\\legendre{D_p}{p}=-1$. One easily checks by hand that this includes all primes $p \\leq 41$. For $p > 41$, such a $D_p$ exists unless\n\\begin{center}\n $\\legendre{D}{p}=1$ for all $D \\in \\cD$,\n\\end{center}\nor, equivalently, unless $\\legendre{D}{p}=1$ for all $D$ in the multiplicative subgroup $\\langle \\cD \\rangle \\subset \\Q^*$ generated by $\\cD$. Rewriting $\\cD$ as\n\\[\n\\{5,2^3,2^2 \\cdot 3,13,17,3 \\cdot 7,2^3 \\cdot 3,2^2 \\cdot 7,29,3 \\cdot 11,37,2^3 \\cdot 5,41,2^2 \\cdot 11,2^3 \\cdot 7,3\\cdot 19,3 \\cdot 23,3 \\cdot 5 \\cdot 7\\}\n\\]\none sees that $\\langle \\cD \\rangle=\\langle \\cL \\rangle$, where $\\cL=\\{\\ell \\text{ prime}: \\ell \\leq 41, \\ell \\neq 31\\}$ is the list of prime factors appearing in $\\cD$. So $D_p$ exists unless $\\legendre{\\ell}{p}=1$ for all $\\ell$, concluding the proof.\n\\end{proof}",
    "post_theorem_intro_text_len": 5561,
    "post_theorem_intro_text": "In particular, this realises the group $\\PSL_2(\\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\\ell \\leq 17, \\ell \\neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \\cite{EK} have proved $\\Q$-rationality. For the primes $\\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \\cite{CZ}, and further studied by the first-named author \\cite{Demeio} and by the second-named author in collaboration with Mezzedimi \\cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.\n\n\\begin{theorem}\\label{Thm:K3}\n Let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\\Q)$ is not thin.\n\\end{theorem}\n\nIn fact, we prove an even stronger statement, namely that $X(\\Q(t))$ is non-thin in $X_{\\bar \\Q(t)}(\\bar \\Q(t))$ (Theorem \\ref{Thm:K3regular}), thus producing an abundance of {\\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.\nThe application to the Inverse Galois Problem is found through the following\n\\begin{theorem}\\label{Thm:cover}\n    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\\mathfrak o$, and let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$.\n\\end{theorem}\n\nCompared to the well-known modular curve setting, the existence of the Galois cover which is geometrically integral may be surprising at first, see Remark~\\ref{Rem:shih}.\n\nTheorem \\ref{Thm:IGP} is proven by combining Theorems \\ref{Thm:K3} and \\ref{Thm:cover}. \nThe proofs of Theorems \\ref{Thm:K3} and \\ref{Thm:cover} rely on the theory of canonical models of compactified Shimura varieties as applied to Hilbert modular surfaces. More precisely, Theorem \\ref{Thm:K3} relies on an analysis of the intersection graph of these modular surfaces, while Theorem \\ref{Thm:cover} follows naturally by looking at the Shimura data involved. (Alternatively, Theorem \\ref{Thm:cover} can be inferred by looking at the $p$-torsion representation of the abelian surfaces with real multiplication parametrized by Hilbert Modular Surfaces.) \n\nThe particular Shimura varieties under our consideration {\\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \\to X$ to {\\em these} rational curves does not produce surjective representations $\\Gal(\\overline{\\Q(t)}/\\Q(t)) \\to \\PSL_2(\\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \\emph{not} sufficient to construct Galois representations of $\\PSL_2(\\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \\ref{Thm:K3}.\n\nThe Shimura theory is particularly explicit in our case but it would be interesting to see if a similar approach could apply to other settings.\n\nIt would also be interesting to study the Hilbert Property for honestly elliptic Hilbert modular surfaces, where the multiple fibration method does not apply. If instead of proving non-thinness of rational points the goal is to only specialise a specific Galois cover like in Theorem~\\ref{Thm:cover}, then one may use less than the full Hilbert Property, although knowledge of the intersection graph of the modular surface would still be necessary. We will pursue this approach in a separate forthcoming work.\n\nThe article is organised as follows: In \\S\\ref{Sec:hilbert}, we review and extend the multiple fibration method to nonconstant $\\Q(t)$-points. We develop the necessary Shimura theory of Hilbert modular surfaces in \\S\\ref{Sec:shimura} and construct the cover from Theorem~\\ref{Thm:cover} in \\S\\ref{Sec:cover}. We then study the algebraic cycles on our surfaces in \\S\\ref{Sec:cycles}. Finally, we prove Theorems~\\ref{Thm:IGP} and \\ref{Thm:K3} in \\S\\ref{Sec:main}.\n\n\\paragraph{Acknowledgements}\nThe authors thank Gerard Van der Geer and John Voight for helpful comments. They gratefully acknowledge the hospitality of ICMS Edinburgh, the Center for Mathematical Sciences at Technion, and the University of Bath, where part of this research was carried out, and thank Daniel Loughran for the invitation to the latter. The second author was supported by UKRI award UKRI094.",
    "sketch": "To prove Theorem~\\ref{Thm:IGP}, the authors \"combin[e] Theorems \\ref{Thm:K3} and \\ref{Thm:cover}.\" Theorem~\\ref{Thm:cover} supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a \"Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$.\" Theorem~\\ref{Thm:K3} provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then \"$X(\\Q)$ is not thin\" (indeed they prove the stronger non-thinness statement over $\\Q(t)$ in Theorem~\\ref{Thm:K3regular}, yielding \"an abundance of rational curves\").\n\nFor small $\\ell$ with $\\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they \"apply the classical Hilbert Irreducibility Theorem.\" For the remaining listed primes $\\ell$ where the surfaces are \"not (uni)rational,\" they instead \"prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces\" by establishing that \"the Hilbert Property holds\" for the relevant low-discriminant surfaces via the \"multiple fibration method.\" Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, \"specialising the cover $Y_p \\to X$ to these rational curves does not produce surjective representations\" to $\\PSL_2(\\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature \"allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject,\" with the propagation formalised by the multiple fibration method used in proving Theorem~\\ref{Thm:K3}.",
    "expanded_sketch": "To prove the main theorem, the authors “combin[e] Theorems \\ref{Thm:K3} and \\ref{Thm:cover}.” We first record the two inputs.\n\n\\begin{theorem}\\label{Thm:cover}\n    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\\mathfrak o$, and let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{Thm:K3}\n Let $X/\\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\\Q)$ is not thin.\n\\end{theorem}\n\nThe first theorem supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a Galois geometrically integral cover $Y_p \\to X$ with group $\\PSL_2(\\mathfrak o/p)$. The second theorem provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then $X(\\Q)$ is not thin (indeed they prove the stronger non-thinness statement over $\\Q(t)$ in\n\\begin{theorem}\\label{Thm:K3regular}\n     Let $X/\\Q$ be the Hilbert modular surface of discriminant $D\\in \\cS_+$ (resp.\\ $\\cS_-$) and principal genus (resp.\\ non-principal genus) at base level. Then the natural image of $X(\\Q(t))$ in $X_{\\bar \\Q(t)}$ is non-thin.\n\\end{theorem}\n, yielding “an abundance of rational curves”).\n\nFor small $\\ell$ with $\\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they “apply the classical Hilbert Irreducibility Theorem.” For the remaining listed primes $\\ell$ where the surfaces are “not (uni)rational,” they instead “prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces” by establishing that “the Hilbert Property holds” for the relevant low-discriminant surfaces via the “multiple fibration method.” Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, “specialising the cover $Y_p \\to X$ to these rational curves does not produce surjective representations” to $\\PSL_2(\\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature “allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject,” with the propagation formalised by the multiple fibration method used in proving the theorem above.",
    "expanded_theorem": "\\label{Thm:IGP}\n    For any prime $p$ for which there exists an $\\ell \\in \\cL$ with the Kronecker symbol $\\legendre{\\ell}{p}$ equal to $-1$, the group $\\PSL_2(\\F_{p^2})$ is a \\emph{regular} Galois group over $\\Q$.\\footnote{Concretely, this implies that $\\PSL_2(\\F_{p^2})$ is realisable as a regular Galois group over $\\Q$ for all $p<3,267,289$ (compared to the previous implied bound $p<1,009$).}",
    "theorem_type": [
      "Implication",
      "Existence"
    ],
    "mcq": {
      "question": "Let \\(\\mathcal L=\\{\\ell\\text{ prime}: \\ell\\le 41,\\ \\ell\\ne 31\\}\\). Suppose \\(p\\) is a prime such that there exists some \\(\\ell\\in \\mathcal L\\) with Kronecker symbol \\(\\left(\\frac{\\ell}{p}\\right)=-1\\). Under this hypothesis, which statement about \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) holds over \\(\\mathbb Q\\)? Here, saying that a finite group \\(G\\) is a regular Galois group over \\(\\mathbb Q\\) means that there exists a finite geometrically integral cover \\(C\\to \\mathbb P^1_{\\mathbb Q}\\) whose induced function-field extension is Galois with group \\(G\\) (equivalently, there is a Galois extension \\(E/\\mathbb Q(t)\\) with Galois group \\(G\\) and \\(E\\cap \\overline{\\mathbb Q}=\\mathbb Q\\)).",
      "correct_choice": {
        "label": "A",
        "text": "The group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) is a regular Galois group over \\(\\mathbb Q\\); equivalently, there exists a Galois extension \\(E/\\mathbb Q(t)\\) with Galois group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) such that \\(E\\cap \\overline{\\mathbb Q}=\\mathbb Q\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "The group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) is a Galois group over \\(\\mathbb Q\\); equivalently, there exists a finite Galois extension \\(L/\\mathbb Q\\) with Galois group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\)."
        },
        {
          "label": "C",
          "text": "There exists a finite geometrically integral cover \\(C\\to \\mathbb P^1_{\\mathbb Q}\\) whose induced function-field extension has Galois group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\)."
        },
        {
          "label": "D",
          "text": "The group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) is a regular Galois group over \\(\\mathbb Q\\) for every prime \\(p\\) such that \\(\\left(\\frac{\\ell}{p}\\right)=-1\\) for all \\(\\ell\\in \\mathcal L\\); equivalently, there exists a Galois extension \\(E/\\mathbb Q(t)\\) with Galois group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) and \\(E\\cap \\overline{\\mathbb Q}=\\mathbb Q\\) whenever this stronger condition holds."
        },
        {
          "label": "E",
          "text": "The group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) is a regular Galois group over \\(\\mathbb Q\\); equivalently, for every real quadratic field \\(K\\) in which \\(p\\) is inert, there exists a Galois extension \\(E/\\mathbb Q(t)\\) with Galois group \\(\\operatorname{PSL}_2(\\mathbb F_{p^2})\\) such that \\(E\\cap \\overline{\\mathbb Q}=\\mathbb Q\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "regularity_vs_classical_realisation",
          "template_used": "property_confusion"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "dropped_constant_field_condition_equivalence_clause",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "characteristic",
          "tampered_component": "existential_condition_on_ell_replaced_by_universal_condition",
          "template_used": "quantifier_dependence"
        },
        {
          "label": "E",
          "sketch_hook_type": "characteristic",
          "tampered_component": "specific_discriminant_selection_needed_for_K3_and_cover_input",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and a definition, but the answer still depends on identifying the precise existence claim among several close variants."
      },
      "TAS": {
        "score": 0,
        "justification": "The correct choice is essentially a direct statement of the underlying theorem, with only equivalent phrasing added. The item mainly tests recognition of the exact theorem statement rather than selecting among genuinely competing conclusions derived from reasoning."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning pressure in checking quantifiers, the distinction between regular realization and mere realization over Q, and PSL2 versus PGL2. However, the task is still largely theorem recall/matching rather than substantial generative inference."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing PSL2 with PGL2, weakening regularity to mere realizability, strengthening the quantifier on the Kronecker condition, or adding an unnecessary universal quadratic-field condition."
      },
      "total_score": 5,
      "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the result and only moderately tests genuine generative reasoning."
    }
  },
  {
    "id": "2512.23668v1",
    "paper_link": "http://arxiv.org/abs/2512.23668v1",
    "theorems_cnt": 5,
    "theorem": {
      "env_name": "theorem",
      "content": "[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*}",
      "start_pos": 4898,
      "end_pos": 5226,
      "label": "thm:HiroseSato1"
    },
    "ref_dict": {
      "it:HMSW1": "\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item\\label{it:HMSW1} $\\{w-\\cD(w) \\mid w\\in \\cH^0\\}=\\Ker(Z^{\\diamondsuit})\\subset\\Ker(Z)$,\n\\item\\label{it:HMSW2} $\\cD(w)=w$ for $w\\in\\cH^{\\geq 2}$,\n\\item\\label{it:HMSW3} $\\cD(w)=\\cD(\\tau(w))$ for $w\\in \\cH^0$.\n\\end{enumerate}",
      "thm:HiroseSato1": "\\begin{theorem}[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*}\n\\end{theorem}",
      "thm:HiroseSato2": "\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\cH^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}",
      "eq:Hoffman-conj": "\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}",
      "thm:drop1": "\\begin{theorem}[Hirose--Maesaka--Seki--Watanabe~\\cite{HiroseMaesakaSekiWatanabe2025+}]\nThe drop 1 operator $\\cD$ satisfies the following properties$:$\n\\label{thm:drop1}\n\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item\\label{it:HMSW1} $\\{w-\\cD(w) \\mid w\\in \\cH^0\\}=\\Ker(Z^{\\diamondsuit})\\subset\\Ker(Z)$,\n\\item\\label{it:HMSW2} $\\cD(w)=w$ for $w\\in\\cH^{\\geq 2}$,\n\\item\\label{it:HMSW3} $\\cD(w)=\\cD(\\tau(w))$ for $w\\in \\cH^0$.\n\\end{enumerate}\n\\end{theorem}",
      "def:drop1operator": "\\begin{definition}[Hirose--Maesaka--Seki--Watanabe~\\cite{HiroseMaesakaSekiWatanabe2025+}]\\label{def:drop1operator}\nWe define the \\emph{drop 1 operator} $\\cD\\colon\\cH^0\\to\\cH^{\\geq 2}$ to be a $\\ZZ$-linear map given as follows.\nIt suffices to define $\\cD(w)$ for each word $w\\in\\cH^0$.\nFor $\\bc=(c_1,\\dots, c_{2t})$, we write $\\fD(\\bc)=\\cD(y^{c_1}x^{c_2}\\cdots y^{c_{2t-1}}x^{c_{2t}})$.\nThe image $\\fD(\\bc)$ is defined inductively as follows:\n\\begin{align*}\n\\fD(\\bc)&=\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 1}}(-1)^{\\#B-1}\\fD(\\bc_{(-A)}-\\bdelta_B)x^{\\#A+\\#B}\\\\\n&\\quad+\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 2}}(-1)^{\\#B-1}\\fD(\\bc_{(-A)}-\\bdelta_B)z_{\\#A+\\#B}\\\\\n&\\quad+\\sum_{\\substack{A\\in\\cP^{\\text{oe}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{oe}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 2}}(-1)^{\\#B}\\fD(\\bc_{(-A)}-\\bdelta_B)z_{\\#A+\\#B}.\n\\end{align*}\nHere the initial condition is $\\fD(\\varnothing)=\\cD(1)=1$.\n\\end{definition}",
      "cor:FMZV": "\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\n(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\star w_2).\n\\]\n\\end{corollary}",
      "cor:diamond-lift": "\\begin{corollary}\\label{cor:diamond-lift}\nFor $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\nZ^{\\diamondsuit}(w_1\\tau(w_2))=Z^{\\diamondsuit}(w_1\\star w_2).\n\\]\n\\end{corollary}"
    },
    "pre_theorem_intro_text_len": 938,
    "pre_theorem_intro_text": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.",
    "context": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.",
    "full_context": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.\n\nMore generally, applying the same procedure to Corollary~\\ref{cor:diamond-lift} (see Section~\\ref{sec:finite} for more details), we have the following.\nFor a word $w=z_{k_1}\\cdots z_{k_r}$, we set $\\overleftarrow{w}\\coloneqq z_{k_r}\\cdots z_{k_1}$, and $\\wt(w)$ denotes the total degree of $w$, that is, $\\wt(w)=k_1+\\cdots + k_r$.\nA $\\ZZ$-linear map $Z_{\\cA}\\colon \\ZZ+y\\cH=\\ZZ\\langle z_1,z_2,\\dots\\rangle\\to\\cA$ is defined by $Z_{\\cA}(z_{k_1}\\cdots z_{k_r})=\\zeta^{}_{\\cA}(k_1,\\dots, k_r)$ and $Z_{\\cA}(1)=1$.\n\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\n(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\star w_2).\n\\]\n\\end{corollary}\nThis is a family of relations among FMZVs in which only 2 and 3 occur as entries.\nSince the shuffle relation for FMZVs $(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\sh w_2)$ holds (see \\cite{KanekoZagier2025+,Ono2017}), Corollary~\\ref{cor:FMZV} can also be rewritten as\n\\[\nZ_{\\cA}(w_1\\sh w_2-w_1\\star w_2)=0\n\\]\nwhich is of the form of a certain double shuffle relation.\n\\section*{Acknowledgements}\nThe author would like to thank Professor Minoru Hirose for kindly answering his questions about the previous works by Hirose and Sato.\nThis research was inspired by a question that came to mind when the author got married, and the author would like to thank his wife, Yui, for marrying him.\n\\section{The drop 1 operator}\nWe introduce the following notation to define the main operator.\nLet $t$ be a positive integer, let $\\bc=(c_1,\\dots, c_{2t})$ be a tuple of positive integers, and let $S\\subset\\ZZ$.\n\\begin{itemize}[leftmargin=2.5em]\n\\item $\\#A$ denotes the cardinality of a finite set $A$.\n\\item $[n]\\coloneqq\\{1,2,\\dots,n\\}$ for $n\\in\\ZZ_{\\geq 0}$. $[0]=\\varnothing$.\n\\item $[2t]_{\\bc}^1\\coloneqq\\{i\\in[2t] \\mid c_i=1\\}.$\n\\item $[2t]_{\\bc}^{>1}\\coloneqq\\{i\\in[2t] \\mid c_i>1\\}.$\n\\item $\\cP^{\\text{eo}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\cap 2\\ZZ \\Longleftrightarrow i+1\\in A\\setminus 2\\ZZ\\}.$\n\\item $\\cP^{\\text{oe}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\setminus 2\\ZZ \\Longleftrightarrow i+1\\in A\\cap 2\\ZZ\\}.$\n\\item $\\cP^{\\smallxcancel{\\text{eo}}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\cap 2\\ZZ \\Longrightarrow i+1\\not\\in A\\setminus 2\\ZZ\\}.$\n\\item $\\cP^{\\smallxcancel{\\text{oe}}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\setminus 2\\ZZ \\Longrightarrow i+1\\not\\in A\\cap 2\\ZZ\\}.$\n\\item $\\bc_{(-A)}\\coloneqq (c_{i_1},\\dots, c_{i_r})$, where $[2t]\\setminus A=\\{i_1,\\dots, i_r\\}$ with $i_1<\\cdots<i_r$, for $A\\subset[2t]$.\n\\item $\\bc_{(-\\varnothing)}=\\bc$, $\\bc_{(-[2t])}=\\varnothing$.\n\\item $\\delta_{i\\in B}\\coloneqq\\begin{cases}1 & \\text{if } i\\in B, \\\\ 0 & \\text{otherwise}\\end{cases}$ for a set $B$.\n\\item $\\bc_{(-A)}-\\bdelta_B\\coloneqq(c_{i_1}-\\delta_{i_1\\in B},\\dots, c_{i_r}-\\delta_{i_r\\in B})$ for the above $\\bc_{(-A)}$ and $B\\subset [2t]$.\n\\end{itemize}\n\nWe consider the case $c>1$.\nThen $\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc})=\\cP([2t]^{>1}_{\\bc})$ holds.\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$, decompose it as $A=A_1\\cup A_2 \\cup A_3$, where\n\\[\nA_1\\coloneqq A \\cap [2l_r-1],\\quad A_2\\coloneqq A\\cap ([2l_r+2c-2]\\setminus [2l_r]),\\quad A_3\\coloneqq A\\setminus [2l_r+2c-1].\n\\]\nWe define a map $\\iota_1\\colon\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ by\n\\[\n\\iota_1(A)\\coloneqq (A_1-1)\\cup (A_2-1)\\cup\\{2l_r+2c-3,2l_r+2c-2\\}\\cup(A_3+1)\n\\]\nand a map $\\iota_2\\colon\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ by\n\\[\n\\iota_2(A)\\coloneqq (A_1-1)\\cup (A_2-1)\\cup(A_3+1).\n\\]\nA bijection $\\cP^{\\text{eo}}([2t]^1_{\\bc})\\sqcup\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ is obtained by sending an element $A$ in the first copy of $\\cP^{\\text{eo}}([2t]^1_{\\bc})$ to $\\iota_1(A)$, and an element $A$ in the second copy of $\\cP^{\\text{eo}}([2t]^1_{\\bc})$ to $\\iota_2(A)$.\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$, we have $\\bc_{(-A)}=\\bc_{(-\\iota_2(A))}$.\nTherefore, if we compute the defining recurrence for the drop 1 operator using this bijection, every term in the second sum is canceled by a term in the third sum, and we are left with\n\\begin{align*}\n\\fD(\\bc)&=\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 1}}(-1)^{\\#B-1}\\biggl(\\fD(\\bc_{(-A)}-\\bdelta_B)-\\fD(\\bc_{(-\\iota_1(A))}-\\bdelta_B)z_2\\biggr)x^{\\#A+\\#B}\\\\\n&\\quad+\\fD(\\bc_{(-\\{2l_r+2c-3,2l_r+2c-2\\})})z_2.\n\\end{align*}\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$ or $\\cP^{\\text{oe}}([2t]^1_{\\bc})$, there exist $a'_1$, $\\dots$, $a'_r$, $b'_1$, $\\dots$, $b'_s$, and $c'$ such that\n\\[\n\\bc_{(-A)}=[a'_1,\\dots, a'_r;b'_1,\\dots, b'_s;c'].\n\\]\nHere $a'_1\\in[a_1]$, $\\dots$, $a'_r\\in[a_r]$, $b'_1\\in[b_1]$, $\\dots$, $b'_s\\in[b_s]$, $c'\\in[c]$, and $a'_1+\\cdots+a'_r+b'_1+\\cdots+b'_s+c'=t+1-\\#A/2$.\nMoreover, upon adding $-\\bdelta_B$, addition occurs between the corresponding components; for example, when\n\\begin{align*}\n\\bc&=[2,1,3;4,1,1,2;3]\\\\\n&=(\\stackrel{1}{1},\\stackrel{2}{1},\\stackrel{3}{1},\\stackrel{\\color{red}{4}}{2},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{8}{1},\\stackrel{9}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{14}{1},\\stackrel{15}{1},\\stackrel{16}{1},\\stackrel{\\color{red}{17}}{2},\\stackrel{18}{1},\\stackrel{19}{1},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{22}{1},\\stackrel{23}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{red}{27}}{2},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{30}{1},\\stackrel{31}{1},\\stackrel{32}{1}),\n\\end{align*}\n\\[\nA=\\{2,3,8,9,14,15,18,19,22,23,30,31\\}\\in\\cP^{\\text{eo}}([32]^1_{[2,1,3;4,1,1,2;3]}),\n\\]\nand\n\\[\nB=\\{4,17,27\\}\\subset[32]^{>1}_{[2,1,3;4,1,1,2;3]}=\\{4,6,12,17,25,27,29\\},\n\\]\nwe have\n\\begin{align*}\n&[2,1,3;4,1,1,2;3]_{(-\\{2,3,8,9,14,15,18,19,22,23,30,31\\})}=[1,1,2;2,1,1,1;2]\\\\\n&=(\\stackrel{1}{1},\\stackrel{\\color{red}{4}}{2},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{16}{1},\\stackrel{\\color{red}{17}}{2},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{red}{27}}{2},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{32}{1})\n\\end{align*}\nand\n\\begin{align*}\n&[2,1,3;4,1,1,2;3]_{(-\\{2,3,8,9,14,15,18,19,22,23,30,31\\})}-\\bdelta_{\\{4,17,27\\}}\\\\\n&=[1+1,2;1+1,1;2+2]=[2,2;2,1;4]\\\\\n&=(\\stackrel{1}{1},\\stackrel{\\color{blue}{4}}{1},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{16}{1},\\stackrel{\\color{blue}{17}}{1},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{blue}{27}}{1},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{32}{1}).\n\\end{align*}\nTherefore, for any $A, B$ with $\\#A+\\#B\\geq 1$, the induction hypothesis implies that there exist words $v_1, v_2$ in $\\cH^{2,3}$ such that $\\bc_{(\\iota_1(-A))}-\\bdelta_B$ corresponds to $v_1\\tau(v_2)$, and we have\n\\[\n\\fD(\\bc_{(-A)}-\\bdelta_B)-\\fD(\\bc_{(-\\iota_1(A))}-\\bdelta_B)z_2=\\cD(v_1z_2\\tau(v_2))-\\cD(v_1\\tau(v_2))z_2=v_1z_2\\star v_2-(v_1\\star v_2)z_2=0.\n\\]\nSince we are in the case $c>1$, we can write $w_1=wz_2$ with a certain word $w$ in $\\cH^{2,3}$.\nThen we obtain\n\\begin{align*}\n\\fD(\\bc)&=\\fD(\\bc_{(-\\{2l_r+2c-3,2l_r+2c-2\\})})z_2=\\fD([a_1,\\dots, a_r;b_1,\\dots, b_s;c-1])z_2=\\cD(w\\tau(w_2))z_2\\\\\n&=(w\\star w_2)z_2=wz_2\\star w_2=w_1\\star w_2.\n\\end{align*}",
    "post_theorem_intro_text_len": 7570,
    "post_theorem_intro_text": "Their proof can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation} later formulated in \\cite{HiroseSato2019B}.\n\nHirose and Sato gave a further generalization in \\cite{HiroseSato2022+}, and we set up notation in order to state their result.\nLet $\\mathcal{H}\\coloneqq\\mathbb{Z}\\langle x,y\\rangle$ be the non-commutative polynomial ring over $\\mathbb{Z}$ in two variables $x$ and $y$.\nLet $z_k\\coloneqq yx^{k-1}$ for each positive integer $k$.\nSubrings $\\mathcal{H}^{2,3}\\subset\\mathcal{H}^0\\subset\\mathcal{H}$ are defined by $\\mathcal{H}^{2,3}\\coloneqq\\mathbb{Z}\\langle z_2,z_3\\rangle$ and $\\mathcal{H}^0\\coloneqq\\mathbb{Z}+y\\mathcal{H} x$, respectively.\nA $\\mathbb{Z}$-linear map $Z\\colon\\mathcal{H}^0\\to\\mathbb{R}$ is defined by $Z(z_{k_1}\\cdots z_{k_r})=\\zeta(k_1,\\dots, k_r)$ and $Z(1)=1$.\nA product $\\star$ on $\\mathcal{H}^{2,3}$ is defined inductively by \n\\begin{itemize}[leftmargin=2.5em]\n\\item $w\\star 1=1\\star w=w$,\n\\item $wz_2\\star w'=w\\star w'z_2=(w\\star w')z_2$,\n\\item $wz_3\\star w'z_3=(w\\star w'z_3)z_3+(wz_3\\star w')z_3+(w\\star w')z_2^3$\n\\end{itemize}\nfor any words $w$ and $w'$ in $\\mathcal{H}^{2,3}$, and then extended by $\\mathbb{Z}$-bilinearity.\nLet $\\tau$ be an anti-automorphism on $\\mathcal{H}$ defined by $\\tau(x)=y$ and $\\tau(y)=x$.\nUsing this notation, we can now state the following theorem.\n\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\mathcal{H}^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}\nBy specializing Theorem~\\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$, we recover Theorem~\\ref{thm:HiroseSato1}.\nHirose and Sato derived Theorem~\\ref{thm:HiroseSato2} from their \\emph{block shuffle identity} (\\cite[Theorem~10]{HiroseSato2022+}).\n\nHoffman conjectured in \\cite{Hoffman1997} that any multiple zeta value can be expressed as a $\\mathbb{Q}$-linear combination of the elements of\n\\[\n\\{\\zeta(k_1,\\dots, k_r) \\mid r\\in\\ZZ_{>0}, k_1,\\dots, k_r\\in\\{2,3\\}\\}.\n\\]\nThis conjecture was proved by Brown in \\cite{Brown2012}.\nAlthough the linear independence of these elements remains open, they are called the \\emph{Hoffman basis}.\nTheorem~\\ref{thm:HiroseSato2} provides a family for which the expansion in the Hoffman basis can be described explicitly, and, even more remarkably, it shows that the expansion can be taken with \\emph{integer coefficients}.\nIn general, the coefficients are not integral; for instance,\n\\[\n\\zeta(3,1,4)=-\\frac{2048}{4125}\\zeta(2,2,2,2)-\\frac{17}{275}\\zeta(2,3,3)+\\frac{2}{275}\\zeta(3,2,3)+\\frac{113}{275}\\zeta(3,3,2).\n\\]\n\nIn contrast, Hirose, Maesaka, Seki, and Watanabe recently proved in \\cite{HiroseMaesakaSekiWatanabe2025+}, together with an explicit algorithm, that any multiple zeta value can be expressed as a $\\mathbb{Z}$-linear combination of\n\\[\n\\{\\zeta(k_1,\\dots, k_r) \\mid r\\in\\ZZ_{>0}, k_1,\\dots, k_r\\geq 2\\}.\n\\]\nTheir explicit expansions give rise to a family of relations among MZVs, which we refer to as the \\emph{drop 1 relation}.\nSince their expansions have integer coefficients, it is natural to ask whether Theorem~\\ref{thm:HiroseSato2} can be viewed as an expansion arising from the drop 1 relation.\nHowever, in general, expansions coming from the drop 1 relation involve MZVs with entries greater than $3$, so it is not obvious whether the expansion in Theorem~\\ref{thm:HiroseSato2}, in which only 2 and 3 appear, can be obtained from the drop 1 relation.\nThe main result of this paper is that this is indeed the case.\n\nA subring $\\mathcal{H}^{\\geq 2}\\subset\\mathcal{H}^0$ is defined by $\\mathcal{H}^{\\geq 2}\\coloneqq\\mathbb{Z}+z_2\\mathbb{Z}\\langle x,z_2\\rangle=\\mathbb{Z}\\langle z_2, z_3, z_4,\\dots\\rangle$.\nA linear map $\\mathcal{D}\\colon\\mathcal{H}^0\\to\\mathcal{H}^{\\geq 2}$, called the \\emph{drop 1 operator}, is defined in Definition~\\ref{def:drop1operator}, and its properties are summarized in Theorem~\\ref{thm:drop1}.\n\\begin{theorem}\\label{thm:main}\nFor $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\n\\mathcal{D}(w_1\\tau(w_2))=w_1\\star w_2.\n\\]\n\\end{theorem}\nNot only does this theorem yield a new proof of Theorem~\\ref{thm:HiroseSato2}, but it also implies, by Theorem~\\ref{thm:drop1}~\\ref{it:HMSW1}, that the same relations hold for \\emph{multiple zeta diamond values}.\nWe omit the definition of $\\zeta^{\\diamondsuit}(k_1,\\dots, k_r)$ here; see \\cite[Definition~2.3]{HiroseMaesakaSekiWatanabe2025+}.\nA $\\mathbb{Z}$-linear map $Z^{\\diamondsuit}\\colon\\mathcal{H}^0\\to\\mathbb{Q}^{\\mathbb{N}}$ is defined by $Z^{\\diamondsuit}(z_{k_1}\\cdots z_{k_r})=\\zeta^{\\diamondsuit}(k_1,\\dots, k_r)$ and $Z^{\\diamondsuit}(1)=1$.\n\n\\begin{corollary}\\label{cor:diamond-lift}\nFor $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\nZ^{\\diamondsuit}(w_1\\tau(w_2))=Z^{\\diamondsuit}(w_1\\star w_2).\n\\]\n\\end{corollary}\nFor example, the relation corresponding to \\eqref{eq:Hoffman-conj}, \n\\[\n\\zeta^{\\diamondsuit}(3,\\{2\\}^{c-1},1,2)=\\zeta^{\\diamondsuit}(\\{2\\}^{c+2})+2\\zeta^{\\diamondsuit}(3,3,\\{2\\}^{c-1})\n\\]\nmeans that, for any positive integer $N$,\n\\begin{align*}\n&\\sum_{0<n_1<n_2<\\cdots<n_{c}<n_{c+1}<n_{c+2}<N}\\frac{1}{n_1^3n_2^2\\cdots n_{c}^2n_{c+1}^{}n_{c+2}^2}\\\\\n&+\\sum_{0<n_1<n_2<\\cdots<n_{c}<n_{c+1}\\leq n_{c+2}<N}\\frac{1}{n_1^3n_2^2\\cdots n_{c}^2(N-n_{c+1}^{})n_{c+2}^2}\\\\\n&=\\sum_{0<n_1<\\cdots < n_{c+2}<N}\\frac{1}{n_1^2\\cdots n_{c+2}^2}+2\\sum_{0<n_1<n_2<n_3<\\cdots<n_{c+1}<N}\\frac{1}{n_1^3n_2^3n_3^2\\cdots n_{c+1}^2}\n\\end{align*}\nholds.\nLetting $N\\to\\infty$ yields \\eqref{eq:Hoffman-conj}.\nOn the other hand, if we take $N=p$ for a prime $p$ and take modulo $p$, then we have the following relation among \\emph{finite multiple zeta values} (FMZVs):\n\\[\n\\zeta^{}_{\\mathcal{A}}(3,\\{2\\}^{c-1},3)+2\\zeta^{}_{\\mathcal{A}}(3,3,\\{2\\}^{c-1})=0.\n\\]\nHere, for positive integers $k_1,\\dots, k_r$, the FMZV $\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)$ is defined as an element of $\\mathcal{A}\\coloneqq\\left.\\prod_p\\mathbb{Z}/p\\mathbb{Z}\\right/\\bigoplus_p\\mathbb{Z}/p\\mathbb{Z}$ by\n\\[\n\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)\\coloneqq (\\zeta^{}_p(k_1,\\dots,k_r)\\bmod{p})_p,\n\\]\nwhere $p$ runs over all prime numbers, and\n\\[\n\\zeta^{}_p(k_1,\\dots,k_r)\\coloneqq\\sum_{0<n_1<\\cdots<n_r<p}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nThis definition is due to Kaneko and Zagier (\\cite{KanekoZagier2025+}).\nWe have used a well-known formula: $\\zeta^{}_{\\mathcal{A}}(\\{2\\}^{c+2})=0$.\n\nMore generally, applying the same procedure to Corollary~\\ref{cor:diamond-lift} (see Section~\\ref{sec:finite} for more details), we have the following.\nFor a word $w=z_{k_1}\\cdots z_{k_r}$, we set $\\overleftarrow{w}\\coloneqq z_{k_r}\\cdots z_{k_1}$, and $\\mathrm{wt}(w)$ denotes the total degree of $w$, that is, $\\mathrm{wt}(w)=k_1+\\cdots + k_r$.\nA $\\mathbb{Z}$-linear map $Z_{\\mathcal{A}}\\colon \\mathbb{Z}+y\\mathcal{H}=\\mathbb{Z}\\langle z_1,z_2,\\dots\\rangle\\to\\mathcal{A}$ is defined by $Z_{\\mathcal{A}}(z_{k_1}\\cdots z_{k_r})=\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)$ and $Z_{\\mathcal{A}}(1)=1$.\n\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\n(-1)^{\\mathrm{wt}(w_2)}Z_{\\mathcal{A}}(w_1\\overleftarrow{w_2})=Z_{\\mathcal{A}}(w_1\\star w_2).\n\\]\n\\end{corollary}\nThis is a family of relations among FMZVs in which only 2 and 3 occur as entries.\nSince the shuffle relation for FMZVs $(-1)^{\\mathrm{wt}(w_2)}Z_{\\mathcal{A}}(w_1\\overleftarrow{w_2})=Z_{\\mathcal{A}}(w_1\\sh w_2)$ holds (see \\cite{KanekoZagier2025+,Ono2017}), Corollary~\\ref{cor:FMZV} can also be rewritten as\n\\[\nZ_{\\mathcal{A}}(w_1\\sh w_2-w_1\\star w_2)=0\n\\]\nwhich is of the form of a certain double shuffle relation.",
    "sketch": "Their proof (of Theorem~\\ref{thm:HiroseSato1}) “can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation}.”\n\nThe text then indicates that Theorem~\\ref{thm:HiroseSato1} is recovered by “specializing Theorem~\\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$,” and that Hirose--Sato “derived Theorem~\\ref{thm:HiroseSato2} from their \\emph{block shuffle identity}.”",
    "expanded_sketch": "Their proof (of the main theorem) “can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation}.”\n\nThe text then indicates that the main theorem is recovered by specializing the following theorem to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$.\n\n\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\cH^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}\n\nIt is also stated that Hirose--Sato “derived the theorem above from their \\emph{block shuffle identity}.”",
    "expanded_theorem": "[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*},",
    "theorem_type": [
      "Universal",
      "Equivalence"
    ],
    "mcq": {
      "question": "For positive integers $k_1,\\dots,k_r$ with $k_r\\ge 2$, the multiple zeta value is defined by\n\\[\n\\zeta(k_1,\\dots,k_r)=\\sum_{0<n_1<\\cdots<n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nFor an integer $n\\ge 0$, let $\\{2\\}^n$ denote the string consisting of $n$ copies of $2$ (so $\\{2\\}^0$ is the empty string). Which identity holds for every choice of positive integers $a,b,c$?",
      "correct_choice": {
        "label": "A",
        "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+2\\,\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1}).\n\\]"
        },
        {
          "label": "C",
          "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+R_{a,b,c},\n\\]\nwhere\n\\[\nR_{a,b,c}=\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"
        },
        {
          "label": "D",
          "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{c-1},3,\\{2\\}^{a-1}).\n\\]"
        },
        {
          "label": "E",
          "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c+1})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "B"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "other",
          "tampered_component": "symmetrized two-term contribution collapsed to a doubled single term",
          "template_used": "wildcard"
        },
        {
          "label": "C",
          "sketch_hook_type": "other",
          "tampered_component": "explicit decomposition of the remainder into the two displayed zeta terms",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "precise placement of the blocks in the second 3-term on the right-hand side",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "total weight balance between the left-hand side and the pure even zeta term",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem only gives the definition of multiple zeta values and notation; it does not reveal or strongly hint at the specific identity in choice A."
      },
      "TAS": {
        "score": 2,
        "justification": "The item is not a direct restatement of a theorem from the stem. The student must choose among several structurally similar candidate identities."
      },
      "GPS": {
        "score": 1,
        "justification": "There is some reasoning/knowledge required to distinguish the precise placement, symmetry, and signs in A/B/D/E. However, choice C is a weak existential statement that is effectively easier to endorse without knowing the exact identity, which reduces genuine generative pressure."
      },
      "DQS": {
        "score": 1,
        "justification": "B, D, and E are plausible theorem-variant distractors involving coefficient, sign, and block-order mistakes. But C is a poor distractor because it is a weaker true statement, making the item ambiguous rather than cleanly discriminative."
      },
      "total_score": 6,
      "overall_assessment": "A mathematically sophisticated and non-leaky item, but weakened by an ambiguous weaker-true distractor (C), which reduces its effectiveness as a single-answer reasoning question."
    }
  },
  {
    "id": "2512.24115v1",
    "paper_link": "http://arxiv.org/abs/2512.24115v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "proposition",
      "content": "If $G$ is a sun graph on $2n\\ge 6$ vertices, then $\\displaystyle \\zeta(G)=\\sum_{i=0}^n{n\\choose i}=2^n. $",
      "start_pos": 8748,
      "end_pos": 8886,
      "label": null
    },
    "ref_dict": {
      "Ex": "\\begin{example}{\\label{Ex}}\n\\end{example}",
      "fig1": "\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five"
    },
    "pre_theorem_intro_text_len": 4834,
    "pre_theorem_intro_text": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover}  the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}. \n\nDominating sets, in particular $\\gamma$-sets, have been extensively researched. We recommend to the reader Haynes et al.'s book \\cite{Hay2} and several other research works \\cite{FB, BR, EC, GD, MH, FH, Hay1, ML} on variants and applications of dominating sets.\nIn this literature we found various versions of $\\gamma$-set such as the perfect dominating set, the double dominating set, the strong dominating set, and the restrained dominating set. In general, each such type of $\\gamma$-set has some specified properties. Here, we list five of the most commonly studied $\\gamma$-sets along with their properties. Later in Example \\ref{Ex}, we illustrate each such $\\gamma$-set.\n\n\\begin{enumerate}\n\t\\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n\t\\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n\t\\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n\t\\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nBecause a given graph may have multiple $\\gamma$-sets, this article addresses a natural but fundamental question:\\textit{ How many  $\\gamma$-sets does a given graph have?} To answer this question, we introduce the notion of \\textbf{dominion}.\n\nThe \\textit{dominion (number)} of  a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words, \n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a  complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{tikzpicture}[scale=1]\n\t\\clip(0,-1) rectangle (4,4);\n\t\\draw [line width=2pt] (2.62,1)-- (2,2);\n\t\\draw [line width=2pt] (2,2)-- (1.98,2.84);\n\t\\draw [line width=2pt] (2.62,1)-- (3,0.34);\n\t\\draw [line width=2pt] (2,2)-- (1.48,1.02);\n\t\\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n\t\\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n\t\\begin{scriptsize}\n\t\\draw [fill=black] (2,2) circle (2.5pt);\n\t\\draw[color=black] (2.2,2.1) node {$u$};\n\t\\draw [fill=black] (2.62,1) circle (2.5pt);\n\t\\draw[color=black] (2.92,1) node {$v$};\n\t\\draw [fill=black] (1.98,2.84) circle (2.5pt);\n\t\\draw[color=black] (1.94,3.1) node {$x$};\n\t\\draw [fill=black] (3,0.34) circle (2.5pt);\n\t\\draw[color=black] (3.2,0.15) node {$y$};\n\t\\draw [fill=black] (1.48,1.02) circle (2.5pt);\n\t\\draw[color=black] (1.14,1) node {$w$};\n\t\\draw [fill=black] (1.06,0.34) circle (2.5pt);\n\t\\draw[color=black] (0.9,0.15) node {$z$};\n\t\\end{scriptsize}\n\t\\end{tikzpicture}\n\t\\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five commonly studied, as  previously mentioned.\n\n\\begin{itemize}\n\n\t\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n\t\\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n\t \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n\t\\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are  $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.",
    "context": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover} the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}.\n\n\\begin{enumerate}\n \\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n \\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n \\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n \\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nThe \\textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words,\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\begin{itemize}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five",
    "full_context": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover} the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}.\n\n\\begin{enumerate}\n \\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n \\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n \\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n \\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nThe \\textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words,\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\begin{itemize}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\ds \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\begin{proof}\nFor each vertex on the cycle, and for each dominating set $S\\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives $\\ds \\sum_{i=0}^n{n\\choose i}$. Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$. This gives $\\zeta(G)=2^n$.\n\\end{proof}\n\n\\begin{claim}{\\label{c4}}\n When $n=3k, k\\ge 1$, $\\zeta(P_n)=1$ and the unique $\\gamma$-set in $P_n$ contains $v_{n-1}$. Moreover, when $n=3k+2, k\\ge 1$, there is only one $\\gamma$-set in $P_n$ containing $v_{n}$. \n\\end{claim}\n\\begin{proof}\n When $n=3k, k\\ge 1$, we have $\\zeta(P_n)=k$ and the conclusion follows by the pigeon-hold argument alluded in Claim \\ref{c1}. Each of the sets of $3$ consecutive vertices $\\{v_{3r+1},v_{3r+2},v_{3r+3}\\}$, $r=0,\\ldots,k-1$, must contain at least one vertex of any $\\gamma$-set $S$, and since $\\gamma=k$, each of those sets must contain exactly one element of $S$. Since $S$ is dominating, the element of $S$ in $\\{v_1,v_2,v_3\\}$ must be $v_1$ or $v_2$. If $v_1$ then the ``next'' element of $S$ must be $v_4$, which forces $S=\\{v_1,v_4,\\ldots,v_{3k-2}\\}$, leaving $v_{3k}$ uncovered. Therefore, the first element of $S$ must be $v_2$, which forces $S=\\{v_2,v_5,\\ldots,v_{3k-1}, v_{3k+2}\\}$.\n\n\\begin{proposition}{\\label{Prop2}}\n For any connected graph $G$ on $n$ vertices, $1\\le \\zeta(G)\\le {n\\choose \\gamma(G)}$.\n\n\\end{proof}\n\\begin{corollary}\nFor any connected graph $G$ with $\\gamma(G)=1$, $\\zeta(\\underbrace{G\\vee \\ldots \\vee G}_r)=r\\zeta(G)$, $r\\ge 1$.\n\\end{corollary}\n\\begin{proof}\nFrom Part (a) of Theorem \\ref{T4} (when $\\gamma(G_1)=1=\\gamma(G_2)$), the result follows by induction on $r\\ge 1$.\n\\end{proof}\nWe note that in the case when $G=K_1$, it is clear that $\\underbrace{G\\vee \\ldots \\vee G}_r:=K_r$ and $\\zeta(K_r)=r=r\\zeta(K_1)$. Here, we present a more general result for the special case when $G=\\overline{K_{m}}$, the complement of a complete graph on $m\\ge 1$ vertices. Note that $\\overline{K_{m_1}} \\vee \\ldots \\vee \\overline{K_{m_k}}$ is a complete $k$-partite graph often written as $K(m_1,m_2,\\ldots, m_k)$, with $m_i\\ge 1$ and $k\\ge 2$.\n\n\\end{theorem}\n\\begin{proof}\nLet $V_i$ denote each disjoint partite (vertex) set of $G$, with $|V_i|=m_i$, $i=1,\\ldots, k$. If, for some $j=1,\\ldots, r$, $m_j=1$ and $m_i>1$ for $i>r$, then it is clear that $\\gamma(G)=1$ and the $\\gamma$-sets in $G$ are the singleton subsets of $V_1\\cup\\ldots \\cup V_r$. It follows that $\\zeta(G)=r$. \nAssume $m_i>1$, for $i=1,\\ldots, k$. It is clear that $\\gamma(G)=2$. Any vertex $v\\in V_i$ covers all vertices $v'\\in V(G)\\setminus V_i$ except those in $V_i$. Thus, for each $v\\in V_i$ and $v'\\in V_j$, $i\\ne j$, $\\{v,v'\\}$ is a $\\gamma$-set of $G$. It follows that there are $\\ds \\sum_{i\\ne j}^{{k}}m_im_j$ total such sets. Further, if $|V_i|=2$ for $i=1,\\ldots, r$, then each such sets, $V_i$, is an additional $\\gamma$-set, giving the result.\n\\end{proof}",
    "post_theorem_intro_text_len": 511,
    "post_theorem_intro_text": "\\begin{proof}\nFor each vertex on the cycle, and for each dominating set $S\\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives  $\\displaystyle \\sum_{i=0}^n{n\\choose i}$.  Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$. This gives $\\zeta(G)=2^n$.\n\\end{proof}",
    "sketch": "For each vertex on the cycle and each dominating set $S\\subseteq V(G)$, \"either the vertex or its leaf neighbor must be in $S$.\" Hence all $\\gamma$-sets can be formed by choosing a subset of the $n$ leaves, and then including \"the cycle neighbors of unchosen leafs,\" yielding $\\sum_{i=0}^n \\binom{n}{i}$. Conversely, \"every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$,\" so $\\zeta(G)=\\sum_{i=0}^n \\binom{n}{i}=2^n$.",
    "expanded_sketch": "No expanded sketch found.",
    "expanded_theorem": "If $G$ is a sun graph on $2n\\ge 6$ vertices, then $\\displaystyle \\zeta(G)=\\sum_{i=0}^n{n\\choose i}=2^n. $,",
    "theorem_type": [
      "Implication",
      "Inequality or Bound"
    ],
    "mcq": {
      "question": "Let b3(G) denote the domination number of a graph G, i.e. the minimum size of a dominating set, and let b6(G) be the number of b3-sets of G, where a b3-set is a dominating set of cardinality b3(G). A sun graph on 2n vertices (with 2n \\ge 6) is the graph obtained from a cycle C_n by attaching exactly one pendant leaf to each cycle vertex. Which quantitative statement holds for b6(G)?",
      "correct_choice": {
        "label": "A",
        "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\sum_{i=0}^{n}\\binom{n}{i}=2^n\\)."
      },
      "choices": [
        {
          "label": "B",
          "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\binom{2n}{n}\\)."
        },
        {
          "label": "C",
          "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)\\ge 2^n\\)."
        },
        {
          "label": "D",
          "text": "If G is a sun graph on 2n \\ge 6 vertices, then every \\(\\gamma\\)-set is obtained by choosing exactly one vertex from each leaf-cycle pair, and hence \\(\\zeta(G)=n\\)."
        },
        {
          "label": "E",
          "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\sum_{i=1}^{n}\\binom{n}{i}=2^n-1\\)."
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "D"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "independent binary choices at n cycle-leaf pairs replaced by unrestricted n-subset count from 2n vertices",
          "template_used": "stronger_trap"
        },
        {
          "label": "C",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "replaced exact enumeration by the weaker lower bound only",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "other",
          "tampered_component": "local one-from-each-pair structure kept but global count collapsed from 2 choices per pair to one canonical choice",
          "template_used": "wildcard"
        },
        {
          "label": "E",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "excluded one valid endpoint case in the binomial sum",
          "template_used": "boundary_range"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem defines the sun graph and the counting function zeta(G), but it does not explicitly state or trivially reveal the value 2^n. The correct answer must still be derived from the graph structure."
      },
      "TAS": {
        "score": 2,
        "justification": "This is not a mere restatement of a definition from the stem. The solver must determine gamma(G) and then characterize all minimum dominating sets, so the conclusion is not tautological."
      },
      "GPS": {
        "score": 1,
        "justification": "Some genuine reasoning is required: one must see that every leaf-cycle pair forces at least one selected vertex, conclude gamma(G)=n, and then count the binary choices. However, once that structure is noticed, the answer becomes fairly direct rather than deeply generative."
      },
      "DQS": {
        "score": 1,
        "justification": "Several distractors reflect plausible counting mistakes (off-by-one, overcounting, confusing all subsets of a given size with minimum dominating sets). However, choice C is actually true as a weaker statement, which makes the option set ambiguous in a single-answer MCQ and reduces distractor quality."
      },
      "total_score": 6,
      "overall_assessment": "A mathematically meaningful item with no answer leakage and a non-tautological setup, but its discriminative quality is weakened by the presence of a weaker true distractor, making the single-correct format imperfect."
    }
  },
  {
    "id": "2512.24158v1",
    "paper_link": "http://arxiv.org/abs/2512.24158v1",
    "theorems_cnt": 2,
    "theorem": {
      "env_name": "theorem",
      "content": "\\label{theorem:main}\n\tLet $\\lambda\\in DP_n$ and $(\\rho_\\lambda,V_\\lambda)$ be the corresponding irreducible spin representation of $\\tilde S_n$.\n\tLet $g\\in \\tilde S_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and let $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_\\lambda(g)$ of $\\rho_\\lambda(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 8$, $(\\lambda,\\mu)$ is one of the pairs listed in Table~\\ref{table:exceptional_cases_1_8}.\n\n\tFor $n>8$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $\\lambda=(n-1,1)$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}",
      "start_pos": 7272,
      "end_pos": 8778,
      "label": "theorem:main"
    },
    "ref_dict": {
      "table:exceptional_cases_1_8": "\\begin{table}[h] \n\\caption{Exceptions for $1\\leqslant n\\leqslant 8$ in $\\tilde S_n$}\n\\label{table:exceptional_cases_1_8}\n    \\centering\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{|c|cc|}\n    \\hline\n        & $(\\mu, p_\\mu^\\lambda(x))$ &  \\\\\n        \\hline\n        $n=2$  &   &  \\\\\n        $\\lambda=(2)^\\pm$ & $((2)^+, \\frac{x^2+1}{x\\pm i})$, & $((2)^-, \\frac{x^2+1}{x\\mp i})$ \\\\\n        \\hline\n        $n=3$  &   &  \\\\\n        $\\lambda=(3)$ & $((3)^+, \\frac{x^3+1}{x+1})$, & $((3)^-, \\frac{x^3-1}{x-1})$ \\\\\n        $\\lambda=(2,1)^\\pm$ & $((2,1)^+, \\frac{x^2+1}{x\\pm i})$, & $((2,1)^-, \\frac{x^2+1}{x\\pm i})$ \\\\\n        & $((3)^+, \\frac{x^3+1}{x^2-x+1})$ & $((3)^-, \\frac{x^3-1}{x^2+x+1})$ \\\\\n        \\hline\n        $n=4$  &   &  \\\\\n        $\\lambda=(4)^\\pm$ & $((3,1)^+, \\frac{x^3+1}{x+1})$, & $((3,1)^-, \\frac{x^3-1}{x-1})$ \\\\\n        & $((4)^+, x^2\\mp \\sqrt{2}x+1)$, & $((4)^-, x^2\\pm \\sqrt{2}x+1)$ \\\\ \\hline\n        $n=5$  &   &   \\\\\n        $\\lambda=(5)$ & $((3,1^2)^+, \\frac{x^3+1}{x+1})$, & $((3,1^2)^-, \\frac{x^3-1}{x-1})$, \\\\\n        & $((3,2)^\\pm, \\frac{x^6+1}{x^2+1})$ & $((5)^\\pm, \\frac{x^5\\pm1}{x\\pm1})$ \\\\\n        $\\lambda=(3,2)^\\pm$ & $((5)^+, \\frac{x^5+1}{x+1})$, & $((5)^-, \\frac{x^5-1}{x-1})$ \\\\ & $((3,2)^+, x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)$ & $((3,2)^-, x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)$.  \\\\ \\hline\n        $n=6$  &   &   \\\\\n        $\\lambda=(6)^\\pm$ & $((6)^+, x^4\\pm\\sqrt{-3}x^3-2x^2\\pm\\sqrt{-3}x+1)$,\n        & $((6)^-, x^4\\mp\\sqrt{-3}x^3-2x^2\\mp\\sqrt{-3}x+1)$, \\\\\n        & $((3,1^3)^+, \\frac{x^3+1}{x+1})$, & $((3,1^3)^-, \\frac{x^3-1}{x-1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n        & $((3,2,1)^\\pm, \\frac{x^6+1}{x^2+1})$. & \\\\\n        $\\lambda=(3,2,1)^\\pm$ & $((3,2,1)^+, x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)$, & $((3,2,1)^-, x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)$, \\\\\n        & $((3,3)^+, \\frac{x^3-1}{x-1})$, & $((3,3)^-, \\frac{x^3+1}{x+1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n         & $((6)^\\pm, \\frac{x^6-1}{x^2-1})$. & \\\\ \\hline\n\t    $n=7$ & & \\\\\n\t    $\\lambda=(7)$ & $((3,1^4)^\\pm, \\frac{x^3\\pm1}{x\\pm1})$, & $((5,1^2)^\\pm, \\frac{x^5\\pm1}{x\\pm5})$,  \\\\\n\t      & $((3,2,1^2), \\frac{x^6+1}{x^2+1})$, & $((3,2,2), \\frac{x^6+1}{x^2+1})$, \\\\\n\t      & $((4,3)^\\pm, \\frac{x^{12}+1}{x^4+1})$, & $((5,1^2)^\\pm, \\frac{x^{5}\\pm1}{x\\pm1})$, \\\\\n\t      & $((5,2)^\\pm, \\frac{x^{10}+1}{x^2+1})$. & \\\\\n\t    \\hline\n\t    $n=8$ & & \\\\\n\t    $\\lambda=(8)^\\pm$ & $((5,1^3)^+, \\frac{x^{5}+1}{x+1})$, & $((5,1^3)^-, \\frac{x^5-1}{x-1})$,  \\\\\n\t    & $((3,1^5)^+, \\frac{x^3+1}{x+1})$, & $((3,1^5)^-, \\frac{x^3-1}{x-1})$,  \\\\\n\t    & $((5,3)^+, \\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})$, & $((5,3)^-, \\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})$,  \\\\\n        & $((8)^+, \\frac{x^{8}-1}{x\\pm1})$, & $((8)^-, \\frac{x^{8}-1}{x\\mp1})$,  \\\\\n        & $((5,2,1)^\\pm, \\frac{x^{10}+1}{x^2+1})$, & $((3,2^2,1), \\frac{x^6+1}{x^2+1})$, \\\\\n        & $((3,2,1^3), \\frac{x^{6}+1}{x^2+1})$, & $((4,3,1)^\\pm, \\frac{x^{12}+1}{x^4+1})$. \\\\\n        $\\lambda=(7,1)$ & $((5,3)^\\pm, \\frac{x^{15}\\mp1}{x\\mp1})$. &   \\\\   \\hline\n    \\end{tabular}\n\\end{table}",
      "table:exceptional_cases_1_8_alt": "\\begin{table}[h] \n\\caption{Exceptions for $1\\leqslant n\\leqslant 8$ in $\\tilde A_n$}\n\\label{table:exceptional_cases_1_8_alt}\n    \\centering\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{|c|cc|}\n    \\hline\n        & $(\\mu, p_\\mu^\\lambda(x))$ &  \\\\\n        \\hline\n        $n=3$  &   &  \\\\\n        $\\lambda=(3)^\\pm$ & $((3)^+, x-\\omega_3^{\\pm 1})$, & $((3)^-, x-\\omega_3^{\\mp 1})$ \\\\\n        $\\lambda=(2,1)^\\pm$ \n        & $((3)^+, \\frac{x^3+1}{x^2-x+1})$ & $((3)^-, \\frac{x^3-1}{x^2+x+1})$ \\\\\n        \\hline\n        $n=4$  &   &  \\\\\n        $\\lambda=(4)^\\pm$ & $((3,1)^+, \\frac{x^3+1}{x+1})$, & $((3,1)^-, \\frac{x^3-1}{x-1})$ \\\\\n\t\t$\\lambda=(3,1)^\\pm$ & $((3,1)^+, \\frac{x^3+1}{x+1})$, & $((3,1)^-, \\frac{x^3-1}{x-1})$\\\\\n       \\hline\n        $n=5$  &   &   \\\\\n        $\\lambda=(5)^\\pm$ & $((5)^+, (x \\mp \\omega_5)(x \\mp \\omega_5^2))$, & $((5)^-,(x \\mp \\omega_5^3)(x \\mp \\omega_5^4) )$, \\\\\n\t\t& $((3,1^2)^+, \\frac{x^3+1}{x+1})$, & $((3,1^2)^-, \\frac{x^3-1}{x-1})$, \\\\\n        \\hline\n        $n=6$  &   &   \\\\ $\\lambda=(6)^\\pm$\n        & $((3,1^3)^+, \\frac{x^3+1}{x+1})$, & $((3,1^3)^-, \\frac{x^3-1}{x-1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n        $\\lambda=(3,2,1)^\\pm$ \n        & $((3,3)^+, \\frac{x^3-1}{x-1})$, & $((3,3)^-, \\frac{x^3+1}{x+1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n\\hline\n\t    $n=7$ & &  \\\\ $\\lambda=(7)^\\pm$\n\t\t& $((7)^+, (x\\mp1)(x\\mp \\omega_7)(x\\mp \\omega_7^2)(x\\mp \\omega_7^3))$, & $((7)^+, (x\\mp1)(x\\mp \\omega_7^4)(x\\mp \\omega_7^5)(x\\mp \\omega_7^6))$,  \\\\\n\t    & $((3,1^4)^\\pm, \\frac{x^3\\pm1}{x\\pm1})$, & $((5,1^2)^\\pm, \\frac{x^5\\pm1}{x\\pm1})$,  \\\\\n\t    & $((3,2,2), \\frac{x^6+1}{x^2+1})$, & $((5,1^2)^\\pm, \\frac{x^{5}\\pm1}{x\\pm1})$.  \\\\\n\t    \\hline\n\t    $n=8$ & & \\\\\n\t    $\\lambda=(8)^\\pm$ & $((5,1^3)^+, \\frac{x^{5}+1}{x+1})$, & $((5,1^3)^-, \\frac{x^5-1}{x-1})$,  \\\\\n\t    & $((3,1^5)^+, \\frac{x^3+1}{x+1})$, & $((3,1^5)^-, \\frac{x^3-1}{x-1})$,  \\\\\n\t    & $((5,3)^+, \\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})$, & $((5,3)^-, \\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})$,  \\\\\n        & $((3,2^2,1), \\frac{x^6+1}{x^2+1})$. & \\\\\n        $\\lambda=(7,1)$ & $((5,3)^\\pm, \\frac{x^{15}\\mp1}{x\\mp1})$. &   \\\\   \\hline\n    \\end{tabular}\n\\end{table}",
      "theorem:main_alt": "\\begin{theorem}\\label{theorem:main_alt}\n\tLet $(\\rho,V)$ be an irreducible spin representation of $\\tilde A_n$ indexed by a partition $\\lambda$.\n\tLet $g\\in \\tilde A_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_V(g)$ of $\\rho(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 10$ and $((\\rho,V),g)$ is one of the pairs listed in Tables~\\ref{table:exceptional_cases_1_8_alt} and \\ref{table:exceptional_cases_9_10_alt}.\n\n\tFor $n>10$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $V=V_{(n-1,1)}$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}\n\\end{theorem}",
      "table:exceptional_cases_9_10_alt": "\\begin{table}[h]\n\\caption{Exceptions for $n=9,10$ in $\\tilde A_n$}\n\\label{table:exceptional_cases_9_10_alt}\n    \\centering\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{|c|cc|}\n    \\hline\n        & $(\\mu, p_\\mu^\\lambda(x))$ &  \\\\\n        \\hline\n        $n=9$  & & \\\\\n\t\t$\\lambda=(9)^\\pm$ & $((9)^+, \\frac{(x^9-1)(x-1)(x-\\omega_3^{\\pm1})}{(x^3-1)})$,  & $((9)^-, \\frac{(x^9-1)(x-1)(x-\\omega_3^{\\mp1})}{(x^3-1)})$, \\\\\t\n\t\t& $((3,1^6)^\\pm, \\frac{x^3\\pm1}{x\\pm1})$,  & $((3,2,1^4), \\frac{x^6+1}{x^2+1})$, \\\\\n\t\t& $((3,2^2,1^2), \\frac{x^6+1}{x^2+1})$,  & $((3,2^3), \\frac{x^6-1}{x^2-1})$ \\\\\n\t\t& $((4,3,1^2), \\frac{x^{12}+1}{x^4+1})$, & $((4,3,2), \\frac{x^{12}+1}{x^4+1})$, \\\\\n\t\t& $((5,1^4)^\\pm, \\frac{x^{5}\\pm1}{x\\pm1})$, & $((5,2,1^2), \\frac{x^{10}+1}{x^2+1})$,  \\\\\n\t\t& $((5,2^2), \\frac{x^{10}+1}{x^2+1})$, & $((5,3,1)^\\pm, \\frac{(x^{15}\\mp1)(x\\mp1)}{(x^5\\mp1)(x^3\\mp1)})$,   \\\\\n\t\t& $((5,4)^+, \\prod_{i=1,3,5,7} (x-\\omega_8^i \\omega_5^{\\pm 1})(x-\\omega_8^i \\omega_5^{\\pm 2}))$, & $((5,4)^-, \\prod_{i=1,3,5,7} (x-\\omega_8^i \\omega_5^{\\mp 1})(x-\\omega_8^i \\omega_5^{\\mp 2}))$. \\\\\n\t\t$\\lambda=(8,1)^\\pm$ & $((5,3,1)^+, \\frac{x^{15}-1}{x-1})$,  & $((5,3,1)^-, \\frac{x^{15}+1}{x+1})$, \\\\\n\t\t\\hline\n        $n=10$  &   & \\\\\n        $\\lambda=(10)^\\pm$  & $((7,3)^+, \\frac{x^{21}+1}{x^3+1})$, & $((7,3)^-, \\frac{x^{21}-1}{x^3-1})$,  \\\\\n        & $((5,4,1)^\\pm, \\frac{x^{20}+1}{x^4+1})$, & $((5,3,2)^\\pm, \\frac{x^{30}+1}{(x^{10}+1)(x^4-x^2+1)})$, \\\\\n        & $((5,3,1^2)^+, \\frac{x^{15-1}}{(x^5-1)(x^2+x+1)})$, & $((5,3,1^2)^-, \\frac{x^{15}+1}{(x^{5}+1)(x^2-x+1)})$, \\\\\n        & $((5,2^2,1), \\frac{x^{10}+1}{x^2+1})$, & $((4,3,2,1), \\frac{x^{12}+1}{x^4+1})$, \\\\\n        & $((5,2,1^3), \\frac{x^{10}+1}{x^2+1})$, & $((4,3,1^3), \\frac{x^{12}+1}{x^4+1})$, \\\\\n        & $((5,1^5)^+, \\frac{x^5+1}{x+1})$, & $((5,1^5)^-, \\frac{x^5-1}{x-1})$, \\\\\n        & $((3,2^3,1), \\frac{x^6-1}{x^2-1})$, & $((3,2^2,1^3), \\frac{x^6+1}{x^2+1})$,  \\\\\n        & $((3,1^7)^+, \\frac{x^3+1}{x+1})$, & $((3,1^7)^-, \\frac{x^3-1}{x-1})$,  \\\\\n        & $((3,2,1^5), \\frac{x^6+1}{x^2+1})$,  & \\\\\n        $\\lambda=(9,1)^\\pm$  & $((5,3,2)^\\pm, \\frac{x^{30}+1}{x^2+1})$, & $((5,3,1^2)^\\pm, \\frac{x^{15}\\mp1}{x\\mp1})$  \\\\\n        \\hline\n\n    \\end{tabular}\n\\end{table}",
      "theorem:main": "\\begin{theorem}\\label{theorem:main}\n\tLet $\\lambda\\in DP_n$ and $(\\rho_\\lambda,V_\\lambda)$ be the corresponding irreducible spin representation of $\\tilde S_n$.\n\tLet $g\\in \\tilde S_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and let $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_\\lambda(g)$ of $\\rho_\\lambda(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 8$, $(\\lambda,\\mu)$ is one of the pairs listed in Table~\\ref{table:exceptional_cases_1_8}.\n\n\tFor $n>8$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $\\lambda=(n-1,1)$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}\n\\end{theorem}"
    },
    "pre_theorem_intro_text_len": 2602,
    "pre_theorem_intro_text": "Let $G$ denote a double cover of either the symmetric group or the alternating group.\nIn this article, we determine the minimal polynomials of elements of $G$ in its irreducible complex spin representations.\nWe find that (see Theorems~\\ref{theorem:main} and~\\ref{theorem:main_alt}), for most elements $g\\in G$, and most irreducible spin representations $\\rho$ of $G$, the minimal polynomial of $\\rho(g)$ is $x^k\\pm 1$, where $k$ is the order of $g$ modulo the center of $G$. The exceptions fall into four infinite families, and finitely many sporadic examples.\n\nThe question of finding minimal polynomials of elements in representations for a given group goes back to the famous work of Hall and Higman~\\cite{Hall_Higman}, as well as Shult~\\cite{Shult}.\nThe general problem was formulated explicitly by Tiep and Zalesskii: given an element $g$ of a finite group $G$ and an irreducible representation $\\rho$ of $g$, determine the eigenvalues of $\\rho(g)$~\\cite{Tiep_Zalesski}.\nIn his survey articles~\\cite{survey_Zalesski_algebraic_Chevalley_2009,survey_Zalesski_eigenvalue_1_2025}, Zalesskii discusses the importance of this problem and its applications to arithmetic and algebraic geometry, topology, game theory, combinatorics, group theory, etc.\nThe only groups $G$ where this problem is solved for all $g\\in G$ and every irreducible complex representation $\\rho$ of $G$ are the symmetric groups and the alternating groups, due to the work of Swanson~\\cite{Swanson} (for longest cycles in symmetric groups), Yang--Staroletov~\\cite{Yang_Staroletov} and Velmurugan~\\cite{vel_dec_2024} (for certain families of permutations in symmetric and alternating groups), Amrutha--Prasad--Velmurugan~\\cite{p2024cyclic,Inv_vectors,fpsac2024} (for the occurrence of the eigenvalue $1$ for all permutations in all representations of symmetric and alternating groups), and finally completed by Staroletov~\\cite{Staroletov_all_eigenvalues}.\n\nLet $\\tilde S_n$ and $\\tilde A_n$ denote the double covers of $S_n$ and $A_n$, respectively.\nTo each partition $\\lambda$ in $DP_n$, the partitions of $n$ with distinct parts, there is an associated irreducible spin representation $(\\rho_\\lambda,V_\\lambda)$ of $\\tilde S_n$.\nWhen $\\lambda$ has an odd number of even parts, then $V_\\lambda$ can be twisted by the sign character of $\\tilde S_n$ to obtain a non-isomorphic representation.\nIn this case, we sometimes denote $V_\\lambda$ by $V_\\lambda^+$ and its twist by $V_\\lambda^-$.\nIt suffices to determine the minimal polynomial of $g\\in G$ in $V_\\lambda$ for $\\lambda\\in DP_n$.\nWe now state our main theorem for $\\tilde S_n$:",
    "context": "Let $G$ denote a double cover of either the symmetric group or the alternating group.\nIn this article, we determine the minimal polynomials of elements of $G$ in its irreducible complex spin representations.\nWe find that (see Theorems~\\ref{theorem:main} and~\\ref{theorem:main_alt}), for most elements $g\\in G$, and most irreducible spin representations $\\rho$ of $G$, the minimal polynomial of $\\rho(g)$ is $x^k\\pm 1$, where $k$ is the order of $g$ modulo the center of $G$. The exceptions fall into four infinite families, and finitely many sporadic examples.\n\nThe question of finding minimal polynomials of elements in representations for a given group goes back to the famous work of Hall and Higman~\\cite{Hall_Higman}, as well as Shult~\\cite{Shult}.\nThe general problem was formulated explicitly by Tiep and Zalesskii: given an element $g$ of a finite group $G$ and an irreducible representation $\\rho$ of $g$, determine the eigenvalues of $\\rho(g)$~\\cite{Tiep_Zalesski}.\nIn his survey articles~\\cite{survey_Zalesski_algebraic_Chevalley_2009,survey_Zalesski_eigenvalue_1_2025}, Zalesskii discusses the importance of this problem and its applications to arithmetic and algebraic geometry, topology, game theory, combinatorics, group theory, etc.\nThe only groups $G$ where this problem is solved for all $g\\in G$ and every irreducible complex representation $\\rho$ of $G$ are the symmetric groups and the alternating groups, due to the work of Swanson~\\cite{Swanson} (for longest cycles in symmetric groups), Yang--Staroletov~\\cite{Yang_Staroletov} and Velmurugan~\\cite{vel_dec_2024} (for certain families of permutations in symmetric and alternating groups), Amrutha--Prasad--Velmurugan~\\cite{p2024cyclic,Inv_vectors,fpsac2024} (for the occurrence of the eigenvalue $1$ for all permutations in all representations of symmetric and alternating groups), and finally completed by Staroletov~\\cite{Staroletov_all_eigenvalues}.\n\nLet $\\tilde S_n$ and $\\tilde A_n$ denote the double covers of $S_n$ and $A_n$, respectively.\nTo each partition $\\lambda$ in $DP_n$, the partitions of $n$ with distinct parts, there is an associated irreducible spin representation $(\\rho_\\lambda,V_\\lambda)$ of $\\tilde S_n$.\nWhen $\\lambda$ has an odd number of even parts, then $V_\\lambda$ can be twisted by the sign character of $\\tilde S_n$ to obtain a non-isomorphic representation.\nIn this case, we sometimes denote $V_\\lambda$ by $V_\\lambda^+$ and its twist by $V_\\lambda^-$.\nIt suffices to determine the minimal polynomial of $g\\in G$ in $V_\\lambda$ for $\\lambda\\in DP_n$.\nWe now state our main theorem for $\\tilde S_n$:",
    "full_context": "Let $G$ denote a double cover of either the symmetric group or the alternating group.\nIn this article, we determine the minimal polynomials of elements of $G$ in its irreducible complex spin representations.\nWe find that (see Theorems~\\ref{theorem:main} and~\\ref{theorem:main_alt}), for most elements $g\\in G$, and most irreducible spin representations $\\rho$ of $G$, the minimal polynomial of $\\rho(g)$ is $x^k\\pm 1$, where $k$ is the order of $g$ modulo the center of $G$. The exceptions fall into four infinite families, and finitely many sporadic examples.\n\nThe question of finding minimal polynomials of elements in representations for a given group goes back to the famous work of Hall and Higman~\\cite{Hall_Higman}, as well as Shult~\\cite{Shult}.\nThe general problem was formulated explicitly by Tiep and Zalesskii: given an element $g$ of a finite group $G$ and an irreducible representation $\\rho$ of $g$, determine the eigenvalues of $\\rho(g)$~\\cite{Tiep_Zalesski}.\nIn his survey articles~\\cite{survey_Zalesski_algebraic_Chevalley_2009,survey_Zalesski_eigenvalue_1_2025}, Zalesskii discusses the importance of this problem and its applications to arithmetic and algebraic geometry, topology, game theory, combinatorics, group theory, etc.\nThe only groups $G$ where this problem is solved for all $g\\in G$ and every irreducible complex representation $\\rho$ of $G$ are the symmetric groups and the alternating groups, due to the work of Swanson~\\cite{Swanson} (for longest cycles in symmetric groups), Yang--Staroletov~\\cite{Yang_Staroletov} and Velmurugan~\\cite{vel_dec_2024} (for certain families of permutations in symmetric and alternating groups), Amrutha--Prasad--Velmurugan~\\cite{p2024cyclic,Inv_vectors,fpsac2024} (for the occurrence of the eigenvalue $1$ for all permutations in all representations of symmetric and alternating groups), and finally completed by Staroletov~\\cite{Staroletov_all_eigenvalues}.\n\nLet $\\tilde S_n$ and $\\tilde A_n$ denote the double covers of $S_n$ and $A_n$, respectively.\nTo each partition $\\lambda$ in $DP_n$, the partitions of $n$ with distinct parts, there is an associated irreducible spin representation $(\\rho_\\lambda,V_\\lambda)$ of $\\tilde S_n$.\nWhen $\\lambda$ has an odd number of even parts, then $V_\\lambda$ can be twisted by the sign character of $\\tilde S_n$ to obtain a non-isomorphic representation.\nIn this case, we sometimes denote $V_\\lambda$ by $V_\\lambda^+$ and its twist by $V_\\lambda^-$.\nIt suffices to determine the minimal polynomial of $g\\in G$ in $V_\\lambda$ for $\\lambda\\in DP_n$.\nWe now state our main theorem for $\\tilde S_n$:\n\n\\noindent For $n\\leq 10$ and $((\\rho,V),g)$ is one of the pairs listed in Tables~\\ref{table:exceptional_cases_1_8_alt} and \\ref{table:exceptional_cases_9_10_alt}.\n\nIn this section we shall prove the following theorem.\n\\begin{theorem}\n \\label{theorem:cycle_double_S_n}\n Let $(\\phi_{\\lambda}, V_\\lambda)$ be an irreducible spin representation of $\\tilde S_n$ indexed by a partition $\\lambda\\vdash n$ with distinct parts.\n Let $g=w_n$ be an element of $\\tilde S_n$ with cycle type $(n)$.\n Then the minimal polynomial $p_\\lambda(g)$ of $\\phi_{\\lambda}(g)$ has $n$ distinct roots except in the following cases:\n \\begin{enumerate}\n \\item $\\phi_{(3)}$ and $p_\\lambda(\\sigma_3^\\pm)= x^2 \\mp x+1$.\n \\item $\\phi_{(4)}^\\pm$ and $p_\\lambda(\\sigma_4^\\pm)= x^2 \\pm \\sqrt{2} x +1$.\n \\item $\\phi_{(5)}$ and $p_\\lambda(\\sigma_5^\\pm)= \\frac{x^5\\pm 1}{x\\pm 1}$.\n \\item $\\phi_{(6)}^\\pm$ and $p_\\lambda(\\sigma_6^\\epsilon)= \\frac{x^6 -1}{(x-\\eta_1)(x-\\eta_2)}$, where $\\eta_1= \\frac{1\\pm \\epsilon \\sqrt{3}i}{2}$ and $\\eta_2= \\frac{-1\\pm \\epsilon \\sqrt{3}i}{2}$.\n \\item $\\phi_{(8)}^\\pm$ and $p_\\lambda(\\sigma_8^\\epsilon)= \\frac{x^8 -1}{(x\\pm \\epsilon 1)}$.\n \\item $\\phi_{(2,1)}^\\pm$ and $p_\\lambda(\\sigma_{(3)}^\\pm)= x \\pm 1$.\n \\end{enumerate}\n\\end{theorem}\n\n\\begin{lemma}\\label{lemma:basic_spin_representation_eigenvalues}\nSuppose that $\\rho$ ( or $\\rho^\\pm$) is the basic spin representation of $\\tilde S_n$ and $\\sigma\\in C_\\alpha^\\pm$.\nDenote the least common multiple of elements of $\\alpha$ by $k$.\nThe following statements are true.\n\\begin{enumerate}\n \\item If $3$ is the unique part of $\\alpha$ that is divisible by $3$, and it is not true that $5$ is the unique part of $\\alpha$ that is divisible by $5$, then the minimal polynomial of $\\rho(\\sigma)$ equals $\\frac{x^{k}\\pm1}{x^{k/3}\\pm1}$.\n\\item If $5$ is the unique part of $\\alpha$ that is divisible by $5$, and it is not true that $3$ is the unique part of $\\alpha$ that is divisible by $3$, then the minimal polynomial of $\\rho(\\sigma)$ equals $\\frac{x^{k}\\pm1}{x^{k/5}\\pm1}$.\n\\item If $\\alpha$ contains $3$ and $5$, while other parts of $\\alpha$ are not divisible by $3$ and $5$, then the minimal polynomial of $\\rho(\\sigma)$ equals $\\frac{(x^{k}-\\tau1)(x^{k/15}-\\tau1)}{(x^{k/3}-\\tau1)(x^{k/5}-\\tau1)}$, where $\\tau\\in\\{\\pm1\\}$.\n\\item In all other cases, the minimal polynomial of $\\rho(\\sigma)$ equals $x^{k}\\pm1$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIf $\\sigma$ is an $n$-cycle, then the assertion follows from Theorem~\\ref{theorem:cycle_double_S_n}.\nTherefore, we can assume that $l(\\alpha)\\geqslant 2$.\nNote that we can assume that $n\\geqslant12$. We prove the statement using induction on $n$.\n\nThen we have\n \\begin{align*}\n w_p w_q (v \\otimes v_1 \\otimes v_2)\n &= w_p ( v \\otimes v_1 \\otimes w_q v_2 )\\\\\n &= \\xi_1 v \\otimes w_p v_1 \\otimes S w_q v_2\\\\\n &= \\kappa_V v \\otimes w_p v_1 \\otimes S w_q v_2.\n \\end{align*}\n Since $V_\\beta$ is self-associate, by Theorem~\\ref{theorem:cycle_double_S_n}, the set of eigenvalues of $w_q$ are $\\{\\zeta \\mid \\zeta^q=\\varepsilon_q\\}$.\n Clearly, the set of eigenvalues of $S w_q$ is $\\{\\pm I\\}.\\{\\zeta \\mid \\zeta^q=\\varepsilon_q\\}$.\n Therefore, the set of eigenvalues of $w_\\mu$ contains\n \\begin{align*}\n \\{\\kappa_V\\}.\\{\\eta \\mid \\eta^p=\\varepsilon_p\\}.\\{\\pm I\\}.\\{\\zeta \\mid \\zeta^q=\\varepsilon_q\\}\\\\\n = \\{\\pm I\\}.\\{\\eta \\mid \\eta^p=\\varepsilon_p\\}.\\{\\zeta \\mid \\zeta^q=\\varepsilon_q\\}\\\\\n = \\{\\eta \\mid \\eta^{\\lcm(p,q)}=\\varepsilon_{\\lcm(p,q)}\\}.\n \\end{align*}\n This completes the proof.\n\\end{proof}\n\\begin{lemma}\\label{lemma:all_even_cycles_at_least_two_parts}\n Let $\\mu$ be a strict partition of $n$ with at least two parts such that all parts of $\\mu$ are even.\n Then the minimal polynomial of $w_\\mu$ in the spin representation $V_\\lambda$ is $x^{\\lcm(\\mu)} - \\varepsilon_\\mu$.\n\\end{lemma}\n\\begin{proof}\n Let us verify the lemma for $n\\leq 16$ by direct computation and hence we assume that $n\\geq 17$.\n If $\\mu$ has exactly two parts, then the result follows from Lemma~\\ref{lemma:two_even_cycles}.\n Now we assume that $\\mu$ has at least three parts.\n Write $w_\\mu = w_{\\tilde\\mu} w_{\\mu_l}$ where $\\tilde\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_{l-1})$.\n Let $\\lambda$ be a strict partition of $n$.\n As in the proof of Lemma~\\ref{lemma:two_even_cycles}, there exist strict partitions $\\alpha,\\beta$ of $|\\tilde\\mu|$ and $\\mu_l$ respectively such that the shifted LR coefficient $f_{\\alpha\\beta}^\\lambda>0$ and $\\beta\\neq (3,2,1)$.\n Then by induction, we may assume that the set of eigenvalues of $\\rho_{\\alpha}(w_{\\tilde\\mu})$ is $\\{\\eta \\mid \\eta^{\\lcm(\\tilde\\mu)}=\\varepsilon_{\\tilde\\mu}\\}$.\n If $w_{\\tilde\\mu}$ is odd, then the proof proceeds exactly as in the proof of Lemma~\\ref{lemma:two_even_cycles}.\n\nWe have, in fact, one more very useful information similar what we we have in the alternating group case which is as follows.\nFor a partition $\\lambda \\in DP_n^+$, we define the subtraction of those characters as\n\\begin{displaymath}\n \\Delta_\\lambda := \\phi_\\lambda^+ - \\phi_\\lambda^-.\n\\end{displaymath}\n\\begin{theorem}\\label{theorem:Delta_details_double_alternating}\\cite[Theorem 7.4]{Stembridge_Shifted}\n Let $\\lambda\\in DP_n^+$. Then we have\n $\\Delta_\\lambda(\\sigma)= \\pm i^{(n-l(\\lambda))/2} \\sqrt{z_\\lambda} $ if $|\\sigma|=\\lambda$ and $0$ in all other cases.\n\\end{theorem}\nWe have the following obvious lemma.\n\\begin{lemma}\\label{lemma:double_A_n_not_DP_n^-}\n Let $g$ be an element of $\\tilde A_n$ with cycle type $\\mu$ and $\\lambda$ be a strict partition of $n$.\n Then for any irreducible spin representation $V_\\lambda$ of $\\tilde A_n$, we have that the minimal polynomial of $g$ in $V_\\lambda$ is same as in the irreducible representation $V$ of $\\tilde S_n$ which contains $V_\\lambda$ upon restriction to $\\tilde A_n$ if $\\lambda \\notin DP_n^+$ or $\\lambda \\in DP_n^+$ with $\\lambda \\neq \\mu$.\n\\end{lemma}\n\\begin{proof}\n Let $\\phi_\\lambda$ (resp. $\\phi_\\lambda^\\pm$) be the character of the irreducible representation $V_\\lambda$ (resp. $V_\\lambda^\\pm$) of $\\tilde A_n$.\n Using Lemma~\\ref{lemma:conjugacy_classes_A_n}, Lemma~\\ref{lemma:restriction_to_A_n}, and Theorem~\\ref{theorem:Delta_details_double_alternating}, we see that $\\phi_\\lambda(g^i)$ (resp. $\\phi_\\lambda^\\pm(g^i)$) is either equal to $\\phi_\\lambda(g^i)$ or $\\phi_\\lambda(g^i)/2$ for all $i$.\n Computing the inner product of characters, we see that the minimal polynomial of $g$ in $V_\\lambda$ (or $V_\\lambda^\\pm$) is same as the minimal polynomial of $g$ in $V_\\lambda$.\n This proves the lemma.\n\\end{proof}\n\nThe following results essentially follow from our proof of Theorem~\\ref{theorem:cycle_double_S_n} and Lemma~\\ref{lemma:Prime_cycle_sym}.\n\\begin{theorem}\\label{theorem:cycle_double_A_n_odd}\nLet $n\\geq 3$ be an odd integer.\nLet $\\lambda \\in DP_n^+$.\nThen the minimal polynomial of $w_{(n)}$ in the irreducible spin representation $\\phi_\\lambda^\\pm$ has $n$ distinct roots except in the following cases:\n\\begin{itemize}\n \\item $n=3$ and $\\lambda = (3)$,\n \\item $n=5$ and $\\lambda = (5)$ or $(3,2)$,\n \\item $n=7$ and $\\lambda = (7)$,\n \\item $n=9$ and $\\lambda=(9)$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nThe theorem uses induction on $n$.\n Let us verify the theorem for $n\\leq 63$.\n Therefore, we may from now on assume that $n>63$.\n Write $n=mp$, where $p$ is the smallest prime dividing $n$.\n In particular, $m>9$.\n Let $\\Res^{\\tilde A_n}_{\\tilde A_m^{\\times p}} V_\\lambda\\geq V \\times V_1 \\otimes V_2 \\otimes \\dotsb \\otimes V_p$, for some irreducible spin representation $V \\otimes V_1 \\otimes V_2 \\otimes \\dotsb \\otimes V_p$ of $\\tilde A_m^{\\times p}$.",
    "post_theorem_intro_text_len": 2698,
    "post_theorem_intro_text": "The irreducible spin representations of $\\tilde A_n$ are indexed by $DP_n$.\nMore precisely, for each $\\lambda\\in DP_n^-$, we have a unique irreducible spin representation $V_\\lambda$ and for each $\\lambda\\in DP_n^+$, we have two non-isomorphic irreducible spin representations $V_\\lambda^\\pm$ of $\\tilde A_n$.\nThen the analogous theorem for $\\tilde A_n$ is the following.\n\\begin{theorem}\\label{theorem:main_alt}\n\tLet $(\\rho,V)$ be an irreducible spin representation of $\\tilde A_n$ indexed by a partition $\\lambda$.\n\tLet $g\\in \\tilde A_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_V(g)$ of $\\rho(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 10$ and $((\\rho,V),g)$ is one of the pairs listed in Tables~\\ref{table:exceptional_cases_1_8_alt} and \\ref{table:exceptional_cases_9_10_alt}.\n\n\tFor $n>10$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $V=V_{(n-1,1)}$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}\n\\end{theorem}\n\nThe article is organized as follows. In Sections~\\ref{section:Preliminaries for the double cover of the symmetric group} and ~\\ref{section:combinatprics_on_tableaux}, we provide the necessary background while developing some new results on the structure theory of  $\\tilde S_n$.\nIn Section~\\ref{section:minimal_polynomial_of_n_cycles}, we prove Theorem~\\ref{theorem:main} for the elements of $\\tilde S_n$ with cycle type $(n)$. \nIn Section~\\ref{section:minimal_polynomial_some_family}, we prove Theorem~\\ref{theorem:main} for some specific families of elements and irreducible spin representations.\nSections~\\ref{section:minimal_polynomial_of_any_element} and~\\ref{section:minimal_poly_of_any_element_in_A_n} are devoted to the proofs of Theorems~\\ref{theorem:main} and~\\ref{theorem:main_alt}, respectively.",
    "sketch": "The post-theorem introduction does not give a step-by-step proof argument for Theorem~\\ref{theorem:main}, but it does outline where the proof is carried out in the paper: Section~\\ref{section:minimal_polynomial_of_n_cycles} proves Theorem~\\ref{theorem:main} for elements of $\\tilde S_n$ with cycle type $(n)$; Section~\\ref{section:minimal_polynomial_some_family} proves Theorem~\\ref{theorem:main} for \"some specific families of elements and irreducible spin representations\"; and Sections~\\ref{section:minimal_polynomial_of_any_element} and~\\ref{section:minimal_poly_of_any_element_in_A_n} are \"devoted to the proofs\" of Theorems~\\ref{theorem:main} and~\\ref{theorem:main_alt}, respectively.",
    "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof argument for the main theorem, but it does outline where the proof is carried out in the paper: next the paper proves the main theorem for elements of $\\tilde S_n$ with cycle type $(n)$; later it proves the main theorem for \"some specific families of elements and irreducible spin representations\"; and finally it is devoted to the proofs of the main theorem and the following theorem, respectively. We first prove the following theorem. \n\n\\begin{theorem}\\label{theorem:main_alt}\n\tLet $(\\rho,V)$ be an irreducible spin representation of $\\tilde A_n$ indexed by a partition $\\lambda$.\n\tLet $g\\in \\tilde A_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_V(g)$ of $\\rho(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 10$ and $((\\rho,V),g)$ is one of the pairs listed in Tables~\\ref{table:exceptional_cases_1_8_alt} and \\ref{table:exceptional_cases_9_10_alt}.\n\n\tFor $n>10$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$.\n\t\t\\item $V=V_{(n)}$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $V=V_{(n-1,1)}$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}\n\\end{theorem}",
    "expanded_theorem": "\\label{theorem:main}\n\tLet $\\lambda\\in DP_n$ and $(\\rho_\\lambda,V_\\lambda)$ be the corresponding irreducible spin representation of $\\tilde S_n$.\n\tLet $g\\in \\tilde S_n$ be an element with cycle type $\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_k)$ and let $k=\\lcm(\\mu)$.\n\tThen the minimal polynomial $p_\\lambda(g)$ of $\\rho_\\lambda(g)$ has $k$ distinct roots except in the following cases:\n\n\t\\noindent For $n\\leq 8$, $(\\lambda,\\mu)$ is one of the pairs listed in the following table.\n\n\\begin{table}[h] \n\\caption{Exceptions for $1\\leqslant n\\leqslant 8$ in $\\tilde S_n$}\n\\label{table:exceptional_cases_1_8}\n    \\centering\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{|c|cc|}\n    \\hline\n        & $(\\mu, p_\\mu^\\lambda(x))$ &  \\\\\n        \\hline\n        $n=2$  &   &  \\\\\n        $\\lambda=(2)^\\pm$ & $((2)^+, \\frac{x^2+1}{x\\pm i})$, & $((2)^-, \\frac{x^2+1}{x\\mp i})$ \\\\\n        \\hline\n        $n=3$  &   &  \\\\\n        $\\lambda=(3)$ & $((3)^+, \\frac{x^3+1}{x+1})$, & $((3)^-, \\frac{x^3-1}{x-1})$ \\\\\n        $\\lambda=(2,1)^\\pm$ & $((2,1)^+, \\frac{x^2+1}{x\\pm i})$, & $((2,1)^-, \\frac{x^2+1}{x\\pm i})$ \\\\\n        & $((3)^+, \\frac{x^3+1}{x^2-x+1})$ & $((3)^-, \\frac{x^3-1}{x^2+x+1})$ \\\\\n        \\hline\n        $n=4$  &   &  \\\\\n        $\\lambda=(4)^\\pm$ & $((3,1)^+, \\frac{x^3+1}{x+1})$, & $((3,1)^-, \\frac{x^3-1}{x-1})$ \\\\\n        & $((4)^+, x^2\\mp \\sqrt{2}x+1)$, & $((4)^-, x^2\\pm \\sqrt{2}x+1)$ \\\\ \\hline\n        $n=5$  &   &   \\\\\n        $\\lambda=(5)$ & $((3,1^2)^+, \\frac{x^3+1}{x+1})$, & $((3,1^2)^-, \\frac{x^3-1}{x-1})$, \\\\\n        & $((3,2)^\\pm, \\frac{x^6+1}{x^2+1})$ & $((5)^\\pm, \\frac{x^5\\pm1}{x\\pm1})$ \\\\\n        $\\lambda=(3,2)^\\pm$ & $((5)^+, \\frac{x^5+1}{x+1})$, & $((5)^-, \\frac{x^5-1}{x-1})$ \\\\ & $((3,2)^+, x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)$ & $((3,2)^-, x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)$.  \\\\ \\hline\n        $n=6$  &   &   \\\\\n        $\\lambda=(6)^\\pm$ & $((6)^+, x^4\\pm\\sqrt{-3}x^3-2x^2\\pm\\sqrt{-3}x+1)$,\n        & $((6)^-, x^4\\mp\\sqrt{-3}x^3-2x^2\\mp\\sqrt{-3}x+1)$, \\\\\n        & $((3,1^3)^+, \\frac{x^3+1}{x+1})$, & $((3,1^3)^-, \\frac{x^3-1}{x-1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n        & $((3,2,1)^\\pm, \\frac{x^6+1}{x^2+1})$. & \\\\\n        $\\lambda=(3,2,1)^\\pm$ & $((3,2,1)^+, x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)$, & $((3,2,1)^-, x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)$, \\\\\n        & $((3,3)^+, \\frac{x^3-1}{x-1})$, & $((3,3)^-, \\frac{x^3+1}{x+1})$, \\\\\n        & $((5,1)^+, \\frac{x^5+1}{x+1})$, & $((5,1)^-, \\frac{x^5-1}{x-1})$, \\\\\n         & $((6)^\\pm, \\frac{x^6-1}{x^2-1})$. & \\\\ \\hline\n\t    $n=7$ & & \\\\\n\t    $\\lambda=(7)$ & $((3,1^4)^\\pm, \\frac{x^3\\pm1}{x\\pm1})$, & $((5,1^2)^\\pm, \\frac{x^5\\pm1}{x\\pm5})$,  \\\\\n\t      & $((3,2,1^2), \\frac{x^6+1}{x^2+1})$, & $((3,2,2), \\frac{x^6+1}{x^2+1})$, \\\\\n\t      & $((4,3)^\\pm, \\frac{x^{12}+1}{x^4+1})$, & $((5,1^2)^\\pm, \\frac{x^{5}\\pm1}{x\\pm1})$, \\\\\n\t      & $((5,2)^\\pm, \\frac{x^{10}+1}{x^2+1})$. & \\\\\n\t    \\hline\n\t    $n=8$ & & \\\\\n\t    $\\lambda=(8)^\\pm$ & $((5,1^3)^+, \\frac{x^{5}+1}{x+1})$, & $((5,1^3)^-, \\frac{x^5-1}{x-1})$,  \\\\\n\t    & $((3,1^5)^+, \\frac{x^3+1}{x+1})$, & $((3,1^5)^-, \\frac{x^3-1}{x-1})$,  \\\\\n\t    & $((5,3)^+, \\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})$, & $((5,3)^-, \\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})$,  \\\\\n        & $((8)^+, \\frac{x^{8}-1}{x\\pm1})$, & $((8)^-, \\frac{x^{8}-1}{x\\mp1})$,  \\\\\n        & $((5,2,1)^\\pm, \\frac{x^{10}+1}{x^2+1})$, & $((3,2^2,1), \\frac{x^6+1}{x^2+1})$, \\\\\n        & $((3,2,1^3), \\frac{x^{6}+1}{x^2+1})$, & $((4,3,1)^\\pm, \\frac{x^{12}+1}{x^4+1})$. \\\\\n        $\\lambda=(7,1)$ & $((5,3)^\\pm, \\frac{x^{15}\\mp1}{x\\mp1})$. &   \\\\   \\hline\n    \\end{tabular}\n\\end{table}\n\n\tFor $n>8$, the following four exceptions occur:\n\\begin{itemize}\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}$.\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}$\n\t\t\\item $\\lambda=(n)$ and $\\mu$ contains $3$ and $5$ and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}$.\n\t\t\\item $\\lambda=(n-1,1)$ and $\\mu$ contains $3$ and $5$, and all other parts of $\\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\\frac{(x^{k}\\pm1)}{(x^{k/15}\\pm1)}$.\n\t\\end{itemize}",
    "theorem_type": [
      "Universal",
      "Implication"
    ],
    "mcq": {
      "question": "Let \\(\\tilde S_n\\) be the double cover of the symmetric group \\(S_n\\). For a strict partition \\(\\lambda\\in DP_n\\) of \\(n\\) (that is, a partition of \\(n\\) with distinct parts), let \\((\\rho_\\lambda,V_\\lambda)\\) be the corresponding irreducible complex spin representation of \\(\\tilde S_n\\); when \\(\\lambda\\) has an odd number of even parts, the two associate spin representations are denoted by \\(\\lambda^\\pm\\). Let \\(g\\in \\tilde S_n\\) project to a permutation of cycle type \\(\\mu=(\\mu_1,\\mu_2,\\dotsc,\\mu_r)\\), and set \\(k=\\operatorname{lcm}(\\mu_1,\\mu_2,\\dotsc,\\mu_r)\\). In the exceptional list, notation such as \\((\\mu)^\\pm\\) refers to one of the two lifted classes in \\(\\tilde S_n\\) lying over the cycle type \\(\\mu\\). Which statement holds for every such \\(\\lambda\\) and \\(g\\) about the minimal polynomial \\(p_\\lambda(g)\\) of \\(\\rho_\\lambda(g)\\)?",
      "correct_choice": {
        "label": "A",
        "text": "For every \\(\\lambda\\in DP_n\\) and every \\(g\\in \\tilde S_n\\) of cycle type \\(\\mu\\), the minimal polynomial \\(p_\\lambda(g)\\) has \\(k\\) distinct roots, where \\(k=\\operatorname{lcm}(\\mu)\\), except in the following cases.\n\nFor \\(n\\le 8\\), if \\((\\lambda,\\mu)\\) is one of the listed pairs, then \\(p_\\lambda(g)\\) is the indicated polynomial:\n- \\(n=2\\):\n  - \\(\\lambda=(2)^\\pm\\): \\(((2)^+,\\frac{x^2+1}{x\\pm i})\\), \\(((2)^-,\\frac{x^2+1}{x\\mp i})\\).\n- \\(n=3\\):\n  - \\(\\lambda=(3)\\): \\(((3)^+,\\frac{x^3+1}{x+1})\\), \\(((3)^-,\\frac{x^3-1}{x-1})\\).\n  - \\(\\lambda=(2,1)^\\pm\\): \\(((2,1)^+,\\frac{x^2+1}{x\\pm i})\\), \\(((2,1)^-,\\frac{x^2+1}{x\\pm i})\\), \\(((3)^+,\\frac{x^3+1}{x^2-x+1})\\), \\(((3)^-,\\frac{x^3-1}{x^2+x+1})\\).\n- \\(n=4\\):\n  - \\(\\lambda=(4)^\\pm\\): \\(((3,1)^+,\\frac{x^3+1}{x+1})\\), \\(((3,1)^-,\\frac{x^3-1}{x-1})\\), \\(((4)^+,x^2\\mp \\sqrt{2}x+1)\\), \\(((4)^-,x^2\\pm \\sqrt{2}x+1)\\).\n- \\(n=5\\):\n  - \\(\\lambda=(5)\\): \\(((3,1^2)^+,\\frac{x^3+1}{x+1})\\), \\(((3,1^2)^-,\\frac{x^3-1}{x-1})\\), \\(((3,2)^\\pm,\\frac{x^6+1}{x^2+1})\\), \\(((5)^\\pm,\\frac{x^5\\pm1}{x\\pm1})\\).\n  - \\(\\lambda=(3,2)^\\pm\\): \\(((5)^+,\\frac{x^5+1}{x+1})\\), \\(((5)^-,\\frac{x^5-1}{x-1})\\), \\(((3,2)^+,x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)\\), \\(((3,2)^-,x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)\\).\n- \\(n=6\\):\n  - \\(\\lambda=(6)^\\pm\\): \\(((6)^+,x^4\\pm\\sqrt{-3}x^3-2x^2\\pm\\sqrt{-3}x+1)\\), \\(((6)^-,x^4\\mp\\sqrt{-3}x^3-2x^2\\mp\\sqrt{-3}x+1)\\), \\(((3,1^3)^+,\\frac{x^3+1}{x+1})\\), \\(((3,1^3)^-,\\frac{x^3-1}{x-1})\\), \\(((5,1)^+,\\frac{x^5+1}{x+1})\\), \\(((5,1)^-,\\frac{x^5-1}{x-1})\\), \\(((3,2,1)^\\pm,\\frac{x^6+1}{x^2+1})\\).\n  - \\(\\lambda=(3,2,1)^\\pm\\): \\(((3,2,1)^+,x^4\\mp\\sqrt{3}x^3+2x^2\\mp\\sqrt{3}x+1)\\), \\(((3,2,1)^-,x^4\\pm\\sqrt{3}x^3+2x^2\\pm\\sqrt{3}x+1)\\), \\(((3,3)^+,\\frac{x^3-1}{x-1})\\), \\(((3,3)^-,\\frac{x^3+1}{x+1})\\), \\(((5,1)^+,\\frac{x^5+1}{x+1})\\), \\(((5,1)^-,\\frac{x^5-1}{x-1})\\), \\(((6)^\\pm,\\frac{x^6-1}{x^2-1})\\).\n- \\(n=7\\):\n  - \\(\\lambda=(7)\\): \\(((3,1^4)^\\pm,\\frac{x^3\\pm1}{x\\pm1})\\), \\(((5,1^2)^\\pm,\\frac{x^5\\pm1}{x\\pm5})\\), \\(((3,2,1^2),\\frac{x^6+1}{x^2+1})\\), \\(((3,2,2),\\frac{x^6+1}{x^2+1})\\), \\(((4,3)^\\pm,\\frac{x^{12}+1}{x^4+1})\\), \\(((5,1^2)^\\pm,\\frac{x^5\\pm1}{x\\pm1})\\), \\(((5,2)^\\pm,\\frac{x^{10}+1}{x^2+1})\\).\n- \\(n=8\\):\n  - \\(\\lambda=(8)^\\pm\\): \\(((5,1^3)^+,\\frac{x^5+1}{x+1})\\), \\(((5,1^3)^-,\\frac{x^5-1}{x-1})\\), \\(((3,1^5)^+,\\frac{x^3+1}{x+1})\\), \\(((3,1^5)^-,\\frac{x^3-1}{x-1})\\), \\(((5,3)^+,\\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})\\), \\(((5,3)^-,\\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})\\), \\(((8)^+,\\frac{x^8-1}{x\\pm1})\\), \\(((8)^-,\\frac{x^8-1}{x\\mp1})\\), \\(((5,2,1)^\\pm,\\frac{x^{10}+1}{x^2+1})\\), \\(((3,2^2,1),\\frac{x^6+1}{x^2+1})\\), \\(((3,2,1^3),\\frac{x^6+1}{x^2+1})\\), \\(((4,3,1)^\\pm,\\frac{x^{12}+1}{x^4+1})\\).\n  - \\(\\lambda=(7,1)\\): \\(((5,3)^\\pm,\\frac{x^{15}\\mp1}{x\\mp1})\\).\n\nFor \\(n>8\\), the only exceptions are the following four infinite families:\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) with the remaining parts not divisible by \\(3\\), and \\(5\\) is not the unique part divisible by \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(5\\) with the remaining parts not divisible by \\(5\\), and \\(3\\) is not the unique part divisible by \\(3\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}.\n  \\]\n- \\(\\lambda=(n-1,1)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm1}{x^{k/15}\\pm1}.\n  \\]"
      },
      "choices": [
        {
          "label": "B",
          "text": "For every \\(\\lambda\\in DP_n\\) and every \\(g\\in \\tilde S_n\\) of cycle type \\(\\mu\\), the minimal polynomial \\(p_\\lambda(g)\\) has \\(k\\) distinct roots, where \\(k=\\operatorname{lcm}(\\mu)\\), except in the following cases.\n\nFor \\(n\\le 10\\), if \\((\\lambda,\\mu)\\) is one of the listed pairs, then \\(p_\\lambda(g)\\) is the indicated polynomial. For \\(n>10\\), the only exceptions are the following four infinite families:\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) with the remaining parts not divisible by \\(3\\), and \\(5\\) is not the unique part divisible by \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(5\\) with the remaining parts not divisible by \\(5\\), and \\(3\\) is not the unique part divisible by \\(3\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}.\n  \\]\n- \\(\\lambda=(n-1,1)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm1}{x^{k/15}\\pm1}.\n  \\]"
        },
        {
          "label": "C",
          "text": "For every \\(\\lambda\\in DP_n\\) and every \\(g\\in \\tilde S_n\\) of cycle type \\(\\mu\\), if \\((\\lambda,\\mu)\\) is not one of the exceptional pairs for \\(n\\le 8\\) and does not belong to one of the four infinite families for \\(n>8\\), then the minimal polynomial \\(p_\\lambda(g)\\) has \\(k\\) distinct roots, where \\(k=\\operatorname{lcm}(\\mu)\\)."
        },
        {
          "label": "D",
          "text": "For every \\(\\lambda\\in DP_n\\) and every \\(g\\in \\tilde S_n\\) of cycle type \\(\\mu\\), the minimal polynomial \\(p_\\lambda(g)\\) has \\(k\\) distinct roots, where \\(k=\\operatorname{lcm}(\\mu)\\), except in the following cases.\n\nFor \\(n\\le 8\\), if \\((\\lambda,\\mu)\\) is one of the listed pairs, then \\(p_\\lambda(g)\\) is the indicated polynomial. For \\(n>8\\), the only exceptions are the following four infinite families:\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) with exactly one part divisible by \\(3\\), and \\(5\\) is not the unique part divisible by \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/3}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(5\\) with exactly one part divisible by \\(5\\), and \\(3\\) is not the unique part divisible by \\(3\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/5}\\pm 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{(x^{k}\\pm1)(x^{k/15}\\pm 1)}{(x^{k/3}\\pm1)(x^{k/5}\\pm 1)}.\n  \\]\n- \\(\\lambda=(n-1,1)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm1}{x^{k/15}\\pm1}.\n  \\]"
        },
        {
          "label": "E",
          "text": "For every \\(\\lambda\\in DP_n\\) and every \\(g\\in \\tilde S_n\\) of cycle type \\(\\mu\\), the minimal polynomial \\(p_\\lambda(g)\\) has \\(k\\) distinct roots, where \\(k=\\operatorname{lcm}(\\mu)\\), except in the following cases.\n\nFor \\(n\\le 8\\), if \\((\\lambda,\\mu)\\) is one of the listed pairs, then \\(p_\\lambda(g)\\) is the indicated polynomial. For \\(n>8\\), the only exceptions are the following four infinite families:\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) with the remaining parts not divisible by \\(3\\), and \\(5\\) is not the unique part divisible by \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/3}\\mp 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(5\\) with the remaining parts not divisible by \\(5\\), and \\(3\\) is not the unique part divisible by \\(3\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm 1}{x^{k/5}\\mp 1}.\n  \\]\n- \\(\\lambda=(n)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{(x^{k}\\pm1)(x^{k/15}\\mp 1)}{(x^{k/3}\\mp1)(x^{k/5}\\mp 1)}.\n  \\]\n- \\(\\lambda=(n-1,1)\\) and \\(\\mu\\) contains \\(3\\) and \\(5\\), and all other parts of \\(\\mu\\) are not divisible by \\(3\\) or \\(5\\). In this case,\n  \\[\n  p_\\lambda(g)=\\frac{x^{k}\\pm1}{x^{k/15}\\mp1}.\n  \\]"
        }
      ],
      "meta": {
        "weaker_true_label": "C",
        "false_labels": [
          "B",
          "D",
          "E"
        ],
        "wildcard_false_label": "E"
      },
      "sketch_usage_meta": [
        {
          "label": "B",
          "sketch_hook_type": "finiteness",
          "tampered_component": "small_n_cutoff_8_to_10",
          "template_used": "boundary_range"
        },
        {
          "label": "C",
          "sketch_hook_type": "finiteness",
          "tampered_component": "explicit_description_of_exceptional_polynomials_and_families_removed",
          "template_used": "weaker_true"
        },
        {
          "label": "D",
          "sketch_hook_type": "counting_estimate",
          "tampered_component": "remaining_parts_not_divisible_by_3_or_5_replaced_by_unique_part_divisible_condition",
          "template_used": "property_confusion"
        },
        {
          "label": "E",
          "sketch_hook_type": "other",
          "tampered_component": "sign_coupling_in_exceptional_rational_formulas",
          "template_used": "wildcard"
        }
      ]
    },
    "qa quality eval": {
      "ALS": {
        "score": 2,
        "justification": "The stem gives definitions and setup but does not explicitly or implicitly reveal the correct choice; the answer is not leaked from the wording alone."
      },
      "TAS": {
        "score": 0,
        "justification": "This is essentially a theorem-recall item: the correct option restates the full theorem almost verbatim rather than asking for an inferred consequence or comparison of competing mathematical conclusions."
      },
      "GPS": {
        "score": 1,
        "justification": "Some care is needed to distinguish the exact theorem statement from subtly altered variants and a weaker-true summary, but the task is mainly recognition/checking details, not genuine mathematical generation or derivation."
      },
      "DQS": {
        "score": 2,
        "justification": "The distractors are plausible and distinct: they use realistic failure modes such as boundary shifts, formula tampering, weakening the claim, and adding an unjustified converse."
      },
      "total_score": 5,
      "overall_assessment": "Strong distractors, but the item is largely a verbatim theorem-identification question rather than a generative reasoning problem."
    }
  }
]